Carole Couhert
email: couhert@enstimac.fr

Sylvain Salvador

J-M Commandré

Impact of torrefaction on syngas production from wood

Keywords:

teaching and research institutions in France or abroad, or from public or private research centers.

Introduction

Thermo-chemical conversion of biomass is a means to produce energy and to reduce greenhouse gases. Rapid gasification of biomass at high temperature is one of the most promising technologies for the production of syngas. It can be achieved in several seconds in an entrained flow reactor. At high temperature (1400 °C), it is possible to obtain a syngas (CO, H 2 and CO 2 ) containing very small amounts of residual hydrocarbons and solid carbon (char). Heat and mass transfers are very effective in this kind of reactor, but the biomass must be dried and ground to particles measuring several hundreds of micrometers prior to injection. These preliminary steps, especially grinding, consume large amounts of energy and represent obstacles that need to be overcome in order to expand the use of biomass in thermochemical processes.

Torrefaction is a technology which allows moisture and low weight organic volatile components of biomass to be removed, producing a hydrophobic solid product with an increased energy density (on a mass basis) and greatly increased grindability [START_REF] Svoboda | Pretreatment and feeding of biomass for pulverized entrained flow gasification[END_REF][START_REF] Bridgeman | Torrefaction of reed canary grass, wheat straw and willow to enhance solid fuel qualities and combustion properties[END_REF][START_REF] Arias | Influence of torrefaction on the grindability and reactivity of woody biomass[END_REF][START_REF] Phuong | Effects of heat treatment on brittleness of styrax tonkinensis wood[END_REF][START_REF] Repellin | Evaluation of heat treated beech by non destructive testing[END_REF]. Electricity requirements for size reduction of torrefied wood are 50-85% smaller in comparison with fresh wood [START_REF] Bergman | Torrefaction for entrained flow gasification of biomass[END_REF][START_REF] Bergman | Torrefaction for biomass upgrading[END_REF].

To reduce solid mechanical properties, torrefaction is generally achieved in the temperature range of 220-300 °C, under an inert atmosphere. It forms three phases: a solid product of a dark colour, an acidic aqueous phase and permanent gases such as CO and CO 2 [START_REF] Bridgeman | Torrefaction of reed canary grass, wheat straw and willow to enhance solid fuel qualities and combustion properties[END_REF][START_REF] Prins | Torrefaction of wood. Part 2. Analysis of products[END_REF], typically 20% of CO and 80% of CO 2 [START_REF] Prins | More efficient biomass gasification via torrefaction[END_REF].

The kinetics of torrefaction reactions in the temperature range of 230-300 °C can be described accurately by a two-step mechanism. The first step is representative of hemicellulose decomposition, while the second step represents cellulose decomposition [START_REF] Prins | Torrefaction of wood. Part 1. Weight loss kinetics[END_REF]. At temperatures above 300-320 °C, fast thermal cracking of cellulose may cause tar formation, so operation below 300 °C is recommended.

Despite torrefaction is an energy consuming treatment, several works can be found on the advantages -in terms of the energy efficiency of a process -of torrefaction as a pre-treatment prior to gasification [START_REF] Prins | More efficient biomass gasification via torrefaction[END_REF]. The concept of torrefaction -aided gasification is similar to two-stage pyrolysis-gasification but the advantage is that the heat requirements for torrefaction are smaller than those for pyrolysis at higher temperatures [START_REF] Prins | More efficient biomass gasification via torrefaction[END_REF]. Some studies on torrefied wood reactivity have been realised under air in TGA or thanks to a burner [START_REF] Bridgeman | Torrefaction of reed canary grass, wheat straw and willow to enhance solid fuel qualities and combustion properties[END_REF][START_REF] Arias | Influence of torrefaction on the grindability and reactivity of woody biomass[END_REF]. However, studies carried out on gasification concern only raw biomass [START_REF] Klose | On the intrinsic reaction rate of biomass char gasification with carbon dioxide and steam[END_REF][START_REF] Blasi | Combustion and gasification rates of lignocellulosic chars[END_REF]. To our knowledge, no works can be found on the impact of torrefaction on the behaviour of wood during gasification by steam at high temperature in an entrained flow reactor, neither in terms of syngas quantities and composition, nor in terms of solid gasification kinetics.

In this paper, gas yields and reaction kinetics obtained during gasification of torrefied woods were studied. Beechwood was submitted to light torrefaction (240 °C) and severe torrefaction (260 °C), using a specially designed crossed fixed bed reactor. After characterisation, initial wood and torrefied woods were gasified in a new high-temperature entrained flow reactor at 1400 °C and 1200 °C for 2 s in an atmosphere containing 20 vol% steam in N 2 . The syngas produced was analysed. The experiments conducted at 1400 °C were modelled using a thermodynamical approach. Experiments carried out at 1200 °C gave information on chemical reaction kinetics.

Experimental

Characterisation of beechwood

Beechwood was supplied by Lignex, France, as 50-500 lm particles. They were stored in plastic bags in a dark place at the temperature of the laboratory. The 80-200 lm fraction was selected by sieving prior to any analysis or experiment. The ultimate analysis of the selected particles was compared to that of the supplied sample and did not deviate by more than 0.5%, attesting to a non-selective sampling.

The wood contained 10% moisture. As indicated by ultimate analysis performed on the dry wood, and reported in Table 1, the wood contained 50.3% of C, 5.3% of H and 44.4% of O on a dry ash free basis (daf). Proximate analysis performed on dry wood indicated a quantity of volatile matter of 84.2% and 0.3% of ash, resulting in a fixed carbon amount of 15.5%.

Torrefaction reactor and operating conditions

Wood particle torrefaction was carried out in a specially designed reactor enabling accurate control of the temperature inside the sample body and rapid removal of the product gas. As illustrated in Fig. 1, the reactor consisted of a 60 mm i.d. quartz tube in which 25 g of wood particles were deposited on a ceramic wood layer. The reactor was surrounded by a 2 kW electrical furnace. The inert atmosphere gas, N 2 , was fed from the bottom at a flowrate of 0.3 l min À1 at STP, heated along the lower part of the reactor and crossing the sample bed. It was expected that this method would ensure a good thermal contact between gas and all particles, while taking the gas products of torrefaction away from the bed (at a Darcy velocity of 0.0034 m s À1 at 250 °C). At the top of the reactor, a ceramic wool swab prevented air from entering the reactor by natural convection effects. The gas temperature upstream of the sample was recorded.

Particles were deposited in the cold reactor. The inert gas flow was started and maintained. The reactor was then heated to the stage temperature, following a ramp at 10 K min À1 . The temperature was maintained for 1 h alternatively at 240 °C and 260 °C; natural cooling was then allowed to occur.

When collecting the torrefied samples from the reactor, a uniform colour was observed inside the sample body -which was darker than the initial wood -attesting to a uniform torrefaction progression. Indeed, other tests performed even on much smaller samples but without inert gas circulation resulted in clearly nonuniform conversion inside the sample body.

Gasification reactor and operating conditions

The gasification of wood and torrefied woods was carried out in a High Temperature Entrained Flow Reactor (HT-EFR) specifically set up to achieve high heating-rate wood gasification. This HT-EFR is electrically heated by a 18 kW three-zone electrical furnace, and is able to reach 1600 °C in a 1 m long isothermal reaction zone. As illustrated in Fig. 2, wood particles were fed at a very constant mass flowrate of 0.5 g min À1 thanks to a new in-house-designed gravimetric feeding system, and continuously injected in the reactor through a hot-oil cooled feeding probe. They were uniformly dispersed on the section of a 75 mm i.d. alumina reactor swept by 22.4 l min À1 at STP of an atmosphere gas containing 20 vol% of steam in N 2 . With such solid and gas flow rates, steam is present in largely stoichiometric conditions. Atmosphere gas was generated by feeding the controlled flows of water and nitrogen in a 2 kW electrical steam generator. This atmosphere gas was then preheated using a 2.5 kW electrical battery of heating elements before reaching the 1 m long isothermal reaction zone, itself heated by the electrical furnace. Gasification took place in the reactor during a controlled residence time (typically several seconds). At 1760 mm downstream of the injection point, gases and particles were sampled by a hot-oil cooled probe, and separated using a settling box and a filter, both heated to avoid water condensation. The water was first condensed in a heat exchanger, and non-condensable gases were forwarded to: a non dispersive IR analyser and FTIR analyser to quantify CO, CO 2 , CH 4 , C 2 H 2 and C 2 H 4 , a TCD to quantify H 2 . During experiments, the atmosphere gas and the reactor walls were heated to 1200 °C or 1400 °C. Measurements of the gas phase temperature inside the reactor indicated temperatures close to those of the reactor walls, with a difference always lower than 30 °C.

Results

Characterization of torrefied woods

Wood was dried overnight at 105 °C, and then torrefied. After torrefaction at 240 °C and 260 °C, the observed mass losses were 11.5% and 21.0%, respectively of the mass of dry wood. The experiments were repeated and the results deviated by less than 1%. These results are in agreement with the literature [START_REF] Bridgeman | Torrefaction of reed canary grass, wheat straw and willow to enhance solid fuel qualities and combustion properties[END_REF][START_REF] Prins | Torrefaction of wood. Part 2. Analysis of products[END_REF][START_REF] Prins | More efficient biomass gasification via torrefaction[END_REF][START_REF] Prins | Torrefaction of wood. Part 1. Weight loss kinetics[END_REF]. But it is interesting to note that the composition of biomass fuels has a significant impact on the amount of solid residue after torrefaction [START_REF] Bridgeman | Torrefaction of reed canary grass, wheat straw and willow to enhance solid fuel qualities and combustion properties[END_REF].

As illustrated in Fig. 3, the torrefied wood showed darker brown colours than the initial wood, tending towards black after severe torrefaction. After the two treatments, the wood particles -observed with an optical microscope -preserved their external shape aspect.

The volatile matter content decreased from 84.2% for wood down to 80.6% and 75.7% for the torrefied woods (Table 1). This is in agreement with the results of Bridgeman et al. who showed that the trend is towards a reduction in volatile matter and an increase in fixed carbon and ash content [START_REF] Bridgeman | Torrefaction of reed canary grass, wheat straw and willow to enhance solid fuel qualities and combustion properties[END_REF]. From the ultimate analysis also reported in Table 1, it can be seen that the O content decreased from 44.4% for wood down to 42.9% and 40.4% for the torrefied woods, which represents a decrease of 3.4% and 9.0% after torrefaction at 240 °C and 260 °C respectively. In terms of the O/C ratio, the value decreased from 0.88 for wood down to 0.83 and 0.74 for the torrefied woods; which is in agreement with the literature [START_REF] Bridgeman | Torrefaction of reed canary grass, wheat straw and willow to enhance solid fuel qualities and combustion properties[END_REF][START_REF] Prins | Torrefaction of wood. Part 2. Analysis of products[END_REF][START_REF] Prins | More efficient biomass gasification via torrefaction[END_REF].

These results attest that torrefaction is a way to increase the energy density of the wood by removing oxygen. This is confirmed by the LCV of the products that were measured in a IKA C5000 bomb calorimeter: 19532 kJ kg À1 and 20742 kJ kg À1 for the torrefied wood at 240 °C and the torrefied wood at 260 °C, respectively, to be compared with 17782 kJ kg À1 for the initial wood, i.e. an increase of 17% (in the case of torrefied wood at 260 °C). Prins et al. showed that the energy density increases with torrefaction temperature and found an increase of 20% at a torrefaction temperature of 280 °C [START_REF] Prins | Torrefaction of wood. Part 2. Analysis of products[END_REF].

Gasification experiments and results

Gasification experiments and modelling at 1400°C

The two torrefied woods -and the initial wood as a referencewere gasified at a temperature enabling a nearly complete conversion to H 2 , CO and CO 2 : 1400 °C. This value was retained from previous experiments. The gas volume fractions in the flue gas were converted into yields, on a dry ash free (daf) basis, and reported in Fig. 4. Each experiment was repeated two or three times enabling repeatability error bars to be drawn on the graph. The analysis indicates that the gasification was actually complete: the concentration of CH 4 and other hydrocarbons like C 2 H 2 and C 2 H 4 were inferior to 30 ppm, the detection limit of the analysers. This corresponds to quantities of less than 0.2% of the dry wood. The two torrefied woods produced more H 2 and CO than the parent wood, respectively 7% and 20% more, which was an expected result since their O content was lower than that of wood. CO 2 yields were quite similar for the wood and the torrefied woods. As far as the solid residue is concerned, only grey coloured ash was recovered after gasification, in such small quantities that collecting and weighing them would not have been accurate.

Given the small size of the particles and the high operation temperature, a thermo-dynamical equilibrium type model was selected to describe the previous experiments. The FactSage Ò software with the ''FACT 5.3" data base (4429 species) was used to this purpose. The inputs of the model were:

-the quantities of C, H and O contained in the dried wood as determined from ultimate analysis; -the quantities of H and O contained by steam and of N in the atmosphere; -the temperature of gasification; -the pressure of the reactor (atmospheric).

The results are reported in Fig. 5 together with experimental results. They show that model and experiments are in agreement concerning H 2 and CO 2 yields. The CO yields obtained with the model are higher than those obtained during experiments. This model shows that carbon mass balance is not closed -with an error of 8% -during experiments. This error can be attributed to the sum of uncertainties resulting from ultimate analysis of the woods and of gas sampling and analysis.

As regards CH 4 and hydrocarbon amounts produced, the model predicts no CH 4 no C 2 H 4 and no C 2 H 2 . This is in agreement with the results obtained experimentally.

It can be concluded that the thermo-dynamical equilibrium approach can be retained in such conditions to describe the gas yields from woody and torrefied wood biomass.

Gasification experiments at 1200 °C -kinetic aspects

The following section aims at answering a question: does torrefied wood require more severe conditions for gasification to operate? Indeed, the previous experiments of complete gasification did not provide information on this point. The gasification was therefore also operated at a lower temperature, 1200 °C, to compare the gasification progress in the cases of wood and of torrefied woods.

The results reported in Fig. 6 show that gas yields of wood and torrefied woods are quite similar. However, CO and H 2 yields of torrefied woods are slightly higher than those of wood, as was the case with experiments at 1400 °C where equilibrium was reached. It can be noted that CO 2 , CO and H 2 yields are lower at 1200 °C than at 1400 °C, showing that gasification is not finished at 1200 °C.

During rapid high temperature pyrolysis of thin wood particles, CH 4 is produced in quantities of about 6% of the dry wood [START_REF] Couhert | Pyrolyse flash à haute température de la biomasse lignocellulosique et de ses composés-production de gaz de synthèse[END_REF][START_REF] Couhert | Is it possible to predict gas yields of any biomass after rapid pyrolysis at high temperature from its composition in cellulose, hemicellulose and lignin[END_REF][START_REF] Dupont | Vapogazéification de la biomasse: contribution à l'étude phénoménologique entre 800 et 1000[END_REF]. CH 4 is then cracked and reformed by steam and converted to CO and H 2 , but this species is the most thermally stable hydrocarbon species and takes the longest time to reform by steam. However, it can be observed in Fig. 6 that the residual amounts of CH 4 are similar after gasification of wood and torrefied woods, which confirms that chemical reaction kinetics in gas phase are quite similar.

As regards the amounts of residual carbon, i.e. carbon that was not gasified into CO and H 2 by steam, the quantities collected were approximately 1.1% of the initial mass of wood and 4.8% of the initial mass of torrefied wood. Sampling of solid residue in the entrained flow reactor was not isokinetic. Therefore, these results are not very accurate but clearly show that more char is collected after gasification of torrefied woods. Moreover, proximate analysis performed on all solid residues showed that the char from wood contained 66% of ash, whereas the chars from torrefied woods contained 26% of ash. Chars from torrefied woods contained more residual carbon, which confirms that they are less reactive than the char from wood. One could think of using the ash tracer method to calculate the conversion progress of char gasification in the two cases of wood and torrefied wood. Nevertheless, the calculation would not be accurate enough, regarding the very low ash content of the parent wood.

Conclusions

The gasification for 2 s at 1400 °C of a wood and of its product after torrefaction with two levels of severity indicated that the torrefied woods produce approximately the same quantities of CO 2 , 7% more H 2 and 20% more CO than the parent wood. In these conditions where thermo-dynamical equilibrium is reached, this Gas yield (g/g dry wood)

beechwood torrefied beechwood 240°C torrefied beechwood 260°C Gas yield (g/g dry wood)

beechwood torrefied beechwood 240°C torrefied beechwood 260°C Fig. 6. Gas yields obtained during gasification of beechwood and torrefied beechwoods at 1200 °C under 20% of steam.

result is consistent with the increase in the C and H content induced by torrefaction.

From gasification experiments performed at a lower temperature, 1200 °C, it can be stated that the kinetics of gasification in gas phase of torrefied wood is comparable to that of the parent wood. However, the chars from torrefied woods are less reactive towards steam than the char from wood.

Fig. 1 .

 1 Fig. 1. Torrefaction reactor.

Fig. 2 .

 2 Fig. 2. Gasification reactor.

Fig. 3 .

 3 Fig. 3. Photography of particles of beechwood (a), torrefied beechwood at 240 °C (b) and torrefied beechwood at 260 °C (c).

Fig. 4 .Fig. 5 .

 45 Fig. 4. Gas yields obtained during gasification of beechwood and torrefiedbeechwoods at 1400 °C under 20% of steam.

Table 1

 1 Ultimate analysis and proximate analysis of wood and torrefied woods.

		Ultimate analysis (daf mass %)				Proximate analysis (dry mass %)
		C	H	N	S	O	Ash	Volatile matters	Fixed carbon
	Beechwood	50.3	5.3	0	0	44.4	0.3	84.2	15.5
	Torrefied beechwood 240 °C	51.7	5.4	0	0	42.9	0.35	80.6	19.2
	Torrefied beechwood 260 °C	54.4	5.2	0	0	40.4	0.4	75.7	24.2

Acknowledgement

We wish to thank the Agence Nationale de la Recherche for having funded this project.


Sylvain Salvador

Jean-Henry Ferrasse

A Practical Method to Derive Sample Temperature during Nonisothermal Coupled Thermogravimetry Analysis and Differential Scanning Calorimetry Experiments

Nonisothermal thermogravimetry differential scanning calorimetry (TG-DSC) mounting is intensively used for the determination of kinetic parameters and reaction heat along the chemical transformation of a solid. Nevertheless, when tests are performed with heating rates as high as those encountered in industrial processes, e.g., several tens of K min -1 , there is great uncertainty in the knowledge of the exact sample temperature. In this work, a method to derive a simple mathematical expression is proposed and fully described in order to calculate the real sample temperature throughout a temperature-ramped test on a commercial apparatus. The furnace temperature and the heat flow signals were used, together with the crucible specific heat and the heating rate. A number of validation tests were performed to derive similar reaction rates for a reference. Firstorder kinetic reactions were presented and reconciled over a large range of heating rates from 3 to 50 K min -1 .

Introduction

Temperature-ramped thermogravimetry (TG) analysis is an efficient way to derive, from a single experiment, kinetic parameters for chemical reactions such as thermal degradation over a wide range of temperature. Coupling mass information to heat flow information additionally facilitates the derivation of the reaction heat associated with the transformation (differential scanning calorimetry (DSC)). For technical reasons, it is difficult in coupled TG-DSC mountings to measure the sample temperature while weighing it continuously with high precision. The temperature is, therefore, usually measured from a thermocouple placed either inside the atmosphere gas or inserted into the furnace wall. However, it is established that there can exist a temperature gap between the furnace and the crucible as high as several tens of degrees. This causes a problem since the relevant temperature for kinetic parameter determination is the sample temperature [1]. High heating rates favor this difference.

A number of technical procedures have been proposed to solve this problem. They are mainly based on different experiments, one for mass recording and some others for tem-perature recording [2,3]. Subramanian [4] proposed a mounting with optical measurement of temperatures, without direct contact of the probe with the crucible. The effect of emissive properties of materials now needs to be investigated more thoroughly to check the reliability of measurements.

In contrast to purely technical solutions, some authors tried mathematical corrections. Sigrist [5] developed a black-box type of model for the heat flowmeter crucible mounting. The author proposes an apparatus transfer function that links melting points of standards (inputs) with the recorded signals during experiments (outputs). The author, nevertheless, notices himself that the apparatus function varies according to the data set used to calculate them. Dong et al. [START_REF] Dong | Proc. of the 14 th Symp. on Thermophysical Properties[END_REF] proposed a numerical model using enthalpy as a function of temperature as an input. A temperature is then scanned or modulated. The response of the calorimeter is calculated. This predicts the sample thermocouple temperature and the temperature difference between the sample and reference thermocouples as a function of time and temperature.

In this paper, a practical method is proposed for establishing a simple mathematical expression dedicated to the TG-DSC apparatus used, to calculate throughout an experiment the temperature difference, TD, between the sample and the furnace. The method is based on a description of the net heat flow recorded as the sum of a crucible inertia heat flow, a sample inertia heat flow, and a chemical reaction heat flow. The expression is determined from a heat balance over the heat flowmeter, on the one hand, and from a heat balance over the crucible, on the other hand. A simplified heat transfer model is used. As detailed below, it is possible to fit the parameters appearing in the mathematical expression through the reconciliation of the kinetic parameters determined at two different heating rates for a reference chemical reaction.

Deriving Mathematical Expression

SETARAM TG-DSC 111 Mounting

The general arrangement of the TG-DSC coupling system used is shown in Fig. 1 (for details, see manufacturer (www.setaram.com)). A similar mounting is placed beside the one represented here (sample mounting), and generally operates with an empty crucible or a reference material (reference mounting). The recorded signals are the temperature measured inside the furnace wall, T f , the crucible plus sample mass, m C + m S , and the raw heat flow, f R-REC . As detailed further, this flow signal is the difference between the sample heat flow thermopile signal, f R , and the reference heat flow thermopile signal, f REF .

Assumptions

a) It is assumed that the temperature of the crucible and the temperature of the sample remain identical and equal to T C . For a detailed calculation based on the thermal Biot number, the reader can refer to [START_REF] Ferrasse | [END_REF]. b) Convection heat transfer between the crucible and the furnace is generally supposed to be more efficient than radiation as long as the temperature remains below approximately 500 °C. This was confirmed with the calculation of the global heat transfer coefficient (see annex 1). c) For the low gas flow rates typically used, the flow inside the furnace is laminar. The heat exchange between the crucible and the furnace can be described as heat conduction through the atmosphere gas film that separates them. The validity of this assumption is stated (see also annex 1). Indeed, the Nusselt number for cocylindrical flow only depends on the diameter ratio and wall boundary [START_REF] Incropera | Fundamentals of Heat and Mass Transfer[END_REF]. d) The heat flowmeter presents a pure time delay, e.g., its flow signal is time delayed, by s HFM seconds, as compared with the actual heat flow leaving the crucible [START_REF] Calvet | Microcalorimétrie. Applications Physico-Chimiques et Biologiques[END_REF].

Mathematical Modeling

The mathematical modeling is only based on several enthalpy balances.

The enthalpy balance for the flowmeter (domain F in Fig. 1) gives 1) :

f R M D M s dT c dt f Ch f g (1)
where f R is the raw heat flow (measured by the sample thermopile), the first term on the right-hand side is the thermal inertia of the crucible + sample, f Ch is the chemical reaction heat flow, and f g is the heat flow resulting from the enthalpy variation of the sweeping gas entering and leaving the domain. The term f g is suppressed by running a blank test in which an empty crucible is heated under the same temperature ramp. During this blank test, the heat flow of the sample thermopile can be expressed as:

f B = M C b + f g (2)
In Eqs. ( 1) and ( 2), f g can be considered as equal. It should be borne in mind that an error might be introduced by the fact that the total exit gas flow changes when some gas is produced by the reaction. With the assumption made above, the heat flow due to the product gas enthalpy is incorporated into f Ch .

The flow signal recorded by the apparatus, f R-REC , is the difference between the sample thermopile signal f R and the reference thermopile signal f REF . Thus, the recorded raw signal is: List of symbols at the end of the paper.

f R-REC = f R -f REF
and the recorded blank signal is:

f B-REC = f B -f REF
Substracting the recorded blank flow f B-REC from the recorded raw flow f R-REC gives the "recorded" net flow:

f N-REC f R-REC À f B-REC M C M s dT c dt f Ch À M C b (3)
since the reference flow signal f REF is the same during the experiment and during the blank test.

An enthalpy balance for the crucible + sample (domain C in Fig. 1) gives:

M C M s dT c dt f Ch À f C-F 0 (4)
where f C-F is the exchanged flow between the crucible and the furnace.

Following assumption (c), the term f C-F will be expressed as:

f C-F = Ak f (T W -T C ) ( 5
)
where A is a constant that characterizes the TG-DSC apparatus under the atmosphere flow conditions used, and k f is the thermal conductivity of the atmosphere gas. In the following, this conductivity will be assessed as a polynomial function of the furnace temperature. Eq. ( 5) shows that there exists a temperature difference due to heat transfer, TD HT , between the crucible and the furnace. This difference can be expressed from Eqs. ( 3), (4), and (5):

TD HT f N-REC M c b A k f (6)
In addition, the heat flow measurement system globally introduces a response time s HFM (see assumption (d)) which will be responsible for another temperature difference between the crucible and the furnace, TD HFM [START_REF] Calvet | Microcalorimétrie. Applications Physico-Chimiques et Biologiques[END_REF]. This term can be expressed from the flowmeter time delay s HFM and from the heating rate b:

TD HFM = s HFM b (7)
Thus, the total temperature difference between the crucible and the furnace is: TD = T f -T c = TD HT + TD HFM [START_REF] Incropera | Fundamentals of Heat and Mass Transfer[END_REF] Moreover, the chemical reaction heat flow can be expressed as follows if the reaction heat DH is in J kg lost -1 :

f Ch dm S dt DH (9)
Finally, the net recorded flow can be rewritten from Eqs. (3) and ( 9):

f NÀREC M C M s dT c dt dm S dt DH À M C b (10)
In Eq. ( 10), it appears that the net recorded flow is the sum of three separate heat flows. The method consists in calculating each of these flow contributions and in summing them to calculate a net flow f N-c . The unknown parameters (DH, cp, A) are determined by fitting f N-REC and f N-c on a standard experiment.

Then, TD can be calculated by adjusting s HFM with a second standard experiment.

Experiments

In the proposed method, the standard experiments are conducted with a reference material. 30 mg of calcium oxalate are heated at 3 K min -1 (first experiment) and 30 K min -1 (second experiment) under a nitrogen atmosphere; the flow rate is 54 mL min -1 at STP conditions.

In the temperature range from 150 to 350 °C, calcium oxalate undergoes a decomposition to carbonate, responsible for approximately 12 % mass loss. The reaction heat is reported as -400 kJ kg oxalate -1 , which is equivalent to 3.33 MJ kg lost -1 .

Validation of the Model

Determination of Constants in Expression of TD

The recorded net heat flow f N-REC and the DTG signal (derived directly from the mass signal by the apparatus software) are plotted in Fig. 2; they reveal a time shift of approximately 20 s between the peak values of the two signals. According to Eq. ( 9), the time shift between the two signals is physically unsatisfactory, confirming that f N-REC includes inertia terms. It is also possible to calculate and to plot in Fig. 2 two of the three terms on the right-hand side of Eq. ( 10) since: -If A is fixed (initialization), the crucible temperature can be calculated from Eq. ( 6). M C is known; M S can be calculated from the sample mass m S recorded during the test, if cp S if fixed (initialization). The first term can then be calculated. -The chemical reaction heat, the second term in Eq. [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF], can be calculated if a reaction heat DH is fixed. -For the third term, the thermal inertia of the crucible is a known constant.

A problem may arise due to the fact that three unknowns are to be fitted: A, DH, and cp S . Nevertheless, they can be determined through a series of easy trials and adjusted for the three terms. Indeed: -When the chemical reaction is over, the net flow f N-c is sensitive only to cp S . This specific heat is directly determined by fitting f N-c and f N-REC in the pre-or postreaction zone, which is a common way of measuring specific heat when no reaction occurs. -DH acts on the peak intensity of f N-c and on the left-right position of the peak. -A only acts on the left-right position of the f N-c peak.

It is, therefore, possible to fix DH, A, and cp S separately. Several iterations lead to the f N-c curve in Fig. 2, deriving the values:

A = 0.140 m DH = -3.76 MJ kg -1 cp S = 1100 J kg -1 K -1
These values could be fitted without ambiguity within ±2 %. The experimental net flow f N-REC was satisfactorily reconstructed from the sum of the three terms in Eq. [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF] (see Fig. 2).

It is noteworthy that the inertia heat flow of the crucible + sample is important as compared with the chemical reaction heat flow f Ch , which makes it important to run a high quality blank test. Calculating M C and M S indicates that the sample is responsible for only approximately 1 % of the total inertia.

Concerning oxalate, it is assumed that the chemical reaction follows first-order kinetics. It is then possible to calculate the reaction rate k at each temperature from Eq. ( 11) since dm/dt and m are known:

dm dt À k m À A 0 exp À E R T m ( 11
)
where m is the reactive remaining mass, or m sm final .

The values of k along the test performed at 30 K min -1 , and those for the test performed at 3 K min -1 , using the furnace temperature in the abscissa (1/T f ) are plotted in Fig. 3a). The plot exhibits straight lines, which indicates that the first-order kinetics is a suitable model to describe the reaction. Nevertheless, large differences in values of k for a given temperature are observed, due to the error introduced by the use of the furnace temperature instead of the actual sample temperature. As f N-c is not sensitive to s HFM , (that appears only in Eq. ( 7)), it is possible to completely reconcile the reaction rates obtained at two heating rates (see Fig. 3b)), using a value of 12.0 s for s HFM .

The temperature correction that finally has to be applied is:

TD T f À T c f N-REC M c b 0:140 k g 12:0 b (13)
The temperature difference TD of the crucible towards the furnace during the two experiments can reach a value as high as 35 °C during the test at 30 K min -1 . The TD during the test at 3 K min -1 is much lower but still not negligible. It has to be noticed that, assuming the chemical reaction heat flow f Ch is set equal to zero, TD significantly decreases with increasing furnace temperature, this being only due to the increase in k f with temperature.

Validation of Method

At first, the constants determined through the procedure have a physical meaning: -The calculated chemical reaction heat flow signal f Ch is time-synchronized with the mass derivative, which satisfies the basic physical concept of reaction heat in Eq. ( 9). -The reaction heat derived is equal (within the fitting error, or ±2 %) to that given by the classical integration of the surface between the net flow curve f N-REC and the base line represented in Fig. 2. -The value of 1100 for the sample-specific heat cp S is a suitable value.

To check the validity of the calculation of TD, two additional tests with heating rates of 15 and 50 K min -1 were performed. In Fig. 4, the values of the reaction rate k from the experiments at 3, 15, 30 and 50 K min -1 were plotted versus the reverse of the crucible temperature calculated using the expression of TD in Eq. ( 13). Whatever the heating rate operated, the values of k are very similar; this attests for the good description of TD.

Finally, another validation test was performed with potassium perchlorate. This calibration material is reported to undergo a transition at the reference temperature of 300 °C. The transformation occurs without mass change of the sample. The product was heated at 30 K min -1 , under a 54 mL min -1 flow of N 2 . Fig. 5 illustrates the evolution of the furnace temperature and that of the crucible, calculated using the expression of TD derived previously. The net recorded heat flow has also been plotted. It is generally admitted that the period where a transition occurs or where the smelting of a metal is isothermal, is the period during which the heat flow curve is linear. This zone is indicated in the figure thanks to two vertical dotted lines. During this period, the calculated temperature of the crucible indicates a plateau around 305 °C. This value is very close to the reference temperature of 300 °C, while, during the same period, the furnace temperature is between 325 and 332 °C. The correction using TD appears to give a much better estimation of the actual sample temperature.

Conclusion

It is possible to derive a simple mathematical expression to calculate the actual crucible. This expression requires the recorded net heat flow data and the heating rate as inputs plus a number of constants that characterize the TG-DSC mounting, whose method of determination has been detailed. This determination necessitates conduction of two simple experiments with different heating rates, and a simple posttreatment of the data in a series of manual trials to adjust four parameters whose independence has been checked. For clarity, the procedure is summarized in the appendix.

The method can be generalized to other TG-DSC mountings in which the crucible is separated from the heating source by an air film. In the case of the apparatus used here and a standard 5 mm outer diameter platinum crucible, the expression for the temperature difference between the crucible and the furnace is: enable the physical meanings to be ascribed to the derived mathematical expression.

TD T f À T c f N-
For practical purposes, Fig. 6 established from the mathematical expression gives the temperature difference between the crucible and the furnace as a function of the net heat flow (given by the apparatus along an experiment) for different heating rates. It confirms the necessity in most of the cases to operate a furnace temperature correction such as the one studied here.

Finally, for oxalate, it is shown that the expression of TD to calculate the sample temperature allows derivation of similar reaction rates for a first-order kinetic reaction over a wide range of heating rates: 3 to 50 K min -1 . • adjust s HFM to reconcile the two curves for k.

Received: January 17, 2006

Symbols used

A [m] apparatus constant cp [J kg -1 K -1 ] specific heat D [m] diameter k [s -1 ] reaction rate l [m] height of crucible m [kg] mass M [J K -1 ] heat capacity = m*cp S [m 2 ] surface T [K] temperature TD [K] temperature difference Greek letters f [W] heat flow b [K s -1 ] heating rate k [W m -1 K -1 ]
The expression

TD T f À T c f N-REC M c b A k g s HFM b
is fully determined.

Annex 1

To calculate the equivalent heat transfer coefficient at the surface of the crucible h, the resistance between the crucible and the surroundings can be expressed as follows, using the crucible outside surface S c (wall + bottom):

R CÀf 1 h S c (16)
The two expressions of the resistance in Eqs. ( 15) and (16) lead to h = 31.5 W m -2 K -1 .

The validity of the assumption of conductive heat transfer between the crucible and the furnace wall was checked by running an experiment at 30 K min -1 with a different N 2 flow: 20 mL min -1 at STP conditions. The value of constant A fitted from this experiment was again 0.14. The fact that the heat transfer coefficient does not depend upon the flow velocity attests for the mainly conductive -and not convective -nature of the heat transfer.

The contribution of radiation to the heat exchange between the crucible and the furnace wall can be estimated at this stage. The global heat transfer coefficient h can be expressed as the sum of a conduction heat transfer h cond and of a radiation heat transfer h rad :

h = h cond + h rad ( 17
)
The value of h rad can be estimated from the following two expressions of the radiative heat flow between the furnace and the crucible:

e f e c r T 4 f À T 4 c À Á h rad T f À T 4 c À Á (18)
where the emissivities e f and e f are taken from [START_REF] Siegel | Thermal Radiation Heat Transfer[END_REF], and r is the Boltzmann constant. At the reaction peak, for instance, the furnace temperature is T f = 275 °C and the crucible temperature is T c = 311 °C. This gives a value for h rad of 3.09 W m -2 K -1 . This shows, by comparison with the global heat transfer coefficient h = 31.5 W m -2 K -1 , that the contribution of radiation in this range of temperature is small. As far as the heat transfer between the crucible and the furnace is concerned, a reference can be found in terms of the equivalent conduction resistance to cross the atmosphere gas film separating the crucible and the furnace wall. This resistance can be expressed in cylindrical geometry:

R cond 1 2 p k f l ln D f D C ( 14
)
At the reaction peak intensity, for example, where T f = 275 °C and k f = 0.0436 W m -1 K -1 , its value is 143 K W -1 . On the other hand, Eq. ( 5) is equivalent to:

R 1 A k f ( 15
)
With the fitted value A= 0.140, this gives R = 164 K W -1 . The similarity in the values of R and R cond attests for the physical significance of the value of A that was fitted.

Figure 1 .

 1 Figure 1. Arrangement of TG-DSC apparatus. (1) crucible, (2) sample, (3) thermopile, (4) furnace, (5) atmosphere gas, (6) heating elements, (7) weighing system, (8) furnace thermocouple. A similar mounting (reference crucible) is located beside this one. -1)List of symbols at the end of the paper.

Figure 2 .

 2 Figure 2. Time evolution for oxalate of (+) the recorded net heat flow f N-REC (W), (×) the sample mass derivative, (-) the calculated net heat flow f N-c (W), () the chemical reaction heat flow f Ch (W), (᭺) the crucible + sample inertia heat flow (M C + M S )dT C /dt (W), (...) arbitrary base line for heat integration.

Figure 3 .

 3 Figure 3. Reaction rate for decomposition to carbonate of calcium oxalate reaction during experiments with heating rates of (-) 3 K min -1 and (-) 30 K min -1 . (a) Abscissa is 1/T f , (b) 1/T C with T c = T f -TD and s HFM = 12 s.

  REC M c b 0:140 k g 12:0 b where f N-REC is the blank corrected heat flow and b the heating rate. The decomposition of the net heat flow into a sum of identified heat flows also has the advantage of leading to the determination of the reaction heat and of the sample specific heat. This and other considerations

Figure 4 .

 4 Figure 4. Reaction rate for monoxide release from calcium oxalate reaction during experiments with heating rates of 3, 15, 30, and 50 K min -1 versus sample corrected temperature.

Figure 5 .

 5 Figure 5. Temperature of furnace (-) and calculated temperature for crucible (-) during transition reaction of potassium perchlorate, under nitrogen, at 30 K min -1 . (-) Net recorded heat flow f N-REC versus time. Vertical line indicates linear zone of heat flow.

3 -

 3 This paper's data m c = 9.66 10 -4 kg cp C = 133 J kg -1 K -1 S c = 1.770 10 -4 m 2 (crucible belt + bottom) e C = 0.25 e f = 0.M C M s dT c dt where dT c dt is obtained deriving numerically T C . with T C = T F -TD where TD TD HT TD HFM f N-REC M c b A k g s HFM b in which A, cp S (in M S ), and s HFM are initialized (see results presented in this paper); f Ch dm S dt DH in which DH is initialized; -M C b is a known constant. • Adjust cp S , A and DH to fit with f N-c and f N-REC . • Plot on the same graph the values of k versus 1/T C (see Fig. 3(b)) for the two experiments at 3 and 30 K min -1 , (T C = T f -TD).

Appendix

Summary of the procedure

• Run two experiments with 30 mg of calcium oxalate under nitrogen at 3 and 30 K min -1 .

• Run the blanks.

• Calculate the recorded net heat flows f N-REC (classical correction of an experiment with a blank test).

• Plot on the same graph f N-REC for the experiment at 30 K min -1 and f N-c , sum of the terms on the right-hand side of Eq. ( 10), e.g.:


Prof Alain Daidie
email: alain.daidie@insa-toulouse.fr

Dr Manuel Paredes
email: manuel.paredes@insa-toulouse.fr

Clément Chirol
email: clement.chirol@airbus.com

Victor Achard
email: victor.achard@insa-toulouse.fr

Prof. Dr Victor Achard Corresponding-Author

Optimization of the Cold Expansion Process for Titanium Holes

Cold expansion of metallic holes is widely used in the aerospace industry as an efficient

industrial process for improving the fatigue performance of mechanical assemblies. To increase the efficiency of the process, it is worth considering the possibility of carrying out the expansion twice in succession in the same hole. This paper discusses the consequences of such an operation on a single titanium hole through a numerical and experimental study.

More precisely, dedicated finite element modeling of the cold expansion process is developed in order to simulate cases where the expansion is carried out twice: in the same direction and in opposite directions. The residual fields generated are analyzed and the results compared with experimental results. Furthermore, in this study, we propose to assess a new methodology for measuring the intensity of the compressive residual stresses generated at the hole edge. Specifically, we use the incremental hole drilling technique to characterize the expanded holes, with or without a reaming operation of the cold worked hole. Results show that double expansion has a very positive impact on the homogeneity of the compressive fields generated as long as the direction of expansion is properly selected to guarantee the efficiency of the process.

Introduction

Aeronautical structures consist of large numbers of riveted or bolted sub-assemblies that require the presence of fastening holes. These sections are often critical in the design and act as major sources of stress concentration where fatigue damage starts. [1] This study focuses on optimizing the fatigue performance of bolted joints using hole cold expansion. Cold expansion involves the generation of compressive residual stress fields close to the hole edge.

The material is thus subjected to compressive and tensile residual stresses that are in overall equilibrium, the compressive stresses generally being balanced by a tensile field away from the hole. [2] To be optimal, radial expansion of the hole should induce circumferential stresses close to the compressive limit at which the material yields. They thus compensate for the mean value of future tensile fatigue loading and reduce the criticality of these joints. Today, the cold expansion process is used almost exclusively for aluminum holes. However, optimization of the primary structure of aircraft inevitably leads to the use of titanium alloys. As a result, the number of parts and assemblies made from this material is observed to be increasing exponentially in modern aircraft. As they are very expensive and used in critical areas where loading levels are high, it would be highly desirable to provide processes and tools to improve their fatigue performance. In the aeronautical industry, Ti-6Al-4V alloys are found in the great majority of titanium structural applications and are therefore considered in this study. In the literature, few studies describe the implementation of expansion processes for hard alloys. No precise analysis was made of the response of hard metals subjected to expansion and only studies of specific cases can be found, whereas generic processes designed for the expansion of holes in aluminium alloys were considered.

In addition, it is difficult to quantify the fatigue gains obtained in these studies as the various configurations tested involved many parameters influencing fatigue behaviour. However, the previous studies that exist have demonstrated their effectiveness, in particular for titanium alloys where fourfold increases may be observed in the lifespan of cold expanded holes. [3,4,5,6,7] Among the existing processes designed to generate extensive and controllable residual stress fields at the edge of the hole while minimizing damage to the hole surface, split sleeve cold expansion remains the most widely used in industry for aluminum applications. [6,[START_REF] Reid | European Conference for Aerospace Sciences (EUCASS)[END_REF][START_REF] Karabin | [END_REF] This process greatly improves the fatigue performance of the hole. For example, it is commonly noted that the lifespan of a hole that has been cold expanded using the split sleeve process is three to ten times that of an "as drilled" hole in aluminum alloys such as 2024 T351. [10,11] From a technological point of view, this process can basically be divided into six key steps.

First of all, after dimensional checking of the original hole, a pre-lubricated split sleeve is placed on an oversized tapered mandrel, which is connected to the hydraulic actuator. The split sleeve is slid along the maximum diameter of the mandrel and positioned on the minimum diameter so that the tooling can be inserted from one side of the hole. The actuator is subsequently triggered and the mandrel is pulled through the sleeve and the hole, performing the expansion. In the split sleeve process, the sleeve is then removed from its housing. Here, a pre-lubricated sleeve is useful to reduce the axial effort involved in pulling the mandrel and to protect the contact surface. Finally, the hole is reamed to its functional diameter and checked. It is then ready to receive a fastener, for example. The theoretical expansion ratio applied (IE) is defined by relation (1).

= ∅ ∅ ∅ (1)
Where the tooling diameter corresponds to the maximum diameter of the mandrel and sleeve assembly and the initial diameter is the one measured after initial drilling of the specimens.

Despite the numerous advantages of this method, e.g. its rapidity of execution, the low cost involved and the high reliability achieved, a study of the residual field generated by expansion reveals significant heterogeneity of the circumferential stress in the thickness of the hole treated as the one-sidedness of the expansion application induces major axial loads and causes significant side effects. Unfortunately, it also turns out that this circumferential component is responsible for the moderation of stress concentration existing in a loaded drilled plate. More precisely, this heterogeneity is mainly characterized by a decrease in the residual circumferential stresses at the side where the mandrel enters. These poorly expanded areas may reduce the efficiency of the process, the areas exerting the least compression being the most vulnerable. This phenomenon is generally identified and quantified using numerical studies, which are numerous in the literature concerning the split sleeve expansion process. [12,13] Numerical methods such as finite element modeling appear to be good tools for predicting the response of the material and particularly the residual fields generated (stress, strain), on condition that the modeling strategies are sound. The modeling choices must be clearly selected in order to simulate highly non-linear behaviors such as high plastic flows and high strain rates. In the case of hard alloys, we found only three numerical studies in the literature. [5,14,[START_REF] Reid | European Conference for Aerospace Sciences (EUCASS)[END_REF].

This heterogeneity of stresses in the thickness of the hole can be linked theoretically to a reduced fatigue benefit provided by cold expansion. Given the high service loads supported by titanium aerospace components, even a small decrease in the intensity of the compressive stress can lead to significant reductions of lifespan. Considering titanium alloy, the combination of low modulus and high elastic strength is likely to aggravate the phenomenon of reduction of stresses at the mandrel entrance. Thus, in order to achieve even greater benefits, it appears interesting to find ways to improve the homogeneity of these compressive fields while considering the industrial advantages of the split sleeve process. In this study, a simple, low cost methodology is evaluated, which is intended to achieve two successive expansions of the same hole, before its final reaming to the functional diameter. The hope is that the radial expansion of titanium holes will thus be optimized. This kind of methodology has already been studied in cold expansion of aluminum holes, where a study is particularly noteworthy. [14] These authors were able to show the beneficial effect of double expansion in opposite directions using conical pins. On the other hand, other studies have shown the benefit of multiple cold expansions on the stress fields generated to be variable. [16,17,18] In the field of expansion, the characterization of stress fields is a very complex challenge.

According to our experience, the stress fields and the areas affected by cold expansion are undetectable by micrograph and microstructural analysis. The stress fields are not "visible" within the material and are the result of a series of balancing/unbalancing effects of the residual stress and strain fields in the material. Furthermore, machining of the parts may involve additional residual fields that can significantly affect those generated by cold expansion. The main objective is to ascertain the distribution and intensity of hoop stresses from the edge of the expanded hole and determine the potential improvement in the fatigue strength of the hole. Validation of predictive models of the residual fields is also very important and can enable fatigue models to be validated for cold expanded holes. Today there is a real need to develop a dedicated method for the study of cold expanded holes. This interest is shared by both the scientific and industrial communities. The main objectives are to limit the amount of expensive fatigue testing (that guarantees and quantifies the effectiveness of a process), to test numerous experimental configurations (new materials, new geometries) and environments (measurement of the relaxation during fatigue, after exposure to temperature or after reaming), and finally to reach a precise characterization of the expanded holes in terms of the stress fields generated. On the other hand, these methodologies can be used to assess the effectiveness of a cold expansion process and think about the best methodologies for extending the process to hard metals, as proposed in this study.

We can distinguish among the measuring methods according to whether they are nondestructive, destructive or semi-destructive (that do not imply the destruction or significant alteration of the structure where the stress is measured). It appears that these methods are interesting for the measurement of the deformation in the material, stress being a mathematical concept that should be deduced from these measurements. In the elastic domain, the values given by the deformation tensor can be connected with the internal stress tensor via the generalized Hooke's law. It is important to note that the destructive and semi-destructive methods measure the through-thickness stress/strain values, while the non-destructive methods provide information on the state of the residual fields at the surface or at shallow depth (few µm). Therefore, except for neutron diffraction technology or ultrasonic scanning, they cannot account for the stresses prevailing in the material at depths sufficient to eliminate surface effects that are not representative of the overall behavior of the hole. In addition, in some alloys, X-ray methods are highly dispersive because they are very dependent on the orientation of the grain. On the other hand, optical methods are very useful to describe the distribution of strain fields in the plane of the part to be studied. One study, in particular, showed the capacity, in terms of results and feasibility, of the stereo-correlation hole edge for stamping and expansion. [19,20] Other studies have used Speckle or Moiré interferometry, or photoelasticimetry. [21,[START_REF] Doyle | Manual on Experimental Stress Analysis[END_REF] One last original methodology proposed by consists of setting up an expanded grid around the area to measure deformations in the plane of the expanded hole and thus work back to the stress undergone by the periphery of the expanded bore through continuum mechanics. [START_REF] Beaver | EIS 1 Measurement and Fatigue[END_REF] In other circumstances, destructive methods may be of interest. We first identified the "contour" method. This method aims to achieve residual stresses in one direction. First, the section is cut and the residual stresses are released. The surface profile is measured accurately and the 3D residual profile is finally translated into a two-dimensional stress state through finite element software. [START_REF] Ismonov | [END_REF] In view of the high complexity of the remaining fields after expansion (in plane and out of plane), a precise reconstruction of the "released" face seems complex. Another technique called the "curvature method" or "deflection method" aims to work back to the residual stress state in a solid via the measurement of the change in its macroscopic deformation after successive removal of material performed on strategic faces. [25] This method is particularly suitable for studying thin sheet metal, for example after shot-peening. [26] Among semi-destructive methods, we identified the "Sachs method" which is a measure of residual stress relaxation around a hole by progressive reaming of the perimeter. The specific application to the case of a cold worked hole includes installation of strain gauges on the outer surface of the treated hole. The hole is gradually reamed by machining techniques and deformation values are recorded. However, the generation of additional residual stress by drilling is problematic with the use of such techniques. This method has been applied to the study of expansion processes and stress measurement on a thin cylindrical wall. [27,28] The incremental hole technique, however, appears very interesting to determine the surface and subsurface stress fields at surface of a part. This technique is used to analyze the orientation and intensity of residual fields. [START_REF] Kelsey | Proc. SESA XIV 1956[END_REF] The method can be used on all types of finished parts without cutting and without dimensional restriction (if the surface is reasonably plane around the hole). All types of materials and microstructure can be tested, on condition that they satisfy elasticity and isotropy. In addition, instrumentation and measurement is relatively fast.

Surface preparation, installation and wiring of the gauge and setting of the software may last about 20 minutes, and drilling and measurement may take about 1h (automatic process).

However, this technique does not measure the residual stresses at large depths in the part (typically to a depth value close to the diameter of the measurement hole). Some variations, such as the deep hole drilling technique, can be used to address this problem but are much more expensive and complex. [START_REF] Kitano | [END_REF] Each of the methods described above has its limitations and fields of application can be very different. Here, in the study of high stress gradients, the incremental hole drilling technique is chosen. Thus, by combining the numerical characterization with the results provided by this method at specific measuring points, we will propose an innovative method for the study of expanded holes.

The aim of the present work was to identify the influence of multiple successive expansion in a titanium hole and see whether it could improve the homogeneity of the residual stress fields in the hole edge. We first introduce and describe the modeling strategies employed to study this issue. Specifically, we explain the models that were used in association to simulate successive expansions of the same hole, conserving the tooling of the split sleeve process and considering various expansion directions. Then, we analyze the results achieved via axisymmetric FE modeling of a single split sleeve cold expansion in Ti-6Al-4V annealed holes. Taking the intensity and distribution of circumferential stresses into consideration allows the efficiency of the process to be estimated for single and multiple successive expansions. These multiple expansions could be carried out according to two different scenarios: twice in the same direction or twice in opposite directions. The numerical results will be compared with experimental results to assess the representativeness of the model, considering two different hardening laws. Finally, the measurement of the actual stress fields at the hole edge is assessed via the incremental hole drilling technique. The principle of the method employed is presented and various cases of cold expansion of a titanium alloy are treated, i.e. single/successive expansion and the influence of reaming on the intensity and homogenization of the stress fields.

Modeling strategy for simulation of multiple cold expansion in a titanium hole

Description of the Axisymmetric Model

This section describes the numerical model established to simulate split sleeve cold expansion in a titanium hole. The modeling choices generally associated with the expansion processes are the use of a 3D model with an isotropic or kinematic elasto-plastic constitutive material law and a Von Mises yield criterion. This strategy was particularly justified by the localized phenomena occurring on the hole during expansion because of the presence of the split in the sleeve. However, simulations show that the residual stress field is no longer disturbed by the presence of the split when we move slightly away from its zone of influence along the circumference. [14] In the present problem, axisymmetric modeling appears to be sufficiently representative of the phenomena operating over the vast majority of expanded sections, at least those that will be the most loaded during subsequent use. In previous work using these models, we obtained results that were very reliable compared with experimental results. So, in this study, we decided to develop other axisymmetric models of the process. The model presented (Fig. 1) was set up using the commercial program ABAQUS 6.12 and results were obtained using the ABAQUS Standard implicit solver. The geometries, loading environments and boundary conditions were finely selected for each main step of the expansion process.

Expansion (pulling of the mandrel through the hole) was followed by the removal of the sleeve from the specimen, which was then reamed to the nominal functional diameter by removal of a calibrated set of elements from the hole edge. Finally, a relaxation step was included to reach the final equilibrium of the part. This reaming simulation remains simplified with respect to the complexity of the phenomena involved in reality, where there are problems associated with machining, such as integrity of the surface generated, the displacement of the cutting tool or the heat generated and its influence on stress relaxation. Nevertheless, this strategy provides valuable information about the re-distribution of stress fields due to the removal of the contact zones between the expansion tooling and the part.

The geometry was chosen according to the geometry of the critical section of an elementary Type-T fatigue coupon (Fig. 1), with a hole diameter after reaming of 9.52 mm, a width-todiameter ratio of 3 and a thickness of 5 mm. Reaction to the axial loads generated during the expansion was obtained by axial locking of the base of the jaw ( = 0) in order to be representative of the actual loading environment. Contact modeling between parts was selected with a penalty friction formulation and considering a hard contact. Friction coefficient between the sleeve and the part is 0.1 in the light of the materials in contact, the excellent surface integrity of the different parts and the high lubrication during the process.

Meshing of the part and the sleeve was performed using 8-node biquadratic axisymmetric quadrilateral elements, including full integration in the high plasticity rates areas. Meshing of the mandrel and the jaw used 4-node bilinear axisymmetric quadrilateral elements. The minimal size of the elements was 8.10 -3 mm close to the hole edge. A parametric study determined the maximum size of the mesh that gave good contact and no interference between the components at the corners (Fig. 1). A first assumption was made concerning the modeling of the sleeve. It was considered as an axisymmetric part whereas, in real conditions, the split allowed slight opening of the sleeve, which may have significantly reduced the hoop stresses within its section.

Regarding the material law used, a standard linear kinematic hardening law was considered (one hardening slope) in order to account for the subsequent traction-compression of the material during expansion, a scenario that may involve kinematic hardening. This law will be compared to a multi-linear isotropic hardening law. All values are derived from characterization data obtained from quasi-static tests on Ti-6Al-4V annealed specimens. The material of the sleeve is martensitic steel with isotropic hardening, while the mandrel and jaws are simulated as elastic linear steel parts.

Modeling Multiple Successive Cold Expansion

This section studies the influence of successive expansions of a hole. To simulate this process, modeling considered the crossing of several mandrels in the same hole before the final reaming of the hole was performed (Fig. 2). At each step of expansion, a new sleeve was inserted into the hole. Two successive expansion scenarios are thus considered. First of all, the expansion obtained in two opposite directions (OD) is simulated. The idea here is that the opposite direction may restore an effective radial expansion rate at the mandrel entrance (of the first expansion). This scenario is then compared with a double expansion carried out in the same direction (SD), which is easier to achieve industrially. The purpose is to study the impact of these scenarios on the residual state of the hole, analyzing, for example, the distribution and intensity of tangential compressive fields and residual strain. A first model simulates two successive expansions according to the z and -z directions (Fig. 2, left) while a second simulates two successive expansions in the z direction (Fig. 2, right). Note that, in both scenarios, the boundary conditions are representative of actual experimental conditions.

The expanded part is kept axially free so that it can move and creep along the sleeve. We thus hope to capture all effects associated with the direction of travel during tooling displacement.

Numerical results -circumferential stress analysis

Single split sleeve cold expansion

This section reports the results obtained through FE modeling of the split sleeve process performed once within a titanium hole. Results are taken from a simulation considering a 6% initial expansion ratio. Because of its predominance in the onset and propagation of cracks in the hole section, only the circumferential stress will be analyzed in this paper. The simulations show the compressive stress close to the hole section. The results shown (Fig. 3 (a)) compares the circumferential residual fields generated after cold expansion and (Fig. 3 (b)) shows these fields after cold expansion followed by reaming to the functional diameter. Stresses are plotted over three paths corresponding to the entrance face of the mandrel, the midsection of the specimen and the exit face of the mandrel. For the entrance and exit sections, paths were chosen at the first row of elements under the free surface to avoid some singularities existing near the hole surface.

A good way to analyze the effectiveness of the expansion is to observe the intensity of the compressive peak and the extent of the compressive area. We find the well-known phenomenon of reduced compressive stresses at the entrance side of the mandrel, "the weak point" (Fig. 3 (a)). We note also that the compressive fields are more extensive in the mandrel entrance and exit sections than in the midsection (compressive boundary is 4.2 mm instead of 3 mm). (Fig. 3 (b)) shows the strong benefit of reaming and relaxation on the homogeneity of the stresses in the thickness (arrow 2). We note that the reaming induced homogenization of the stress along the section. Stress peak intensities were closer at the edge of the hole, especially at the mandrel entrance position, where the stress became more compressive (about -150 MPa).

In order to improve the efficiency of the process and, more precisely, the intensity of stresses at the mandrel entrance, we now check the influence of performing the double expansion in different directions. The results will then be compared with those obtained after a double expansion performed in the same direction.

Successive Double Cold Expansion Performed in Opposite Directions (OD)

The results presented in this section were obtained from simulations of successive double expansion carried out in two different directions (z and -z). The expansion ratios were fixed at 6% relative to the initial hole (the same tooling dimension is used). (Fig. 4) shows the distribution of circumferential residual fields at different steps of the process. For convenience, we will always call the same side the "exit" face: it corresponds to the exit face of the mandrel at the first expansion step. Similarly, the entrance face of the first step will still be called the entrance face in the following steps.

We thus find that, following the first expansion, the compressive stresses are reduced at the mandrel entrance, as shown in section 3.1. Thereafter, the second expansion in the opposite direction induces a very interesting balance of the stress fields. This weakness is now seen to affect the mandrel entrance of the second mandrel. This result is remarkable and can be connected with the out of plane displacement (axial direction) of the material surrounding the hole and kinematic hardening effects, inducing a lowering of the compressive yield strength.

We note that, after reaming and relaxation, the homogeneity of the stresses is very good and residual circumferential fields are almost symmetrical with respect to the midsection of the hole.

The previous observations were confirmed by the analysis of the stress curves plotted at the entrance, exit and midsection of the part. (Fig. 4 (a,b)) shows how the compressive peak switches between the two outer surfaces of the part after successive expansions (arrows n°1 and n°2). (Fig. 4 (c)) shows that, after reaming, this scenario gives a homogeneous expansion of the hole and no significant differences are observed in curve shapes between the entrance and the exit of the mandrel (only a slight weakness persists at the entrance of the second mandrel). In addition, no change is observed in the value and distribution of the tensile stress generated away from the hole. These results show the effectiveness of this expansion scenario on the process efficiency. In cases where no damage is found in the titanium alloy microstructure after a multiple expansion, it seems that performing a double expansion could contribute to higher fatigue performance and/or the elimination of weak points that allow fatigue cracks to grow.

Successive Double Cold Expansion Performed in the Same Direction (SD)

We now present the results obtained from simulations of successive expansions carried out in the same direction. Again, initial expansion ratios were fixed at 6% and kinematic hardening was used. (Fig. 5) shows the distribution of circumferential residual fields at different steps of the process. It is observed that expansion efficiency is improved at the mandrel entrance whereas, at the exit side, the stress intensity and distribution appears less important. We note that, after reaming and relaxation, homogeneity of the stresses is improved throughout the thickness.

Analysis of the stress curves confirmed this tendency. More precisely, (Fig. 5 (a,b)) shows that the second expansion involves only a slight increase in stress intensity at the mandrel entrance whereas it involves a marked reduction of the stress intensity at the mandrel exit, of about 200 MPa (arrows n°1 and n°2). On the other hand, (Fig. 5 (c)) shows that reaming and relaxation allowed intensities and shapes of compressive areas to be reached that were close to those observed after a single expansion (Fig. 3 (b)). Note that the successive double expansion led to a slight increase in the size of the compressive area as well as the stress intensities. It can be deduced that this scenario of successive expansion (SD) is less efficient than the previous one (OD) according to the homogeneity of the compressive fields generated in the hole section.

Assessment of stress at the edge of a cold expanded hole

Principle of the measurement method

This section presents the evaluation of an experimental technique adapted for stress measurement at the edge of cold expanded holes. The incremental hole drilling technique consists in measuring the deformations recorded by a gauge after drilling of a volume of material containing residual stresses. The internal stress removed from within the part causes a new mechanical equilibrium around the drilled hole, which results in deformations measured in the gauges located at its periphery. The drilling can be done particularly in the center of a rosette specially designed for these applications and the procedure is standardized. [START_REF]American society for testing and materials[END_REF] According to the magnitude and direction of the deformation recorded during drilling, the geometric parameters of the experiment and the mechanical properties of the materials used (Young's modulus and Poisson's ratio), it is possible to reconstruct the stress field that existed before the drilling of the hole. This method provides a value of "first order" residual stresses, i.e. the macroscopic stresses.

In previous sections, the numerical study of the expansion process resulted in a good theoretical understanding of the circumferential stress fields generated by the split sleeve process. We have seen that the stress fields generated by expansion are first known to be biaxial (radial and circumferential) in the plane of the plate (i.e. orthogonal to the axis of the expanded hole). To measure this stress field, it appears sufficient to use the hole drilling method (HDM) that enables the reconstruction of a biaxial stress field that is homogeneous in the hole thickness. This method is based on the measurement of the biaxial strain field relaxed after the drilling of a small hole in the study area. Lame's equations then allow a rapid reconstruction of the initial stress field. However, the stresses generated by the split sleeve process are also known to be heterogeneous in the thickness of the treated hole. We have seen that stress fields generated at the mandrel entrance should be less intense than at the mandrel exit. Thus, in order to capture all the tri-axial variation of the stress, the incremental hole drilling technique (IHDM), a more complicated version of the previous one, is used. Here, the variation of deformation is also captured through drilling increments (in depth), by successively removing a well-defined volume of material (coring). The drilling is carried out according to a fixed number of increments where strain values are recorded and where stresses will be deduced. This method requires the use of a finite element model of the experimental process to identify the laws connecting the value read at the surface of the part (where the gauge is stuck) with the change in residual stress at the bottom of the drilled hole.

It allows the well-known coefficients of influence to be determined. [32] Correlation coefficients are specific for each experimental configuration (hole diameter, plate thickness, rosette dimensions, etc.) and for each increment. All the coefficients are computed using a dedicated numerical modeling of the experimental configuration. When a drilling increment is completed, the drilling tool is released from the hole. The deformations are then recorded over a fixed time. The software deduces the mean and standard deviation values stored in order to interpret them later to calculate the nominal stress value and an uncertainty associated with the stabilization of the deformation. The main assumptions of the method are the use of the mechanics of linear elastic continuum: constant shear between the layers of increments, absence of shear between the layers of material removed and non-consideration of out-ofplane stresses.

For cold expansion issues, the basic idea is shown (Fig. 6). Thanks to the small diameter of the hole that is drilled for the deformation measurement, we are likely to capture the value of the hoop stress that is then theoretically almost constant at the compressive peak (close to the compressive yield strength). As seen in section 3.1, the compressive threshold at the hole edge may propagate quite far (≈ 2 mm), at a constant stress intensity in the mandrel entrance, mid and mandrel exit sections. As will be seen in the experimental procedure, the diameter of the measurement hole is smaller than the width of this threshold and allows it to be measured.

Beyond these 2 mm, the compressive stress decreases greatly and its variation will be difficult to capture precisely. The objective is to drill a small hole that is almost tangent to the perimeter of the cold expanded hole. The measurement hole is then surrounded by the deformation gauges, which can record the variation in deformation generated during the drilling of the hole. Thus, if the orientation of the measurement gauges is mastered, the method seems quite suitable to determine the maximum value of the circumferential compressive threshold, which is theoretically very constant and extensive at the exit face of the mandrel. On the other hand, the radial stress value is theoretically not constant close to the hole edge, so an averaged value will be measured. The software used is calibrated to calculate the plane stress state (S11, S22 and S12) in the direction 1,2 of the strain gauges (3).

̿ , ) = . . . . . . (3)
Thus the calculated stresses are not the principal stress but the stresses in the "gauge" coordinate system and the associated shear in the measuring direction. As shown in (Fig. 6), the direction 11 is collinear with the gauge a (0°) and direction 22 is collinear with the gauge c (90°). The center of the coordinate system is the center of the drilling target. The rosette was bonded with the 0° direction tangential to the hole edge in the area of the net-section to allow comparison with our numerical results. The 90° gauge was thus oriented to measure radial strain. With this approach, direct conversion is possible between the polar coordinate system , and the gauge coordinate system 1 , 2 . For now, we make a first assessment of the feasibility of the measurement method and a preliminary analysis of the results for the study of the expanded hole. In fact, despite all the theoretical advantages of this method, some limitations are still present on the quantitative confirmation of the results obtained. In particular, it is necessary to check the possible existence of a correction that has to be performed on the coefficients of influence in the case of a free edge close to the measuring hole as the loss of stiffness caused by the decrease in the volume of material surrounding the hole may influence coefficients, mainly in the radial direction. To limit this potential effect on the value of the circumferential stress, the 0° gauges (direction 1) are oriented in the direction of the desired circumferential stress. Finally, attention must be paid to the possible reserve hardening of the measurement hole, which exhibits a high stress concentration.

Experimental procedure

In order to perform the drilling and data acquisition, a MELIAD RMSHD40 device was used to control the drilling increments and also to calculate the influence coefficients and perform post-processing of the recorded deformation data. To measure strain, rosettes with three strain directions (0°, 90° and 45°) were used to define the plane stress state. These gauges have a target, 1.8 mm in diameter, in their center to achieve drilling. The largest drilling diameter was used as it provided higher precision on the strain value recorder (diameter of 1.8 mm).

Smaller drilling diameters (1 mm) are available for better resolution to capture gradients on the shape of the residual stress fields.

In the present study, we chose HBM K-Ry6 rosettes, which are very small and pre-wired, in order to simplify the instrumentation of the cold worked specimens. The temperature compensation factor was not considered because the gauges used exhibited negligible variations in their characteristics over the range 15-30°C, which included the experimental conditions. The challenge was to achieve the bonding of gauges near the hole, always in the same position, as selected (Fig. 6). Apart from a thorough cleaning, no surface preparation was performed before bonding the gauge. Furthermore, in order to avoid erroneous results, measurements at the entrance and exit face were never made face to face. This avoided the risk of relaxation in the same ligament of the specimens. Once the gauge was bonded, the instrumented specimen was placed on the table of the controlled drilling device. The specimen was then connected to the acquisition system and the gauge was balanced. The whole was connected to a computer workstation with dedicated software, which, in particular, controlled the drilling device.

The drilling was initiated in carefully chosen cutting conditions in order to generate the desired flat-bottomed and square-cornered hole shape. To not add further residual stresses during cutting and thus affect the results, a new "flat bottom" carbide end mill was used for each measurement point. The tool and machining parameters selected for the incremental drilling operation are discussed Table 1. To reduce cutting forces, the feed rate was very low.

Given the very small diameter, the spindle speed was very high in order to maintain a sufficient cutting speed. The biggest technical difficulty was the precise centering of the measuring hole with respect to the target of the rosette. This was done manually on a micrometric table, using magnifiers. The worst experimental defect that occurred was a 0.1 mm discrepancy in coaxiality between the target and the hole. The hole was then divided into a fixed number of depth increments, where the stress was calculated. We defined 19 increments for a drilling depth of 1.05mm.

IHDM analysis -Stress fields after single and OD cold expansion

We now present the results obtained using this experimental technique. We were able to treat various cold working configurations that were simulated in the numerical section. The measurements were generally performed twice for each configuration in order to check the consistency and repeatability of the results. We first observed the results obtained on a Ti-6Al-4V sheet immediately after machining and tempering in order to find the initial stress state. The gauge was then bonded and oriented with respect to the rolling direction of the sheet. Values of residual stress close to 0 MPa were measured in all directions on the plate (rolling direction and transverse direction) (Fig. 7). However, the two increments located below 50 µm in depth showed very significant compressive peak: less than -600 MPa on machined and tempered plates. These stresses were also balanced by a less intense but more extensive (200 µm) tensile field. An additional measurement was made at the edge of a "virgin" hole. This measurement also showed negligible stresses as soon as we penetrated slightly below the surface.

Next, we measured the stress in titanium specimens that were cold expanded according to the split sleeve process, with an initial expansion ratio of 6%. These specimens were identical to those considered section 3.4 and corresponded exactly to the conditions simulated in the numerical study. The measurement at the edge of an expanded hole proved very interesting (Fig. 7). At the mandrel exit, we observed a circumferential field that was very intense and relatively constant through the thickness, ranging between -800 and -1100 MPa. At the mandrel entrance, we observed a residual stress close to 0 MPa at the surface, which quickly became highly compressive in depth, where it reached a value close to -1100 MPa.

As the numerical study had indicated that a good way to optimize the expansion process of a titanium alloy was to carry out a second expansion in the direction opposite to that of the first (OD process), the measurement methodology described above was used to verify the impact of this theoretical "optimization" experimentally. (Fig. 7) shows also the results obtained at the mandrel exit and entrance faces after the two successive expansions. Several points are worth noting. First, a clear increase can be seen in the intensity of the compressive area at the entrance face of the mandrel. Thus the circumferential stress quickly reached -1000 MPa (z = 0.1 mm) while, with a single expansion, these compression levels occurred at greater depth, close to 1 mm. Moreover, the stresses were always highly compressive as depth in the section increased (>1100 MPa). These measurements prove that the OD successive expansion can have a very positive impact on the homogeneity of the stresses in the expanded sections and is likely to solve the problem of the weak point localized at the entry face of the mandrel. On the other hand, a very interesting phenomenon of lowering of the stress level is visible at the output face of the mandrel, the stress intensity being within the range [-750; -500] MPa. This result, already forecasted by the numerical simulations, can be specifically explained by the significant Bauschinger effect of this alloy. These results indicate that, despite the improvement in the homogeneity of the stresses within the section, there is a risk of compensation of the benefit related to the overall reduction in stress level after kinematic hardening. Only fatigue tests would reveal which parameter has the most influence on fatigue performance.

IHDM analysis after reaming -Stress fields after single and OD cold expansion

For the standard industrial process of cold expansion, reaming is generally carried out after cold working in order to remove the expansion defects and allow correct installation of fasteners. In the present case, we wanted to check the impact of this reaming on the distribution and intensity of residual stress fields. Additional specimens were therefore expanded using the split sleeve process (6% cold expansion) and reamed on machining centers. The reaming process was orbital (biaxial kinematics of the cutting tool), using a flatnose end mill. The tool and machining parameters selected for the reaming operation are discussed Table 1. This process reduced heating and cutting forces and was therefore more "neutral" regarding the residual stresses at the hole edge. The holes were reamed to a diameter 9.52 mm and chamfered. Localization of the measure point for IHDM is slightly modified.

The incremental hole measurements revealed a notable impact of the reaming process on the stress field in the hole edge (Fig. 8). Thus, it can be seen that, at the exit face of the mandrel, the overall intensity of the circumferential stress was reduced, meaning that a stress redistribution had taken place in the section or relaxation had occurred during the reaming operation. While the stress remained relatively constant over the measured depth, the measured values were within the [-750; -500] MPa range. On the other hand, the results observed at the face where the mandrel entered were very interesting and it could be seen that the weak point observed post expansion was removed by the reaming operation. A much more homogeneous stress was thus found in the depth, the stress intensity being within the range [-750; -500] MPa. These results were highlighted by the numerical study, which explained the phenomenon by a simple redistribution of stress after reaming. These measurements suggest that the reaming process can be very beneficial for the distribution of stresses and thus the fatigue performance of the hole. Reaming was also performed on specimens that were expanded twice (OD). Again, the compressive circumferential fields were well balanced on the exit face, where the stress level stabilized at -1000 MPa (Fig. 8). This balance was accompanied by a slight reduction in circumferential stress at the entrance face, then the stress came within the range [-1000; -750] MPa. The impact of reaming on the "final" circumferential stress fields at the hole edge can thus be observed. However, the solution of double successive expansion still appears to be beneficial for optimizing the process and extra compression at the hole edge can be generally observed. In fact, it can be concluded from this study that the expansion process in titanium alloys cannot be dissociated from the reaming process, which necessarily takes place immediately after the expansion. Representation of stress measurements in the thickness of the expanded sections helps to realize the impact of the double expansion (Fig. 9). The measurement intervals are here connected using extrapolation assuming linear evolution of the stress within the section z ∈ [1mm; 4mm].

Conclusion

This paper discusses the consequences of the successive split sleeve cold expansion of a single hole in titanium through a numerical study. In order to resolve the issue, dedicated finite element modeling of the cold expansion process was developed to simulate cases where the expansion was carried out twice in the same direction and in two opposite directions. The residual fields generated were analyzed and the results compared with experimental feasibility results and stress measurements using an incremental hole drilling technique. From the results obtained, several conclusions can be drawn:

1) Single split sleeve cold expansion process induced compressive circumferential stress fields at the edge of a titanium hole. These fields were significantly heterogeneous through the thickness of the hole treated. Heterogeneity concerned the extent and intensity of the fields. "Weak points" were mainly in the area of the mandrel entrance.

2) A first model allowed two successive expansions to be simulated according to the z and -z direction (opposite directions). It was observed that, after reaming, the homogeneity of the compressive fields was improved and that weak points were no longer observed over the hole section.

3) A second model simulated two successive expansions in the z direction (same sense). The homogeneity of the compressive fields after reaming was slightly improved. However, weak points remained at the mandrel entrance where the extent and intensities of the fields were still lower than in other sections.

4) A new experimental assessment of the residual stress at the edge of the cold expanded hole has been proposed. The technique allows the circumferential stress to be measured at the mandrel entrance and exit faces. The results show that heterogeneity exists in "real" cold expanded holes and that successive expansion in opposite directions improves the efficiency of expansion. Moreover, the existence of the Bauschinger effect was highlighted experimentally and the influence of the reaming on the redistribution of the stresses in cold expanded sections was assessed.

After finding that it was possible to improve the process efficiency on these alloys, a multiple expansion methodology was proposed, which improved the homogeneity of the stress field in the hole edge. The numerical study presented in this paper validates the effectiveness of this strategy and tends to confirm that the fatigue performance of holes could be improved. The next step will be to validate these conclusions by experimental fatigue tests on elementary fatigue specimens, and also to check its effectiveness on assembled (i.e. bolted) specimens.

Supporting information

In order to judge the quality of the model, the previous results were compared with experimental measurements. First of all, the values obtained from the measured residual expansion ratios (before reaming) were compared as these very specific results showed very little spread. Residual expansion rate was obtained by measuring the cold worked hole (residual hole) using an internal micrometer (out of the affected zone). The residual expansion rate (RE) was calculated for each specimen and defined by relation (2).

= ∅$ % &' ∅ ∅ (2)
Experimentally, these residual diameters were measured at the midsection of the coupons.

Coupons were made of annealed Ti-6Al-4V. Maximum tolerance on the hole diameter was ± 0.02 mm, which ensured good accuracy of the expansion ratios actually applied. Expansion was performed using split sleeve cold expansion tooling on "5 hole" feasibility specimens, always in the plane of the sheets (longitudinal-longitudinal/transverse orientation). These normalized specimens had a width equal to three times the nominal hole diameter (free-edge distance) and the distance between two holes was equal to five nominal diameters so as to avoid reciprocal influence. These values can be compared with the numerical radial coordinates measured on the hole section before reaming. (Fig. 10) shows the numerical residual profile after a single expansion and after successive expansions. It can be seen that successive expansions tend to increase the radial displacement and thus the residual ratio.

Moreover, the classical "inverted barrel" shape of the hole observed after a single expansion is modified. The final profile is also dependent on the direction of expansion. From these profiles, we can extract the values measured at the midsection of the hole (axial coordinate z=2.5mm), deduce the residual diameters and thus compare them with experimental results. A comparison of the numerical and experimental expansion ratios is shown in (Fig. 11 (a)).

Comparison was made for a single 6% expansion and for two successive 6% expansions in opposite directions (OD). Moreover, we decided to compare the results obtained using kinematic hardening and isotropic hardening.

The results (Fig. 11 (b)) show that, for a single and for a double successive expansion (OD), the value of the compressive peak is highly dependent on the law chosen for the material.

Thus, if the law used is sensitive to the decrease in yield strength after bidirectional hardening, a decrease of the intensity of the compressive peak can be seen. However, there is no change in the general trends relating to the residual fields generated and, in particular, there is no change in the compressive boundaries in the entrance, exit and median sections. As long as a means can be found to measure the value of the compressive peak precisely, it is possible to refine these elasto-plastic laws to obtain a reliable model for predicting the stress distribution in the expanded section of hard alloys. Concerning the residual ratio measured, whatever the material laws chosen, there is an excellent correlation of these variables. In fact, the model was able to predict the residual expansion with great accuracy. Thus, by using the isotropic and kinematic hardening laws, the residual expansion could be predicted with an accuracy within the experimental dispersion. Moreover, the values obtained were extremely close to the average values of the experimental tests (<1%).

Received: ((will be filled in by the editorial staff)) Revised: ((will be filled in by the editorial staff)) Published online: ((will be filled in by the editorial staff))

_[ 1

 1 ] R.E. Peterson, in Stress concentration design factors, Wiley, New York 1953. _[2] M. Paredes, R. Canivenc, M. Sartor, Proc. of the Institution of Mech.Eng. Part G: J. of Aero. Eng. 2013. _[3] L. Phillips Joseph, AFML-TR-74-10 1974. _[4] D.L. Rich, L.F. Impellizzeri, ASTM STP 637 1977. _[5] W.Z. Yan, X.S. Wang, H.S. Gao, Z.F. Yue, Eng. Fract. Mech. 2012, 88. _[6] A.C. Rufin, J. Eng. Gas Turbines Power 1993, 115.

Fig. 1 .

 1 Fig. 1. Axisymmetric modeling of split sleeve expansion process and location of the critical section modeled (taken from a fatigue specimen).

Fig. 2 .

 2 Fig. 2. Modeling multiple successive cold expansions through axisymmetric FEM. Two expansions in opposite mandrel directions (OD) (left), and two expansions in the same mandrel direction (SD) (right).

Fig. 3 .

 3 Fig. 3. Overview of residual circumferential stress fields and associated curves after Expansion (6%) of the hole (a), followed by reaming (b).

Fig. 4 .

 4 Fig. 4. Scenario OD: overview of residual circumferential stress fields and associated curves after first Expansion (6%) of the hole (a), second expansion (b), followed by reaming (c).

Fig. 5 .

 5 Fig. 5. Scenario SD: overview of residual circumferential stress fields and associated curves after first Expansion (6%) of the hole (a), second expansion (b), followed by reaming (c).

Fig. 7 .Fig. 8 .

 78 Fig. 7. Stress field measured by IHDM: milled and stress relieved, 6% and 2 × 6% in opposite directions, 9.52 mm αβ Ti-6Al-4V hole.

Fig. 9 .Fig. 10 .Fig. 11 .

 91011 Fig.9. Stress analysis at the hole edges after cold expansion (top) and after cold expansion and reaming (bottom). Entrance (z=0) and exit face (z=5) measurement connected using linear extrapolation.

Table . 1. Tools and machining parameters selected for the incremental drilling and reaming operations.