Quang-Tien Ngo

Long

Width variation problematic in steel rolling

] studied a quadratic velocity family that includes the "simple" one and leads to better results with a higher computing time.

 but they are all time-consuming. Thus, the idea is to decompose the plastic width variation into three terms: total, elastic and thermal width variations through the plastic zone that are determined by three new models. The simplified roll-bite entry & exit models allow estimating the elastic and plastic width variations before and after the roll-bite. They give equally the longitudinal stresses defining the boundary conditions for the roll-bite model which is indeed the 3D "simple" UBM approximating the total width variation term. Moreover, with the plastic deformation and friction dissipation powers given by the same model, the thermal width variation term is also obtained. The width variation model, called UBM -Slab combined is very fast (0.05s) and predicts accurately the width variation in comparison with Lam3-Tec3 (<6%). I express, finally my all appreciation to my wife and my two daughters for supporting all my endeavours. They are my daily inexpiable sources of motivation.

Thanks

I would like to acknowledge everyone who has assisted me throughout my doctoral studies over the years.

Firstly, I would like to give special thanks to my thesis director, professor Alain EHRLACHER (Navier laboratory -ENPC, Paris-Tech) who accepted me to perform this thesis in a particular financial condition. Then, he always directed and advised me with excellently creative ideas allowing me to advance on the topic. In addition, I am extremely grateful to him for his the encouragements at numerous difficult moments during the thesis. Without such supports I could have never completed the thesis successfully as it is.

A sincere thank to my industrial advisor Nicolas LEGRAND (ArcelorMittal Global Research & Development) for his guidance and orientation that inspired my works and developments. He is also the person who initiated, constructed the subject and supported me to perform this thesis.

I would like to appreciate Pierre-Stéphane MANGA, manager of Cold plant service and Marie-Christine REG-NIER, manager of Downstream Processes department who always made the best conditions for me to work on the thesis. My sincere thanks to my previous managers Akli ELIAS and Daniel LAUNET as well as to ArcelorMittal direction for their agreement and support to this project. Thanks to Camille ROUBIN, Maxime LAUGIER and Eliette MATHEY and all my other colleagues for their prestigeful technical exchanges which strongly enriched me about rolling process culture. I highly appreciated professor Ahmed BENALLAL at LMT -ENS Cachan, CNRS Research Director and professor Habibou MAITOURNAM, Mechanical Unit Director at ENSTA for accepting to be my thesis rapporteurs as well as for their careful lecture and very worthy recommendations. I would like to acknowledge professor Pierre MONTMITONNET, CNRS Research Director, Cemef -MINES ParisTech for agreeing to be the committee president.

Very special thanks to my parents, my sisters and my brother as well as all my friends for their inducement and expectation that became my great motivation to become doctor.

Long abstract Objective

Cold rolling of flat products is nowadays one of the main forming processes in metallurgical industry. The process geometry seems simple but its control requires the understanding of intricate thermo-mechanical aspects. In cold rolling process, due to the reduction in thickness the strip width is also changed. It is observed that in most of the cases the strip width decreases. And this width variation an be up to more than 10mm while it is not predicted in the production plants. By consequence, the plants produce usually an over-width in order to ensure the customer requirement and this over-width leads to an important over-cost of the production.

One solution to reduce the over-width consists in using a predictive model of width variation for each process in general and for cold rolling in particular. Such a model enables to determine the necessary width at the entry of cold rolling process to produce the required width. However, the over-width is not eliminated. It is still necessary to take an over-width due to the uncertainty of the model. Therefore, the more accurate the model the lower the unavoidable over-width. That means, even with a predictive model the production is set up to get an over-width in most of the cases and an under-width otherwise. In addition, products provided by hot rolling mill have usually different width in comparison with what the cold rolling mill requires and it varies all along the product. It is, thus also benefiting to adjust online the width in cold rolling process by varying some rolling parameters. For these reasons, the thesis aims at developing a predictive width variation model for cold rolling process which is accurate and fast enough to be used in real-time production.

Bibliographic reviews 2D simplified models in literature A

In the beginning of the 20th century when the computer science notion had never ever existed, there were many efforts to develop analytical rolling models. The two famous families of models are slab method based on the wellknown equilibrium equations pioneered by [START_REF] Karman | Beitrag zur theorie des walzvorgangesbeitrag zur theorie des walzvorganges[END_REF] in 1925 and Orowan's theory [START_REF] Orowan | The calculation of roll pressure in hot and cold flat rolling[END_REF] considering the inhomogeneity of stresses across strip thickness. Around the 1950s, simplified methods were introduced in order to take into account the strip elastic deformation [START_REF] Bland | The calculation of roll force and torque in cold strip rolling with tensions[END_REF][START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] before that Cosse [START_REF] Cosse | Mathematical study of cold rolling[END_REF] proposed the first complete elasto-plastic model in 1968.

In parallel with the developments of models for strip deformation, there exist equally divers models for the workroll deformation. The very first one, Hitchcock's model [START_REF] Hitchcock | Roll neck bearings[END_REF] considers that the deformed work-roll remains circular with a higher diameter. This model is still used today by a large number of models for industrial rolling preset. It is modified in 1952 by [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] to take into account influence of the elastic deformation areas. Later, in 1960 [START_REF] Jortner | An analysis of cold strip rolling[END_REF] proposed to determine the deformation of each point on the surface of the roll by a sum of influence function of each finite element with a specific pressure distribution. The method approaches accurately the work-roll profile. Fleck and Johnson [START_REF] Fleck | Towards a new theory of cold rolling thin foil[END_REF] investigated on thin strip rolling and were the firsts who consider the existence of elastic deformation areas inside the roll-bite (the strip deformation is elastic and plastic alternatively) as well as the existence of a neutral zone instead of a neutral point previously. This model is a significant progress to approach rolling process of very thin strip such as packaging products.

Rigid-plastic 3D UBM model for flat strip A

Applying the method proposed in the chapter 4 any 3D velocity field is also composed of 3D "simple" one and an additional term. The characteristics of this additional term are analyzed. Nevertheless, in order to keep a fast computing time of the power optimization resolution, we carried out the study of width variation using the 3D "simple" velocity field. The integrations are done using Gauss's method allowing accelerating the computing time. A comparison with experiments has showed a very good coherence. These experiments have been performed in ArcelorMittal pilot mill with relatively narrow strips (w e ≃ 60 -70mm).

In addition, with this 3D "simple" UBM model, an analysis has been realized and pointed out the effect of rolling parameters on the strip width spread. As results for a narrow strip, the width spread increases strongly with an increase in the reduction and falls down exponentially as a function of the strip entry width. It grows almost linearly as the roll radius increases and decreases with an increase in the entry or exit tensions. These results are coherent with existing works in the literature.

Rigid-plastic 3D UBM models for crown strip A

Some studies using a mixture of analytical and numerical methods [START_REF] Tozawa | Predicting the profile of rolled strip[END_REF][START_REF] Matsumoto | Mechanism of material deformation related to shape and crown phenomena[END_REF][START_REF] Allwood | A hybrid 2.5 dimensional elastoviscoplastic model of hot strip rolling for practical applications[END_REF][START_REF] Dixon | An analytical model for the lateral spread and shape defects from the rolling of a ridge profile or edge drop[END_REF][START_REF] Cozijnsen | Development of a shape model that includes edge spread for cold rolling[END_REF] pointed out that for thin strip rolling, the spread is small. Nevertheless, these studies showed out that the exit thickness profile of the strip (closely linked to the strip flatness) can influence the strip lateral spread.

Interested in this phenomenon, a new UBM approach is developed for cold rolling of strip with initial thickness crown while the work-roll is considered rigid and perfectly cylindric. First, an analysis is proposed to study kinematically admissible velocity fields in supposing some hypotheses. As the geometry of the strip is more complex than the case of flat strip rolling, the roll bite is divided into three areas in which the velocity field is different. The model shows that the width variation decreases with an increase in the strip initial crown. These results match very well those obtained with Lam3-Tec3 . Moreover, as can be noted, the strip crown increases the strip thickness reduction is higher at the strip center than at the edges that leads to a flatness defect called "center wave". Thus, the more the strip crown, the more "center wave" and the smaller the strip spread that can even be negative (necking).

UBM -Slab combined model to predict thermo-elasto-plastic width variation in industrial conditions A

As previously seen, the UBM model for flat and crowned strip in rolling process match well the experiments on pilot mill as well as Lam3-Tec3 . However, it is worth to highlight that the UBM assumes a rigid-plastic behavior of the strip that is justified for narrow strip rolling because the elastic width variation are negligible. On the opposite, in automotive industrial rolling condition the strip is large and the elastic width variation which is proportional to the strip width is no longer negligible. This elastic deformation is reversible but it has important impact on the plastic one. In addition, friction and plastic deformation powers heat up the strip. The material is, thus dilated in the width direction

Steel applications

Steel is one of the materials the most used over the world. Talking about steel induces thinking of strength, durability, safety and cleanliness. That is why we can find steel anywhere in our daily life. The steel applications field can be classified into five main domains.

1. Automotive industry: Car white body is composed of thin, flat carbon steel. High strength steel and stainless steel are used for structure, reinforcement and safety parts. Wheel and suspension parts are also made by strength steel while engine is long product steel. We can also find steel in many other pieces of a car as: tyre reinforcement (steel cord), exhaust system and decoration (stainless, aluminized or chromium steel). More than a half of car weight is made from steel.

2. Packaging: food containers, drink cans, liquid and gas containers. These consumable products are made from steel partly because of the steel high recyclability. It is incredibly true that drink cans made from steel could be recycled infinite of times. For this application, we use mostly thin law carbon steel resisting to high pressure.

High quality of surface is required because steel is coated (health safety) and painted.

3. Household appliances: Many kitchen objects as oven, refrigerator, washing machine, sink... are made from painted low carbon steel, enameling steel and stainless steel. And thanks to its health safety, cleanliness and very high strength, stainless steel is most used for cooking utensils and cutlery.

Construction and mechanical industry:

Thanks to very interesting ratio strength/weight and high durability, steel always keep its place in construction market among numerous number of materials. Many bridges, offshore platforms, boats and sheet pilings are made from heavy steel plates, high strength beam and wires. All rails are high carbon long steel product. Steel is also used to produce: tubes, pipes, tanks for petrol, chemical and food industries as well as transportation or specific products like pressure vessels and springs...

5.

Building: There are more and more building with steel structure using steel beam (long product), flat panels, roofs. Different from other material stainless steel or painted and coated steel are used for decoration.

In the present thesis, we are interested in rolling process of flat carbon steel for automotive and packaging applications. Thus, after a brief history of metallurgy, the production route of these kinds of steels, considered representative for general steel production, will be presented.

Steel and metallurgy history

The discovery of steel: By the 11th century BC it has been discovered that iron can be much improved. If it is reheated in a furnace with charcoal (containing carbon), some of the carbon is transferred to the iron. This process hardens the metal. In addition this effect is considerably greater if the hot metal is rapidly reduced in temperature, usually achieved by quenching it in water. The new material is steel. It can be worked just like softer iron, and it will keep a finer edge, capable of being honed to sharpness. Gradually, from the 11th century onwards, steel replaces bronze weapons in the Middle East, birthplace of the Iron Age.

The first cast iron: Thus far in the story iron has been heated and hammered, but never melted. Its melting point (1528°C) is too high for primitive furnaces, which can reach about 1300°C and are adequate for copper (melting at 1083°C). This limitation is overcome when the Chinese develop a furnace hot enough to melt iron, enabling them to produce the world's first cast iron -an event traditionally dated in the Chinese histories to 513 BC. In this they are a thousand and more years ahead of the western world. The first iron foundry in England, for example, dates only from AD 1161. By that time the Chinese have already pioneered the structural use of cast iron, using it sometimes for the pillars of full-size pagodas.

Ironmasters of Coalbrookdale:

Until the early 18th century the working of iron has been restricted by a practical consideration. The melting of iron requires large quantities of charcoal, with the result that ironworks are usually sited inaccessibly in the middle of forests. And charcoal is expensive. In 1709 Abraham Darby, an ironmaster with a furnace at Coalbrookdale on the river Severn, discovers that coke can be used instead of charcoal for the smelting of pig iron. This Severn region becomes Britain's centre of iron production in the early stages of the Industrial Revolution. Its pre-eminence is seen in the Darby family's own construction of the first iron bridge, and in the achievements of John Wilkinson.

Ironbridge 1779: In the space of a few months in 1779 the world's first iron bridge, with a single span of over 100 feet, is erected for Abraham Darby (the 3rd of that name) over the Severn just downstream from Coalbrookdale. Work has gone on for some time in building the foundations and casting the huge curving ribs. But in this new technology little time need be spent in assembling the parts -which amount, it is proudly announced, to 378 tons of metal.

Puddling and rolling 1783-1784: In successive years Henry Cort, an ironmaster with a mill near Fareham in Hampshire, patents two processes of lasting significance in the story of metallurgy. One is the technique which becomes known as puddling. Cort's innovation is a furnace which shakes the molten iron so that air mingles with it. Oxygen combines with carbon in the metallic compound, leaving almost pure iron. Unlike the brittle pig iron (or cast iron), this purer metal is malleable. Capable of being hammered and shaped, it is a much more useful metal in industrial processes than cast iron.

In the previous year Cort has also patented a machine for drawing out red-hot lumps of purefied metal between grooved rollers, turning them into manageable bars without the laborious process of hammering. His device is the origin of the rolling mills which subsequently become the standard factories of the steel industry.

Steel growth since the last century:

The world-wide steel industry has tremendous growth during the 20th century, from an annual production of 20 million tons of steel in 1900 to more than 1.2 milliard tons nowadays. The most important growth rate has been performed in the 50s and 60s after the Second World War when the reconstruction as well as the economy and military concurrence in many countries required more and more steel. Another fast growth period is since 2000 when emerging countries as China, India... realize incredible economic growth. During this period the steel technology has been developed with a drastic rate as the evolution of sciences, engineering technologies and computer science. It is uncountable the number of scientific articles, books, thesis as well as numerous number of patents about the steel compositions and production processes.

Typical steel production route 1.2.1 Liquid steel

There are two ways to produce liquid steel, a classic way called "primary steel route" using iron ore and the other using steel scrap called "recycling route". This section has objective to introduce typical processes of steel production, their roles and particularities explaining their existence. The physical or chemical principle of certain processes are only shortly and roughly explained but not detailed.

1.2.1.a Primary route

Coke oven and sinter plant A

In a classic production route, raw iron ore follows first a sintering process to be purer. Today, after sintering the average iron ore charge varies from 70% up to 90%. About coke, most of coke is man-made and obtained by pyrolysis of coal in regrouped furnaces in absence of air. This process, realized at about 1000°C, is called coke-making providing coke with high carbon content and few impurities. The coke is essential fuel for blast furnace thanks to its solidity, able to support charge and porosity allowing the transfer of gas and liquid through.

Blast furnace (BF) -Pig iron A

BF operates on the principle of chemical reduction whereby carbon monoxide, having a stronger affinity for the oxygen in iron ore than iron does, reduces the iron to its elemental form. Chemical reactions, control of temperature and the circulation of materials in the furnace are in fact complicated. Avoiding details of what happens, a very short description can be given as follows.

The main chemical reaction producing the molten iron, that might be indeed divided into multiple steps, is:

Fe 2 O 3 + 3CO = 2Fe + 3CO 2 .
(1.1)

The iron ore and coke are supplied through the top of furnace in successively forming alternative layers while the gas is flowed into furnace at the bottom (see Figure 1.1). Going up, the gas is efficiently in contact with solids all along the furnace height. As output, the pig iron obtained and extracted at the bottom of the furnace contains generally Fe (93-95%), C (3-5%), Si (0.2-0.8%), Mn (0.2-2%) and also Al, S, P... Limestone (CaCO 3 is provided into the top side in order to remove some impurities contained in iron ore notably silica Si. At the middle of the furnace, limestone is decomposed by reaction with CO 2 and then the calcium oxide obtained reacts with various acidic impurities (silica for example), to form a fayalitic calcium silicate.

CaCO 3 → CaO + CO 2 SiO 2 + CaO → CaSiO 3 (1.2)
Slag is the liquid mainly composed of remaining of limestone decomposition and impurities of iron CaO, SiO 2 , Al 2 O 3 and MgO as well as silicates of calcium (CaSiO 3 )of other metals ... The liquid slag floats on top of the liquid iron since it is less dense and is removed continuously from the furnace bottom. At the furnace top side, the monoxide and dioxide of carbon are evacuated in the waste gas. Figure 1.1 gives approximated quantity of main inputs and outputs corresponding to a ton of hot metal (pig iron).

Basic oxygen furnace (BOF) -converter A

BF pig iron, containing 3-5% C, can be used to make cast iron but more often refined further to make steel (much less C content. Liquid steel needs to contain lower contents of C, Mn, Si. Table 1.1 shows an example of composition of input pig iron and desired composition of liquid steel. To do that, the molten pig iron is poured into a Basic Oxygen Furnace (BOF) where most of the remaining carbon will be removed. In a BOF, pure oxygen is blown through a long tube, or lance inserted into the furnace top side. Sometimes, for higher stirring (to get lower C content), oxygen is blown from the bottom side (see Figure 1.

Width variation problematic in steel rolling

2).

As can be seen in Figure 1.3 that the affinity for oxygen of C at high temperature is much higher than that of Fe. Because other elements present in pig iron as Al, Mn, Si, Cr have also higher affinity for oxygen than Fe, they are equally almost removed from liquid pig iron by combining with oxygen to form oxide and stay in slag floating on the steel liquid such as:

2C + O 2 = 2CO Si + O 2 = (SiO 2 ) Mn + O 2 = (MnO 2 ).
Since S and P have similar affinity for oxygen as Fe (see Figure 1.3) they are more difficult to be removed. The solution is to add limestone to supply CaO born after decomposition of CaCO 3 (see equation 1.2). The basic reaction to eliminate P is: (1.4)

2P + 5O + n(CaO) = (nCaO -P 2 O 5 ) (1.

Width variation problematic in steel rolling

1.2 Typical steel production route

1.2.1.b Recycling route

For a recycling route, the mail raw material is no longer iron ore (oxide of Fe) but steel scrap with various compositions. Reducing oxides of Fe in BF and reducing C (and other impurities) in BOF are no longer necessary. Thus, instead of BF and BOF, Electric Arc Furnace (EAF) is used to melt scrap and produce directly liquid steel.

The cost of an EAF is about 8MC(Alternating Current -AC EAF) or 23MC(Direct Current -DC EAF) much lower than that required to build a BF (about 300MC). With lower productivity (about 0.7Mt per year), EAF is suitable for mini-mills. Invented in 1900, it had been used mainly for long products (bars, rods and small profiles) production. Today, EAF is also used to make flat carbon (thin slab casting) or heavy section products. Typically, better grades of steel products come from virgin iron ore and are rolled a great deal to fully develop internal quality and grain structure. In the market for midquality steels, the integrated can offer perhaps more than enough quality but often at too high cost. The mini-mills, in contrast, may have the right cost structure but not necessarily the right quality. By choosing suitable production route, companies can achieve a better pairing of cost and quality.

Raw materials and elaboration A

For the production of steel in a EAF, the following principal raw materials are used as feedstock:

• Recycled steel scrap • Hot metal • Pig iron • Reduced iron
In general, three first materials (scrap, hot metal, pig iron) are charged into EAF before the actual elaboration starts and the reduced iron is continuously fed into the vessel during elaboration to adjust the target composition.

Heating A

The EAF principle is to heat material by electric arc through the metal between a graphite electrode inside and another at the bottom of the furnace thanks to a high voltage of about 1300V ((see Figure 1.4). The heating can be divided into two steps: meltdown and super-hearing. During the first phase, the electrode starts from the top of the scrap charge, goes down close to the bottom to melt little quantify of scrap and form small metal bath. The electric 1. Width variation problematic in steel rolling 1.2 Typical steel production route energy continues to melt scrap until obtaining flat bath (end of meltdown). After this meltdown phase, there remains some unmelted scrap, the superheating (refining) consists in melt them by moving the electrode inside the furnace and use lower level of power.

Almost all materials charged into the furnace can be oxidized and hence release some energy. Iron and tramp elements when oxidized are transferred from the steel bath to the slag while carbon containing elements are converted to furnace off-gas. Chemical energy is also entered using external divides as natural gas burners to assist meltdown avoiding cold spots in the vessel and oxygen & carbon lances providing oxygen for oxidation and post combustion.

Metallurgical results obtained at the EAF A

The remaining tramp elements (Cu, Sn, Ni, Mo, Sb...) in liquid steel depends on the scrap composition (scrap yard) while carbon and phosphorus content is result of the process (oxygen injection). Table 1.2 gives a rough idea about the composition of liquid steel obtained by EAF.

Process

Secondary metallurgy and casting

Secondary metallurgy A After BOF or EAF, the liquid steel enters into the refining process called secondary metallurgy which has primary objective to finely adjust chemical composition of steel in controlling impurities and metallic inclusions. This secondary steelmaking process is most commonly performed in ladles. The necessary alloying elements are added while the impurities are removed by deoxidation (Al, Si), metal/slag reaction and vacuum degassing. Tight control of ladle metallurgy is associated with producing high grades of steel in which the tolerances in chemistry and consistency are narrow.

The second objective of the process is to prepare the right temperature of liquid steel just before casting process typically about 20°C more than liquidus temperature. A too low temperature may cause risk of solidification in the ladle or tundish while a too high temperature would make uncomplete solidification producing break at the casting machine exit. The ladles are commonly equipped of small electric arc furnace that is used to regulate the liquid steel temperature, for instance to re-heat the liquid steel when the casting process is delayed.

Continuous casting A

Figure 1.5 shows standard components of a casting machine. The ladles containing liquid steel are charged on the top side of the machine (feeding zone) where a rotating system allows replacing a full ladle into the position of an empty "in casting" ladle at the end of each casting "sequence". The liquid steel flows from the ladle to a tundish where the steel is well protected thermally and chemically. The tundish aims at feeding several strands in liquid steel and is a buffer tank during ladle change. It enables to control and regulate the steel flow rate in molds.

After the feeding zone, the steel comes in the molds, head zone, where heat is extracted to form primary solidified shell and give suitable geometry. Out of mold, the steel comes into solidification zone with many rolls guiding the change of direction from vertical to horizontal. The steel is solidified completely with position of the end of solidification depends on steel grade and process parameters. After an oxy-cutting process, solid steel slabs are produced with a dimension required by hot rolling plant. After casting machine, automotive and packaging product slabs have generally a dimension of about 5-15m (Length) x 800-2000mm (Width) x 220-260mm (Thickness) weighting from 20 to 35tons. In a same casting sequence, the slabs have same width and thickness. It is necessary to note that casting machine only produces certain values of width, for example 800mm, 1200mm, 1600mm and 2000mm but not any desired one.

Hot rolling plant

The objective of a Hot Strip Mill (HSM) is to reduce the product thickness and width while controlling the product surface quality and mechanical properties. Different installations of a HSM can be described as follows (see Figure 1.6). Hot rolling is also called hot metalworking process which consists in deforming product above the phase transformation temperature of the material because:

• at austenite phase, material is much softer • at higher temperature, the grains deform during rolling, they recrystallize, which maintains an equiaxed microstructure and homogenous grains size.

• the phase transformation needs to be precisely performed to get desired mechanical properties. This is done the most commonly during the natural cooling at the coil park after coiling process.

In general, to maintain a safety factor a finishing temperature (end of finishing mill) is usually defined about 100°C above the phase transformation temperature.

1.2.3.a Reheating furnaces

The reheating furnace function is to heat slabs up to enough high temperature (about 1100-1300°C) depending on the steel grade by using natural and coke furnace gases. A HSM can work with 2,3 or 4 furnaces in function of the productivity. A furnace has a power of about 120MW and a capacity of about 350t/h (heating time of a slab is about 2-3h).

Width variation problematic in steel rolling

1.2.3.b Roughing mill

Before the rolling process, a descaling is necessary to remove the scale (oxide layer) forming on the slab surface in the reheating furnace. Because a thick scale layer would be broken and inserted into the steel causing surface quality defect. The slab passes firstly under two pairs of powerful spray headers that blast high-pressure water to remove the 3mm-thick scale layer. Shortly after, a relatively small 2-High rolling mill called a scalebreaker reduces slightly slab thickness to break up any scale that remains. Then sweep sprays clean away any loosened scale that remains on the slab surfaces. The transfer bar will be descaled once or twice more during roughing to remove the scale that has grown back over the some minutes spent in the roughing mill.

The roughing mill can compose of more or less five stands through which the slab goes in keeping a same direction. Sometimes, it is a reversible stand where the slab passes an impair number of times (five times for example). In any case, a roughing stand is a combination of a vertical rolling stand called edger aiming at reducing the slab width and a horizontal stand reducing the slab thickness. After roughing mill, the slab thickness usually decreases to about 60mm and is elongated to about 40m. We remark that the product width is only rolled in roughing mill when the product is thick enough and in the later rolling processes variation of width is a consequence but not an objective.

After the roughing mill, the slab is transferred to the finishing mill with a low velocity on a segment of free or motorized small rolls that is called waiting table or rollers table. And the slab is now called a transfer bar. In some necessary cases, in order to prevent the slab from radiation thermal lost we switch down a tunnel to cover entirely or partly this segment.

1.2.3.c Cropping machine

Then, the transfer bar is descaled once more to eliminate most of scale grown during transfer time and just before the finishing mill it passes into a cropping machine. Because a bad quality head-end (oval form or with ski-effect) is critical to properly threading the finishing mill and the downcoiler, and an fish-tail tail-end can mark work-roll surface, the head and tail-ends of nearly every transfer bar are cropped by a pair of large steel drums each with a shearblade extending along its length. With the bar crawling along the roller table at around 0.5-0.7m/s, some sensors detect its position and speed in order to time the crop shear drums to optimize the amount cropped.

1.2.3.d Finishing mill

One of a finishing mill functions is to reduce the product thickness to a predefined (targeted) value. As the transfer bar is enough long and it will be longer and longer (for example 600m for an exit thickness of about 3mm), the product 1. Width variation problematic in steel rolling 1.2 Typical steel production route can be rolled at the same time in several stands increasing the productivity. In general, a finishing mill is a system including seven (more or less) successive 4-High stands forming a tandem. Between stands, there are tensiometers which allows to control the strip tension between stands making rolling process stable. A measurement of thickness is usually available at the exit of finishing mill allow to control and regulate the thickness by acting on the roll force (if hydraulic technology stand) or screw position (mechanical technology stand) to vary thickness reduction of one or more stands.

Another objective of finishing mill is to get a right temperature at its exit (entry of run-out table). That is why the exit strip temperature is measured and enables to adjust the rolling speed. In average, the rolling time of a strip is about 60-100s. It is easy to remark that the tail-end of strip is therefore colder than the head-end of about 50-70°C. In order to obtain a homogenous temperature along the strip, the finishing mill increases continually its rolling speed. This is a very common practice to compensate the temperature lost.

1.2.3.e Run-out table

Metallurgically critical to the properties of hot-rolled steel is the coiling temperature, as the coil will cool from this temperature to ambient over the course of three days, a heat treatment comparable to annealing. Coiling temperature is specified by product metallurgists to search optimal mechanical properties. Therefore the objective of the run-out table is to cool the strip from the temperature at the exit of the finishing mill to optimal coiling temperature.

The run-out table is composed of many water valves regrouped in different segment spraying the water at low pressure on the strip. Because the temperature at the exit of finishing mill can be fluctuating all along the strip. At the same time the strip speed is not controlled in the run-out table but by the finishing mill, an automatic system opening/closing the valves enables to regulate the number of opened valves to meet targeted temperature through the coil length.

1.2.3.f Coiling process

Out of the HSM, the strip is about 400-700m long and is coiled to be easily transported. Coiling temperature is a key element for metallurgical properties of material and can vary from 550°C to 800°C depending on grade.

Commonly a HSM relies on two coilers working alteratively avoiding long waiting time. A coiler begins with a pair of pinch rolls that catch the strip head-end. The head-end is deflected by a gate down to a mandrel and is guided around the mandrel, laps begin to build around the mandrel, forcing away the wrapper rolls. Once the head-end is cinched and friction and tension prevent the wraps of steel from slipping relative to the mandrel, the wrapper rolls disengage from the growing coil of steel. Before the strip tail is pulled through the pinch rolls, the wrapper rolls are reengaged. A hydraulic coil car moves into place beneath the coil, and, after rising up to support the coils bulk, strips the coil from the mandrel and places it in position. The coil is ready to be pickled and sent to customer (hot rolled product) or transported to cold rolling process (cold rolled product).

Cold rolling plant

Cold rolling mill has main objectives to reduce the product thickness with high surface quality, good flatness and mechanical properties. Figure 1.7 describes different processes and necessary installations of a cold rolling plant which allow to obtain these objectives.

1.2.4.a Pickling line

At the end of HSM products, coiled at high temperature (550°C-800°C) develop scale layer during the cooling time in air. Depending on the coiling temperature that the scale thickness can vary from 5µm to 20µm. In order to avoid incrustation of this scale into the steel during rolling, it is necessary to move it out thanks to acid tanks. The usual acids in pickling line are HCl at about 85°C or H2SO4 at about 100°C. At the beginning of the pickling process, a tension leveller aims at breaking the scale layer facilitating efficiency of the acid action. The tension leveller introduces tensions and alternative bending movements to deform plastically the strip. In general, the strip is elongated of about 0.5-2.0% barking more or less scale layer. An elongation of about 2% can allow to decrease twice the necessary pickling time.

1.2.4.b Side trimming

In order to eliminate edge defects that potentially make strip break in cold rolling, strip is often side trimmed before the rolling mill. And depending on steel grade and product dimension, high quality of strip edges is required to reduce the risk of work-roll mark in rolling process. The side trimming lets equally to obtain homogenous width along the strip length. However, this operation requiring a minimum cut-off width is a significant material yield source. Side trimming operation can be therefore skipped off when the risks mentioned above are estimated negligible. For automotive steel production about 60% of products are side trimmed before cold rolling mill.

1.2.4.c Cold rolling

Main objective -Thickness reduction A

The main functionality of cold rolling mill is to reduce the strip thickness to the final one while providing high surface quality. The most common flat product cold rolling mills contain from 4, 5 or 6 4-High or 6-High rolling stands. For automotive product, the entry thickness varies from 2mm to 6mm for a total reduction of 40-85%.

For packaging product, the reduction in cold rolling needs to be well defined in order to reduce the planar anisotropy after annealing. This anisotropy generates a famous type of defect, called earning defect, in deep drawing process as can drawing. The anisotropy increases as a function of cold rolling reduction, then decreases and becomes zero at very high reduction. So depending on grade (especially on Carbon content) the suitable cold rolling reduction is defined, usually between 86 and 92%. After the annealing, if the cold rolled thickness is still higher than the commanded one, the strip thickness is reduced once more at the skin-pass process (see section 1.2.4.e). Many packaging products follow this production route that is called double reductions. The 1st reduction is done at the tandem cold rolling mill (before annealing process) and the 2nd reduction is done at the skin-pass rolling mill (after the annealing).

Flatness A

In rolling, the strip can be deformed heterogeneously in width direction, meaning that the reduction is not homogenous. In this case, it is elongated more or less at the strip center and edges and after rolling the strip can contain important residual stress and and flatness defects. There may be many reasons for these defects:

The first one is the very important roll force, 1000 to 3000tons that deforms the work-rolls, which are in contact with the strip, in deflexion mode, and can make reduce more thickness at strip edges than at the strip center. The material is, hence more elongated at the edge than the center which generates flatness defect called "long edge". To limit amplitude of flatness defects, bigger work-rolls should be a solution. However, that increase the contact area with the strip and increase the necessary roll force for a same thickness reduction. More clever solution is to use back-up rolls which are bigger and in contact with the work-rolls to prevent them from deflexion deformation. A stand with only a pair of work-rolls is called 2-High stand, with a pair of back-up rolls likewise is 4-High stand. The 4-High

Width variation problematic in steel rolling

1.2 Typical steel production route stand is able to use smaller work-roll and therefore needs lower roll force than the 2-High one. That is the reason why in industrial flat rolling, the common stand technology is 4-High or 6-High stands. For stainless steel (very hard steel), there may be used 20-High stand, called Sendzimir stand, with several time smaller work-rolls than a typical automotive 4-High stand.

Additionally, to correct the "long-edger" flatness defect, bending force could be used to separate the ends of top and bottom work-rolls (see Figure 1.8). On the contrary, negative bending exerted to bring the work-rolls ends together should be used to correct "long-center" defect.

Another solution consists in using work-rolls designed initially with small positive (higher diameter at the center than two ends) or negative crown allowing to correct "long-edge" and "long center" defects. More recently, smart crown technology is developed to control faster and more efficiently the flatness defect. That consists in designing an intelligent work-roll profile: continues variable crown (CVC) as shown in Figure 1.9. By shifting the work-rolls, it is possible to change the gap between them and that enables to control the flatness efficiently.

Lubrication A

In cold rolling, lubrication is an essential factor allowing to obtain a good strip surface. An insufficient quality or quantity of lubricant could create scratch defect. By decreasing the roll-strip contact friction the lubrication reduces the un-useful energy dissipated by contact friction and slows down the work-roll wear. In cold rolling process, two common techniques to apply the lubricant are direct and recirculated applications. The direct application uses less stable oil and reject it after while the recirculated system uses more stable oil and reuse it after a retreating process.

Cooling A

During rolling, the electricity consumed is mostly transformed into heat distributed to the strip and the tools (workroll, back-up roll...). In average, a cold rolling mill consumes from 15 to 20MW and is able to heat the strip and work-roll of several hundred °C. That is the reason why it is very important to cool down the work-rolls as well as the strip. Most common technology is water nozzles sprays. The work-roll cooling can be at the entry (before the roll-bite) or/and at the exit (after the roll-bite). The strip cooling is done between two stands (interstand). A bad cooling system leads to too high temperature degrading work-roll surface and creating heating mark defect on strip. That is also origins of work-roll thermal crown (more dilatation at medium of work-roll) causing "long-center" flatness defects.

Roughness control A

One of qualities required by customers or by next process (galvanizing for example) is that the strip surface roughness need to be in a certain range. A too low roughness (smooth surface) makes the strip not adherent enough to paint layer. A too high roughness could increase paint consumption. In order to obtain roughness, the last stand work with rough work-rolls with a roll force defined to obtain right roughness. This stand does not aim at making reduction but at 1. Width variation problematic in steel rolling 1.2 Typical steel production route producing strip roughness and regulating strip flatness. In the contrary, in a tin-plate tandem mill (packaging product), the last stand uses low roughness work-roll and makes high reduction.

Coupling line A

The pickling and cold rolling lines can be completely separated. The coil coming from HSM is uncoiled, pickled and recoiled at the pickling line before being transported to the cold rolling mill (CRM). Then in the CRM it is uncoiled again, cold rolled and recoiled. Or the two processes are sometimes coupled where the coils are uncoiled, welded successively, head to tail, and then continually pickled and then cold rolled before being recoiled. As coupling line allows to increase productivity and decrease the transport, waiting time and other management cost, it becomes more and more frequent since the last decades.

1.2.4.d Annealing -material drawability

After being deformed in the cold rolling, the strip material is strongly work-hardened and the microstructure grains are sharply reduced in thickness direction and elongated in rolling direction. The material becomes anisotropic, hard and fragile. However, customers need high drawability steels supporting forming processes to make car pieces. Therefore, after cold rolling, the annealing is necessary to recrystallize the material making it less brittle and more workable.

Batch annealing (BA) A

In BA, the coils are heated intact in small furnaces over approximately 3 days. They are usually stacked four or five high on fixed bases, covered, as shown in Figure 1.10. To prevent oxidation of the strip, the atmosphere around the strip inside the furnaces is a controlled mixture of H2 and N2 although hydrogen only is sometimes used because of its increased conductivity. It is usually used for packaging steels.

Continuous annealing line (CAL) A

The CAL subjects rolled strip product to a sequence of furnaces to elevate and profile the strip temperature according to grade and dimension. Unlike BA, in CAL the strip is uncoiled, treated and recoiled in approximately 15 minutes. Figure 1.11 shows a typical example of CAL installation. Accumulators provide storage areas between static steel coils and continuous strip running through the furnace sections. Different furnaces are necessary to give steel the desired properties by heating to particular temperatures. In the heating furnace, the product is heated to the highest temperature. Then the soaking furnace is required to maintain strip temperature that allows to finish recrystallization of the material forming more homogenous and bigger grains size. The strip is cooled down slowly and fast in the first and secondary primary cooling sections before going into the overaging chamber where the strip is maintained at an intermediate temperature eliminating carbon precipitates.

Width variation problematic in steel rolling

1.2 Typical steel production route

1.2.4.e Skin-pass rolling

After the annealing process, many products have abnormal work-harding behavior with an yield plateau as shown in Figure 1.12. At the beginning of plastic deformation, the material is work-softened and not work-hardened up to certain plastic deformation. If the steel is sent directly to customers, during their forming process, the material will be deformed. In the steel, some local points will be deformed first and because of its work-softening, there will be a strong local concentration of deformation at these points. That creates heterogenous thickness in the structure and apparent surface defects when painted. Lüders defect is the demonstration [START_REF] Fouratier | Control of sheet surface defects and deep drawing in final strips production steps[END_REF].

Therefore, the primary functionality of a skin-pass mill called also temper mill is to suppress the yield point plateau of annealed steel by doing relatively small but enough plastic deformation in the strip. This elongation is to be defined as a function of steel grade and annealing cycle. It varies commonly from 0.5 to 2.5%. Figure 1.12: Tensile yield stress curve of an annealed steel before (left) and after skin-pass process (right).

The second objective of skin-pass rolling is to provide a strip roughness required by the customer varying from 1 to 3µm. The work-rolls roughnesses are quite important and the roll force is calculated coil-to-coil (it can not be too low or too high) to succeed the roughness transfer. A temper mill aims equally at improving strip flatness thanks to small deformation that its offers to the strip and a bending system.

For automotive steel, skin-pass mill can be 1-stand mill called also stand-alone. The skin-pass mill can roll one by one the coils coming from BA and is called offline skin-pass line. And when it is combined with and continually feeded by a CAL, it is called CAL skin-pass line. The skin-pass is also used in galvanizing line after galvannealing process.

On the other side, packaging products are often highly drawable after annealing. It is, thus possible to reduce their thickness to reach lower thickness range and work-harden them to get higher yield stress by having always a good final drawablitity. That is why the packaging skin-pass mill is usually 2-stand mill able to make from 1 to 55% of reduction to obtain the final customer thickness and provide good roughness and flatness.

1.2.4.f Coating

This is the last process giving additional properties to products as anti-corrosion, adhesion and aesthetic appearance required by every steel market. There exist different kinds of coating lines adapted to different steels and customers. Most of automotive products are treated in two common Hot Dip Galvanizing GI (Zn with 0.2 to 0.3% of Al) and GA (ZnFe with 0.1 to 0.135% of Al) are to deposit Zn -Al layer on the steel surface. Packaging products are electro-coated, organic coated or tinned. Strip width decrease called width necking, according to Legrand, Becker and Roubin [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF], may reach over 20mm through different steel production processes of a cold plant, which are introduced in the section 1.2.4. Continuous annealing and galvanizing (or annealing) furnaces, temper mills and tension levellers are known to have influence on strip width variations. Between these processes, tandem cold mill is considered as the most influent factor on that phenomenon. For instance, Figure 1.13 shows the strip width variation of about 6000 automotive steel products rolled at the 4-stand cold rolling tandem mill of ArcelorMittal Florange. It can be seen that in this tandem mill, strip width variations are mostly negative (width necking) and can be up to 15mm or more. These width contractions (showed in Figure 1.13) are scattered and badly estimated nowadays. Consequently, that could lead to producing under-width coils unsatisfying customer requirements and the coils are therefore downgraded or rejected. To compensate for these width variations, cold plants use to order to hot strip mills coils with significant over-widths, these over-widths are often overestimated. The hot finishing strip mill, with its own width scatter, tends to increase also the width ordered by cold mills. All these over-widths and the associated side trimming operation produce an important and non optimized yield all along the production route.

Width variation problematic in steel rolling

1.3.1.b Heterogenous width profile at entry of cold plant

It is usually observed that the width is not homogenous along a coil. In particular, at the entry of the cold plant, the width profile has a minimum which can be up to 10mm less than the width average and concentrate locally near to the tail end of the coil (about 90m in the example given by 1.14). A typical width profile obtained after HSM is given in the graphic 1.14. We remark that the tail end at pickling line of the cold plant corresponds to the hot coil head end. It remains being the tail at CRM for a coupling (pickling -CRM) line or becomes the head end for uncoupled line.

Width specification using prediction models 1.3.2.a Taking into account width variation in each process

The width specification is an indusial terms meaning the determination of targeted width of each product along the production route from the known customer one. The schema 1.15 shows typical methodology about how a cold plant 1. Width variation problematic in steel rolling 1.3 Width variation problematic in cold rolling process determines the necessary entry width allowing to produce final customer width. The targeted width at the entry of each process is based on targeted one at the entry of the next downstream process and an estimation of width variation in this process itself. In the other words, firstly the entry width of the last process (skin-pass rolling in this example) is calculated from customer width and an estimation of width variation in the skin-pass. Similarly, the entry thickness of annealing is that at the entry of the skin-pass adjusted by an estimated width variation of the annealing process. The same operation is repeated for cold rolling mill, tension leveller (at entry of pickling line), until the side trimmer to obtain the targeted cold plant entry width. The cold plant orders the HSM to provide the concerned product with this width. For the HSM, the cold plant is customer. The first over-width in HSM is to compensate the thermal contraction related to the difference of temperature, about 600°C at HSM coiler, and ambient temperature at the entry of cold plant. And the second is an over-width for width variation in the finishing mill. Up to the roughing mill, the edgers, mentioned in the section 1.2.3.b, where the processes are controlled to produce the targeted width for the Finishing Mill. It is important to highlight that, in a process if the width variation is positive or in other words a width spread, the necessary entry width is smaller than the targeted exit one of the process. However, in reality almost every process produces a width necking for most of products passing through. That are the reasons why an over-width for each process is to be determined during specification.

The side trimmer is particular where the width is trimmed to be homogenous at both edges. To ensure good cutting quality this operation requires a minimum trimmed width that is more or less 5mm per edge (fixed or varied as a function of strip thickness) meaning 10mm of width trimmed. It is an important material yield compared to other processes. Thus, only a special part of production passe though this process.

1.3.2.b Taking into account width variation along the coil

As discussed in the section 1.3.1.b the HSM produces today varying strip width profile with a local minimum. Therefore, in the specification strategy, the HSM takes into account this fact simply by targeting to provide the coils with the minimum width equal or higher than that required by the cold plant. That means, the HSM specified over-width is higher (than if width profile is homogenous) to compensate in addition the local under-width.

1.3.2.c Safety margin and uncertainty of predictive models

For cold plant as well as for hot plant, a width compensation is needed to compensate width decrease in different processes. This is estimated by predictive models that have certain level of accuracy. The uncertainty of prediction 1. Width variation problematic in steel rolling 1.3 Width variation problematic in cold rolling process models is not only due to the own quality of the models themselves but also the uncertainty of the models inputs. In other words, the processes parameters (models inputs) forecasted during specification phase are also different from the real production ones. Consequently, because of models uncertainty the production line would provide some times over width and other times under width.

The over width could increase the production cost and reduce benefice but it is usually accepted by customer or can be easily side trimmed to right width just before expedition. In the contrary, the under width is usually unacceptable. The customer refuse the coils and there may be even consequence of commercial agreement. Therefore, the production line avoids under-width by taking a safety margin.

This safety margin can be taken constant or calculated in function of the predictive models error following the strategy of plants. To facilitate the discussion we take an example where the safety margin of each process is equal to three times the standard deviation of models error denoted σ: Safety margin = 3σ. This strategy promotes over-width strategy in order to avoid under-width problem. If the prediction error follows the normal Gaussian distribution, such a safely margin results to under width for 0.135% and over width for 99.865% of products. We note that the average error of predictive models is small because the models are tuned and adapted to the measured data. Thus the models average error is negligible before their standard deviation, meaning the predictions are well centered. The specification method illustrated in Figure 1.15 can be summed up the schema 1.16. The HSM calculate the HSM width compensation to compensate the width variation at all Hot Plant process from the roughing mill exit width to finishing mill and thermal contraction for the local minimum width point. And the HSM safety margin is equal to three times the total of standard deviation of predictive models of all these HSM processes. Similarly, the cold plant width compensation is the total of width necking in all cold plant process calculated by predictive models and cold plant safety margin is also equal to three times the standard deviations of all these models.

Over-width material yield 1.3.3.a Material yield due to predictive models uncertainty

As can be seen in previous section, a such specification strategy leads to about 99.865% of products having overwidth and 0.135% having under-width. This is true for every process if each process adopt this same strategy. The under-width products are usually rejected and becomes scrap to come back to liquid steel production. Sometimes, they are only downgraded and sale at lower price. The other products, most of production, with over-width are side trimmed before being sent to the customer. They are also in some cases sent to the customer without being paid. Thus,

Width variation problematic in steel rolling

1.3 Width variation problematic in cold rolling process in any case, the over-width is a material yield. These over-width products (99.865% of the production) have an average over-width a little higher than but very closed to 3σ, the safety margin. Therefore, it is reasonable to consider that the average over-width is equal to the safety margin.

Hence, the material yield related to predictive models error is equal to 3σ ColdPlant for cold plant and 3σ HSM for hot plant (including thermal contraction between HSM and cold plant).

1.3.3.b Material yield due to width heterogenous profile

As discussed above, the strip width provide by the HSM is heterogenous and the specification aims at ensuring that the minimum width is higher than the width required by cold plant (see Figure 1.16). That results to the fact that the material at the edgers where the strip width is higher than the minimum width is not serviceable. And this is and represent material yield for HSM. All along of a coil, the useless part has an average width equal to the intra-coil under-width defined as the difference of the coil nominal width (average width) and the minimum one. And for the whole production (many coils) the average of this useless part is equal to the average of intra-coil under-width.

Finally, if the material yield is defined as the useless part width averaged on a many-coil production, it is approximated by the sum of cold plant safety margin, HSM safety margin and the average intra-coil under-width of products provided by the HSM to the cold plant. It is necessary to note that, this estimation is true if the safety margin is high enough so that most of products are produced with an over-width. This hypothesis is verified as shown in the example above because the plants aim at avoiding providing under-width products to customer.

Two ways reducing material yield 1.3.4.a Improving accuracy of predictive models

According to a strategy described above the safety margin and material yield depend directly on the uncertainty of width variation predictive models, obviously improving accuracy of the predictive model of any process allows to save the material yield.

1.3.4.b Using process actuators

Coil-to-coil width correction A Following the specification strategy described above, arriving at the cold plant, a product provided by the HSM has a higher width than that required in most of the times. This entry over-width varies from a product to another. As the width variation in a cold plant is about several millimeters for a width of more than one meter, meaning less than one percent. The entry over-width will be changed of about less than one percent. In other word, the entry over-width is almost kept unchanged and is transformed to an final over-width. If there exists an actuator of which an variation will cause a change of the width variation in the cold plant, it should be able to play with this actuator to make more width necking in the cold plant and therefore reduce the material yield. The new value of this actuator parameter is chosen in function of the entry over-width measured to compensate it. An process actuator is efficient when its variation makes sensible change of the width variation in the process.

The actuator is even more important in the case of under-width because it generates a significant lost. Moreover, as an under-width is usually small, it is more possible to be adjusted by an actuator.

Intra-coil width correction A

At the coil-to-coil scale, the actuator is chosen for the minimum width point of each coil. At the scale of a coil, the other point all along a coil will be in over-width, the interest is of course to reduce this over width by setting varying value of the actuator parameter. This material is elongated and transformed in to the coil length direction and is thus saved.

Width variation problematic in steel rolling

1.4 Thesis objective and approaches

Thesis objective and approaches 1.4.1 Thesis objective -predictive model for cold rolling process

The primary objective of the thesis is to develop a rapid and accurate predictive model of width variation in cold rolling process.

The phenomena contributing to the width variation of strip in cold rolling will be presented in detail in the chapter 7. It will be shown that the subject has been studied by different studies such as [START_REF] Hwang | An investigation into the plastic deformation behavior at the roll gap during plate rolling[END_REF], [START_REF] Komori | An upper bound method for analysis of three-dimensional deformation in the flat rolling of bars[END_REF], [START_REF] Richelsen | 3d analysis of coldrolling using a constitutive model for interface friction[END_REF], [START_REF] Jiang | A 3-d finite element method analysis of cold rolling of thin strip with friction variation[END_REF], [START_REF] Kazeminezhad | A theoretical and experimental investigation on wire flat rolling process using deformation pattern[END_REF], [START_REF] Wanga | Analysis of 3d rolling forming with generalized rigid-plastic boundaries approach[END_REF]... Many of them brought out phenomena involving in the width variation for flat bar rolling with small ratio width-thickness (less than 10 in general). For flat sheet rolling where the width-thickness ratio varies from 200 to 1500 for automotive and up to more than 5000 for packaging products, there are very few existing studies. Between them the two typical ones are [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] and [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF]. In this rolling configuration, the width variation is a thermo-mechanical problem that Finite Elements Method (FEM ) calculations like Abaqus , Lam3-Tec3 are able to simulate and give satisfying predictive results. However, these methods are very expensive in terms of computing time (from several hours to few days). Another method named stream lines method [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF] allows also to model the width variation problem accurately but the computing time remains in the order of few hours. These models are of course not applicable to the width specification and even less serviceable for online control of strip width.

On the other hand, some statistic and neural networks models exist and can be easily developed and tuned to industrial database [START_REF] Legrand | Rfcs rapport de l'etude sur la variation de largeur en usine à froid[END_REF], [START_REF] Chun | Application of neural networks to predict the width variation in a plate mill[END_REF]. They are extremely rapid but poor in physical comprehension and accuracy. Thus, the present thesis aims at developing a physical model by introducing some simplifying hypotheses and analytical methods as far as possible to obtain accurate predictive results with reasonable calculation time.

Success criterion: Such a model is considered successful when its accuracy is higher than that of statistical models, closed as much as possible to that of FEM models and has a computing time of order of a second.

Approaches

Issued from a bibliographic study about width variation of flat product in rolling process which is presented in the next chapters, the Upper Bound Method, an approach based on the velocity filed seems to be an adequate method to develop fast model of width variation. Hence, the present thesis is based on this method to start with. According to this model, it is the plastic lateral flow of material that creates the width variation. The obtained results are in excellent agreement with other existing models and experiments performed in within ArcelorMittal laboratory pilot rolling mill.

However, it is necessary to highlight that the strips width the experiments is quite narrow. The width-thickness ratio is lower than 60. Very quickly, when applying this model to automotive product with typical width-thickness ratio of about 1000, the width variation obtained is always positive (width spread) unlike the industrial measurements showing negative width variation. The FEM calculations with Abaqus and Lam3-Tec3 are therefore performed in this rolling condition in order to bring out physical phenomena involving in the width variation. Despite of it reversibility, the elastic deformation has an important impact on the final width variation. That is also proved that the thermal deformation influence in a similar way the plastic deformation generated in the roll-bite and thus contributes to the final width variation. The amplitude of both phenomena increases while the lateral flow decreases sharply with the strip width. Therefore, the influencing phenomena in the width variation and above all their coupling become really complicated. Developing rapid or analytical model is therefore extremely delicate.

Thus, the methodology is to analyze width variation phenomena by FEM in order to find good simplifying hypotheses and appropriate way to introduce them to the new predictive model. The FEM results are also used for validation of the rapid model.

Chapter 2

Rolling process modeling reviews

To answer the needs for research and industrial applications, there have been developed numerous of rolling models with various levels of complexity and rapidity. This chapter firstly aims at giving a general point of view on rolling process and modeling. Afterwards, some typical models are presented separately for work-roll elastic deformation, tribology of roll-strip contact and strip elasto-plastic deformation. We privilege the discussions on analytical or semi-analytical models. The main physical understandings and equations are detailed for the two famous families of rolling models, slab method for homogeneous deformation and Orowan theory taking into account inhomogeneity of strip deformation across the thickness. The developments concerning introduction of elastic deformation are as well mentioned. These elements help to develop a new elastic-plastic plane deformation model introduced in a later chapter. This new model aims at approximating the width variation.

General description of rolling problem

Main characteristics of rolling process

In the section 1.2 of the previous chapter, all processes of a production route are presented. That brings out the main aims of hot rolling as well as of cold rolling processes. The hot rolling (see 1.2.3) aims at reducing strip width and thickness and controlling outlet temperature in order to get targeted mechanical properties. On the other hand, during cold rolling (see 1.2.4) only strip thickness is reduced while the width variation is a consequence. And unlike the hot rolling, the cold rolling objective is to obtain a very good surface and flatness.

A definition of rolling process:

There exist many definitions of this process in the literature. Here following is presented one of them. According to [START_REF] Montmitonnet | Laminage -objectifs et modélisation[END_REF] this operation is defined as follows: In metallurgic industry, rolling is an operation having objective to reduce one or more dimensions of a long product thanks to a system of two or several axisymmetric tools rotating around their axis. It is the rotation of these tools which drives the product move toward the roll-bite thanks to the presence of work roll -strip contact friction, see Figure 2.1.

Figure 2.1: Rolling process usually consists in reducing strip thickness and elongate its length by consequence.

2.1.1.a Steady state or transient rolling and modeling

Industrial rolling process can be performed constantly in transient phase, especially in hot slab rolling. In a roughing mill for instant, product is rolled one by one separately and rolling of product extremities has transient nature, i.e rolling parameters (strip deformation, roll force, velocity...) vary during time. In this process, the product is not long enough to establish steady state rolling.

Unlikely, in a cold rolling, the product is very long compared to its width and work-roll size, the rolling is mostly performed at a steady state. The establishment of mechanical steady state is quite fast. In general, after a rolled length of several times of the widths mechanical fields becomes constant during time. In particular, for 2D (plane strain hypothesis) it is even faster. Rolling several times of roll-bite length is already sufficient to get steady state. Nevertheless, the thermal steady state requires longer time. Just after being changed, work-rolls start with homogenous temperature equal to ambient one. It is heated by exchanged heat with products and by friction dissipation. This heating process requires many rotations, sometimes only after rolling several product, the thermal steady state is established.

Time-dependent or incremental methods A

Correspondingly to these possible phases of rolling process, there exist two categories of models. The first one contains time-dependent models allowing to understand what happens during transient phases. The incremental or Lagrangien models integrate the equations in function of time. The product geometry is known at the given time, and its evolution in time is calculated by integrating the velocity field in a same way for the strip and the tool (see Figure 2.2). These methods allow to model not only the transitory states but also the equilibrium ones by integrating the equations long time enough until obtaining stable situation. However the calculation time is important. Typical applications of incremental method are for the roughing mill where the extremities (head and tail) deformations and the threading issues should be studied. These models are also used to model unstable phases as flatness issues or vibrations problems for both hot and cold rolling.

Rolling process modeling reviews

Stationary or steady state methods A

Stationary models are constructed by eliminating the time in all equations sometime called the Eulerian models. This method represent certainly an attempt to develop but in return, many substantial advantages. In particular, they decreases the computing time by more than ten or even hundred times in some cases. The principal difficulty is that the calculation domain i.e the geometry of the product and tool during and after deformation become an additional unknown of the problem because of the free surfaces where no displacement boundary conditions are present [START_REF] Chenot | A method for determining free surfaces in steady state finite element computations[END_REF]. In general, this difficulty is translated into a calculation by iteration (see Figure 2.2.

Remark:

The main objective of the thesis is to model the width variation during cold rolling. Of course, during this process, the product thickness and mechanical properties are not homogenous from at head to the tail end. But very commonly these variations are small and negligible. And when these variations occur on a long part of the product (low frequent variations) the process is considered as quasi steady state rolling, meaning that the time depending solution is a succession of many steady state solutions of the inputs that are time-dependent. That is why henceforth only steady state models will be mentioned.

2.1.1.b Symmetric or asymmetric rolling

Operator-motor sides asymmetry: An example of asymmetries is tilting problem when roll gap is not constant along the rolls axe (the strip width direction). The strip is therefore deformed more at one edge side than the other. By consequence, one of the strip edges is more elongated making the strip direction be curved at the exit of rolling stand and causing strip steering problem. Many other reasons could also make asymmetries in width direction. For example, inhomogeneous lubrication can create inhomogeneous friction along the strip width direction. Or an heterogenous work-roll cooling can also make an varying temperature distribution along the work-roll axe. And as a result of thermal dilatation, the work-roll apparent diameter is not constant and can make a various roll-gap... Anyway, every asymmetry in width direction is not desired. Therefore, automatic control system attempts to eliminate apparent asymmetric defects on the strip (anti-tilting for example). In other words, the rolling process is controlled to be as symmetric in width direction as possible.

Top-bottom asymmetry:

The rolling can be top-bottom asymmetric due to many reasons such as a difference of top and bottom work-rolls diameter, surface or roughness, a difference of their speed during rolling... One of the obvious origins for the top-bottom asymmetry is the gravity. Because the strip is horizontal, the lubricant and coolant are stored more easily on the top strip surface than on the bottom side. Thus, the roll-bite friction and the heat exchange coefficients are often top-bottom asymmetric. This kind of asymmetry could generate appearing defects on the strip like ski-effect in hot rolling or bow-flatness defect in cold rolling (the strip is curved in rolling direction). Rarely but there exist desired top-bottom asymmetries in rolling unlike the operator-motor sides asymmetries. T.Hoang [START_REF] Hoang | Asymmetric rolling analysis -energy saving and ski effect[END_REF] develops an allowing to quantify the ski-effect amplitude du to top-bottom asymmetries. The author brings out that the rolling with different work-rolls diameter

Rolling process modeling reviews

2.1 General description of rolling problem should allow to decrease both roll force and power compared to symmetric rolling with a same average work-roll diameter.

Remark: Only symmetric rolling is considered in this thesis because asymmetric defects are not what the present study searches for. Even though asymmetric defects may influence the width variation, our objective, they are not the principle motivations.

Main objects to be modeled in rolling

The rolling process contains essentially a product, tools (including two work-rolls, two back-up rolls...), a lubrication system and a cooling system as can be shown in a simplified way by Figure 2.3.

Strip -Roll-bite and out of roll-bite:

As the objective of rolling is to reduce the thickness, the strip is deformed plastically due to the pressure exerted by the work-rolls. The strip part under contact is called the roll-bite. It is obvious that what happens in the roll-bite is the main goal of any model but the elastic deformation out-side the roll bite can be also important. For example the distribution of tension in width direction representing flatness varies significantly out side of the roll-bite. According to Saint Venant's principle, the length of extending parts on both sides is comparable to the strip width. In brief, it is necessary to model the strip before, in side and after the roll-bite as shown in Figure 2.3.

Tool:

The tool is usually modeled as a system of a pair of work-rolls and/or another of back-up roll. Obviously, only elastic deformation are expected in the tool at least in macroscopic scale. The micro plastic deformation is inevitable because of the rough surfaces contact between strip and work-roll, and between work-roll and backup-roll. In macroscopic scale, the work-roll is deformed in two different modes called respectively flattening (radial and tangential direction) and deflexion (roll axes direction). The modeling of the work-roll deformation will be more explained in a later section 2.4.

Contact surface -lubricant:

The contact nature depends much on the presence or absence of lubricant. In many hot rolling processes, no lubricant is injected to the strip and work-roll contact. In these cases, the roll-bite is lubricated by the work-roll scale layer and the strip scale layer due to oxidation of their surfaces at high temperature. Otherwise, the hot rolling process is sometimes lubricated. Certainly due to the strip high temperature, the oil is burn rapidly once in contact with the strip and only a very thin film of residual oil particles stays and lubricates the roll-bite. This film is extremely thin but could change completely the contact nature and decrease drastically the friction coefficient. In both cold or hot rolling the contact tribology is complicated and require sophisticated models depending on the study objective. In the section 2.2, some typical models for cold rolling will be mentioned.

Cooling system -Thermal phenomena:

As mentioned in the section (see 1.2.4), the cooling system including work-roll cooling and strip cooling is necessary to keep the temperature of the tool and the strip at reasonable levels. However, it is not in a rolling model that the cooling system is modeled in detail. It is simply modeled as a heat exchange fluxes on the cooling zones. And the heat exchange coefficients are usually identified independently.

A well-posed problem of rolling

The mathematical term well-posed problem stems from a definition given by J.Hadamard [START_REF] Hadamard | Sur les problèmes aux dérivées partielles et leur signification physique[END_REF]. He believed that mathematical models of physical phenomena should have the properties that: A solution exists. The solution is unique and the solution behavior changes continuously with the initial conditions. In continuum mechanics, a problem is call well-posed when:

1. In side modeled objects, the volume charges are given 2. On all external surfaces of the modeled objects the boundary conditions are given and complete. On a surface, the boundary conditions are considered as complete when there are three boundary conditions on the velocity or (and) the stress corresponding to the three space directions. In other word, the velocity and stress boundary conditions are complementary on this surface.

3. Constitutive equations related to the materials behavior are defined. That includes elastic constitutive equation, plasticization criterion and flow equations.

And in general, a well-posed mechanical problem results to a unique stress solution while the velocity solution is not necessarily when the rigid body movements are possible.

Hereby, we will describe the rolling problem in a very general case and with all necessary conditions to be a well-posed one.

Lubrication O c Lubrication Work roll ) ( y h e ) ( y h s Z Entry part Roll bite Exit part ) ( y s X Y Z O X O Figure 2.
3: For symmetric rolling, only a quarter geometry, a half of the thickness and a half the width, is necessary to be modeled.

2.1.3.a Equilibrium

By neglecting the volume gravity force before the stress and variation of stress, the equilibrium equations in both the strip and the roll are:

∇ σ = 0 (2.1)

2.1.3.b Boundary conditions

Thanks to the symmetry hypotheses in both strip thickness and strip width directions, only a quarter of the total geometry is necessary to be modeled as shown in Figure 2.3.

Work-roll boundary conditions A

The work roll is driven by a motor. Its peripheral velocity V c is given and controlled while the driving torque provided by the motor, denoted Torque, is a consequence. Not only in contact with the strip, the work-roll is also in contact with the back-up roll for a 4-High stand. In point of view of the work-roll, the stresses on the contact surface with the strip are given, they are consequence of the strip behavior being deformed. The remaining surface is free mechanically. The work-roll is blocked in rolling direction Ox and strip width direction Oy. Its center position is adjusted in Oz by an automatic gauge control (AGC) in order to obtained the targeted strip exit thickness h s .

The thermal conditions are the heat exchange of different natures in the whole surface of the roll: steel-steel contact heat exchange with the strip and back-roll, heat exchange with work-roll coolant (water or emulsion 1 ), heat exchange 2. Rolling process modeling reviews 2.1 General description of rolling problem with air otherwise. The radiation is negligible. On the contact surface, the friction is a heat generation source. In general, the friction power is shared one part to the work-roll, the other to the strip with a given sharing coefficient. This coefficient, very difficult to be identified is commonly equal to 0.5.

Strip boundary conditions

Strip entry section: The entry section is the initial form of the strip before rolling. It can be rectangular with given thickness h e and given width b e . And if the strip has an initial crown, its thickness profile is given instead of h e . On this section the material moves with a homogenous velocity in rolling direction, meaning:

u| S e = V e .e x .
V e is an unknown of the problem but the total entry tension (resulting on S e ) is given as T e .S e where T e is the average specific entry tension. All other stresses components are null, in other words:

S e
σ.e x dS = T e S e e x .

Strip exit section: Similarly to the entry section, the velocity at the exit section is also homogeneous and unknown. The nominal (average) exit thickness is given but unlike the entry section, the exit thickness profile and exit width are unknown. And the exit total tension is given:

   u| S s = V s .e x S s
σ.e x dS = T s S s e x .

It is notable to remind that both velocities V e and V s are not given and to be determined as results of the problem.

Boundary condition on symmetry plane z = 0: If the whole strip is modeled, by definition, a top-bottom symmetric velocity field verifies three followings conditions for every strip material point:

     u x (x, y, z) = u x (x, y, -z) u y (x, y, z) = u y (x, y, -z) u z (x, y, z) = -u z (x, y, -z) (2.2)
∀x, y, z. These equations imply that ∀x, y when z -→ 0:

           ∂u x ∂z (x, y, z = 0) = 0 ∂u y ∂z (x, y, z = 0) = 0 u z (x, y, z = 0) = 0 (2.3)
When only top half of the geometry is modeled, the velocity needs to verify 2.3 and the velocity of the bottom half is simply deduced using 2.2. In other words, the symmetry conditions on symmetry plane z = 0 is given by 2.3.

It is remarkable that the first two equations of 2.3 can be also obtained by symmetry conditions on the rotation rate saying that on this symmetric surface the rotation rate in x and y direction are null. To demonstrate that, it is enough to rewrite the dedition of rotation rate tensor and multiply it to the normal vector of symmetry surface -e z to obtain the rotation rate vector. The first two components of this vector must be 0. Then using the third equation of 2.3, we obtain the same equations as 2.3.

Boundary conditions on symmetry plane y = 0: The same, the boundary condition on y = 0 symmetric plane can be written as: ∀x, z,

             ∂u x ∂y (x, y = 0, z) = 0 u y (x, y = 0, z) = 0 ∂u z ∂y (x, y = 0, z) = 0 (2.4)
Strip edge surface y = b(x, z): From the entry to the exit section through the roll-bite the width function changes and in a general case, strip edge can be different from a straight section, meaning that b is a function of both x and z. This is a free surface on which that all stress field components are null:

σ (x, y = b(x, z), z) .n = 0 ∀x, z.
(2.5) Furthermore, at steady state the width surface is also stream lines. Thus, in terms of velocity field, the boundary conditions can be written as:

     u y (x, y = b(x, z), z) = ∂b ∂x (x, z).u x (x, y = b(x, z), z) u z (x, y = b(x, z), z) = ∂b ∂z (x, z).u x (x, y = b(x, z), z) (2.6)

Strip top surface z = h(x, y):

The thickness profile at the entry section is given as h e (y), then h(x, y) becomes unknown and varies during the area before the roll-bite. In the contact area h(x, y) is imposed by the work-roll shape (within a constant because the work-roll position needs to be determined to obtain the strip exit thickness). After the contact h(x, y) becomes free again and varies to the exit section where its nominal value is equal to targeted exit thickness h s .

The boundary condition on this surface is as follows: before and after the contact, the strip top surface is free: σ (x, y, h(x, y)) .n = 0 ∀x, y.

(2.7)

Contact surface: In the contact area, the first boundary condition is that the strip thickness is imposed by the work-roll shape h(x, y) issued from a work-roll deformation model (see the section 2.4 for typical ones). This condition fixes the deformation of the strip in the normal direction but is not enough because it does not describe what happens in the tangential directions. Therefore, complementary boundary conditions take over the two tangential stresses. These conditions are given by a tribological model where the friction coefficient can be isotropic or not, dependent of normal stress or not or can vary as a function of sliding velocity or not... (see some typical model in the section 2.2).

2.1.3.c Material constitutive equations

For the roll, the constitutive equation is simple because there is only elastic deformation. For the strip, many models build with rigid-plastic (RP) or slightly-compressive rigid-plastic [START_REF] Mori | Simulation of three-dimensional deformation in rolling by the finite element method[END_REF] to avoid difficulties related to the incompressibility in flow formulation. Some others use elastic-viscoplastic (EVP) or thermal-EVP behavior. The thermal effects are not only important for hot rolling through thermo-mechanical coupling (influence of temperature on yield stress) and also its influence on microstructure. For cold rolling, these effects are less primordial.

The constitutive equation is thus a key point of each model allowing to take into account or to simplify the physic of strip deformation. We will see in the section 2.3 some typical strip models.

Rolling process modeling reviews

Rolling model -A coupling of separated models

As described above, a rolling problem contains mainly three objects to be modeled with very different natures of involving physical phenomena. Until today there exist no coupled model that resolves all coupling equations at the same time. But each object is modeled separately and the whole model is a weak coupling of them. Figure 2.4 shows the principle of coupling. Being more or less sophisticated but almost every models in the literature are built in this way. The strip model needs the entry thickness profile h e (y), exit targeted nominal thickness h s , entry and exit average tensions T e , T s , material rheology and behavior law but also the work-roll shape and friction coefficient. For the first iteration, we can assume a simple work-roll shape and a constant friction coefficient. This model computes the contact geometry, the contact stresses and temperature, velocity, stress and temperature fields inside the strip. Then, the work-roll model uses the contact geometry, stresses and temperature given by the strip model to evaluate the work-roll shape. The tribological model uses also the contact pressures, contact velocity fields and temperature to estimate the macroscopic friction coefficient along the roll-bite. We can hence feed the strip model with this newly obtained work-roll shape and friction coefficient and start the second iteration... The convergence will be reached when the evolution of work-roll shape, contact geometry and stresses, friction coefficient are enough stable from one interaction to the next one.

Typical tribological models

Let start with tribological models for two reasons. The first is that the equation giving the shear stress on the contact surface is necessary for the understanding of the resolution of strip models. Thus, for each presented strip model a friction one will be cited. The second reason is that, tribological model can become very rapidly complicated if we want to determine the friction stress based on the what happens on the contact surface in the roughness scale. These problems can be in no way treated shortly in this section. Therefore, only some typical facts about friction models will be cited here without much details.

Dry friction 2.2.1.a Coulomb friction model

The classic rules of sliding friction were discovered by Leonardo da Vinci (1452-1519), but remained unpublished in his notebooks. They were then rediscovered by G. Amontons (1699) and the understanding of friction was further developed by C-A. De Coulomb (1785). The sliding or kinetic friction were expressed as three empirical laws:

1. Amontons' first law: The force of friction is directly proportional to the applied load.

2. Amontons' second law: The force of friction is independent of the apparent area of contact.

3. Coulomb's law of friction: Kinetic friction is independent of the sliding velocity.

These laws led to popular Coulomb dry friction model for sliding contact as follows:

(2.8)

where the "-" means that the friction force is in the opposite direction of the relative sliding velocity denoted v. The Coulomb friction model can, when the sliding happens be written by a relation between the normal and tangential stresses σ n and τ as follows:

τ = -µσ n sign(v) (2.9)
where µ is the Coulomb friction coefficient. This friction model is most commonly used and usually referred to as dry friction although it is used for dry contacts as well as boundary and mixed lubricated contacts.

2.2.1.b Asperity contact theory of adhesive friction

Inelastic adhesion concept of friction -Bowden and Tabor 1954 Considering that the classical frictional law of Amontons was based on the projected or apparent area, Bowden and Tabor (1954) were concerned with the real area over which the two sliding bodies are in contact. The real area of contact is made up of a large number of small regions of contact, called asperities or junctions of contact, where atom-to-atom contact takes place. Bowden and Tabor showed that the force of friction between two sliding surfaces is strongly dependent on the real area of contact. Assuming during a frictional sliding process a fully plastic flow situation of all asperities, friction is found to change linearly with the applied load as demanded by Amontons 1st Law.

Elastic adhesion concept of friction Archard 1953

Bowden and Tabor investigated on friction for a purely elastic sliding process. They used a simplified single asperity model of contact based on the Hertzian elastic theory, and found a non-linear friction-load dependence (F = N2/3 ), which clearly contradicted Amontons 1st Law and the experiments. It was Archard (1953), who recognized that there was indeed no contradiction. Instead of assuming a constant number of asperities as Bowden and Tabor did, Archard assumed a load dependent number of asperities. With this assumption the controversy between the elastic multiple asperity hypothesis and Amontons 1st Law could be resolved.

Lubricated friction 2.2.2.a Viscous friction model

Until lubrication was studied pragmatically, it was Nikolai Pavlovich Petrov and Osborne Reynolds around 1880, who recognized the hydrodynamic nature of lubrication, and introduced a theory of fluid-film lubrication. Still today, Reynolds' steady state equation of fluid film lubrication given by:

τ = - η h lub v. (2.10)
is valid for hydrodynamic lubrication of thick films where the frictional stress is proportional to both the sliding velocity v and the bulk fluid viscosity η and inversely proportional to the film lubricant thickness h lub .

2.2.2.b Stribeck friction model

The hydrodynamic theory breaks down below a critical thickness threshold that is expressed in the Stribeck-Curve [START_REF] Stribeck | Die wesentlichen eigenshaften der gleit-und rollenlager[END_REF]. Stribeck does not only take into account the fact that the static friction is higher than the sliding friction but also the dependence on sliding speed.

τ = -sign(v) [τ c + (τ s -τ c )] e (-γv) i - η h lub v (2.11)
where τ is the friction stress, v the sliding speed, τ c the Coulomb sliding friction stress, τ s the maximum static friction stress, γ the sliding speed coefficient and i an exponent.

Friction in cold rolling

2.2.3.a Cold rolling lubrication

Lubrication in cold rolling is mandatory to obtain high strip surface quality and protection from wear. Thus, most of cold rolling processes of automotive and packaging steels, except some skin-pass rolling, are lubricated using different lubricant types and technologies. A cold rolling lubrication can be classified into three regimes, as illustrated in Figure 2.7:

1. Hydrodynamic regime happens when the oil film is thick enough to separate mostly the roughness of the workroll and that of the strip. In this regime, the strip surface is rather freely deformed so there is an apparition of grain poly-crystallin structure on the strip surface after rolling. The friction coefficient is very low in this regime.

2. Limit regime is the one when the oil film is very thin and there is a strong steel-steel interaction between strip and work-roll leading to very high friction coefficient. In this case, the work-roll roughness is printed or transferred onto the strip surface. After rolling, the grooves appear on the strip surface with roughness pattern similar to that of the work-roll.

3. Mixed regime with intermediate oil film thickness. There exist alternatively the limit and hydrodynamic contacts. The friction coefficient is medium and the obtained strip surface is a mix of groove and bump areas.

In general, the cold rolling process aiming at making quite high reduction and providing good strip quality is in mixed regime for the best compromise between high quality surface and low friction coefficient. At the last stand of automotive steel rolling when the strip roughness need to be controlled, the limit lubrication is required by using very high work-roll roughness. This limit regime allows a good transfer of work-roll roughness onto the strip. Skin-pass rolling is also usually in this regime for the same reason.

Rolling process modeling reviews

2.2.3.b Adopted friction models for cold rolling

As mentioned above, the objective of this study concerns especially the rolled strip and work-roll deformation but less the tribology model. And in the point of view of strip model, the tribology model is an input giving the distribution of the friction stress along the contact area. Then, if someone is interested in modeling of the whole system, with any disposed tribology, it is possible to couple the three models as showed in the section 2.1.4. In the present thesis any friction model can be classified into two families.

Load-linear-dependent friction model

τ(x) = -µ(x)σ n (x)dir (v(x)) (2.12)
that includes the Coulomb's friction law stating that µ is constant as a function of sliding velocity.

Load-independent friction model τ(x) = -mxσ 0 (x)dir (v(x)) (2.13) where dir(v) is the unity direction vector of the sliding velocity v at the point x. This model includes Tresca friction law.

The coefficients distributions µ(x) and m(x) can be more or less complicated. However, they are given by tribological models as an input of our developed models.

Typical strip models 2.3.1 0D and 1D models -Analytical models

In the beginning of the 20th century when the computer science notion had never ever existed, there were many efforts to develop analytical rolling models with objective to estimate macroscopic rolling parameters such as roll force and torque. The 0D models are those with analytical solution of roll pressure and friction distribution. This solution is sufficiently simple that the roll force and torque calculated by the integration of the roll pressure and friction can be obtained analytically. Otherwise, when it is necessary to solve numerically the differential equations concerning roll pressure along the roll-bite, they are 1D models. In those models, the same physical simplifications such as plane strain deformation, rigid-plastic behavior with constant stress, dry slipping friction are generally used. Many of them use also the hypothesis of homogeneous deformation (slab method base) except the Orowan's theory [START_REF] Orowan | The calculation of roll pressure in hot and cold flat rolling[END_REF] and derived models. And it is then in the nature of mathematical approximations that these various slab method based models differ.

In order to understand the simplifications adopted by different analytical models, let's study in the first place the 1D theory based on slab method.

Rolling process modeling reviews

2.3 Typical strip models

2.3.1.a Homogeneous deformation -Slab model

Hypotheses and simplified constitutive equation: A

The first hypothesis is plane strain deformation implying that there is no deformation in width direction and other strain components as well as strain rate and stress fields are constant in strip width direction Oy. This hypothesis includes of course that the roll profile is straight or the roll-gap is constant in width direction. The problem is thus a 2D one where all mechanical fields depend on x and z direction. The plane strain condition can be written as follows, ∀x, z: The second hypothesis considers that the deformations are homogeneous in the thickness direction all along the roll-bite which implies therefore all mechanical fields are constant strip thickness direction. In other words, a material vertical slab will stay vertical all along the roll-bite. And this is the reason why the method is called "slab method". Furthermore, combining with plane strain hypothesis, we deduce that all mechanical fields are only x-dependent. This model becomes, thus a 1D model.

ǫ yy (x, z) = 0 d yy (x, z) = 0 (2.14) h z h e n ) (x xx ) ( dx x h s O ) ( dx x xx x O ) (x zz
The third hypothesis consists in neglecting all shear stresses. The stress tensor is, hence:

σ =   σ xx (x) 0 0 0 σ yy (x) 0 0 0 σ zz (x)   . (2.15)
The hydrostatic and deviatoric tensors related to this stress tensor are:

P = 1 3 σ xx (x) + σ yy (x) + σ zz (x) I (2.16) S = 1 3   2σ xx (x) -σ yy (x) -σ zz (x) 0 0 0 2σ yy (x) -σ xx (x) -σ zz (x) 0 0 0 2σ zz (x) -σ xx (x) -σ yy (x)   .
(2.17)

The forth hypothesis says that the strip behavior is rigid-plastic with an yield stress depending on the strain and strain rate. As these fields depend only on x, then σ 0 (ǫ, ǫ) = σ 0 (x) or k(ǫ, ǫ) = k(x). The hypothesis announces also that the material behavior verifies Von-Mises plastic criterion. Thus, under the roll-bite we can write: This equation and the 2nd equation of 2.14 (plane strain deformation hypothesis) imply that S yy = 0, thus:

S : S = 2 3 σ 2 0 (x) = 2k 2 (x). ( 2
σ yy (x) = σ xx (x) + σ zz (x) 2 .
(2.20)

Finally, under these forth hypothesis, the stress tensor can be simplified as:

σ(x, y, z) =   σ xx (x) 0 0 0 1 2 [σ xx (x) + σ zz (x)] 0 0 0 σ zz (x)   (2.21)
and the deviatoric tensor becomes:

S = 1 2   σ xx (x) -σ zz (x) 0 0 0 0 0 0 0 σ zz (x) -σ xx (x)   . (2.22)
By consequence, the Von Mises plastic criterion (2.18) deduces that:

[σ xx (x) -σ zz (x)] 2 = 4k 2 (x)
And noting that the strip material is compressed more in thickness direction than longitudinal one, i.e σ xx (x)σ zz (x) > 0, we obtain as follows the simplified constitutive equation of the slab method:

σ xx (x) -σ zz (x) = 2k(x).
(2.23)

The stress tensor can be finally expressed by only one 1-variable function as follows:

σ =   σ xx (x) 0 0 0 σ xx (x) -k(x) 0 0 0 σ xx (x) -2k(x)   . (2.24)
Equilibrium equations A

The equilibrium in x direction of a slab under contact (see Figure 2.8) is written as:

d [σ xx (x).h(x)] = (σ n (x) sin γ -τ(x) cos γ) dx cos γ then d dx [σ xx (x).h(x)] = σ n (x) tan γ -τ(x) (2.25)
where γ is the angle between the local normal vector n of contact surface and the vertical direction Oz, positive by convention. As the roll shape h(x) is given, γ can be also calculated as:

tan γ = -h ′ (2.26)
Therefore, we can rewrite 2.25 as follows:

d dx [σ xx (x).h(x)] = -σ n (x)h ′ (x) -τ(x) (2.27)
The equilibrium in z direction of the same slab leads to:

σ zz (x) dx = -(σ n (x) cos γ + τ(x) sin γ) dx cos γ that deduces: σ zz (x) = -σ n (x) + τ(x)h ′ (x).
(2.28) By substituting 2.28 into 2.23 we obtain:

σ xx (x) = 2k(x) -σ n (x) + τ(x)h ′ (x) (2.

29)

Friction law A As discussed previously, two families of friction models are studied: load dependent represented by 2.12 including Coulomb's dry model and load independent given by 2.13 containing Tresca's friction model. By definition, the neutral point is the point where the tangential velocity of strip surface velocity is equal to the roll one. Thus, before the neutral point, the roll velocity is higher than that of the strip and the friction is conventionally positive and inversely it is negative after.

Load dependent friction model: A Using 2.12 and the convention above, the friction can be written as:

τ(x) = ±µ(x).σ n (x) (2.30)
where: + before andafter the neutral point. A friction coefficient issued from a general tribological should be in general form as µ(x, y). Nevertheless in plane strain deformation, all mechanical fields are independent of y, the friction does either: µ = µ(x). We will keep this expression of µ(x) in the latter equations even if in practice, all 0D and 1D existing models consider a constant friction along the roll-bite due to the complexity of contact lubrication modeling (see section 2.2).

Using the equation 2.30 and by eliminating then σ n in the equations 2.27 and 2.29, we can obtain the following ordinal differential equations of σ xx (x): before neutral point:

d dx [σ xx (x).h(x)] = [σ xx (x) -2k(x)] µ(x) + h ′ (x) 1 -µ(x)h ′ (x)
after neutral point :

d dx [σ xx (x).h(x)] = [σ xx (x) -2k(x)] -µ(x) + h ′ (x) 1 + µ(x)h ′ (x) (2.31)
Load independent friction model A Similarly, with 2.13 and the convention above, the friction is given by: τ(x) = ±m(x).σ 0 (x).

(2.32)

Also by eliminating σ n in the equations2.27 and 2.29, the final ordinal differential equations of σ xx (x) for loadindependent friction model (Tresca's friction law) 2.32 are obtained as follows:

before neutral point:

d dx σ xx (x) = 2k(x) h(x) -h ′ (x) -m(x) 1 + h ′ 2 (x) 2 after neutral point : d dx σ xx (x) = 2k(x) h(x) -h ′ (x) + m(x) 1 + h ′ 2 (x) 2 (2.33)

Boundary conditions and resolution algorithm A

With the hypothesis that the stresses are homogeneous in the thickness in the roll-bite, we deduce that at the entry and exit of the roll-bite:

σ xx (X e ) = T e σ xx (0) = T s . (2.34)
The equations 2.31 or 2.33 and the boundary conditions 2.34 allow to determine completely the two curves of σ xx (x), one before and the other after the neutral point. The intersection of these two curves determines the neutral point.

This resolution is in our days very easy thanks to finite different method and the obtained calculation time could be much less than few milliseconds with hundreds of slabs in the roll-bite. Nevertheless, the numerical resolution of the previous equations seemed to be impossible during many years before the existence of computer. For this reason, there were many attempts to simplify them in order to develop analytical solutions.

2.3.1.b Rigid-plastic slab based models

The very first investigation in rolling process is Karman's study [START_REF] Karman | Beitrag zur theorie des walzvorgangesbeitrag zur theorie des walzvorganges[END_REF] in 1925. He is the first who wrote out the equilibrium of slabs under a form of differential equation as 2.25. Karman is then considered as the pioneer and founder of slabs method. Various simplification methods of von Karman equation lead to various solutions accordingly. Using small angle approximation, Tselikov [START_REF] Tselikov | Effect of external friction and tension on the pressure of the metal on the roll in rolling[END_REF] deduced a simplified integration considering entry, exit tensions and a rigid perfectly plastic material without work-hardening. Unfortunately, when comparing the roll force with the measurements, he used too high Coulomb friction coefficient of 0.6. Nadai [START_REF] Nadai | The forces required for rolling steel strip under tension[END_REF] also used small angle assumption but applied for different friction laws: Coulomb, constant and slip velocity dependent.

We can obviously remark that, depending on the complexity of the function h ′ (x) that the differential equations 2.31 and 2.33 may have or not analytical solutions. Hence, one of the most common simplification assumption is to consider that the roll is circular so that the function h ′ (x) can be expressed simply as: h ′ (x) =tan θ where θ is the angle position (see Figure 2.8). Before 1935, Karman [START_REF] Karman | Beitrag zur theorie des walzvorgangesbeitrag zur theorie des walzvorganges[END_REF], Siebel [100] and Ekelund [34] considered rigid roll with initial radius R while other authors, after 1935 mostly used Hithcock's circular roll deformed radius R de f [START_REF] Hitchcock | Roll neck bearings[END_REF] depending on the roll force (see section 2.4). In some cases, as the strip model is simplified it can be coupled with Hitchcock's model analytically or otherwise by iteration method as explained in the section 2.1.4.

To understand more in detail the foundation of these models, let's study the two analytical models proposed by Bland & Ford [START_REF] Bland | The calculation of roll force and torque in cold strip rolling with tensions[END_REF] and by Alexander [START_REF] Alexander | On the theory of rolling[END_REF] correspondingly to Coulomb's and Tresca's friction laws.

Bland and Ford model test

In 1948, Bland and Ford [START_REF] Bland | The calculation of roll force and torque in cold strip rolling with tensions[END_REF] proposed an analytical solution in the case of circular work-roll, homogenous deformation, small angle assumption and Coulomb's law with a constant friction coefficient all along the roll bite. Under the circular work-roll deformation and small angle assumptions, we note that:

             x = -R sin θ dx = -R cos θ dθ ≃ -R dθ h = h s + R(1 -cos θ) ≃ h s + R 2 θ 2 h ′ = -tan θ ≃ -θ (2.35)
And the equation 2.28 implies that:

σ zz = -σ n + τ.h ′ (x) ≃ -σ n (1 + µθ) ≃ -σ n .
Then, by replacing σ xx from 2.23 into the final equation 2.31 corresponding to Coulomb's friction law, we obtain (only the equation before the neutral point is studied because the one after will be deduced easily then):

d -R dθ [(2k + σ zz ) .h] = σ zz µ -θ 1 + µθ (2.36)
The left side term of this equation can be developed as:

d dθ 2kh 1 + σ zz 2k = d dθ (2kh) 1 + σ zz 2k + 2kh d dθ σ zz 2k .
(2.37)

In the condition of small angle, the author highlighted that the first term of the right hand side of the equation 2.37 is negligible before the second one. By the way, as the term µθ ≪ 1, it is neglected in the equation 2.36 and this equation is then simplified as:

2kh d dθ σ zz 2k = -Rσ zz (µ -θ) .
We can rewrite this equation as:

d dθ σ zz 2k = σ zz 2k R (θ -µ) h s + R 2 θ 2 (2.
= Rθ dθ h s + R 2 θ 2 - Rµ dθ h s + R 2 θ 2 .
(2.39)

Finally: after neutral point:

d σ zz 2k σ zz 2k = dh h -2µ R 2h s R 2h s dθ 1 + R 2h s θ 2 . ( 2
σ zz = -2kh.K s e µ.H with H = 2 R 2h s arctan θ R 2h s .
(2.41)

K e and K s are two constants defined by the two boundary conditions at entry and exit as a function of entry and exit tensions and flow stresses. The neutral point is then determined by the equality of the two curves of sigma zz before and after neutral point given by 2.41. As can be seen, Bland and Ford solution is analytical for both longitudinal and vertical stresses as well as the contact pressure distribution along the roll bite. However, the roll force and torque can not be obtained analytically and need approximating numerically.

Alexander model test

Always keeping the circular work-roll hypothesis but unlike Bland and Ford, Alexander [START_REF] Alexander | On the theory of rolling[END_REF] discards the assumption of small angle and considers Tresca's friction law (also with constant friction coefficient along the roll bite) and a rolled material with constant yield stress (without work-hardening). Using 2.35 but without small angle approximation, the equation 2.33 becomes (before neutral point):

dσ xx -R cos θ dθ = 2k h tan θ - m 2 cos 2 θ which implies: dσ xx dθ = - 2kR h sin θ + mkR h cos θ or dσ xx dθ = - 2k h dh dθ + mk 1 + h s R cos θ -cos 2 θ . (2.42)
Finally by integrating analytically from the entry and similarly from the exit, Alexander obtains an analytical solution as follows:

before neutral point:

σ xx = T e -2k ln h h e + mk [G(θ) -G(θ e )]
after neutral point :

σ xx = T s -2k ln h h s -mkG(θ) with G(θ) = R R + h s        ln tan θ 2 + π 4 + 2 1 + h s R 2 -1 arctan tan θ 2 1 + 2R h s        (2.43)
The contact pressure can be calculated from this solution and the equation 2.29.

2.3.1.c Inhomogeneous deformation -Orowan theory

Unlike previous models, Orowan [START_REF] Orowan | The calculation of roll pressure in hot and cold flat rolling[END_REF] discarded constant yield stress assumption and introduced strain-stress curve through σ 0 (θ). Especially, he attempted to consider inhomogeneity of deformation in thickness direction based on Prandtl's [START_REF] Prandtl | [END_REF] and Nadai's theory [START_REF] Nadai | Plasticity[END_REF]. According to Prandtl's analytical stresses solution for the problem of plane strain compression between two parallel plates with sticking friction, the shear stress is distributed linearly in the slab thickness and therefore the relation between longitudinal and vertical components is:

       σ xz = - σ 0 √ 3 z h σ xx = σ zz + 2 √ 3 σ 0 1 - z 2 h 2 .
(2.44) And Nadai's solution for plane strain compression between inclined plates with a sticking friction is quite similar but he used polar co-ordinates instead of cartesian ones:

         σ xz = - σ 0 √ 3 ϑ θ σ xx = σ zz + 2 √ 3 σ 0 1 - ϑ 2 θ 2 .
(2.45) Figure 2.9: Illustration of Orowan model.

Orowan assumed that the rolling deformation resembled the compression between inclined plates to take into account heterogeneity of deformation in rolling. Moreover, he adapted Nadai's solution for a general, slipping-sticking friction law that is defined as follow:

         λ = min 1, √ 3µσ n σ 0 τ = ±λ σ 0 √ 3 .
(2.46)

He obtained finally:

       d dθ (σ xx .h) = R de f σ n sin θ ± R de f τ cos θ σ xx = σ n - 2σ 0 √ 3 W(λ, θ) ± τ 1 θ - 1 tan θ (2.47)

Rolling process modeling reviews

Typical strip models

where σ xx is defined as the average of σ xx in the strip thickness and W(λ, θ) is a geometrical function given by:

W(λ, θ) = 1 sin θ θ 0 1 -λ 2 ξ θ 2 cos ξ dξ.

Discussions A

It is important to note that with this friction law, Orowan's theory is able to model both slipping or sticking areas alternatively along the roll-bite and that makes Orowan theory advantageous compared to previous ones. In general, when the sticking friction occurs, it is first at the neutral point corresponding to the maximum pressure and expands with increasing of friction coefficient to the entry and exit. Further, as can be seen, with the inhomogeneity of the plastic flow taken into account the differential equations 2.47 are not more complicated than with assumption of homogeneous compression (slab method). The resolution algorithm is therefore the same. The model showed excellent results compared to Siebel and Lueg's measurements [START_REF] Siebel | [END_REF].

However, the computing time is significantly higher due to the integration computation of W(λ, θ). That is why during several decades, being used as a standard against which many other models are often compared, Orowan's theory has not been applied to mill design and operation because of expensive time needed for calculations.

Regarding inhomogeneity of deformation, it is certainly difficult to judge if the rolling compression is similar to that of inclined compression plates problem. It is however interesting to just highlight that in the chapter 4, a new model of inhomogeneous and non-linear plastic deformation will be introduced. The deformation is supposed oscillation mode, implying that the shear stress is strongly non-linear and oscillates with a period equal to strip thickness function h(x). These oscillation are obtained by careful Lam3-Tec3 , finite-element method calculations as well as by UBM . The shear stress variation in thickness direction is indeed, more complicated than a linear relation.

2.3.1.d Orowan-based models

A

Since the publication of the Orowan model in 1943, a lot of work was carried out to explore or simplify its application for gains of computing time [START_REF] El-Kalay | Factors affecting friction and their effect upon load, torque and spread in hot flat rolling[END_REF][START_REF] Venter | Modelling of the rolling process -i: Inhomogeneous deformation model[END_REF][START_REF] Venter | Modelling of the rolling process -ii: Evaluation of the stress distribution in the rolled material[END_REF][START_REF] Freshwater | Simplified theories of flat rolling -i: The calculation of roll pressure, roll force and roll torque[END_REF][START_REF] Oda | Adaptive technology for thickness control of finisher set-up on hot strip mill[END_REF][START_REF] Okamoto | Measurement and estimation of mean flow stress and coefficient of friction in tandem mills[END_REF]. Here following are presented two examples. The frist one, Sims's model, with full sticking friction is very famous and largely used in industrial rolling process control. The other, recently developed attempts to keep more originalities of Orowan theory concerning mixed friction and work-hardening effect.

Sims model A

Assuming full sticking with constant yield stress and small angle approximation, Sims [START_REF] Sims | The calculation of roll force and torque in hot rolling mills[END_REF] solved Orowan's differential equation analytically and derived algebraic formulae for roll pressure distribution, neutral angle, roll force and torque as follows 2 :

                     F = 2 √ 3 σ 0 2R de f δQ f Q f = h s δ   π 2 arctan δ hs - R de f 2h s 1 2 ln h e h s -ln h n h s   - π 4 Torque = 2 √ 3 σ 0 R de f R (θ e -2θ n ) .
(2.48) 2 The formula is slightly different from that of [START_REF] Sims | The calculation of roll force and torque in hot rolling mills[END_REF] because of different notations

Li's model A

In 2006, based equally on the Orowan theory but unlike Sims, Li et al [START_REF] Li | Roll force model for online application in hot strip rolling with varying friction conditions[END_REF] discards the assumption of sticking friction. They also use the small angle hypothesis and by assuming that the function W(λ, θ) is independent of θ that seem to be justified (see the graphic of W in the p. 20 of [START_REF] Orowan | The calculation of roll pressure in hot and cold flat rolling[END_REF]). This function becomes a simple function of λ, W = W(λ). By consequence algebraic equations of roll pressures in the forward and backward slip zones have been derived by solving the two Orowan equations 2.47 analytically. The roll force and torque are then integrated numerically and the model is already fast because no integration of w is needed. Further, by approaching the roll pressure distribution as a quadratic function of θ by part (before and after the neutral point), its is only necessary to compute the roll pressure at 5 points (2 before, 2 after and the neutral point) to get a quadratic approximation. The roll force and torque are then analytical integration from this approximated solution. Finally this quadratic approximation requires extremely low calculation time while its errors compared to Orowan's exact solution are lower than 1% for rolling conditions of a 7-stand and 3-stand aluminium mills.

2.3.1.e Taking into account elasticity -Bland and Ford 1952

After the Orowan theory and the simplification developments, some latter studies brought out the inaccuracy of these models despite of their sophistication allowing for inhomogeneous deformation and slipping-sticking friction. This inaccuracy is indeed due to the elastic deformation and can be important for estimation of both roll force and much more in the case of roll torque. It is predominately significant for small thickness reduction ratio, less than 10%. Ford et al [37], Bland & Ford [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] and Bland & Sims [15] made considerable modifications to the simplified theory of cold rolling developed by Bland & Ford [START_REF] Bland | The calculation of roll force and torque in cold strip rolling with tensions[END_REF]. Until the work of Bland and Force in 1952 [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF], non of the theories of rolling, nor the calculations of roll force and torque have taken into account the effect of the elastic zones. While these elastic zones do not only have direct influence on the roll force and torque but also change the entry and exit tensions of the plastic zone. Bland and Ford proposed a method for approximating these contributions of entry and exit elastic zones to the roll force and torque.

Elastic recovery: Let study first their analysis for the elastic recovery zone at the exit. At first, the authors assume plane strain deformation for all plastic and elastic zones, even if this hypothesis leads to a discontinuity of stresses due to the difference of Poisson's coefficient ν = 0.3 in elastic zone and 0.5 in yield criterion. They consider in elastic recovery zone the longitudinal stress (tension) changes but not much and can be approximated constant and equal to T s . The vertical stress varies from a certain value at the lowest point of the work-roll and becomes 0 at the last point of contact x = a. The lowest point is also the last point of plastic deformation. All shear stresses are neglected.

Bland and Ford consider in this elastic zone that: σ yy = ν (σ xx + σ zz ) then by Hooke's generalized law:

ǫ zz = - ν(1 + ν) E σ xx + 1 -ν 2 E σ zz σ zz = E 1 -ν 2 ǫ zz + ν 1 -ν σ xx .
We remark that in fact this hypothesis is too strict because at the last point of contact x = a the plane strain deformation is not really verified. This point will be discussed more in the latter chapter 7, section 7.4.1. In fact, in this zone only the variations of stresses verify plane strain hypothesis, meaning:

∆ǫ zz = - ν(1 + ν) E ∆σ xx + 1 -ν 2 E ∆σ zz .
If the reference is chosen at the point x = a where σ zz = 0 and reminding that ∆σ xx ≃ 0, we have:

∆ǫ zz = 1 -ν 2 E σ zz .
(2.49)

We will continue to use this weaker hypothesis instead of plane strain one because indeed it does not change the results of Bland and Ford analysis.

If ∆u z denotes the variation of vertical displacement on the upper surface with respect to the reference point x = a, z = h s , as the upper surface is in contact with the work-roll we have:

∆u z = - 1 2R de f a 2 -x 2 .
(2.50)

And by definition,

∆u z = h 0 ∆ǫ zz dz. (2.51)
By substituting the equations 2.49 and 2.50 into 2.51 and using h ≃ h s , σ zz is constant in the thickness, we get:

u z = - 1 2R de f a 2 -x 2 = h 0 1 -ν 2 E σ zz ≃ h s 1 -ν 2 E σ zz
which deduces finally:

σ zz = - E 2 (1 -ν 2 ) R de f h s a 2 -x 2 .
(2.52)

Now, the yield criterion at the x = 0 implies:

σ xx (x = 0) -σ zz (x = 0) = k(0) = k s ⇒ σ zz (x = 0) ≃ T s -k. (2.53)
By comparing two equations 2.52 and 2.53, we can determine a as follows:

a 2 = 2 1 -ν 2 R de f h s E (k -T s ) . (2.54)
The contribution of this elastic zone to the roll force, torque and tension can be estimated by:

P elas s = - a 0 σ zz dx = Ea 3 3 (1 -ν 2 ) R de f h s G elas s = -µRP elas s T plas s = T s - µP elas s h s .
(2.55)

Elastic compression at entry: With a similar analysis the authors obtain equally a parabolic distribution of vertical stress (similar to 2.52) as follows:

σ zz = - E 2 (1 -ν 2 ) R de f h e L 2 -x 2 ≃ - E.L (1 -ν 2 ) R de f h e δx.
(2.56)

The first point of yield criterion is determined by σ zz = T ek, thus:

δx e = 1 -ν 2 R de f h e E.L (k -T e ) (2.57)
where L is the contact length. The contribution of this elastic compression zone to the roll force, torque and tension are given by:

P elas e = - δx e 0 σ zz dδx = E.L 2 (1 -ν 2 ) R de f h e δx 2 e = 1 -ν 2 R de f h e 2E.L (k -T e ) 2 .
G elas e = µRP elas e T plas e

= T e -µP elas e h e .

(2.58)

Discussions: This very first work taking into account the elasticity in rolling showed out that the elastic zones (especially the exit one) have an appreciative effect on the total roll force and torque. And these results of Bland and Force being completely analytical are largely used for online applications. Nevertheless, this is not the exact solution of slab method for elastic behavior because of many approximations (small angle even for entry elastic compression zone, small perturbations...). And the most important point that the authors also brought out is that in the plastic zone the elastic component of strain is not taken into account by their model and it has never been done previously. Until 1968, Cosse et al [START_REF] Cosse | Mathematical study of cold rolling[END_REF] solved this problem by building the most complete model based on slab method for elasto-plastic behavior.

2.3.1.f Elasto-plastic slab model -Cosse et al 1968

Cosse et al [START_REF] Cosse | Mathematical study of cold rolling[END_REF] assume the main hypotheses of slab method such as plane strain, homogenous deformation in thickness. They considers Coulomb slipping friction (no sticking) law with a constant friction coefficient along the roll bite. On the other hand, they introduce elasto-plastic behavior of rolled material by using the complete Prandtl-Reuss equations [START_REF] Hill | The Mathematical theory of Plasticity[END_REF]. The material behavior law depends equally on strain rate allowing to model influence of rolling speed. Moreover, the authors attempted to develop the most general theory as possible by using non-circular work-roll deformation model based on the influence functions method allowing good performances even in the conditions small reduction rolling.

Material mechanical behavior: test

Based on literature studies, the authors use an analytical form that takes into account work-hardening effect as follows:

ǫ = σ 0 E + σ 0 B n = σ 0 E + |σ 0 | n-1 σ 0 B n . (2.59)
The dynamic effect can be taken into account through the parameter B without impact on the resolution of equations system.

Equilibrium equation: test

The equilibrium equations for a slab are the same as 2.27 and 2.28. As a reminder, the definition of the angle γ representing the local normal vector of strip surface as tan γ = h ′ . According to us Cosse et al made small mistake in their equilibrium equations using θ instead of γ (γ = θ for non-circular work-roll, see Figure 2.10) as follows:

d dx [σ xx .h] = σ n tan θ -τ = σ n (tan θ -µ) σ zz = -σ n -τ tan θ = -σ n (1 + µ tan θ).
(2.60) However, this is not a big issue and could be easily corrected. Indeed, this fact does not influence the great interest of this model which is how to deal with elasto-plastic behavior. Let follow the authors original equations.

The roll profile is represented by a various radius function R ′ (θ) which, unlike the Hitchcock curved radius R de f , is the real distance from a point on the roll surface to its center. To understand the calculation of thickness, let h 2 is the targeted value of the minimum height of the roll gap (in this article, h 2 is given but not the outgoing height h s as in other ones). The question is how to set the work-roll center vertical position in order to get this targeted minimum gap as the roll is deformed. At first let fix the center so that the non-deformed half of roll gap is equal to h 2 , meaning the distance between the two rolls centers is 2(h 2 + R) and as the roll is deformed the minimum height is equal to h 4 . So to get the minimum height of h 2 we need to set a new position of the roll so that the half of roll gap with non-deformed roll is:

h 0 = 2h 2 -h 4 . (2.61)
. Finally, the strip thickness can be calculated as follows: By neglecting µ tan θ in the 2nd equation of 2.60 and substituting it into the 1st one, the authors obtain following equilibrium equation:

h = 2h 2 -h 4 + R -R ′ (θ) cos θ. ( 2
dσ xx dθ = 1 h R ′ σ zz (sin θ -cos θ) -σ xx (R ′ sin θ -cos θ dR ′ dθ ) . (2.63)

Elasto-plastic compression: test

This zone extends from the entry of the strip between the rolls (θ 1 ) and the minimum height in the gap (θ 4 ). The general equations relating stresses and strains are those of Prandtl-Reuss [START_REF] Hill | The Mathematical theory of Plasticity[END_REF]:

δǫ xx = 1 E δσ xx -ν δσ yy + δσ zz + δǫ p σ 0 σ xx - 1 2 σ yy + σ zz δǫ yy = 1 E δσ yy -ν (δσ zz + δσ xx ) + δǫ p σ 0 σ yy - 1 2 (σ zz + σ xx ) δǫ zz = 1 E δσ zz -ν δσ xx + δσ yy + δǫ p σ 0 σ zz - 1 2 σ xx + σ yy (2.64)
The plane strain condition is:

δǫ yy = 0. (2.65)
In addition to the Prandtl-Reuss relationships, Von Mises plasticity criterion is chosen. This criterion is, in the condition of no shear stresses, given by:

σ 0 = (σ xx -σ yy ) 2 + (σ yy -σ zz ) 2 + (σ zz -σ xx ) 2 2 .
(2.66)

The strip material behavior 2.59 implies that:

δλ = δǫ σ 0 = n B n |σ 0 | n-1 dσ 0 σ 0 . (2.67)
By definition:

ǫ zz = ln h h e (2.

68)

Not showing here the details of calculations, we highlight only that the authors obtain from the equations above the three final differential equations as follows:

dS x dθ = R ′ (S z + S y )(sin θ -µ cos θ) h -A(S x + S y ) - dS y dθ dS y dθ = A G 1 - G 1 -FS x (2S z -S x ) 3G 1 dS z dθ + G -G 1 + FS x (2S x -S z ) 3G 1 dS x dθ dS z dθ = A G + FS z (S z -S x ) - FS z (2S x -S z ) G + FS z (2S z -S x ) dS x dθ (2.69) where                                          G = (1 + ν)E G 1 = (1 -2ν)E S x = σ xx -σ yy S y = σ yy S z = σ zz -σ yy F = 3n 4B n S 2 x + S 2 z -S x S z n-3 2 A = R ′ sin θ -dR ′ dθ cos θ h S 2 x + S 2 z -S x S z = S 2 y = σ 2 0 (2.70)
The equations 2.69 are solutions for the elasto-plastic zone corresponding to the part before the neutral point. For the other part, the sign of friction coefficient inverses.

Elastic recovery: This zone extends from θ 4 to the exit of the contact θ 2 . In this zone, the material behavior follows Hooke's law:

dǫ xx = 1 E dσ xx -ν dσ yy + dσ zz dǫ yy = 1 E dσ yy -ν ( dσ zz + dσ xx ) dǫ zz = 1 E dσ zz -ν dσ xx + dσ yy (2.71)
and the plane strain condition implies dǫ yy = 0, hence:

dσ yy = ν ( dσ zz + dσ xx ) . (2.72)
Then, substituting 2.72 into the third equations of 2.71, we have:

dǫ zz dθ = 1 E 1 -ν 2 dσ zz dθ -ν (1 + ν) dσ xx dθ . (2.73)
Noting that the strip thickness is imposed by the work-roll shape along this elastic zone, then:

dǫ zz dθ = 1 h dh dθ . (2.74)
The equation 2.73 becomes:

dσ zz dθ = E 1 -ν 2 1 h dh dθ + ν 1 -ν dσ xx dθ . (2.75)
The equilibrium equation 2.63 and the equation 2.75 form a system of two differential equations which represent the solution in the elastic recovery zone.

Rolling process modeling reviews

Typical strip models

Roll flattening by influence functions method:

The authors use the method proposed by Jortner [START_REF] Jortner | An analysis of cold strip rolling[END_REF], see the section 2.4.2.b.

Resolution: Here following is the principles of numerical resolution:

1. In the first iteration, the roll is considered to be non-deformed and the h 0 = h 2 . In subsequent iterations, the deformation of the roll corresponding to the pressure distribution found in the preceding iteration is calculated. The angle θ 4 as well as the minimum height h 4 are then determined. The new position of the roll is estimated with

h 0 = 2h 2 -h 4 .
2. Determine entry angle θ 1

3. Chose initial value of neutral angle θ N .

4. Solve the system of differential equations 2.69 for the elasto-plastic compression zone using initial conditions at θ = θ 1 as σ xx = T e and σ yy = σ zz = 0. The system is solved numerically according to an improved Runge-Kutta method [START_REF] Chai | [END_REF]. In each step, the instantaneous strain rate is calculated first and then the material law parameters B and/or n are deduced. The θ is compared to θ N to decide the sign of the friction. This sequence of calculations is repeated until θ 4 .

5. Solve the differential equations 2.63 and 2.75. The initial conditions at θ = θ 4 are the stresses σ xx and σ zz issued from the exit of the elasto-plastic zone. The neutral point can also be present in this zone so the thickness h is compared to the thickness h N to decide the sign of friction in the equation 2.63. This operation is repeated until the value of σ zz reaches 0, to within certain accuracy. The angle θ 2 as well as σ xx and σ zz are thus defined.

6. Compare the tension σ xx (θ 2 ) obtained to the exit tension T s . If they are not identical, it means that the value θ N is not correct. Chose a new value of θ N and restart the calculations from the step 4 (elasto-plastic zone) to step 6 until σ xx (θ 2 ) = T s within a given accuracy.

7. Calculate the foll force and compare to the values obtained by two previous iterations. If they are less different than a given precision, the solution is obtained. If not, restart from step 1 with new contact stresses distribution.

Discussions: test

Cosse and al proposed a very general theory for cold rolling with minimum as possible of assumptions both in the strip and the roll. This elasto-plastic solution allows to overcome many questions previously put out, such as the discontinuity of σ yy at the limit between elastic and plastic zones because of the Poisson's coefficient of 0.3 (for elastic zone) and 0.5 (for plastic zone) in plane strain conditions. Moreover, unlike Bland & Ford, the plane strain is correctly written using the variation of strain is equal to zero. The model can also be accommodated for any material behavior law as well as any friction law (Tresca or Coulomb, contact or variable along the roll bite...). However, it is important to highlight that both Bland & Ford and Cosse et al models neglect shear stress (issued from assumption of homogeneous deformation) in the entry elastic compression zone. That should not be correct. Indeed, at the first contact point du to the brutal change of thickness the shear stress can be very important despite it can be concentrated in a really small zone. It could even be the main reason for the plastic yield at this point. That is why we will propose a very new and simple model for this in the section 7.3. The elastic recovery will be re-studied in the same chapter because these elastic zones have primordial influence on the width variation of the strip that is the objective of the thesis. Cosse model will be the base for our developments concerning the influence of elasticity on the width variation.

2D models

In cold temper rolling processes (skin-pass), as thickness reduction is usually very small, deformation is strongly z-dependent with a concentration near the strip/roll contact). The same is true in hot strip rolling, in particular in the roughing mill because of significant ratio thickness to contact length. In the finishing section, the strains and stresses are more or less z-independent in the last stands, but the strong temperature coupling brings the z-coordinate back in. 2D models are particularly required when we are interested in flatness defects like longbow and crossbow in a top-bottom asymmetric rolling (see Figure 2.11). The longbow defect corresponding to a curvature along the strop is caused by inhomogeneous longitudinal heterogenous residual stress σ xx and crossbow representing by a curvature across its width is due to heterogenous residual stress σ yy . In such cases, 1D approaches are inadequate and must be replaced by plane strain 2D models, whereby fields depend on 2 space variables x and z. In the literature, several types of methods are able to make these requirements met. The first one is Finite Difference Methods (FDM ) [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF] and [START_REF] Orban | Shape control in cold rolling[END_REF]. Of course FEM [START_REF] Lee | New solution to rigid-plastic deformation problems using a matrix method[END_REF][START_REF] Lahoti | Computer-aided analysis of deformations and temperatures in strip rolling[END_REF] are useful to quote pioneering work. FEM models are generally quasi-static implicit [START_REF] Li | Rigid-plastic finite element analysis of plane strain rolling[END_REF], sometimes dynamic explicit [START_REF] Lau | An explicit time integration elastic-plastic finite element algorithm for analysis of high speed rolling[END_REF]. And recently, the Element-Free Galerkin method [START_REF] Xiong | Application of the element-free galerkin method to the simulation of plane strain rolling[END_REF]. These methods may use any kind of constitutive equations. The last commonly used method for 2D rolling is the Upper-Bound or Upper-Bound-like methods with velocity fields based on streamlines generally, using either visco-plastic or rigid-plastic slightly compressible. UBM -based 2D models will be presented in the chapter 3.

Unlike 1D models, 2D ones could approach physical solutions of shear strains and stresses without any assumption as proposed by Orowan. The z-dependence of velocity, strain, stress and temperature are equally modeled and this fact enables to predict profiles of residual stress distribution in strip thickness as well as profiles of temperature and microstructure. Moreover, they are able to deal with extremity effects as well as unsteady state while 1D models are confined to stabilized state. On the other hand, these models are more time-consuming and there is not really significant development of simplified solutions of 2D models except some UBM -based ones.

3D models

When objective of the study concern residual stresses, flatness defects or when we are interested in the distribution of any parameters in width direction Oy, 3D models become necessary. These models are also required otherwise for any case where the plane strain deformation is no longer valid, for example rolling process of relatively narrow and thick products. Obviously, for the width variation, 3D models should be inevitable. The methods include generalized Upper Bound Methods, such as Finite Strip Methods with rigid plastic behavior [START_REF] Liu | Strip layer method for simulation of the 3d deformations of plate and strip rolling[END_REF]. 3D models are often FDM [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF][START_REF] Orban | Shape control in cold rolling[END_REF], and widely FEM with dominantly implicit ( [START_REF] Liu | Analysis of stress and strain distributions in slab rolling using an elastic-plastic finite-element method[END_REF] and many others) or sometimes dynamic explicit formulations.

Typical work-roll deformation models

For a 4-High stand, the most commonly used in automotive and packaging steel rolling, the roll force is usually applied at the ends of the backup-rolls. The work-rolls are at the same time under the contact pressure with the backuprolls and the strip. The roll torques are very often driven directly on the work-rolls. The work-roll deformation can be separated into two modes. The first one, called bending is the roll deformation in the transverse-vertical plane Oyz, as can be seen in Figure 1.8. The deformation of the roll axis generally can be calculated by beam theory. That can result to a deformed strip thickness profile with a maximum at the center and a slight decreasing toward the edges.

The second mode caused by strip contact pressure is called flattening. That is the local deformation of the roll in the plane Oxz reducing in fact the distance of the work-roll surface material point to its axis. In addition, at the strip edges the roll profile is locally reflected in the transverse vertical plane Oyz because of the discontinuity of contact 2. Rolling process modeling reviews 2.4 Typical work-roll deformation models pressure (see Figure 2.12). By consequence, the strip thickness is reduce more at its edges, called often edge drop phenomenon.

Bending and flattening in width direction Oyz -3D modeling

As mentioned above, 2D and 3D modeling is not the main subject of the thesis, then it is so for the work-roll deformation in 3D modeling. That is why although there exist many interesting works using FDM , FEM and analytical methods to model the bending and flattening at the edge of the strip, they are not presented in this document.

On the other hand, we present only in some words about Tec3 a work-roll deformation model developed by Arcelor-Mittal. Beam + influence functions for flattening in both planes xz and yz. The combination of Lam3 and Tec3 becomes referent model and will be used to estimate performance of models developed in this thesis.

Flattening models for 2D modeling 2.4.2.a Hitchcock model

The very first exiting model of roll deformation is Hithcock's one [START_REF] Hitchcock | Roll neck bearings[END_REF] for roll flattening. Based on the analysis of the elastic Hertz contact, he considers the flattened roll stay circular in the roll-bite but with an increased radius. This deformed radius is explicitly given by:

R de f = R 1 + 1 -ν 2 r πE r F δ . (2.76)
Thanks to simplicity, satisfying results and the fact that the deformed roll is always circular allowing many simplifications of strip models, this formula is largely used 1D and 2D models.

Latter, taking into account the elasticity in the deformation of the strip see te section 2.3.1.e, Bland & Ford [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] introduced two additional contact arcs at entry and exit of the roll bite under which the strip is only deformed elastically. Due to these elastic arcs the contact length is higher and by consequence the roll is less deformed for a same given roll force. That is why they proposed to correct Hitchcock's formula as follows:

R de f = R 1 + 1 -ν 2 r πE r F √ δ + δ 1 + δ 2 + √ δ 2 2 .
(2.77)

where δ 1 and δ 2 are defined as follows:

       δ 1 = ν r (1 + ν r ) E r (T s h s -T e h e ) δ 2 = h s 1 -ν 2 r E r (k s -σ s ) (2.78)
Hitchcock's model more and more valid if the rolled material is softer or the ratio thickness over roll diameter is higher or the reduction is larger. This model is very commonly used in hot rolling domain including roughing as well as finishing mills. For cold rolling, it is only valid the firsts stands of automotive tandem mills. For rolling of packaging products (very low thickness down to 0.17mm) the roll is strongly deformed and can not be approached by this model.

An other case where this model reaches is not accurate is skin-pass rolling where the reduction is generally from 0.2% to 3%.

2.4.2.b Jortner model -Influence functions

Jortner model [START_REF] Jortner | An analysis of cold strip rolling[END_REF] closely approximates the condition found in a 4-high mill where the back-up roll supplies the balancing load. Making use of St. Venant's principle, it is assumed that the actual location and distribution of the backup pressure will have a negligible effect on the radial deformations of points on the contact arc. Thus, this model was based on the elementary problem of a cylinder subjected to diametrical concentrated loads . Based on the exact elastic deformation solution of this problem given by Timoshenko and Goodier [START_REF] Timoshenko | Theory of Elasticity[END_REF], Jortner integrated the radial strain along a radius to obtain the radial deformation of an arbitrary point S(R, θ) as follow: Replacing of the concentrate force by a continuous pressure σ n over the small finite angle 2α as shown in Figure 2.13, not only eliminates the infinite stresses and strains under the point load but also permits an arbitrary pressure distribution to be expressed in terms of finite increments. Performing the integration involving equation 2.79 yields the 2. Rolling process modeling reviews 2.4 Typical work-roll deformation models following result when the point S is outside the pressure area, |θ| > α:

u r (R, θ) = P π.E r . 1 -ν 2 r cos θ ln 1 -cos θ 1 + cos θ + 2 -1 -ν r -2ν 2 r sin θ arctan 1 + cos θ sin θ + arctan 1 -cos θ sin θ (2.79) 2 2 . n S O R 2 2 .
                     u r (R, θ, α) = σ n α π.E r . [M(θ + α) -M(θ -α) + N(θ + α) -N(θ -α)]
with:

M(ξ) = 1 -ν 2 r sin ξ ln 1 -cos ξ 1 + cos ξ N(ξ) = 1 -ν r -2ν 2 r cos ξ arctan 1 + cos ξ sin ξ + arctan 1 -cos ξ sin ξ . (2.80)
When the point S is inside the pressure area, |θ| < α, it is necessary to integrate from both sides but not going through the point S. The final result is:

u r (R, θ, α) = u r (R, θ, α) - σ n α π.E r 1 -ν r -2ν 2 r . (2.81)
Finally, with the influence function established the deformation of a point on the surface of the roll due to a specific pressure distribution can be determined by a sum of finite increments.

In the same article, Jortner has coupled this roll deformation model with a strip model based on Bland & Ford [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] by iterative method as explained in the section 2.1.4. By comparing the roll and torque with sixteen experiments, he obtained quite low errors but it is difficult to judge if the accuracy of the prediction is better than previous models. However, it is felt that this method is at least as accurate and considerably more general than other methods in use.

Remark: It is important to note that the author considered that tangential shearing stresses on the contact arc have a negligible effect on the radial deformations. Therefore, only the radial deformation of the work-roll caused by radial stress (main contributing factor

) is considered in this article.

2.4.2.c Fleck and Johnson -Neutral zone existence

In the domain of thin metallic foil cold rolling (typically packaging product), according to slab method 2.3.1.a, the evolution of normal stress along the roll bite is proportional to 1/h(x) and becomes therefore important for thin strip especially around the neutral point. With classic sliding friction, tangential stress is also important and itself makes the normal stress increase more after resolution. Finally, the normal and tangential stresses are artificially increased around the neutral point. After, Fleck and Johnson [START_REF] Fleck | Towards a new theory of cold rolling thin foil[END_REF], this phenomenon does not really exists because the work-roll should be deformed under this concentrated contact stresses. They consider that there exist a region at the middle of the roll bite where the strip thickness is not reduced. This problem was previously recognized by Quan [START_REF] Quan | Deformation characteristics of the cross shear cold rolling (cscr) of ultra thin strip and the theory of the "elastic plug[END_REF] but Quan assumed that the rolls remain circular.

Rolling process modeling reviews

Discussions

In Fleck and Johnson theory, the work roll is allowed to deform to a non-circular profile and the plastic reduction occurs near entry and near exit of the roll bite, separated by a central region where the strip does not suffer reduction and does not slip relative to the work roll, see Figure 2.14.

For the work-roll deformation, based on Johnson and Bentall [START_REF] Johnson | The onset of yield in the cold rolling of thin strip[END_REF] study for the case of zero reduction, Fleck and Johnson postulated that:

σ n (x) = p 0 1 - x L 2 -K n (h e -h(x)) (2.82)
where p 0 is the maximum normal stress. This expression is Hertzian solution adjusted by a "elastic foundation" perturbation due to the local plastic reduction h eh(x) of the foil at considered point. And K n is a "foundation modulus" constant chosen to give a good match with the true deformation. The resolution method consists in writing in each zone the governing equation of stresses and continuous conditions through successive zones. Iterative calculations are required to determine p 0 , position and length of each zone.

Remark: This model is the first that introduce the presence of non-reduction zone allowing to approach thin and very thin rolling. However, it is not really advantageous because the strip thickness evolution is almost imposed and because the model does not take neither into account the entry and exit tensions nor the strip material work-hardening.

2.4.2.d Matsumoto

Matsumoto [START_REF] Matsumoto | Elastic-plastic theroy of cold and temper rolling[END_REF] is an elastic-plastic model with presence of a neutral zone. The difference compared to Fleck and Johnson model is that, Matsumoto considers that the neutral zone is also the non-reduction one. The roll bite is therefore composed of 5 zones instead of 7 by Fleck and Johnson model. In addition, he points out that in this zone, the plastic thickness is constant by the real apparent thickness is variable due to the stress and elastic deformation variation, meaning the work-roll shape is not really constant, see Matsumoto's model based on influence function model for the work-roll deformation and Orowan model for the strip. But allowing to occur the neutral zone, Matsumoto obtained a quite general rolling model for both cold and temper rolling. His model is robust even for very low reduction such as 0.1% or even lower. He introduces equally a numerical method using a fixed number of elements in the roll bite at a coordinate ξ that is standardized always 0 at exit and 1 and the entry. With this method, he obtains an influence matrix (calculated only once) instead of influence function allowing to improve the computing time to less than one hundred ms.

Discussions

Online process control Although being developed long time ago, these analytical models thanks to their advantage rapidly, are very largely used for industrial rolling process preset and control. For example, Sims's model with sticking friction usually coupled with the Hitchcock's model is very commonly used and gives excellent performances in strip or plate hot rolling where a high friction coefficient occurs. However, when lubricant is applied, a significantly lower friction occurs, that has been observed in hot rolling of steel [START_REF] Schrodter | Roll lubrication trials at eko stahl's hot strip mill[END_REF], [START_REF] Steiner | Optimization of work roll lubrication at the hot strip mill[END_REF] and [START_REF] Peretic | Coordinated application of roll gap lubrication, work roll cooling and antipeeling systems in hot rolling mills[END_REF]. There were many developments of simplified models and there will be more allowing to improve more and more quality of online process control.

In cold rolling mill, in order to cover high range of reduction (low reduction for hard and wide products) the elastic spring back (elastic recovery zone) needs taking into account. The models such as Bland & Ford [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF], [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] and Bryant & Osborn [START_REF] Bryant | Derivation and assessment of roll force models[END_REF], [START_REF] Bryant | Derivation and assessment of simplified models for torque, slip and neutral angle[END_REF] are generally used.

For skin-pass rolling with relatively high friction and small reduction (about 0.2-3.0%) the elastic deformation becomes extremely important. Sometime, the elastic deformation may appear inside the roll-bite, i.e the material deformation is elastic and plastic alternatively. In addition, due to a very small contact length the work-roll deformation is strongly non-circular. By consequence, the models taking into account strip elasticity and non-circular deformation of the work-roll such as [START_REF] Fleck | Towards a new theory of cold rolling thin foil[END_REF][START_REF] Matsumoto | Elastic-plastic theroy of cold and temper rolling[END_REF] could be enough accurate for this process.

Offline applications

The 2D and 3D models are of course extremely important understanding of physical phenomena and to explore more in detail where the measurements are not accessible. These model are, therefore generally used for pioneering studies of new subjects, or as a standard to validate simplified models. They are sometimes used for finding the causes of certain defects allowing then to propose corrections process actions.

Chapter 3 Upper Bound Method applied in rolling process

This chapter presents in the first place the extremum principles. The Upper Bound Method (UBM ) is one of these principles that is very largely used to model metal forming processes. The second section presents the typical applications of the UBM to rolling process in plane strain condition (2D). The application pointed out the main advantages of UBM to be analytical (or almost), fast and give interesting results. All along the chapter, the reader will see that the principal difficulty of the method is how to build suitable kinematically admissible velocity field for the considered problem. There exist two families of velocity fields: one with rigid body (slip lines) and the other with continuous velocity field in the plastic deformation zone.

The UBM , allowing to estimate the velocity field, is furthermore frequently used to model the width variation in rolling. Such 3D UBM models will be presented in a later chapter 5.

Principle of the UBM

Extremum principles

Prager and Hodge [START_REF] Prager | Theory of perfectly plastic solids[END_REF] formulated in 1951 the lower and upper bound theorems for perfectly plastic solids obeying two flow laws Mises and Prandtl-Reuss. These theorems became extremely powerful to obtain approximate solution to complicated boundary problems. It is largely used to model metal forming processes such as drawing, extrusion, rolling... As the foundation of the theorems is similar for Mises and Prandtl-Reuss material behaviors and it is the Mises one that is used in this thesis, here will be presented the mentioned extremum principles for Mises material behavior law.

3.1.1.a Description of plastic problem

Let consider a domain Ω deformed plastically with boundary conditions as shown in Figure 3.1: The material particles on the portion S u of the surface S are made to move with given velocity u d while the remainder S T of the surface S is subjected to given surface stress T d . Assuming that the whole domain Ω is in a state of plastic flow and we desire to determine the stress field σ or strain rate ǫ throughout Ω.

A stress field defined throughout Ω is statically admissible if it satisfies • the equilibrium condition ∇ σ = 0 (equation 2.1)

• the plastic yield condition

S : S = 2 3 σ 2 0 in Ω (3.1)
• and the boundary condition

σ.n = T on S T . (3.2)
A strain rate field ǫ defined throughout Ω is kinematically admissible if it is derived from a velocity field which satisfies

• the condition of incompressibility div u = tr ǫ = 0 in Ω (3.3)
• and boundary condition u = u d on S u .

(3.4)

Figure 3.1: The problem on the body Ω deformed plastically under velocity boundary condition on S u and stress (or tension) condition on S T . d = ǫ.

3.1.1.b Extremum theorems

Theorem 1: Among all statically admissible stress fields σ the actual one σ a maximizes the expression

I = S u σ.n.u d dS. ( 3.5)
Demonstration: First of all, we will remind the principle of virtual work. Let consider the integral

Ω σ : ǫ dV. (3.6)
where is a virtual strain derived from a virtual displacement field ξ by

ǫ = 1 2 grad ξ + grad T ξ . (3.7)
Since the stress tensor is symmetric, the integrand of 3.6 can be written as σ : grad ξ. Application of Green's theorem leads to:

Ω σ : ǫ dV = Ω σ : grad ξ dV = S σ.n.ξ dS - Ω ∇ σ.ξ dV. (3.8)
We can deduce from this equation by using the equilibrium condition 2.1 that

Ω σ : ǫ dV = S σ.n.ξ dS. (3.9)
In the analogy of this result we can write also

Ω σ : ǫ dV = S σ.n.u dS. (3.10)
where ǫ is the strain rate associated with any virtual velocity field u.

To prove the Theorem 1, let consider the difference I a -I where I is defined by 3.5 associated with some statically admissible σ and I a associated with the actual stress σ a . Since both stress tensors satisfy 3.2 we can write this difference as follows

I a -I = S u (σ a -σ).n.u d dS = S (σ a -σ).n.u a dS (3.11)
where u a denotes the actual velocity field. Applying the principle of virtual work, 3.10, we obtain

I a -I = Ω (σ a -σ) : ǫa dS (3.12)
with ǫa is the actual strain rate field. Since tr ǫ = 0 (3.3) on account of the incompressibility of the material and according to Mises' material flow law ǫ = λS (3.13) with certain coefficient λ > 0, we obtain

I a -I = Ω (S a -S) : ǫa dS = Ω λ(S a -S) : S a dS = Ω λ(σ 2 0 -S : S a ) dS. (3.14)
Now, by Schwarz' inequality (see for example [START_REF] Courant | Differential and integral calculus[END_REF])

S : S a ≤ S : S S a : S a = σ 2 0 (3.15)
where the equality sign can hold only if S = cS a . Since both S and S a satisfy the yield condition 3.1 we deduce then c = 1. Finally, we conclude that

I a -I ≥ 0 (3.16)
where the equality can hold only if S = S a which implies that the stress fields σ and σ a can differ at most by a constant hydrostatic pressure. The boundary condition 3.2, however, rules out such a difference. Thus, the relation S = S a implies that σ = σ a except in the case where S u comprise the entire S.

Upper Bound Method applied in rolling process

3.1 Principle of the UBM Theorem 2: Among all kinematically admissible strain rate fields ǫ the actual one ǫa minimizes the expression

J = 2 3 σ 0 Ω ǫ : ǫ dV - S T T d .u dS.
(3.17)

Demonstration: Let consider J -J a = 2 3 σ 0 Ω ǫ : ǫ -ǫa : ǫa dV - S T T d . (u -u a ) dS. (3.18)
By transforming this difference in a manner similar to that used in proving Theorem 1, we find

J -J a = 2 3 σ 0 Ω ǫ : ǫ -ǫa : ǫa dV - Ω 1 λ ǫa ǫ -ǫa dV. (3.19)
Since Mises' law 3.13, the coefficient λ can be obtained by

λ = ǫa : ǫa S a : S a = ǫa : ǫa 2 3 σ 2 0 . (3.20)
Substituting this into 3.19 we have

J -J a = 2 3 σ 0 Ω ǫ : ǫ ǫa : ǫa -ǫ : ǫa ǫa : ǫa dV. (3.21)
According to Schwarz' inequality, the numerator of the integrand in 3.21 can not be negative, meaning that

J -J a ≥ 0. (3.22)
The equality can hold only if ǫ = c ǫa where c is a scalar factor of proportionality. In certain cases, such as where the surface S T comprises the entire surface S or u d = 0 on a non-empty S u ... the strain rates are determined only to within an arbitrary constant factor.

Discussion: Theorem 1 was first proved by Hill [START_REF] Hill | A comparative study of some variational principles in the theory of plasticity[END_REF] for the special case where the velocities were prescribed over the entire surface and Theorem 2 is due to Markov [START_REF] Markov | On variational principles in the theory of plasticity (russian)[END_REF]. Both principles were preceded historically by Sadowsky's heuristic principle of "maximum plastic resistance" [START_REF] Sadowsky | A principle of maximum plastic resistance[END_REF] stating that among all statically admissible stress distributions the actual one requires a maximum external effort to maintain the flow. This principle led to correct results in certain special cases. Hill has stated that in these cases, Sadowsky's principle coincides with Hill's principle and in general case it will not lead to correct relists even if ambiguity can be avoided in defining the term "effort".

Combination of both principles:

The two theorems may be combined to yield upper and lower bounds. Indeed, using Mises flow equation 3.13, yield condition 3.1 and 3.20 we find

S a : ǫa = 2 3 σ 0 ǫa : ǫa . (3.23) Thus, 2 3 σ 0 Ω ǫa : ǫa dV = Ω S a : ǫa dV = Ω σ a : ǫa dV = S σ a .n.u a dS (3.24)
which implies then

J a = 2 3 σ 0 Ω ǫa : ǫa dV - S T T d .u a dS = S σ a .n.u a dS - S T σ a .n.u a dS = S u σ a .n.u dS = I a . (3.25)
Theorems 1 and 2 can therefore be combined as follows

I ≤ I a = J a ≤ J. (3.26)

Upper Bound Method applied in rolling process

3.1 Principle of the UBM

UBM applied to rolling process

Theorem 2 is largely used for metal forming processes like extrusion, drawing or rolling in order to find an upper bound estimation of external charge (tension or torque). To apply this extremum principle to rolling process, the following assumptions are necessary.

3.1.2.a Assumptions

Rigid work-roll:

The work-roll is considered rigid and its shape (circular or not) is given as an input of the problem.

Rigid-plastic strip:

The strip mechanical behavior is considered rigid-plastic which implies that all elastic deformations are neglected. This is an important hypothesis that simplifies what happens out of the roll-bite. As a consequence of this assumption, the strip entry and exit parts before and after the roll-bite have no deformation and move as rigid bodies. In other words, the velocity field is homogenous and constant throughout these parts. Thus, there are two surfaces of velocity discontinuity before and after the plastic deformation zone Ω. The surfaces are denoted Γ e and Γ s (see Figure 3.2. Their shape is not predetermined but depends on the choice of plastic velocity field. And the dissipation due to the discontinuity of velocity on these surfaces are added into the plastic deformation of the function J in theorem 2.

Tresca friction law:

The studied problem includes both the strip and the work-roll. The friction between them is then an internal force. The Tresca friction law 2.13 is often used because the friction stress is known as a function of the material yield stress. Therefore, its dissipation can be added into the function J in a similar way as the dissipations on Γ e and Γ s .

Figure 3.2: 2D modeling of rolling with UBM . The plastic area Ω (clear-brown) is limited by the discontinuity surface Γ e , Γ s , contact surface and symmetry plane z = 0. The form of this area and the discontinuity surfaces depends on the choice of velocity field in this area. They are outputs (results) of the model.

3.1.2.b Statement of the UBM for rolling process

With the previous assumptions, theorem 2 (3.17) states that among all kinematically admissible velocity fields u the actual one minimizes the expression:

J(u) = Ω σ 0 ǫ(u). dΩ + S d σ 0 √ 3 . ∆u dS + S c τ. ∆u c dS - S ext T d .u. dS (3.27)

Upper Bound Method applied in rolling process

3.2 Velocity fields with rigid bodies motions where J represents the externally supplied power on the roll, σ 0 is the yield stress, σ 0 √ 3 is shear yield stress, ǫ is equivalent strain rate, ∆u is the velocity discontinuity on the surfaces S d of the strip material including Γ e and Γ s , ∆u c the difference between strip material velocity and work-roll velocity on the contact surface S c . The first term of the right side of 3.27 corresponds to the power of plastic deformation due to the continuous deformation inside Ω, denoted J ǫ. The second term is the power of plastic dissipation on the surfaces of velocity discontinuity, denoted J ∆u . Sum of these two terms is the total plastic deformation power J de f = J ǫ + J ∆u . The third one is the friction dissipation power, J f ric and the last term is the external work power (entry and exit tensions), J ten .

3.1.2.c 2D rolling UBM -based models in literature

As the set of the kinematically admissible forms an infinite-dimension space, the optimization problem (Theorem 2) to find out the optimum velocity field needs simplifying. In practice, a family of velocity fields with few parameters is studied of which the velocity field minimizing the power function defined by 3.27 is the best approximation of the real one. The UBM result depends, thus strongly on the choice and construction of velocity fields. In literature, the authors apply this method to rolling using different families of velocity fields. It is possible to classify them into two categories of velocity fields. The first one consider rigid bodies velocity field (called also slip lines because the deformation concentrates on some shear lines) and the second one analyzes the plastic deformation zone with continuous velocity field.

Velocity fields with rigid bodies motions

Simulated sliding theory and experiments

Most of metals have a crystalline microstructure, the behaviour of which in plastic deformation consists of glide packages having slipped inside a grain. Deformation occurs almost entirely on slip surfaces while the material between them remains practically unaffected. This theory of simulated sliding in microscopic scale has already been applied for a long time in continuous plastic deformation of metal. In 1976, Piispanen [START_REF] Piispanen | Plastic deformation of metal: theory of simulated slidding[END_REF] stated also that the plastic deformation of rigid plastic material is also macroscopically inhomogeneous. The author performed experiments on hot rolling process using miniature rollers. The bar was rolled until continuity is attained in the strain process. After the rolling test, the rolled bar emerging surface was polished but not etched so that only marks from deformation showed. The roller and test bar were then placed on the specimen stage of a metallurgical microscope allowing to take metallographic images continuously during the process.

For the test A (see Figure 3.3 -Fig 7 of [START_REF] Piispanen | Plastic deformation of metal: theory of simulated slidding[END_REF], page 48), the author observed that the configuration of lines begun to appear at points B and G in Figure 3.3(a). As rolling continued the deformation area grew but the signs of deformations appeared in four deformation zones as showed in Figure 3.3(b). Almost no deformation was observed inside the four zones but the deformation concentrated only on the slip lines (limits between the areas) resembling the letter X. Other rolling parameters showed in Table 3.1 were also experimented. Except the test D where two bars were put one on the other and rolled at the same time, the concentration of deformation on the slip lines was confirmed for all the tests B, C, E (see Figure 3.4 for the result of the test C). The same result was observed for the test F which is a second rolling pass using the rolled bar after the test E. Since these interesting results, the author proposed a upper bound model considering that the deformation is concentrated on the shear slip lines, that is called later unitriangular velocity field. The details of this model will be presented in a following section 3.2.3 The test G is quite different where the contact length is twice higher than the strip thickness and Figure 3.5 showed a clear deformation in the zone BAG at the entry side. The first letter X of deformation concentration occurs only a half of the contact length and we may expect another one between C1D1CD. However, the deformation observed was not clear. This difference is not random. We will see later that it is better to apply multitriangular velocity field in this case. Furthermore, in the next chapter a new UBM model with continuous velocity field with shear concentration on certain slip zones allows also to explain why when the contact length is twice the strip thickness we expect to have two letter X.

A, R = 10mm, 2h e = 2.5mm,2h s = 2.08mm, contact length is estimated L = R(2h e -2h s ) = 2.05mm.

Upper Bound Method applied in rolling process

3.2 Velocity fields with rigid bodies motions

Concept for analysis of linear and rotational velocity fields

Avitzur and Pachla [START_REF] Avitzur | The upper bound appoach to plan strain peoblems using linear and rotational velocity fields -part i: Basic concepts[END_REF] investigates an upper bound approach to plan strain deformation of a rigid -perfectly plastic material where the deformation region is divided into a finite number of rigid triangular bodies that slide with respect to one another. By consequence, there is no continuous plastic deformation power and all the plastic deformation power concentrates on the surfaces of velocity discontinuity or in other words the first term of the equation 3.27 is equal to zero and the second one represents the total plastic power. The authors give the concept and specific equation for the three general cases of neighboring rigid body zones where the zones are both in rotational motion (case 1), one in linear, the other in rotational motion (case 2) and both in linear motion (case 3). The shape of the surface of velocity discontinuity and the shear power losses are clearly presented. As a result of this article, when one of the rigid bodies exhibit rotational motion, the surface of velocity discontinuity is found to be cylindrical (circular in the plan strain study) and it is planar in the case with both bodies in linear motion. The velocity discontinuity is found to be constant along the entire surface of velocity discontinuity.

Even though Avitzur and Pachla [START_REF] Avitzur | The upper bound appoach to plan strain peoblems using linear and rotational velocity fields -part i: Basic concepts[END_REF] are not the first who study rolling process by rigid bodies UBM , but their study synthesized the concept and gave powerful tool to develop more complex rigid bodies UBM models for metal forming processes at that time. The concept is applicable to strip drawing, extrusion, forging, rolling, leveling and machining. The same authors presents then the applications of this concept to drawing, cutting and rolling processes in [START_REF] Avitzur | The upper bound appoach to plan strain peoblems using linear and rotational velocity fields -part ii: Applications[END_REF].

Remark: It is assumed that in order to get higher accuracy an attempt should be made to have the assumed velocity field resemble to the actual one as closely as possible. However, even when the assumed velocity is inaccurate, the power consumption calculated based on that field will can approximate the actual power. This family of rigid bodies velocities allows only to have an estimation of power but not velocity field.

Unitriangular velocity field

Johnson and Mellor [51] analyzed strip rolling by the unitriangular velocity field based on the curvilinear triangle, opening a new avenue of approach, the UBM in rolling. As can be seen in Figure 3.6, the strip is divided into three zones. Two zones before and after the roll-bite are in linear motion with a velocity V e and V s . And the triangular zone under the roll-bite is in rotational motion around the work-roll center O c with a rotational velocity w. Applying the concept of Avitzur and Pachla [START_REF] Avitzur | The upper bound appoach to plan strain peoblems using linear and rotational velocity fields -part i: Basic concepts[END_REF], the surfaces of velocity discontinuity AP ⌢ and PB ⌢ are circular corresponding to the center O 1 on the axes Oz at vertical position z 1 and radius r 1 and O 2 on the axes Oz at vertical position z 1 and radius r 2 .

Geometrical parameters:

We will see here below that all geometrical parameters z 1 , r 1 , z 2 and r 2 can be expressed by the only one x P (that we chose arbitrarily). Indeed, the fact that the points A and P belong to the arc (O 1 , r 1 ) implies

AO 1 2 = (x 1 -x A ) 2 + (z 1 -z A ) 2 ⇔ r 2 1 = L 2 + (z 1 -h e ) 2 (3.28) PO 1 2 = (x 1 -x P ) 2 + (z 1 -z P ) 2 ⇔ r 2 1 = x 2 P + z 2 1 . (3.29)
Comparing the right side of these two equations we obtain And by consequence the length of the arcs AP ⌢ and PB ⌢ are complectly determined as a functions of

z 1 (x P ) = L 2 + h 2 e -x 2 P 2h e . ( 3
x P            AP ⌢ = 2r 1 arcsin AP 2r 1 = 2r 1 arcsin (L + x P ) 2 + h 2 e 2r 1 PB ⌢ = 2r 2 arcsin PB 2r 2 = 2r 2 arcsin x 2 P + z 2 2 2r 2 . (3.33)
Velocity discontinuities As stated Avitzur and Pachla [START_REF] Avitzur | The upper bound appoach to plan strain peoblems using linear and rotational velocity fields -part i: Basic concepts[END_REF], the velocity discontinuities are constant all along these arcs. They will be obtained as a function of geometrical parameter x P and the rotational velocity ω of the triangular rigid body. Noting that V A = ωAO c , since the uniformity of the triangles AV A ∆V AP and AO 1 O c (see Figure 3.6) we can obtain the entry velocity and the velocity discontinuity over AP ⌢ as follows

       V e = V A O 1 O c AO c = [z c -z 1 (x P )] ω ∆V AP = V A AO 1 AO c = r 1 (x P )ω. (3.34)
Of course, with similar analysis the exit velocity and the velocity discontinuity over PB

⌢ are V e = [z c -z 2 (x P )] ω ∆V PB = r 2 (x P )ω. (3.35)
Calculations of powers: By definition of the velocity field, it is obvious that the continuous plastic deformation power J ǫ is equal to 0. All plastic power is concentrated on the surfaces of velocity discontinuity. Using the second equations of 3.34 and 3.35, this power can be obtained as

J ∆u = σ 0 √ 3 [AP ⌢ .∆V AP + PB ⌢ .∆V PB ] = σ 0 √ 3 ω [AP ⌢ (x P )r 1 (x P ) + PB ⌢ (x P )r 2 (x P )] . (3.36)
If we note

P ∆u (x p ) = σ 0 √ 3 ω [AP ⌢ (x P )r 1 (x P ) + PB ⌢ (x P )r 2 (x P )]
(3.37)

Upper Bound Method applied in rolling process

3.2 Velocity fields with rigid bodies motions the power of velocity discontinuity can be expressed by

J ∆u (ω, x p ) = P ∆u (x p )ω. (3.38)
The friction power is given by

   J f ric (ω, x p ) = P f ric |ω -ω c | where P f ric = m σ 0 √ 3 AB ⌢ . (3.39)
And using the first equations of 3.34 and 3.35 the power supplied by entry and exit tensions can be written by J ten (ω, x p ) = -T e h e V e + T s h s V e = P ten ω

where

P ten = -T e h e [z c -z 1 (x P )] + T s h s [z c -z 2 (x P )] . (3.40)
Finally the power function defined by the equation 3.27 is, thus obtained

J(ω, x p ) = P ∆u (x p )ω + P f ric |ω -ω c | -P ten (x p )ω (3.41) or J(ω, x p ) =      P ∆u (x p ) -P ten (x p ) -P f ric ω -P f ric ω c if ω < ω c P ∆u (x p ) -P ten (x p ) + P f ric ω + P f ric ω c if ω > ω c (3.42)
Optimum power: The power function J(ω, x p ) given by 3.42 is a function of two independent parameters x P and ω. With respect to ω this function is linear and has a minimum at the point ω = ω c if and only if the slope is negative before and positive after this point. In other words, J(ω, x p ) has a minimum with respect to ω if and only if -P f ric < P ∆u (x p ) -P ten (x p ) < P f ric (3.43) and the minimum is obtained when the rigid body rotates with a same velocity as that of the work-roll. In this case, the power function becomes 1-variable function

J 1 (x p ) = J(ω c , x p ) = P ∆u (x p ) -P ten (x p ) ω c (3.44)
where P ∆u (x p ) and P ten (x p ) are given by 3.37 3.40.

Optimization of this 1-variable function gives an minimum upper bound approach of the supplied power. The results of this model will be compared to the slabs velocity field in the section 4. [START_REF] Avitzur | Maximum reduction in cold strip rolling[END_REF].

Approach of roll force:

On the shear arcs AP and and PB the shear stress amplitude is equal to the material yield shear stress k as showed in Figure 3.7. Let consider first the arc AP ⌢ and consider that the shear stress k is constant along this arc. It is obvious that the total tangential force -→ F t integral of shear stress along AP

⌢ is a vector parallel to -→ PA or -→ F t = PA ⌢ k dl = k -→ PA. (3.45)
Similarly, the normal force -→ F n which is the integral of normal stress is perpendicular to -→ F t .

If now, we consider that the force exerted by the roll on the strip is vertical (because the resulting force along x is, in a real rolling condition, much smaller than the one in z direction), this force must satisfy the condition as showed by three green vectors in Figure 3.7. By uniformity of triangles forming by the three green force vectors and the AA ′ P (A ′ is the projection of A on Ox, we can write following relation

F AP AP = F t h e . ( 3
F AP = k.AP 2 h e = k h e + (L + x P ) 2 h e . (3.47)
The force F PB can be derived in a same way

F PB = k h s + x 2 P h s . (3.48)
And the total force is obtained

F = F AP + F PB . (3.49)

Multitriangular velocity field

For cold rolling, the contact length is generally several times higher than the strip thickness, the unitriangle approach is less realistic. Camurri and Lavanchy [START_REF] Camurri | Application de la théorie des lignes de glissement au laminage à froid de tôle[END_REF] in 1984 applied the upper bound method with slip line velocity field to cold strip rolling by introducing multitriangles slip lines. This model has been reanalyzed by Avitzur, Talbert and Gordon [START_REF] Avitzur | Analysis of strip rolling by the upper bound approach[END_REF] by using the concept presented in [START_REF] Avitzur | The upper bound appoach to plan strain peoblems using linear and rotational velocity fields -part i: Basic concepts[END_REF]. 3.8 shows the application of the multitriangular velocity field to the process of strip rolling. Within the curvilinear triangles with apexes P i on the plane of symmetry, a rotational velocity field prevails. Other triangles with their bases on the plane of symmetry and apexes on the surface of the roll, bound the regions of linear motion parallel to the plane of symmetry. According to the analysis of Avitzur and Pachla [START_REF] Avitzur | The upper bound appoach to plan strain peoblems using linear and rotational velocity fields -part i: Basic concepts[END_REF], the axes of symmetry of all the cylindrical surfaces of velocity discontinuity are situated along the z-axis. Each of the curvilinear triangles can be treated separately as a unitriangle problem. When the upper bounds for each triangle are computed and added together, the total power is obtained.

There are several ways to optimize such a system. Avitzur, Talbert and Gordon [START_REF] Avitzur | Analysis of strip rolling by the upper bound approach[END_REF] propose to chose at the beginning arbitrarily the points A i and B i then the positions of P i is optimized for each triangle separately as for the unitriangular case. For each single triangle, the power function is only optimize with respect to x P i but not to ω i because ω i needs satisfying flow rate conservation condition and so fixed when treating each separated triangle. The powers of each triangle are then added together to obtain the total power. Next, the positions of A i and B i are selected along the roll surface and now positions of P i corresponding to the separated triangle optimum are found and upper bound approach is finally evaluated.

In this case of multitriangular velocity field, the meaning of the neutral region becomes evident. To fulfill the volume constancy requirement, the angular velocities ω i increase as the material moves forwards the exit. When the global optimum occurs, among the treated rigid body triangular regions there is one that rotates with the same rotational velocity as the work-roll. The segment between the point A i and B i f this triangle represents the neutral zone.

2D continuous velocity fields

Continuous eccentric velocity field

Velocity field A Avitzur [START_REF] Avitzur | Maximum reduction in cold strip rolling[END_REF][START_REF] Avitzur | Power analysis of cold strip rolling[END_REF][START_REF] Avitzur | An upper bound approach to cold strip rolling[END_REF] performed several analyses with including UBM to cold strip rolling. The author proposed, for the first time, a continuous velocity field that let us call "eccentric" velocity field. This field is described as: Cylindrical symmetry exists along each arc connecting any two symmetrical points on the opposing rolls. The arcs are eccentric and each one meets the roll surfaces at right angles. The radius of these arcs increases as the material point advances further toward the exit (see Figure 3.9). The velocity of each material point on an arc is on the direction of the radius of the arc (passing through the arc center). On the entry arc (surface of velocity discontinuity), the condition of continuity of normal velocity through the entry arc implies u(θ e , ϑ) = V e cos ϑ.

(3.50)

This velocity decreases when ϑ increases from 0 to θ e . In the same analogy, the velocity at any other material point is given by u(θ, ϑ) = A(θ) cos ϑ.

As the velocity is, by construction, tangential to the work-roll surface, the only condition so that the velocity field is kinematically admissible is the incompressibility or the flow rate constancy written as

C vol = θ 0 A(θ) cos ϑ.r(θ) dϑ = A(θ)r(θ) sin θ = A(θ).h(θ) (3.51)
where r is the radius of the arc and θ is the angle of the arc length (equal to the angular position of the point P, intersection of the arc with the work-roll). We deduce then

A(θ) = C vol h(θ) (3.52)
and therefore

u(θ, ϑ) = C vol h(θ) cos ϑ. (3.53)

Calculating of power functions A

Based on the hypotheses of small angle, the author simplified the velocity as follows: each material point on a same arc has identical speed and forward all to the center of the arc. And here following are the power function obtained.

Power of continuous plastic deformation:

After some mathematical developments the author obtain

J ǫ = 2 √ 3 σ 0 C vol ln h e h s . (3.54)
In fact, as can be seen in Figure 3.9, an arc at the entry of plastic zone will stay an arc. Only its size (height) becomes smaller when the material point moves forward the exit. The plastic power obtained can be understood as the power to deform the material from h e to h s under plane strain condition without shear deformations.

Power of velocity discontinuity at entry:

The velocity discontinuity at the entry arc is equal to V e sin ϑ, using the hypothesis of small angle the power of velocity discontinuity can be computed as

J ∆u = σ 0 √ 3 θ e 0 V e sin ϑr e dθ = σ 0 √ 3 V e r e (1 -cos θ e ). (3.55)
where r e is the radius of the entry arc defined by r e θ e = h e . Thus,

J ∆u = σ 0 √ 3 V e r e ( 1 2 θ 2 e ) = σ 0 2 √ 3 C vol θ e (3.56)
with θ e ≃ 2(h e -h s )

R

.

Power of friction:

The difference of velocity between the strip and the roll on the contact surface is

∆u = u -V c = C vol h s + R 2 θ 2 -V c
3. Upper Bound Method applied in rolling process

2D continuous velocity fields

Therefore, the friction power is

J f ric = θ e 0 mσ 0 √ 3 | C vol h s + R 2 θ 2 -V c | R dθ = mσ 0 √ 3 θ n 0 C vol h s + R 2 θ 2 -V c R dθ + θ e θ n V c - C vol h s + R 2 θ 2 R dθ = mσ 0 √ 3 V c R(θ e -2θ n ) + θ n 0 C vol h s + R 2 θ 2 R dθ - θ e θ n C vol h s + R 2 θ 2 R dθ = mσ 0 √ 3 V c R(θ e -2θ n ) -C vol 2R h s arctan R 2h s θ e -2 arctan R 2h s θ n (3.57)
By noting that

h e = h s + R 2 θ 2 e it can be deduced that R 2h s θ e = h e h s - 1
.

Power of the tensions:

The following formula is valid for all 2D velocity fields

J ten = (T s -T e ) C vol . (3.58)
Optimization of total power functions: By writing the equality of strip and work-roll velocity at the neutral point θ n , the flow rate can be given by

C vol = V c h n = V c h s 1 + Rθ 2 n 2h s . (3.59)
The total power becomes a function of unique parameter θ n and the optimum solution is analytically obtained as follows

R 2h s θ n = 1 3    arctan h e h s -1 -m 2h s R   ln h e h s + 1 4 2h s R h e h s -1 - T s -T e 2 √ 3 σ 0      . (3.60)

Continuous simple velocity field

Bouharaoua and al [START_REF] Bouharaoua | A la recherche d'une modélisation simplifiée du laminage[END_REF] study a symmetric rolling of flat and thin sheets. The authors based on a hypothesis that a material vertical cross section will stay vertical all along the roll-bite. This hypothesis is equivalent to the one of the slab method. The difference is that the UBM approaches by velocity field and the slab method bases on the stress field. The hypothesis of homogenous deformation implies obviously that the longitudinal velocity u x is constant across the thickness direction and depends only on x. Then, the incompressibility of the material deduces that the vertical velocity u z is linear in z. The authors obtain a velocity field as follows

         u x = C vol h(x) u z = h ′ (x)C vol h 2 (x) .z (
(x) = R + h s -R 2 -x 2 (3.62)
and its derivative is obtained

h ′ (x) = -x √ R 2 -x 2 (3.63)
This field is called "simple" because it is deduced quite directly from the hypothesis of homogeneity of longitudinal deformation across the thickness direction and the incompressibility of the material. Nevertheless, we will see later that this field is the same as the elliptical flow lines introduced in 3.3.5.

Since in independence of u x in z, the surfaces of velocity discontinuity at entry and exit, Γ e and Γ s are both vertical. Moreover, at the exit x = 0, as h ′ (0) = 0 the vertical velocity is equal to zero. In other words, the velocity discontinuity does not really occurs on Γ s even if Γ s is always the limits between plastic and elastic zones. This velocity field is illustrated in Figure 3.10.

z V c h e h N V e V s ) , ( z x u h s s x ) , ( z x u

Calculating of power functions A

Bouharaoua and al [START_REF] Bouharaoua | A la recherche d'une modélisation simplifiée du laminage[END_REF] did developed as far as possible to get analytical expression of all the power functions. However, they obtained the continuous plastic deformation power J ǫ as a double integral with respect to x and z. Here will be presented results of our development to get analytical solution for the integral in z and obtain an expression of J ǫ as a single integral with respect to x.

Power of continuous plastic deformation:

After the calculations presented in the appendix B.1, we have:

J ǫ = σ 0 2 √ 3 C vol 0 -L I 2 + 4h ′ 2 I 1 Ln I 1 + I 2 | 2h ′ | dx h (3.64)
where

I 1 =| h.h ′′ -2h ′ 2 | I 2 = 4h ′ 2 + I 2 1 .
Power of velocity discontinuity: By remarking that [|u|] =| u z (x = -L) | power of velocity discontinuity can be written as :

J ∆u = h(-L) 0 σ 0 √ 3 [|u|] dz = h(-L) 0 σ 0 √ 3 | h ′ (-L) | C vol h 2 (-L) zdz = σ 0 2 √ 3 | h ′ (-L) | C vol . (3.65)

Upper Bound Method applied in rolling process

2D continuous velocity fields

Power of friction: At the contact surface, the difference of velocity between the strip and the roll is given by

[|u|] =| V c -u 2 1 (z = h) + u 2 3 (z = h) |=| V c - h e V e h 1 + h ′ 2 | .
Therefore, the friction power is

J f ric = 0 -L mσ 0 √ 3 | V c - h e V e h 1 + h ′ 2 | 1 + h ′ 2 dx = mσ 0 √ 3 0 -L | V c - √ 1 + h ′ 2 h C vol | 1 + h ′ 2 dx.
(3.66)

Power of the tensions:

The following formula is valid for all 2D velocity fields

J ten = (T s -T e ) C vol . (3.67)
As can be seen, the power of velocity discontinuity J ∆u , friction power J f ric and tensions power J ten are analytically obtained. The plastic J ǫ is 1-variable integral and can be performed numerically without difficulty. The total power is finally a function of unique variable of the problem C vol and the minimization is solved thanks to Newton-Raphson method. The results of this model will be compared to some others in the section 4.5.

Otherwise, it should be noted that in their study, S.Bouharaoua et al [START_REF] Bouharaoua | A la recherche d'une modélisation simplifiée du laminage[END_REF] do not take into account the power dissipated by the discontinuity of velocity at the entry of the roll bite.

Improved velocity field

When the ratio thickness over work-roll radius increases, the heterogeneity of the deformation across thickness direction increases and can not be neglected. To take into account this dependence, Bouharaoua and al [START_REF] Bouharaoua | A la recherche d'une modélisation simplifiée du laminage[END_REF] propose to improve their linear form of vertical velocity to a third degree polynomial depending on z and obtain :

   u x = a(x) + f (x)z 2 u z = -a ′ (x)z -f ′ (x) z 3 3 (3.68)
where

f (x) = 3 h 3 (x) [k -a(x)h(x)].
As a result, to describe the velocity field, they need a scalar parameter k and a function, a(x). The calculation becomes very heavy and far to be analytical. By consequence the minimization is no longer simple to compute.

Circular stream lines velocity field

T.Hoang [START_REF] Hoang | Asymmetric rolling analysis -energy saving and ski effect[END_REF] proposes a velocity field by considering that the material stream lines in the roll bite are circular. To simplify the equations, the author introduces two new coordinates s, exit height of the stream line and θ angular position of the considered point (see Figure 3.11). The material is supposed to follow the circular stream lines with the same center as the work roll O c and a radius defined by .69) These relation between these new coordinates and Cartesian ones are And the circular velocity field is given by

R(s) = Rh s s . ( 3
     x = - Rh s s sin θ z = s + Rh s s (1 -cos θ) (3.
u(x, z) = p(s)s 2 Rh s + (s 2 -Rh s ) cos θ cos θ -sin θ (3.71)
where p(s) is the velocity at the roll bite exit as a function of the height s across the thickness direction. For symmetric rolling, this function is constant. And it is taken linear for asymmetric rolling conditions. In this case, this function allows to model the curvature of the strip at the exit of roll bite. This is called ski or longbow defect (see Figure 2.11).

Unlike Bouharaoua and al [START_REF] Bouharaoua | A la recherche d'une modélisation simplifiée du laminage[END_REF], T.Hoang [START_REF] Hoang | Asymmetric rolling analysis -energy saving and ski effect[END_REF] takes into account the dissipation of the discontinuity of velocity at the entry of roll bite. However, he simplifies the calculation of this dissipation in considering that the discontinuity surface is a vertical straight surface. This is not exact because by construction, the longitudinal velocity in the roll bite u x depends on z while it is constant and equal to V e before the roll bite. Although the induced error may be small we can not compare the results of this model to other ones.

Elliptical stream lines velocity field

Method for constructing velocity fields A

In 2001 Dogruoglu [START_REF] Dogruoglu | On constructing kinematically admissible velocity fields in cold sheet rolling[END_REF] introduced a rigorous and systematic method for constructing kinematically admissible velocity fields necessary in the analysis of plastic forming processes by UBM . The objective of the method is build a family of kinematically admissible velocity fields by pre-assuming a certain form of the flow lines. In his applications, the flow lines are chosen being circular and elliptical.

We can sum up the results of this method as follows. If the flow lines in the plastic deformation zone is represented as a one-parameter family of curves

f (x, z) = ξ, 0 ≤ ξ ≤ 1. (3.72)
then the following velocity field

     u x = - ∂ f ∂z F(ξ) u z = ∂ f ∂x F(ξ) (3.73)
verifies automatically the incompressibility condition. So that the velocity field given by 3.73 is kinematically admissible, it is necessary to choose the flow lines that include boundary lines of the problem (for example the work-roll surface in rolling process).

The function F(ξ) is an arbitrary differentiable function which will be determined by the initial conditions of the considered problem. In rolling process, F(ξ) is determined by one of two conditions of the continuity of the velocity components in the direction normal to the boundaries Γ e and Γ s of the plastic deformation zone. In other words, the form of Γ e or Γ s can be chosen arbitrarily. After, F(ξ) is obtained and the velocity field is determined. The other condition will be used to determine the related boundary of the deformation zone.

Elliptical flow lines A

The author applied then this method to determine an elliptical flow lines velocity field. The elliptical flow lines family is defined by

f (x, z) = z R + h s - √ R 2 -x 2 = ξ, 0 ≤ ξ ≤ 1. (3.74)
Now if the exit surface of velocity discontinuity is chose as a vertical line defined by

x = 0, z = z 2 0 ≤ z 2 ≤ h s . ( 3
= z 2 (ξ) = h s ξ. (3.76)
Using the definition of velocity field 3.73, the continuity of normal velocity through Γ s becomes

u x (0, z) = - ∂ f ∂z (0, z)F(ξ) = - 1 h s F(ξ) = V s .
Then,

F(ξ) = -h s V s = -C vol (3.77)
equal to the flow rate (with opposite sign).

Finally, the author obtained following elliptical velocity field

           u x = - ∂ f ∂z F(ξ) = C vol R + h s - √ R 2 -x 2 u z = ∂ f ∂x F(ξ) = C vol xz √ R 2 -x 2 R + h s - √ R 2 -x 2 2 (3.78)
To complete the solution, the entry surface is found to be also a vertical plane defined by

x = (2R + h s -h e )(h e -h s ), z = z 2 0 ≤ z 2 ≤ h s . (3.79)
Remark: Using the expression of thickness function and its derivative given correspondingly by 3.62 and 3.63 to simplify the elliptical flow lines velocity field given by 3.78 we obtain exactly the simple velocity field defined by 3.61. Indeed, these two velocity fields are the same.

Results

A

Dogruoglu [START_REF] Dogruoglu | On constructing kinematically admissible velocity fields in cold sheet rolling[END_REF] computed the power functions by introducing the neutral point position X n (x * according to the authors notation) as the unique parameter instead of C vol . The optimization becomes more simple. The author compared then the obtained minimum power J with the results of the eccentric velocity field model given in a Avitzur's book [START_REF] Avitzur | Metal Forming: Process and Analysis[END_REF]. Dogruoglu introduced, then the ideal plastic deformation power

J I de f = 2σ 0 √ 3 C vol ln h e h s (3.80)
which is in fact the plastic deformation if the strip material is deformed under plane strain condition only in longitudinal and thickness directions without shear deformation. And a comparison is based on the relative difference between the obtained plastic deformation power and this ideal one. However, Dogruoglu did not take into account the power of velocity discontinuity in the plastic deformation power. He calculated the relative difference by

Relative difference = J ǫ -J I de f J I de f . (3.81)
In the considered rolling case (m = 0.6, hs/R = 0.002), the author showed that this relative difference is excellently small, varying from 0.19% down to 0.014% when the reduction increases from 5% to 60%, meaning that the plastic deformation is very closed to the actual one.

Nevertheless, let consider for example a slip lines velocity field such us unitriangular one, there is no continuous deformation power, J ǫ = 0. All plastic deformation power is generated by the velocity discontinuity, J ∆u . In this case the relative difference, defined by the formula of Dogruoglu 3.82is equal to 0 implying that the plastic deformation is lower than the ideal one. This is, of course not true. Indeed, the total plastic deformation power must include both power of continuous plastic deformation and the power of velocity discontinuity. Then, the relative difference should be defined as

Relative difference = J ǫ + J ∆u -J I de f J I de f . (3.82)
With this correction, for the same considered rolling case, the relative difference varies from 3.73% to 1.07%.

Discussions

This chapter shows that the UBM based on the extremum principle formulated by [START_REF] Prager | Theory of perfectly plastic solids[END_REF] is largely used in metal rolling process to evaluate especially the supplied power and torque. There were developed many slip lines (rigid body) velocity fields. The unitriangular is suitable to be used in hot rolling conditions where a high friction coefficient occurs and the ratio thickness over contact length is close to 1. For cold rolling, the contact length is usually much higher than strip thickness, the multitriangular is more efficient. An other advantage of this velocity field is that the sliding between the strip and the work-roll out of the neutral zone is allowed which correspond better to the condition of cold rolling.

Nevertheless, the optimization of solution for multitriangular is not simple mainly due to the unknown number of triangles. It is necessary to vary it and the number of parameters will vary at the same time. Some author solve mathematically the optimization problem by consider the number of triangles as a continuous "real" number but not "integer" number, see [START_REF] Kiuchi | Limit analysis of flow through inclined converging planes[END_REF][START_REF] Pan | The study of distorted grid pattern for flow through conical converging dies by the multi-triangluar velocity field[END_REF][START_REF] Avitzur | Metal Froming: Application of Limit Analaysis[END_REF]. This method allowed to obtain lower upper bound in many cases. Withal, the velocity field obtained by non-integer number of triangles seems to be somewhat illusive. Unlike the slip line or rigid bodies velocity fields the continuous ones give often more interesting information about the velocity field itself. The main difficulty is how to parameterize a good velocity fields. Authors proposed different methods for constructing kinematically admissible velocity field. The most famous one is based on flow functions and flow lines ( [START_REF] Dogruoglu | On constructing kinematically admissible velocity fields in cold sheet rolling[END_REF]). Advantage of the method is that you can imagine any flow lines that verify the boundary condition (work-roll surface must be a flow line), the method will give expression of the velocity field. Yet, this advantage is also an disadvantage because the flow lines need to be pre-assumed and sometimes it is not easy to imagines the complicated one. For example, in the next chapter, the flow lines of rolled material as results of Lam3-Tec3 are complicated. It oscillates somehow around a smooth curve. In the same chapter, we will introduce an other method to build kinematically admissible velocity field that does not pre-assume flow lines but the flow lines will be outputs of the model. There will also be presented a comparison of unitriangular, continuous simple (elliptical) velocity fields with the new oscillating one and Lam3-Tec3 results.

By giving the flow velocity the continuous velocity fields own an other advantage to approach the flow patterns such as 3D models approaching the width variation of the rolled strip. Some UBM models will be presented in the chapter 5.

Chapter 4

Oscillation of mechanical fields in roll bite

The rolling process has been analyzed by various analytical and numerical methods as the slab method, UBM and FEM . The FEM calculation allows having accurate estimation of mechanical solution but computing time is usually long. Beside, the heterogeneity of strain or stress fields across the strip thickness needs understanding and taking into account to explain heterogeneity of metallurgical characteristics and microstructure after rolling or rolled-product apparent defect prediction. This heterogeneity remains very little investigated. Orowan is one of rare authors who take into account the variation of stress field in the thickness. However, the shear stress is imposed linear across the thickness direction. The chapter presents firstly a method for constructing kinematically admissible velocity field for rolling process. This method is not based on pre-assumed flow lines as usual but on the understanding of their physical behavior. Then, a new family called "oscillating" velocity fields allowing to model the non-linear heterogeneity of deformation in the thickness is proposed. And application of UBM results to a final velocity field that corresponds to a lower rolling power in comparison with the unitriangular and the elliptical ones. This obtained velocity field, having characteristics of wave propagation, matches well Lam3-Tec3 results.

Introduction

Although in flat product cold rolling condition, the strip thickness is very small compared to its width and the contact length with the work-roll, the vertical velocity, vertical strain, vertical strain rate fields are not homogenous across the strip thickness. These fields oscillate in both x and z directions. This is an interesting results that may be in contrast to what could be imagined (the thinner the strip, more homogenous the mechanical fields). This chapter introduces a complete application of UBM to analyze this heterogeneity of mechanical field and compare to Lam3-Tec3 .

As a reminder, following the UBM introduced in the previous chapter, the actual velocity field in a rolling process is the one that minimizes the power function 3.27. Usually, a family of velocity fields with few parameters is studied of which the velocity field minimizing the power function is the best approximation of the real one. The lower the power, the better the approximation. The rigid body (or slip lines) velocity fields such as unitriangular, multitriangular are obviously heterogenous velocities across the thickness but the changes between the rigid regions are abrupt. The shear stresses are concentrated only on certain slip lines. In the contrary, the typical continuous velocity field like simple (called also elliptical) has longitudinal and vertical deformation rates ǫxx and ǫzz which are homogeneous in the thickness. The circular velocity field is not so different from the simple one. This chapter presents new family of velocity fields also in plane strain condition. These velocity fields are contiguous and have certain concentration of shear strain rate somehow similar to slip lines ones.

That family is built as an addition of an "oscillating part" to the "simple" (elliptical) velocity field of which the longitudinal component is constant in the strip thickness (see 3.61). We show that the best field of that family gives a better approximation than the simple one. This study proves there is an oscillating part in the velocity field during the rolling process. A careful observation of the fields obtained by a finite-element method Lam3-Tec3 shows that the oscillation phenomenon is in fact really present even for thin strip. And the oscillations predicted by the analytical model (UBM ) match very well the Lam3-Tec3 results.

Method for constructing velocity field 4.2.1 General admissible conditions of velocity field

The model is developed under the same assumptions as presented in the part 3.1.2. Let remind that before the roll-bite the velocity is uniform and equal to V e and it is also uniform and equal to V s after the roll-bite (see Figure 3.2). The material is deformed plastically in the zone Ω delimited by the two surfaces Γ e and Γ s which are also the surfaces of velocity discontinuity because the velocity field in this field is different from V e and V s . Modeling of the velocity in this area is the key point of an UBM model. Once the velocity fields in all the three zones are modeled, the Γ e and Γ s can be determined.

Using the boundary conditions described for a general elasto-plastic previously (section 2.1.3) and simplify them for rigid perfectly plastic condition (especially the conditions of the symmetry on z = 0 given by 2.3), the kinematically admissible conditions of the velocity field in the plastic deformation zone Ω are:

• Incompressibility of rigid plastic material:

div u(x, z) = ∂u x ∂x (x, z) + ∂u z ∂z (x, z) = 0 ∀(x, z) ∈ Ω (4.1)
• Boundary conditions on the surface of symmetry z = 0:

u z (x, 0) = 0 ∀x ∈ [-L, 0] (4.2)
and

∂u x ∂z (x, 0) = 0 ∀x ∈ [-L, 0]. (4.3)
• Boundary condition on the contact surface z = h(x):

u z (x, h(x)) -u x (x, h(x)) .h ′ (x) = 0 ∀x ∈ [-L, 0]. (4.4)
Concerning the flow rate, if the three conditions 4.1, 4.2 and 4.4 are verified, it can be demonstrated that the volume flow rate through a cross section of the strip is conserved (constant). Indeed, if C vol (x) denotes the flow rate through the cross section at x, it is defined as a function of x

C vol (x) = h(x) 0 u x (x, z).dz. (4.5)
We consider now its derivative:

dC vol (x) dx = d dx h(x) 0 u x (x, z). dz = h ′ (x)u x (x, h(x)) + h(x) 0 ∂u x ∂x (x, z). dz.
Using the equations 4.1 then 4.2 and 4.4 we obtain:

dC vol (x) dx = h ′ (x)u x (x, h(x)) - h(x) 0 ∂u z ∂z (x, z).dz = h ′ (x)u x (x, h(x)) -u z (x, h(x)) + u z (x, 0) = h ′ (x)u x (x, h(x)) -u z (x, h(x)) + u z (x, 0) = 0.
This implies the conservation of the flow rate:

C vol = h(x) 0 u x (x, z).dz = const ∀x ∈ [-L, 0]. (4.6)
Thus, the forth conditions 4.1, 4.2, 4.3, 4.4 (or 4.6) are all the kinematically admissible conditions of a velocity field defined throughout the plastic deformation zone Ω.

It is easy to verify that the "simple" velocity field given by 3.61 verifies these conditions. This field describes the average behavior of strain in the thickness. We remark that any kinematically admissible velocity field can be written as a sum of the simple velocity field and an addition term as following:

u(x, z) = C vol .      1 h(x) + f (x, z) h ′ (x) h 2 (x) .z + g(x, z)     
The admissible conditions will be rewritten thanks to the 2 functions f (x, z) and g(x, z). The first condition 4.1 implies that:

∂ f ∂x (x, z) + ∂g ∂z (x, z) = 0 ∀(x, z) ∈ Ω (4.7)

Oscillation of mechanical fields in roll bite

Method for constructing velocity field

There exists always a function denoted F(x, z) so that:

       f (x, z) = ∂F(x, z) ∂z g(x, z) = - ∂F(x, z) ∂x
With this expression of f (x, z) and g(x, z), the condition 4.1 is always verified. F(x, z) is flow function for the addition part determined within a constant.

The condition 4.2 is equivalent to F(x, 0) = const. As the value of this constant has no real importance, it can be chosen equal to 0, then

F(x, 0) = 0 ∀x ∈ [-L, 0]. (4.8)
The condition 4.3 becomes:

∂ 2 F ∂z 2 (x, 0) = 0. (4.9)
And the condition 4.6 gives:

F(x, h(x)) -F(x, 0) = 0.
Finally, any 2D kinematically admissible velocity field can be written as:

u(x, z) = C vol .      1 h(x) + ∂F(x, z) ∂z h ′ (x) h 2 (x) .z - ∂F(x, z) ∂x      (4.10)
with kinematically admissible conditions:

         F(x, 0) = 0 ∀x ∈ [-L, 0] F(x, h(x)) = 0 ∀x ∈ [-L, 0] ∂ 2 F ∂z 2 (x, 0) = 0 ∀x ∈ [-L, 0]. (4.11)
The tensors of strain rate: Since 4.10, all the derivatives of the velocity field are calculated as follow: The components of strain rate tensors ǫ(u) are:

∂u x ∂x (x, z) = C vol . - h ′ h 2 + ∂ 2 F(x, z) ∂x ∂z (4.12) ∂u x ∂z (x, z) = C vol . ∂ 2 F(x, z) ∂z 2 (4.13) ∂u z ∂x (x, z) = C vol .   h ′ h 2 ′ z - ∂ 2 F(x, z) ∂x 2   (4.14) ∂u z ∂z (x, z) = C vol . h ′ h 2 - ∂ 2 F(x,
ǫxx (u) = ∂u x ∂x (x, z) = C vol . - h ′ h 2 + ∂ 2 F(x, z) ∂x ∂z ǫzz (u) = ∂u z ∂z (x, z) = C vol . h ′ h 2 - ∂ 2 F(x, z) ∂x ∂z = -ǫxx (x, z) ǫxz (u) = 1 2 ∂u x ∂z + ∂u z ∂x = C vol 2 .   h ′ h 2 ′ z + ∂ 2 F(x, z) ∂z 2 - ∂ 2 F(x, z) ∂x 2   ǫ(u) = 2 3 ( ǫ2 xx (u) + 2 ǫ2 xz (u) + ǫ2 zz (u)) = 2 √ 3 ǫ2 xx (u) + ǫ2 xz (u) (4.16)

Applications

The expression 4.10 of kinematically admissible velocity is only a different form of flow function method (for example [START_REF] Dogruoglu | On constructing kinematically admissible velocity fields in cold sheet rolling[END_REF]). On the other hand, the kinematically admissible conditions written as 4.11 are relatively simplified and enable to easily propose some family of function F(x, z). For example, any function defined by t(x)z n with n = 1 or n > 2 verifies the first and the third equations of 4.11. These arguments lead naturally to following family of functions

F(x, z) = g 1 (x)z + ∞ ∑ n=3 g n (x)z n . (4.17)
The only condition so that the velocity defined by equations 4.10 and 4.17 is kinematically admissible is the second equation of 4.11. Of course, n can also be real or integer numbers but they are necessarily greater than 2.

Example of third order polynomial velocity field A

If g n (x) are chosen so that g n (x) = 0 for all n > 3, we have

F(x, z) = g 1 (x)z + g 3 (x)z 3 . (4.18)
The only kinematically admissible condition is (second equation of 4.11)

F(x, h(x)) = g 1 (x)h(x) + g 3 (x)h 3 (x) = 0 ∀x ∈ [-L, 0] (4.19)
which implies, then

g 1 (x) = -g 3 (x)h 2 (x) = 0 ∀x ∈ [-L, 0]. (4.20)
Substituting 4.20 into 4.18 we have

F(x, z) = g 3 (x)z(z 2 -h 2 (x)). (4.21)
Finally, any velocity field given by 4.10 where the function F(x, z) defined by 4.21 and the function g 3 (x) is freely chosen, even constant. And the 3-order polynomial velocity field is given by:

u(x, z) = C vol .      1 h(x) + 3z 2 -h 2 g 3 (x) h ′ (x) h 2 (x) .z -z 3 -h 2 z g ′ 3 (x)      (4.22)
We will see in the next section another proposition of the function F(x, y) having a nature of wave propagation that leads to a new velocity field with oscillating characteristics.

An oscillating velocity field proposal

In order to minimize the rolling power in function of velocity knowing that the space of function F(x, z) is an infinite-dimensional space, we need to parameterize it meaning restrict and transform the space of F(x, z) into a finitedimensional one. The parameterizing manner is more interesting if there are less number of parameters and when the optimum power is lower. We introduce in this section a particular way to build the function F(x, z) with few parameters.

First, it can be seen that the first equation of 4.11 implies that

∂ 2 F ∂x 2 (x, 0) = 0 ∀x ∈ [-L, 0].
Therefore, the third equation of 4.11 can be rewritten as

∂ 2 F ∂z 2 (x, 0) - ∂ 2 F ∂x 2 (x, 0) = 0 ∀x ∈ [-L, 0]. (4.23)
The kinematically admissible conditions given by equation 4.11 can be rewritten as follows

         F(x, 0) = 0 ∀x ∈ [-L, 0] F(x, h(x)) = 0 ∀x ∈ [-L, 0] ∂ 2 F ∂z 2 (x, 0) - ∂ 2 F ∂x 2 (x, 0) = 0 ∀x ∈ [-L, 0]. (4.24)
The third equation is similar to a "wave propagation" one will allows to propose similar "oscillation" solutions.

Indeed, let study now following equations similar to the kinematically admissible conditions 4.11:

         F * (x, 0) = 0 ∀x ∈ [-L, 0] F * (x, h(x)) = 0 ∀x ∈ [-L, 0] ∂ 2 F * ∂z 2 (x, z) - ∂ 2 F * ∂x 2 (x, z) = 0 ∀(x, z) ∈ Ω (4.25)
The third equation means that the shear strain rate ǫxz is not only equal to 0 on the surface of symmetry z = 0 but every where in the roll-bite (see the second equation). It is similar to the D'Alambert equation for one-dimensional wave problem. The general solution of this equation is:

F * (x, z) = G * (x + z) + H * (x -z)
with G and H any 1-variable functions, twice differentiable. Hence, the equations 4.25 are strictly equivalent to:

     F * (x, 0) = G * (x + 0) + H * (x -0) = 0 ∀x ∈ [-L, 0] F * (x, h(x)) = G * (x + h(x)) + H * (x -h(x)) = 0 ∀x ∈ [-L, 0] F * (x, z) = G * (x + z) + H * (x -z) ∀(x, z) ∈ Ω
By simplifying, this general solution of 4.25 can be given by:

     F * (x, z) = G * (x + z) -G * (x -z) ∀(x, z) ∈ Ω G * (x + h(x)) = G * (x -h(x)) ∀x ∈ [-L, 0] ∀G ∈ C2(ℜ) (4.26)
Now, let come back to the admissible conditions 4.24. The difference with respect to the equations 4.25 is that the d'Alambert equation is not verified every where in the roll-bite but only on the surface of symmetry z = 0. Of course, all functions verifying 4.25 satisfy also 4.24. Meaning all functions F(x, z) given by 4.26 are also solution of 4.11. However, having no solution to describe the whole admissible space, we propose to study a family of functions verifying the following equation:

     F(x, z) = A e [G e (x + z) -G e (x -z)] .K e (x) ∀(x, z) ∈ Ω G e (x + h(x)) = G e (x -h(x)) ∀x ∈ [-L, 0] ∀G e , K e ∈ C2(ℜ) (4.27)
where A e is a parameter that allow to represent the amplitude of F(x, z) and thus the oscillation part in the velocity field. G e and K e are two one-variable functions, twice differentiable. When A e = 0 the velocity field is identical to the "simplified" or elliptical one.

It is easy to verify that all functions F(x, z) given by 4.27 verify the kinematically admissible conditions 4.11. Even if this family dose not contain all admissible functions F(x, z), we will see later that the velocity solution (best velocity of the family) carries effectively oscillation characteristics and gives lower power than the simple one.

Periodicity of velocity fields and determination of G e

Noting that the equation 4.27 is given for every x in the roll-bite, if this function G is known on a segment [x 0h(x 0 ), x 0 + h(x 0 )], for any x 0 in the roll-bite, it is entirely defined in the whole roll-bite. It seems to be a quasi periodic function with a period equal to 2h(x) -the thickness of the strip -varying along the roll bite.

Therefore, to construct the function G e all along the roll-bite, it is only necessary to do it on a segment, for example the segment at the entry of the roll-bite [x eh(x e ), x e + h(x e )]. This function is denoted G 0 . To have the continuity of the velocity field, the function G e and its derivative need to be continuous. Therefore,

     G 0 (x e + h(x e )) = G 0 (x e -h(x e )) G ′ 0 (x e + h(x e )) = 1 -h ′ (x e ) 1 + h ′ (x e ) G ′ 0 (x e -h(x e )) (4.28)
Choice of function G 0 : the function G 0 can be chosen as

G 0 (x) = cos (g(x)) g(x) = a 0 + a 1 x + a 2 x 2 . (4.29)
With this construction 4.29, it is easy to check that G 0 (x) satisfies the two continuity conditions of 4.28 if g verifies following 3 conditions:

         g(x e -h(x e )) = π g(x e + h(x e )) = -π g ′ (x e + h(x e )) = 1 -h ′ (x e ) 1 + h ′ (x e ) g ′ (x e -h(x e )) (4.30)
The 3 parameters: a 0 , a 1 , a 2 are then completely determined.

Choice of function K e

We chose a function with one parameter, as

K e (x) = x 3 (4.31)
which models the decrease of the oscillation amplitude from the entry towards the exit of the roll-bite.

Choice of function F

Both functions G e and K e are now completely determined. The function F(x, z) given by equations 4.27 contain thefore only one parameter A e . It can be noted that, the function G e is somehow similar to the wave propagation from the entry forward the exit of the roll-bite. Now in the analogy we propose to add a similar function representing the wave propagation from the exit towards the entry of the roll bite. Similarly to G e , G s is also constructed by using the periodicity equation

G s (x + h(x)) = G s (x -h(x)) ∀x ∈ [-L, 0]
but based on the its definition on the exit segment [-h s , h s ] instead of the entry one [x eh e , x e + h e ]. Regarding the function K s , the analogy is understood as follows.

The function K e is maximum (in absolute value) at the entry and vanishes at the exit. The function K s is chosen as 3 (4.32)

K s (x) = (x -x e )

Oscillation of mechanical fields in roll bite

4.3 An oscillating velocity field proposal so that it is maximum at the exit and vanishes at the entry.

Finally F(x, z) = A e [G e (x + z) -G e (x -z)] K e (x) + A s [G s (x + z) -G s (x -z)] K s (x). (4.33)

Summery of velocity fields

From the expression of velocity field 4.10 and the choice of the function F(x, z) by 4.33 the velocity is

u(x, z) = C vol .v(A e , A s , x, z) (4.34)
where v(x, z) is elementary velocity field of which the components are defined as follows

v x (A e , A s , x, z) = 1 h(x) + A e G ′ e (x + z) + G ′ e (x -z) K e (x) + A s G ′ s (x + z) + G ′ s (x -z) K s (x) (4.35)
and

v z (A e , A s , x, z) = h ′ (x) h 2 (x) .z -A e G ′ e (x + z) -G ′ e (x -z) K e (x) + (G e (x + z) -G e (x -z)) K ′ e (x)
-A s G ′ s (x + z) -G ′ s (x -z) K s (x) + (G s (x + z) -G s (x -z)) K ′ s (x) . (4.36)
As being discussed previously, all functions G e , K e and G s , K s are completely defined. The velocity field contents therefore three parameters A e , A s and C vol that need determining by minimizing the power function given by 3.27. Of course, we can chose A e = 0 and A s = 0 if we only want to model what the oscillation from the exit and inversely A e = 0 and A s = 0 if we are only interested in the oscillation from the entry.

By writing the equality of the strip velocity and work-roll velocity at the neutral point, we have

V c = u 2 x (x n , h n ) + u 2 z (x n , h n ) = C vol v x (A e , A s , x n , h n ) 1 + h ′ n (4.37)
where

h n = h(x n ) and h ′ n = h ′ (x n ).
We deduce then

C vol = V c v x (x n , h n ) 1 + h ′ n ( 4

Remark: The elementary velocity is the velocity field if the flow rate is equal to an unity (C vol = 1). It characterizes the stream lines of material flow. Depending on the value of A e and A s that the amplitude of oscillation parts is more or less important and change the stream lines compared to the classic ones corresponding to the classic velocity field. As this elementary velocity does not depend on the neutral point position

x n , the flow patterns defined from this velocity family are independent of x n . The parameter x n is only involved in velocity field through the total flow rate C vol (A e , A s , x n ).

Discussion: The new family of the velocity fields introduced in this section has an oscillation terms. The periodicity characteristic of this oscillation has particularly been brought out and the function G e and/or G s describing this phenomenon is carefully built. Finally, with only three parameters A e , A s and x n this velocity family is able to model the mechanical fields heterogeneity across the strip thickness. In this velocity family, the power function J(u) depending on the velocity field becomes a function of these three parameters A e , A s and x n , J(u(A e , A s , x n )). The minimization of this function with respect to the three parameters is then numerically possible. Hereafter, we will present an optimization strategy efficient in terms of computing time.

UBM with the oscillating velocity field 4.4.1 New strategy for minimization of power function

After introducing this new velocity field family represented by three parameters x n , A e and A s . The power function given by 3.27 becomes a function of these three parameters and the optimization problem can be written as:

               ∂J ∂x n (A e , A s , x n ) = 0 ∂J ∂A e (A e , A s , x n ) = 0 ∂J ∂A s (A e , A s , x n ) = 0. (4.40)
This section presents a new optimization strategy that allows to obtain the optimum neutral point more easily and to decrease significantly the computation time. This strategy is applicable if the power function is analytically derivable with respect to x n . And this is true for every kinematically admissible velocity field expressed as 4.39. The method consisting in optimizing analytically the power function with respect to x n and then numerically with respect to A e and A s .

4.4.1.a Calculation of power functions and derivatives

Power of plastic deformation: Indeed, thanks to the expression 4.39 of the velocity field, the power dissipated by plastic deformation in Ω (the first term of the equation 3.27 right side) can be expressed as follow:

J ǫ(u)( A e , A s , x n ) = C vol (A e , A s , x n ).J ǫ(v)( A e , A s ) where J ǫ(v)( A e , A s ) = Ω σ 0 . ǫ(v)(A e , A s , x, z) dΩ. ( 4.41)
Power of discontinuity of velocity: Also the power dissipated by velocity discontinuity (the second term of the equation 3.27 right side becomes:

J ∆u (A e , A s , x n ) = C vol (A e , A s , x n ).J ∆v (A e , A s ) where J ∆v (A e , A s ) = S d σ 0 √ 3 ∆v(A e , A s , x, z) dS. (4.42)
Power of entry and exit tensions: In the same way, the tensions power (the forth term of the equation 3.27 right side):

J ten (u)(A e , A s , x n ) = C vol (A e , A s , x n )J ten (v)
where J ten (u) =

S T d T d .v(A e , A s , x, z) dS = T s -T e . (4.43)
Thus, their derivatives with respect to x n are nicely represented by:

                 ∂J ǫ(u) ∂x n (A e , A s , x n ) = ∂C vol (A e , A s , x n ) ∂x n .J ǫ(v)( A e , A s ) ∂J ∆u (u) ∂x n (A e , A s , x n ) = ∂C vol (A e , A s , x n ) ∂x n .J ∆v(v) (A e , A s ) ∂J ten (u) ∂x n (A e , A s , x n ) = ∂C vol (A e , A s , x n ) ∂x n . (T s -T e ) (4.44)
Power of contact friction: About the friction power (the third term of the equation 3.27 right side), its dependence on x n is more complicated. However, we will show that its derivative with respect to x n can be equally obtained analytically. The difference of velocity on the contact surface is defined by:

∆u c (x) = u(x, h(x)) -V c . (4.45)
The friction power is then written as:

J f ric (u)(A e , A s , x n ) = S c τ. ∆u c (x, h(x)) . dS = m.σ 0 √ 3 . 0 x n - x n -L ( u(x, h(x)) -V c ) . 1 + h ′ 2 (x) dx.
By definition of neutral point, |∆u c (x n , h n )| = 0 we can deduce then:

∂J f ric (u) ∂x n (A e , A s , x n ) = m.σ 0 √ 3 0 x n - x n -L ∂ u(A, x, h(x)) ∂x n 1 + h ′ 2 (x) dx -2∆u c (x n , h n ) 1 + h ′ 2 n = m.σ 0 √ 3 ∂C vol (A e , A s , x n ) ∂x n 0 x n - x n -L v(A, x, h(x)) . 1 + h ′ 2 (x) dx.
Let denote now

J V c f ric (x n ) = m.σ 0 √ 3 . x n -L - 0 x n V c . 1 + h ′ 2 (x) dx (4.46)
and

J v f ric (A e , A s , x n ) = m.σ 0 √ 3 . 0 x n - x n -L v(A, x, h(x)) . 1 + h ′ 2 (x) dx = m.σ 0 √ 3 . 0 x n - x n -L v x (A, x, h(x)) 1 + h ′ 2 (x) dx. (4.47)
In fact, -J V c f ric (x n ) is the power of the friction stress on the work-roll that rotates at peripheral velocity V c . It depends only on the neutral position but not on the strip speed. And -J v f ric (A e , A s , x n ) is the power of the friction stress acted on the strip corresponding to the strip elementary velocity v(A e , A s , x, z). Then we can rewrite both the friction power and its derivative with respect to x n as:

     J f ric (u)(A e , A s , x n ) = J V c f ric (x n ) + C vol (A e , A s , x n ).J v f ric (A e , A s , x n ) ∂J f ric ∂x n (A e , A s , x n ) = ∂C vol (A e , A s , x n ) ∂x n .J v f ric (A e , A s , x n ). (4.48)
Total rolling power function: The total power function and its derivative with respect to x n becomes:

     J(A e , A s , x n ) = J V c f ric (x n ) + C vol (A e , A s , x n ) J ǫ(v)( A e , A s ) + J ∆v (v)(A e , A s ) + J v f ric (A e , A s , x n ) -T s + T e ∂J ∂x n (A e , A s , x n ) = ∂C vol (A e , A s , x n ) ∂x n J ǫ(v)( A e , A s ) + J ∆v (v)(A e , A s ) + J v f ric (A e , A s , x n ) -T s + T e . ( 4
∂J ∂x n (A e , A s , x n ) = 0 ⇔ J ǫ(v)( A e , A s ) + J ∆v (A e , A s ) + J v f ric (A e , A s , x n ) -T s + T e = 0. (4.50)
This is an equation allowing to determine x n as a function of A e and A s . If x 1 n (A e , A s ) denotes the solution of this equation (4.50) then by substituting 4.50 into the first equation of 4.49 and using 4.46 the minimum power with respect to x n is obtained as following function of A e , A s

J 1 (A e , A s ) = J(x 1 n (A e , A s ), A e , A s ) = J V c f ric (x 1 n (A e , A s )) = mσ 0 √ 3 x 1 n (A e ,A s ) -L - 0 x 1 n (A e ,A s ) V c 1 + h ′ 2 (x) dx = mσ 0 √ 3 V c (AN ⌢ -NB ⌢ ) (4.51)
We can deduce from this equation that

         ∂J 1 ∂A e (A e , A s ) = 2 m.σ 0 √ 3 V c . 1 + h ′ 2 (x 1 n ). ∂x 1 n ∂A e (A e , A s ) ∂J 1 ∂A s (A e , A s ) = 2 m.σ 0 √ 3 V c . 1 + h ′ 2 (x 1 n ). ∂x 1 n ∂A s (A e , A s ). (4.52)
In other words, the optimization of J 1 (A e , A s ) and x 1 n (A e , A s ) with respect to A e , A s are obtained at a same value of A e , A s . This analysis of the power function and its derivatives results finally to the following three-step numerical resolution:

1. For given values of A e , A s , solve 4.50 by Newton algorithm to obtain x 1 n (A e , A s ). This step gives us numerically the function x 1 n (A e , A s ).

Minimize of x 1

n (A e , A s ) issued from the first step by Newton-Raphson algorithm (see section 5.3.5.c). The first and second order derivatives of x 1 n (A e , A s ) are calculated numerically. We obtain at the end of this step the optimum value of A e , A s and minimum value of x n = x 1 n (A e , A s ).

3. Compute the minimum rolling power using 4.51

4.4.1.c Application field of this strategy

This new method of minimization is applicable for 2D approaches where the friction power J f ric is analytically derivable with respect to x n . For 3D approach, we will see in the next chapter that the derivative of J f ric with respect to x n is much more complicated and need to be calculated numerically.

It is an application for all velocity fields of which the elementary part does not depend explicitly on x n . Or, the flow lines are independently of the neutral point (no perturbation of the neutral point as singularity point on the flow lines). This is in fact valid for all velocity fields studied in the literature till our days.

Numerical resolution -programming

The previous section presents a new optimization strategy. In this section the details of numerical calculation of all elementary power functions J ǫ(v)( A e , A s ), J ∆v (A e , A s ) and J v f ric (A e , A s , x n ) given by 4.41, 4.42 and 4.43 will be detailed.

4.4.2.a Surface of velocity discontinuity at roll-bite entry

As a reminder, by construction we propose a differential continuous velocity field in each zone: before, inside and after the roll bite. The second zone (inside the roll-bite) can be understood as the plastic deformation zone. There are two discontinuity surfaces, one at the roll bite entry and one at the roll bite exit, as showed in Figure 4.1. The interesting point is that the form of these discontinuity surfaces is not predefined but is a output of the model. Indeed, during the research of optimum velocity field, the discontinuity surfaces changes as a function of velocity field. Here are presented the details how to determine numerically the discontinuity surfaces as well as how to calculate numerically the elementary dissipation powers J ∆v (A e , A s ). We study first the entry one. Reminding that before the plastic deformation zone Ω, the velocity is uniform and equal to V e , and by definition of the flow rate, we can write:

V e = C vol (A e , A s , x n ). 1 h e .
In the analogy of the elementary velocity field definition 4.39, the elementary velocity field before the roll-bite is v e = 1 h e . With the simple velocity (3.61), as the longitudinal velocity is constant across the thickness, the discontinuity surface of velocity field is therefore a cross section.

However, with the oscillating velocity proposed in previous session which is also rewritten as 4.39, the discontinuity surface Γ e is no longer a cross section because the longitudinal velocity varies in the thickness direction. The difference of elementary velocity between the plastic zone and the entry zone is calculated as:

∆v =   v x (x, z) - 1 h e v z (x, z)   (4.53)
where v x (x, z) and v z (x, z) are given by 4.35 and 4.36 respectively. The kinematically admissible condition of velocity on the discontinuity surface is defined as ∆v.n = 0. In other words, the discontinuity of velocity needs to be tangential to the discontinuity surface at any point of Γ e : ∆v ∆v = t. (4.54)

Based on this statement, the velocity discontinuity surface is determined starting from a chosen point on this surface. The chosen point is the first point of contact between the work-roll and the strip P 1 (x P 1 , z P 1 ) ≡ A with x 1 = -L, z 1 = h(-L). P 1 is the beginning point of the velocity discontinuity surface at roll-bite entry, P 1 ∈ Γ e . If P 2 (x

P 2 = x P 1 + dx 1 , z P 2 = z P 1 + dz 1 )
∈ Γ e is a neighboring point of P 1 and the distance between them is denoted by dl = dx 2 1 + dz 2 1 then the tangential direction is approximated by:

t ≈ 1 dl dx 1 dz 1 .
The admissible condition of velocity discontinuity 4.54 implies that:

dx 1 dz 1 = ∆v x (P 1 ) ∆v z (P 1 ) dl ∆v (4.55)
And

∆v 1 = v x (P 1 ) - 1 h e 2 + v 2 z (P 1 ) (4.56)
In practice, the value of dl is chosen (= h e /20 for example) and the equation 4.55 allows to determine the point P 2 and the dissipation power on this segment P 1 P 2 as

J ∆v 1 = σ 0 √ 3 ∆v 1 dl. (4.57)
Similarly from P 2 , we determine the next point P 3 and the dissipation power on the segment P 2 P 3 . By repeating this operation until P n (x P n , z P n ) with z P n = 0 the discontinuity surface and the corresponding dissipation power is completely determined. This approximation requires that dl is chosen small enough depending on the complexity of the discontinuity surface.

           J Γ e ∆v = σ 0 √ 3 n-1 ∑ i=1 ∆v i dl with ∆v i = v x (P i ) - 1 h e 2 + v 2 z (P i ) (4.58)

4.4.2.b Surface of velocity discontinuity at roll-bite exit

In a general case, the discontinuity surface at the roll-bite exit can also exist and is different from a cross section. If Q 1 (0, h s ) ≡ B denotes the last contact point, it is chosen as the first point of Γ s . And following the same method used to determine Γ e , we can determine

Γ s with Q n (x Q n , z Q n ) denoting the last end of Γ s , z Q n = 0. All calculations are similar.

4.4.2.c Elementary plastic deformation power

As can be seen in the second equation of 4.41, in order to calculate the deformation power J ǫ(v)( A e , A s ), it is necessary to compute the ǫ(v)(A e , A s ). The elementary strain rate tensor ǫ(v) corresponding to the elementary velocity v(A e , A s , x, z) defined by 4.35 and 4.36 can be obtained as follows:

                                         ǫxx (v)(x, z) = - h ′ h 2 + A e G ′′ e (x + z) + G ′′ e (x -z) K e (x) + G ′ e (x + z) + G ′ e (x -z) K ′ e (x) + A s G ′′ s (x + z) + G ′′ s (x -z) K s (x) + G ′ s (x + z) + G ′ s (x -z) K ′ s (x) ǫxz (v)(x, z) = 1 2 h ′ h 2 ′ z -G ′ e (x + z) -G ′ e (x -z) K ′ e (x) - 1 2 [G e (x + z) -G e (x -z)] K ′′ e (x) -G ′ s (x + z) -G ′ s (x -z) K ′ s (x) - 1 2 [G s (x + z) -G s (x -z)] K ′′ s (x) ǫzz (v)(x, z) = -ǫxx (v)(x, z) ǫ(v)(x, z) = 2 √ 3 ǫ2 xx (v) + ǫ2 xz (v) (4.59)

Plastic deformation zone

In order to compute numerically the plastic deformation power, it is necessary to divide the plastic deformation zone Ω into three zones. The first one, Ω 1 defined between the three surfaces: Γ e , z = h(x), x = x P n . The Ω 2 is the area limited by: z = 0, z = h(x), x = x P n and x = x Q n . And the last one, Ω 3 is comprised between x = x Q n , z = h(x) and Γ s . The integral in the expression of J ǫ(v)( A e , A s ) is calculated separately in each area.

Integration in Ω 1

At the same time of the determination of each point P i (x P i , z P i ) and the dissipation by velocity discontinuity on Γ e (see section 4.4.2.a) the plastic dissipation power can be also calculated as:

J Ω 1 ǫ (v)(A e , A s ) = Ω 1 σ 0 . ǫ(v)(A e , A s , x, z). dΩ = σ 0 n-1 ∑ i=1 dx i .J i ǫ(v)( A e , A s )
where J i ǫ is the elementary plastic deformation power in the zone delimited by: z ∈ P i P i+1 , z = h(x), x = x i , x = x i+1 (see Figure 4.2). If the coordinates of the element P i P i+1 are given as

     x ele i = 1 2 (x i + x i+1 ) z ele i = 1 2 (z i + z i+1 ) (4.60)
J i ǫ can be obtained by

J i ǫ(v)( A e , A s ) = h(x ele i ) z ele i ǫ(v)(A, x ele i , z) dz. (4.61)
The integral in this equation is done numerically using Gauss integration method presented in the section A. We can find also in the same section the value of weighting factors ω k and function arguments ξ k used in Gauss Quadrature Formulas. Reminding that, in a given segment the N-point Gauss integration gives the exact value for any polynomial of 2N-1 order. However, for an oscillation function having several periods on the integration segment, this approach is no longer accurate. By construction (see the section 4.3), the velocity field is oscillating with a period equal to 2h(x).

By consequence, the strain rate field ǫ(v) has also the same oscillation period. In the thickness direction, only a half the strip thickness is studied. As a half of thickness corresponds to a half of oscillation period, the field ǫ(v) can be accurately approximated by a 5-order polynomial. So in the thickness direction, only one element is sufficient to integrate. In choosing N = 3 we have a good compromise of calculation time and precision.

Hence, using Gauss integration operator IGauss N=3 1D given by A.12 to approximate the integral of the equation 4.61 we obtain:

J i ǫ(v)( A e , A s ) ≈IGauss N=3 1D ( f i , z ele i , h(x ele i ), 1) with f i = z → ǫ(v)(A e , A s , x ele i , z) therefore J Ω 1 ǫ (v)(A e , A s ) ≈σ 0 n-1 ∑ i=1 dx i .IGauss N=3 1D ( f i , z ele i , h(x ele i ), 1)
with

f i = z → ǫ(v)(A e , A s , x ele i , z) (4.62)
where n is the number of point P i on Γ e . x ele i , z ele i and dx i are given by 4.60. IGauss N=3 1D is the Gauss-Legendre operator approaching 1D integral defined by A.12. ǫ(v) is calculated thanks to the equations 4.59.

Integration in Ω 3

This is done similarly as the integration in Ω 1 .

Integration in Ω 2

By developing the 2D integral in Ω 2 of the elementary plastic deformation power into a double integral, we obtain:

J Ω 2 ǫ (v)(A e , A s ) = Ω 2 σ 0 . ǫ(v)(A e , A s , x, z). dΩ = σ 0 x Qn x Pn h(x) 0 ǫ(v)(A e , A s , x, z). dz dx. (4.63)
These integrals are equally done numerically Gauss-Legendre integration. Similarly as in Ω 1 , for the integral though the thickness, only one element is enough. But in the longitudinal direction, we need to divide the segment [x P n , x Q n ] into a certain number of elements n ele x so that the size of each element is less than a half of period, h(x). This number is chosen as n ele x = trunc L h s . The size of each element is then

∆x = x Q n -x P n n ele .
The i th is between two nodes x node i-1 = x P n + (i -1)∆x and x node i = x P n + i∆x. Thus:

J Ω 2 ǫ (v)(A e , A s ) ≈σ 0 .IGauss N=3 2D ( f 2 , x Γ e n , 0, 0, h(.), n ele x , 1) with f 2 := (x, z) → ǫ(v)(A e , A s , x, z) (4.64)
where IGauss N=3 2D is the Gauss-Legendre operator approaching 2D integral defined by A.15 and ǫ(v)(A e , A s , x, z) given by the equations 4.59.

4.4.2.d Elementary friction dissipation power

Both friction power J V c f ric (x n ) given by 4.46 and J v f ric (A e , A s , x n ) given by 4.47 are necessary to be calculated. With the elementary velocity given by 4.35 and 4.36, the two integrals are already explicit enough. Noting n b ele and n a ele are the number of elements before and after the neutral point. Similarly as above we chose:

n b ele = trunc x n + L h s
and n a ele = trunc -x n h s so that the elements are smaller than a half of oscillation period. The sizes of elements before and after the neutral point are then given by:

∆x b = x n -x e n b
ele and ∆x a = -x n n a ele .

Using Gauss-Legendre integration method, we have:

J V c f ric (x n ) ≈ m.σ 0 √ 3 V c IGauss N=3 1D ( f c , -L, x n , n b ele ) -IGauss N=3 1D ( f c , x n , 0, n a ele ) with f c = x → 1 + h ′ 2 (x) (4.65)
And similarly:

J v f ric (A e , A s , x n ) ≈ m.σ 0 √ 3 . -IGauss N=3 1D ( f v x , -L, x n , n b ele ) + IGauss N=3 1D ( f v x , x n , 0, n a ele ) with f v x = x → v x (A e , A s , x, h(x)) 1 + h ′ 2 (x) (4.66)

4.4.2.e Elementary tensions power

The elementary tensions power is explicitly given by the second equation of 4.43.

Oscillation of mechanical fields in roll bite

4.5 Comparison with Lam3-Tec3 and other UBM models

Comparison with Lam3-Tec3 and other UBM models

This session is a comparison of UBM results obtained with three velocity fields: unitriangular one, elliptical (or simple) one and oscillating -the new one and the Finite Elements Method Lam3-Tec3 . The Lam3-Tec3 is a software developed by ArcelorMittal Group and several partners. Lam3-Tec3 is able to give the solution (strain and stress fields) at stationary state and the final mesh gives the material flow lines.

For this comparison, three rolling conditions corresponding to roughing mill, finishing and cold rolling mill are considered (see details in Table 4

Power and neutral point

Dependence of x 1 n and J 1 on A e and A s

As discussed previously, the equations 4.52 implies that the optimization of J 1 (A e , A s ) and x 1 n (A e , A s ) with respect to A e , A s are obtained at a same value of A e , A s . Indeed, this remark is confirmed for the case of roughing mill (case 1, Table 4.1) by Figure 4.5 showing the dependence of x 1 n , solution of 4.50 and J 1 given by 4.51 as a functions of A e when A s = 0. It can be seen also in Figure 4.4 illustrating the variation of x 1 n and J 1 as a functions of A s for A e = 0 that the functions are optimum at the same value of A s . The same conclusion is observed with the finishing condition (Figures 4.5 Furthermore, these graphics show also that the minimum values of x 1 n and J 1 are reached at A e and A s which are different from 0, i.e different from the simple (elliptical) velocity field solution. By comparing the two graphics of each rolling case (the graphics are not at a same scale), it can be seen that the optimum neutral position x 1 n as well as the optimum power J 1 obtained by varying A e are lower than that obtained by varying A s . That means, the oscillation part from the entry side contributes more efficiently to the reduction of power. 4.1). The dependence of x 1 n and J 1 as functions of A e when A s = 0. 4.1). The dependence of x 1 n and J 1 as functions of A s when A e = 0. 4.1). The dependence ofx 1 n and J 1 as functions of A e when A s = 0. 4.1). The dependence of x 1 n and J 1 as functions of A s when A e = 0. 4.1). The dependence ofx 1 n and J 1 as functions of A e when A s = 0. 4.1). The dependence of x 1 n and J 1 as functions of A s when A e = 0.

Comparison of optimum power to other models

The function J 1 (A e , A s ) as well as x 1 n (A e , A s ) are both convex in both directions (see the graphics above). The optimum values of A e and A s are obtained by 2D Newton-Raphson algorithm without difficulty. The obtained results are given in Tables 4.2, 4.3 and 4.4. The roll torque is evaluated by

T = J.R V c . (4.67)
About the unitriangular velocity, the optimum power is quit interesting in the case 1 -roughing mill condition. It is lower than the that of simple velocity. But in the case 2 -hot finishing mill and case 3 -cold rolling mill, it is much higher than the power given by the simple and oscillating velocity fields. In these cases, the multitriangular model should give significantly better results.

In all the three cases, the oscillating velocity field gives a lower upper bound estimation of rolling power J, torque T and neutral position x n than the simple (elliptical) one. These new estimations are much closer to Lam3-Tec3 results. The three results tables shows that the improvement is mainly thanks to a decrease of plastic deformation power J de f and secondarily to friction one J f ric .

Plastic area -discontinuity surfaces

Unlike the case of simple (elliptical) velocity field, with the oscillating velocity field the entry and exit discontinuity surfaces Γ e and Γ s are not prefixed but they are outputs of the UBM model. The obtained results corresponding to the By consequence, the discontinuity surfaces are no longer vertical planes but curved ones resulting to a smaller plastic deformation domain Ω. This, in addition to that fact that the oscillating velocity requires less deformation power in comparison with the simple one on a same integration domain, allows to decrease the plastic deformation power.

Velocity isovalue surfaces

By construction of the UBM velocity field, the discontinuity surfaces Γ e and Γ s are equally the limits between the non-deformation and plastic deformation zones. As for Lam3-Tec3 , the velocity discontinuity surfaces do not really exist because the velocity field is continuous. The comparison of discontinuity surface between UBM and Lam3-Tec3 is not simple. However, this velocity field of Lam3-Tec3 varies strongly through small zones at the entry and exit of the roll-bite. Hence, that should be interesting to compare the isovalue surfaces of the velocity field which are the isovalue To do this comparison, the isovalue cartography is, at first obtained with "GLview" a post-treatment software allowing to visualize the results of Lam3-Tec3 . Figures 4.15, 4.17 and 4.19 show the Lam3-Tec3 results of longitudinal velocity u x for the 3 rolling cases. The limits between two successive zones are the isovalue curves of u x . Then, using the same value of ratio u x /V e we build the isovalue curves of u x with UBM results (Figures 4.16 If the hypothesis "a cross section will stay a cross section" of the simple velocity field is assumed, the isovalue curves of u x are the straigth vertical lines. Nevertheless, the results of Lam3-Tec3 as well as of UBM show that the isocurve are very different from a straight vertical lines. And the UBM results are relatively closed results to that of Lam3-Tec3 3.

Oscillation of velocity field along the streamlines 4.5.4.a Streamlines

Existing methods for constructing velocity field as [START_REF] Dogruoglu | On constructing kinematically admissible velocity fields in cold sheet rolling[END_REF]... usually based on a the given stream lines. For example, by assuming that the flow lines are circular or elliptical, that the velocity fields will be determined. By this way, these methods introduce also quite important constraints to the velocity fields because it is very difficult to imagine a very From the left to the right hand side, the 9 isovalue curves obtained by UBM correspond to the same ratio u x /V e . In addition, the last curve corresponds to u x = 1.4874V e = 0.99V s . good and complete flow lines. For this reason, until today, except circular and elliptical flow lines, there does not exist any other imagined flow lines pattern to approach rolling process.

The method that we introduce in this chapter discards this constrain. Contrarily to previous methods, the form of the flow lines are not pre-assumed but complete results of the model. In order to simplify the comparison with Lam3-Tec3 where the number of elements in the half thickness is equal to 19, the UBM results will be analyzed at the equivalent streamlines meaning 20 streamlines from the symmetry line (z = 0) to the strip surface (z = h(x). Figure 4.21 shows the streamlines obtained by the UBM with the velocity corresponding to the optimum value of x n and A e , A s for the case 3 -cold rolling first stand. The streamlines are named from 1 (the symmetry line z = 0) to 20 (the strip surface z = h(x)). Before the roll-bite, these streamlines line are equidistant with a distance of h e /19. It can be seen the streamlines oscillate slightly throughout the plastic deformation zone, especially at the beginning of the roll-bite while before and after the roll-bite they are straight and equidistant. Figure 4.22 illustrates that the streamlines obtained by UBM and Lam3-Tec3 are very closed.

For the case 1 (roughing mill) and 2 (finishing mil), the streamlines are not showed here but the numeration of the streamlines is the same as showed in Figure 4.21 for the case 3. In this session, we will study the behavior of the velocity field along each streamline.

4.5.4.b Oscillation of longitudinal velocity

It should be reminded that the new velocity field is the sum of the classic part (simple velocity) and the oscillating part (see the expression of velocity field given by 4.10. first, and progressively the line 15, 10 ... and the line 1 at the center of the strip is deformed last. In this area the velocity of line 1 increases more and more slowly while the longitudinal velocity of other lines increases more and more quickly. And after about a half entry thickness the tendency is inverted, the line 1 is the fastest and the line 20 is the slowest. This longitudinal velocity field along a stream line oscillates around the classic part which is the same for all stream lines. The period is similar to the thickness which varies from h e at the entry to h s at the exit of the roll bite. The amplitude of this oscillation decreases along the roll bite. 4.4), the forward slip obtained by Lam3-Tec3 in the case 1 and 2 is higher than that obtained by UBM (see Tables 4.2 and 4.3). That means the entry and exit velocities are also higher which can be seen by comparing the amplitude of oscillation by Lam3-Tec3 is higher but the order of velocity value of the streamlines is the same (from the fastest one to the slowest one). 4.31 and 4.33 show the vertical velocity field as well as its simple part obtained by UBM along the stream lines 1, 5, 10, 15 and 20 for the three studied rolling cases. Obviously, along the center, line 1 the vertical velocity is equal to 0 and there is no oscillation of vertical velocity. As for the contact surface, line 20, the oscillation is relatively small. Merely along the other streamlines the vertical velocity field oscillate around the simple part and the oscillation can be very important. These oscillations have same period equal to the strip thickness 2h(x) as that of the longitudinal velocity and their amplitude decreases also from the entry to the exit.

4.5.4.c Oscillation of vertical velocity

It can be seen in Figures 4.30, 4.32 and 4.34 that like the longitudinal velocity, the UBM vertical velocity field is quite closed even if the oscillation amplitude is less important than the Lam3-Tec3 result.

Conclusions and perspectives

Conclusions

The present chapter introduces a new approach of the velocity fields in the roll-bite and gives a high understanding about a particular phenomenon: the velocity oscillates spatially in both thickness and longitudinal directions. We presented a new optimization strategy that consists in obtaining an equation determining the optimum neutral point. This result allows to improve the numerical algorithm as well as the calculation time. The UBM using the new proposed velocity family results to an optimum velocity that oscillates spatially throughout the roll-bite with a psudo-period equal to the local strip thickness h(x). This obtained oscillating velocity field matches very well the Lam3-Tec3 results and improves the upper bound of rolling power and torque in comparison with the simple (elliptical) velocity field. This velocity field has characteristics of both elliptical (continuous) and multitriangular (rigid movements) velocity fields.

Perspectives

Polynomial velocity family

As explained previously, the oscillation velocity allows to reduce mainly the deformation power. Consider now the family F(x, z) = ∑ g i (x).z i ∀i = 2 that is introduced in the section 4.2.2. This family would allow to model the heterogeneity of mechanical fields across the thickness. But unlike the oscillating one, it allows especially to have higher strip speed on the contact before the neutral point and lower strip speed on the contact after and this fact would help to reduce the differential velocity between the strip and the work-roll and reduce therefore the friction power. The numerical calculation presented in this chapter enables the study by UBM with this polynomial velocity field family.

As can be remarked from the results showed in Table 4.2, 4.3 and 4.4, the oscillating velocity field gives very closed deformation power to Lam3-Tec3 one while the friction power is significantly higher. Hence, studying this polynomial velocity field may be interesting.

Perturbation of velocity field around the neutral point

By definition, the neutral point is the point where the contact shear stress (friction stress) is discontinuous (positive before and negative after). This discontinuity of friction creates a discontinuity of the shear stress σ xz meaning that for the material with Mise's behavior without viscosity effect (given by 3.13), the ǫxz is also discontinuous. All the previous continuous velocity fields (eccentric, simple-elliptical and oscillating) are not able to take into account this phenomenon. In order to model that, it is necessary to separate the roll-bite into two areas with a discontinuity surface at the neutral point. Furthermore, as the neutral point is not known and needs determining, it is necessary to build a velocity field depending explicitly on the neutral point. Meaning that the neutral point is not only involved in the velocity field via the flow rate as 4.39 but the elementary velocity field depends also on the x n . Moreover, we can observe that the UBM under-estimates the forward slip in comparison with Lam3-Tec3 for all the three rolling cases while neutral point obtained by UBM is quite closed to Lam3-Tec3 . Inversely, the unitriangular over-estimate it. This fact is not a random and can be explained by the existing of a neutral zone -sticking area (but not a point). With much more freedom degree the velocity of Lam3-Tec3 can approach the actual one even around the neutral point and is able to model well the neutral zone. The rigid motion model such as multitriangular one is able to model the neutral zone. The triangular that corresponds to the neutral zone rotates around the work-roll center with the same angular velocity. For this reason, the longitudinal velocity increases from the surface contact z = h(x n ) to the strip center z = 0. While continuous velocity fields studied until our days (eccentricity, simple-elliptical, circular or oscillating ones) have flow patterns (elementary part) that do not depend on the neutral point. Meaning that, at the neutral point, the distribution of longitudinal velocity across the strip thickness can be different from that described above for neutral zone. That is why these models under-estimate very often the flow rate through the cross section at the neutral point.

The oscillating velocity with advantage of low deformation power and the multitriangular with advantage of neutral zone modeling can be combined together to create the new one. The idea is to model the neutral zone as a rigid curvilinear triangular with a base denoted N 1 N 2 on the surface of the roll and with an apex P on the plane of symmetry as shown in Figure 4.35. This triangular rotates around the work-roll center with a same angular velocity. Before this neutral zone the velocity is the oscillating one given by 4.10 with

F(x, z) = A e [G e (x + z) -G e (x -z)] K e (x).
And after the neutral zone with

F(x, z) = A s [G s (x + z) -G s (x -z)] K s (x).
Finally, this model needs equally only three parameters A e , A s and x P position of the neutral triangular apex. The curves PN 1 and PN 2 are two surfaces of velocity discontinuity that could be determined by the same method for determination of entry and exit discontinuity surfaces Γ e and Γ s (see 4.4.2.a).

Rigid-plastic UBM model for width spread

The first two sections of this chapter show a relatively wide study of existing models for width spread. Important effort began in the area of the 1960's in developing empirical formula to predict the spread. The most popular ones are [START_REF] Wusatowski | Hot rolling. Iron and Steel[END_REF][START_REF] Sparling | Formula for spread in hot flat rolling[END_REF][START_REF] Helmi | Geometrics factors affecting spread in hot flat rolling of steel[END_REF][START_REF] Beese | Nomograms for predicting the spread of hot rolled slabs[END_REF]. Later, several 3D analyses using FEM for flat and shape rolling [START_REF] Mori | Simulation of three-dimensional deformation in rolling by the finite element method[END_REF][START_REF] Mori | Finite element simulation of three-dimensional deformation in shape rolling[END_REF][START_REF] Kim | Three-dimensional analysis and computer simulation of shape rolling by the finite and slab element method[END_REF] where the shape and the spread are predicted. Montmitonnet [START_REF] Montmitonnet | A coupled thermomechanical approach for hot rolling bya 3d finite element method[END_REF] gives even a complete thermomechanical solution. However, as the FEM is well-known to be high time-consuming, the UBM is more commonly used thanks to its simplicity and rapidity. Some typical UBM models [START_REF] Komori | An upper bound method for analysis of three-dimensional deformation in the flat rolling of bars[END_REF][START_REF] Serek | Analysis of cold and hot plate rolling using dual stream functions[END_REF]

with interesting velocity fields are then presented. The third section is an analysis of 3D kinematically admissible velocity fields suitable to the flat thin strip rolling that is followed by an UBM analysis. A parametric study brings out the impact of rolling parameters on the width spread of the strip. And the UBM results show a very good agreement in comparison with the experiments performed at ArcelorMittal (AM).

Rigid-plastic UBM model for width spread

5.1 Statistical models for width spread

Statistical models for width spread

Typical empirical formulae

Although many studies on rolling problem allowing to estimate roll force and torque have been carried out, the width variation problem had remained for a very long time rebellious. Considerable effort began in the area of the 1960's. During that period emphasis was placed on developing empirical formula to predict the net spread. The most popular authors of these formulae are Wusatowski [START_REF] Wusatowski | Hot rolling. Iron and Steel[END_REF], Hill, Sparling [START_REF] Sparling | Formula for spread in hot flat rolling[END_REF] and Helmi & Alexander [START_REF] Helmi | Geometrics factors affecting spread in hot flat rolling of steel[END_REF] and Beese [START_REF] Beese | Nomograms for predicting the spread of hot rolled slabs[END_REF]. Table 5.1: Some typical empirical formulas giving the width variation factor S. It is important to note that the authors use, in their original formulae, total initial width w 1 , total initial thickness h 1 while in the present thesis we use half of initial width w e and half of initial thickness h e . That is why these formulae seem to be different from the original ones.

Effect of the thickness reduction ratio

An increase of the thickness reduction ratio r (in keeping a constant entry thickness h e ) leads to a longer contact length. Therefore the resistance to elongation in the longitudinal direction will be increased and the metal will be forced to spread more effectively in the lateral direction. This tendency is well confirmed by Siebel formula, Sparling formula, Hill formula, Helmi & Alexander formula and Beese formula. As can be deduced directly from the formulae shown in Table 5.1, they predict a strong, much more than linear, dependence of width spread coefficient on the reduction. The exception to this trend is the Wusatowski formula that predicts a constant width spread coefficient when the relative reduction r increases.

The absolute spread can be deduced from 5.1 as

∆w = w e h e h s S -1 . (5.2)
This equation implies that the absolute spread ∆w depends in a similar manner as the spread coefficient S on the thickness reduction ratio.

Effect of slab initial width

All empirical models predict a steep decrease of the width spread coefficient with an increase in the slab initial width. This means that the ratio of the transverse elongation to longitudinal one becomes smaller when the initial width increases. It is worth to note that the absolute width spread also decreases with an increase in the initial thickness although not at the same rate as the width spread coefficient.

Rigid-plastic UBM model for width spread

5.2 3D rigid-plastic UBM for width variation analysis

Effect of slab initial thickness

The width spread coefficient S generally increases with an increase in the initial slab thickness if the relative reduction r is kept constant. The exception to this trend is the formula derived by Helmi & Alexander. This model predicts that when the initial thickness increase, width change increases then decreases after reaching a maximum at a certain initial slab thickness.

Effect of slab roll diameter Similar to an increase in the reduction ratio, an increase in the work roll diameter produces a longer contact zone between the roll and the slab. This would increase resistance to the longitudinal elongation thus causing an increase in the metal flow in the lateral direction. The qualitative evaluation if this phenomenon is analytically confirmed by all empirical models presented in this section.

Domain of validity -unsuitable for cold rolling

The empirical approaches introduced in this section are not based on the resolution of equations but on the analysis of experimental results. The experimental results were either extracted from industrial production or issued from laboratory experiments specifically performed to measure and study the strip width variation. In these experiments, the strip are sometimes made of steel but sometimes of simulation materials such as lead or even plasticine. That is a reason using these models for quantitative evaluation of width variation in steels rolling may be degraded due to the fact that the elastic deformation (of plasticine) as well as thermal deformation are negligible. For the same reason, extrapolation of these formulas to the rolling conditions which are different from the experiments ones, the results can be hazardous. Table 5.2: Rolling conditions of trials allowing to tune the empirical models.

Parameter

Table 5.2 shows the rolling experimental conditions of the trials that allow to elaborate the empirical models mentioned above. It can be seen from this table that, these domain of validity of these empirical models correspond to billets (long product) rolling or flat hot rolling. It is therefore not recommended to extrapolate these formulae to the cold rolling domain where the width over thickness ratio varies from 400 to over 2000 for automotive product rolling and up to 5000 for packaging product rolling.

3D rigid-plastic UBM for width variation analysis

Advantage of UBM in comparison to FEM and empirical models

In the literature, there are many analyses using FEM for flat and shape rolling such as [START_REF] Mori | Simulation of three-dimensional deformation in rolling by the finite element method[END_REF][START_REF] Mori | Finite element simulation of three-dimensional deformation in shape rolling[END_REF][START_REF] Kim | Three-dimensional analysis and computer simulation of shape rolling by the finite and slab element method[END_REF] where the shape and the spread are predicted and [START_REF] Montmitonnet | A coupled thermomechanical approach for hot rolling bya 3d finite element method[END_REF] where a complete thermomechanical solution is even found. Some efforts have been made in order to reduce the computing capacity and time. Certain authors, based on the UBM , use numerical methods for integration which are similar to FEM , [START_REF] Komori | Rigid-plastic finite element method for analysis of three dimensional rolling that requires small memory capacity[END_REF][START_REF] Abrinia | Three-dimensional analysis of shape rolling using a generalized upper bound approach[END_REF]. Nevertheless, the FEM stays generally at high timeconsuming level. That is the reason why the UBM is commonly used thanks to its simplicity and rapidity. On the other hand, unlike the empirical models presented above, the UBM is physical and fully predictive. It is, hence applicable for any range of geometrical parameter ratio including cold rolling conditions with high width-thickness ratio. In the present thesis, we are interested in the UBM 3D rolling analyses.

As a reminder, the principle of UBM is based on theorem 2 (3.17) stating that among all kinematically admissible velocity fields the actual one minimizes the rolling power function 3.27. Applying this principle to rolling process require that the work-roll shape is given, the strip material is rigid-plastic and the friction shear stress is known -Tresca friction for example (see more detail in 3.1.2). Thus, to model the width variation problem by UBM , the main point is to build kinematically admissible velocity fields for 3D rolling problem.

Before introducing the typical existing UBM -based models for width spread analyses, it is worth to note a difference between 2D and 3D boundary conditions that may influence the velocity constructing method as well as the resolution of UBM .

Difference between 2D and 3D UBM

Lateral free surface: In a 2D model, there are boundary conditions imposed on the velocity field at all the four boundary surfaces of the roll bite : entry, exit, the contact surface and the surface of symmetry z = 0. However, in a 3D model, there is a free surface, the later surface defined by y = w(x) on which the boundary condition is not imposed on the velocity field but on the stress field (σ ij = 0). The difficulty is that the actual velocity field needs to be, in a steady state, tangential to the free surface but a kinematically admissible one does not necessarily verify this condition.

How can we, thus introduce a family of such kinematically admissible velocity fields?

A solution for this difficulty is: imagine a virtual boundary for the velocity field noted ϕ(x) which can be different from w(x) so that the velocity is tangential to this virtual boundary. Thus, we can easily represent the kinematically admissible velocity field thanks to this virtual boundary function.

Change in resolution strategy:

As already mentioned in the paragraph 2.1.1.a, the principal difficulty of the stationary models is that the geometry of the strip during and after deformation is an additional unknown of the problem. In our problem, the function w(x) is the unknown that we are interested to search.

To find the function w(x) at the equilibrium state, let first initialize it as w(x) = w e for example. Then using the principle of the UBM , search for the velocity field represented by the virtual boundary ϕ(x) that minimizes the rolling power. If the virtual boundary is equal within a certain precision to the function w(x), we have a velocity field that is tangential to the real free surface and that is the stable solution. Otherwise, if ϕ(x) = w(x) within a certain precision, the velocity is not tangential to the real free surface, the material has tendency to flow to have a new boundary that is ϕ(x). Therefore, take w(x) = ϕ(x) and repeat the operations. In general, after few iterations the convergent geometry is obtained. That is the main change in comparison with 2D UBM models.

UBM with 3D "simple" velocity field [83]

5.2.3.a Model description

Oh et Kobayashi [START_REF] Oh | An approximate method for a 3d analysis of rolling[END_REF] are one of the pioneers who applied the UBM to 3D analysis of rolling process. The authors supposed a velocity field as follows

                   u x (x, y) = C vol 1 h(x)ϕ(x) u y (x, y) = C vol ϕ ′ (x)y h(x)ϕ 2 (x) u z (x, z) = C vol h ′ (x)z h 2 (x)ϕ(x) . (5.3)
where ϕ(x) is the velocity virtual boundary which coincides to the width function for the actual velocity at stationary regime. Let call this velocity field "3D simple velocity field" because like the 2D simple one (see 3.3.2), this 3D velocity field states also that a cross section remains a cross section under the deformation along the roll bite.

Rigid-plastic UBM model for width spread

5.2 3D rigid-plastic UBM for width variation analysis

In order to compute and minimize the power function given by 3.27, the authors parameterize the free surface function ϕ(x), that represents the width evolution, as a third-order polynomial in x. The function is chosen so that w(-L) = w e w ′ (0) = 0 (5.4)

With these two equations, the width function can be represented by a third order polynomial of x as follows

ϕ(x) = w e + α + β L - 3α L 2 x 2 + β L 2 - 2α L 3 x 3 .
(5.5)

with two unknown parameters that are α -the total width variation and β = w ′ (-L) -sloop of width function at roll bite entry.

In addition of these two unknowns, there is an other parameter, the flow rate C vol which can be expressed as a function of the neutral line position (neutral point in 2D model) x n . The three parameters are obtained by minimizing the rolling power function. This model is studied in detail in the following section 5.3 of the present chapter. It results are compared to pilot (laboratory) rolling experiments and show a good agreement. The question on the rapidity (computing time) is also investigated.

5.2.3.b Parametric study -application to Hot Finishing Mill

D.Pirus [START_REF] Pirus | Modèle d'élargissement d'une bande laminée en emprise sous traction dans une cage de train finisseur par la méthode borne superieure[END_REF] reproduced successfully this model in fortran and validated it by comparing with the results published by Oh et Kobayashi [START_REF] Oh | An approximate method for a 3d analysis of rolling[END_REF]. Pirus used Newton-Raphson method for the minimization resolution of rolling power. He carried out, then a sensibility study of the width variation for finishing mill conditions. This study led to a conclusion that the width variation is always positive, meaning the strip always spreads out under rolling process. And the spread increases with an increase in the reduction, strip entry thickness, work-roll diameter or strip yield stress. Inversely, it decreases with an increase in strip width, entry and exit tensions, friction coefficient and work-roll Young modulus.

UBM with combination of basic velocity fields by Komori [59]

5.2.4.a Velocity field

In 2002, K. Komori [START_REF] Komori | An upper bound method for analysis of three-dimensional deformation in the flat rolling of bars[END_REF] introduced a new UBM to analyze the 3D deformation for rolling. He proposes to represent the velocity field as a linear combination of predefined fundamental velocity fields u i as u = a i u i .

(5.6) Figure 5.1: Illustrated fundamental kinematically admissible velocity fields: (a) u 1 and (b) u 2 (Fig. 4 in [START_REF] Komori | An upper bound method for analysis of three-dimensional deformation in the flat rolling of bars[END_REF]).

Rigid-plastic UBM model for width spread

5.2 3D rigid-plastic UBM for width variation analysis

In the article, two fundamental fields are mentioned. The first one u 1 is similar to the 2D simple velocity field 3.61, called equally "elliptical velocity field" (3.3.5) representing the deformation in thickness direction (plane strain deformation)

           u 1x = V c h e h u 1y = 0 u 1z = V c h e h zh ′ h .
(5.7)

The second fundamental velocity field u 2 representing the deformation in width direction is given by

             u 2x = V c u 2y = -V c yh ′ h u 2z = V c zh ′ h .
(5.8)

5.2.4.b Calculation and minimization of power function

Using the velocity given by 5.6, the power function defined by 3.27 becomes a function of a i . And the integrations are obtained using similar method as finite-element analysis. The first and second order derivatives of the power function with respect to a i are also derived analytically before being integrated numerically. Then, the minimization of the rolling power is performed with respect to two parameters a 1 and a 2 by the Newton-Raphson method.

5.2.4.c Discussions

Komori's combination method advantages: It is worth to highlight that, with the proposed combination method for constructing velocity field, the structure of the computer program is quite independent to the choice of fundamental velocity fields. It seems, therefore relatively simple to enrich the velocity field and quality of the analysis by assuming more kinematically admissible velocity fields.

Regarding unknown integration domain: However, unlike the resolution proposed in the paragraph 5.2.2, only one iteration is performed by considering that the domain of integration (strip geometry) is the initial configuration without width spread. Then, the optimum velocity gives a new geometry for the strip which is considered as the final geometry of the strip in stable regime. While according to us, it would be necessary to repeat the operations by integrating the power over the new geometry and find the new velocity field, then repeat again until the velocity field is tangential to the free surface.

Surface of velocity discontinuity is assumed vertical:

In the paper, the author considers that the entry surface of velocity discontinuity is a cross section. That is true by coincidence for the chosen velocity based on two fundamental ones 5.7 and 5.8. However, when other fundamental velocities are chosen or added, the surface of velocity discontinuity may be different from a cross section and need to be determined based on the velocity field. See 4.4.2.a, an example of method for numerical determination of surface of velocity discontinuity for a 2D velocity field.

The "3D simple velocity fields" family includes Komori's one: In his paper, the author only presented two velocity fields given by 5.7 and 5.8. It is not difficult to demonstrate that the combination of these velocities following 5.6 is a specific case of the "3D simple velocity" given by 5.3. Indeed, if we chose the virtual boundary function ϕ(x) and C vol as follows

   C vol = (a 1 + a 2 )V c h e w e 1 ϕ(x) = 1 w e + a 2 V c C vol (h(x) -h e )
(5.9)

Rigid-plastic UBM model for width spread

5.2 3D rigid-plastic UBM for width variation analysis then the "3D simple velocity field" can be expressed by

u simple = a 1 u 1 + a 2 u 2 (5.10)
equal to Komori's velocity. In other words, the "3D simple velocity fields" family includes and becomes Komori's for a specific choice of function ϕ(x) given by 5.9.

UBM with 3D polynomial velocity field [99]

5.2.5.a velocity field using dual stream functions

Serek [START_REF] Serek | Analysis of cold and hot plate rolling using dual stream functions[END_REF] proposed, in 2008 a model for plate rolling in hot and cold condition using UBM . The velocity fields are constructed by means of Dual Stream Functions and the rigid-plastic boundary at the entrance of the roll bite is assumed to be a quadratic form instead of a plane and normal to rolling direction. By using Dual Stream Functions as partly given in [START_REF] Yih | Stream functions in three dimensional flows[END_REF], the unknown three velocity components can be reduced to two and the incompressibility condition is satisfied. By defining ψ(x, y, z) and χ(x, y, z) stream functions, the velocity components u x , u y , u z in three dimensional non-compressible flow are written as follows: The volume flow rate at any cross section in three dimensional coordinates is written as Based on previous studies [START_REF] Barata-Marques | The use of dual stream functions in the analysis of three dimensional metal forming process[END_REF][START_REF] Martins | Upper bound analysis of plane strain rolling using a flow function and the wighted residuals method[END_REF], Serek [START_REF] Serek | Analysis of cold and hot plate rolling using dual stream functions[END_REF] chooses a stream function for metal flow in xz plane defined in the roll bite, as follows

               u x = ∂ψ
Q = (ψ 2 -ψ 1 ) (χ 2 -χ 1 ) . (5.12)
ψ = C vol z h + f (x)z [z -h] (5.13)
where f (x) is a gradient of velocity distribution in horizontal direction. Some mistakes are observed the formulae of ψ given by equations 5 and 11 in in [START_REF] Serek | Analysis of cold and hot plate rolling using dual stream functions[END_REF] but the velocities deduced from the stream functions are correct. where w is the width profile at the free surface. This function is chosen as a third order polynomial so that its derivative is equal to zero at entry x = -L and exit x = 0 (our point of reference is different from [START_REF] Serek | Analysis of cold and hot plate rolling using dual stream functions[END_REF]) of the roll bite, then

w = w e + ∆w 1 - 2x 3 L 3 - 3x 2 L 2 .
(5.15)

The rigid-plastic boundary (surface of velocity discontinuity) at the roll bite entry can be determined by equality of flow rate before and inside the roll bite

ψ = C vol z e (x) h + f (x)z [z e (x) -h] = C vol z e (x) h e (5.16)
then

z e (x) = h + 1 h e -1 h f (x) .
(5.17)

By assuming that this entry surface is quadratic and the exit one is vertical, the function f (x) can be deduced as

f (x) = ax 2 .
(5.18)

If L 1 denotes the x-distance between the starting and finishing points of the entry surface Γ e we have

0 = z e (L 1 -L) = h(L 1 -L) + 1 h e - 1 h(L 1 -L) a(L 1 -L) 2 ⇒ a = 1 h(L 1 -L) -1 h e (L 1 -L) 2 h(L 1 -L) . ( 5

.19)

L 1 is an output of the model which is determined by the optimization of the power function. The velocity is finally obtained as

                   u x = C vol 1 h + f (x)(2z -h) 1 w u y = C vol 1 h + f (x)(2z -h) w ′ y w 2 u z = C vol h ′ z h 2 + f ′ (x)z(z -h) -f (x)zh ′ 1 w .
(5.20)

This velocity field verifies all kinematically admissible conditions. Figure 5.2 illustrates the deformation area as well as the parameters of Serek's velocity field.

5.2.5.b Calculation and optimization of power function

The calculation and optimization of the power function is performed numerically in Matlab. We remark however that, according to the paper nomenclature, m denotes "Coulomb" friction but the author does not explain in detail how the friction power can be calculated. The m seems to be actually "Tresca" friction coefficient.

5.2.5.c Discussions

The velocity field given in this paper is interesting. It allows to model a non-vertical surface of velocity discontinuity at the roll bite entry and more general than the 3D simple one (for a same choice of w).

Non-null shear strain rates on surface of symmetry z = 0: It is noted that the shear strain rates ǫxz and ǫyz deduced from the velocity field 5.20 are not equal to zero when z → 0. Meaning that the shear strain ǫ xz and ǫ xz are different from 0 closed to the surface of symmetry, z → 0. This fact seems unrealistic. However, the condition of nullity of strain rates ǫxz and ǫyz when z → 0 (two first equations of 2.3) are not strictly necessary conditions for any kinematically admissible velocity field although the actual velocity field, being continuous and continuously derivable, itself verifies these conditions.

Rigid-plastic UBM model for width spread

Chosen rigid-plastic model of width variation in rolling

Integration domain: The paper does not precise how and what is the geometry of the strip over which the power function is integrated. If the geometry of the strip is considered as the initial one, this is the same mistake as Koromi's model. If the strip geometry is defined by the width function w given by 5.15 that is also used in the velocity expression, the results should not be corrects because in this manner, both geometry of strip and velocity vary during optimization. The correct way is to fixe the strip geometry and to vary only the velocity. That is why, it is necessary to introduce a virtual boundary ϕ and an iteration method as explained in the paragraph 5.2.2 and in Figure 2.2.

Results and comparison to experiments

The rolling experiments were performed using the work rolls of the mill, having a diameter of 200 mm and a length 150 mm, driven by a 35 kW constant torque, dc motor. The roll speed is kept constant at 20 rpm, giving a roll surface velocity of 209 mm/s. Following Serek's UBM model, the rolling force and elongations in horizontal and vertical directions were computed for a bar of rectangular plate with dimensions of 10 x 50 x 150 mm. The optimum velocity field was attained after optimization of the power function and the analysis was carried out with the optimum velocity field.

Nevertheless, there is no detail about the method for approaching the rolling pressure and the rolling force but the results of these parameters were presented. On the other hand, the graphic 4 and 5 show that the rolling force reaches minimum for an optimum value of total spread. This can lead to a misunderstanding "why the rolling force but not the power or torque as expected?". In addition, the results show that a total spread equal to 25 mm, i.e 50% of width spread that is a very high spread ratio. Moreover, the width spread is quite constant as a function of reduction. It is almost equal to 25 mm for all levels of reduction from 10% to 50%. These results of the width spread behavior are quite unexpected and in the contrary in comparison to other models.

Chosen rigid-plastic model of width variation in rolling

UBM is chosen

As already mentioned above, the statistical models are too simple to be representative all the varying rolling condition in rolling. The FEM models require inversely high computing time while recent studies showed relatively high-quality developments of the UBM for width variation analysis. There were advancements in terms of both velocity fields approaches and numerical methods giving advantage of simplicity and rapidity. We propose, hence to use the UBM of which the principle is studied in the previous chapters.

We propose to use the resolution regarding the free surface y = w(x) given in the paragraph 5.2.2 thanks to a virtual boundary of the velocity field ϕ(x). And the resolution is solved by iteration method until the virtual and the real boundary are identical within a chosen precision, ϕ(x) ≃ w(x). In each iteration, the minimization of rolling power is calculated by the Newton-Raphson method.

Concerning the numerical method for integration of power function and its derivatives, the effort is made in order to go as far as possible analytically before using numerical method. That is important point allowing to reduce significatively the number of elements required and therefore the computing time.

Assumptions of the model

This section presents general assumptions that define the conditions cold rolling process, denoted as GA (General Assumptions). Although, some of them have been already mentioned in previous chapters or sections, let list them all together.

5.3.2.a GA1-Symmetry

Only symmetric rolling is considered, i.e the rolling stand is top-bottom and left-right (operator-motor sides) symmetric. This hypothesis is already discussed in the paragraph 2.1.1.b. As a consequence, it is sufficient to study a quarter of total geometry of rolling mill.

Rigid-plastic UBM model for width spread

5.3 Chosen rigid-plastic model of width variation in rolling

5.3.2.b GA2 -Work roll shape

The roll is considered to have no initial bending meaning a straight profile across the width direction. On the other hand, it can be flattened and its radius is determined by Hitchcock's model. In other words, the roll is considered cylindric without bending having the deformed radius. Therefore, the product thickness is constant in along the width direction, denoted h(x).

5.3.2.c GA3 -Tresca friction law

As discussed in the section 3.1.2, the friction between the strip and the roll is an internal force and Tresca friction law 2.13 is assumed because the friction stress is known as a function of the material yield stress.

5.3.2.d GA4 -Strip rigid-plastic behavior

The strip mechanical behavior is considered rigid-plastic satisfying Von-Mises criterion. As a consequence, the material is incompressible under deformation

divu = ∂u x ∂x + ∂u y ∂y + ∂u z ∂z = 0 (5.21)
and the velocity field is homogenous and unidirectional before and after the roll bite.

The three assumptions GA2, GA3 and GA4 are necessary and enable to apply the UBM . They are already mentioned in one of previous chapter (see section 3.1.2).

5.3.2.e GA5 -Average behavior across thickness direction GA5:

In automotive rolling process, the product thickness is much smaller than its width and the contact length. This reality induces to an assumption (GA5) as follows: the vertical deformation rate is constant across thickness direction, meaning the component ǫzz (x, y, z) = ǫzz (x, y). That implies therefore a linear vertical velocity u z as a function of z u z (x, y, z) = z. ǫzz (x, y). (5.22) Further, issued from the equations 5.21 and 5.22 we assume also that u x and u y do not neither depend on z. The fact that transversal velocity u y does not depend on z implies that the strip width profile is independent of z and therefore becomes a function of x, denoted w(x).

Validity: A deep analysis of 2D velocity field by Lam3-Tec3 and UBM is done in the chapter 4. As a result of this analysis, Figure 4.34 shows that even for thin strip cold rolling condition (h e = 1.4mm, L c = 17mm) the velocity field oscillates significantly around its average behavior represented by the 2D simple velocity. The assumption GA5 seems therefore unrealistic. However, the real question is "what is the impact of the velocity oscillations on the width variation". To answer to this question, let compare several Lam3-Tec3 simulations using 5 elements (fine mesh) and 1 element in the strip half thickness. Figure 5.3 shows results in terms of width spread for different meshing. The fine mesh has 5 elements in the half thickness and the others has only 1 element in the half thickness. It can be see that the results with different meshes are practically the same. Moreover, by remarking that with 1 element in the thickness, Lam3-Tec3 considers a homogenous deformation in the thickness.

The hypothesis GA5 is finally confirmed in the case of narrow strip rolling. We may understand the GA5 as follows: in terms of width variation, it is enough to consider the average behavior of the velocity field in the thickness direction.

Choice of velocity

5.3.3.a General velocity field

By using Dual Stream Functions, a 3D velocity of an incompressible material in the plastic deformation zone Ω can be given by 5.11. And the boundary conditions of this velocity field are:

• Boundary conditions on the surface of symmetry z = 0:

ψ(x, y, 0) = const ∀(x, y) ∈ [-L, 0]x[0, ϕ]
(5.23)

• Boundary condition on the contact surface z = h(x):

ψ(x, y, h) = const ∀(x, y) ∈ [-L, 0]x[0, ϕ] (5.24)
• Boundary conditions on the surface of symmetry y = 0:

χ(x, 0, z) = const ∀(x, z) ∈ [-L, 0]x[0, h] (5.25)
• Boundary conditions on the virtual free surface y = ϕ:

χ(x, ϕ, z) = const ∀(x, z) ∈ [-L, 0]x[0, h] (5.26)

5.3.3.b Choice of stream functions

The assumption GA5 implies that the xz plane stream function ψ is linear as a function of z. Thus, thanks to two boundary conditions on z = 0 (5.23) and z = h (5.24), we can choose the xz plane stream function ψ as follows

ψ = C vol z h .
(5.27)

The same assumption GA5 stating further that u x and u y do not depend on z, implies thus a xy plane stream is a solution by verifying both boundary conditions on y = 0 (5.25) and on y = ϕ (5.26). This solution describes a constant strain rate ǫyy across the width direction. We remark that any other xy stream function verifying the assumption GA5 can be written as a sum of this simple solution and an addition term depending on x and y as follows χ = -y ϕ + g(x, y) .

(5.29)

Now, the boundary conditions 5.25 and 5.26 becomes

g(x, 0) = 0 g(x, ϕ) = 0.
(5.30)

Since the equation 5.27, we deduce that the plastic deformation boundary surfaces (surface of velocity discontinuity) at the roll bite entry and exit are vertical. By writing the equality of the stream flow at these surfaces to the uniforms and homogenous flow before and after the roll bite we have

     -ψ(-L, y, z)χ(-L, y, z) = C vol yz ϕ e h e
ψ(0, y, z)χ(0, y, z) = C vol yz ϕ s h s .

(5.31)

The equations imply quite directly that g(-L, y) = 0 g(0, y) = 0.

(5.32)

Finally, by substituting 5.27 and 5.29 into 5.11 we obtain a solution for 3D velocity field as follows

u x (x, y) = C vol 1 h 1 ϕ + ∂g ∂y u y (x, y) = C vol 1 h yϕ ′ ϕ 2 - ∂g ∂x u z (x, z) = C vol zh ′ h 2 1 ϕ + ∂g ∂y (5.33)
with kinematically admissible conditions given by

g(x, 0) = 0 g(x, ϕ) = 0 g(-L, y) = 0 g(0, y) = 0.
(5.34)

5.3.3.c Discussions

Boundary effect at the edge and function g A

When the function g(x, y) = 0, the component u x depends only on x and the component u y is linear in y and so in y. To verify the velocity field, we show in Figure 5.4 the dependence of u x and u y in y calculated by Lam3-Tec3 for a case of rigid roll. For different abscissas, in the central part (from y = 0 to y = 495mm) the longitudinal velocity is constant and the lateral velocity u y is linear in y. That means, g(x, y) is mostly equal to zero in this part and increases exponentially at the edge of the strip. It represents therefore the bound effect.

Two natures of the width spread A

Then, the width spread is composed of two contributions. The first is associated to the central part where the velocity u y is linear in y and the second is associated to the bound effect represented by the function g(x, y). The size of the bound effect area is strongly related to the strip's thickness.

In the following, before presenting the UBM for the lateral spread, we perform a 2D study on the rolling stability which allows to understand the rolling conditions in industry and to validate the UBM for a 2D velocity field compared to a classic method for rolling, the slab method.

5.3.3.d Final choice of velocity field

In this thesis, we propose to study at first the phenomenon corresponding to the spread of the central part without bound effect i.e g(x, y) = 0. The 3D velocity field written by 5.33 becomes the "3D simple velocity" given by 5.3. If necessary, deeper understanding of the function g should be studied to go further.

Computation of power function

In order to reduce the computing time, the power functions are mostly calculated analytically. The integrations are as much as possible done analytically before using the numerical method. These calculations are long and are, thus presented in the appendix B.2. The power consumed by the discontinuity of velocity J ∆u and the power of external forces J ten are completely analytical. However, the two other powers consumed by plastic deformation J ǫ and by friction J f ric are developed analytically in y and z and have form of an unidimensional integral in function of x. We note that these calculations are independent of the width function form i.e the parametrization of this function. where

Power of plastic deformation

J ǫ = σ 0 √ 6 C vol 0 -L P x (x)dx
P x (x) = 2 3 Q 2 + f 2 + g 2 + Q 2 + 1 3 f 2 g Ln g + Q 2 + f 2 + g 2 Q 2 + f 2 + Q 2 + 1 3 g 2 f Ln f + Q 2 + f 2 + g 2 Q 2 + g 2 + 4Q 3 3 f g   arctan   Qg Q 2 + f 2 + f Q 2 + f 2 + g 2 + Q 2 + f 2   -arctan g Q 2 + g 2 + Q  
with the three functions reducing the expression of the power J ǫ given by

Q = 2 h ′ 2 h 2 + 2 ϕ ′ 2 ϕ 2 + 2 h ′ ϕ ′ hϕ f = 1 √ 2 ϕ ′′ -h ′ ϕ ′ h -2 ϕ ′ 2 ϕ b ϕ g = 1 √ 2 h ′′ -h ′ ϕ ′ ϕ -2 h ′ 2 h .

Power of the discontinuity of velocity

J ∆u = σ 0 √ 3 C vol    ϕ ′ 2 e + h ′ 2 e 3 + h ′ 2 e 6 ϕ ′ e Ln    ϕ ′ e + ϕ ′ 2 e + h ′ 2 e h ′ e    + ϕ ′ 2 e 6 h ′ e Ln    h ′ e + ϕ ′ 2 e + h ′ 2 e ϕ ′ e       (5.36)
where the 2 notations h ′ e and ϕ ′ e are defined as

h ′ e = h ′ (-L) ϕ ′ e = ϕ ′ (-L) .

Power of friction

J f ric = mσ 0 C vol 2 √ 3 0 -L b ϕ √ 1 + h ′ 2 h   ϕ ′ 2 + I 2 + I 2 |ϕ ′ | Ln   |ϕ ′ | + ϕ ′ 2 + I 2 I     dx .
(5.37)

with

I = 1 + h ′ 2 - hϕV c C vol .

Power of entry and exit tensions

J ten = -Te + T s b ϕ C vol .
(5.38)

Total rolling power

J = J ǫ + J ∆u + J f ric -J ten . ( 5

Numerical resolution

5.3.5.a Algorithm

The algorithm of the resolution is shown in Figure 5.5 that can be divided into 5 steps as following:

1. First, the programme reads the data file that contents all the necessary parameters concerning the rolling process.

2. Then, if the option of roll's deformation is active, it calculates the deformed radius by the Hitchcock's model. [START_REF] Beese | Nomograms for predicting the spread of hot rolled slabs[END_REF] (see Table 5.1) that gives an approximate width spread of the strip, we initialize our width function for the first iteration. Beese's model is simple, direct and therefore very fast. Nevertheless, in some cases such as crowned strip rolling it is really not accurate and then the program needs more iterations to converge (see chapter 6).

Next, by using an empirical model -Beese's model

4. For each iteration, the width function is given at the end of the previous iteration or by the initiation mentioned above for the first iteration. And here, the most important work of the UBM begins. For example, in iteration n, we calculate the rolling power composed of the powers consumed by plastic deformation, discontinuity of velocity, friction and external forces, depending on the width function w n (x) (given by the previous interation) and equally on function of the velocity field that is represented by the function ϕ n (x). Then, in minimizing this rolling power we obtain a solution for velocity function noted ϕ min n (x). However, noting that space of functions is of infinite dimensions, we need to parameterize the velocity function ϕ n (x) by a limited number of parameters. In the section 5.3.5.b we present two parameterizations, a classical and a new. In many cases, we need only two parameters to well describe the function but three parameters are necessary for more complicated cases like crowned sheet rolling (see chapter 6).

In this space of finite dimensions, the minimization is carried out by the Newton-Raphson method that is presented in section 5.3.5.c.

At the end of each iteration, we test if the difference between the solution of velocity function ϕ min

n (x) and the function w n (x) is smaller than a given precision ǫ related to the precision of the width spread. Until this test is true, we continue next iteration n+1, by taking the width function for this iteration like w n+1 (x) = ϕ min n (x).

Rigid-plastic UBM model for width spread

Chosen rigid-plastic model of width variation in rolling

Remark regarding unknown geometry of the strip (see 5.2.2): It is worth to note from the algorithm showed in Figure 5.5 that for some iteration n, the strip geometry is given w n (x), we search for the optimum velocity field presented by it virtual boundary ϕ min n (x) by minimizing the rolling power. Next, we consider that the material flows following this solution velocity field and has a new geometry which is w n+1 (x) = ϕ min n (x). In this new geometry, the strip has tendency to flow with an other optimum velocity ϕ min n+1 (x) and will have a new geometry. That continues until that the geometries of two successive iterations are identical within a chosen precision. Numerically, the identity of two functions is expressed by ϕ min n+1 (x)w n (x) < ǫ. The strip geometry no longer changes and gives a stable configuration. This is the solution of the stabilized state of the rolling.

No contact areas

Contact areas

Entry Exit

Rolling bite

) (x \ (x)
Figure 5.6: Optimal velocity field associated to a virtual bound ϕ(x) for a given geometry represented by a width function w(x) at an iteration of the resolution.

5.3.5.b Choice of width profile function

First of all, the numerical resolution mentioned above requires a same parameterizing of both real and virtual width functions 2 because at a stabilized solution these two functions will be numerically identical. These functions are both equal to w e at the roll bite entry and at the exit their derivative is equal to zero. In addition, in order to keep the advantage of the UBM in term of computing time, the number of parameters is essentially limited. There are presented here following two forms of the width function using two parameters. In these two cases, the first parameter α is the width spread and the second β is a parameter related to the tangency of the width at the entry of roll gap.

Form 1 : Polynomial in x test

The classical form is a polynomial in x. With 2 conditions at the entry and exit, the width function can be expressed by a polynomial of degree 3 as a function of x and 2 parameters α and β:

w = w e + α + βx 2 + α L 3 + β L x 3 .
(5.40)

Remark : test

This form is useful and practical. Yet, it does not represent the direct relation between the reduction in thickness and the width spread of the strip in rolling. The greater the reduction, the greater the width variation. Consequently, we propose another form which expresses the function w(x) as a function of h(x) instead of x in the form 1.

Form 2 : Polynomial in h(x) test

w = w e + h e -h h e -h s α + β 1 - h h s .
(5.41)

2 virtual width function is called also the velocity function because our velocity field will be expressed in this function

5.

Rigid-plastic UBM model for width spread

Chosen rigid-plastic model of width variation in rolling

And finally, the velocity function ϕ(x) is parameterized in the same way of the width function but with the parameters noted α 1 and β 1 .

ϕ(x, α 1 ,

β 1 ) = w(x, α = α 1 , β = β 1 ) . (5.42)
With these parameterizations, for a given strip geometry (α and β are given) the power function J is a function of three parameters. The first one is V e in the expression of C vol (C vol = h e w e V e ). Two others are α 1 and β 1 in the velocity function ϕ(x). The minimization of the power function is performed by the Newton-Raphson method presented in the following section.

5.3.5.c Newton-Raphson method for minimization

Let present here Newton-Raphson method for the minimization problem of a n-variable real function P : R n → R so that (x 1 , x 2 , .., x n ) → P(x 1 , x 2 , .., x n ). The minimization problem is represented by :

                     ∂P ∂x 1 = 0 ∂P ∂x 2 = 0 ... ∂P ∂x n = 0.
(5.43)

The method consists in solving iteratively. In a given iteration where the equation 5. [START_REF] Helmi | Geometrics factors affecting spread in hot flat rolling of steel[END_REF] is not yet satisfied, the step of change of the variables between two successive iteration is determined as follows where

P i = ∂P ∂x i et P ij = ∂ 2 P ∂x i ∂x j
. In our program the derivatives are calculated numerically using:

                   P i = P(x i + δx i ) -P(x i -δx i ) 2δx i P ii = P(x i + δx i ) + P(x i -δx i ) -2P(x i ) δx 2 i P ij = P(x i + δx i , x j + δx j ) + P(x i -δx i , x j -δx j ) -P(x i -δx i , x j + δx j ) -P(x i + δx i , x j -δx j ) 4δx i δx j (5.45)
with δx i is chosen numerical parameters which are small enough. And the convergence criterion is choose as follows:

P(x 1 , x 2 , ..., x n ) < P(x 1 , x 2 , ..x i ± ǫ i , .., x n ) ∀i = 1..n (5.46)
with ǫ i is the predefined precision related to the parameter x i .

5.3.5.d Computing time

The program is performed in C++ as the main part of a combined model. The computing time of the whole model varies from 0.01s to 0.05s (CPU: Intel Core I5-4200M, 250GHz) that enables online applications (see more details in the section 8.1).

Rigid-plastic UBM model for width spread

5.3 Chosen rigid-plastic model of width variation in rolling

Comparison with ArcelorMittal trials on the pilot rolling mill

This section presents a comparison between the results obtained with the UBM using 3D simple velocity field and the experiments performed on ArcelorMittal pilot rolling mill with narrow strips (about 65 -70mm of width). These experiments were carried out on two types of steel, soft -DWI σ 0 = 280Mpa and hard -soldur σ 0 = 933Mpa. The rolling parameters of the trials corresponding to these types of steel are presented in Tables C. [START_REF] Abrinia | Three-dimensional analysis of shape rolling using a generalized upper bound approach[END_REF] and C.2 of Appendix C. The Lam3-Tec3 calculations for these experiments are also presented in the same appendix. In this modeling, the friction is modeled by Coulomb's law. This friction is, furthermore anisotropic with an anisotropy defined by the ratio between the transversal friction on the longitudinal friction. And we observe that this anisotropy has a significant impact on the exit profile of the strip. Therefore, in comparing the exit profile of the strip calculated by Lam3-Tec3 with the measure, we obtain the good value of friction anisotropy coefficient. On the other hand, in UBM model the friction is modeled as an isotropic Tresca's law. The friction coefficient value is obtained by seeking the good forward slip defined by the difference in % of strip exit velocity and the roll velocity which are both measured during the experiments.

The models results are equally given in Tables C. [START_REF] Abrinia | Three-dimensional analysis of shape rolling using a generalized upper bound approach[END_REF] and C.2. And Figure 5.7 shows two histograms comparing the width variations obtained in the experiments, with UBM model and with Lam3-Tec3 model for the two types of steel:DWI and Soldur. It can be seen that the UBM gives very coherent results in comparison with the measurement as well as with Lam3-Tec3 , especially for DWI steel trials. These results are also presented in Figure 5.8 which shows that for soft steel (DWI), the model matches really well the measurements. It matches, however slightly less for hard steel (Soldur). For this steel, the UBM model underestimates the width spread for most of the cases. This may be explained by the fact that for very hard steel, the edge drop phenomenon due to the local deformation of the work-roll at the edge of the strip is more important leading to a greater width spread. Whilst this phenomenon is not taken into account by the UBM model.

We conclude then, the UBM predicts well the width spread of the pilot experiments with narrow strips. Improving the work-roll deformation model could help to improve the model accuracy especially for hard and very hard material.

Parametric study for using UBM model

In this section, the effect of rolling parameters on the width spread is studied in a theoretical rolling condition defined in Table 5.3. The parameters will be changed one by one to study their effect on the width spread and the rolling power. The rolling parameters of the referent point for the parametric study.

5.3.7.a Influence of rolling parameters on width spread

The dependence of the width spread on the rolling parameters is demonstrated in Figure 5.9. It is observed that the width spread

• increases as a function of: the reduction r, the roll radius R, the friction coefficient m and the entry thickness h e

• is independent of the roll velocity V c

• decreases with an increase in: the entry width w e as well as the entry and exit tensions T e , T s .

In the first order, the width spread increases rapidly with the thickness reduction and falls down exponentially with an increase in the strip entry width. This is a very common result observed by many works in the literature (see previous sections 5.1 and 5.2). In the second order, it grows almost linearly as a function of the roll radius because of the increase of the contact length and linearly decreases with the entry and exit tensions. The dependence of the width spread on the entry tension is slightly more important than the exit tension. This could due to the fact that the entry section is larger than the exit one. Finally, in the third order, the entry thickness and the friction coefficient make the width spread increase but more slightly. In some other cases, the friction coefficient may have negative influence on the width spread.

In addition, two extremity phenomena can be observed also in Figure 5.9. First, for an extremely narrow strip i.e w e < 8mm, the width spread increases with the entry width w e which is in contrary to the general case. Indeed, the smaller the w e , the less material to spread out therefore the smaller the width spread. Secondly, for a very thin strip i.e h e is very small, we observe an inverse impact of the entry thickness on the width spread. Anyway, these phenomena are two really special cases that are hardly met in flat rolling.

5.

Rigid-plastic UBM model for width spread 5.3 Chosen rigid-plastic model of width variation in rolling

5.3.7.b Influence of rolling parameters on rolling power

Rolling power and rolling torque test

As a reminder, the minimum of the rolling power function gives an upper bound estimation of the real one provided by the motor on the work-roll. The associated roll torque can be evaluated from the power by equation 4.67. Figure 5.10 points out that the rolling power J increases strongly with the reduction r and depends almost linearly in the other parameters. It increases also as a function of the roll radius R, the friction coefficient m, the strip entry thickness h e , the strip entry width w e , the entry tension T e except only the exit tension T s . As the exit tension makes the strip move in rolling direction, the necessary rolling power is smaller when this tension increases. Or it can be seen from the equation 5.38 that the dependence of the rolling power on T e and T s is related to the term of power of external forces J ten . At stable state, w = ϕ, we have J ten = (T s -T e )C vol . That is why the rolling power J increases with T e and decreases with T s with almost the same sensitivity.

Comparison between two parameterizations of the width function A

Moreover, it should be interesting to compare the efficiency of the two forms of the width function presented in the paragraph 5.3.5.b in order to choose the best one. As a reminder, the best form is the one which gives the lowest rolling power.

Let firstly choose the form 1, polynomial in x as the reference and study the difference in terms of rolling power of the form 2, polynomial in h(x) in comparison with the first one. Figure 5.11 shows that this difference is negative for r <= 45%. That means if the reduction r <= 45%, the polynomial in h(x) form is better. But inversely the form 1 is better for very high reduction. In addition, Figures 5.9 and 5.10 show that the width spread as well as the rolling power are almost identical. Therefore, for us the two forms of width function can be considered equivalent. The form 1 is slightly advantageous in terms of computing time.

Volume specific rolling energy test

Let now introduce a quantity called specific rolling energy that represents the energy provided by the motor to roll an unity volume of material. It is defined as .47) This parameter may help to study how it is possible to save rolling energy by well choosing values of the rolling parameters. It can be seen in Figure 5.12 that this quantity increases as a function of the thickness reduction. Moreover, unlike the rolling power which linearly grows up as a function of the entry width, the specific rolling energy is practically constant except for very narrow strip -w e < 20mm. Indeed, the higher the strip width, the higher the sliding velocity in y direction and thus the higher the friction power which contributes to an increase of the specific rolling energy as a function of the strip width. However, the width spread tends to zero very rapidly when the strip width increases, the lateral velocity field is asymptotic to zero. That explains why the specific rolling energy becomes a constant as a function of the width for large strip.

E vol = J C vol . ( 5
Beside, it can be noted that the deformation dissipation energy per volume unit increases with the reduction r but is almost independent on the strip thickness h e if r is constant. And the tensions power per volume unit (deduced from the equation 5.38) is strictly independent of h e . On the other hand, the friction power is proportional to the contact length L which is proportional to √ Rh e r while the volume flow rate C vol is proportional to h e . Therefore, the friction energy per volume unit is proportional to 1/ √ h e . Thus, the friction energy per volume unit decreases as a function of entry thickness. That is why the specific energy decreases when entry thickness increases as it can be seen in the graphic E volr of Figure 5.12.

When h e , w e and r are constant, the specific rolling energy depends similarly as the rolling power on the other rolling parameters. Actually, Figure 5.12 shows that E vol decreases when: T s or h e increase and when r, m, R, T e or σ 0 decrease. However, in a rolling process, the strip initial geometry, the desired reduction and the product characteristics are the given constraints, meaning w e , h e , r and σ 0 are fixed. The only three parameters which allow to optimize the specific rolling energy are m, R and T s -T e . But an increase in T s -T e will require equally a certain energy. Thus, 5. Rigid-plastic UBM model for width spread 5.4 Conclusions and perspectives increase T s -T e to decrease E vol is a solution that just allows to transfer the energy consumed by the rolling motor to other machines but does not save the total consumed energy. It is possible to save energy by using small rolls and by lubricating the contact surface to reduce to friction coefficient between the roll and the strip.

Nevertheless, it is complicated in developing more in this subject because there are a lot of constraints concerning the roll and the lubrication system. For example, the roll must not be too small to assure a certain rigidity in flexion. The friction coefficient needs to be enough to keep the rolling stability. The choice of roll radius and the lubricant is an optimization problem which is not studied further in this thesis.

Conclusions and perspectives

In this chapter, we study a velocity field family which explains the width variation of a rigid-plastic strip. This width variation can be composed of two parts: the homogenous lateral flow in the central part of the strip as well as the apparition of the boundary effect near to the strip edge. This complex effect is modeled by a function g. However, in a first order approach, let neglect this phenomenon and consider therefore that g = 0. As a result, the velocity field becomes the "3D simple one" as given by [START_REF] Oh | An approximate method for a 3d analysis of rolling[END_REF]. It is demonstrated also in this chapter that the 3D simple velocity is, for a specific choice of width function (see the last paragraph of section 5.2.4.c) equivalent to the one proposed by [START_REF] Komori | An upper bound method for analysis of three-dimensional deformation in the flat rolling of bars[END_REF] based on a combination of some fundamental velocity fields.

It is worth to highlight, in particular that using UBM the free surface y = w(x) is an additional unknown of the problem (see section 5.2.2). A solution for this difficulty is employed. The idea is to fix the strip geometry and search for the optimum velocity, update the strip geometry so that the new width is tangential to that optimum velocity field and repeat these operations until the geometry and the optimum velocity are unchanged and coincided. However, [START_REF] Komori | An upper bound method for analysis of three-dimensional deformation in the flat rolling of bars[END_REF] only uses one iteration and [START_REF] Serek | Analysis of cold and hot plate rolling using dual stream functions[END_REF] does not precise how he deals with this issue.

With the chosen velocity field (3D simple one), the UBM gives a good coherence in terms of the width spread in comparison with experiments performed on ArcelotMittal pilot mill. In addition, the UBM analysis allows studying effect of rolling parameters on the strip width spread. The results show that width spread increases strongly with an increase in the reduction and falls down exponentially with an increase in the strip entry width. It grows almost linearly as the roll radius increases and decreases with an increase in the entry or exit tensions. The entry thickness and the friction coefficient have less important impact on the width spread. These are very common results observed by previous works in the literature (see previous sections 5. [START_REF] Abrinia | Three-dimensional analysis of shape rolling using a generalized upper bound approach[END_REF]

and 5.2).

The results obtained in this chapter raise up a further UBM analysis for a crowned strip rolling. The lateral boundary function (ϕ(x) and w(x)) is parameterized differently but the method principle is essentially unchanged. This study is presented in the chapter 6 as an attempt to study the relation between strip spread and strip thickness profile or flatness.

Rigid-plastic UBM model for width spread

Rigid-plastic UBM model for width spread

Conclusions and perspectives

Volumic rolling energy depends on reduction 0.0E+00

UBM for crowned strip rolling Introduction

The previous chapter [START_REF] Avitzur | Maximum reduction in cold strip rolling[END_REF] shows the typical models in the literature, from statistical to FEM and UBM ones predicting the width spread for flat strip rolling. These models consider a rectangular form of the strip. On the other hand, for certain long and shape rolling processes, an important number of 3D models were developed to predict the shape variation including the width spread. Some are based on FEM [START_REF] Mori | Finite element simulation of three-dimensional deformation in shape rolling[END_REF][START_REF] Mori | Non-steady-state simulation of three-dimensional deformation around front and rear ends in shape rolling by the finite element method[END_REF][START_REF] Kim | Three dimensional finite element analysis of non-isothermal shape rolling[END_REF], others on UBM [START_REF] Saito | Deformation analysis in shape rolling using an upper bound method[END_REF][START_REF] Abrinia | Three-dimensional analysis of shape rolling using a generalized upper bound approach[END_REF].

For flat automotive product rolling, some efforts were made in the subject of the strip shape (thickness profile). Such studies aim at analyzing the flatness and roll force distribution across the width direction. The width spread is also analyzed and the results show out that the downstream shape influences closely the lateral spread. This mechanism has been modelled by various authors using a mixture of analytical and numerical methods [START_REF] Tozawa | Predicting the profile of rolled strip[END_REF][START_REF] Matsumoto | Mechanism of material deformation related to shape and crown phenomena[END_REF][START_REF] Allwood | A hybrid 2.5 dimensional elastoviscoplastic model of hot strip rolling for practical applications[END_REF][START_REF] Dixon | An analytical model for the lateral spread and shape defects from the rolling of a ridge profile or edge drop[END_REF][START_REF] Cozijnsen | Development of a shape model that includes edge spread for cold rolling[END_REF]. These analyses also point out that for thin strip rolling there is a very small lateral spread, with the exception near the edge where the lateral stress is small allowing the material to spread, rather than being compressive (in the lateral direction) as is the case for plane strain rolling conditions. This is the width spread caused by "edge drop phenomenon".

In this chapter, a new UBM approach is developed for cold rolling of strip with an initial thickness crown that might be created by the earlier rolling process (hot rolling or previous stand). The model allows to understand how the strip spread depends on the change of the strip thickness profile in rolling. First, an analysis is proposed to study 6. UBM for crowned strip rolling 6.1 Velocity field proposition kinematically admissible velocity fields by supposing some simplifying hypothesis. As the geometry of the strip is more complex than the case of flat strip rolling, the roll bite is divided into three areas and the velocity field is different throughout each of them. Then, the rolling power are calculated correspondingly to the three zones. All the powers calculations are more complicated and need being approximated numerically. The Gauss's method is used and the computing time is well improved. On the other hand, the numerical resolution is the same as in the previous chapter. The only change is that there are more parameters in the rolling power function to minimize because the real and virtual width functions are parameterized using three parameters instead of two in the case of flat rectangular strip. The same method, Newton-Raphson is used for optimization problem.

Velocity field proposition 6.1.1 Assumptions

Firstly, the general assumptions presented in the previous chapter (see subsection 5.3.2) are also assumed. In addition, some specific assumptions (denoted SA) which are associated only to a crowned strip rolling will be necessary to simplify the velocity field construction.

6.1.1.a SA1:

The first SA mentions that in the upstream, the strip has a positive crown, i.e the strip thickness is maximum at the center and decreases along the width direction. In other words, the strip entry thickness h e (y) is a mono-decreasing function. In industry, this type of crown is observed almost all the time. Hence, the center of the strip is in the contact with the roll before the edges. The roll-bite is divided into two parts: partial and total rolled parts. It is possible, thus to distinguish three areas associated with theses two parts as shown in Figure 6.1. In each area, the velocity field is different and will be studied one by one in the next section by considering the kinematically admissible conditions. This assumption allows determining the function ψ(x) representing the interface between the areas I and II. To begin with, let imagine that the strip has no lateral spread in the way that lateral velocity is equal to zero everywhere in the strip. To simplify the reading of the thesis, this no-spread configuration is conventionally called imaginary configuration. The interface between the areas I and II is presented by a function ψ 0 (x) that satisfies the following equation:

h e (ψ 0 (x)) = h(x) ⇔ ψ 0 (x) = h -1 e (h(x)) (6.1)
where h(x) is the thickness imposed by the work-roll profile. As a reminder,

h(x) = R + h s - √ R 2 -x 2
for a circular work-roll. To determine the function ψ(x), let return to the real configuration in which the strip spreads and the lateral velocity may be different from zero anywhere. To simplify further, an additional hypothesis is assumed as follows.

UBM for crowned strip rolling

6.1 Velocity field proposition

6.1.1.b SA2:

The displacement of material elements in the area II in comparison with the imaginary configuration is just a translation in horizontal plane. In other words, this area is not affected by the roll but only by the lateral spread of the area I. Thus,

       u I I x = V e u I I y is independent on y u I I z = 0. (6.2)
In consequence, the fact that u I I y = constant allows to determine the function ψ(x) in the real configuration as:

ψ(x) = ψ 0 (x) + ϕ(x) -w e (6.3)
and the fact that

u I I z = 0 implies that h(x, y) = h e (y -ϕ(x) + w e ) (6.4)
everywhere in the area II.

Kinematically admissible conditions and velocity field in area II

6.1.2.a Boundary conditions

On contact surface z = h(x): the velocity must be tangential to the surface. Then :

u I,I I I z (x, y, z = h(x)) = u I,I I I x (x, y)h ′ (x) . (6.5)
From equations 5.22 and 6.5 we have:

u I,I I I z (x, y, z = h(x)) = h ′ (x) h(x) u I,I I I x (x, y)z . (6.6)
On the virtual lateral surface y = ϕ(x): the velocity is tangential to the virtual lateral surface represented by the function ϕ(x) that may be different from real lateral surface w(x) in a current iteration of numerical resolution. That means

u I I,I I I y (x, ϕ(x)) = ϕ ′ (x)u I I,I I I x (x, ϕ(x)) . (6.7)
In this state, the velocity in the area II is completely known

u I I x = V e u I I y = V e ϕ ′ u I I z = 0. (6.8)
On plan of symmetry y = 0: the condition of symmetry of lateral velocity is written as u y (x, y = 0) = 0 . (6.9)

6.1.2.b Volume flow rate conservation

The volume flow rate is calculated easily in the upstream part by : In area I and II: In this part, the volume flow rate is composed of two terms corresponding to the two areas I and II :

C vol = V e S e = V e
C I vol = h ψ 0 u I x dy (6.11)
and

C I I vol = V e ϕ ψ
h(x, y)dy (6.12)

In using equations 6.27, 6.4 and in changing of variable y 0 = yϕ + w e , we obtain

C I I vol = V e w e ψ 0
h e (y 0 )dy 0 (6.13) Thus, the flow rate conservation can be expressed by :

C vol = C I vol + C I I vol ⇒C I vol = h ψ 0 u I x dy = V e ψ 0 0 h e (y 0 )dy 0 . (6.14)
In area III: the flow rate conservation condition is given by

C vol = h ϕ 0 u I I I x dy = V e w e
0 h e (y 0 )dy 0 (6.15)

6.1.2.c Conditions of incompressibility

In continuous velocity field areas test

As can be seen previously, the material is considered incompressible, i.e the velocity must satisfy the equation 5.21 everywhere. In the area II, the velocity given by the equation 6.

On the surfaces of velocity discontinuity test

The same as the case of flat strip rolling, on the exit surface, there is no discontinuity of velocity. On the other hand, a crowned strip rolling geometry is more complicated and may have plural surfaces of velocity discontinuity instead of one. At the entry between upstream part and area II, there is no discontinuity. This will be a condition concerning the width function w(x). The first surface of discontinuity of velocity is the interface between the areas I and II, and the second is between the areas I and III. In general, the condition of pertinence of a velocity field on a surface of discontinuity is that the normal velocity through this surface is to be continuous. Study now this condition for these two surfaces of velocity discontinuity.

Surface of velocity discontinuity between the areas I and II :

u I I -u I n I-I I = 0 ⇔ u I I y (x) -u I y (x, ψ(x)) = ψ ′ u I I x (x) -u I x (x, ψ(x)) .
(6.17)

Surface of velocity discontinuity between the areas I and III:

u I I I x (-L, y) = u I x (-L, y) for ∀y ∈ [0, ϕ] . (6.18)
6. UBM for crowned strip rolling 6.1 Velocity field proposition

Velocity field in area I

In the previous section, a kinetically admissible velocity field in the area II was proposed as given in equation 6.8. In this section, we search for a kinetically admissible velocity field in area I. The equation 6.14 can be rewritten as :

ψ 0 u I x dy = V e h ψ 0
0 h e (y 0 )dy 0 .

By using a simple change of integration variable y 0 = ψ 0 ψ y we have :

ψ 0 u I x dy = V e h ψ 0 h e ( ψ 0 ψ y) ψ 0 ψ dy ⇔ ψ 0 u I x - V e h h e ( ψ 0 ψ y) ψ 0 ψ dy = 0 .
We propose thus

u I x = V e h h e ( ψ 0 ψ y) ψ 0 ψ . (6.19)
The component u I z is easily determined by the equation 6.6. In addition, the equation 6.16 becomes :

∂u I y ∂y = - ∂u I x ∂x - ∂u I z ∂z or ∂u I y ∂y = - V e h h ′ e ( ψ 0 ψ y) ψ 0 ψ ′ ψ 0 ψ y + h e ( ψ 0 ψ y) ψ 0 ψ ′
Furthermore, in using the condition of symmetry with respect to the plan y = 0, equation 6.9, we obtain :

u I y = - V e h h e ( ψ 0 ψ y) ψ 0 ψ ′ y . (6.20)
Hence, the velocity field in this area is completely determined while the condition on the surface of discontinuity between area I and II has not been used yet. We will demonstrate that this condition is satisfied by the velocity field proposed. In substituting the equations 6.19 and 6.20 into the equation 6.17, we have :

V e ϕ ′ + V e ψ 0 ψ ′ ϕ = ψ ′ V e -V e ψ 0 ψ ⇔ ϕ ′ + ϕ ′ 0 = ψ ′ .
Indeed, this equation is implied by the equation 6.27. The velocity field in area I is finally kinematically admissible and given by:

u I x = V e h h e ( ψ 0 ψ y) ψ 0 ψ u I y = - V e h h e ( ψ 0 ψ y) ψ 0 ψ ′ y u I z = u I x h ′ h z .
(6.21)

Velocity field in area III

A velocity field similar to the one of the area I in which ψ is replaced by ϕ and ψ 0 by w e , is a possible solution for the area III. However, in this way, at the roll bite exit, the longitudinal velocity depends always in y while it is observed 6. UBM for crowned strip rolling 6.1 Velocity field proposition not. Lam3-Tec3 gives a solution that is homogenous in lateral direction at the exit, see Figure 6.2. For this reason in our model, the velocity is homogenized by a function noted ζ(x) depending only on x. This function is equal to 1 at the begin and 0 at the end of the area III. This function is freely modeled and in this study, it is chosen as

ζ = x L I I I 2 (6.22)
where L I I I is the contact length of the totally laminated part (area III). We propose finally:

u I I I x = V e h h e ( w e ϕ y) w e ϕ ζ(x) + ξ(x) .
The function ξ(x) is added to ensure the volume flow rate conservation. Thus, it is identified by the equation 6.15 as

ξ = V e h S e ϕ (1 -ζ)
where S e = w e 0 h e (y)dy denotes the entry across section of the strip.

x(mm) K10 K8 K6 K4 K2 C.2 -with a 0.06mm-crown over strip half thickness. The longitudinal velocity is non-homogenous along the roll-bite but is homogenous at the roll bite exit.

Moreover, the condition on the contact surface (equation 6.6) gives directly u I I I z . And u I I I y is determined by the condition of incompressibility (equation 6.16). Finally, a solution of velocity filed in the area III can be: In these areas, the velocity fields are similar. It is thus possible to generalize the calculation by studying a velocity field as follows

u I I I x = V e hϕ w e h e ( w e ϕ y)ζ + S e (1 -ζ) u I I I y = u I I I x ϕ ′ ϕ y + V e hϕ ζ ′ yS e -ϕ
     u x (x, y) u y (x, y) u z (x, y, z) = d 33 (x, y)z (6.24)
in a given domain Ω. This is a general form for the two velocity fields presented previously corresponding to the two areas. For the area I, the domain is

Ω I = [x = (-L, -L I I I ), y = (0, b I (x)]
and Ω I I I = [x = (-L I I I , 0), y = (0, w(x)] for the area III.

The plastic deformation power is computed by

J ǫ = Ω σ 0 2 3 d : d dΩ = Ω σ 0 2 3 d 2 xx + d 2 yy + d 2 zz + 2d 2 xy + 2d 2 xz + 2d 2 yz dΩ = Ω 2D dxdy h 0 2 3 σ 0 Q 2 + g 2 z h 2 dz = σ 0 √ 6 Ω 2D dxdy 1 0 2 Q 2 + g 2 z * 2 hdz * .
with Ω 2D is the projection of Ω on the xy plan. Then, by using the formula B.1, we obtain

J ǫ = σ 0 √ 6 Ω 2D h Q 2 + g 2 + Q 2 g ln g + Q 2 + g 2 Q dxdy . (6.25)
where

Q(x, y) = d 2 xx + d 2 yy + d 2 zz + 2d 2 xy = ∂u 2 x ∂x + ∂u 2 y ∂y + d 2 zz + 1 2 ∂u x ∂y + ∂u y ∂x 2 g(x, y) = 1 h 1 2 ∂d 2 zz ∂x + 1 2
∂d 2 zz ∂y .

6.2.1.b Area II

In this area, the velocity field is much more simple. The only component that is different from zero is :

d I I xy = d I I yx = 1 2 V e ϕ ′′ . (6.26)
Therefore, the plastic deformation power is easily calculated :

J I I ǫ = σ 0 √ 3 V e -L I I I -L w(x) w I (x)
ϕ ′′ h I I (y)dxdy 6. UBM for crowned strip rolling 6.2 Calculation of the powers where w I (x) is the width of the area I. This function can be determined by the following equation

w I (x) = ψ 0 (x) + w(x) -w e (6.27)
which is in the same analogy as the equation 6.27. By changing the variable y 0 = yw I (x) + ψ 0 (x) = yw(x) + w e , we obtain finally

J I I ǫ = σ 0 √ 3 V e -L I I I -L ϕ ′′ dx w e ψ 0
h e (y 0 )dy 0 (6.28)

Power of discontinuity of velocity 6.2.2.a Between the areas I and II

The discontinuity of velocity is previously proved tangential to the surface. It is given by :

∆V I-I I =     ∆u I-I I x ∆u I-I I y ∆u I-I I z     =      u I I x (x, ψ) -u I x (x, ψ) u I I y (x, ψ) -u I y (x, ψ) 0 -u I x (x, ψ)h ′ z h      Then, J I-I I ∆u = σ 0 √ 3 S I-I I dis ||∆V|| I-I I ds = σ 0 √ 3 B A h 0 ∆u I-I I x 2 + ∆u I-I I y 2 + u I x 2 h ′ 2 z h 2 dzdl = σ 0 √ 3 B A 1 0 h Q I-I I 2 + g I-I I 2 z * 2 dz * dl
Using again the formula B.1, we have

J I-I I ∆u = σ 0 √ 3 -L I I I -L h   Q I-I I 2 + g I-I I 2 + Q I-I I 2 g I-I I ln g I-I I + Q I-I I 2 + g I-I I 2 Q I-I I   1 + ψ ′ 2 dx (6.29)
where

Q I-I I (x) = ∆u I-I I x 2 + ∆u I-I I y 2 (x, ψ(x)) g I-I I (x) = u I x 2 h ′ 2 (x, ψ(x)) .

6.2.2.b Between the areas I and III

The discontinuity of velocity between area I and III is more simple because it has an unique component different from zero which is given by ∆u I-I I I y = u I I I yu I y (x = -L I I I , y). The power consumed by this discontinuity of velocity is:

J I-I I I ∆u = σ 0 √ 3

Power of friction

As explained before, the velocity fields in areas I and III are similar and we can use the form 6.24 to present both of them. We will find a formula which is applicable to calculate the friction power on the contact surface with the roll of these two areas. The area studied is noted

S contact = [X 1 , X 2 ] × [0, Y 2 (x)]. For the area I, X 1 = -L, X 2 = -L I I I , Y 2 (x) = w I (x) and for the area III, X 1 = -L I I I , X 2 = 0, Y 2 (x) = w(x).
The difference of velocity between the roll and the strip on these contact surfaces is:

∆V = u x 1 + h ′ 2 -V c 2 + u 2 y .
Thus,

J f ric = mσ 0 √ 3 S contact ∆V dS = mσ 0 √ 3 X 2 X 1 dx Y 2 (x) 0 u x 1 + h ′ 2 -V c 2 + u 2 y 1 + h ′ 2 dy (6.31)

Power of entry and exit tensions

The power of the entry and exit tensions is given by :

J ten = - S e
T e u I I x (-L, y, z)dS + S s T s u I I I x (0, y, z)dS .

In using the expressions of velocity fields studied before, we obtain :

J ten = V e S e -T e + T s b ϕ (x = 0) . (6.32)

Numerical resolution

Numerical integral: The integrations in expressions of all these powers are performed using the Gauss's method (see Appendix A.2). This attractive method of integration helps to reduce significatively the computing time of the program.

Algorithm of resolution:

As a reminder, the UBM principle for the crowned strip rolling is the same as for the flat strip rolling presented in the previous chapter. Therefore, the numerical resolution is also the same, shown in Figure 5.5.

Width function parameterizing

In the previous chapter, there were presented two parameterizing forms for width function w(x) (and virtual width function ϕ(x)). The first is a 3rd-order polynomial of x and the second one is a polynomial of h(x). We, then, demonstrated that the second form relating directly the width function to the reduction in thickness via the function h(x) is better than the first form for a reduction r < 45% in giving smaller rolling power, see section 5.3.7.b. By consequence, in this case of crowned strip rolling, let study only the second form. However, in this case, the width function must have a null derivative at the first point of contact at x = -L because the width of the laminated area (area I) tends to zero at this point. This condition is represented in the following form of the width function by a new parameter k:

w(x) = w e + h -h centre e h s -h centre e k [α + β(h -h s )] . (6.33)
for k > 1.

6. UBM for crowned strip rolling 6.4 Comparison between UBM and Lam3-Tec3

Minimization

The minimization of the power function to find the optimal velocity field for each iteration is also performed by Newton-Raphson method presented in the subsection 5.3.5.c. Nevertheless, there are 4 parameters instead of 3 with the new parameter k in addition. The calculation time increases certainly but the method is unchanged.

Remark:

The optimization program for 4 parameters has not been built yet while it is possible to fix the parameter k, k = 2 for example, and minimize the rolling power in 3 parameters by using the 3-parameter optimization program already done in previous chapter. Therefore, by varying the value of k and by observing the variation of the corresponding minimum rolling power we can choose the best value of k that gives the minimum rolling power.

Comparison between UBM and Lam3-Tec3

This section presents the width spread calculated by the UBM model for a crowned strip rolling, in comparison with Lam3-Tec3 . The strip is narrow enough so that all elastic deformations are negligible. The rolling parameters correspond to the trial E16 of the experiments with hard steel presented in Table C.2. Nevertheless, the entry thickness is modified so that the average entry thickness is the same and the strip crown is a polynomial of degree 2 or 4, two typical forms of strip crown. 75 for a crown of polynomial of degree 2. It is observed that the three curves have a same negative slope, implying the width spread decreases almost linearly as a function of strip crown and that the UBM results match very well those of Lam3-Tec3 independently on the value of k in terms of the tendency. In absolute value, with k = 1.75 the UBM curve is closer to the Lam3-Tec3 curve than with k = 2. In addition, Figure 6.4 shows that the minimum rolling power corresponding to k = 1.75 is lower than that with k = 2. Knowing that the smaller the rolling power, the better width spread calculated by the UBM .

Similarly, for a crown of polynomial of degree 4, it can be seen in Figure 6.5 that the width spread decreases also linearly with an increase in the strip entry crown but more slightly. The UBM and Lam3-Tec3 have a good coherence in terms of tangency of this decrease. And in comparing the minimum rolling power calculated by the UBM for k = 2, k = 1.75 and k = 1.5 (see Figure 6.6) we conclude, an other time, that the more minimized the rolling power, the better width spread the UBM gives. Therefore, we need in a near future a 4-parameter to give better results.

UBM for crowned strip rolling

Conclusion

The previous chapter (5) presented existing models from statistical to FEM and UBM ones predicting the width spread for flat strip rolling. These models consider a rectangular cross section of the strip at the entry. For long and shape product rolling processes, an important number of 3D models were developed to predict the shape variation including the width spread. In this chapter, a new UBM approach is developed for cold rolling of a strip with initial thickness crown while work-roll is considered straight and perfectly cylindric. The model allows to understand how the strip spread depends on the initial thickness profile. As the geometry of the strip is more complex than the case of flat strip rolling, the roll bite is divided into three areas in which the velocity field is different. The optimization of rolling power is performed by using Newton-Raphson algorithm. As a result, the model shows that the width variation decreases with an increase in the strip initial crown and the UBM results match very well those obtained with Lam3-Tec3 . This model development showed that the UBM is applicable even when the geometry seems to be complicated. Furthermore, similarly to this UBM model, it is possible to develop another one for a rectangular strip and crown work-rolls. When the work-roll crown is positive, i.e the center diameter is more important than the work-roll ends, the strip center is also in contact with the work-roll before the strip edges. The geometry problem would be treated in a resembling way. This is an interesting perspective of this chapter.

Chapter 7 A thermal-elastic-plastic width model

The previous chapters present the UBM -based models for width variation of flat and crowned strip in rolling process. The results match well to the experiments on pilot mill using narrow strips. However, it is important to note that the UBM assumes a rigid-plastic behavior of the strip that is justified for a narrow strip because the elastic width variation are negligible. On the opposite, in industrial rolling the strip is large and the elastic width variation which is proportional to the strip width is no longer negligible. This elastic deformation is reversible but it has important impact on the plastic one. In addition, friction and plastic deformation powers may heat the strip about [START_REF] Johnson | The onset of yield in the cold rolling of thin strip[END_REF]

-100°C up. The material is, thus dilated in the width direction but it can not because of the contact friction with the roll. That creates compression plastic deformation -called thermal contraction. In this chapter, after a bibliography review a new width variation model for online applications is developed taking into account effects of elastic and thermal deformations in addition to the width variation of UBM model proposed in the previous chapters.

Introduction

In the first section, an analysis based on Lam3-Tec3 results allow to bring out the contribution of elastic deformation in the total plastic width spread in a large strip rolling case. Following, a new assumption is proposed to build a new model taking into account elastic deformation effect. In this model, as the elastic deformation is important, it is necessary to determine stress fields in the strip. In the roll-bite, the material is plasticized in a almost-plane strain deformation condition, the slab method (see section 2.3.1.a) gives a good prediction of average values over thickness direction of the stress fields. Then, two following sections present simplified methods to model two phenomena: elastoplastic compression at the entry and elastic spring back at the exit of roll bite. Finally, in the last section, comparisons of width spread with Lam3-Tec3 as well as with industrial rolling observations are presented to validate the new model.

Bibliographic review on width variation in industrial cold rolling 7.1.1 Main phenomena involved in width variation

Based on some studies existing perviously in literature, N.Legrand et al [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] give a global view about phenomena involved in the width variation of the strip in flat automotive rolling. According to the authors, the width spread in rolling is a thermo-mechanic problem. They bring out three main physical phenomena contributing to the width variation as follows. This phenomenon is due to a 3D plastic flow near strip edges (approximately 20 to 30 mm from strip edges). Especially, at the strip edges the roll is significantly deformed elastically (see Figures 1.8 and 2.12) decreasing locally the strip thickness. By consequence, after rolling the strip thickness is lower at the edges than at the center (Figure 7.1). This phenomenon promotes an increase of the strip width. The problem of strip edge drop in cold rolling has been extensively investigated [START_REF] Kitamura | Edge-drop control of hot and cold rolled strip by tapered-crown roll shifting mill[END_REF][START_REF] Mitai | Analysis of strip rolling by three-dimensionnal rigid plastic fem[END_REF][START_REF] Campas | New online gage for edge drop measurement and effect of tapered work rolls[END_REF][START_REF] Hartung | A new way to reduce the edge drop[END_REF] and certain number of other works.

7.1.1.a Edge drop

7.1.1.b Elastic spring back

According to Legrand et al [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] the strip "elastic spring back" at the roll-bite exit contributes to strip width increase in cold rolling. In roll bite, the transverse stress is negative, i.e the material is compressed in width direction. And the elastic recovery mainly corresponds to transverse stress σ yy relaxation just at the roll-bite exit. The authors give an estimation of this transverse stress based on Lam3-Tec3 FE ( [START_REF] Hacquin | Modélisation thermomécanique 3D du laminage: couplage bande-cylindres (3D thermomechanical modelling of rolling processes strip-roll coupling)[END_REF]) simulations. They point out that the contribution of transverse stress relaxation to width broadening depends on steel strip work hardening which can be significant in cold rolling. Due to this work hardening, strip elastic recovery at the roll bite exit (relaxation of transverse stress σ yy ) is greater than strip elastic compression that takes place at the roll bite entry, thus strip width increases.

In this chapter (section 7.2.2.f), a further study of width variation phenomena shows that the elastic stress evolution in the roll-bite influences and changes the plastic transversal deformation. Therefore, its contribution in width variation

A thermal-elastic-plastic width model

7.1 Bibliographic review on width variation in industrial cold rolling depends not only on material work hardening but also strongly on entry and exit tensions values as well as friction coefficients...

7.1.1.c Thermal contraction

Edwards [START_REF] Edwards | Width contraction in cold rolling[END_REF] explains that successive strip thermal contraction in each inter-stand (due to cooling) combined with no possibility of strip thermal expansion in each roll bite (due to friction) is one possible cause of strip width decrease in cold rolling. During cold rolling, strip temperature increases in the roll bite due to plastic strain and friction. However, it is assumed that this temperature increase occurs without any possibility of thermal expansion due to roll-strip friction forces in the roll bite. Then outside the roll-bite, strip is cooled down by coolant spray and natural cooling with a free thermal contraction. As a consequence, the thermal irreversible expansion/contraction tends to decrease strip width continuously through the successive stands.

Indeed according to us, more precisely the thermal deformation itself is reversible but in the roll-bite due to the contact constrain the thermal dilatation generates a negative plastic deformation across the width that is irreversible and contributes to the width necking of the strip. More details will be pointed out in the section 7.2.2.f.

Effect of rolling parameters on width variation

Width variation is mostly negative in industrial rolling

Lafontaine [START_REF] Lafontaine | Première approche sur le rétreint au cours du laminage à froid[END_REF] investigated on strip width variation in cold rolling using a database from Arcelor Florange's 5stand tin-plate and 4-stand sheet cold tandem mills. He observed that strip width variation is usually negative (necking) and this width necking is statistically higher on tin plate than on sheet product. He showed that the wider the strip, the higher the width necking and that width contraction is statistically higher with IF (soft) steels than with harder ones.

Using width measurement devise existing on the industrial plants combined with a new width measurement devise developed by ArcelorMittal research, Legrand and Ngo [START_REF] Legrand | Rfcs rapport de l'etude sur la variation de largeur en usine à froid[END_REF] analyse database of several cold rolling mills. They observe equally that the width variation in both tin plate and sheet cold rolling mills is usually negative. They perform also specific trials to analyse width variation problem using ArcelorMittal research Maizières pilot mill and some ArcelorMittal industrial rolling mills. The results of this work as well as those of a few other articles are presented hereafter to point out the effect of rolling parameters on the width variation.

7.1.2.a Anisotropic friction effect

On a study of edge drop phenomenon, Legrand et al [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] perform cold rolling trials on ArcelorMittal Research Maizieres pilot mill with a 50m/min rolling speed and strip width varying from 60 to 70 mm, initial thickness from 1.5 to 3.0 mm, work-roll diameter of 400 mm. They measure the entry and exit profile in order to bring out the edge drop phenomenon. In addition, the authors use Lam3-Tec3 ( [START_REF] Hacquin | Modélisation thermomécanique 3D du laminage: couplage bande-cylindres (3D thermomechanical modelling of rolling processes strip-roll coupling)[END_REF])to simulate these trials to get better understanding of-the phenomena. By comparing the measured strip exit thickness profiles and that obtained by Lam3-Tec3 simulation, the authors bring out the anisotropy of friction coefficient: friction coefficient in the transverse direction (across the strip width) µ y is higher than that in the rolling direction µ x . Figure 7.2 show an example of the trial named E16 where the anisotropic factor is equal to 2. They obtain anisotropy facto varying from 2 to 4 depending on the trials conditions.

According to the authors, such an anisotropic friction can be explained by the work-roll circumferential striations due to grinding process which generates an anisotropy of the work-roll roughness as showed by Figure 7.3.

By comparing width variations obtained from pilot mill trials and from numerical simulations, they show that the FE model Lam3-Tec3 is able to predict strip width with an accuracy of +/-0.25 mm if anisotropic friction is taken into account (see Figure 7.4). They conclude that transverse friction has a significant influence on strip width: a variation of transverse friction µ y by a factor of 2 to 4 makes a strip width modification by 0.5 to 1.2 mm. Therefore, transverse friction appears as a possible actuator for strip width control.

7.1.2.b Effect of bending is indeed effect of strip flatness

Schwarz [START_REF] Schwarz | Etude de la variation de largeur au tandem 4 cages de sollac biache[END_REF] analyzed strip width variation on Biache's cold tandem mill (France) where a strip width sensor was implemented at the tandem exit. He showed that roll bending, strip tension, rolling pressure and thermal effects have a significant influence on strip width. However no model was used or developed. The results presented were essentially statistical and not predictive.

By doing similar simulations using Lam3-Tec3 ( [START_REF] Hacquin | Modélisation thermomécanique 3D du laminage: couplage bande-cylindres (3D thermomechanical modelling of rolling processes strip-roll coupling)[END_REF]) as [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] (described above) Legrand and Ngo [START_REF] Legrand | Rfcs rapport de l'etude sur la variation de largeur en usine à froid[END_REF] analyse the effect of bending on width variation on stand 1 and 5 of a tin plate rolling mill. They show that the higher the bending the lower the edge drop and by consequence the smaller the width variation (or more necking if the width variation is negative).

An increase of bending gives long center tendency in strip flatness by distancing the heads of top and bottom workrolls and lightening the reduction at the strip edges. Indeed, the strip width variations and flatness are determined by the work-rolls in-charged profile whatever this profile is a results of a bending or a designed inimical profile. In other words, the strip width variation depends on its flatness and the change in flatness during rolling. The bending is a parameter influencing the strip flatness like many others such as the work-roll initial profile, thermal crown (related to a non-homogeneous distribution temperature along the rolls)...

7.1.2.c Effect of tensions

In addition, Legrand and Ngo [START_REF] Legrand | Rfcs rapport de l'etude sur la variation de largeur en usine à froid[END_REF] performed trials on Florange 4-stand automotive rolling mill by rolling 2 coils with same rolling conditions except inter-stand tensions. The rolling parameters are given in Table (7.1). Both coils thickness is reduced from 2.8mm to 0.66mm thanks to the 4 stands. T 1 and T 5 are the entry and exit tensions of the tandem mill and T ij is the interstand tension between stands i and j. It can be seen that the coil "37870-07" is rolled with higher tensions and gets 2mm more necking than the coil "37870-01". The authors conclude hence that the strip tensions have important influence on the width variation and can be a potential width actuator. 7.1: Industrial trials to analyse tensions effect on width variation: thicknesses, tensions and width variations of the 2 tested coils (negative width variation means width decrease).

Coil

7.1.2.d Effect of reduction

Figure 7.5 shows strip width variation produced by Florange 5-stands cold tin plate mill for two different levels of strip reduction (86.5%: D&I food and 92.0%: D&I drink), for the same steel grade D&I. Each point of the graphic corresponds to one coil. It can be seen that an increase of strip thickness reduction by from 86.5% to 92.0% decreases strip width from 5mm to 10mm. Thus, the higher the total reduction the higher the strip width decrease possibly due to a stronger thermal effect (higher heating in the bite) that promotes a width decrease.

7.1.2.e Effect of nominal strip width

As a reminder Lafontaine [START_REF] Lafontaine | Première approche sur le rétreint au cours du laminage à froid[END_REF] has already concluded in his study that the higher the strip width the more important the width necking. Legrand and Ngo [START_REF] Legrand | Rfcs rapport de l'etude sur la variation de largeur en usine à froid[END_REF] confirmed this tendency based on a database analysis of width variation at Mardyck 5-stand and Florange 4-stand sheet cold rolling mills -see Figures 7.7 and 7.8. They explained the tendency by the fact that the higher the strip width the higher the thermal contraction contribution as it is proportional to the strip nominal width. It can be seen more in detail from Figure 7.7 that, indeed the strip width variation amplitude 7. A thermal-elastic-plastic width model 7.1 Bibliographic review on width variation in industrial cold rolling increases with an increase in strip nominal width meaning that if the width variation is negative (necking) it becomes more negative and inversely if it is positive (spread) it gets more positive. It can be seen in Figure 7.9 that the strip width increases with an increase in strip entry thickness but the effect is relatively weak. On the other hand, Figure 7.10 shows that on the Mardyck 5-stand sheet tandem mill the harder the strip the higher the strip width which is possibly due to a higher elastic broadening. However, it is worth to note that, in industrial rolling the harder the steel the lower total reduction. That means the graphic 7.10 does not shows the only effect of strip yield stress and the tendency could be that of the total reduction. Indeed, for soft steel as the total reduction is higher there is more width necking due to reduction effect.

7.1.2.g Summary of rolling parameters effect on strip width variation

Table 7.2 shows a recapitulation of the experiment trials and the industrial database analyses existing in the literature about the impact of rolling parameter on strip width variation. Only in some cases, the corresponding authors give explanations (or share their point of view about possible explanations) because the phenomena remains. These results will be used to validate the simplified model which is developed afterward. Counhaye [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF] developed a multidimensional model based on finite differences numerical resolution, called "Streamlines" model. This method allows to simulate stationary problem formed by equilibrium equations combined with elasto-plastic behavior. The model is coupled to a tool deformation model based on influence functions method and a Coulomb friction model with possibility to take into account friction anisotropy. The principle of the method is to integrate the elasto-plastic laws and equilibrium equations along material streamlines. The streamlines are adjusted iteratively at the same time with shear stresses. This resolution gives stationary solution very similar to that obtained by finite elements methods. The streamlines approach is an alternative solution to FEM .

A thermal-elastic-plastic width model

7.1 Bibliographic review on width variation in industrial cold rolling Using the "Streamlines" model, Counhay [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF] was the first to quantitatively predict strip width variation in flat automotive cold rolling conditions. The author interpret width variation mechanism by three phenomena given perviously in the section 7.1.1. The strip width variations predicted by the model is in correct agreement with measured widths (accuracy +/-1mm) -see Figure 7 As already mentioned above, Legrand et al [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] developed a model for width variation in large strip industrial rolling conditions and pointed out the three phenomena involved in the strip width variation as can be seen previously (section 7.1.1) and state that the total width is given by: ∆w = ∆w EdgeDrop + ∆w SpringBack -∆w Thermal (7.1) where ∆w EdgeDrop , ∆w SpringBack and ∆w Thermal are respectively the width variation parts due to edge drop, spring back and thermal contraction phenomena. The first two terms ∆w EdgeDrop and ∆w SpringBack are obtained by mechanical simulation using Lam3-Tec3 . As for the thermal term, it is given by:

∆w thermal = α∆T (7.2)
where ∆T is the variation of strip temperature from entry to exit of the roll-bite. The authors use a thermal model developed by Dusser et al [START_REF] Dusser | Improved cooling on cold tandem mills[END_REF].

Important contribution of elastic and thermal deformation

The authors applied this model to simulate the trials that they performed on ArcelorMittal Maizières Research pilot mill to interpret these trials and bring out the anisotropy of friction (see 7.1.2.a). They also applied the model to analyse the contribution of each of the three terms in sheet and tin plate mills. The results are given in Tables 7.3 and 7.4. It can be seen that the three terms are in a same order in both tandem mill configurations. The thermal contraction in the case of tin plate mill is even very important because of very high reduction. The width variation due to elastic spring back is also important. It is twice greater the edge drop term in the case of sheet tandem mill and equivalent to edge drop in the case of tin plate. 7.4: Results of contribution of width variation phenomenon using Lam3-Tec3 and a thermal model [START_REF] Dusser | Improved cooling on cold tandem mills[END_REF] on Cockerill tin plate mill for a nominal width w = 910mm, h e = 2.50mm and h s = 0.087mm, red = 84.5% (see [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] for more details). Importance of elastic and thermal deformation: According to the studies presented above for industrial rolling process of flat and large product, there are three mechanisms contributing to strip width variation: edge drop, elastic spring-back (relaxation of transversal stress at exit of the stand) and thermal contraction. Edge drop and elastic springback promotes an increase of strip width while the thermal contraction makes it decrease which explains why in industrial database the width variation can be positive and negative. Furthermore, to quantify the part of each mechanism, [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF][START_REF] Legrand | Rfcs rapport de l'etude sur la variation de largeur en usine à froid[END_REF][START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF] developed models based on FEM (Lam3-Tec3 ) and FDM ("Streamlines method") and illustrated that the width variation due to elastic spring-back and the thermal contraction is important in general flat cold rolling condition. And they can be greater than the width variation due to edge drop in tin-plate mill condition with high reduction level.

In the chapter 5, some statistical models (see 5.1) as well as UBM -based models (5.2) for width variation analysis are presented. The width variation predicted by these models is always positive while the industrial observations in automotive cold rolling mills show that the width variation is mostly negative -called necking. These models assuming a rigid-plastic behavior of the strip are not able to model the spring back phenomena.

No existing model is fast enough for online applications In addition, there exist very few attempts to develop width variation models for flat automotive cold rolling. In our knowledge, only FEM (Lam3-Tec3 , Abaqus ) and Streamlines Finite Difference (see [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF][START_REF] Legrand | Rfcs rapport de l'etude sur la variation de largeur en usine à froid[END_REF][START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF]) are able to describe width variation problem but the computing time is too high (several hours) to be used for preset and online control applications.

Conclusion:

Those are the reason why we aim at developing a rapid model able to predict the width variation in industrial flat large strip cold rolling. The approach is to use Lam3-Tec3 to analyse and understand in details width variation phenomena in order to adopt suitable hypotheses allowing to develop this simplified model in keeping the key phenomena and influence parameters.

Assumptions -analysis of width variation using Lam3-Tec3

Three general assumptions

Firstly, here are the assumptions that are not based the analysis of Lam3-Tec3 results. They are, nevertheless necessary to fixe the working domain of the model to develop.

7.2.2.a Symmetry:

The considered rolling process is both top-bottom and operator-motor side symmetric. Therefore, only one quarter of the geometry needs to be studied (see 2.1.1.b).

7.2.2.b Coulomb friction model:

As we are interested in elastic deformation, especially in the elastic spring back area where the strip is deformed elastically under contact with the work-roll at the exit of the roll-bite, Coulomb's law seem to be more realistic to model the friction. As a reminder, the Tresca's friction depending on the material yield stress is not suitable for elastic deformation areas where the yield stress does not involves in the problem.

7.2.2.c Work roll deformation:

The work-roll shape is considered circular and straight meaning that only the flattening can be taken into account by Hitchcock's model (see 2.4.2.a) and the deflection deformation is not considered. The edge drop is therefore not modelled by this model. The reader of this thesis will see later that a UBM -based model is one part of the width variation global model that we develop in this chapter. And the UBM -based model is the one presented in the section 5.3 which considers a circular and straight work-roll. By the way, as the chapter 6 presents an UBM -based model of width variation for a crowned strip, it is possible to develop an UBM -based model for a non-straight work-roll shape. This potential development is one of perspectives of this work.

Three specific assumptions -Lam3-Tec3 simulations

Secondly, follows are the assumptions which are issued from the understanding of physical phenomena involved in width variation in cold rolling by analyzing Lam3-Tec3 results. Let study the Lam3-Tec3 simulation results for a industrial rolling conditions given in Table 7 Lam3-Tec3 simulations: For each stand, several Lam3-Tec3 simulations are done taking into account different hypotheses as follows:

• elasto-plastic strip with 10 elements in the half thickness • elasto-plastic strip with 1 element in the half thickness • rigid-plastic strip with 1 element in the half thickness (indeed this calculation is performed using a elasto-plastic strip with Young modulus multiplied by 1000)

• thermo-elasto-plastic strip with 1 elements in the half thickness with heat exchange between WR and strip

• thermo-elasto-plastic strip with 1 elements in the half thickness without heat exchange between WR and strip

• thermo-elasto-plastic strip with 10 elements in the half thickness without heat exchange between WR and strip As mentioned above, the strip flatness influences also the width variation. In order to compare these simulations, they are performed so that the strip flatness is the same. In fact, the thermal-elasto-plastic simulation give a same flatness as elasto-plastic one but the rigid-plastic does not. Thus, we choose work-roll initial profile (crown)to get a similar flatness as that of elasto-plastic.

7.2.2.d Average behavior across strip thickness -Slab method

The graphics in Figure 7.12 illustrate the profile of strip width variation from upstream to downstream of each of the four stands. It can be seen that the width spread is almost the same with 1 or 10 elements in half thickness of the strip. The only exception is the first stand but we do not have explanation for this exception. Hence, to model the width variation it is enough to model an average behavior (stress, strain, strain rate...) across strip thickness. This observation leads us to an assumption which is similar to that of slab method: To model the width variation, we can assume an homogeneous deformation across the thickness (as slab method -see 2. This assumption allows to simplify drastically our analysis and model development. From now on, the analysis will be based on Lam3-Tec3 simulations with 1 element in the strip half thickness. As a reminder, the existing models in the literature such as [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] and [START_REF] Cosse | Mathematical study of cold rolling[END_REF] (see sections 2.3.1.e and 2.3.1.f correspondingly) consider that there is a zone where the strip is already under contact with the strip and is deformed elastically before being deformed plastically. This zone is called elastic entry zone. However, in the contrary to this assumption, it can be seen in Figure 7.13 showing a comparison of contact and yield fields obtained with Lam3-Tec3 for the stand 2 of ArcelorMittal Florange mill, that the strip is deformed before being in contact with the work-roll. Except the stand 4 where the reduction is very small red = 1.66%, the strip is deformed plastically before contact.

7.2.2.e

Figure 7.14 shows Lam3-Tec3 results about the contact and yield area for the four stands. It can be seen that for the first three stands 1, 2 and 3 the strip is deformed plastically before being in contact with the roll. But for the stand 4 with small reduction (red=1.66%), the entry compression is purely elastic as supposed by [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] and [START_REF] Cosse | Mathematical study of cold rolling[END_REF]. The "Simplified entry compression model" presented in the section 7.3 is not good for the last stand. However, the assumption stating that the last point of plastic deformation is the lowest point of the roll C is quit verified. On the other hand this figure shows existence of an elastic contact area at the exit that is in a good agreement with the simplified model.

Finally, we consider that the strip deformation can be divided into three zones given in Figure 7.15 as follows:

1. Elasto-plastic compression before contact: The strip is deformed elastically and then plastically just before its contact with the roll.

Roll-bite:

The roll-bite is defined as the zone from the first point of contact B to the last point of plastic deformation C. It is different to both the plastic zone as well as the contact zone.

Exit elastic spring back:

This zone starts at the last plastic deformation point C and is divided itself into two zones, one under contact (CD) and the other is after last contact point D where the strip is free.

7.2.2.f Assumption of material flow in strip width direction

Impact of elastic deformation: Figures 7.16 show comparison of width variation profiles obtained with elastoplastic and rigid-plastic strip behavior correspondingly to the four stands. The simulations with rigid-plastic behavior give width variations much lower than those given by the elasto-plastic ones. This comparison illustrates that the elastic deformation has an important influence on the plastic one and need taking into account to well estimate the width variation of strip in industrial rolling. In the roll-bite, the plastic and elastic spreads have an opposite sign and almost the same absolute value, implying that by action of friction the elastic spread is transformed to plastic spread to keep the same flow as the case of rigid-plastic behavior. In addition, observing that the two curves, X (mm) Figure 7.17: Comparison of width variation of Lam3-Tec3 simulations using elasto-plastic and rigid-plastic strip behavior laws for ArcelorMittal Florange cold rolling mill stands.

elasto-plastic and rigid-plastic width variation profiles are not very different one from the other we propose therefore a new hypothesis to simplify impact of elastic deformation in the plastic width variation as follows:

Independent lateral flow assumption 1 -elasticity influence:

In the roll bite the total elasto-plastic width variation profile is same as the rigid-plastic one. It means that the elastic deformation in the roll bite is completely transformed into plastic deformation or in other words, the total material flow in width direction is almost the same in the two cases: elasto-plastic and rigid-plastic simulations.

Impact of thermal deformation -Thermal contraction:

Similarly to the study of the impact of elastic deformation, thermal-elasto-plastic simulations with Lam3-Tec3 are performed considering that 100% of plastic deformation and friction dissipations are transformed into heat. The friction dissipation is shared 50%-50% between the roll and the strip. The heat exchange coefficient between strip and roll is htc = 2.10 4 w m 2 K . As can be seen in Figure 7.18 that the total strip width variation of thermo-mechanical simulations is the same as that of mechanical ones. The assumption of independent lateral flow (announced above) is also verified. Like the elastic deformation the thermal width dilatation in the roll bite is also completely transformed into plastic one.

Independent lateral flow assumption 2 -thermal dilation influence:

The total strip width variation profile of thermo-mechanical simulations is the same as that of mechanical one. is the total width variation of the strip that is modeled as thermal-elasto-plastic material.

Proposal of thermo-elasto-plastic width variation model

• ∆w elasto-plastic entry is the width variation before entry of roll-bite of elasto-plastic strip (before point B -Figure 7.15).

• ∆w

elasto-plastic roll-bite is the strip width variation in the roll-bite.

• ∆w rigid-plastic is the strip width variation for a rigid-plastic behavior -This variation only occurs inside the roll-bite because there is no elastic deformation outside the roll-bite.

• ∆w elasio-plastic exit is the elastic width variation after exit of roll bite for an elasto-plastic strip. This term is purely elastic and is indeed the elastic spring back term ∆w elasic exit already mentioned above.

7.2.3.b Formula of plastic width variation

The formula 7.5 give us a zone-by-zone breakdown of the thermo-elasto-plastic strip width variation from entry zone through the roll-bite to the exit zone. The impact of elastic and thermal deformations are implicit. In order to better point out these phenomena, let study the "final" strip width variation which corresponds to the plastic one given by: ∆w plastic total = ∆w thermo-elasto-plastic total -∆w elastic total -∆w thermal total (7.6) where

• ∆w plastic total is the total plastic-deformation part of the strip width variation. This is the final permanent width variation after being cooled down to initial temperature (equal to entry temperature) and relaxed of stress.

• ∆w elastic total is the total elastic-deformation part of the strip width variation. This term is determined by the difference between the entry and exit the stress tensors. Far from the roll-bite, the stress tensor contents only longitudial component that is equal to entry and exit tensions. Therefore,

∆w elastic total = ν T e σ e 0 - T s σ s 0 . (7.7)
• ∆w thermal total is the total thermal-deformation part of the strip width variation. This term only depends on the variation of strip temperature given by: ∆w thermal total = wα th ∆T (7.8)

where α th is strip material dilatation coefficient. By substituting 7.5 into 7.6 we obtain:

∆w plastic total = ∆w elasto-plastic entry + ∆w rigid-plastic + ∆w elasto-plastic exit
-∆w elastic total -∆w thermal total (7.9) This formula explicitly shows how the plastic spread in the case of a thermo-elasto-plastic strip is different from that of a rigid-plastic while the formula 7. (7.11) meaning that the total plastic deformation is only caused by the entry and the roll-bite areas.

7.2.3.c Discussions

More complete and clearer formula compared to previous studies [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF][START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] Since the equation 7.9, it seems obvious that first two terms ∆w elasto-plastic entry , ∆w rigid-plastic are not mentioned by Counhay [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF] and Legrand et al [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF]. That means, the new model consider in addition to the phenomena studied by [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF][START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] (see 7.1.1) two other phenomena:

1. Lateral flow of the roll-bite ∆w rigid-plastic : The material flow in roll-bite of the strip is not mentioned in [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF] and [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF], implicitly it is neglected. In the industrial rolling condition, the strip is very large and the this part is negligible but it can be very important for a narrow strip (see more in the previous chapters).

A thermal-elastic-plastic width model

7.3 A simplified entry elasto-plastic compression model 2. Entry compression: Conversely to the way of the exit elastic spring back, at the roll-bite entry, the strip is compressed and has a variation of its width. The article does not mentions this phenomenon but in many cases, it has an important distribution to the total width spread. At the entry, in typical automotive cold rolling conditions the strip is generally deformed plastically before being in contact with the work-roll (see Figure 7.13). In the section 7.3, a new model for the entry compression is proposed in order to estimate the width variation due to the plastic deformation before the contact.

Edge drop is not taken into account

It is also important to highlight that in this thesis, the edge drop is not studied because such a study requires a model for roll deformation. This study can be carried out at the same time with the influence of strip flatness on width variation -considered as a perspective of this thesis. In this case, the "edge drop" is indeed a part of lateral flow term ∆w rigid-plastic which needs to take into account the work-roll in-charged profile especially near to the strip edge (see Figure 2.12 for explanation of this phenomenon).

Development of simplified model

According to the new model 7.9 the stress field needs to be determined. As the strip is large, the 3D mechanical problem may be modeled as 2D -in other words the strip variation is a small perturbation and does not make changes on the global stress field. Therefore, we are interested in a typical 2D model for stress solution such as the slab method. In a simple model considering the behavior of the strip as rigid-plastic, this method is easy to program and gives a good stress field in the roll bite. However, for an elasto-plastic behavior strip, it holds some difficulties particularly at the entry and exit of roll bite.

To solve the whole problem, it is necessary to consider all elasto-plastic equations in three areas: before, inside and after the roll-bite and the two limit conditions on the stress given by entry and exit tensions. These equations are coupled and have to be solved together. The program is heavy and calculation time becomes considerable (see more [START_REF] Cosse | Mathematical study of cold rolling[END_REF] or 2.3.1.f and [START_REF] Stephany | Contribution à l'étude numérique de la lubrification en régime mixte en laminage à froid[END_REF]). [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF] (see 2.3.1.e) propose an alternative simplified resolution allowing to get faster calculation. However, all authors assume that at the entry of roll-bite, the strip is plasticized after being in contact with the roll. Then there is an elastic compression under contact at the entry of which the equations are written similarly to the spring back at the exit. This is in contrary to what we observed with Lam3-Tec3 simulations (see subsection 7.2.2.e and Figure 7.15). These are two reasons why we develop here a simplified model based on "slab" hypothesis stating that it is enough to consider homogeneous behavior in the strip thickness to well model the strip width variation (see 7.2.2.d). Three following sections present respectively simplified models for entry compression, the exit spring back and roll-bite thermal heating.

A simplified entry elasto-plastic compression model

Shear stress assumption

At the first point of contact with the work roll, the strip deformed, more exactly sheared in xz plane. This entry area is modelled as a slipping surface (a discontinuity surface of strip material velocity) by the numerous upper bound methods existing in literature. For example, the upper bound method with rigid bodies motions velocity field [START_REF] Avitzur | The upper bound appoach to plan strain peoblems using linear and rotational velocity fields -part i: Basic concepts[END_REF] -see 3.2.3 or continuous eccentric velocity field [START_REF] Avitzur | Maximum reduction in cold strip rolling[END_REF][START_REF] Avitzur | Power analysis of cold strip rolling[END_REF][START_REF] Avitzur | An upper bound approach to cold strip rolling[END_REF]] -see 3.3.1 or continuous simple or elliptical velocity field [START_REF] Bouharaoua | A la recherche d'une modélisation simplifiée du laminage[END_REF] and [START_REF] Dogruoglu | On constructing kinematically admissible velocity fields in cold sheet rolling[END_REF] -see 3.3.2 and 3.3.5 or circular velocity field [START_REF] Hoang | Asymmetric rolling analysis -energy saving and ski effect[END_REF] -see 3.3.4.

Moreover, as a reminder of the subsection 7.2.2.e, for the typical rolling conditions of Florange 4-stand mill, we observe a plastic deformation area just before contact. This is illustrated in Figure 7.13 for the stand 2. This is true for a case where the thickness reduction is quite high (stands 1, 2 and 3) and not true for the stand 4 where the reduction is only 1.66% (see Figure 7.14). Therefore, let consider the following assumption. We assume moreover that the other shear stress components are equal to zero σ A xy = σ A yz = 0 and there is always no stress in the lateral vertical directions

σ A yy = σ A zz = 0.

Homogeneous stress in thickness assumption

The analysis of Lam3-Tec3 calculations resulted to the assumption that: To model the width variation, we can assume an homogeneous deformation across the thickness (see 7.2.2.d). Now, by writing the equation of equilibrium in longitudinal direction of the strip part before the first point of plastic deformation IAA'I' we deduce that σ A xx = T e . Therefore, the stresses tensor at AA' is:

σ A =   T e 0 σ A xz 0 0 0 σ A xz 0 0   (7.12)
The deviatoric stresses tensor is:

S A =   2 3 T e 0 σ A xz 0 -1 3 T e 0 σ A xz 0 -1 3 T e   (7.13)
As A is the first point of plasticization, the Von Mises criteria is verified:

3 2 S A : S A = 3 2 4 9 T 2 e + 1 9 T 2 e + 1 9 T 2 e + 2σ A2 xz = σ 0
Then:

σ A xz = - σ 2 0 -T 2 e 3 (7.14)
Then the stresses at the point A is:

σ A =     T e 0 - σ 2 0 -T 2 e 3 0 0 0 - σ 2 0 -T 2 e 3 0 0     (7.15)

Plane strain deformation assumption under contact

Plane strain deformation assumption: Once the strip is under contact with the work-roll, we assume that it is in plane strain deformation. This assumptions means that we consider the strip width variation is a very small perturbation and negligible in terms of stress approaches. Moreover, we assume that at the first point of contact B, the shear stresses are equal to zero. By consequence, at B all hypotheses if slab method are verified and therefore the stress tensor at this point can be given by the equation 2.24. In addition, similarly to the point A, the equilibrium in longitudinal direction implies that: σ B xx = T e . Finally, we have: Plane strain deformation between A and B AB plane strain assumption: We assume that the point A is so closed to the point B that the total width of the strip does no changes between these two points.

σ B =    T e 0 0 0 T e -σ 0
This assumption induces that on AB:

∆ǫ elastic yy + ∆ǫ plastic yy = 0 or ∆ǫ plastic yy = -∆ǫ elastic yy = σ A yy -σ B yy -ν σ A xx -σ B xx + σ A zz -σ B zz E (7.17)
By substituting the equations 7.16 and 7.16 in to 7.17, we obtain: illustrating that the plastic width variation is only impacted by the elastic and thermal deformation between A and C (the first and last point of plastic deformation). This formula is valid when the following assumptions are verified: Independent lateral flow assumptions 1 & 2 (see 7.2.2.f and Plane strain deformation between A and B presented previously.

∆ǫ plastic yy = (1 -2ν) σ 0 √ 3 -(1 -ν)T e E ( 7

A simplified elastic spring back model -elastic slab method

As mentioned previously, the exit spring back is the width variation of the strip at the roll-bite exit due to a relaxation of strip stresses. Actually, the spring back is realized in two steps (see Figure 7.19 for illustration). The first step is the relaxation of the vertical stress σ zz between the lowest point of the roll C and the last point of contact D. And in the second step, the lateral stress is released within a longer part of the strip. In general, the length of convergence of lateral stress relaxation is approximately equal to the strip width.

The second step appears after contact and the problem is much more simple than the first one because the strip is free. The only unknown for this step is the stress at D. Therefore, the first relaxation step is the principal one of the exit spring back problem. For a 2D problem, the spring back problem is considered to be the first step ( [START_REF] Bland | Part iii -an approximate treatment of the elastic compression of the strip in cold rolling[END_REF][START_REF] Cosse | Mathematical study of cold rolling[END_REF]) -relaxation of the compression in thickness within the area CD. In this area, the relaxation of vertical stress drives to a variation of the stress in all other directions. Moreover, unlike the entry area AB the tangency of the roll surface at the exit of roll bite is near to zero and therefore the size of the spring back area CD is considerable. It can be about 10% (even much more in the case of small reduction rolling) of contact length. Its great influence on roll force and torque is observed.

Inputs of model

Exit thickness and tension: At the exit far from the roll bite F, the strip thickness is equal to h s and the stress is given by the exit tension T s as follows

h relax s ≃ 1 + ν T s E h s . (7.24)
Work-roll shape: The roll shape is an input of the problem that gives the strip thickness function h(x). Even though the resolution of spring back problem does not depend on the form of this function, the roll is modeled to be circle with a deformed radius R de f calculated by Hitchcock's model. Then, the strip thickness within contact area CD is given by the work-roll shape and h C strip thickness at C representing the minimum thickness of the strip:

h(h C , x) = h C + R de f -R 2 de f -x 2 . (7.25)

Outputs of model

The thickness and tension at the exit for from the roll-bite F are known but the strip thicknesses h C at the lowest point C and x D position of the last point of contact D are unknowns of the problem. They are the outputs of the problem. The strain and stress as well as the strip width variation through out CDF are equally the outputs. As a reminder of the Lam3-Tec3 calculations analysis (see 7.2.2.d), to model the width variation, it is enough to consider a homogeneous deformation across the strip thickness. Hence, we develop in this section a simplified slab method to approach the exit spring back problem. Some assumptions are the same as those of rigid-plastic slab method already presented in 2.3.1.a except the hypothesis on rigid-plastic behavior. Two of them stating that deformations are homogeneous across the thickness and shear stresses are negligible imply that the stress tensor only depends on x but not on y and z, σ(x). By consequence, the elastic deformation strain tensor is also a function of x only. As in this spring back area the plastic strain is unchanged, let use ǫ * * to denote the elastic strain (which normally denote the total strain). This notation is of course only valid in this section of the thesis.

As the strip thickness is given by 7.25 as a function of h C and x, the elastic deformation in thickness direction is also a function of h C and x:

ǫ zz (h C , x) = h(h C , x) h relax s -1 (7.26)
where h relax s is the exit strip haft thickness after release the stress T s given by 7.24.

7.4.1.b Assumption of plane strain deformation

Elastic plane strain deformation within CD: As the first step of spring back is always under contact and the length of this area CD is much smaller than the length of the second step, we can suppose that there is no width variation in the first step. This assumption implies that along this vertical stress spring back area, the lateral deformation is unchanged and equal to ǫ yy (x) = const = ǫ yy .

Plastic plane strain deformation at C: We suppose firstly that the last point of plastic deformation is C, the lowest point of the roll surface. It is considered in addition that the deformation of the strip is also plane at this point.

7.4.1.c Assumption of total relaxation of vertical stress

This assumption states that after the first step,i.e at D, the stress i.e σ zz is totally released.

Basic equations

7.4.2.a Equations of elastic behavior

In elastic domain, the mechanical behavior is modeled by the Hooke law. As the shear stress and strain components are neglected and the other components are functions of x only, the Hooke law is written as follows

                 σ xx (x) = E (1 + ν)(1 -2ν) (1 -ν)ǫ xx (x) + νǫ yy + νǫ zz (h C , x) σ yy (x) = E (1 + ν)(1 -2ν) νǫ xx (x) + (1 -ν)ǫ yy + νǫ zz (h C , x) σ zz (x) = E (1 + ν)(1 -2ν) νǫ xx (x) + νǫ yy + (1 -ν)ǫ zz (h C , x) (7.27)
where E s and ν are the Young modulus and Poisson coefficient of strip material and ǫ are the elastic strain within CD.

By subtracting the second to the third equation of 7.27, we obtain And by eliminating ǫ xx from the first and third equation of 7.27 we deduce that

σ yy (x) -σ zz (x) = E 1 + ν ǫ yy (x) + νǫ zz (x) . ( 7
σ zz (x) = ν 1 + ν σ xx (x) + E 1 -ν 2 νǫ yy (x) + ǫ zz (x) .
(7.29)

7.4.2.b Boundary conditions at C

As the stress at C satisfies at the same time condition of plastic plane strain deformation as well as Von-Mises criterium like the rigid-plastic slab method, it can be given by 2.24, meaning:

σ C =     T C 0 0 0 T C - σ C 0 √ 3 0 0 0 T C - 2σ C 0 √ 3     . (7.30)
where T C denotes the longitudinal stress at C, i.e σ C xx = T C . We deduce from 7.30 that

σ C yy = T C - σ C 0 √ 3 (7.31) σ C zz = T C -2 σ C 0 √ 3 . (7.32)
As C is the limit between the plastic and elastic deformation areas, the stress at C satisfies also the elastic behavior equations, in particular 7.28 and 7.29. By substituting 7.31 and 7.32 into 7.28 we obtain:

ǫ yy = ǫ C zz (h C ) + 1 + ν √ 3 σ C 0 E (7.33)
allowing to determine ǫ yy as a function of h C and only h C : ǫ yy (h C ).

Similarly, by substituting 7.31 into 7.29 we can calculate T C as a function of h C as follows

T C (h C ) = E (1 + ν)(1 -2ν) ǫ yy (h C ) + νǫ C zz (h C ) + 1 -ν 1 -2ν σ C 0 E . ( 7
.34)

7.4.2.c Equilibrium equations within CD -Slab method

The equilibrium equations of a slab in x and z directions are respectively given by 2.27 and 2.28 (see details in subsection 2.3.1.a). To make it easier for the readers, these equations are rewritten here as

• In x direction : d dx (h(x)σ xx (x)) = -σ n (x)h ′ (x) -τ(x) . (7.35)
• In z direction : .36) In this spring back area the strip is elastically deformed and the friction is modeled by Coulomb model. It is important to note that the longitudinal velocity of the strip is usually greater than the roll velocity in this area. The exceptions cases are those where the neutral point is very closed to the point C. In these cases there are two neutral points, one within BC and the other within CD. In this thesis, these exceptions are not studied because of several reasons. Firstly, these cases are not frequent in industrial rolling because they are closed to the stability limit -neutral point is at C or does not exist. Secondly, these cases can be treated in a similar way as the case without neutral point in CD which will be studied hereafter because the position of the neutral points can be determined by the flow conservation equation. We assume therefore, that there is no neutral point in the spring back area and the strip velocity is higher than that of the roll. The friction acted on the strip is, by consequence given by 2.30 with negative sign: τ = -µσ n .

σ zz (x) = -σ n (x) + τ(x)h ′ (x) . ( 7
(7.37)

By eliminating τ(x) and σ n (x) from three equations 7.35, 7.36 and 7.37 we obtain :

d dx σ xx (x) = 1 h(x) h ′ (x) -µ 1 + h ′ (x)µ σ zz (x) -h ′ (x)σ xx (x) . (7.38)
Finally, by combining the equilibrium equation 7.38 with 7.29, we obtain a differential equation allowing to determine the evolution of σ xx as a function of x: Obviously, the stress at D follows the elastic behavior equations. Thus, by substituting 7.41 and 7.42 into 7.29 we obtain

d dx σ xx (x) + A(x)σ xx (x) = B(h C , x) . (7.39) with A(x) = 1 h -ν 1 -ν h ′ -µ 1 + h ′ µ + h ′ B(h C , x) = 1 h h ′ -µ 1 + h ′ µ E 1 -ν 2 νǫ yy (h C ) + ǫ zz (h C , x) .
νǫ yy (h C ) + ǫ D zz (h C , x D ) = - ν(1 + ν) E T s (7.43)
where the deformation ǫ D zz (h C , x D ) is given by 7.26 for x = x D . 5. using finite difference method to determine σ xx (x = x D ) (x D is obtained in the previous step) from the differential equation 7.38 where A(x) and B(h C , x) are given by 7.40. The initial condition is at C where x = 0 and σ xx (x = 0) = T C determined in the step 3.

Resolution algorithm

6. calculate the difference between σ xx (h C , x = x D ) and T s Indeed, the difference between σ xx (h C , x = x D ) and T s determined by this way is an 1-variable function of h C . We choose, therefore Newton algorithm to determine h C so that this difference between σ xx (h C , x = x D ) and T s is equal to 0. The initial value of h C can be chosen as that of Bland & Ford model (given in the subsection 2.3.1.e).

7.4.3.b Determination of width variation

Once h C is obtained, repeat the first two operations mentioned above to determine ǫ yy (h C ). The width variations in spring back areas are then calculated by:

∆w CD = 0 ∆w thermo-elasto-plastic exit = ∆w DF = ǫ s yy -ǫ yy (h C ) w (7.44)
where ǫ s yy is yy component of the strain tensor given by 7.23.

A simplified model for roll-bite 7.5.1 Total width variation in roll-bite -rigid-plastic UBM

It should be recalled that thanks to two assumptions Independent lateral flow assumption 1 and 2 for elastic deformation and thermal dilation influences (presented in the section 7.2.2.f, the total material flow in the roll-bite (between B and C) of a thermal-elasto-plastic simulation is relatively similar to that of a rigid-plastic model. Moreover, keeping in mind that the edge drop phenomena is not considered in this thesis. In addition, we highlight once more that the strip flatness variation has important impact on the width variation. Our study is limited for the case where strip is flat is at both the entry and exit of the roll-bite (no flatness defect). The strip is thus deformed quite homogeneously across the width direction i.e the strip thickness reduction is homogeneous across the strip width. In other words, we can consider a rectangular strip at the entry and a straight under-charge profile of the roll (without deflexion).

In this condition, the rigid-plastic material flow is completely predicted by the model introduced in the section 5.3 using the simple or elliptical 3D velocity field. In this section, the comparison of UBM results obtained using two form of the width function (polynomial in x and polynomial in h(x)) shows that the results are very similar so we choose the polynomial in x for its simplicity. See all details of the model in 5.3.

In summary, the total width variation in the roll-bite is finally appreciated by

∆w thermo-elasto-plastic roll-bite ≃ ∆w rigid-plastic roll-bite = α (7.45)
where α is the rigid-plastic width variation that one of two parameters of the width function 5.40 determined by rigidplastic UBM model.

Elastic deformation -Stress approach

According to the assumption Independent lateral flow assumption 1 -influence of elastic deformation, the elastic deformation in the roll-bite induces a plastic deformation of a same amplitude but with an opposite sign. Indeed, it is the elastic strain variation between B and C that need to be determined. As a result of the entry compression model (see 7.3), the stress tensor is completely determined at B. It is given by 7.16. On the other hand, the stress tensor at C is 7. A thermal-elastic-plastic width model

∆w elastic roll-bite = w 1 -2ν E (T C -T e ) - 1 √ 3 σ C 0 -σ B 0 . (7.47)
In the particular case where the material behavior is modelled without work-hardening effect

∆w elastic roll-bite = w (1 -2ν) (T C -T e ) E . ( 7

Assumption of heat generation

Generally in cold rolling, strip is heated in the roll-bite due to the energy dissipated by plastic deformation and friction between the work-roll and the strip. We assume that these energies are completely transformed into heat and at the contact the heat generated by friction is shared equally to the roll and the strip.

Adiabatic thermal process in roll-bite

As a reminder, the assumption Independent lateral flow assumption 2 -thermal dilatation influence (see 7.2.2.f)states that the thermal dilatation of the strip in the roll-bite induces plastic deformation of a same amplitude but with an opposite sign. That means it is the strip temperature variation inside the roll-bite that is important but not outside the roll-bite. On the other hand, inside the roll-bite the only heat exchange is due to the contact with the roll. The heat transfer coefficient between the strip and the roll in cold rolling condition (with lubrication) is approximated of about htc = 2.10 4 w/(m 2 K). However, as the rolling speed is quite high (>6000mm/s) while the contact length is relatively small (<20mm), let assume that the exchanged heat is negligible in comparison to that generated by plastic deformation and friction.

In order to verify this assumption, consider now the Lam3-Tec3 simulations with and without heat exchange between the strip and the roll. The simulations are performed for the same rolling conditions given in Table 7.5. Table 7.6 shows the strip temperature variation from the roll-bite entry to the exit. The results obtained with or without heat exchange with the work-roll are very closed for all the four stands conditions. The adiabatic assumption leads to an error on ∆T lower than 1°K which corresponds to an relative error lower than 1% for stand 1, 2 and 4. For stand 4, as the temperature variation is quite low because of a small reduction in thickness. These simulations conclude, hence in typical automotive products cold rolling conditions, the adiabatic assumption is valid.

Remark: Unlike cold rolling, in hot rolling conditions (roughing and finishing mills) the strip should be considered in isotherm condition. Indeed, the strip temperature (∼1000°C in roughing mill and about ∼900°C for finishing mill) is much higher than the that of the roll (∼80°C), the strip-to-roll transferred heat is important and usually equivalent to the heat generated by plastic deformation and friction dissipation.

7.5.3.b Simplified thermal model

The rigid-plastic model (5.3) used to estimate the width variation in the roll-bit is, as a reminder based consists in searching for the optimum velocity field by minimizing the rolling power. It does not only give an estimation of width variation along the roll-bite thanks to the velocity field but also a very good approach of plastic deformation, friction and tensions powers. Therefore, the strip temperature variation in the roll-bite can be determined based on powers given by the rigid-plastic UBM by considering the assumptions described above:

∆T = γ J de f + c share J f ric C vol ρC p (7.49)
where J de f is the plastic deformation power. J f ric is the friction power which is shared at a proportion c share to the strip and 1c share to the roll. C vol is the volumic flow rate which is equally given by the rigid-plastic model, ρ is density and C p is the masse specific heat capacity of the strip. It is worth to remind that the plastic deformation power J de f is actually the sum of two terms, volumetric continuous deformation power J ǫ and discontinuity of velocity one J ∆u . The first term J ǫ corresponds to the plastic deformation dissipation through the roll-bite area (between B and C) and the second one J ∆u refers to the plastic shearing deformation at the entry of the roll-bite (area between A and B). And γ is a coefficient that we introduce to model the thermal conditions. Typically γ = 1 for automotive cold rolling condition where we consider that 100% of dissipation transformed into heat and the strip thermal condition is adiabatic. γ ≃ 0 for hot rolling where the strip is considered in isotherm condition.

Finally, the thermal width dilatation in the roll-bite can be determined thanks to 7.8 with ∆T given by 7.

Remark:

The friction in elastic spring back area (between C and D) contributes actually to the strip temperature variation. However, as the the spring back contact length (BC) is relatively small in comparison to the total contact length and as the contact pressure decreases down to 0 at D, the average friction stress is also lower. Furthermore, the difference of velocity between the strip and the roll is much smaller than that inside the roll-bite. The friction dissipation in spring back area is eventually negligible in comparison to that inside the roll-bite and especially to the plastic deformation dissipation.

Summary

Bibliography: In the first part of the chapter, a bibliographic study shows that in industrial automotive rolling (flat and large strip) the elastic and thermal deformations have important impact on the final width variation. However, there exist very few width variation models that are applicable for this kind of rolling process. Legrand [START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] and Counhaye [START_REF] Counhaye | Modélisation et contrôle industriel de la géométrie des aciers laminés à froid[END_REF] are the only two existing model in our knowledge. One is based on Lam3-Tec3 and the other on "streamline" finite difference method. Both requires high computing time. Moreover, these models state both that the total width variation is given by the combination of three terms: edge drop, thermal contraction and elastic spring back (equation 7.1). The two studies give important base for the studies of width variation in automotive cold rolling process. However, the 7. A thermal-elastic-plastic width model 7.6 Summary explanations are quite general. Indeed, the total width variation term is not explicitly defined. It may be understood as the total elasto-plastic width variation for a thermo-elastoc-plastic model without the thermal term? It may be referred as the plastic one but in this case the total elastic width variation needs to be extracted or not? Moreover, this formula seems to implicitly suppose that at the entry of the roll-bite? The width variation due to the lateral flow in the roll-bite seems to be neglected equally? New width variation formula: These are the motivations to go more in details. This chapter presents, thus the new model (see section 7.2) based on some assumptions adequately chosen thanks to an analysis of Lam3-Tec3 simulations. The most important are two assumptions on lateral flow: Independent lateral flow assumptions 1elasticity influence and 2 -thermal dilation influence (see 7.2.2.f). These assumptions lead to two basic formula 7.11 for the plastic width variation which is the permanent width change.

Simplified model for the roll-bite entry: By assuming a homogeneous stress in thickness and by writing the Von Mises plastic criteria at A, the stress tensor at this point is entirely determined and given by 7.15. Further, the plane strain deformation between A and B assumption leads to rewrite the plastic width variation formula 7.11 by another 8.2.2 that states that the plastic width variation is only impacted by the elastic and thermal deformation between A and C.

Simplified model for the roll-bite exit: is developed based on a slab method, the four assumptions: plane strain deformation within CD, the friction is negative on this segment, the last point of plastic deformation C coincides to the lowest point of the work-roll and the vertical stress is totally relaxed at the last point of contact D. This model allows to approach the stress at C by 7.30 with T C determined at the same time as h C by an iterative algorithm 7.4.3. Using 7.15 and 7.30 the elastic width variation within AC -the second term of the right hand side of 8.2.2 is obtained by

∆w elastic AC = ǫ elatic-C yy -ǫ elatic-A yy w = 1 -2ν E (T C -k C ) + ν E T e w. ( 7.51)
Simplified model for the roll-bite: As a reminder, the first term of the right hand side of the equation 8.2.2, ∆w rigid-plastic represents indeed the total width variation of the thermo-elasto-plastic strip between B and C that is approximately estimated by the width variation of a rigid-plastic one. This term can be, thus determined by the rigidplastic UBM developed in the section 5.3. The boundary conditions (stress tensor) at B and C are given by the roll-bite entry and exit models instead of entry and exit tensions initially imposed. In other words, the roll-bite entry and exit model do not only allow to estimate the elastic term by 7.51 but also boundary conditions for the roll-bite model. Moreover, it should be recalled that the UBM is based on an optimization of the rolling power, the plastic deformation and friction dissipation powers are also determined by this model. In addition, for cold rolling processes, as the contact time is small it is usually reasonable to assume that the rolled strip is in adiabatic condition under the roll-bite. This assumption allows to evaluate the increase of strip temperature and therefore the thermal width variation term ∆w thermal total of the equation 8.2.2. The width variation model is hence completed.

Let call this simplified thermo-elasto-plastic width variation model the UBM-Slab combined model.

Chapter 8

The UBM-Slab combined model validation

The previous chapter presents the new width variation model for a thermal-elastoplastic strip in rolling process which is called UBM-Slab combined model. The programming of this model is detailed in the first section of the present chapter. The results show a very interesting computing time -less than 0.05s enabling online applications including dynamic control. In the second section, a comparison of the width variation obtained by the UBM-Slab combined model with those obtained by Lam3-Tec3 for the four stands of Florange cold rolling mill. It is observed a good agreement between the two models. The total plastic width variations obtained with the two models are very closed. Finally, a parametric study using the UBM-Slab combined model is done and shows clearly how the width variation depends on the rolling. It is important to highlight that these dependencies are very different for narrow and large strips. And once more, the results match really well the tendencies observed in industrial data presented by some studies existing in literature.

Contents

Simplified model algorithm and programming

Algorithm

As can be noted from the previous sections, the entry compression as well as the exit elastic spring back models are completely independent from each other and from the roll-bite one. They are calculated first. In contrast, the roll-bite model, between B and C, has actually limits conditions which are outputs of the two other models. Indeed, following the simplified entry model the longitudinal stress at B is always equal to the entry tension T e thanks to the fact that the strip is deformed before the contact. Otherwise, this stress can be different from T e . And at the exit, the tension at C, an output of the exit elastic spring back model T C is an input for both roll-bite total width variation model (rigid-plastic UBM ) and roll-bite elastic width variation one. The algorithm of the whole model is therefore illustrated in Figure 8 The entry model is simple and the plastic width variation ∆w plastic entry is calculated directly using 7.19. Concerning the elastic spring back model, the algorithm is described in the section 7.4.3. The determination of h C is done by Newton method where the target function is the difference between the longitudinal stress at D and the exit tension σ D xx -T s = 0. We obtain finally h C , T C , ∆w elastic exit and evolution of stress fields throughout CD.

As for the roll-bite, the elastic model is simple. The roll-bite elastic width variation ∆w elastic roll-bite is resulted from σ B and σ C outputs of the entry compression and elastic spring back models respectively. The rigid-plastic UBM for the total width variation in the roll-bite is feed by h C and T C given by elastic spring back model as inputs instead of h S and T s . Some outputs of UBM model, J de f and J f ric are used by the thermal model to determine the temperature variation ∆T and the thermal width variation ∆w thermal . Finally, all the width variation terms in entry, roll-bite and exit obtained by these models becomes inputs of the width model presented in the section 7.2 to determine the total and plastic width variations ∆w total and ∆w plastic total .

C++ programming and fast computing time

The model is entirely programmed in C++. Most of the models described above such as entry compression, rollbite elastic, thermal models are analytical and require, hence almost no calculation time. The exit elastic spring back contain iterative calculation but the convergence is obtained generally after 2 or 3 iterations. It is also very fast, less than 1.10 -3 s. The roll-bite UBM require obviously a computing time the most important due to the calculation of powers and iterative calculation to minimize the rolling power. Indeed, thanks to the fact that the calculation of all power functions are obtained analytically (see section 5.3.4). In addition, using Gauss's method to approximate some integrals as the 2D integral of deformation power and the 1D integral of friction power, help to reduce effectively the number of points where the integrated functions need to be calculated. Moreover, the rolling power optimization by Newton-Raphson algorithm requires few iterations, generally 2 for very large strip (as the width variation is small and easy to be found) and 3 or 4 for narrow one. The computing time is finally very fast. The total computing time for Florange 4-stand mill conditions varies from 0.01s to 0.05s (CPU: Intel Core I5-4200M, 250GHz). Table 8.1 shows in details the results obtained by the UBM-Slab combined simplified model for the 4 stands of Florange cold rolling mill conditions given in Table 7

Validation by comparison with Lam3-Tec3

Very good prediction of final plastic width variation

.2 gives the profile of elastic, thermal, plastic and total width variation obtained with Lam3-Tec3 for the four stands. For first three stands the total (thermo-elasto-plastic) width variation is relatively small while the thermal one is important. That leads finally to a large negative plastic width variation (width necking). The stand 4 with particular working conditions (very low reduction) is an exception. The plastic deformation is positive meaning a width spread. All these width variations due to elastic, thermal, plastic and total deformations are summed up in Table 8.2 in comparison with those obtained by the simplified model. It can be seen very good agreement between Lam3-Tec3 and the simplified model regarding all these width variations terms. As the plastic one is the real final width variation of the strip, let take a look on a comparison of this term between Lam3-Tec3 and the simplified model through graphic 8.3. The four points of the graphic corresponding to four stands are almost perfectly on the bisector line. ) estimated by the new model is generally similar to Lam3-Tec3 except for the stand 4 because with very small reduction level the slab assumption is not well verified. The elastic spring back model is, hence relatively accurate when the reduction is important enough (as stands 1, 2 and 3 for example). The most important error comes from the total width variation between AC, estimated by rigid-plastic UBM model. This model gives always underestimated value in comparison with Lam3-Tec3 .

The simplified model can be improved?

As conclusion although the strip plastic width variation predicted by the simplified model is in a very good agreement with Lam3-Tec3 there exist some further potential improvements. The entry compression model can be improved by better estimating the stress at C. The elastic spring back model is good when the reduction level is high enough. The rigid-plastic model provides an accurate prediction of dissipation powers and therefore of thermal width variation. But it underestimates the total width variation in the roll-bite. In this section the new model will be used to evaluate effect of each rolling parameter on the width variation for an industrial rolling condition in order to compare to the observations already presented in literature (see the section 7.1.2). These previous studies are done on the total width variation from the entry to exit of tandem mill, meaning through 4 or 5 stands. The reason is that nowadays to install in a good industrial condition the strip width measurement devise between the stands is impossible. Therefore, our analysis consisting in studying how width variation depends on each rolling parameter will be only interpreted relatively but the absolute value of width is an objective.

The UBM-Slab combined model validation

We choose to study the stand 1 conditions (given in Table 7.5) because the stands 2 and 3 are with very low friction (the forward slip is already close to 0 and the neutral point is closed to the lowest point C). By consequence, when one of some other inputs varies the neutral point can go out of the roll-bite and there is no equilibrium solution. The model does not converge. Discussions: At first, that seem obvious that the entry and exit tensions do not impact significantly the dissipation power which depends essentially on the strip yield stress and the reduction. That is why the thermal width variation is almost unchanged with a variation of tensions. Moreover, as the rigid-plastic UBM width variation is negligible (∼ 0) because the strip is very large, it is almost independent on all input parameters (except the strip width). According to the formula of the plastic width variation 8.2.2 the only term that changes as a function of entry and exit tensions is ∆w elastic AC . The equation 7.51 implies that when T e increases the ∆w elastic AC increases and leads to a decrease of plastic width variation. When T s increases, T C grows and therefore the w elastic AC increases equally and the plastic width variation drops down. As 1 -2ν > ν, the exit tension seems to have more important impact on the width than the entry one.

8.3.1.a Effect of tensions

8.3.1.b Effect of reduction

It can be seen from the 8.6 that the effect of reduction (for a same entry thickness) on the width variation is very important: the more reduction the more necking. The rigid-plastic width variation (representing total width variation in the roll-bite is always null approximately and the elastic width variation between A and C is quit constant that can be explained by the equation 7.51. And it is the thermal width variation that changes shapely with an increase in reduction because the plastic deformation as well as the friction powers increase as a function of the reduction.

8.3.1.c Effect of nominal strip width

Unlike the others graphic, the 8.7 shows that the rigid-plastic width variation term changes when the nominal width varies. This term ∆w rigid-plastic roll-bite is important and even dominant against thermal and elastic terms ∆w thermal total and ∆w elastic AC when the width is small (w e <50mm or the total width 2w e <100mm). This terms decreases drastically and becomes negligible as the half width is higher than about 300mm. That is why for the width of w e = 585mm of the Stand 1, this term is always almost nullified.

For automotive industrial rolling, the strip width varies from about 800mm to over 2000mm, meaning w e >400mm. For this range of width, the rigid-plastic term is closed to 0 and the thermal term increases while the elastic term decreases as linear functions of the width. In this case, as the thermal increases faster than the elastic term decreases the plastic width decreases meaning more necking. Nevertheless, in an other case (small reduction for example) it is possible that the thermal term is smaller than the elastic one in absolute value, the plastic width variation is then positive and it would increases with an increase in the nominal with. This tendency is perfectly coherent with the industrial observations presented in the section 7.1.2.e.

The UBM-Slab combined model validation

8.3.1.d Effect of strip yield stress and friction coefficient

According to the equation 7.51, when the yield stress increases the elastic term ∆w elastic AC decreases linearly. That is also showed in Figure 8.8. In addition, as both plastic and friction dissipation powers are proportional to the strip yield stress the thermal width variation term grows also linearly as a function of the yield stress. In the stand 1 rolling condition, as the rigid-plastic width variation is negligible and the thermal term is more important than the elastic one, the plastic width variation is negative and decreases (more necking) for harder material. This dependence of the width variation on the yield stress is, however opposite to the industrial data observation presented in the section 7.1.2.f. As can be seen from the 8.9 that the friction has similar but less important influence on all the width variations terms as the yield stress. Firstly, the thermal terms increases because the friction power increases almost linearly in the friction coefficient. However as the plastic dissipation is unchanged unlike the case when the yield stress varies, the impact of friction coefficient on the thermal term is thus weaker. Secondly, it is important to mention that the simulations are done by varying not only the Tresca friction coefficient m b in the roll-bite but also the Coulomb friction in the elastic spring back area µ which is changed proportionally to the Tresca one. As the difference of longitudinal stress between C and D is proportional to the contact friction stress on this segment, i.e T s -T C increases with an increases in µ. By consequence, the tension T C decreases a function of friction coefficient and that leads, following 7.51, to a decrease of the elastic width variation term. Once more, as showed in Figure 8.9 the increase of the thermal term is more important than the decrease of the elastic one, the plastic width variation decreases in incurring more necking. Figures 8.10 and 8.11 shows that the entry and work-roll diameter have very small impact on the width variation. For classic automotive rolling mills, the work-roll diameter can vary between about 400mm to 600mm. In this range of variation, the work-roll diameter has negligible influence on the width variation.

8.3.1.e Effect of strip thickness and work-roll diameter

Similarly for the strip thickness, for automotive rolling condition the strip entry thickness of one stand can vary from about 0.4mm (at last or before last stands) to 6mm (at first stand), meaning h e varies from 0.2mm to 3mm. The impact of entry thickness is also negligible. This conclusion matches well what was observed by industrial data analyses (see section 7.1.2.f). However, it can be seen in Figure 8.10 there is an increase of the width necking when the strip thickness decreases down to 0.2mm. This range of strip thickness correspond to that of packaging rolling mills last stands where the half strip thickness can be down to less than 0.09mm. In this condition, the lower the strip thickness, the more and more important the ratio contact area over roll-bite volume. By consequence the friction dissipation power may become more and more important in comparison to the plastic one and may lead to an increase in thermal width variation. The strip necking could be significantly more for thin and very thin strips.

Parametric study for a narrow strip rolling -E16 trial

This subsection presents a similar parametric study of the width variation using the simplified model but a narrow strip. The rolling conditions are those of the trial E16 done on ArcelorMittal pilot mill with a strip of 60.2mm wide (see all parameters in Table C.2). The results of the UBM-Slab combined model for this case are presented in Table 8.1. Unlike the industrial strip rolling, for E16 pilot rolling trial the rigid-plastic width variation ∆w rigid-plastic = 0.1116mm is not negligible and even more important than the elastic and thermal terms ∆w elastic AC = -0.0294mm and ∆w thermal total = 0.0523mm. The final plastic width variation is, in this case positive ∆w plastic total = 0.0887mm, i.e the strip is widened.

8.3.2.a Effect of different parameters on width variation -narrow strip

Figures numbered from 8.12 to 8.19 present effect of different parameters on the plastic width variation for this pilot trial rolling condition. The results are summed up in comparison to those for industrial rolling conditions as follows:

• Similar tensions effect meaning the the width variation decreases with an increases in entry or exit tensions. In this case of narrow strip, in addition to the fact that the elastic term ∆w AC elastic increases, the rigid-plastic one ∆w rigid-plastic decreases as a function of tensions amplifying the decrease of the plastic width variation.

• Figure 8.14 shows that, like for Stand 1 condition, the thermal width variation term increases rapidly as the reduction grows while elastic term is remained constant. However, for a narrow strip the rigid-plastic term is no longer negligible and increases even more shapely as a function of reduction leading to an increase of the plastic width variation. The effect of reduction is therefore opposite (i.e width variation grows up while it decreases in the case of large strip). As the width variation is positive for narrow strip and negative for large strip, the reduction amplify the width variation value in both cases.

• Same nominal width effect. Despite of the difference in reduction, thickness, tensions... the effect of nominal width on the plastic width variation in the case of E16 (Figure 8.15) is very similar to the case of Stand 1 (Figure 8.7).

• Same as the case of large strip, an increase in yield stress or in friction coefficient make grow up the thermal width variation term (because of an increase in dissipation power) and diminish the elastic one (see equation 7.51). But as the increase of thermal term is equivalent to the decrease of the elastic one and the rigid-plastic term is not sensible as a function of yield stress and friction coefficient, the plastic width variation is almost unchanged as a function of yield stress or friction. Whilst for large strip, the thermal term is more important than the elastic one, the plastic width variation decreases with an increase in yield stress or friction coefficient.

• More important effect of strip thickness and roll diameter. Indeed, similar to the case of large strip both thermal and elastic width variation terms are almost constant when the strip thickness (for a same reduction level) or roll-diameter vary. But contrary to the case of large strip where the rigid-plastic term is not negligible any more and increases as a function of strip thickness and roll-diameter. That lead to an increase of the plastic width variation as a function of these two parameters.

Summary of parametric studies

Parameter Nomenclature

Industrial observation [ Table 8.4 sums up the existing industrial experiments and statistical observations about effect of rolling parameters on the width variation (see more details in the previous chapter, 7.1.2) as well as the results of parametric studies using the new simplified model for two rolling cases with a large and a narrow strip. The dependence of width variation on rolling parameters obtained by the UBM-Slab combined model is in a very good agreement with that observed statistically on industrial data. For narrow strip, the plastic width variation is positive (widening) and the effects of rolling parameters on the plastic width variation are relatively different. Therefore, the influence of rolling parameters on the width variation

Conclusions

Fast computing time enables online applications: As the model for roll-bite entry is completely analytical and the exit one is quasi-analytical requiring only one computation loop when searching h C (see the algorithm in section 7.4.3.a) the computing time of these models are very small (less than a millisecond programmed in C++ code). The main time consuming factor is related to the roll-bite model which is indeed the rigid-plastic UBM . Nevertheless, as the powers computation is developed analytically as far as it can be (see section 5.3.4) and the integrals in powers function are computed numerically using Gauss's method, the final computing time (in C++) is less than 0.05s. That enables online applications such as preset and dynamic control of the width.

Good prediction of plastic width variation:

The comparison of the plastic width variation obtained by the UBM-Slab combined model with that obtained by Lam3-Tec3 for the four stands of Florange cold rolling mill shows a good agreement between the two models. The difference of plastic width variations between the two models is less than 6% of for stands 1, 2 and 3 about 10% for the last stand (which only makes very small reduction in strip thickness).

Good prediction of influence of rolling parameter on width variation:

Parametric study is done using the UBM-Slab combined model for two rolling conditions, one with large strip (stand 1 condition) and the other with narrow strip (pilot trial E16 condition). The model is able to predict clearly how the width variation depends on the rolling in each case. In the case of industrial condition (stand 1) with large strip, except the yield stress effect, the results match really well the tendencies observed in industrial data presented by some studies existing in literature. It is interesting to highlight that these dependencies are very different for narrow and large strips.

Moreover, as the UBM-Slab combined model is able to explain the contribution of each phenomena involved into the plastic width variation (see formula 8.2.2): the total width variation in the roll-bite, the elastic width variation between the first and last points of plastic deformation A and C. It allows to understand how and how much each of these three terms varies as function of each rolling parameter.

Key improvement of the model is to take into account influence of flatness: The limit of the UBM-Slab combined model is that it considers only a straight (non-deflexion) work-roll. The flatness of the strip is by consequence not considered. Whilst, the literature highlighted that the flatness (strip thickness profile) has important influence on the strip width variation. This is also confirmed by the study of a crown strip by UBM in the chapter 6. It is necessary to remind that the UBM-Slab combined model is valid when the rolling condition allows to obtain good strip flatness. In industrial reality, this condition that is desired but not always obtained. Hence, taking into account this phenomenon is a future improvement of the model open a very high potential opening largely the application domain of the model.

Chapter 9

General conclusions and perspectives

Conclusions

The development of predictive width variation model for automotive cold rolling is achieved

In the current industrial context where the width variation in cold rolling is important but not predicted, the main objective of the present thesis is to develop a predictive width variation model for automotive cold rolling process. Such a model need to be accurate and rapid to be used in real-time process control. For bar rolling process, the width variation topic has been studied largely and an important number of developed models were based on FEM , UBM as well as empirical methods. The UBM provides a good comprise between the accuracy and computing time. Nevertheless, in automotive rolling condition, the strip is large and the elastic width variation is no longer negligible. This elastic deformation is reversible but it has important impact on the plastic one. Moreover, due to the friction and plastic deformation powers the strip is heated up significantly. It is, therefore dilated in the width direction but it can not because of the contact friction with the roll. That creates compression plastic deformation -called thermal contraction. The modelling of width variation becomes more complicated and requires thermo-mechanical understanding. In this domain, there exist very few models. They are based on FEM and stream lines FDM and by consequence have important computing time.

We have, in the present thesis, carried out a deep analysis based on Lam3-Tec3 simulations and brought out all the phenomena involved in the width variation. Since the analysis, the most important simplifying assumptions have been found. According to these assumptions, the elastic deformation as well as the thermal dilatation of the strip in the roll-bite create a plastic deformation of a same amplitude but with an opposite sign. In other words, the total width variation in the roll-bite of a thermo-elasto-plastic strip is the same as that of a rigid-plastic one. By consequence, we proposed a formula that computes the plastic width variation as a combination of three terms: the total width variation in the roll-bite, the elastic and thermal width variations between the first and last points of plastic deformation. In order to determine these three terms, we developed simplified models for the entry, exit and inside the roll-bite.

The simplified models for the entry and exit of the roll-bite are based on the assumption of a homogeneous stress and deformation across the strip thickness (slab method). They give approximations of the stress solution before and after the roll-bite allowing to determine the elastic width variation term between the first and the last points of the plastic deformation zone. As for the roll-bite model, it is the rigid-plastic UBM with 3D "simple" velocity field. The boundary conditions (longitudinal stress tensor) at the roll-bite entry and exit are given by the roll-bite entry and exit models instead of entry and exit tensions initially imposed. In addition, as the model allows to determine the plastic deformation and friction dissipation powers, the increase of strip temperature and the thermal width variation term can be computed. The simplified width variation model is thus completed and called the UBM-Slab combined model.

A comparison has been performed and showed a very good agreement between the UBM-Slab combined model and Lam3-Tec3 . The difference of the total plastic width variations obtained with the two models is less than 6% for a stand that does relatively high reduction and 10% for a stand with very small reduction. The UBM-Slab combined model allows predicting the influence of rolling parameters on the final width variation and the results match really 9. General conclusions and perspectives 9.2 Perspectives well the tendencies observed in industrial data presented by some studies existing in the literature. Furthermore, as the model for roll-bite entry is completely analytical and the exit one is quasi-analytical, the main computing time is related to the roll-bite model -the rigid-plastic UBM . Thanks to the analytical development of the powers computation the total computing time of the width variation model (in C++) is less than 0.05s (CPU: Intel Core I5-4200M, 250GHz) enabling online applications such as preset or dynamic control.

The UBM is always efficient method to develop rapid model for rolling process

The UBM formulated by [START_REF] Prager | Theory of perfectly plastic solids[END_REF] have been for a very long period a method that was largely used to obtain approximate solutions strip drawing, extrusion, forging, rolling, drawing, cutting processes. Indeed, the method require to preassume a velocity field pattern (family) that can be described thanks to a certain number of parameters which are as well the unknowns of the optimization problem. In comparison to FEM where the number of unknowns depends in the freedom degree and number of elements which is usually very important, the UBM is thus advantageous. The quality of the UBM results depends, therefore strongly on the choice and the construction of velocity fields. During the second haft of 20th century, there were studied many rigid bodies motion velocity fields such as unitriangular and multitriangular. Several other continuous velocity fields have been as well proposed as the "eccentric", elliptical or "simple" and circular ones. But indeed, all these mentioned continuous velocity fields are closed each from the other. That makes an impression that despite the existence of sophisticated mathematical methods (Dual Stream Function...) for constructing kinematically admissible velocity fields, it is still difficult to go further than the simple velocity field to describe more deeply the mechanical fields in rolling process. Nevertheless, throughout the thesis we proved that the UBM remains powerful to approach the rolling process if we based on a FEM to better understand the velocity field behavior.

The first example is the 2D oscillation velocity field that allows to take into account the heterogeneity of the velocity, strain rate fields across the strip thickness, a domain very little investigated. We presented a method for constructing kinematically admissible velocity fields based on the DSF method where any kinematically admissible velocity field is expressed as a sum of the "simple" (or elliptical) one and an additional term. By observing that the equations of kinematically admissible conditions of the additional term are closely similar to the wave propagation ones, we proposed a new family of "oscillating" velocity fields. And the UBM using this new velocity family results to an optimum velocity that oscillates spatially throughout the roll-bite with pseudo-period equal to the local strip thickness. The rolling power obtained is smaller than the one with the "simple" (elliptical) velocity field. The results of this model match very well those obtained by Lam3-Tec3 in terms of velocity field, plastic deformation zone and flow lines. As a result of the UBM model as well as Lam3-Tec3 , the mechanical fields heterogeneity is non-linear, quasi-sinusoidal across the strip thickness.

The second example illustrating the interest of the UBM for rapid approach of rolling is the 3D width variation model for a crown strip. We developed a new UBM approach for cold rolling where the strip initial thickness has non-constant profile while the work-roll is considered rigid and perfectly cylindric. As the geometry of the strip is more complex than the case of flat strip rolling, the roll bite is divided into three areas in which the velocity field is different. As a result, the model shows that the width variation decreases with an increase in the strip initial crown and the UBM results match very well those obtained with Lam3-Tec3 .

Perspectives

Industrial applications

Having now a rapid model of width variation, it is possible to apply it to predefine the necessary width at the entry of the cold rolling mills that would give the customer desired width at the exit -specification. The only necessary thing is to build a good database which requires two width measurements at the entry and the exit of the concerned tandem mill. In addition, the temperature and the tension need to be measured the same places (as closed as possible to the width measurement places) allowing to quantify the real plastic width variation of the strip. The off-line and online collection of data database will allow to tune and correct (online adaptation) the model in order to get and keep a good 9. General conclusions and perspectives 9.2 Perspectives predictive performance during time. An analysis of the error of the predictive model needs to be done to define a overwidth strategy that ensure the minimum total over-cost due to over-width for the most of cases and the under-width for the few remaining cases. Furthermore, as a result the model is fast enough to be used for online control. Nevertheless many works need accomplishing. The choice of control parameters requires not only an analysis of the efficiency of all process parameters (similar to the parametric study done in the section 8.3) as well as an analysis of all industrial constrains concerning each parameter. The control strategy should also be suitable to all other existing controls.

Other proposals of velocity fields for 3D UBM analysis 1

Polynomial velocity fields family: The new method for construction the kinematically admissible velocity fields presented in the section 4.2 opens actually divers solutions of kinematically admissible velocity fields for rolling. In addition to the oscillating velocity field, another family has been introduced in the section 4.2.2 -the polynomial one. This velocity family would allow especially to have higher strip speed on the contact before the neutral point and lower strip speed on the contact afterward. That would reduce the friction power and may lead to interesting results in term of rolling power optimization.

Perturbation of velocity field around the neutral point:

The neutral point is the point where the contact shear stress (friction stress) is discontinuous which should create a discontinuity the strain rate. All the previous continuous velocity fields (eccentric, simple-elliptical and oscillating) are not able to model this phenomenon. In order to model that, it is necessary to separate the roll-bite into two areas with a discontinuity surface at the neutral point. On the other hand, it was observed that the UBM under-estimates the forward slip in comparison with Lam3-Tec3 although the neutral point obtained by UBM is quite closed to Lam3-Tec3 . Inversely, the unitriangular over-estimate it. This fact is not a random and can be explained by the existing of a neutral zone -sticking area (but not a point).

The oscillating velocity with advantage of low deformation power and the multitriangular with advantage of neutral zone modeling can be combined together to create the new one. The idea is to model the neutral zone as a rigid curvilinear triangular. This triangular rotates around the work-roll center with a same angular velocity. Before and after this neutral zone the velocity is modelled by the two oscillating ones. This idea is described in Figure 4.35.

3D UBM with deformed work-roll:

Similarly to the UBM model approaching the width variation for the crown strip with the straight work-roll, it is possible to develop another one for a rectangular strip and crown work-rolls. When the work-roll crown is positive, i.e the center diameter is more important than the work-roll ends, the strip center is, same as the case with positive strip crown and straight roll, in contact with the work-roll before the strip edges. The geometry problem would be treated in a resembling way.

A. Numerical Gauss-Legendre integration

A.2 Applications which gives exact expressions for integrating any 2n -1-order polynomial.

A.1.3 Values arguments and weighing factors for n-point Gaussian Rule

The table A.1 gives the coefficients and arguments given for n-point (with n=2, 3, 4, 5, 6) Gauss Quadrature Rule given for integrals I =

A.2 Applications

Gauss-Legendre integration formulas are extremely accurate to approximate the integrals. They are usually recommended when many integrals of a similar nature are to be evaluated. In this case, we can proceed as follows. Pick a few representative integrals, including some with the worst behavior that is likely to occur. Determine the number of sample points N that is needed to obtain the required accuracy. Then fix the value N, and use the Gauss-Legendre rule with N sample points for all the integrals.

There will be appeared in this thesis many integrals of a certain function on arbitrary bounds: where z * = z h .

I 1 =
We introduce an useful mathematic formula which is used frequently in our calculations : where

Q = 2 h ′ 2 h 2 + 2 ϕ ′ 2 ϕ 2 + 2 h ′ ϕ ′ hϕ f = 1 √ 2 ϕ ′′ - h ′ ϕ ′ h -2 ϕ ′ 2 ϕ b ϕ g = 1 √ 2 h ′′ - h ′ ϕ ′ ϕ -2 h ′ 2
h .

(B.5)

We calculate now the P x in using the formula B.1 for a = Q 2 + f 2 y * 2 et c = g :

P x = 2 b ϕ 1 0 1 0 Q 2 + f 2 y * 2 + g 2 z * 2 dz * dy * = b ϕ 1 0 Q 2 + f 2 y * 2 + g 2 + Q 2 + f 2 y * 2 g ln g + Q 2 + g 2 + f 2 y * 2 Q 2 + f 2 y * 2 dy *
For the first term in the integral, we could also apply the formula B.1 for a = Q 2 + g 2 et c = f . And for the second, we use two following formulas : We obtain finally

I 1 (a, b, c) = c 0 ln a 2 + x 2 + b dx = cln b + a 2 + c 2 -c + ln c + √ a 2 + c 2 a + 2 a 2 -b 2 arctan a -b a + b √ a 2 + c 2
J ǫ = σ 0 √ 6 C vol 0 -L P x (x)dx (B. 8
)
with The calculation is performed similarly to that of P x and we receive :

P x = b ϕ 2 3 Q 2 + f 2 + g 2 + Q 2 + 1 3 f 2 g Ln g + Q 2 + f 2 + g 2 Q 2 + f 2 + Q 2 + 1 3 g 2 f Ln f + Q 2 + f 2 + g 2 Q 2 + g 2 + 4Q 3 3 f g   arctan   Qg Q 2 + f 2 + f Q 2 + f 2 + g 2 + Q 2 + f 2   -arctan g Q 2 + g 2 + Q      B.
J ∆u = σ 0 √ 3 C vol    ϕ ′ 2 e + h ′ 2 e 3 + h
J f ric = mσ 0 C vol 2 √ 3 0 -L b ϕ √ 1 + h ′ 2 h   ϕ ′ 2 + J 2 + J 2 |ϕ ′ | Ln   |ϕ ′ | + ϕ ′ 2 + J 2 J     dx . (B.

C.2 Lam3-Tec3 modeling

In order to verify the validity of Lam3-Tec3 we model these trials with Lam3-Tec3 and compare the results with the measurement. For the soldur steel, the strips are rectangular i.e flat. The old modeling Lam3-Tec3 previously performed is credible. The results are presented in the tables C.2. However, for the DWI steel, they are crowned i.e their entry profiles are not rectangular. Therefore, we remodel these trails in taking into account the entry profile of the strips. We note that, the rheology of the steel is measured and the only unknown is the friction which can be isotropic or not.

In observing that the longitudinal friction coefficient µ x influence only the forward slip and the transversal one µ y has an important impact on the exit profile of the strips. Therefore, the µ x is obtained when it gives a good forward slip and the µ y is determined by comparison of the exit profile calculated by Lam3-Tec3 with the measured. We present here the entry and exit profile in modeling Lam3-Tec3 in compare with the measured ones for all the trials on the DWI steel.

In modeling Lam3-Tec3 , the entry profile is devised into two parts. At the centre of the strip, the thickness is modeled constant and equal to the average of the measured thickness in this part. While at the edge part the profile is complicated, coming from the operation of edge cutting, and therefore, we have to approach the measured points by a polynomial of degree 8 and use this polynomial in Lam3-Tec3 calculation. They are showed in the left hand side of the following figures.

We determine next the longitudinal friction coefficient µ x by the way mentioned previously in concerning that the friction is isotropic. The details are not necessary to be presented. Only the results obtained are given the tables C.1 and C.2. Then we vary the transversal friction coefficient µ y to obtain the best exit profile of the strips. The figures in the right hand side show the relative thickness, i.e the centre thickness is normalized to 1, calculated by Lam3-Tec3 with the best value of the µ y which give the best exit profiles in comparison with experiment. The final results on theses trials are presented in the table C.1.

Entry profile

1 . Width variation problematic in steel rolling 1

 11 Strip yield stress and shear yield stress σ n Contact normal pressure (positive value by convention) τ Contact shear stress -friction stress C vol Material volume flow rate L Contact length F Roll force by an unit of width (N/mm) f s Forward slip (%) Tq Roll torque by an unit of width (N.mm/mm=N.m/m) ξ, u Vector of displacement and vector of velocity σ Stress tensor ǫ, ǫ Strain and strain rate tensors J f ric Power consumed by friction J de f Power consumed by plastic deformation J ǫ Power consumed by plastic deformation in the continuous velocity zones J ∆u Power consumed in the surfaces of discontinuity of velocity J ten Power of entry and exit tensions J Total power Γ e , Γ s Surface of velocity discontinuity at roll-bit entry and exit x, y, z 3 Direction coordinates : longitudinal, lateral and vertical

Figure 1 . 1 :

 11 Figure 1.1: Blast furnace scheme and an approximated balance to obtain 1 ton of pig iron.

Figure 1 . 2 :

 12 Figure 1.2: Schema of BOF principle components. BOF allows refining C content of pig iron at high temperature using oxygen.

Figure 1 . 3 :

 13 Figure 1.3: Ellingham diagram: dependence of affinity for oxygen of different elements. Carbon affinity for oxygen increases with an increase in temperature while that of metals decreases.

3 )a

 3 stable oxide in slag. And the desulphurization is based two reactions S + (CaO) = (CaS) + O S + 2O = SO 2 .

Figure 1 . 4 :

 14 Figure 1.4: Illustrating the Electric Arc Furnace which uses scrap steel to produce pure steel.

1 .

 1 Figure 1.5: Continuous casting machine: 1-Ladle, 2-Tundish, 3-Mold, 4-Plasma torch, 5-Stopper, 6-Straight zone.

1. 2

 2 Figure 1.6: Schema of a Hot Strip Mill.

1 .Figure 1 . 7 :

 117 Figure 1.7: Different processes of a cold rolling plant.

Figure 1 . 8 :

 18 Figure 1.8: Positive bending forces are exerted to separate work-rolls ends.

Figure 1 . 9 :

 19 Figure 1.9: CVC rolls allow to control strip flatness.

Figure 1 . 10 :

 110 Figure 1.10: Typical batch annealing base.Figure 1.11: Typical continuous annealing line.

1. 3

 3 Width variation problematic in cold rolling process

Figure 1 . 13 :Figure 1 . 14 :

 113114 Figure 1.13: Width variation observed for 6000 coils rolled at ArcelorMittal Florange 4-stand cold rolling tandem.

Figure 1 . 15 :

 115 Figure 1.15: Schema of the determination of targeted width for different processes before manufacturing.

Figure 1 . 16 :

 116 Figure 1.16: Summary of width specification of HSM and cold plant.

Contents 2 . 1

 21 General description of rolling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.

Figure 2 . 4 :

 24 Figure 2.4: Illustration of the coupling of strip, work-roll and tribology models.

Figure 2 . 5 :

 25 Figure 2.5: Coulomb and viscous friction as a function of sliding speed (source [4]).

Figure 2 . 6 :

 26 Figure 2.6: Relation between friction stress and sliding speed according to Stribeck model (source [4]).

Figure 2 . 7 :

 27 Figure 2.7: Three regimes of cold rolling lubrication correspond to three strip surface qualities.

Figure 2 . 8 :

 28 Figure 2.8: 2D illustration of slab method for a general work-roll shape.

. 40 )

 40 This equation has following analytical solution: before neutral point: σ zz = -2kh.K e e -µ.H

Figure 2 . 10 :

 210 Figure 2.10: Cosse et al slab model for elasto-plastic strip behavior and non-circular work-roll deformation.

Figure 2 . 11 :

 211 Figure 2.11: Crossbow (left) and longbow (right) defects due to homogeneous residual stress in strip thickness.

Figure 2 .

 2 Figure 2.12: Work-roll flattening respectively in plane xz called non circular deformation along the roll-bite and yz called edge drop.

  n

Figure 2 . 13 :

 213 Figure 2.13: Plane-strain problem of a cylinder subjected to diametrically applied pressures over finite angle.

Figure 2 . 14 :

 214 Figure 2.14: The roll bite is divided into 7 zones: Entry elastic zone (A), entry plastic reduction zone (B), plastic contained without reduction & slipping (C), elastic without slipping (D), elastic with slipping (E), exit plastic reduction zone (F) and exit elastic unloading (G).

  Figure 2.15.

Figure 2 . 15 :

 215 Figure 2.15: In this figure, the roll bite entry is in the right side. According to Matsumoto, the roll bite is divided into 5 zones: Entry elastic (D), entry plastic (C), neutral zone without slip and without plastic deformation, exit plastic (B) and exit elastic (A).

Contents 3 . 1 55 3. 2 56 3. 2 . 1 62 3. 3 . 1

 3155256216231 Principle of the UBM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.1 Extremum principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.1.2 UBM applied to rolling process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Velocity fields with rigid bodies motions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Simulated sliding theory and experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2.2 Concept for analysis of linear and rotational velocity fields . . . . . . . . . . . . . . . . . . . 58 3.2.3 Unitriangular velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2.4 Multitriangular velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 2D continuous velocity fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Continuous eccentric velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.3.2 Continuous simple velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.3.3 Improved velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3.4 Circular stream lines velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3.5 Elliptical stream lines velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.4 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3. Upper Bound Method applied in rolling process 3.1 Principle of the UBM

Figure 3 .

 3 Figure 3.3: Test A, R = 10mm, 2h e = 2.5mm,2h s = 2.08mm, contact length is estimated L = R(2h e -2h s ) = 2.05mm. The test bar was recrystallized annealed 0.15% C mild steel.

Figure 3 . 4 :

 34 Figure 3.4: Test C, R = 23.84mm, 2h e = 3.44mm,2h s = 3.33mm, L = 1.62mm. The test bar was a recrystallization annealed 1% Cr, 1% Si steel.

Figure 3 . 5 :

 35 Figure 3.5: Test G, R = 100mm, 2h e = 4.86mm,2h s = 4.22mm, L = 8.00mm. The test bar was an annealed 0.15% C mild steel.

Figure 3 . 6 :

 36 Figure 3.6: Strip rolling -The unitriangular velocity field.

Figure 3 . 8 :

 38 Figure 3.8: The multitriangular velocity field for strip rolling.

Figure

  Figure 3.8 shows the application of the multitriangular velocity field to the process of strip rolling. Within the curvilinear triangles with apexes P i on the plane of symmetry, a rotational velocity field prevails. Other triangles with their bases on the plane of symmetry and apexes on the surface of the roll, bound the regions of linear motion

Figure 3 . 9 :

 39 Figure 3.9: Continuous eccentric velocity field.

Figure 3 . 10 :

 310 Figure 3.10: Continuous simple velocity field with homogenous deformation across the thickness. As h ′ (0) = 0 the velocity is continuous at the exit.

70 ) 3 .

 703 Figure 3.11: Continuous circular velocity field. The velocity is also continuous at the exit. Figure taken from [47].

Contents 4 . 1

 41 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Method for constructing velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.1 General admissible conditions of velocity field . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.3 An oscillating velocity field proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.4 UBM with the oscillating velocity field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4.1 New strategy for minimization of power function . . . . . . . . . . . . . . . . . . . . . . . . 79 4.4.2 Numerical resolution -programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.5 Comparison with Lam3-Tec3 and other UBM models . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5.1 Power and neutral point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.5.2 Plastic area -discontinuity surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5.3 Velocity isovalue surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 4.5.4 Oscillation of velocity field along the streamlines . . . . . . . . . . . . . . . . . . . . . . . . 89 4.6 Conclusions and perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.6.2 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4. Oscillation of mechanical fields in roll bite 4.1 Introduction

.49) 4 . 4 . 1 .

 441 b Numerical resolution strategy Thanks to 4.49, the first equation of 4.40 -the optimization of the power function with respect to x n -is now rewritten as

Figure 4 . 1 :

 41 Figure 4.1: The plastic area Ω is limited by the surfaces of velocity discontinuity Γ e , Γ s . This plastic deformation zone is divided into three zone Ω 1 , Ω 2 and Ω 3 .

Figure 4 . 2 :

 42 Figure 4.2: Zoom of Ω 1 , entry plastic zone.

  and 4.6) and cold rolling condition (Figures 4.7 and 4.8).

Figure 4 . 3 :

 43 Figure 4.3: Roughing Mill condition (case 1, Table4.1). The dependence of x 1 n and J 1 as functions of A e when A s = 0.

Figure 4 . 4 :

 44 Figure 4.4: Roughing Mill condition (case 1, Table4.1). The dependence of x 1 n and J 1 as functions of A s when A e = 0.

Figure 4 . 5 :

 45 Figure 4.5: Finishing Mill 1st stand condition (case 2, Table4.1). The dependence ofx 1 n and J 1 as functions of A e when A s = 0.

Figure 4 . 6 :

 46 Figure 4.6: Finishing Mill 1st stand condition (case 2, Table4.1). The dependence of x 1 n and J 1 as functions of A s when A e = 0.

Figure 4 . 7 :

 47 Figure 4.7: Cold Mill 1st stand condition (case 3, Table4.1). The dependence ofx 1 n and J 1 as functions of A e when A s = 0.

Figure 4 . 8 :

 48 Figure 4.8: Cold Mill 1st stand condition (case 3, Table4.1). The dependence of x 1 n and J 1 as functions of A s when A e = 0.

Figure 4 . 9 :

 49 Figure 4.9: Case 1 -Hot roughing mill. Plastic deformation area by Lam3-Tec3 .

Figure 4 . 10 :

 410 Figure 4.10: Case 1 -Hot roughing mill. Discontinuity surfaces Γ e and Γ s obtained by the UBM with oscillation velocity field.

Figure 4 . 11 :

 411 Figure 4.11: Case 2 -Hot finishing first stand. Plastic deformation area by Lam3-Tec3 .

Figure 4 . 12 :

 412 Figure 4.12: Case 2 -Hot finishing first stand. Discontinuity surfaces Γ e and Γ s obtained by the UBM with oscillation velocity field.

Figure 4 . 13 :

 413 Figure 4.13: Case 3 -Cold rolling first stand. Plastic deformation area by Lam3-Tec3 .

Figure 4 . 14 :

 414 Figure 4.14: Case 3 -Cold rolling first stand. Discontinuity surfaces Γ e and Γ s obtained by the UBM with oscillation velocity field.

  , 4.18 and 4.20).

Figure 4 . 15 : 5243 Figure 4 . 16 :

 4155243416 Figure 4.15: Case 1 -Hot roughing mill. From the left to the right hand side, the 9 isovalue curves (limits between 2 successive colors) obtained by Lam3-Tec3 correspond to the ratio u x /V e =: 1.0464, 1.1062, 1.1659, 1.2256, 1.2854, 1.3451, 1.4048, 1.4645, 1.5243

Figure 4 . 17 :Figure 4 . 18 :

 417418 Figure 4.17: Case 2 -Hot finishing first stand. From the left to the right hand side, the 9 isovalue curves obtained by Lam3-Tec3 correspond to the ratio u x /V e =: 1.0507, 1.1010, 1.1513, 1.2015, 1.2518, 1.3021, 1.3523, 1.4026, 1.4529

Figures 4. 23

 23

  , 4.25 and 4.27 show the optimum velocity field obtained by UBM along the stream lines 1, 5, 10, 15 and 20 for the three rolling cases. In all the cases, at the entry of the roll-bite, the stream line 20 (in contact surface) is the first that starts to increase its velocity, meaning it is deformed

Figure 4 . 19 :

 419 Figure 4.19: Case 3 -Cold rolling first stand.From the left to the right hand side, the 9 isovalue curves obtained by Lam3-Tec3 correspond to the ratio u x /V e =: 1.0983, 1.1679, 1.2375, 1.3070, 1.3766, 1.4462, 1.5157, 1.5853, 1.6549.

Figure 4 . 20 :

 420 Figure 4.20: Case 3 -Cold rolling first stand. From the left to the right hand side, the 9 isovalue curves obtained by UBM correspond to the same ratio u x /V e .

Figure 4 . 21 :

 421 Figure 4.21: Case 3 -Cold rolling first stand. Streamlines obtained by the UBM with oscillating velocity.

Figure 4 . 22 :

 422 Figure 4.22: Case 3 -Cold rolling first stand. Comparison of streamlines between UBM and Lam3-Tec3 .

Figure 4 . 23 :

 423 Figure 4.23: Case 1 -Hot roughing mill. Longitudinal velocity obtained by UBM .

Figure 4 . 24 :

 424 Figure 4.24: Case 1 -Hot roughing mill. Longitudinal velocity obtained by Lam3-Tec3 .

Figures 4. 24

 24 Figures 4.24, 4.26 and 4.28 show the longitudinal velocity obtained by Lam3-Tec3 . While in the case 3, Lam3-Tec3 and UBM give a very similar forward slip (see Table4.4), the forward slip obtained by Lam3-Tec3 in the case 1 and 2 is higher than that obtained by UBM (see Tables 4.2 and 4.3). That means the entry and exit velocities are also higher which can be seen by comparing Figure4.23 to 4.24 and 4.25 to 4.26. Nevertheless, it can be seen in these figures that the velocity fields obtained by Lam3-Tec3 oscillate with a same periodicity as those obtained by UBM . Even though

  Figures 4.24, 4.26 and 4.28 show the longitudinal velocity obtained by Lam3-Tec3 . While in the case 3, Lam3-Tec3 and UBM give a very similar forward slip (see Table4.4), the forward slip obtained by Lam3-Tec3 in the case 1 and 2 is higher than that obtained by UBM (see Tables 4.2 and 4.3). That means the entry and exit velocities are also higher which can be seen by comparing Figure4.23 to 4.24 and 4.25 to 4.26. Nevertheless, it can be seen in these figures that the velocity fields obtained by Lam3-Tec3 oscillate with a same periodicity as those obtained by UBM . Even though

Figure 4 .

 4 Figures 4.24, 4.26 and 4.28 show the longitudinal velocity obtained by Lam3-Tec3 . While in the case 3, Lam3-Tec3 and UBM give a very similar forward slip (see Table4.4), the forward slip obtained by Lam3-Tec3 in the case 1 and 2 is higher than that obtained by UBM (see Tables 4.2 and 4.3). That means the entry and exit velocities are also higher which can be seen by comparing Figure4.23 to 4.24 and 4.25 to 4.26. Nevertheless, it can be seen in these figures that the velocity fields obtained by Lam3-Tec3 oscillate with a same periodicity as those obtained by UBM . Even though

Figure 4 . 25 :

 425 Figure 4.25: Case 2 -Hot finishing first stand. Longitudinal velocity obtained by UBM .

Figure 4 . 26 :

 426 Figure 4.26: Case 2 -Hot finishing first stand. Longitudinal velocity obtained with Lam3-Tec3 .

Figure 4 . 27 :

 427 Figure 4.27: Case 3 -Cold rolling first stand. Longitudinal velocity obtained by UBM .

Figure 4 . 28 :

 428 Figure 4.28: Case 3 -Cold rolling first stand. Longitudinal velocity obtained by Lam3-Tec3 .

Figure 4 . 29 :

 429 Figure 4.29: Case 1 -Hot roughing mill. Vertical velocity obtained by UBM and its simple part.

Figure 4 . 30 :

 430 Figure 4.30: Case 1 -Hot roughing mill. Comparison of vertical velocity between UBM and Lam3-Tec3 .

Figure 4 . 31 :

 431 Figure 4.31: Case 2 -Hot finishing first stand. Vertical velocity obtained by UBM and its simple part.

Figure 4 . 32 :

 432 Figure 4.32: Case 2 -Hot finishing first stand. Comparison of vertical velocity between UBM and Lam3-Tec3 .

Figure 4 . 33 :

 433 Figure 4.33: Case 3 -Cold rolling first stand. Vertical velocity obtained by UBM and its simple part.

Figure 4 . 34 :

 434 Figure 4.34: Case 3 -Cold rolling first stand. Comparison of vertical velocity between UBM and Lam3-Tec3 .

Figures 4. 29

 29 Figures 4.29, 4.31 and 4.33 show the vertical velocity field as well as its simple part obtained by UBM along the stream lines 1, 5, 10, 15 and 20 for the three studied rolling cases. Obviously, along the center, line 1 the vertical velocity is equal to 0 and there is no oscillation of vertical velocity. As for the contact surface, line 20, the oscillation is relatively small. Merely along the other streamlines the vertical velocity field oscillate around the simple part and the oscillation can be very important. These oscillations have same period equal to the strip thickness 2h(x) as that of the longitudinal velocity and their amplitude decreases also from the entry to the exit.

  Figures 4.29, 4.31 and 4.33 show the vertical velocity field as well as its simple part obtained by UBM along the stream lines 1, 5, 10, 15 and 20 for the three studied rolling cases. Obviously, along the center, line 1 the vertical velocity is equal to 0 and there is no oscillation of vertical velocity. As for the contact surface, line 20, the oscillation is relatively small. Merely along the other streamlines the vertical velocity field oscillate around the simple part and the oscillation can be very important. These oscillations have same period equal to the strip thickness 2h(x) as that of the longitudinal velocity and their amplitude decreases also from the entry to the exit.

Figure 4 . 35 :

 435 Figure 4.35: Proposition of combining model: oscillating velocity with neutral rigid triangular.

Figure 5 . 2 :

 52 Figure 5.2: Illustration of deformation zone corresponding to Serek's polynomial velocity field (Fig. 1 in [99]).

5 .

 5 Metal flow in yz plane of the roll bite presented by the stream function as follows χ = -y w (5.14) Rigid-plastic UBM model for width spread 5.2 3D rigid-plastic UBM for width variation analysis

5 .

 5 Rigid-plastic UBM model for width spread 5.3 Chosen rigid-plastic model of width variation in rolling z (1 ele in z) Less fine in z and x Less fine in z,x and y Less fine in z,x,y and in the roll

Figure 5 . 3 :

 53 Figure 5.3: Comparison of width spread between different meshes with 2R = 298.3mm, 2h e = 1.31mm, 2w e = 70mm, 2h s = 0.872mm, Coulomb friction µ = 0.0744.

5. Rigid-plastic UBM model for width spread 5 . 3

 53 Chosen rigid-plastic model of width variation in rolling function χ independent of z. It is easy to verify that the following function χ = -y ϕ (5.28)

5 .Figure 5 . 4 :

 554 Figure 5.4: Longitudinal and lateral velocities distribution across the width direction. Result Lam3-Tec3 for 2R = 298.3mm, 2h e = 1.31mm, 2w e = 70mm, 2h s = 0.872mm, Coulomb friction µ = 0.0744.

(5. 35 ) 5 .

 355 Rigid-plastic UBM model for width spread 5.3 Chosen rigid-plastic model of width variation in rolling

.39) 5 .Figure 5 . 5 :

 555 Figure 5.5: General numerical resolution algorithm for 3D rigid-plastic UBM .

Figure 5 . 7 :

 57 Figure 5.7: Comparison of width spread obtained by UBM and Lam3-Tec3 with experiments results for DWI (left) and Soldur (right) steels.

5. 4

 4 Conclusions and perspectivesInfluence of reduction in thickness on width spread

Figure 5 . 9 :

 59 Figure 5.9: Influence of rolling parameters on width spread.

Figure 5 . 10 :

 510 Figure 5.10: Influence of rolling parameters on rolling power.

Figure 5 . 11 :

 511 Figure 5.11: The difference in rolling power obtained with (%) of form 2 in comparison with that obataind with form 1.

Figure 5 . 12 :

 512 Figure 5.12: Influence of rolling parameters on specific rolling energy.

Figure 6 . 1 :

 61 Figure 6.1: Comparison of the roll-bite geometry between a flat (left) and a crowned (right) strip rolling.

w e 0 h 6 . UBM for crowned strip rolling 6 . 1

 0661 e (y 0 )dy 0 . (6.10) Velocity field proposition

  8 is automatically verified this condition. Let consider this equation for area I and III : ∇u I,I I I = ∂u I

Figure 6 . 2 :

 62 Figure 6.2: Illustration with Lam3-Tec3 result for E16 rolling conditions -TableC.2 -with a 0.06mm-crown over strip half thickness. The longitudinal velocity is non-homogenous along the roll-bite but is homogenous at the roll bite exit.

6 . UBM for crowned strip rolling 6

 66 ζ(x) is given by equation 6.22.

6 . UBM for crowned strip rolling 6 . 3

 663 h(-L I I I ) w(-L I I I ) 0 ∆u I-I I I y dy (6.30) Numerical resolution

Figure 6 . 3 : 75 Figure 6 . 4 :

 637564 Figure 6.3: Crown of degree 2. Comparison of width spread between UBM and Lam3-Tec3 .

Figure 6 .

 6 Figure 6.3 shows the width spread calculated by Lam3-Tec3 and by the UBM model with k = 2 and k = 1.75 for a crown of polynomial of degree 2. It is observed that the three curves have a same negative slope, implying the width spread decreases almost linearly as a function of strip crown and that the UBM results match very well those of Lam3-Tec3 independently on the value of k in terms of the tendency. In absolute value, with k = 1.75 the UBM curve is closer to the Lam3-Tec3 curve than with k = 2. In addition, Figure6.4 shows that the minimum rolling power corresponding to k = 1.75 is lower than that with k = 2. Knowing that the smaller the rolling power, the better width spread calculated by the UBM .

5 Figure 6 . 5 : 5 Figure 6 . 6 :

 565566 Figure 6.5: Crown of degree 4. Comparison of width spread between UBM and Lam3-Tec3 .

Figure 7 . 1 :

 71 Figure 7.1: Strip cross section after rolling.

7 .Figure 7 . 2 :

 772 Figure 7.2: Comparison calculated/measured strip thickness profile: evidence of an anisotropic roll-bite friction (µ x = 0.0496, µ y = 2*µ x ).

Figure 7 . 3 :Figure 7 . 4 :

 7374 Figure 7.3: Illustration of grinding striations inducing anisotropic friction in cold rolling.

Figure 7 . 5 :Figure 7 . 6 :

 7576 Figure 7.5: Important effect of reduction on width variation is showed by a comparison of two series of coils rolled at two reduction levels at Florange 5-stand tin plate mill.

Figure 7 . 7 :

 77 Figure 7.7: Effect of strip nominal width on width variation observed statistically on database of Mardyck 5stand sheet tandem mill.

Figure 7 . 8 :Figure 7 . 9 :Figure 7 . 10 :

 7879710 Figure 7.8: Same observation on Florange 4-stand sheet tandem mill.

Figure 7 . 11 :

 711 Figure 7.11: Comparison of predicted width necking to the measurements done at Feblatil 4-stand tandem mill.

7. A thermal-elastic-plastic width model 7 . 2 7 . 2 7 . 2 . 1

 7272721 Analytical thermal-elastic-plastic width variation model Analytical thermal-elastic-plastic width variation model Why develop new model?

Figure 7 . 12 :

 712 Figure 7.12: Comparison of width variation of Lam3-Tec3 simulations using 10 and 1 element in strip half thickness for each of Florange cold rolling mill stands.

  The strip is deformed plastically before contact with the work-roll B: first point of contact D: last point of contact A: first point of plastic zone C: last point of plastic zone

Figure 7 . 13 :

 713 Figure 7.13: Stand 2: Comparison of contact and yield fields.

7 .Figure 7 . 14 :

 7714 Figure 7.14: The contact and yield fields obtained with Lam3-Tec3 for the four stands of ArcelorMittal Florange mill.Except the stand 4 where the reduction is very small red = 1.66%, the strip is deformed plastically before contact.

Figure 7 . 15 :Figure 7 . 16 :

 715716 Figure 7.15: Roll-bite of an elasto-plastic strip in typical rolling conditions.

Figures 7. 17

 17 Figures 7.17show a zoom in the roll bite of Figure7.[START_REF] Bland | The calculation of roll force and torque in cold strip rolling with tensions[END_REF]. In the roll-bite, the plastic and elastic spreads have an opposite sign and almost the same absolute value, implying that by action of friction the elastic spread is transformed to plastic spread to keep the same flow as the case of rigid-plastic behavior. In addition, observing that the two curves,

  Figures 7.17show a zoom in the roll bite of Figure7.[START_REF] Bland | The calculation of roll force and torque in cold strip rolling with tensions[END_REF]. In the roll-bite, the plastic and elastic spreads have an opposite sign and almost the same absolute value, implying that by action of friction the elastic spread is transformed to plastic spread to keep the same flow as the case of rigid-plastic behavior. In addition, observing that the two curves,

7 .

 7 A thermal-elastic-plastic width model 7.2 Analytical thermal-elastic-plastic width variation model

7. 2 . 3 .Figure 7 . 18 :

 23718 Figure 7.18: Comparison of width variation of Lam3-Tec3 mechanical and thermo-mechanical simulations using 1 element in strip half thickness -ArcelorMittal Florange cold rolling mill.

7 .

 7 A thermal-elastic-plastic width model 7.3 A simplified entry elasto-plastic compression model Shear stress assumption: The strip material is deformed plastically just before the contact (point A -Figure 7.15) due to an apparition of an enough important xz plane shear stress component σ A xz .

7 . 4 A

 74 simplified elastic spring back model -elastic slab method

. 18 )√ 3 -

 183 To sum up, at the entry part IAB the strip width variations are given as follows∆w elasto-plastic entry = 0 ∆w plastic entry = -∆w elastic AB = (1 -2ν) σ 0 (1ν)T e Ew (7.19) By consequence, the total plastic width variation formula 7.11 can be rewritten as follows where ∆w elastic roll-bite = ∆w elastic BC : ∆w plastic total = -∆w elastic AB + ∆w rigid-plastic -∆w elastic BC rigid-plastic -∆w elastic AC -∆w thermal total . (7.21)

Figure 7 . 19 :

 719 Figure 7.19: At the exit, the exit thickness h s and tension T s are known but the thickness h C and tension T C at C as well as the position x D and strip thickness h D at D are unknown.

  strain associated to this stress can be obtained by the linear Hooke law (7.27) as follows strip half thickness after release the stress, the equation 7.23 implies that

7 .

 7 A thermal-elastic-plastic width model 7.4 A simplified elastic spring back model -elastic slab method Homogeneous stress in thickness assumption -slab method:

.28) 7 .

 7 A thermal-elastic-plastic width model 7.4 A simplified elastic spring back model -elastic slab method

7 .

 7 A thermal-elastic-plastic width model 7.4 A simplified elastic spring back model -elastic slab method

  Thanks to the assumption of total relaxation of vertical stress, the vertical stress at D is equal to zero (see 7.41) σ D zz = 0 . (7.41) Moreover, the equilibrium equation of the strip exit part DD'F'F in x direction induces that σ D xx = T s . (7.42)

  49. And the plastic width variation in the roll-bite -between B and C is ∆w plastic roll-bite = ∆w rigid-plastic -∆w elastic roll-bite -∆w thermal roll-bite (7.50)

Figure 8 . 1 :

 81 Figure 8.1: Algorithm of the simplified model for width variation of a thermal-elasto-plastic strip in cold rolling.

8 .

 8 The UBM-Slab combined model validation 8.2 Validation by comparison with Lam3-Tec3

8 .Figure 8 . 2 :Figure 8 . 3 :

 88283 Figure 8.2: Width variations profiles due to elastic, thermal, plastic and total deformations obtained by Lam3-Tec3 for the four stands of ArcelorMittal Florange mill.

8. 3

 3 Validation by comparison with industrial observations 8.3 Validation by comparison with industrial observations 8.3.1 Parametric study for a large strip rolling -Stand 1

Figures 8 .

 8 Figures 8.4 and 8.5 show the that width variation predicted by the simplified model decreases when one of both entry and exit tensions goes up. This tendency matches well the observation on industrial data presented in 7.1.2.c.

8. 3

 3 Validation by comparison with industrial observations

Figure 8 . 4 :Figure 8 . 5 :Figure 8 . 6 :

 848586 Figure 8.4: Effect of T e on ∆w -Stand 1 condition.

Figure 8 . 7 :

 87 Figure 8.7: Effect of w e on ∆w -Stand 1 condition.

Figure 8 . 8 :Figure 8 . 9 : 8 Figure 8 . 10 :Figure 8 . 11 :Figure 8 . 12 :

 88898810811812 Figure 8.8: Effect of σ 0 on ∆w -Stand 1 condition.

Figure 8 . 13 :Figure 8 . 14 :

 813814 Figure 8.13: Effect of T s on ∆w for a narrow strip.

Figure 8 . 15 :

 815 Figure 8.15: Effect of w e on ∆w -narrow strip (E16).

Figure 8 . 16 :Figure 8 . 17 :

 816817 Figure 8.16: Effect of σ 0 on ∆w -narrow strip (E16).

Figure 8 . 18 :

 818 Figure 8.18: Effect of h e on ∆w -narrow strip (E16).

Figure 8 . 19 :

 819 Figure 8.19: Effect of 2R on ∆w -narrow strip (E16).

1 = 4 c 1 = 5 c 1 6 c 1 =

 1415161 0.555555556 x 1 = -0.774596669 c 2 = 0.888888889 x 2 = 0 c 3 = 0.555555556 x 2 = 0.774596669 0.347854845 x 1 = -0.861136312 c 2 = 0.652145155 x 2 = -0.339981044 c 3 = 0.652145155 x 3 = 0.339981044 c 4 = 0.347854845 x 1 = 0.861136312 = 0.236926885 x 1 = -0.906179846 c 2 = 0.478628670 x 2 = -0.538469310 c 3 = 0.568888889 x 3 = 0 c 4 = 0.478628670 x 4 = 0.538469310 c 5 = 0.236926885 x 5 = 0.906179846 0.171324492 x 1 = -0.932469514 c 2 = 0.360761573 x 2 = -0.661209386 c 3 = 0.467913935 x 3 = -0.2386191860 c 4 = 0.467913935 x 4 = 0.2386191860 c 5 = 0.360761573 x 5 = 0.661209386 c 6 = 0.171324492 x 6 = 0.932469514 Table A.1: Values of the coefficients and arguments for 2-point, 3-point, 4-point, 5-point and 6-point Gauss Quadrature Rules.

′ 2 +

 2 factors and function arguments of Gauss-Legendre rule are defined for interval of integration [-1, 1]. We need a variable change for each such integral. In order to simplify the utilization of Gauss-Legendre integration in usual problems, we introduce an operator denoted IGauss N 1D ( f , x min , x max , n ele ) is N-point Gauss integration of a function f on the interval [x min , x max ] divided into n elements on which the integral is approximated by N-point Gauss integration. The size of each element is ∆x = x max -x min n ele . The i th elements is between two nodesB. Calculation of powersB.1 Calculation of power of plastic deformation J ǫ for the simple 2D velocity fieldThe power of plastic deformation in roll bite is given byJ ǫ = hh ′′ -2h ′ 2 2 z * 2 dz *

Q 2 + f 2 y b 2 + g 2 Q 2 +B. 2

 22222 f 2 y * 2 + g 2 z * 2 dz * dy * dx Calculation of powers for the simple 3D velocity field

a 2 + x 2 + b dx = 1 3 c 3 3 + bc 2 a 2 + c 2 - 2 (a 2 -√ a 2 + c 2 -a c - b 2 (3a 2 -

 233222222 ln b + a 2 + c 2 + (a 2b 2 )c -c 3 2b 2 )ln c + √ a 2 + c 2 a (B.7)

2 . 2 2 y 2 (

 2222 Power of the discontinuity of velocity J∆u J ∆u = (x = -L, y) + u 2 z (x = -L, z) dydz -L)y * 2 + h ′ 2 (-L)z * 2 dy * dz * .

2 x 2 + u 2 y′ 2 y

 2222 (x) + u 2 z (x, z = h) -V c (x, y) dy 1 + h ′ 2 dx * 2 dy * dx In noting J = √ 1 + h ′ 2 -hϕV c C voland in using the formula B.1 for a = J et c = b ′ , we have

Figure C. 1 :Figure C. 2 :Figure C. 3 :Figure C. 4 :Figure C. 5 :

 12345 Figure C.1: Sample E1. Comparison between Lam3-Tec3 and experiment.

Table 1 .

 1 Liquid pig iron 1370 4.70 0.23 0.26 0.08 0.02 0.00 Liquid steel 1650 0.05 0.10 0.00 0.02 0.02 0.05

	1.2 Typical steel production route

1: Example of composition of pig iron supplied to BOF and liquid steel obtained.

Table 1 .

 1

		Scrap charge	Cu	Sn	S	P	N	C
			ppm ppm ppm ppm ppm	ppm
		Low quality	250	50	50	40	70-90
	EAF	Standard quality 150 High quality 50	20 10	30 10-20 <10 70-90 15 70-90	40-50
		Scrap + few iron <50 <10 10-20 <10 40-50
	BOF	Pig iron	20	<5	10-20 <10	<20	10-40

2: Comparison of chemical composition of liquid steel obtained by EAF and BOF

28 2.2 Typical tribological models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 2

1 Main characteristics of rolling process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.2 Main objects to be modeled in rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.3 A well-posed problem of rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.1.4 Rolling model -A coupling of separated models . . . . . . . . . . . . . . . . . . . . . . . . .2.1 Dry friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.2 Lubricated friction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.2.3 Friction in cold rolling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3 Typical strip models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.1 0D and 1D models -Analytical models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.2 2D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.3.3 3D models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.

4 Typical work-roll deformation models

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.4.1 Bending and flattening in width direction Oyz -3D modeling . . . . . . . . . . . . . . . . . 46 2.4.2 Flattening models for 2D modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.5 Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . 49 2. Rolling process modeling reviews

	2.1 General description of rolling problem

  The Stribeck friction model is illustrated in Figure 2.6. The Stribeck friction model can provide very good representation of the friction between sliding surfaces. It covers everything from Coulomb friction to viscous depending on the choice of parameter values.

.18) 2. Rolling process modeling reviews

		2.3 Typical strip models
	The Von Mises flow criterion says that:
	S = λd.	(2.19)

38) 2. Rolling process modeling reviews

		2.3 Typical strip models
	or	d σ zz 2k σ zz 2k

2. Rolling process modeling reviews

	2.3 Typical strip models

3. Upper Bound Method applied in rolling process

  The test bar was recrystallized annealed 0.15% C mild steel.

								3.2 Velocity fields with rigid bodies motions
	Test	A	B	C	D	E	F	G
	R(mm)	10.0 23.84 23.84 10.34 24.0 24.0 100
	2h e (mm) 2.50 3.374 3.440 1.800 3.43 3.00 4.86
	2h s (mm) 2.08 3.330 3.330 1.380 3.00 2.80 4.22
	L(mm)	2.05 1.020 1.619 2.084 3.21 2.19 8.00

Table 3 .

 3 1: Series of experiments performed by Piispanen[START_REF] Piispanen | Plastic deformation of metal: theory of simulated slidding[END_REF].

Upper Bound Method applied in rolling process

  .30) Thus, the position of O 1 and the radius r 1 are expressed as functions of x P thanks to 3.30 and 3.29. Similarly for the arc PB

	⌢ we obtain
	r 2 (x P ) =	h 2 s + x 2 P 2h s	(3.31)

z 2 (x P ) = h sr 2 (x P ) (3.32)

3.

3.2 Velocity fields with rigid bodies motions

Upper Bound Method applied in rolling process

		3.2 Velocity fields with rigid bodies motions
		Figure 3.7: Calculating of roll force for unitriangular velocity field approach.
	Using 3.45 and AP	2 = h 2 e + (L + x P ) 2 the equation 3.46 implies

.46) 3.

Upper Bound Method applied in rolling process 3

  .3 2D continuous velocity fields where C vol denotes the flow rate of the rolled material through an across section. The flow rate must be constant and equal to C vol = V e h e = V s h s . It is unique parameter to determine. The thickness function h(x) in the case of circular work-roll is defined as h

3.61)

3.

Table 4 .

 4 .1).

	Case Mill	2h e	2h s	red	2R	T e	T s	σ 0	m	V c
			mm	mm	%	mm	Mpa Mpa	Mpa		m/s
	1	Hot roughing	126.4 79.0 37.5 1043.0 0.0	0.0	76.000 0.6000 1.73138
	2	Hot finishing 1st stand 36.0 24.0 33.3 680.00 0.0	5.0	150.000 0.3000 1.50000
	3	Cold tandem 1st stand	2.8	1.7 40.3 538.65 51.0 150.0 516.619 0.0974 6.45600

1: Rolling cases for comparison of UBM models and Lam3-Tec3 .

Table 4 .

 4 2: Case 1 -Roughing Mill. Comparison of the obtained powers of UBM using different velocity fields: unitriangular, simple (elliptical) and oscillating with Lam3-Tec3 .

	Case 1	xn	fs	Torque	J ǫ	J Γ e ∆u	J Γ s ∆u	J de f	J ten	J f ric	J	Re-Diff
	Hot roughing	mm	%	kNm/m kw/m kw/m kw/m kw/m kw/m kw/m kw/m	%
	Unitriangular		16.68	1229	0	2987 1095 4082	0	0	4082	24.00
	Simple	-31.62 2.241	1298	2934	479	0	3413	0	896	4309	18.35
	Oscillating
	A e =8.172e-9	-35.10 2.976	1202	2642	562	56	3259	0	733	3992	12.19
	A s =-8.6e-10
	Lam3-Tec3		8.429	1168				3494	0	383	3876.5	14.24
	Case 2	xn	fs	Torque	J ǫ	J Γ e ∆u	J Γ s ∆u	J de f	J ten	J f ric	J	Re-Diff
	Hot finishing	mm	%	kNm/m kw/m kw/m kw/m kw/m kw/m kw/m kw/m	%
	Unitriangular		14.3	435.0	0.0	1372	645	2022.1 102.9	0.0	1919.2	39.9
	Simple	-10.89 1.40	372.6	1295.4 150.5	0.0	1445.9 91.3 289.2 1643.8	12.8
	Oscillating
	A e =6.320e-8	-12.07 1.80	351.8	1252.3 114.4 11.6 1378.3 91.6 265.4 1552.1	7.1
	A s =-4.927e-9
	Lam3-Tec3	-12.30 4.62	340.3				1378.0 94.2 217.5 1501.3	4.2

Table 4 .

 4 3: Case 2 -Finishing Mill. Comparison of the obtained powers of UBM using different velocity fields: unitriangular, simple (elliptical) and oscillating with Lam3-Tec3 .

	Case 3	xn	fs	Torque	J ǫ	J Γ e ∆u	J Γ s ∆u	J de f	J ten	J f ric	J	Re-Diff
	Cold rolling	mm	%	kNm/m kw/m kw/m kw/m kw/m kw/m kw/m kw/m	%
	Unitriangular		17.1	310	0	5163 2898 8061.0 626.1	0.0	7434.9	314.9
	Simple	-4.212 3.93	70.4	1726.0 54.2	0.0	1780.3 555.6 463.5 1688.1	3.2
	Oscillating
	A e =8.06e-7	-4.287 4.03	69.2	1723.3 31.5	0.4	1755.1 556.2 460.9 1659.9	1.7
	A s =-8.41e-9
	Lam3-Tec3	-4.325 4.50	68.2				1745.0 558.7 447.4 1633.7	0.6

Table 4 . 4

 44 cases of Table4.1 are given in Figure4.10, 4.12 and 4.14. We can see that, at the entry side the first contact point A is deformed at first then the plastic deformation advances progressively to the strip center. Inversely at the exit side, the strip center stops being deformed at first and the contact surface (point B) is the last plastically deformed point. The Lam3-Tec3 results given in Figures 4.9, 4.11 and 4.13 confirm also this remark.

: Case 3 -Cold rolling. Comparison of the obtained powers of UBM using different velocity fields: unitriangular, simple (elliptical) and oscillating with Lam3-Tec3 . three

Table 5 .

 5 1 gives some typical empirical formulae giving the width spread factor S defined by

	S =	ln w s w e h s ln h e	(5.1)

5. Rigid-plastic UBM model for width spread 5

  .3 Chosen rigid-plastic model of width variation in rolling Figure 5.8: UBM width spread measurements for DWI and Soldure steels.

	Parameters	Notation Value	Unit
	Half entry thickness	h e	2.5	mm
	Half entry widht	w e	25	mm
	Reduction in thickness	r	30	%
	Yield stress of the strip	σ 0	600	Mpa
	Roll's radius	R	250	mm
	Roll's velocity	V c	1000 mm/s
	Entry and exit tensions	T e , T s	150	Mpa
	Tresca's friction coefficient m	0.4

Table 5 . 3 :

 53

5. Rigid-plastic UBM model for width spread

															5.4 Conclusions and perspectives
			Rolling power depends on reduction					Rolling power depends on roll's radius
		1.0E+05										7.0E+04
			Polynomial_in_h & he=1mm								Polynomial_in_h & red=15%
	1/4 rolling power (W)	3.0E+04 4.0E+04 5.0E+04 6.0E+04 7.0E+04 9.0E+04 8.0E+04	Polynomial_in_h & he=2mm Polynomial_in_h & he=3mm Polynomial_in_h & he=4mm Polynomial_in_x & he=1mm Polynomial_in_x & he=2mm Polynomial_in_x & he=3mm Polynomial_in_x & he=4mm						1/4 rolling power (W)	6.0E+04 2.0E+04 3.0E+04 4.0E+04 5.0E+04	Polynomial_in_h & red=30% Polynomial_in_h & red=45% Polynomial_in_h & red=60% Polynomial_in_x & red=15% Polynomial_in_x & red=30% Polynomial_in_x & red=45% Polynomial_in_x & red=60%
		2.0E+04
												1.0E+04
		1.0E+04
		0.0E+00										0.0E+00
		0	10	20	30	40	50	60	70	80	90	0	100	200	300	400	500
					reduction in thickness (%)						roll's radius (mm)
			Rolling power depends on friction					Rolling power depends on entry thickness
		9.0E+04										8.0E+04
			Polynomial_in_h & red=15%								Polynomial_in_h & red=15%
	1/4 rolling power (mW)	2.0E+04 3.0E+04 4.0E+04 5.0E+04 6.0E+04 7.0E+04 8.0E+04	Polynomial_in_h & red=30% Polynomial_in_h & red=45% Polynomial_in_h & red=60% Polynomial_in_x & red=15% Polynomial_in_x & red=30% Polynomial_in_x & red=45% Polynomial_in_x & red=60%						1/4 rolling power (W)	7.0E+04 2.0E+04 3.0E+04 4.0E+04 5.0E+04 6.0E+04	Polynomial_in_h & red=30% Polynomial_in_h & red=45% Polynomial_in_h & red=60% Polynomial_in_x & red=15% Polynomial_in_x & red=30% Polynomial_in_x & red=45% Polynomial_in_x & red=60%
		1.0E+04										1.0E+04
		0.0E+00										0.0E+00
		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0	0	1	2	3	4	5
					Tresca's friction coefficient						half entry thickness (mm)
			Rolling power depends on entry width
		5.0E+05
		4.5E+05
		4.0E+05
		3.5E+05
		3.0E+05
		2.5E+05
		2.0E+05
	1/4	1.5E+05
		1.0E+05
		5.0E+04
		0.0E+00
		0	50		100		150	200		250	300
					half entry width (mm)

rolling power (W)

	Polynomial_in_h & red=15%
	Polynomial_in_h & red=30%
	Polynomial_in_h & red=45%
	Polynomial_in_h & red=60%
	Polynomial_in_x & red=15%
	Polynomial_in_x & red=30%
	Polynomial_in_x & red=45%
	Polynomial_in_x & red=60%

Rolling power depends on entry & exit tensions

		3.5E+04
		3.0E+04
		2.5E+04
		2.0E+04
		1.5E+04
	1/4	1.0E+04
		5.0E+03
		0.0E+00
		0	50	100	150	200	250	300
				entry & exit tensions' variation (Mpa)

rolling power (W)

	Polynomial_in_h & Ts=0Mpa
	Polynomial_in_h & Te=0Mpa

Volumic rolling energy depends on roll's radius

		7.0E+00										1.8E+00
			Polynomial_in_h & he=1mm									Polynomial_in_h & red=15%
	Volumic rolling energy (J/mm3)	1.0E+00 2.0E+00 3.0E+00 4.0E+00 5.0E+00 6.0E+00	Polynomial_in_h & he=2mm Polynomial_in_h & he=3mm Polynomial_in_h & he=4mm Polynomial_in_x & he=1mm Polynomial_in_x & he=2mm Polynomial_in_x & he=3mm Polynomial_in_x & he=4mm						Volumic rolling energy (J/mm3)	2.0E-01 4.0E-01 6.0E-01 8.0E-01 1.0E+00 1.2E+00 1.4E+00 1.6E+00		Polynomial_in_h & red=30% Polynomial_in_h & red=45% Polynomial_in_h & red=60% Polynomial_in_x & red=15% Polynomial_in_x & red=30% Polynomial_in_x & red=45% Polynomial_in_x & red=60%
												0.0E+00
		0	10	20	30	40	50	60	70	80	90	0	100	200	300	400	500
					reduction in thickness (%)						roll's radius (mm)
			Volumic rolling energy depends on friction
		3.0E+00
			Polynomial_in_h & red=15%
	Volumic rolling energy (J/mm3)	5.0E-01 1.0E+00 1.5E+00 2.0E+00 2.5E+00	Polynomial_in_h & red=30% Polynomial_in_h & red=45% Polynomial_in_h & red=60% Polynomial_in_x & red=15% Polynomial_in_x & red=30% Polynomial_in_x & red=45% Polynomial_in_x & red=60%
		0.0E+00
		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9	1.0
					Tresca's friction coefficient

Volumic rolling energy depends on entry thickness

		4.5E+00
					Polynomial_in_h & red=15%
	Volumic rolling energy (J/mm3)	1.0E+00 1.5E+00 2.0E+00 2.5E+00 3.0E+00 3.5E+00 4.0E+00			Polynomial_in_h & red=30% Polynomial_in_h & red=45% Polynomial_in_h & red=60% Polynomial_in_x & red=15% Polynomial_in_x & red=30% Polynomial_in_x & red=45% Polynomial_in_x & red=60%
		5.0E-01
		0.0E+00
		0	1	2	3	4	5
				half entry thickness (mm)

Volumic rolling energy depends on entry width

	1.8E+00
	1.6E+00
	1.4E+00
	1.2E+00
	1.0E+00
	8.0E-01
	6.0E-01
	4.0E-01
	2.0E-01
	0.0E+00
	0	50	100	150	200	250	300
			half entry width (

mm) Volumic rolling energy (J/mm3)

	Polynomial_in_h & red=15%
	Polynomial_in_h & red=30%
	Polynomial_in_h & red=45%
	Polynomial_in_h & red=60%
	Polynomial_in_x & red=15%
	Polynomial_in_x & red=30%
	Polynomial_in_x & red=45%
	Polynomial_in_x & red=60%

Volumic rolling energy depends on entry & exit tensions

	8.0E-01
	7.0E-01
	6.0E-01
	5.0E-01
	4.0E-01
	3.0E-01
	2.0E-01
	1.0E-01
	0.0E+00
	0	50	100	150	200	250	300
			entry & exit

tensions' variation (Mpa) Volumic rolling energy (J/mm3)

	Polynomial_in_h & Ts=0Mpa
	Polynomial_in_h & Te=0Mpa

Table 7 .

 7 ∆w Total = ∆w EdgeDrop + ∆w SpringBack -∆w Thermal Parameter Nomenclature Edge drop ∆w Spring back ∆w Thermal ∆w Total ∆w Impact level 2: Summary of bibliographic review about how rolling parameters influence the strip width variation with explanations if existing. (*) means that the effect of strip yield stress is not clear due to the correlation between strip yield stress and the total reduction in industrial database.

	Nominal width Reduction Entry thickness Strip yield stress Entry & exit tensions Work-roll radius Bending Friction anisotropy	w ր red ր h e ր σ 0 ր T e , T s ր R ր B ր α µ ր	? ? ? ր ? ր ց ց	ր ? ? ր ? ? ? ?	ր ր ? ր ? ? ? ?	|∆w|ր ց ր ր ց ր ց ց	high high low ?(*) average low average average

7.1.

3 Existing models for industrial rolling width variation 7.1.3.a Streamlines finite difference method -[23]

	Calculated width necking
	Mesured width necking

Table 7 .

 7 3: Results of contribution of width variation phenomenon using Lam3-Tec3 and a thermal model[START_REF] Dusser | Improved cooling on cold tandem mills[END_REF] on Mardyck tandem mill for a nominal width w = 1232mm, h e = 5.54mm and h s = 2.54mm, red = 54% (see[START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF] for more details).

		Stand 1 Stand 2 Stand 3 Stand 4 Stand 5 Total
	∆w EdgeDrop (mm)	0.26	0.31	0.16	0.11	0.01	0.85
	∆w SpringBack (mm)	0.16	0.24	0.30	0.26	1.00	1.96
	∆T (°C)	34.8	16.1	30.6	23.7	6.4	111.6
	∆w Thermal (mm)	0.6	0.28	0.53	0.41	0.11	1.92
	∆w Total (mm)	-0.18	0.27	-0.07	-0.04	0.9	0.89
		Stand 1 Stand 2 Stand 3 Stand 4 Stand 5 Total
	∆w EdgeDrop (mm)	0.14	0.09	0.38	0.30	0.79	1.70
	∆w SpringBack (mm)	0.24	0.17	0.14	0.26	0.61	1.40
	∆T (°C)	32.2	24.0	50.2	52.3	82.4	241.1
	∆w Thermal (mm)	0.41	0.31	0.64	0.67	1.05	3.07
	∆w Total (mm)	-0.03	-0.05	-0.12	-0.11	0.35	0.03
	Table

Table 7 .

 7 .5. 5: Typical industrial rolling conditions of flat automotive product -ArcelorMittal Florange 4-stand cold rolling mill.

	Stand 2w e (mm) 2h e (mm) 2h s (mm) red(%) 2R(mm) σ 0 (Mpa)	µ	T e (Mpa) T s (Mpa)
	1	1170.0	2.800	1.677	40.11	538.65	516.619 0.036412	51.0	150.0
	2		1.677	1.007	39.95	517.24	555.480 0.038186	150.0	156.0
	3		1.007	0.663	34.16	601.48	620.368 0.022206	156.0	199.2
	4		0.663	0.652	1.66	550.81	640.000	0.110	199.2	39.0

  7.5 A simplified model for roll-bite entirely determined thanks to the elastic spring back model (see7.4). This tensor is given by 7.30 with T C is an output of elastic spring back model calculated in the step 3 of 7.4.3.a. Using the Hooke's law 7.27, the elastic width variation in the roll-bite can be determined by

	∆w elastic roll-bite = w∆ǫ BC yy = w	∆σ BC yy -ν ∆σ BC yy + ∆σ BC yy E	(7.46)
	Finally, by substituting 7.16 and 7.30 into 7.46 we obtain

7. A thermal-elastic-plastic width model

								7.6 Summary
			With heat exchange Without heat exchange	Error
	Stand Entry T(°K) Exit T(°K) ∆T(°K) Exit T(°K)	∆T(°K)	°K	%
	1	300.0	400.27	100.27	400.82	100.82	0.55 0.55
	2	333.0	439.96	106.96	440.96	107.96	1.00 0.94
	3	353.0	447.65	94.65	448.49	95.49	0.84 0.89
	4	373.0	375.95	2.95	376.06	3.06	0.11 3.63

Table 7 .

 7 6: Results of thermo-mechanical Lam3-Tec3 simulations with and without heat exchange between the strip and the roll for typical industrial cold rolling conditions in Table7.5.

8.1 Simplified model algorithm and programming . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8

.1.1 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 8.1.2 C++ programming and fast computing time . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

8.2 Validation by comparison with Lam3-Tec3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 8

.2.1 Very good prediction of final plastic width variation . . . . . . . . . . . . . . . . . . . . . . . 165 8.2.2 The simplified model can be improved? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

8.3 Validation by comparison with industrial observations . . . . . . . . . . . . . . . . . . . . . . . . 168

  8.3.1 Parametric study for a large strip rolling -Stand 1 . . . . . . . . . . . . . . . . . . . . . . . . 168 8.3.2 Parametric study for a narrow strip rolling -E16 trial . . . . . . . . . . . . . . . . . . . . . . 169 8.3.3 Summary of parametric studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 8.

4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 8. The UBM-Slab combined model validation

	8.1 Simplified model algorithm and programming

Table 8 .

 8 .5.

	Parameter	Unit Stand 1	Stand 2	Stand 3	Stand 4	E16
	J de f + c share J f ric	w 1115370 1190020 1058860	37426	6594
	∆T		°C	102.1	109.3	96.9	3.6	144.8
	X n		mm -2.2081	-0.4738	-0.3350 -0.1320 -1.6076
	h C		mm 0.83488 0.50244 0.33075 0.32502 0.43414
	T C		Mpa	138.29	137.95	184.32	-35.53	-21.20
	x D		mm	0.9408	0.7480	0.6740	0.7602	0.8828
	L		mm	17.438	13.180	10.192	1.889	9.376
	∆w	plastic entry	mm	0.2329	0.0649	0.0949	0.0233	0.0260
	∆w elastic BC ∆w elastic exit ∆w thermo-elasto-plastic = ∆w DF total ∆w rigid-plastic	mm mm mm mm	0.0973 0.0529 0.0675 0.0146	-0.0134 0.0733 0.0782 0.0049	0.0316 0.0272 0.0296 0.0024	-0.2616 -0.0034 0.4187 0.0294 0.4191 0.1410 0.0004 0.1116
	∆w elastic AC ∆w thermal total ∆w plastic total Computing Time	mm -0.1356 mm 0.7171 mm -0.5668 ms 30	-0.0783 0.7676 -0.6844 20	-0.0633 -0.2848 -0.0294 0.6799 0.0254 0.0523 -0.6142 0.2598 0.0887 20 10 30

1: Results obtained by the simplified model -ArcelorMittal Florange 4-stand cold rolling mill.

Table 7

 7

	.5.

which is valid when the assumptions Independent lateral flow assumptions 1 & 2 (see 7.2.2.f) and Plane strain deformation between A and B are verified. In this formula, the three terms ∆w rigid-plastic , ∆w elastic AC and ∆w thermal total are, 8.

The UBM-Slab combined model validation

						8.2 Validation by comparison with Lam3-Tec3
		Elastic	Thermal		Plastic		Total
	Lam3	New Model Lam3 New Model	Lam3	New Model Lam3 New Model
	Stand 1 -0.0820	-0.0827	0.7083	0.7171	-0.5271	-0.5668	0.0989	0.0675
	Stand 2 -0.0040	-0.0050	0.7585	0.7676	-0.6402	-0.6844	0.1140	0.0782
	Stand 3 -0.0350	-0.0361	0.6706	0.6799	-0.5815	-0.6142	0.0542	0.0296
	Stand 4 0.1349	0.1339	0.0233	0.0254	0.2337	0.2598	0.3919	0.4191

Table 8 .

 8 2: Comparison of elastic, thermal, plastic and total width variations obtained by Lam3-Tec3 and the simplified model -ArcelorMittal Florange 4-stand cold rolling mill.indeed estimations of the total, elastic and thermal width variations of the strip between A and C, respectively.

	It can be, nevertheless rewritten as
	∆w	plastic total = ∆w rigid-plastic + ∆w elastic I A	-∆w elastic IC	-∆w thermal
	∆w total AC Lam3 Model	Err	∆w elastic I A Lam3 Model	Err	∆w elastic IC Lam3 Model	Err	∆w thermal total Lam3 Model	Err
	Stand 1 0.095 0.015 -0.080 -0.069 0.000	0.069 -0.158 -0.136 0.022 0.708 0.717 0.009
	Stand 2 0.058 0.005 -0.053 -0.025 0.000	0.025 -0.088 -0.078 0.010 0.759 0.768 0.009
	Stand 3 0.044 0.002 -0.041 -0.023 0.000	0.023 -0.070 -0.063 0.006 0.671 0.680 0.009
	Stand 4 0.009 0.000 -0.008 0.015	0.000 -0.015 -0.238 -0.285 -0.047 0.023 0.025 0.002

total which shows that there are four sources of error in the calculation of plastic width variation of the simplified model:

1. the total width variation between AC estimated by rigid-plastic one ∆w rigid-plastic 2. the elastic width variation at A (compared to the initial width at I) 3. the elastic width variation at C (compared to the initial width at I)

[START_REF] Andersson | Friction models for sliding dry, boundary and mixed lubricated contacts[END_REF]

. the thermal width variation ∆w thermal total (indeed depends on the accuracy of plastic and friction powers).

Table 8 .

 8 3: Comparison of elastic, thermal and total width variations between AC obtained by Lam3-Tec3 and the simplified model -ArcelorMittal Florange 4-stand cold rolling mill.

Table 8 .

 8 [START_REF] Allwood | A hybrid 2.5 dimensional elastoviscoplastic model of hot strip rolling for practical applications[END_REF] shows a comparison of the four mentioned terms between the simplified model and Lam3-Tec3 . It can be observed that the thermal width variation is quit well estimated by the new model meaning that the rigid-plastic UBM is good enough to approximate the plastic and friction dissipation powers. The width at C (or ∆w elastic IC

Table 8 .

 8 4: Comparison of parametric studies using the UBM-Slab combined model and bibliographic review on the influence of rolling parameters on strip width variation in two cases: large and narrow strips.

				64]		UBM-Slab model
			Large strip	Large strip (stand 1)	Narrow strip (E16)
			Tendency	Impact	Tendency	Impact	Tendency Impact
	Nominal width Reduction Entry thickness Strip yield stress Entry & exit tensions Work-roll radius Friction coefficient Bending Friction anisotropy	w red h e σ 0 T e , T s R µ & m B α µ	|∆w|ր ց ր ր ց ր ց ց	high high low average low average average	ց ց → ց ց → ց	high high negligible average average negligible low	ց ր ր ր ց ր ց	high high high low low high low

2 Calculation of powers for the simple 3D velocity field

  + c 2 = 4h ′ 2 + I 2 .2 Calculation of powers for the simple 3D velocity field As a reminder, the simple 3D velocity field is expressed by 5.[START_REF] Allwood | A hybrid 2.5 dimensional elastoviscoplastic model of hot strip rolling for practical applications[END_REF] as followsTensor of deformation rate ǫ testCorresponding to this velocity filed, all components of ǫ are :

	B. Calculation of powers
										u x (x, y) = C vol	1 h(x)ϕ(x)
										u y (x, y) = C vol	ϕ h(x)ϕ 2 (x) ′ (x)y	(B.3)
											u z (x, z) = C vol	h h 2 (x)ϕ(x) ′ (x)z	.
	B.2.1 Power of plastic deformation J ǫ
		ǫxx =	∂u x ∂x	= -	C vol hϕ	h h ′	+	′ ϕ ϕ	(B.4a)
		ǫyy =	∂u y ∂y	=	C vol hϕ	′ ϕ ϕ	(B.4b)
		ǫzz =	∂u z ∂z	=	C vol hϕ	′ h h	(B.4c)
		1 ǫxy = ǫyx = 0 a 2 + c 2 t 2 dt = 1 2 ∂u x ∂y ǫxz = ǫzx = 1 2 ∂u x ∂z	1 2 + +	a 2 + c 2 + ∂u y ∂x = C vol 2hϕ ∂u z ∂x = C vol 2hϕ	a 2 c h ln ϕ ′′ -c + h √ ′ ϕ h a 2 + c 2 ′ -2ϕ ϕ ′ 2 a ′′ -h ′ ϕ ′ ϕ -2h ′ 2 h	.	y ϕ z h	(B.4d) (B.1) (B.4e)
		ǫyz = ǫzy = 0.				(B.4f)
	In using B.1 for a = 2|h Thus, the power of plastic deformation is calculated as ′ |, c = hh ′ 2 and in noting ′′ -2h
		J ǫ =	Ω em prise	σ 0	   2 3	I 1 = c = h.h I 2 = √ a 2 1 ′′ -2h ′ 2 ǫ : ǫ dΩ
	we obtain	=	Ω em prise	σ 0	2 3			ǫ2 xx + ǫ2 yy + ǫ2 zz + 2 ǫ2 xy + 2 ǫ2 xz + 2 ǫ2
							I =	1 2	I 2 +	4h I 1 ′ 2	ln	I 1 + I 2 2|h ′ |	.
	Therefore,
				J ǫ =	σ 0 2 √	3	C vol	0 -L	I 2 +	4h I 1 ′ 2	Ln	I 1 + I 2 | 2h ′ |	dx h	.	(B.2)

B

B.

Table C .

 C 10) 2: Experiment and results obtained by different methods for Soldur steel trials.

	Soldur
	Roll diameter	400	mm
	Limit of elasticity	933	Mpa
	Roll speed	25	m/min
	EXPERIMENT
	Samples	E16	E18	E19	E20	E22	E23	E24
	Entry tension (kg/mm 2 )	3.804	7.620	7.843	7.551	2.534	3.170	5.860
	Exit tension (kg/mm 2 )	3.735	5.786	7.794	8.902	5.226	5.009	21.359
	Entry thickness measured (mm)	1.310	1.308	1.312	1.320	1.310	1.309	1.304
	Exit thickness measured (mm)	0.872	1.139	0.987	0.824	1.133	0.981	0.898
	Entry width measured (mm)	60.20	60.20	60.25	60.20	60.25	60.25	60.20
	Exit width measured (mm)	61.40	60.70	61.10	61.35	60.80	61.05	61.00
	Width spread measured (mm)	1.20	0.50	0.85	1.15	0.55	0.80	0.80
	Forward slip measured (%)	4.86	1.79	3.85	4.86	2.38	3.84	3.79
	Lam3-Tec3
	COULOMB longitudinal friction coefficient	0.04961 0.0564 0.0569 0.04879 0.05698 0.05272
	COULOMB transverse friction coefficient	0.0992	0.1128 0.1138 0.09758 0.1139	0.0701
	Width spread calculated by Lam3-Tec3 (mm) 1.29	0.87	1.06	1.36	0.89	1.13
	UBM
	Tresca isotropic friction coefficient	0.085	0.060	0.085	0.075	0.070	0.080	0.043
	Width spread calculated by UBM (mm)	1.136	0.204	0.587	1.343	0.228	0.655	0.914

RésuméDans le but d'optimiser la mise aux milles au laminage à froid, la thèse consiste à développer un modèle prédictif de variation de largeur à la fois précis et rapide pour être utilisable en temps réel. Des nombreux d'efforts ont commencé en 1960s en développant des formules empiriques permettant d'estimer la variation de largeur au laminage. Mais par la suite, la Méthode des Bornes Supérieures (MBS) est devenue la plus connue grâce à sa simplicité et efficacité. A ce sujet, il sera un manque de ne pas parler de[START_REF] Oh | An approximate method for a 3d analysis of rolling[END_REF]] avec le champ de vitesse 3D "simple",[START_REF] Komori | An upper bound method for analysis of three-dimensional deformation in the flat rolling of bars[END_REF]] avec une méthode de combinaison des champs de vitesse et[START_REF] Serek | Analysis of cold and hot plate rolling using dual stream functions[END_REF]] avec le champ de vitesse quadratique.En approfondissant la méthode, le premier résultat obtenu dans la thèse est un modèle 2D (MBS) avec des champs de vitesse oscillante. Ce champ de vitesse particulier a abouti à des résultats (puissance, vitesse...) plus proches de ceux de Lam3-Tec3 que d'autres champs de vitesse étudiés dans la litérature. Pour une modélisation de variation de largeur, j'ai choisi la MBS avec la vitesse 3D "simple" et obtenu un très bon accord avec les expériences réalisées sur des produits étroits à ArcelorMittal[START_REF] Legrand | Towards a better width contral in cold rolling of flat steel strips[END_REF]]. En outre, un nouveau modèle MBS est développé pour une bande bombée et des cylindres droits. Les résultats montrent que la variation de largeur diminue avec la bombée de la bande et correspondent bien à ceux de Lam3-Tec3 . Cependant, la MBS admet un comportement rigide-plastique tandis qu'au laminage des bandes larges les déformations élastique et thermique ont des impacts importants sur la déformation plastique. Afin d'obtenir un modèle rapide, l'idée a été de décomposer la variation de largeur plastique en trois termes: les variations de largeur totales, élastique et thermique et les déterminer par trois nouveaux modèles simplifiés. Les deux premiers permettent d'estimer les variations de largeur élastique et plastique en amont et en aval de l'emprise. Ils donnent aussi les conditions aux limites au modèle d'emprise qui est en effet la MBS avec le champ de vitesse 3D "simple" permettant d'estimer la variation de la largeur totale. En outre, avec les puissances de déformation et de dissipation plastique de frottement données par le même modèle, la variation de largeur thermique est également obtenue. Le modèle de variation de largeur est donc appelée UBM-Slab combiné, très rapide (0,05 s) et prédit avec précision la largeur de variation par rapport à Lam3-Tec3 (<6%).

Quang-Tien Ngo -2015

Width variation problematic in steel rolling1.4 Thesis objective and approaches

Rolling process modeling reviews 2.1 General description of rolling problem

A mix of water with oil at more or less 1-2% of oil concentration, used also to lubricate the roll-bite.

Rolling process modeling reviews 2.2 Typical tribological models

Rolling process modeling reviews 2.3 Typical strip models

Rolling process modeling reviews 2.4 Typical work-roll deformation models

Rolling process modeling reviews2.5 Discussions

Upper Bound Method applied in rolling process 3.1 Principle of the UBM

Upper Bound Method applied in rolling process3.3 2D continuous velocity fields

Upper Bound Method applied in rolling process3.4 Discussions

Upper Bound Method applied in rolling process3.4 DiscussionsQuang-Tien Ngo -2015

Oscillation of mechanical fields in roll bite 4.2 Method for constructing velocity field

Oscillation of mechanical fields in roll bite 4.3 An oscillating velocity field proposal

Oscillation of mechanical fields in roll bite 4.4 UBM with the oscillating velocity field

Oscillation of mechanical fields in roll bite 4.5 Comparison with Lam3-Tec3 and other UBM models

Oscillation of mechanical fields in roll bite 4.6 Conclusions and perspectives

Oscillation of mechanical fields in roll bite

Oscillation of mechanical fields in roll bite4.6 Conclusions and perspectivesQuang-Tien Ngo -2015

Rigid-plastic UBM model for width spread 5.2 3D rigid-plastic UBM for width variation analysis

Rigid-plastic UBM model for width spread5.4 Conclusions and perspectivesQuang-Tien Ngo -2015

UBM for crowned strip rolling6.5 ConclusionQuang-Tien Ngo -2015

A thermal-elastic-plastic width model 7.6 Summary Quang-Tien Ngo -2015

The UBM-Slab combined model validation 8.3 Validation by comparison with industrial observations

The UBM-Slab combined model validation8.4 ConclusionsQuang-Tien Ngo -2015

See more details in section 4.6.2

General conclusions and perspectives9.2 PerspectivesQuang-Tien Ngo -2015

where f (x) is integrand, a and b are the lower and upper limits of integration. The two-point Gauss Quadrature Rule is an extension an approximation where the arguments of the function are not predetermined but as unknowns x 1 and x 2 and the integral is approximated as

The four unknowns x 1 , x 2 , c 1 and c 2 are found by assuming that the formula gives exact results for integrating a general third order polynomial

of which the integral is

On the other hand, the Gauss's integration is

(A.5) So that the two previous expressions are equal for any arbitrary constants a 0 , a 1 , a 2 and a 3 the following conditions need be verified:

A. Numerical Gauss-Legendre integration

(A.7)

A.1.2 Higher-point Gaussian Quadrature Rule

Similarly, the 3-point Gaussian Quadrature Rule is defined by

The coefficients c 1 , c 2 , and c 3 , and the functional arguments x 1 , x 2 , and x 3 are calculated by assuming the formula gives exact expressions for integrating a fifth order polynomial

And the general n-point rules would approximate the integral by

A. Numerical Gauss-Legendre integration

A.2 Applications

x node i-1 = x min + (i -1)∆x and x node i = x min + i∆x. The operator IGauss N 1D ( f , x min , x max , n ele ) is defined by:

where

An other application needed in this thesis is the 2D integral in a area where the bound of the second variable can depend on the first one. We are interested in the following integral of a function f on the area delimited by x = x min , x = x max , y = y min (x) and y = y max (x): and use approach A.12 the integral I 2 can be rewritten as:

(A.14) By applying A.12 for g(x ij ), noting that the l th element in y direction is defined two nodes y node l-1 (x ij ) = y min (x ij ) + (l -1)∆y(x ij ) and y node l (x ij ) = y min (x ij ) + l∆y(x ij ) we have:

Finally, I 2 can be approximately evaluated by the following operator

and y lk = y node l-1 (x ij ) + ∆y(x ij )

Power of plastic deformation