V Sauvant-Moynot

J Kittel

D Melot

M Roche

Three layer polyolefin coatings: how the FBE primer properties govern the long term adhesion

Loss of adhesion has been identified on pipelines coated with three layer polyolefin coatings (3LPO) over the past few years. This study was launched by TOTAL to correlate the wet behaviour of commercial fusion bonded epoxy (FBE) primers to the long-term adhesion of related 3LPO systems aged in water. The first part of this paper examines the water uptake behaviour of six FBE grades (free standing films and coatings), and characterises their wet Tg and adhesion. The second part of this paper investigates the long-term adhesion of plant applied 3LPO and related bilayer systems (3LPO without top coat to accelerate the water permeation to sublayers). Adhesion and durability of FBE materials in a wet environment are pointed out as key factors governing the long term adhesion of 3LPO systems.

BACKGROUND

Polymeric coatings are being used classically for external pipeline protection in the oil and gas industries over years in complement to cathodic protection (CP). The direct benefit of the use of organic coatings impairing good barrier properties in addition to the cathodic protection is the resulting low demand in CP current.

Conditions for sustainable coating barrier properties

Let first summarise the main parameters governing the performance of an organic coating barrier. i) The quality of steel surface preparation and coating application are key parameters influencing both short and long term coating performances [START_REF] Kehr | Three-Layer FBE-Polyolefin Coatings[END_REF][START_REF] Norman | Excellent pipeline coating require excellent pipeline substrate[END_REF]. Steel preparation includes metal grit-blasting to a given physical roughness (enhanced FBE wettability and anchoring to steel) and degree of cleanliness ensuring that there are no rust nor polluting species (dust, salts...) remaining on the surface.

ii) The coating application is critical to avoid defects in the coating (porosity or inhomogeneous coating thickness with areas with thinner coating) but also to offer proper cure and post-cure conditions for the reacting polymers prior to final quenching.

iii) The ability of coatings to withstand damage is also an important property because it will affect the number of coating defects [START_REF] Edmondson | Measuring the ability of Dual powder systems and three layer polyolefin coatings to withstand damage[END_REF]. Mechanical damages may be hazardous (due to pipe transportation, installation or inspection) but can also affect a pipe in service (tool penetration). iv) The permeability of corrosive species (water, oxygen, ions) through the organic coating is another key factor governing its corrosion protection ability [START_REF] Sangaj | Permeability of polymers in protective organic coatings[END_REF][START_REF] Tsuri | Effect of primer composition on cathodic disbonding resistance and adhesion durability of three layer polyethylene coated steel pipe[END_REF]. However it is not systematically evaluated in service conditions since it is not part of classical prequalification tests. Mathematically the permeability is defined by [START_REF] Crank | Diffusion in polymers[END_REF]:

Pe = D × S ( equation 1)
where Pe, D and S are, respectively, coefficients of permeability (in m 2 .Pa -1 .s -1 ), diffusion (in m 2 .s -1 ) and solubility (in m 3 .m -3 .Pa -1 ).

The permeation flux through an organic coating is proportional to the permeability and inversely proportional to the coating thickness (in the case of a classical diffusion behaviour modelled with a Fick law). In a general way, the permeability of small molecules through polymers is an intrinsic property and varies with the nature of penetrant and polymer, reflecting its molecular structure and the potential degree of interaction between molecules and polymer chains [START_REF] Klopffer | Permeability of gases in polymer materials[END_REF]. Note that diffusion process is thermally activated (temperature dependence modelled with the Arrhenius law), therefore permeability also depends on the operating conditions. v) Differential swelling between steel pipe and coating due to polymer plasticization by water could be detrimental for the long term adhesion of the coating to the metal. This phenomenon is a consequence of the coating permeability towards water and may be enhanced by temperature cycling.

vi) The chemical resistance of the outer coating material to its external environment is obviously a critical factor. In case of multilayer coatings, the durability of sub-layer materials and interfacial boundaries to steel in the presence of permeating molecules may also play a key role in the coating long-term efficiency and was not particularly investigated so far. The durability of a polymer under given environmental conditions is an intrinsic property depending on the chemical composition, e.g. some chemical functions such as dicyandiamines are prone to hydrolysis in the presence of water above 60°C [START_REF] Ivanova | Hygrothermal Aging of Rubber-Modified and Mineral-Filled Dicyandiamide-Cured DGEBA Epoxy Resin. III: Dielectric Spectroscopy Investigation[END_REF]. Hence for multilayer coatings, hydrolysis could occur in sub-layers as soon as water has permeated through the top-coat, even undefectuous.

1.2

Disbonding of 3LPO coatings Three-layer polyolefin systems feature: -an epoxy primer layer (Fusion Bonded Epoxy) providing both adhesion and corrosion protection to steel, -an adhesive tie layer, -a polyolefin outer layer (polyethylene or polypropylene). This top-coat offers outstanding mechanical resistance to damage occurring during both pipe manufacture and installation.

As a consequence, such multi-layer coatings are supposed to provide efficient and sustainable corrosion protection to the pipe steel over 20 years service or more, including exposition to humidity and temperature (cycling conditions). However, loss of adhesion has been identified over the past few years on pipelines coated with 3LPO systems exhibiting no defect [START_REF] Roche | Field experience and laboratory testing of pipeline coatings -An updating[END_REF][START_REF] Roche | Recent experience with pipeline coating failures[END_REF]. Since polyolefins have a very large electric resistance, they can lead to shielding of the cathodic protection, when the coating is disbonded and presents no macroscopic defect for the current flow, especially in onshore applications with a low conductivity environment (soils). Corrosion risks under disbonded coatings urges the need for a better understanding of the degradation mechanisms resulting in the poor adhesion of the FBE primer layer to the pipe steel.

A first study launched by IFP on tube sections industrially coated with 3LPO systems has pointed out that the water permeability of an undefectuous polyethylene top-coat does not prevent the water ingress into sub-layers (adhesive and primer ones) within a few months time, depending on the internal temperature [START_REF] Sauvant-Moynot V | Contribution to a better FBE selection for 3 layer polyolefin coatings[END_REF][START_REF] Sauvant-Moynot V | Nouvelle stratégie de sélection des primaires d'adhésion pour revêtements tricouches, 4èmes Journées d'Aix[END_REF]. Experimental results were in good agreement with unidirectional modelling of the water permeation at 60°C and 20°C, showing that the saturation level in the FBE primer could be reached after one and two years, respectively [START_REF] Sauvant-Moynot V | Use of EIS for the monitoring and modelling of multi-layer anticorrosion coatings[END_REF]. Effects of polyolefin permeability to water were fully highlighted because they were not superimposed to classical disbonding phenomena occurring on scarified coating, due to water interfacial diffusion and cathodic protection. As main consequences of the water permeation through the top-coat material, the adhesion to steel measured by peeling tests was significantly diminished. Bilayer systems (3LPO without top-coat) applied on tubes in similar conditions were also aged and peeled at the same time. This method revealed as an efficient accelerated test since the time of water diffusion through the polyolefin top-coat is saved.

1.3

Specific methodology Previous studies aforementioned were performed with a unique FBE grade, thus the influence of the primer properties on the adhesion performance of 3LPO coatings was not discussed yet. The objective of the present study is to correlate the long-term adhesion of related 3LPO systems, plant applied on steel tubes and aged in water at both room and maximum operating temperatures to the wet properties of commercial FBE materials. Attention will be paid to:

-The glass transition temperature (Tg) in the wet state, referred as wet Tg, lower than the dry Tg due to FBE plasticization by water molecules. The glass transition between a glassy state (T < Tg) to a rubber state (T > Tg for thermosets) can be explained at the molecular scale by the development of generalised cooperative chain motions above Tg, whereas chain mobility remains local below Tg. The glass transition has very important practical consequences for FBE coatings, such as a significant drop of the Young modulus from glassy to rubber state which can explain the loss of adhesion classically observed in the dry state by increasing the temperature in the vicinity of Tg or following immersion [START_REF] Kinloch | Adhesion and adhesives: Science and technology[END_REF]. Another main consequence is the electrical behaviour of epoxies, capacitive below Tg becoming conductive above Tg which favours corrosion. Indeed, there are ionic species within epoxy materials remaining from monomer synthesis [START_REF] Ellis | Chemistry and technology of epoxy resins. chap 1: Introduction to the chemistry, synthesis, manufacture and characterization of epoxy resins[END_REF]. Hence it is of main interest for FBE coatings to determine the wet Tg and to define the upper temperature limit sufficiently below this value to keep FBE glassy in service. -The durability in the wet state since water entrance in FBE coatings favours differential swelling between polymer and metal and may cause hydrolysis, given the chemical nature of FBE and the operating conditions. In turn, polymer hydrolysis can induce osmotic blistering according to mechanisms well described in the literature, where water is continuously entering due to differing osmotic pressure between external and internal media. Hence, it is of main interest to study the long-term water uptake of FBE materials at ambient and maximum service temperatures to reveal unsuitable swelling or hydrolysis phenomena through abnormal behaviours (e.g. non stabilised water uptake).

The first part of this paper examines the long-term water uptake behaviour of six commercial FBE taken as free standing films and applied as coatings to steel plates, and characterises the FBE physico-chemical properties such as the wet Tg and the adhesion after one week and three months immersion in water. Results obtained after ageing are compared to initial properties and discussed. Among the tested FBE grades, two were selected for their differing behaviours towards water and were plant applied on tubes as primers in complete 3LPO systems and related bilayer systems. Laboratory evaluation of the long-term adhesion during ageing were performed on both 3LPO and bilayer systems. The influence of intrinsic FBE properties on the bonding strength measured on tubes during ageing is discussed in the second part of the paper.

EXPERIMENTAL SECTION

Preparation of FBE coatings and free standing films

Five commercial FBE grades were selected among the most widely used for industrial applications as primers for 3LPO systems in the past two decades. A sixth grade of FBE was considered, this latter being suitable for use as 3LPO primer or monolayer anticorrosion coating. They are referred with letters from A to F. FBE coated panels (150µm Dry Film Thickness specified) were prepared in the same industrial laboratory to minimise the differences during the preparation stage. FBE powders were spray applied on 50×100×3 mm 3 steel panels which were freshly gritblasted to SA2 ½ according to ISO 8501 (Rz=70µm). Cure cycles were performed according to each data sheet recommendation and polymerisation was checked by DSC (differences in Tg between first and second run lower than 5°C). Tg values were also measured at IFP on coatings as received as well as DFT (5 measurements for each coated face).

Free standing films (500µm DFT specified) were prepared by their respective manufacturers to ensure the best cure and post-cure conditioning. Panels (50 × 50 mm 2 ) were cut out of films as received for testing purpose. DFT was measured on each panel prior to testing.

Preparation of three and bi-layer coating systems on tubes

The steel preparation and coating application by longitudinal extrusion were performed in a plant according to TOTAL GSGRCOR220 and NFA49-710 on 8 steel pipes (0.114 m diameter and 12 m long) preheated to 220°C. Two pipes were coated with FBE material B with 80 µm and 250 µm nominal thicknesses and over-coated by adhesive material to reach an overall thickness around 500 µm. Two other pipes were coated with complete 3LPO systems (using MD type PE) including 80 µm and 250 µm of FBE B primer. Four other pipes were coated the same way using FBE primer F. Coated pipes were cut into short rings (0.25 m long) for the purpose of IFP study. Coating thickness values were checked on each ring as received. Bi-layer and three-layer coatings are called, respectively, 2L and 3L in the paper.

2.3

Testing of the FBE coatings and free standing films Adhesion. Adhesion on low carbon steel was evaluated at room temperature by the cross-hatched test method (ISO 2409) before and during ageing. For ageing at 60°C, tests were performed on coatings freshly taken out of the water and cooled to room temperature. The disbonding was quantified by the percent of disbonded squares (adhesive rupture) relative to the total amount of squares intended with a knife to the metal. Differential Scanning Calorimetry. DSC scan were carried out from -40°C to 250°C at 10°C/min on 10 mg specimen with a Q100 apparatus from Waters™. Commercial sealed specific pressured cells (upper pressure limit = 30 bar) were used to avoid the water desorption during heating. Dry Tg was measured at the inflection on the heat flow curves obtained during the first run and second run (NFA 49-710). For soaked specimen, DSC measurements were performed on samples freshly taken out of the ageing medium and "wet Tg" was measured at the inflection on the heat flow curve obtained during the first run. The lowering in Tg can be calculated using following equation:

∆Tg = Tg wet -Tg dry (equation 2)
where Tg wet and Tg dry refer to the Tg values measured in the first run for dry and wet states.

Gravimetry. One coating and two film panels per formulation were immersed in deionised water at two temperatures (20°C and 60°C) during 3 months. Samples were periodically removed from the solution, blotted dry and weighed to 0.1 mg with a Mettler AE 163. The water content in weight percent, M, was measured as the mass change with respect to the initial mass of the dry film.

2.4

Testing of the 2L and 3L coating systems on tubes Ageing. 3L coated rings were aged in NaCl 1% at 20°C and 60 °C: the first series of tests was performed on complete coatings, the second series was performed on coatings with artificial defects (2 circumferential scarifications to the steel, 2 mm large and distant of 20 mm) and submitted to cathodic protection by potentiostatic polarisation (-1,5 V vs. Ag/AgCl). Graphite counter electrodes were placed in a separate cell filled with sodium sulphate and isolated by a cationic membrane to avoid any anodic dichloride formation by chloride oxidation. pH adjustment with HCl were conducted regularly to maintain pH values between 8 and 10.

2L coated rings were aged at 20°C and 60 °C by immersion in deionised water to simulate the electrolyte filtered by the thick PE top-coat acting as permselective membrane, where water molecules are supposed to diffuse much faster than ionic species [START_REF] Stratmann | Corrosion Stability of Polymer-Coated Metals -New Concepts Based on Fundamental Understanding[END_REF].

Peel test. 2L and 3L coating peel strengths were measured on pipe rings at ambient atmosphere on a tensile machine at constant peeling velocity of 8.33x10 -5 m.s -1 (5 mm.min -1 ). Mean strength values were calculated from the peeling data collected over almost the all circumference. The average peeling force expressed in N/m was also calculated in each case by dividing the mean peeling strength by the band width. The failure mode was recorded in each case following ISO 4624 criteria: A/B refers to an adhesive rupture between polymer and metal, B refers to a cohesive rupture within a given polymeric layer.

RESULTS

3.1

Ageing of FBE coated and film panels Adhesion to steel and physico-chemical properties (Tg) were evaluated on coatings after 7 days and 90 days aging in hot water and compared to initial values in the dry state. The short term sampling corresponds to the FBE materials in the plasticized state due to reversible water diffusion/solubilisation within epoxies (according to gravimetric data the fickian plateau or pseudo plateau were reached after 7 days for all coatings). The long term sampling corresponds to FBE materials in the aged state -irreversible damage have possibly occurred.

Adhesion of FBE coatings.

Results are reported in Table 1. Tests performed at 20°C on dry FBE materials reveals no disbonding, pointing out that there is no adhesive rupture in the dry state whatever the nature of FBE. Adhesion measured at 20°C after 7 days ageing points out a significant loss of adhesion for A and B FBE materials, where rupture became almost adhesive, whereas others keep their fully non adhesive failure mode. After 90 days ageing, FBE materials A and B still exhibit an almost full adhesive failure, as expected. Only FBE materials C and F keep an intact non adhesive failure mode whereas FBE D and E have a mixed rupture mode including partly adhesive disbonding. 2) are quite similar whatever the FBE material -with somewhat increasing values between first and second run pointing out that infinite cure was not totally achieved. Wet Tg values measured on coatings after 7 days immersion at 60°C (plasticized state) and after 90 days immersion at 60°C (aged state) are also given in Table 2, as well as the shifts in Tg from dry to plasticized state and from dry to aged state calculated using equation 2. One can observe that wet Tg values are significantly lower than dry values, whatever the FBE, which stresses that the upper temperature limit must be defined according to the determination of wet Tg values to have the maximum operating temperature sufficiently lower than the wet Tg values and keep FBE coating materials in the glassy state during operating conditions (see § 1.3).

Table 1 -Adhesion loss measured by cross-hatched tests in dry and soaked states (% disbonded area)

The comparison between plasticized and aged states show almost unchanged wet Tg values, suggesting that damage occurring during long term ageing (if any) are localized and do not affect the overall chain mobility. In terms of Tg lowering, aged FBE B coating exhibits the highest shift (around -30°C) whereas FBE F exhibits the lowest one (around -15°C), revealing that B and F materials, respectively, are the more plasticized and the less plasticized by water. These observations can be correlated to the respective extreme adhesion behaviours of B and F coatings, as stated previously. However, wet Tg values and degree of plasticization cannot directly explain all the differences observed in the long term adhesion behaviours, e.g. FBE E and F have similar dry and wet Tg values but very differing long term adhesion behaviours. Thus, the inspection of other FBE properties is necessary to account for the long term adhesion behaviours. Gravimetry. Among numerous data collected during ageing and possible drawings, it was decided to present the water content (in weight %) measured after 7 and 90 days ageing at 20°C and 60°C for films and coated panels (Figure 1). Data are reported in Table 3 for free films and in Table 4 for coatings. From the comparison of water contents at 7 and 90 days at a given temperature, whatever coated or free film panel, it appears that there are two kinds of water uptake behaviours: (i) a classical fickian behaviour if the water content reaches a saturation level after one week immersion, corresponding to M 7days almost equal to M 90 days ; (ii) an 'abnormal' behaviour for other cases when the water content steadily increases with time; then the water content at 7 days is taken as fickian pseudo-plateau.

According to these definitions, C, D, and F FBE materials exhibit a fickian behaviour at 20°C whereas only FBE F keeps a fickian behaviour at 60°C. The analysis of the short term behaviour of FBE materials leads to following remarks: -The water uptake curves can be modelled in the short term stage using the Fick's law with the water content at 7 days as (apparent) saturation value. Diffusion coefficients reported in Table 3 for free films and Table 4 for coatings are in very close agreement at a given temperature for each FBE. Thus, from a practical point of view, gravimetry data collected on coated materials can be interestingly used to study the water uptake behaviour of the FBE material if no free standing films are available. -

The water contents measured after 7 days are almost independent of the temperature but vary significantly with the FBE material, confirming that the FBE nature governs the water solubility in the plasticized state. The less hydrophilic material is the FBE F whereas FBE B appears as the most hydrophilic. These results support the previous conclusion on the degree of plasticization by water.

The analysis of the long term behaviour of FBE materials leads to following remarks: -The non stabilized water uptake curves could be satisfactorily modelled using Langmuir or Berens and Hopfenberg approaches, adapted from the Fickian model. -

The water contents measured after 90 days are very depending on the FBE material and, more precisely, on its durability at the ageing temperature; water contents after 3 months immersion at 20°C remain below 2% whereas results are very dispersed at 60°C, with high values for A and B FBE materials. These results suggest that those later may be particularly sensitive to hydrolysis in hot water due to their chemical nature.

Additional understanding can be achieved when plotting ∆Tg in function of the water content for the six FBE materials (Figure 2). As main trend, one can observe that the Tg is shifted proportional to the water content (-11 degrees per wgt. percent) when pure plasticization by water occurs, as reported in the literature [START_REF] Ellis | Interaction of epoxy resins with water: the depression of glass transition temperature[END_REF][START_REF] Sauvant-Moynot V | Epoxy coatings for anticorrosion challenges: a link between chemistry and performance?[END_REF]. But some points escape from this trend revealing possible osmotic degradation (water content increases in local blisters whereas wet Tg remains stable) or advanced network hydrolysis (Tg decreases whereas water content remains stable or even diminishes due to small mass loss). Evidence of osmotic degradation. Optical microscopy observations (× 200) were performed on dry and aged films to characterise the possible development of blistering due to osmotic degradation which often accompanies polymer hydrolysis [START_REF] Verdu | Action de l'eau sur les plastiques[END_REF]. Pictures given in Table 5 clearly point out the presence of porosities with diameters larger than 10µm after ageing for FBE A (∼70 porosities/cm 2 ) and B (∼20 porosities/cm 2 ). Osmotic degradation may also have started in FBE D. Blisters within FBE material and related ones developed at the coating/metal interface will obviously affect the long term adhesion to steel for FBE materials A, B and D. For the second part of the study, two epoxy powders were selected for their differing behaviours in terms of water content and durability while exposed to deionised water at 60°C: -FBE B: very hydrophilic material and sensitive to osmotic blistering, -FBE F: with little water uptake and a stable behaviour after three months exposition.

3.2

Ageing of coating systems Main results of peeling tests performed on complete 3L coatings (no artificial defect) are presented before and after ageing in Figure 3 and illustrated in Table 6. It can be pointed out that adhesion is almost unchanged after 3 months immersion at 20°C and ruptures remain cohesive within tie layer. On the contrary, there is a significant loss of adhesion after 3 months immersion at 60°C and ruptures become adhesive, whatever the FBE. Interestingly, the loss of adhesion is clearly stronger for systems based on powder B, which may be directly related to their higher water content in the plasticized state. Note that there is no corrosion found below the 3L coatings under consideration, only white metal. 4 and illustrated in Table 7. There is a significant loss of adhesion after 3 months ageing at 20°C for both powders, however only FBE B exhibits an adhesive failure. The loss of adhesion is greater after ageing at 60°C and adhesive failure are observed in any cases. Besides, numerous corrosion points can be observed under FBE B aged at 60°C whereas metal underneath FBE F is just slightly grey. Thus interfacial corrosion seems prone to develop under powder B. It may be concluded that blisters which are likely to develop within FBE B after 3 months ageing at 60°C could act as preferential areas for the initiation of corrosion [START_REF] Funke | Blistering of paint films and filiform corrosion[END_REF]. From abovementioned results on 3L and 2L systems, major trends highlighted by the peel tests within 3 months are the massive adhesion loss due to water entrance and the cohesive rupture becoming adhesive. The higher disbonded rate noted in 3L systems with powder B may be related to the faster development of corrosion under 2L coatings with powder B. Thus testing on 2L system used for accelerating the water ingress through a 3L undefectuous system is also validated for accelerating the hydrolysis attack and the corrosion development.

CONCLUSIONS

The long term adhesion of FBE is a critical functionality required from 3LPO coatings along the pipe lifetime but water permeability through a 3LPO system without defect cannot be neglected. Thus focus was made on the FBE behaviour in a wet environment. A comparative study was launched by TOTAL to link the wet behaviours of commercial FBE materials to the long term adhesion to steel of complete 3LPO systems which were plant applied on tubes to represent undefectuous industrial coatings. The peeling results obtained on complete 3L systems on tubes pointed out substantial losses of adhesion after three months immersion, even at 20°C since the peel strengths were almost divided by two. Peel tests on complete 3L and 2L coatings showed that systems with powder B are more sensitive to wet ageing than systems with powder F since loss of adhesion was higher in 3L-FBE B systems and metal corrosion occurs faster under 2L-FBE B systems. These results could be related to the intrinsic properties of FBE as standalone coatings and films ; in particular powder B was shown to be highly hydrophilic and sensitive to hydrolysis in hot water. Then it was concluded that the blisters which developed within FBE B after 3 months ageing at 60°C could act as preferential areas for the initiation of corrosion in 3LPO systems.

From the work, it was also concluded that two criteria must be considered for a proper selection of FBE powders, given a maximum operating temperature, to ensure the long term adhesion in service conditions:

-the wet Tg must be higher than max operating temperature to operate FBE material in the glassy state and keep proper mechanical properties (good adhesion) and electrical properties (good barrier); mind that the wet Tg is always inferior to the dry Tg due to FBE plasticization by water uptake, in some cases a fifty degrees lowering can be observed; -the durability in water at the maximum operating temperature must be good; a poor resistance to hydrolysis will undoubtedly affect the mechanical anchoring to steel but may also lead to osmotic blistering which can act as corrosion promoter on steel.

The next step will be to study coated tubes taken out from production pipes operating in various places.

  Tg values measured on coatings as received (Table

Figure 1 -

 1 Figure 1 -Water content (wgt. % ) measured in FBE films (left) and coatings (right) during ageing in deionised water after 7 days and 90 days at 20°C and 60°C

Figure 2 -

 2 Figure 2 -Delta Tg versus water content for the 6 FBE materials during ageing.

Table 5 -

 5 Photographs of FBE films in dry and aged states given by optical microscopy (

Figure 3 -

 3 Figure 3 -Peeling force of 3L coatings aged in NaCl 1% at 20°C (left); 60°C (right)

Figure 4 -

 4 Figure 4 -Peeling force of 2L coatings aged in deionised water at 20°C (left); 60°C (right).

Table 2 -DSC measurements of dry and wet Tg values (°C) and related

 2

	∆ ∆ ∆ ∆Tg

Table 3 -Water content (wgt. % ) measured in free films and coefficients of diffusion according to Fick law.

 3

	FBE film		20°C			60°C
		M 7 days	M 90 days	D (m 2 /s)	M 7 days	M 90 days	D (m 2 /s)
		(%)	(%)		(%)	(%)
	A	1,20 ±0,05 1,60 ±0,10	13 1,8 ±0,2.10 -	1,5 ±0,05	13,7 ±0,10 10±2.10 -13
	B	1,30 ±0,05 1,90 ±0,10	13 1,8 ±0,2.10 -	1,5 ±0,05	8,0 ±0,10	10±2.10 -13
	C	1,00 ±0,05 1,10 ±0,10	13 1,6 ±0,2.10 -	1,2 ±0,05	4,5 ±0,10	15±2.10 -13
	D	1,10 ±0,05 1,20 ±0,10	13 1,5 ±0,2.10 -	1,3 ±0,05	1,9 ±0,10	14±2.10 -13
	E	1,10 ±0,05 1,60 ±0,10	13 2,0 ±0,2.10 -	1,2 ±0,05	5,6 ±0,10	12±2.10 -13
	F	0,85 ±0,05 0,85 ±0,10	13 2,5 ±0,2.10 -	1,1 ±0,05	1,4 ±0,10	15±2.10 -13

Table 4 -Water content (wgt. % ) measured in coatings and coefficients of diffusion according to Fick law.

 4

	FBE		20 °C			60 °C
	coating	M 7 days	M 90 days	D (m 2 /s)	M 7 days	M 90 days
		(%)	(%)		(%)

Table 6 -Photographs of peeled areas for 3L coatings before and after 3 months ageing in NaCl 1% at 20°C and 60°C.

 6 Main results of peeling tests performed on complete 2L coatings (no artificial defect) are presented before and after ageing in Figure

	3L	Before ageing	90 days ageing 20°C	90 days ageing 60°C
	coatings
	FBE B	728 ± 125N	410 ± 75N	12 ± 1N
	80µm	rupture B in tie layer	rupture B in tie layer	rupture A/B
	FBE F	± 96N	398 ± 73N	68 ± 30N
	80µm	rupture B in tie layer	rupture B in tie layer	rupture A/B

Table 7 -Photographs of peeled areas for 2L coatings before and after 3 months ageing in deionised water at 20°C and 60°C.

 7

	2L	Before ageing	90 days ageing 20°C	90 days ageing 60°C
	coatings
	FBE B	Impossible	28 ± 9N	7 ± 1N
	80µm		rupture A/B	rupture A/B
	FBE F	Impossible	13 ± 8N	7 ± 1N
	80µm		rupture B in FBE	rupture A/B

ACKNOWLEDGMENTS

The authors wish to gratefully acknowledge A. Gerbaulet (ARKEMA) for the processing of the FBE coated plates, J. Grenier (IFP) for its implication to the experimental part of this work and TOTAL for financial support.


Valérie Sauvant-Moynot

Jacky Grenier
email: jacky.grenier@ifp.fr

Jean Kittel
email: jean.kittel@ifp.fr

NOUVELLE STRATEGIE DE SELECTION DES PRIMAIRES D'ADHESION POUR REVETÊMENTS TRICOUCHES

Keywords: 3-LPE, FBE, accelerated method, water permeation

Three layer organic coatings are being widely used to protect steel pipelines from external corrosion, in addition to cathodic protection. They consist of an epoxy primer layer (FBE) providing adhesion to steel, overlaid by an intermediate modified polyolefin layer ensuring the proper anchoring of the top-coat to the epoxy primer, and a thick polyolefin external topcoatpolyethylen or polypropylen.

Since the permeability of polyolefin materials to water is not negligible, polyolefins do not prevent water ingress into sub-layers after a few months service. Then soaked water in the epoxy primer layer directly affects its adhesive properties to the steel pipe. Recent case studies reported large scale disbondments of three-layer polyolefin (3-LPE) coatings, which could favour corrosion beneath the coating and lead to possible major damage if cathodic protection is shielded. Therefore, it is necessary to better investigate the adhesion strength of fusion-bonded epoxy primer to steel taking into account the water diffusing through the topcoat within the lifetime of pipeline.

A methodology was developed at IFP to examine the adhesion of FBE primer to steel under wet exposure: peel tests were carried out on pipe rings industrially coated with FBE & intermediate polyolefin layers only, aged in water to simulate the water ingress through the topcoat. This new method is proposed to contribute to the selection of FBE for anticorrosion purpose or could be further developed as accelerated method in pre-qualification tests. This paper examined the adhesive behaviour of sub-layers (FBE+intermediate polyolefin on the one hand, FBE on the other hand) aged in liquid water at 60°C. Results are compared to those obtained in vapor at the same temperature and discussed. In addition, long term ageing of complete and undamaged 3-LPE coatings were performed in liquid water at 20°C and 60°C to illustrate the effective loss of adhesion of the coating due to the water permeation and to highlight the benefits of an accelerated methodology.

INTRODUCTION

Les canalisations en acier pour le transport de gaz ou de fluides pétroliers nécessitent d'être protégées contre la corrosion. Une technique assez courante consiste à combiner deux méthodes de protection complémentaires :

-la protection cathodique, mode de protection électrochimique active, -l'utilisation d'un revêtement organique externe, mode de protection passive. L'utilisation du revêtement a pour but de réduire autant que possible la surface d'acier exposé au milieu extérieur, et donc la demande en courant électrique pour la protection cathodique. Les vitesses de corrosion élevées de l'acier dans les conditions de service des conduites enterrées soulignent la nécessité d'utiliser un revêtement pour constituer une barrière efficace et durable au passage des espèces corrosives. Néanmoins, les zones endommagées (métal mis à nu) sont, en contrepartie, protégées de la corrosion par la protection cathodique.

Rappelons les paramètres principaux dont dépendent les performances d'un revêtement. i) En premier lieu, la qualité de la préparation de surface et de l'application conditionnent l'absence de pollution à l'interface métallique, l'absence de défauts dans le revêtement (lacunes ou zones revêtues sur de très faibles épaisseurs), mais aussi le bon déroulement des réactions de cuisson & post-cuisson. ii) Un autre facteur clé qui gouverne les propriétés barrières est la perméabilité du revêtement aux espèces moléculaires auxquelles il est exposé. En effet, ces espèces moléculaires diffusent à travers la membrane que constitue le revêtement suivant un processus thermiquement activé. La perméabilité d'un matériau à une espèce moléculaire est une propriété intrinsèque, produit du coefficient de diffusion et de la solubilité. Le flux de perméation à travers un revêtement organique est proportionnel à la perméabilité et inversement proportionnel à l'épaisseur du revêtement (comportement diffusionnel classique au sens de Fick). iii) L'efficacité à long terme du revêtement dépend de la durabilité des matériaux constitutifs du revêtement et des liaisons adhésives revêtement/métal, en particulier de leur résistance aux endommagement d'origine chimique, comme l'hydrolyse en présence d'eau. La durabilité d'un matériau est également une propriété intrinsèque, liée à sa nature chimique. iv) Enfin, les endommagements d'origine mécanique (accidentels lors de la pose ou d'une inspection) affectent également les performances à long terme d'un revêtement ; les phénomènes de gonflement différentiel entre le substrat et le revêtement pendant le service (lors de cyclage en température, ou de la plastification du polymère par entrée d'eau) sont également à classer dans cette catégorie comme de possibles et redoutables facteurs d'endommagement mécanique du revêtement et des liaisons adhésives revêtement / métal. Concernant la perméation des espèces corrosives à travers les revêtements tricouches, aucune donnée officielle de perméabilité à l'eau n'est disponible auprès des fournisseurs. IFP a proposé une première simulation de prise en eau des systèmes tricouches sur la base de données génériques, qui estime à une année environ le temps nécessaire à la saturation du primaire si le système est maintenu continûment à 60°C [START_REF] Sauvant-Moynot | Contribution to a better FBE selection for 3 layer polyolefin coatings, 16 th Pipeline Protection Conference[END_REF]. Certes, la détermination des coefficients de diffusion et de solubilité à 60°C pour chacun des constituants permettrait sans aucun doute d'affiner la simulation. Toutefois, il est clair que la perméabilité à l'eau des polyoléfines ne permet pas d'éviter la pénétration de l'eau -rapide, devant la durée de vie du revêtement -jusqu'à la couche du primaire FBE. Ceci affecte immanquablement les propriétés d'adhésion du FBE sur l'acier qui conditionnent les performances du système de revêtement.

En effet, quelques cas de décollement massifs des revêtements tricouches ont été rapportés récemment, sans que des défauts macroscopiques et traversant aient été mis en évidence [START_REF] Roche | External corrosion of pipelines: What risk ?[END_REF][START_REF] Carpentiers | Corrosion under disbonded cotings of cathodically protected pipelines[END_REF][START_REF] Portesan | Cases of massive disbondment with three layer PE pipeline coatings, 3 èmes journées CEFRACOR : Protection cathodique et revêtements associés -Ouvrages enterrés ou exposés au milieu marin[END_REF] Basé sur la méthodologie "du tricouche écorché", cet article présente des résultats expérimentaux récents obtenus sur des portions de canalisation revêtues d'une couche unique du primaire d'adhésion ou d'un bicouche primaire + couche intermédiaire, soumis à un vieillissement en immersion dans l'eau déionisée ou en phase vapeur. L'évolution de l'adhérence a été évaluée par des mesures de force de pelage (bicouche) et arrachement de plots (monocouche). L'objectif est de discuter la pertinence de cette nouvelle méthodologie expérimentale pour sélectionner les primaires FBE et donner lieu au développement d'une procédure de vieillissement accéléré pour la qualification des tricouches.

PROTOCOLE EXPERIMENTAL

Système de revêtement L'étude porte sur des systèmes de revêtement externe tricouches (primaire époxy + PE adhésif + top-coat PE), bicouches (primaire époxy + PE adhésif) et monocouches (primaire époxy), appliqués sur un même tube acier (diamètre 0,14 m ; longueur 12 m) par la société SOCOTUB suivant la spécification technique SPEC PC Rv 06 de Gaz de France. Le tube a été ensuite découpé en viroles de 25 cm de longueur. Il est à noter, en écho au paragraphe i de l'introduction, que la procédure d'application prend bien en compte les points critiques reconnus dans le métier tels que : -les conditions opératoires (degré d'humidité, température, ventilation) durant les phases de préparation de la surface d'acier et d'application ; -l'état de préparation de surface visé (degré de propreté & rugosité, en préconisant systématiquement l'usage d'une grenaille référencée neuve) qui conditionne l'adhésion ; -le cycle de cuisson de la résine qui conditionne l'avancement de la polymérisation; -l'inspection du revêtement après application (porosité à l'éponge).

Les caractéristiques des systèmes de revêtement tricouches, bicouches et monocouche à l'étude sont groupées dans le Tableau 1. Dans les études 1 et 2, les revêtements monocouches et bicouches ont été vieillis, respectivement, en humidité relative et en eau déionisée liquide afin d'analyser l'influence de la concentration en eau sur la perte d'adhérence du FBE (ces deux cas d'exposition à l'eau pure simulent l'eau filtrée par le top-coat PE). Dans les études 3 et 4, des vieillissements long terme sur tricouches non endommagés ont été réalisés en complément des vieillissements court terme sur systèmes écorchés pour démontrer que l'eau diffusant à travers le top-coat induit effectivement une perte d'adhérence.

Méthodes de caractérisation

Adhérence du tricouche FBE L'adhérence des revêtements tricouches est classiquement évaluée par pelage (Figure 1). La bande à peler (largeur 0.028 m) est réalisée par rainurage du revêtement vieilli jusqu'au métal afin d'évaluer la perte d'adhérence causée par l'eau diffusée à travers le top-coat. Un montage de pelage inspiré par les travaux de GDF [START_REF] Coeuille | Revêtements polymères de canalisation de fluide : caractérisation et évolution de l'adhésion en milieu agressif[END_REF][START_REF] Meyer | Impact of aging process of three layers extruded polyethylene coatings on their adhesion on pipelines steels[END_REF] a été développé afin de maintenir un angle de pelage contrôlé de 90°. La virole est maintenue à chacune de ses extrémités par des flasques en libre rotation sur un axe confondu avec l'axe de symétrie de la virole, afin de minimiser les frottements liés au montage. Cet ensemble est fixé sur la traverse mobile de la machine de traction. La bande pelée est fixée entre les mors de la machine, reliés à une cellule de charge de 1 ou 10 kN. La mesure de la force nécessaire au pelage (vitesse de 5 mm/min à 50 mm/min selon le cas ) permet de déterminer l'énergie de pelage, g (J/m²) :

RESULTATS Vieillissement de tricouches non endommagés

Des vieillissements longs termes dans l'eau déionisée à 60°C et 20°C ont été menés dans les études 3 et 4 sur des portions de canalisation revêtues d'un tricouche non endommagé. Au terme des essais, une mesure d'adhérence a été effectuée quand cela était possible.

En effet, le pelage du revêtement tricouche non vieilli est quasi impossible (difficulté à amorcer une bande sans endommager le revêtement et amorcer une zone de striction au sein du polymère durant l'essai). Une mesure de force de pelage sur une faible longueur (environ 6 mm) avait permis de déterminer une énergie de pelage de l'ordre de 26 kJ/m 2 [START_REF] Sauvant-Moynot | Contribution to a better FBE selection for 3 layer polyolefin coatings, 16 th Pipeline Protection Conference[END_REF].

Adhérence après 21j de vieillissement à 20°C & 60°C dans l'eau déionisée (Figure 2)

Le pelage des tubes après 21 jours de vieillissement est impossible à réaliser, que ce vieillissement ait été effectué à 20 ou à 60°C : l'adhérence du revêtement est trop importante. La perte en eau du polymère à l'interface métal/revêtement a été évaluée à 0,7 % par Analyse Thermo Gravimétrique (ATG) immédiatement après pelage de la bande. En considérant que l'eau ne se solubilise pas dans l'adhésif en première approximation, cela correspond à 3,3 % de prise en eau massique dans le FBE.

Ce résultat démontre que la pénétration de l'eau jusqu'à la couche du primaire FBE est effective après quelques semaines à 60°C compte tenu de la perméabilité à l'eau des polyoléfinesce qui est rapide devant la durée de vie du revêtement. Ce résultat valide également les simulations de perméation [START_REF] Sauvant-Moynot | Contribution to a better FBE selection for 3 layer polyolefin coatings, 16 th Pipeline Protection Conference[END_REF], cf. Figure 4 pour le tricouche à l'étude.

Adhérence après 119j de vieillissement à 20°C dans l'eau déionisée

Après un vieillissement de 119 jours dans l'eau déionisée à 20°C, le revêtement tricouche présente toujours une très bonne adhérence. Le pelage reste impossible à réaliser. Ce résultat souligne la nécessité de définir une procédure de vieillissement accéléré pour sélectionner et qualifier les FBE.

Vieillissement de revêtements bicouches non endommagés

Les résultats de l'étude 2, présentés dans ce papier, seront discutés par rapport aux résultats de l'étude 1 déjà publiés [START_REF] Sauvant-Moynot | Contribution to a better FBE selection for 3 layer polyolefin coatings, 16 th Pipeline Protection Conference[END_REF].

Adhérence avant vieillissement : pelage sur bicouche impossible.

Adhérence après 10j de vieillissement dans l'eau déionisée à 60°C (Figure 5) Le pelage de la bande se fait très facilement, la force de pelage varie entre 6 et 10 N ce qui donne une énergie de l'ordre de 400 J/m 2 . Le pelage de la bande laisse le métal à nu, sans trace de corrosion. Adhérence après 21j de vieillissement dans l'eau déionisée à 60°C (Figure 6) La force de pelage varie entre 6 et 10 N soit une énergie moyenne stable de 400 J/m 2 . Le pelage fait apparaître le métal à nu, brillant, sans trace de corrosion. Adhérence après 28j de vieillissement dans l'eau déionisée à 60°C (Figure 7) La force de pelage varie typiquement entre 5 et 9 N soit une énergie moyenne de 350 J/m 2 . Cette fois, des particules métalliques de couleur grise sont visibles sur la bande pelée. Adhérence après 42j de vieillissement dans l'eau déionisée à 60°C (Figure 8) La force de pelage varie de 5 à 10 N pour une énergie moyenne de 375 J/m 2 . Le métal mis à nu est gris-noir (signe de corrosion) et la bande pelée est couverte de particules métalliques. Adhérence après 49j de vieillissement dans l'eau déionisée à 60°C La force de pelage varie typiquement de 15 à 25 N, ce qui correspond à une énergie moyenne de 1000 J/m 2 . Le pelage laisse apparaître le métal noir, signe manifeste de corrosion du substrat entraînée lors de l'arrachement. La résistance à l'arrachement du métal corrodée de plus en plus profondément permet d'expliquer l'augmentation de l'énergie de pelage observée.

La perte en eau du bicouche vieilli 49 jours en eau déionisée à 60°C a été évaluée par ATG à 1,7 %, ce qui correspond à une prise en eau dans le FBE de 8 % environ. Cette valeur relativement élevée suggère que le FBE subit un endommagement hydrolytique. Concernant les revêtements bicouches vieillis à 60°C dans 95% d'humidité relative, il avait été observé [START_REF] Sauvant-Moynot | Contribution to a better FBE selection for 3 layer polyolefin coatings, 16 th Pipeline Protection Conference[END_REF] que l'énergie de pelage moyenne décroît dans le temps mais laisse le métal brillant durant les 42 jours du vieillissement. Pour le vieillissement en eau déionisée, si les résultats obtenus sont comparables après 21 jours, on note en revanche l'apparition de corrosion sur le métal dès 28 jours d'immersion ; cette corrosion se développe avec le temps mais semblerait agir comme un promotteur d'adhérence (ancrage mécanique), en limitant la perte d'adhérence du bicouche observée après 28 j de vieillissement.

Analyse des résultats et discussion

Des phénomènes analogues ont pu être observés pour les revêtements monocouches FBE :

-la rupture devient adhésive (type A/B) en cours du vieillissement, permettant une évaluation quantitative de l'adhérence résiduelle du FBE sur le métal ; -l'adhérence moyenne diminue avec le temps au cours du vieillissement en humidité relative, le métal sous le revêtement restant brillant durant les 42 j d'exposition ; -la corrosion du substrat se développe après 28 jours d'immersion en eau liquide. La perte d'adhérence reste limitée mais on peut noter l'apparition de cloquage dans le FBE après 49 jours d'exposition à l'eau liquide, signe manifeste de la dégradation du primaire (endommagement hydrolytique).

D'après la thermodynamique, la quantité d'eau absorbée à l'équilibre de saturation dans un polymère exposé à de l'eau dépend de la fugacité de l'eau dans le milieu extérieur. La fugacité de l'eau liquide à 60°C et 1 bar est égale, en première approximation, à la pression de vapeur saturante soit 0,20 bar. Pour une humidité relative de 95% obtenue dans une enceinte fermée contenant une solution saturée en K 2 SO 4 , la fugacité correspondante est de 0,19 bar (0,95*0,2). Aussi, étant données les valeurs voisines de fugacité de l'eau dans les deux milieux étudiés, on attend une prise en eau à l'équilibre similaire dans les revêtements pour les études 1 et 2, donc la perte d'adhérence du revêtement humide par gonflement/plastification des liaisons adhésives [START_REF] Leidheiser | Water disbondment and wet adhesion of organic coatings on metals: A review and interpretation[END_REF] sera a priori similaire dans les deux étudesce qui est observé.

Ces similarités, en revanche, ne permettent pas d'expliquer facilement les différents comportements observés pour le développement de la corrosion du substrat. D'après la littérature [START_REF] Greenfield | The protective action of organic coatings on steel: A review[END_REF], la corrosion est initiée en présence de microcavités à l'interface métalpolymère, lorsque l'eau condense, stimulant alors le transport des ions et la corrosion. Une explication pourrait être d'origine cinétique, dans la mesure où le mouillage entre un revêtement et l'eau liquide ne limite pas le flux de diffusion, alors que celui-ci est beaucoup plus faible en phase gaz. Des travaux sont en cours pour confirmer ces hypothèses.

CONCLUSION

Les points forts de la méthodologie "du tricouche écorché" développée pour étudier le vieillissement des primaires d'adhérence FBE sous l'effet de la diffusion d'eau à travers le topcoat (non endommagé) sont les suivants :

 L'étude s'effectue sur des portions de canalisation revêtues industriellement.

 Le principal facteur d'accélération est la suppression du top-coat, de façon à mettre en contact directement l'eau liquide ou vapeur avec le système bicouche (primaire + adhésif) non endommagé ; ainsi, les essais ne seront pas réalisés à une température plus élevée que la température maximum de service afin d'éviter l'activation de réactions chimiques dont les effets sont moindres dans les conditions de service.

 L'adhérence résiduelle est évaluée quantitativement par pelage du revêtement bicouche en l'état humide vieilli, rainuré juste après la sortie du conditionnement ; cela est largement plus satisfaisant qu'une mesure d'adhérence sur FBE monocouche vieilli par arrachement de plot (où le temps de prise de la colle à température ambiante est souvent de 24 heures) ou par quadrillage.

REMERCIEMENTS

Les auteurs tiennent à remercier madame V. Lachet pour son aide en thermodynamique, monsieur X. Lefèbvre pour la modélisation de la prise en eau et monsieur S. Duval pour les nombreux échanges sur le thème des revêtements.

Figure 1 :

 1 Figure 1 : Schéma de principe du montage de pelage des viroles

Figure 2 :Figure 3 :

 23 Figure 2 : Pelage de revêtement tricouche non rainuré vieillis 21 jours dans l'eau déionisée à 20°C (gauche) ; à 60°C (droite)

Figure 4 :

 4 Figure 4 : Profils de concentration en eau simulés dans un revêtement tricouche vieilli à 60°C dans l'eau

Figure 5 :

 5 Figure 5 : Pelage d'un revêtement FBE + adhésif vieilli 10 j dans l'eau déionisée à 60°C

FBEFigure 6 :

 6 Figure 6 : Courbe de pelage d'un revêtement FBE + adhésif vieilli 21 j dans l'eau déionisée à 60°C

FBE+adhésifFigure 7 :

 7 Figure 7 : Pelage d'un revêtement FBE + adhésif vieilli 28 j dans l'eau déionisée à 60°C

Figure 8 :

 8 Figure 8 : Pelage d'un revêtement FBE + adhésif vieilli 42 j dans l'eau déionisée à 60°C

Il est donc important d'étudier la durabilité de l'adhésion du primaire époxy sur le métal dans des conditions représentatives du service.

  . Dans ces situations, il existe un risque de corrosion sous le revêtement, la protection cathodique pouvant être écrantée par le revêtement externe isolant. Un tel phénomène est particulièrement à craindre lorsque l'électrolyte est peu conducteur.

	Dans les normes actuelles de pré-qualification industrielle des tricouches, typiquement les
	documents NF A 49-710 ou DIN 30670 pour le PE et NF A 49-711 ou DIN 30768 pour le PP,
	seuls des essais pour mesurer l'adhérence des revêtements tricouches neufs sont spécifiés ainsi
	que des essais de résistance au décollement cathodique. Ces derniers permettent d'évaluer
	l'adhérence d'un revêtement endommagé jusqu'au métal et placé sous protection cathodique :
	la perte d'adhérence est la conséquence de l'eau diffusée à travers le revêtement non
	endommagé, de l'eau diffusée à l'interface FBE/acier via le trou, et de la polarisation
	cathodique. Aucun essai n'évalue la tenue au vieillissement du FBE affecté par la diffusion
	d'eau à travers les couches supérieures du revêtement.

Aussi, une méthodologie dite "du tricouche écorché" a été proposée par IFP afin d'évaluer rapidement et spécifiquement la perte d'adhérence du FBE sur le métal en raison de la perméation de l'eau à travers un top-coat non endommagé [1] .

Tableau 1 : Caractéristiques des systèmes tricouche / bicouche/ monocouche à l'étude

  Des cellules de vieillissement en verre ont été spécialement conçues pour réaliser les vieillissements sur tubes revêtus prototypes dans des conditions les plus représentatives de celles rencontrées sur le terrain : surface extérieure du pipe exposée à un électrolyte (potentiellement mis en circulation et refroidi) et intérieur du pipe chauffé. A ce stade, les essais ont été conduits en isotherme. Les différentes conditions de vieillissement à l'étude sont récapitulées dans le Tableau 2. Hygrométrie contrôlée (95%) dans une enceinte fermée contenant une solution saturée en K 2 SO 4.

	Epaisseur moyenne	Tricouche	Bicouche	Monocouche
	(µm)
	Primaire FBE	94	94	94
	Adhésif		351	351	/
	Top-coat PEMD	2893	/	/
	Conditions de vieillissement
		Tableau 2 : Récapitulatif des conditions de vieillissement
	Revêtement	Etude	Température	Durée (jours)	Milieu
	Monocouche	1	60°C	0, 21, 28, 42	95% HR a [1]
	FBE	2	60°C	0, 10, 21, 28, 42, 49	Eau déionisée
	Bicouche	1	60°C	0, 21, 28, 42	95% HR a [1]
	FBE+adhésif	2	60°C	0, 10, 21, 28, 42, 49	Eau déionisée
	3LPE	3	60°C	0, 21, 42, 49	Eau déionisée
	non rainuré	4	20°C	0, 21, 42, 49, 119	Eau déionisée
	a

Récapitulatif des énergies moyennes de pelage sur revêtements bicouche (FBE+adhésif) et mesures d'adhérence moyennes sur monocouche (FBE) vieillis à 60°C en eau liquide et comparaison aux résultats obtenus au cours d'un vieillissement à 95% d'humidité relative [1].

  Le Tableau 3 récapitule les mesures d'adhérence sur revêtements bicouches vieillis à 60°C en eau déionisée et donne les résultats des mesures d'adhérence sur les tubes revêtus de monocouche FBE vieillis dans les mêmes conditions. Pour comparaison sont inclus les résultats des mesures d'adhérence sur les mêmes tubes revêtus de monocouche et bicouches et vieillis à 60°C à 95% d'HR[START_REF] Sauvant-Moynot | Contribution to a better FBE selection for 3 layer polyolefin coatings, 16 th Pipeline Protection Conference[END_REF].

	Tableau 3 : revêtement	milieu	10 j	21 j	28 j	42 j	49 j
		95% HR		375 J/m 2	375 J/m 2	250 J/m 2
	FBE/adhésif bicouche	étude 1 60°C	_	brillant métal	brillant métal	brillant métal	_
	pelage impossible	Eau déionisée 60°C étude 2	400 J/m 2 métal brillant	400 J/m 2 métal brillant	350 J/m 2 métal gris	375 J/m 2 métal gris noir	750 J/m 2 métal noir particules
	FBE > 17 MPa B/Y	95% HR 60°C étude 1 Eau déionisée	_ 5 ± 1 MPa A/B	> 7 MPa B/Y métal brillant 5 ± 2 MPa A/B	6 ± 2 MPa A/B métal brillant 5 ± 1 MPa A/B	5 ± 1 MPa A/B métal brillant 5 ± 1 MPa A/B	_ 5 ± 1 MPa A/B
		60°C	métal	métal	métal	métal	métal noir
		étude 2	brillant	brillant	gris noir	gris noir	fbe cloqué


Francesco Dell'isola
email: francesco.dellisola.me@gmail.com

Alessandro Della Corte
email: alessandro.dellacorte.memocs@gmail.com

Antonio Battista
email: antonio.battista@uni-lr.fr

Emilio Barchiesi
email: barchiesiemilio@gmail.com

A Della

Extensible Beam Models in Large Deformation Under Distributed Loading: a Numerical Study on Multiplicity of Solutions

Keywords: Timoshenko beam, Large deformation of beams, Extensional beam model, Shooting technique

In this paper we present numerical solutions to a geometrically nonlinear version of the extensible Timoshenko beam model under distributed load. The particular cases in which: i) extensional stiffness is infinite (inextensible Timoshenko model), ii) shear stiffness is infinite (extensible Euler model) and iii) extensional and shear stiffnesses are infinite (inextensible Euler model) will be numerically explored. Parametric studies on the axial stiffness in both the Euler and Timoshenko cases will also be shown and discussed.

Introduction

In the recent literature the behaviour of a clamped-free nolinear inextensible Euler Elastica introduced in Euler (1952); [START_REF] Bernoulli | Quadratura curvae, e cujus evolutione describitur inflexae laminae curvatura[END_REF][START_REF] Bernoulli | Quadratura curvae, e cujus evolutione describitur inflexae laminae curvatura[END_REF]; see [START_REF] Luongo | On the possible effective elasticity tensors of 2dimensional and 3-dimensional printed materials[END_REF]; [START_REF] Eugster | Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes sive solutio problematis isoperimetrici latissimo sensu accepti[END_REF]; [START_REF] Steigmann | Variational theory for spatial rods[END_REF] for general reference works, has been mathematically investigated under distributed load [START_REF] Dell'isola | Synthesis of fibrous complex structures: designing microstructure to deliver targeted macroscale response[END_REF]. In particular, the set of stable equilibrium configurations has been completely characterized in Della [START_REF] Corte | Large deformations of Timoshenko and Euler beams under distributed load[END_REF]. On the other hand, extending such rigorous results to the extensible Euler beam model, or to the Timoshenko beam model is not straightforward. Indeed, the presence of the additional kinematical descriptor accounting for extensibility gives rise to various new mathematical difficulties. The most basic one is that it changes the functional set in which the problem is naturally collocated and in particular prevents it from being a vector space, since a strictly positive local axial deformation has to be prescribed. Of course one can obtain this making suitable assumptions on the energy, but in any case the nonautonomous variational problem that arises in the case of a distributed load will present new difficulties with respect to the inextensible case [START_REF] Battista | Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams[END_REF].

The problem of existence and stability of equilibrium configurations for extensible Euler and Timoshenko beams in large deformations under distributed load is therefore an open one. Because of this reason, it is interesting to study the behavior of solutions by means of a systematic collection of parametric studies starting from the inextensible Elastica and approaching more general beam models. The present work is aimed at performing such kind of investigation. Our main effort will be to show the variety of different (and at times rather exotic) equilibrium configurations that can arise when the value of the load is large enough. Moreover, we want to numerically investigate how fast the number of possible equilibrium solutions increases with the load and how this particular feature is affected by allowing shear deformation.

The study of these exotic configurations is particularly important, nowadays, due to the enhancement of computational methods that make nonlinearity more practically relevant for structural members (Fertis, 2006;Ladevèze, 2012;[START_REF] Antman | Nonlinear problems of elasticity[END_REF][START_REF] Rezaei | Analysis of nonlinear free vibration and damping of a clamped-clamped beam with embedded prestrained shape memory alloy wires[END_REF][START_REF] Eugster | Director-based beam finite elements relying on the geometrically exact beam theory formulated in skew coordinates[END_REF][START_REF] Steigmann | Finite Elasticity Theory[END_REF] as well as in machine mechanics applications [START_REF] Pepe | Variational feedback control for a nonlinear beam under an earthquake excitation[END_REF][START_REF] Giorgio | Non-linear lumped-parameter modeling of planar multi-link manipulators with highly flexible arms[END_REF]. Moreover, they are becoming fashionable as an elementary constituent of microstructured objects manufactured with computer-aided techniques [START_REF] Atai | On the nonlinear mechanics of discrete networks[END_REF]dell'Isola et al, 2016b;[START_REF] Ravari | A computationally efficient modeling approach for predicting mechanical behavior of cellular lattice structures[END_REF][START_REF] Ravari | Numerical investigation on mechanical properties of cellular lattice structures fabricated by fused deposition modeling[END_REF]dell'Isola et al, 2016a;[START_REF] Turco | Large deformations induced in planar pantographic sheets by loads applied on fibers: experimental validation of a discrete lagrangian model[END_REF][START_REF] Turco | Pantographic lattices with nonorthogonal fibres: Experiments and their numerical simulations[END_REF][START_REF] Luongo | On the possible effective elasticity tensors of 2dimensional and 3-dimensional printed materials[END_REF][START_REF] Spagnuolo | Qualitative pivot damage analysis in aluminum printed pantographic sheets: numerics and experiments[END_REF]. These objects are potentially advantageous for their mechanical characteristics, as shown in recent literature [START_REF] Boutin | Linear pantographic sheets: Asymptotic micro-macro models identification[END_REF][START_REF] Giorgio | Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis[END_REF][START_REF] Scerrato | Three-dimensional instabilities of pantographic sheets with parabolic lattices: numerical investigations[END_REF][START_REF] Eremeyev | On characterization of an elastic network within the six-parameter shell theory[END_REF][START_REF] Golaszewski | Metamaterials with relative displacements in their microstructure: technological challenges in 3D printing, experiments and numerical predictions[END_REF][START_REF] Turco | Quantitative analysis of deformation mechanisms in pantographic substructures: experiments and modeling[END_REF][START_REF] Franciosi | Mean Green operators of deformable fiber networks embedded in a compliant matrix and property estimates[END_REF] and their theoretical study requires tools from homogenization theory [START_REF] Boubaker | Discrete models of woven structures. macroscopic approach[END_REF][START_REF] Ravari | Effects of asymmetric material response on the mechanical behavior of porous shape memory alloys[END_REF][START_REF] Reis | Construction of micropolar continua from the asymptotic homogenization of beam lattices[END_REF][START_REF] Reda | Wave propagation in 3D viscoelastic auxetic and textile materials by homogenized continuum micropolar models[END_REF], discrete mechanics [START_REF] Turco | Discrete is it enough? the revival of Piola-Hencky keynotes to analyze threedimensional Elastica[END_REF][START_REF] Jawed | A Primer on the Kinematics of Discrete Elastic Rods[END_REF][START_REF] Andreaus | A Ritz approach for the static analysis of planar pantographic structures modeled with nonlinear Euler-Bernoulli beams[END_REF], theory of generalized continua [START_REF] Misra | A review of presentations and discussions of the workshop computational mechanics of generalized continua and applications to materials with microstructure that was held in Catania 29-31 October 2015[END_REF]Altenbach et al, 2013;Altenbach and Eremeyev, 2013;[START_REF] Placidi | Identification of two-dimensional pantographic structure via a linear d4 orthotropic second gradient elastic model[END_REF][START_REF] Placidi | Reflection and transmission of plane waves at surfaces carrying material properties and embedded in second-gradient materials[END_REF][START_REF] Placidi | A second gradient formulation for a 2D fabric sheet with inextensible fibres[END_REF] as well as developments in nonlinear elasticity of beams. In this last regard, certainly a full understanding of the onset and the characteristics of multiple solutions for the static problem under distributed load would be an important step forward, and possibly more general beam models will also have to be considered [START_REF] Diyaroglu | Peridynamics for bending of beams and plates with transverse shear deformation[END_REF][START_REF] Challamel | Variational formulation of gradient or/and nonlocal higher-order shear elasticity beams[END_REF][START_REF] Challamel | Nonlocal equivalent continua for buckling and vibration analyses of microstructured beams[END_REF] because of the exotic properties of microstructured continua [START_REF] Misra | Pantographic metamaterials show atypical poynting effect reversal[END_REF][START_REF] Dell'isola | Pantographic metamaterials: an example of mathematically driven design and of its technological challenges[END_REF][START_REF] Diyaroglu | An Euler-Bernoulli beam formulation in an ordinary state-based peridynamic framework[END_REF].

The paper is organized as follows: we introduce the general model for a nonlinear version of the Timoshenko beam; we deduce the equilibrium equations by means of the Lagrange multipliers method. Then, imposing stationarity to the energy functional, we deduce an expression for the total energy in stationary points which depends only on the angles formed by the tangent to the deformed shape. Then we show and discuss numerical results on the multiplicity of solutions with large value of the load and on the effect of releasing extensional stiffness in both the Euler and the Timoshenko case. Finally we propose some future research directions.

The Model

Kinematics and Deformation Energy

Let {D 1 , D 2 } be an orthogonal reference system in which the beam lies in the unstressed configuration along D 1 . We will denote by s the abscissa along the beam, by the apex the differentiation with respect to the reference abscissa1 and by χ(s) the placement function. The tangent vector to the current configuration of the beam is then:

χ = α(s) [cos(θ(s))D 1 + sin(θ(s))D 2 ] := α(s)e(θ(s)) (2.1)
where e(θ(s)) represents the unit vector parallel to χ . Therefore α(s) describes the local elongation of the beam:

χ (s) = α(s) (2.2)
while θ(s) is the angle between χ (s) and D 1 . We will assume the following energy functional:

E def (χ, ϕ) := L 0 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ k e 2 ( χ (s) -1) 2 extensional energy + k b 2 (ϕ (s)) 2 flexural energy + k t 2 (ϕ(s) -θ(s)) 2 shear energy ⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭ ds (2.
3) Here ϕ is the angle between sections of the beam (supposed rigid) and the normal to the neutral axes, while k b , k e and k t are respectively the bending, extensional and shear stiffness. The energy can be rewritten as:

E def = L 0 k e 2 (α -1) 2 + k b 2 (ϕ (s)) 2 + k t 2 (ϕ(s) -θ(s)) 2 ds.
(2.4)

We introduce now a uniformly distributed load b(s) and a concentrated load and couple in the endpoint of the beam s = L, denoted respectively by R(L) and M (L). The total energy of the system is then:

E tot = L 0 k e 2 (α -1) 2 + k b 2 (ϕ (s)) 2 + k t 2 (ϕ(s) -θ(s)) 2 -b • χ ds -R • χ(L) -M ϕ(L)
(2.5)

The beam model described by the functional (2.5) is a geometrically nonlinear version of the Timoshenko beam model (introduced in [START_REF] Timoshenko | Lxvi. on the correction for shear of the differential equation for transverse vibrations of prismatic bars[END_REF][START_REF] Timoshenko | X. on the transverse vibrations of bars of uniform cross-section. The London[END_REF], which is a particular case of Cosserat continuum introduced in [START_REF] Cosserat | Equilibria of a clamped Euler beam (Elastica) with distributed load: Large deformations[END_REF]; for general references and interesting results see e.g. [START_REF] Altenbach | On generalized Cosserat-type theories of plates and shells: a short review and bibliography[END_REF]; [START_REF] Birsan | Deformation analysis of functionally graded beams by the direct approach[END_REF]Forest (2005); [START_REF] Eremeyev | Material symmetry group and constitutive equations of micropolar anisotropic elastic solids[END_REF]). Interesting generalizations of the Timoshenko beam model have been proposed (see e.g. [START_REF] Romano | Shear deformability of thin-walled beams with arbitrary cross sections[END_REF][START_REF] Serpieri | A frame-independent solution to Saint-Venant's flexure problem[END_REF], while a periodic mechanical system whose homogenized limit is the model (2.5) (in the particular case α ≡ 1) is shown in [START_REF] Battista | Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams[END_REF]. It is in fact a microstructured 1D system whose unit cell is an articulated parallelogram and equipped with suitably placed rotational springs, and it can be easily obtained by means of 3D printing.

Of course also other (possibly more complex) microstructured systems can have a similar homogenized version (on microstructured continua see e.g. [START_REF] Barchiesi | Mechanical metamaterials: a state of the art[END_REF][START_REF] Engelbrecht | Reflections on mathematical models of deformation waves in elastic microstructured solids[END_REF][START_REF] Engelbrecht | Waves in microstructured materials and dispersion[END_REF][START_REF] Barchiesi | A 1D continuum model for beams with pantographic microstructure: Asymptotic micro-macro identification and numerical results[END_REF].

Lagrange Multipliers Method

Given the total energy of the system, equilibrium configurations are found as stationary points of the energy functional (2.5). A synthetic formulation of the problem, taking into account together the constrain of Eq. (2.1) and the total energy, is obtained with the introduction of a Lagrange multiplier Λ(s). We get the following functional formulation:

E tot = L 0 k e 2 (α -1) 2 + k b 2 (ϕ (s)) 2 + k t 2 (ϕ(s) -θ(s)) 2 -b • χ+ + Λ • (χ -α [cos(θ(s))D 1 + sin(θ(s))D 2 ]) ds -R • χ(L) -M ϕ(L) (2.6)
i.e., the energy is a function of the fields α(s), ϕ(s), θ(s), χ(s), Λ(s). The first variation of the energy with respect to these fields (considered independent) gives the two boundary value problems (BVPs):

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ -k b ϕ + k t (ϕ -θ) = 0 k b ϕ (L)δϕ(L) = M δϕ(L) k b ϕ (0)δϕ(0) = 0 (2.7) and ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ -Λ -b = 0 Λ(L) • δχ(L) = R • δχ(L) Λ(0) • δχ(0) = 0, (2.8)
as well as the algebraic relations:

k t (ϕ -θ) + αΛ • e ⊥ (θ) = 0 (2.9) k e (α -1) -Λ • e(θ) = 0 (2.10)
and the kinematic constraint Eq. (2.1). In Eq. (2.9) we introduced:

e ⊥ (θ(s)) = -sin(θ(s))D 1 + cos(θ(s))D 2
and the two boundary conditions in 0 are imposed considering a cantilever beam (lying along D 1 in the reference configuration and clamped in the extreme s = 0).

Numerical Simulations

Numerical Methods

An increasingly popular approach for the numerical study of nonlinear beams is isogeometric analysis (see e.g. [START_REF] Balobanov | Locking-free variational formulations and isogeometric analysis for the timoshenko beam models of strain gradient and classical elasticity[END_REF][START_REF] Niiranen | Variational formulations, model comparisons and numerical methods for Euler-Bernoulli micro-and nano-beam models[END_REF][START_REF] Cazzani | Isogeometric analysis of plane-curved beams[END_REF][START_REF] Greco | An efficient blended mixed b-spline formulation for removing membrane locking in plane curved Kirchhoff rods[END_REF][START_REF] Dortdivanlioglu | Computational aspects of morphological instabilities using isogeometric analysis[END_REF], which is a suitable variant of the finite element method. This method is very powerful and relatively light from a computational point of view, but just like every energy-related method it is not very suitable to study the multiplicity of arising solutions. For this reason, the numerical technique used here is the same as in [START_REF] Battista | Large deformations of 1D microstructured systems modeled as generalized Timoshenko beams[END_REF]. Indeed, the boundary value problem for the clamped-free Euler and Timoshenko beams has been solved by means of a shooting technique. We introduce a family of Cauchy problems 2:

P k = ⎧ ⎪ ⎨ ⎪ ⎩ θ = -b(1 -s) cos θ θ(0) = 0 θ (0) = k (2.11)
depending on the parameter k. Then we selected the solutions of (2.11) which satisfy (with prescribed accuracy) θ (1) = 0, so as to obtain a numerical solution of the equilibrium condition δE tot = 0. Clearly the solution for the Cauchy problem exists and is unique for every initial datum k.

In Fig. 2.1, we show the plot of θ (1) as a function of k := θ (0) for an inextensible Euler beam model for b = 250. The graph intersects in five different points the horizontal axis, which means that in this case we have five different solutions of the boundary value problem with θ(0) = 0 and θ (1) = 0. In the next section, the solutions will be shown in the same order as they appear as intersections between θ (1) and the horizontal axis. Therefore the absolute minimum of the total energy will be always the rightmost configuration and, of course, the only solution for b small enough. Remark 2.1. Of course this numerical technique can only capture regular equilibrium configurations. It is well-known that the minima (and maxima) of nonautonomous functionals may be not regular enough to satisfy the Euler-Lagrange equations (even in the one-dimensional case as in [START_REF] Ball | One-dimensional variational problems whose minimizers do not satisfy the Euler-Lagrange equation[END_REF]. This, together with the assessment of the stability of the solutions, is the main reason for which a rigorous study of the problem will be crucial.

The Number of Equilibrium Configurations when the Load Increases

It is generally very difficult to address theoretically the problem of evaluating how fast the number of solutions of a nonlinear parametric dynamical system increases with the parameter [START_REF] Guckenheimer | Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields[END_REF], which in our case is the external load. It is therefore interesting, as a preliminary step, to address the problem numerically. In Fig. 2.2 the number of solutions for an inextensible (left) and extensible (right) Euler beem model is plotted as a function of the non-dimensional external load. As expected, the behavior is that of a step-function, as it is clear that new branches of solutions arise only when the external load overcomes specific thresholds. The number of equilibrium configurations increases significantly with the load, and again as expected it reaches slightly larger values in case an additional kinematic degree of freedom (i.e. α) is included. Figure 2.3 is consistent with this. Indeed, in this case inextensible (left) and extensible (right) Timoshenko beam models are considered, which produces an even more rapid increase of the possible equilibrium configurations. It has to be remarked, however, that a straight comparison with the Euler case is difficult because adimensionalizing the load is a different procedure in the two cases.

Equilibrium Configurations

In the first gallery of equilibrium solutions we will show the full set of equilibrium configurations when the nondimensional load is b = 60, b = 250 and b = 500. For the chosen values of the other parameters, we have the same number of solutions (when applying the same load) for the four beam models. It has to be remarked that no branch of solutions appear to bifurcate. Instead, when the load increases, new branches appear at some thresholds. It can be seen that θ(s) is a positive monotonic function for the rightmost solution in all the cases. For b = 60 the central equilib- rium shape is non monotonic, while the leftmost is a negative monotonic function. This is a general trend as the load increases. Indeed, branches that arise with increasingly large values of the load will be made of progressively more numerous monotonic pieces. Of course in the Timoshenko case in general it is θ(0) = 0. As the boundary datum for θ in 0 can have more than one solution, we have always chosen the smallest one in absolute value (see also the Appendix). The axial elongation relative to Figs. 2.9 and 2.11 are shown respectively in Figs. 2.16 and 2.18. The shear It is not clear whether the solutions shown herein for the extensible Euler and Timoshenko models can be stable-while in Della Corte et al ( 2019) it has been proved that for the inextensible Euler case only the left and right configurations of Figs. 2.4 and 2.8 can be stable. shown for the two previous cases respectively. The elongation reaches its maximum (minimum) value where the beam lies parallel to the load with the same (opposite) direction. Instead, it is close to 1 where the beam lies orthogonal to the load. While in the Euler case the change in k e causes a minimal change in the deformed shape, in the Timoshenko case there is a much more relevant influence of k e on the configuration. In particular, decreasing k e allows a much larger maximum value of the local geometrical curvature γ := θ /α of the beam. This maximum is attained when θ(s) = -π/2; we will define s 0 the point at which this occurs. It has to be noted, however, that the Chebyshev curvature θ (s 0 ) takes similar values in the two cases. For instance, in the rightmost simulation of Fig. 2.24 we have θ (s 0 ) ≈ -16.4 and γ(s 0 ) ≈ -17.9, while in the rightmost simulation of Fig. 2.26 we have θ (s 0 ) ≈ -16.3 and γ(s 0 ) ≈ -111.9 (we point out that s 0 ≈ 0.303 in the right panel of Fig. 2.24 and s 0 ≈ 0.288 in the right panel of Fig. 2.26).

Parametric Study on the Extensional Stiffness

Conclusions

In this paper we numerically studied clamped-free Euler and Timoshenko beams in large deformation under distributed load. Extensibility has been taken into account and results on the static behavior of the beam under different values of the load and of the axial stiffness has been shown. The main interest of the results consists in the multiplicity of solutions that arise as the load increases, not as a bifurcation of ex- isting branches of solutions but as new branches that arise when the load overcomes a series of progressively larger threshold-values. Future investigations are required to establish whether these multiple solutions can be stable. In this regard, an analysis of the small oscillations of the beam around candidate stable equilibria would be useful to assess numerically the question. Moreover, addressing theoretically the Apart from the models introduced here, an independent and novel approach to understand the large deformation of beams to be investigated in the future to regard a beam as the boundary curve of a two-dimensional manifold in a threedimensional space. In doing so, not only is the curve endowed with its own energy similar to that in the context of lower-dimensional energetics (Javili et al, 2013b) but also in a geometrically nonlinear framework [START_REF] Javili | A unified computational framework for bulk and surface elasticity theory: a curvilinear-coordinate-based finite element methodology[END_REF] and in accordance with higher gradient elasticity accounting for boundary energetics elaborated in Javili et al (2013a). The advantage of this approach, particularly from a computational viewpoint, is that the bulk material acts to regularize the behavior of the beam especially important to analyze the instabilities associated with thin beams similar to the instabilities of thin films on an elastic foundation [START_REF] Javili | Computational aspects of growth-induced instabilities through eigenvalue analysis[END_REF]. Obviously, in the limiting case of the vanishing bulk one would recover exactly the beam theory.

Appendix

We show here the Euler-Lagrange boundary value problem associated with the functional (2.5). We recall that the solutions θ(s) of this BVP are the scalar fields of angles formed by the tangent to the deformed configuration and a reference axis at the equilibrium for a clamped-free, extensible Timoshenko beam, in large deformation regime, under distributed load. (2.13)

Remark 2.2. The boundary conditions in the previous problem have in general more than one solution. In the numerical simulations, we always considered the value θ(0) which was smaller in absolute value.

Fig. 2

 2 Fig. 2.1 θ (1) as a function of k := θ (0) for an inextensible Euler beam with a transverse applied load b = 250. Solutions to the boundary value problem (2.11) correspond to the intersections of the curve with the horizontal axis.

  Fig. 2.2 Number of solutions, n, as a function of the transversal applied load, b. Left: inextensible Euler model. Right: extensible Euler model (ke = 1.3 × 10 3 ).

  Fig. 2.4 Clamped inextensible Euler beam with a transversal applied load b = 60

Fig. 2 .

 2 Fig. 2.7 Clamped extensible Timoshenko beam with a transverse applied load b = 60 (kt = 7000, ke = 3000)

Fig. 2 .

 2 Fig. 2.12 Clamped inextensible Euler beam with a transverse applied load b = 500.

Fig. 2 .

 2 Fig. 2.14 Clamped inextensible Timoshenko beam with a transverse applied load b = 500 and kt = 7000.

Fig. 2 .

 2 Fig. 2.15 Clamped extensible Timoshenko beam with a transverse applied load b = 500 and kt = 7000 and ke = 5000.

Fig. 2 .Fig. 2 .Fig. 2 .

 222 Fig.2.17 Above: plot of φ (dotted) and θ for the five configurations shown in Fig.2.10. Below: the corresponding plot of φ -θ.

Fig. 2 .

 2 Fig. 2.20 Plot of α for the configurations in Fig. 2.13.

Fig. 2 .Fig. 2 .

 22 Fig. 2.21 Above: plot of φ (dotted) and θ for the five configurations shown in Fig. 2.14. Below: the corresponding plot of φ -θ.

Fig. 2 .Fig. 2 .

 22 Fig. 2.23 Above: plot of φ (dotted) and θ for the five configurations shown in Fig. 2.15. Below: the corresponding plot of φ -θ.

Fig. 2 .Fig. 2 .Fig. 2 .

 222 Fig. 2.25 The local elongation α(s) relative to the equilibrium shapes shown in Fig. 2.24.

θ

  = k b bk e (1s) sin(θ) -(b(1s)) 2 (cos(2θ)) + k e k t -1 bk b (1s)θ 2 cos(θ)(4b(1s) sin(θ) + k e ) -2bk b θ(2b(1s) cos(2θ)+ -k e sin(θ)) + cos(θ) 2b 2 k b sin(θ)bk t (1s)(b(1s) sin(θ) + k e )(2.12) with boundary conditions:θ -b 2 (1s) 2 sin(θ) cos(θ) k e k e + b(1s) cos(θ)

Notice that this means that, denoting by θ the angle formed by the tangent to the deformed shape and a reference axis, θ does not coincide with the geometrical curvature but with the so-called Chebyshev curvature (see[START_REF] Chebyshev | Sur la coupe des vetements[END_REF].

We consider here the inextensible Euler beam for simplicity, but everything is analogous for the general case.