Dorick Ballat-Durand
email: dorick.ballat-durand@utc.fr

Salima Bouvier
email: salima.bouvier@utc.fr

Marion Risbet
email: marion.risbet@utc.fr

Wolfgang Pantleon

Through Analysis of the Microstructure Changes During Linear Friction Welding of the Near-α Titanium Alloy Ti-6Al-2Sn-4Zr-2Mo (Ti6242) Towards Microstructure Optimization

Keywords: Linear Friction Welding, titanium, texture, EBSD, microstructure, β reconstruction, recrystallization Linear Friction Welding, titanium, texture, EBSD, microstructure, β reconstruction, recrystallization

Linear Friction Welding (LFW) is a solid-state joining process producing narrow joints mainly developed for the aircraft industry. The thermo-mechanical loads involved in LFW lead to significant local microstructural changes. This study aimed at identifying the mechanisms impacting these changes in order to develop a Post-Weld Heat Treatment (PHWT) optimizing the joint microstructure. The temperature fields showed that a zone of 1 mm on either side of the weld center line experienced thermo-mechanical processing in the β-domain for 2 s followed by a rapid cooling to 400 °C. Inspection of the weld by

2

Optical Microscopy (OM) and Scanning Electron Microscopy (SEM) revealed a strongly affected microstructure characterized by a sharp microstructural refinement and the presence of defects at the interface. The joint consists of: 1) the Welding Line (WL) which underwent a complete α→β transformation accompanied by the recrystallization of the prior-β grain and the development of a {110}<111> texture followed by intragranular precipitation of textured α' Hexagonal Close-Packed (HCP) martensitic laths; 2) the Thermo-Mechanically Affected Zone (TMAZ) characterized by a partial α→β transformation resulting in a microstructure refinement by α variant selection upon cooling. A third zone, the Heat Affected Zone (HAZ), was revealed as having a microstructure indistinguishable from the base material (BM) but being slightly harder.

The texture analysis of the reconstructed β phase in the joint core showed that the local deformation conditions were asymmetrical between the forging and the oscillating part and that the WL may have experienced a complex material stirring with turbulent flow. These microstructural changes generate an increase in hardness in the joint with a maximum increase of HV0.3 by 40% in the WL. The PWHT consisting of an α+β annealing followed by ageing resulted in an α'→α+β decomposition and α globularization in the TMAZ leading to a gradual microstructure refinement from the BM to the WL. A rather homogenous hardness was obtained across the assembly after the PWHT.

INTRODUCTION

Near-α Ti-alloys such as the Ti-6Al-2Sn-4Zr-2Mo (Ti6242) are of interest for gas turbine components as they offer an advantageous compromise between tensile strength, creep resistance and toughness [START_REF] Boyer | Materials properties handbook: titanium alloys, 4. printing[END_REF], [START_REF] Lütjering | Titanium[END_REF].

The near-α alloys are known to have a fairly good weldability using conventional fusion welding techniques. Nevertheless the weld pool generated by those processes could lead to solidification defects and/or oxygen and nitrogen pollution into the joint if the protective environment is deficient [START_REF] Huang | On the mechanism of porosity formation during welding of titanium alloys[END_REF], [START_REF] Donachie | Titanium A Technical Guide Second Edition[END_REF]. This pollution could be responsible for a substantial embrittlement of the weld [START_REF] Liu | Effects of oxygen and heat treatment on the mechanical properties of alpha and beta titanium alloys[END_REF], [START_REF] Ando | Microstructure and mechanical properties of a high nitrogen titanium alloy[END_REF]. As a result, solid-state joining processes are considered as encouraging alternative techniques for joining Ti-alloys because they prevent the joint from solidification defects and environment/surface pollution [START_REF] Mcandrew | Modelling of the workpiece geometry effects on Ti-6Al-4V linear friction welds[END_REF]. Tialloys are excellent candidates for Linear Friction Welding (LFW) as the local friction heating in combination with their low thermal conductivity leads to the formation of thin, sound welds.

Preliminary studies on LFW Ti-alloys focused on determining indicators based on the main process parameters (amplitude, frequency, axial pressure and axial shortening) to define whether a sound joint was achieved [START_REF] Vairis | High frequency linear friction welding of a titanium alloy[END_REF], [START_REF] Wanjara | Linear friction welding of Ti-6Al-4V: processing, microstructure, and mechanical-property inter-relationships[END_REF]. Following studies aimed at providing microstructure and mechanical characterizations of LFW joints mainly focused on α+β Ti-6Al-4V (Ti64) assemblies, however only few papers were dedicated to joining near-α alloys such as Ti6242 or IMI834 [START_REF] Baeslack | Characterization of Solid-Phase Welds Between Ti-6A1-2Sn-4Zr-2Mo-0.1Si and Ti-13.5A1-21.5Nb Titanium Aluminide[END_REF]- [START_REF] Frankel | Comparison of residual stresses in Ti-6Al-4V and Ti-6Al-2Sn-4Zr-2Mo linear friction welds[END_REF]. The approaches adopted in the previous studies for characterizing the microstructures relied on basic observations as their main scope was mechanical characterization. In fact, the microstructures formed in the joint appeared quite similar between α+β and near-α LFW assemblies [START_REF] Li | Linear and rotary friction welding review[END_REF], [START_REF] Mcandrew | A literature review of Ti-6Al-4V linear friction welding[END_REF]: highly textured ultra-fine martensitic α' laths precipitating within recrystallized-like prior-β grains in the core of the joint (100 μm in width) with squeezed α+β layers on the sides (1 mm in width). Such a microstructure led to a substantial hardening of the joint resulting in failure in the base material (i.e. far from the weld) upon tensile testing. As a result the authors developed PWHTs mainly for stress relieving purposes using intermediate temperature ranges (600-700 °C) [START_REF] Meshram | Influence of matrix microstructure on aging response of near alpha titanium alloy (IMI834) parent metal and welds on toughness[END_REF], [START_REF] Frankel | Comparison of residual stresses in Ti-6Al-4V and Ti-6Al-2Sn-4Zr-2Mo linear friction welds[END_REF]. However, reshaping the microstructures resulting from LFW by means of high temperature PWHTs might be of interest as the as-welded microstructures in the joint core are in a metastable state due to the displacive transformation leading to discontinuous mechanical properties across the assembly.

The present study follows on from previous investigations on LFW of the βmetastable Ti17 [START_REF] Ballat-Durand | Multi-scale and multi-technic microstructure analysis of a linear friction weld of the metastable-β titanium alloy Ti-5Al-2Sn-2Zr-4Mo-4Cr (Ti17) towards a new Post-Weld Heat Treatment[END_REF] aiming at: 1) characterizing the very local microstructures and crystallographic configurations; 2) identifying the mechanisms impacting the microstructural changes and the hardness properties within the weld. A similar quantitative approach has been performed in this paper on the near-α Ti6242. However the article will focus on using the crystallographic information from the martensitic aswelded (AW) microstructures to reconstruct the high temperature β microstructure before cooling. The information gathered will permit to design a PWHT that ensures equilibrium in the joint, smooth microstructure transitions without any changes in the as-received microstructure, or in the hardness of the base material. The characteristic LFW macro-zones will be investigated by Optical Microscopy (OM), Scanning Electron Microscopy (SEM), Electron Back-Scattered Diffraction (EBSD), X-Ray Diffraction (XRD) and micro-hardness measurements. The results of monotonic tensile tests and fatigue behavior characterization on the as-welded configuration are reported by others in [START_REF] Garcia | Strength and fatigue strength of a similar Ti-6Al-2Sn-4Zr-2Mo-0.1Si linear friction welded joint[END_REF].

MATERIAL AND METHODS

The LFW equipment used was a MDS-30 developed by the company ACB, Nantes, France. The samples were welded using: a forging pressure of 90 MPa, an oscillating motion amplitude and frequency of 2 mm and 50 Hz, respectively, and a threshold in the axial shortening of 3 mm. The temperatures experienced during the process were measured by nine INCO sheathed thermocouples on each part (forging and oscillating) placed centrally (i.e. at a depth of 7.5 mm) and initially located at 2, 4, 6 and 8 mm from the contact surface.

Blocks of 15*80*70 mm 3 were obtained by electrical discharge machining from a forged billet (R=250 mm ; L=570 mm) of TIMETAL®6-2-4-2 with the 80 mm edge parallel to the billet axis. The material was provided in the solution treated then aged configuration (STA) by TIMET Savoie S.A., France. The specimen were welded on the raw 15*80 mm 2 surface, the 80 mm edge being aligned with the oscillation direction (see Figure 1a). In Figure 1b the flash of expelled material along <x> and <y> can be clearly seen. The samples dedicated to microstructural characterization were cut across the weld ensuring a sample height about 10 mm on either side of the weld center line (Figure 1a). The specimens were polished using the following papers at 100 rpm: P80, P220, P1200, P2400, and P4000; the papers above grade 1000 were used twice for 30 s each. Surface finishing required using polyester cloth polishing, first with a 6 µm diamond suspension then with a Struers oxide polish suspension (OPS) for 2 min each at 100 rpm. Vibratory polishing was performed for EBSD investigations using a solution composed by 50 ml of OPS and 50 ml of H2O on a Buehler VibroMet 2 for 48 h under a load of 200 g. Optical microscopy required the use of an etchant for 12 s at room temperature with the following composition: 2 ml HF, 98 ml H2O [START_REF]Metallography and microstructures[END_REF].

Macro-scale analysis was performed using both an OM Olympus BX41M and a micro-hardness machine Zwick-Roell ZHUµ-S. Micro-scale observations were performed on a Zeiss Σigma SEM (at 15 kV) coupled with an EBSD system (at 25 kV).

The crystallographic investigations were carried out on a Bruker D8 ADVANCE Twin X-Ray diffractometer using a Cu source at 20 kV/40 mA through a 0.1 mm slit with symmetrical beam path. Electrical-tape masks were used to irradiate only the areas of interest. The PWHTs were performed using a Nabertherm P330 oven under an Argon protective atmosphere.

The Matlab© tool box Mtex 5.1.1 was used for post-processing the EBSD data and for reconstructing the β orientations at high temperatures prior to the martensitic transformation with self-developed subroutines. The XRD diffractograms were postprocessed using the software Diffrac.Eva©.

Figure 1 : a) Sketch of the geometry of the welded parts and the sampling method for observations and diffraction analysis [START_REF] Ballat-Durand | Multi-scale and multi-technic microstructure analysis of a linear friction weld of the metastable-β titanium alloy Ti-5Al-2Sn-2Zr-4Mo-4Cr (Ti17) towards a new Post-Weld Heat Treatment[END_REF] with the forging direction <z> and the oscillating direction <y>; b) picture of the whole assembly. The forging part is the upper part (z>0) and the oscillating part is the lower part (z<0). 6

RESULTS AND DISCUSSIONS

MICROSTRUCTURE OF THE BASE MATERIAL

The base material near-α Ti-6Al-2Sn-4Zr-2Mo in the as-received, fully-equiaxed state is characterized by predominant textured nodular primary α (αp) grains (90 vol.%) within a β matrix (10 vol.%) (Figure 2). This microstructure results from β thermomechanical processing forming large recrystallized β grains followed by lamellar α precipitation upon cooling; then α+β forging steps are performed to obtain globularization through lamellar break down of the α colonies [START_REF] Banerjee | Perspectives on Titanium Science and Technology[END_REF]. The fully-equiaxed Ti6242 in this study is characterized by a β-transus of 990 °C, a yield strength and ultimate tensile stress of 895 MPa and 1018 MPa, respectively, with an elongation at fracture of 19%. Quenching the fully-equiaxed Ti6242 from the β domain forms entangled α' martensitic laths precipitated within prior-β grains; fast cooling from the α+β domain forms a bimodal microstructure made of shrunk αp nodules and lamellar secondary α (αs) precipitated within transformed β (βt) zones [START_REF] Boyer | Materials properties handbook: titanium alloys, 4. printing[END_REF], [START_REF] Lütjering | Titanium[END_REF], [START_REF] Semiatin | Alpha/Beta Heat Treatment of a Titanium Alloy with a Nonuniform Microstructure[END_REF].

MACROSCALE ANALYSIS OF A WELDED SPECIMEN

An overall micrograph of the Ti6242 LFW joint is displayed in Figure 3. Friction heating combined with the forging pressure made the viscous material to flow out of the assembly, into the flash, mainly on the long sides of the sample (as also seen in Figure 1b). A significant curvature of the joint can be noted which is characterized by a point symmetry at the weld center. This particular joint morphology is associated with a remarkable asymmetry between both transversal flashes. Both features could be resulting from an initial misalignment of the two parts prior to welding (seen by the offset of the parts in the welded state) which might have affected considerably the material flow during the process. The total LFW process time lasted 1,35 s with an axial shortening of 3,31 mm. The temperature data in Figure 4 showed that solely the thermocouples TC1+5+9, the closest to the weld center line, experienced a local β thermo-mechanical processing characterized by a brisk heating to 1200 °C (1000 K/s) followed by fast cooling (-80 K/s). However one can notice important differences between TC5 at the center and TC1+9 at the edges in terms of their cooling rates as the core cooling was much slower than the ones on the sides. This heterogeneity could have a major impact on larger parts concerning microstructure changes and residual stresses. The second and third rows of thermocouples (namely TC3+7 and TC2+8) recorded similar symmetric maximum temperatures delayed from the end of the oscillations. This delay increases and the local maximum temperature decreases when getting farther from the interface as expected from heat conduction. Furthermore, a rather pronounced {xz} mirror symmetric behavior can be noted between the opposite thermocouples TC1+9, TC3+7 and TC2+8 in both welded parts indicating a probable {xz} mirror symmetry in the heat flux. Yet both fourth rows of thermocouples (TC4+6) were subject to anomalies which might result from the deterioration of the thermocouple connection during the process. Due to the axial shortening by material extrusion, the z position of the thermocouples changed during the process resulting in the TC1+5+9 of the front rows to be approximately at 0.5 mm from the weld center line at the end of the process. A rather symmetrical axial shortening have been noted between the forging part and the oscillating part. Combining the temperature data with the axial shortening enabled the identification of the local temperature maxima depending on the final distance of a material point from the weld center line. These data in Figure 5 revealed that a band of approximately ±1 mm from the interface reached the β-transus of 990 °C and above with a dwell time in the β domain about 2.5 s. A band of approximately ±2 mm experienced temperatures above 700 °C critical for the start of a (partial) α→β transformation [START_REF] Semiatin | Alpha/Beta Heat Treatment of a Titanium Alloy with a Nonuniform Microstructure[END_REF], [START_REF] Malinov | Synchrotron X-ray diffraction study of the phase transformations in titanium alloys[END_REF]. The joint observation by polarized-light OM before etching in Figure 6 highlighted the presence of a strong microstructure heterogeneity between the base material and the joint. A sharp transition has been identified from the equiaxed initial microstructure to a region with microstructure refinement starting 800 μm from the weld center line. This central zone named the Thermo-Mechanically Affected Zone (TMAZ) seemed to exhibit a continuous grain refinement similar to nodule fragmentation. However, such an abrupt microstructure change could not be observed any longer after etching, but it allowed to reveal the subdivision of this central fragmented-like zone into three subzones: the narrow Welding Line (WL) of 60 μm in width made of fine laths within recrystallized-like equiaxed grains (with grain diameter about 5 µm); highly deformed α nodules in the near-TMAZ (NTMAZ) of 300 μm in width, and the far-TMAZ (FTMAZ) up to a width of 800 μm. An antagonism exists with the FTMAZ looking just plastically affected on the etched sample while exhibiting a remarkable microstructure refinement under polarized-light. These microstructure changes result in a significant gradual increase in hardness in the joint with a sharp maximum in the WL (a 40% increase). A third weld zone characterized by a microstructure similar to the one of the base material but displaying slightly higher hardness values has also been identified as the Heat Affected Zone (HAZ) of 800 μm in width connecting the base material and the TMAZ. The total Process Affected Zone (PAZ) width is thus about ±1.6 mm on either side of the weld center line and slightly thinner (by 20%) than expected from the temperature fields.

The temperatures recorded in this study were in the same order of magnitude of 1000 °C as identified in [START_REF] Wen | Mathematical modelling of joint temperature during linear friction welding of dissimilar Ti-6.5 Al-3.5 Mo-1.5 Zr-0.3 Si and Ti-5Al-2Sn-2Zr-4Mo-4Cr alloys[END_REF], [START_REF] Owen | The evolution of Ti-6Al-4V following extreme thermo-mechanical processing[END_REF] for LFW of Ti-6Al-4V. Comparable microstructures have been identified in [START_REF] Baeslack | Characterization of Solid-Phase Welds Between Ti-6A1-2Sn-4Zr-2Mo-0.1Si and Ti-13.5A1-21.5Nb Titanium Aluminide[END_REF]- [START_REF] Frankel | Comparison of residual stresses in Ti-6Al-4V and Ti-6Al-2Sn-4Zr-2Mo linear friction welds[END_REF] on LFW welds of near-α Ti-alloys with a similar weld hardening. Cross-checking the results from OM and micro-hardness data was fundamental to precisely identify the positions of the characteristic macro-zones of this joint despite the temperature data. However, it only suited preliminary identifications of the major microstructural changes. The limited information that can be extracted from those methods solely do not allow any proper characterization of the thermo-mechanical load accommodation mechanisms, nor of the crystallographic changes that happened in the joint during LFW. The following sections are thus dedicated to the microscale characterization of the three macro-zones forming the PAZ with the use of SEM, EBSD and XRD. The information gathered in these sections will be crucial for designing the specific PWHT.

MICROSTRUCTURAL AND CRYSTALLOGRAPHICAL ANALYSIS OF THE CHARACTERISTIC ZONES OF THE JOINT

High magnification observations by SEM in Figure 7 showed the important changes in the microstructure across the weld. The WL is characterized by thin laths precipitated within fine equiaxed grains without any traces of plastic deformation (the discussion will be the subject of the specific section 3.4.2.). On the contrary, the NTMAZ exhibits a severely deformed prior-nodular microstructure, this aspect is diminished when approaching the FTMAZ as well as a gain in contrast with respect to the retained internodular β matrix. Furthermore, the image contrasts suggest that the α grains are progressively fragmented and/or transformed in the TMAZ with traces of fine-lath networks formed within the former β matrix in the NTMAZ. Just deformed fragmentedlike former αp within a clear β matric can be observed in the FTMAZ. Attentive observations at the core of the weld revealed the presence of fine defects and/or porosities at the supposed center of the WL shown in Figure 8. These porosities, highly detrimental to fatigue resistance, have already been reported in [START_REF] Garcia | Strength and fatigue strength of a similar Ti-6Al-2Sn-4Zr-2Mo-0.1Si linear friction welded joint[END_REF] and were attributed to the shrinkage of microscopic liquid pockets. A local enrichment in copper and zinc from electrical discharge machining was believed to cause this liquefaction. Additional EBSD analysis of the HCP phase across the joint in Figure 9 revealed a considerable microstructure refinement involving different mechanisms in the three macro-zones of the joint: the WL, the NTMAZ and the FTMAZ. The zoom in the WL in Figure 9b showed the formation of fine HCP entangled laths in the WL corroborating the SEM observations; no traces of prior-β grains were identifiable here. The FTMAZ fragmented-like microstructure in Figure 9d displayed fine α precipitates together with coarse α grains slightly smaller than the α nodules in the base material coexisting with residual β matrix. The NTMAZ microstructure in Figure 9c appeared to be a hybrid between the two previously depicted microstructures by exhibiting a very fine acicular microstructure interspersed with highly deformed residual nodules. However, the super fine laths formed within the former β matrix in the NTMAZ could not be indexed. Grains formed by less than five measurement points were considered as not indexed.

A specific feature could be noted in the FTMAZ as some fine α precipitates appeared to share clear Burgers Orientation Relationship (BOR) with the adjacent β phase. The results in Figure 10 showed that notable <0001>α//<110>β and <112 0>α//<111>β alignments could be identified between the tested β grain and its α neighbors (40 α grains). Such a BOR between the α nodules and the β matrix could not be found in the base material due to lamellar breakdown. One can also notice an occurring texture in the HCP orientation map in Figure 9a which has been confirmed by plotting the 0001 and 112 0 pole figures in Figure 11. This texture is characterized by the gradual formation of one dominating texture component when approaching the WL from the oscillating FTMAZ. The most frequent texture component in the WL is {0001}<112 0> with the {0001} plane slightly misaligned with the transversal plane {xz} and the <112 0> direction along the main material expulsion direction <x>. Nevertheless no particular texture was observed past 600 μm from the weld center line.

THERMO-MECHANICAL LOAD ACCOMODATION MECHANISMS ACROSS THE WELD

PHASE TRANSFORMATIONS AND MICROSTRUCTURE

REFINEMENT

The main consequence of the LFW process on the joint is the heterogeneous partial/complete α→β phase transformation due to gradients of the thermo-mechanical processing conditions leading to significant microstructural changes. These gradients resulted in the formation of macro-zones distinguishable by sharp microstructure transitions as well as different hardness properties. The thermocouple data suggested that a central band about ±1 mm along <z> experienced temperatures above the βtransus (990 °C) followed by quenching. One could expect from the literature [START_REF] Boyer | Materials properties handbook: titanium alloys, 4. printing[END_REF], [START_REF] Lütjering | Titanium[END_REF] that such a heat treatment should have turned the whole central band into a fully transformed martensitic microstructure. However the phase identification by EBSD and XRD showed solely the joint core (WL+NTMAZs) to exhibit strong correlations with the α' martensitic microstructure obtained in the water quenched sample. Indeed it has been proven by [START_REF] Elmer | Phase transformation dynamics during welding of Ti-6Al-4V[END_REF], [START_REF] Ismail | In-situ investigation of phase transformation kinetics in Ti-6Al-4V under rapid heating condition using high-energy synchrotron diffraction[END_REF] that a substantial superheat above the conventional β-transus is required to complete the α→β transformation in zones that endured high heating rates (500-1000 K/s) during gas tungsten arc welding of Ti-6Al-4V. Furthermore the identification of coarse deformed former αp-like nodules in the NTMAZ could be a marker of a partial α+β→α+α' transformation, so that solely the WL might have undergone a complete α→β transformation. Thus the authors, with the help of EBSD and β reconstruction algorithms, will provide crucial information about phase transformation and microstructural refinement in LFW Ti6242. A quantitative analysis of the EBSD data concerning the features of α/α' grains and the β phase area fraction summarized in Figure 13 Complementary analysis on the cumulated misorientation angle distribution at the α/α' grain boundaries plotted in Figure 14 also revealed a significant microstructure change in the joint from the base material features, the trends appeared in fact quite similar to the α' of the water quenched sample. Indeed the data from the WL, NTMAZ and FTAMZ also showed steps on the curves at the characteristic angles of 60° and 90°.

Such distributions are analogous to the ones obtained from the β→α variant selection during the β→α'/αs transformation within a single β parent grain in [START_REF] Wang | Effect of self-accommodation on α/α boundary populations in pure titanium[END_REF], thus indicating a probable microstructure refinement by β→α variant selection relying on BOR. However the differences with the water quenched sample with respect to the occurrence of small misorientations might be due to two effects in the joint: 1) the fragmentation of prior-β grains which increases the frequency of common boundaries between α laths precipitated from different β parent grains; 2) an increase in poorly indexed regions jeopardizing the theoretical stepped curve. The HAZ exhibited a microstructure and cumulated misorientation angle distribution comparable to the ones of the base material as-received (BM AR) corroborating the threshold effect of the microstructural refinement. Local orientation observations in the FTMAZ revealed that most "fragments" constituting different former αp nodules had orientations whose specific <0001> and <112 0> directions were collinear to <110> and <111> directions of the adjacent β matrix respectively. This indicated probable orientation relationships inherited from a common β parent grain through the BOR. As a result the authors proposed here to reconstruct the prior-β orientations of clusters that share a common β parent to identify the contribution of the α→β→α'/αs phase transformation to the microstructure refinement. To do so, an algorithm based on the work of [START_REF] Tari | Back calculation of parent austenite orientation using a clustering approach[END_REF]- [START_REF] Germain | Reliability of reconstructed β-orientation maps in titanium alloys[END_REF] for prior-austenitic grain reconstruction from martensitic clusters has been developed which relies on the following major steps: 1) finding each local α/α' cluster containing at least three neighboring laths separated by grain boundaries with misorientations above 15°; 2) computing each possible β parent orientation for each HCP orientation ensuring the BOR {0001}α//{110}β and <112 0>α//<111>β; 3) comparing the β parent orientations of a reference lath with the orientations of the other laths of the cluster; 4) assigning to this reference lath the β orientation ensuring a minimum average misorientation among the solutions that have at least two neighbors with misorientation angles lower than 4°; 5) repeat steps 3-4 for each lath in a cluster and 6) steps 2-5 for each cluster. The α/α' laths and fragments dissatisfying these conditions were classified as unreconstructable grains.

The Matlab© scripts using the Mtex 5. The result of the reconstruction shown in Figure 15 revealed that, contrary to the base material as-received and the HAZ, the algorithm identified a significant area fraction of α/α' grains sharing common β parents in the joint. Different features across the WL/NTMAZ and the FTMAZ can be noticed: the WL and the NTMAZ are characterized by much fewer unreconstructable α fragments (6% vs. 20% of the indexed data for the WL/NTMAZ and the FTMAZ, respectively) and the development of a strong texture with <111> along <x> associated with the visible formation of subgrains within deformed large prior-β grains. A retro analysis of the misorientation angle frequency over the α/α' grain boundaries in the joint separating the reconstructable fragments from the unreconstructable ones is shown in Figure 16. The distribution for the unreconstructable α grains revealed the absence of kinks, but a rather continuous increase remarkably different from the distribution of the reconstructable ones.

Furthermore the distribution of the reconstructable fragments appeared remarkably similar to the angle distribution for random variant selection from a single β parent grain thus corroborating the microstructural refinement by β→α variant selection.

Figure 16 : Cumulated correlated misorientation angle distributions excluding the reconstructable fragments (solid red) from the unreconstructable ones (solid black) compared to uncorrelated distribution of random orientations (dashed black). 650 unreconstructable α grains were used in the analysis.

Comparing the SEM and XRD observations with the reconstructed high temperature β microstructure confirmed that actually both the WL and the NTMAZs underwent a complete α→β transformation during the process. The few remaining unreconstructable grains were assumed to be due to a 3D effect where the α'/α lath matching would be found above or below the observed surface. The combined results from the β reconstruction, the boundary misorientation data and the clear BOR between the β matrix and neighboring α fragments suggest that the FTMAZ experienced local thermomechanical loads that might be similar to a conventional α+β processing (T>700 °C).

The latter is known to refine the fully-equiaxed microstructure into a duplex/bimodal one as depicted in [START_REF] Saxena | Processing map-microstructure evolution correlation of hot compressed near alpha titanium alloy (TiHy 600)[END_REF]- [START_REF] Balasundar | Modeling the hot working behavior of near-α titanium alloy IMI 834[END_REF]. Furthermore the shear strains experienced in the TMAZ during LFW associated with the local shear induced by the βt→αs transformation upon cooling might have enhanced the fragmentation of the remaining αp nodules into unreconstructable fragments. The presence of retained β matrix after fast cooling in the FTMAZ might be due to a combination of two effects: the lack of time at high temperatures associated with a poor material mixing. These would lead to a negligible redistribution of β-stabilizing elements into the bulk thus jeopardizing the common duplex-microstructure formation (αs lamellae interspersed with scattered β spots).

MATERIAL FLOW AND RECRYSTALLIZATION

The traces of fine equiaxed grains revealed by SEM observations combined with the misorientation angle distribution of the α' laths in the WL led to the proposal that the martensitic laths precipitated within fragmented prior-β grains in the joint core. This suggestion was corroborated by the computation in Figure 15 showing a visible subgrain formation within deformed prior-β grains in the WL. A preliminary analysis was conducted on the misorientation angles (correlated between neighboring grains and uncorrelated between randomly chosen grains). The results are displayed in Figure 17a combined with the computed β texture within the WL and the NTMAZ presented as pole figures in Figure 17b and Figure 17c respectively. andc) the oscillating NTMAZ. With decreasing distance from the weld center, i.e. from the FTMAZ to the WL, first an increase in the frequency of low misorientation angles at the expense of higher misorientation angles is observed, followed by a slight decrease in favor of boundaries with very high angles. These high fraction of boundaries with very high angles are a consequence of the developing texture from one dominating component towards two dominating components. A threshold of 5° was used for grain boundary detection and halfwidth of 5° was set for the orientation distribution function calculation.

The analysis in Figure 17a revealed a significant increase in the frequency of low misorientation angles between the oscillating FTMAZ and the oscillating NTMAZ. This effect might be due to re-orientation of the β grains during deformation towards an ideal orientation properly aligned with the deformation conditions [START_REF] Sedlacek | Subgrain Formation during Deformation: Physical Origin and Consequences[END_REF]. Indeed a single {110}<111> texture component (with a <111> direction along <x> and a {110} plane as {xy} plane) can be observed in the oscillating NTMAZ in Figure 17c corroborating the increase in frequency of low misorientation angles at the expense of the higher angles. The WL, on the contrary, appeared to be defined by two main twin-symmetric {110}<111> texture components as seen from the pole figures showed in Figure 17b.

However solely one could be identified as coming from the NTMAZ of the oscillating part requiring a detailed local analysis for revealing the origin of the twin-symmetric component (βTWIN). To do so, a preliminary approach was to compare the ideal reconstructed orientations of the β phase from the WL to the ones from the forging and the oscillating NTMAZs in the pole figures displayed in Figure 18. A sharp border can be observed in Figure 19 between the two sides of the weld (i.e forging and oscillating) and the local misorientations to the ideal parent orientation showed the absence of any gradual crystal rotation from one part to the other. Solely a random orientation spread from the ideal orientations could be observed (10° and 8° for the forging and the oscillating part, respectively). The twin component βTWIN which is twin-symmetric to βWL appeared to be mostly present in the oscillating part in the form of large clusters and as scattered fine grains in both NTMAZs. An analysis along the WL from a band of 700x50 μm 2 identified an even blending of βWL and βTWIN (as defined in Figure 18e within the joint center line characterized by an area fraction of 45% of βWL, 44% of βTWIN and 11% of unrelated orientations. This texture blending could be caused by a local interpenetrating material flow mixing of the two singlecomponent textures coming from the forging and the oscillating NTMAZs. Such a material mix could be of major importance as: 1) it could enhance the βstabilizing element redistribution in the bulk explaining the differences of microstructure noted by SEM between the WL and the NTMAZ ("Ghost" β in the NTMAZ); 2) it might have an influence on the defect redistribution impacting the weld quality. Material stirring and the related plastic deformation could also be responsible for the significant increase in frequency observed at the misorientation angle of 60° in the WL compared to the NTMAZ (in Figure 17a) through recrystallized nuclei formation in twin orientation and the generation of βWL/βTWIN boundaries. Indeed large deformation could lead to such an apparently recrystallized microstructures due to a phenomenon called continuous dynamic recrystallization (CDRX) [START_REF] Sakai | Overview no. 35 dynamic recrystallization: mechanical and microstructural considerations[END_REF], [START_REF] Perdrix | Comportement mecanique et evolution structurale de l'aluminium au cours d'une deformation a chaud de grande amplitude[END_REF] by grain subdivision or fragmentation forming large misorientation boundaries as depicted in [START_REF] Hughes | High angle boundaries formed by grain subdivision mechanisms[END_REF], [START_REF] Pantleon | Deformation structure versus grain structure after severe plastic deformation[END_REF].

The recrystallized nuclei drawn in Figure 19 within the stirred microstructure and the corresponding 110 and 111 pole figures for each weld side plotted in Figure 20 revealed the following major features: 1) the nuclei are scattered across the WL and the NTMAZs; 2) The nuclei orientations are mainly related to the local ideal orientations but slightly rotated; 3) a recrystallized area fraction of 16% was identified with a mean nuclei diameter of 9 μm ± 3 μm. In addition to that the asymmetric behavior identified through local texture development is also visible in terms of recrystallization features as: 1) the nuclei rotation from the ideal orientations appeared to be much more remarkable in the oscillating part compared to the forging side; 2) the oscillating part exhibits a higher recrystallized area fraction (17% vs. 14%) with slightly smaller nuclei diameter (8.8 μm vs. 9.2 μm). Preceding studies on LFW of near-α Ti-alloys proposed that dynamic recrystallization might have happened in the WL and in the NTMAZs resulting in a similar grain size [10]- [START_REF] Li | Linear and rotary friction welding review[END_REF]. Yet slightly different α textures were identified in previous studies on LFW joints of Ti-alloys mainly characterized by the prismatic plane being coplanar with the theoretical friction plane and the <112 0> direction being parallel to the oscillation direction which was also the main material expulsion direction [START_REF] Attallah | Microstructural development during linear friction welding of titanium alloys[END_REF]- [START_REF] Romero | Effect of the forging pressure on the microstructure and residual stress development in Ti-6Al-4V linear friction welds[END_REF].

Furthermore a slight rotation in texture in the TMAZ was also observed in [START_REF] Karadge | Texture development in Ti-6Al-4V linear friction welds[END_REF] compared to the weld center corroborating the results shown in this study. The differences between those and the {0001}<112 0> texture reported here with <112 0> parallel to the main expulsion direction, but perpendicular to the oscillation direction might be caused by: 1) the welding configuration (i.e. process parameters, aspect ratios of the welded blocks, the oscillating direction …); 2) the experimental method chosen for texture identification (XRD, EBSD …) and the area investigated which affects the fraction of NTMAZ incorporated in the analysis. Nevertheless it has been shown that the {0001}<112 0> texture of the α/α'phase revealed here resulted from the {110}<111> texture of the β phase developing during LFW in the joint core. Similar texture of the β phase have been reported for LFW β Ti-alloys in [START_REF] Ballat-Durand | Multi-scale and multi-technic microstructure analysis of a linear friction weld of the metastable-β titanium alloy Ti-5Al-2Sn-2Zr-4Mo-4Cr (Ti17) towards a new Post-Weld Heat Treatment[END_REF], [START_REF] Dalgaard | Linear friction welding of a near-β titanium alloy[END_REF]; yet the authors are presently unable to explain the tendency of those precipitates towards developing specific variants rather than a random distribution among all the possible variants from the prior-β grain with {110}<111> orientations. This could be caused by concurrent developing residual stresses or temperature gradients during cooling. Furthermore, the proposal of an asymmetric texture development in the β phase towards different plastic behavior in the forging and the oscillating part (caused by forging and oscillating motion) stated in this study has not been reported in the literature yet, a preceding study on LFW Ti17 also showed an asymmetric behavior between the forging and the oscillating part with the oscillating part being more affected by the heat flux [START_REF] Ballat-Durand | Multi-scale and multi-technic microstructure analysis of a linear friction weld of the metastable-β titanium alloy Ti-5Al-2Sn-2Zr-4Mo-4Cr (Ti17) towards a new Post-Weld Heat Treatment[END_REF].

POST-WELD HEAT TREATMENT

Previous studies [START_REF] Frankel | Comparison of residual stresses in Ti-6Al-4V and Ti-6Al-2Sn-4Zr-2Mo linear friction welds[END_REF], [START_REF] Romero | Effect of the forging pressure on the microstructure and residual stress development in Ti-6Al-4V linear friction welds[END_REF] highlighted the fact that LFW of Ti-alloys causes important residual stresses within the joint in the as-welded configuration which could be detrimental to the assembly fatigue resistance. It has been shown in [START_REF] Frankel | Comparison of residual stresses in Ti-6Al-4V and Ti-6Al-2Sn-4Zr-2Mo linear friction welds[END_REF] that ageing near-α Ti-alloy LFW joint at 700 °C permitted to lower the residual stresses drastically. However, authors in [START_REF] Meshram | Influence of matrix microstructure on aging response of near alpha titanium alloy (IMI834) parent metal and welds on toughness[END_REF] identified that an ageing at 700 °C for 2 h led to a drop in toughness due to interface weakening resulting from β and silicide precipitation along the α lath boundaries. Homogenizing PWHT through β annealing followed by α+β ageing as realized in [START_REF] Ballat-Durand | Multi-scale and multi-technic microstructure analysis of a linear friction weld of the metastable-β titanium alloy Ti-5Al-2Sn-2Zr-4Mo-4Cr (Ti17) towards a new Post-Weld Heat Treatment[END_REF] has been tested by the authors, but led to a loss of the fullyequiaxed microstructure of the base material in favor of a lamellar microstructure (1000 °C/3 h ; 1000 °C → 400 °C at 2 °C/min). Such a microstructure was characterized by highly-textured α macro-zones with significant hardness fluctuations (α phase anisotropy) known to be detrimental to cold dwell fatigue resistance [START_REF] Lütjering | Titanium[END_REF]. The PWHT shown here consisted in an α+β annealing followed by ageing (910 °C/2 h ; 910 °C → 635 °C /2 h ; 635 °C/8 h) in order to allow the α'→α+β decomposition and the α globularization in the TMAZ to form a gradual microstructure refinement across the TMAZ.

OM and SEM observations on the joint after this PWHT are shown in Figure 21 depicting the local microstructure. The annealing in the α+β domain at 80 K below the β-transus for 2 h permitted to decompose the α' into equilibrium α+β in the WL, in the TMAZ this led to αs→β transformation and partial α→β in the rest of the assembly without abnormal β grain growth due to grain pinning by remaining partially transformed αp nodules. Slow cooling followed by ageing at 635 °C allowed to form coarse entangled α laths within a β matrix in the WL and equiaxed α grains in the TMAZ. Furthermore no porosities/defects formerly present in the joint were noticeable after PWHT which might be due defect dissolution into the bulk as observed in diffusion bonding processes [START_REF] Kazakov | Diffusion Bonding of materials[END_REF]. In the SEM images, the α phase in dark grey is distinguishable from the β phase in light grey.

The effect of the PWHT on the microstructural features (i.e. β phase fraction, α grain diameter/aspect ratio and micro-hardness) of the former LFW macro-zones and the base material are shown in Figure 22 in comparison with the ones of the base material as-received. It can be seen from these results that the PWHT ensured: 1) quasiuniform levels of β phase fraction and micro-hardness were obtained in the whole assembly; 2) a gradual microstructure refinement from the HAZ to the WL instead of the as-welded sharp TMAZ/HAZ border; 3) microstructural features of the treated base material similar to the ones of the as-received configuration.

CONCLUSION

A sound weld has been achieved by LFW on the near-α alloy Ti6242. The microstructures and hardness properties resulting from the process have been investigated in the as-welded condition as well as after a PWHT and the following conclusions can be drawn:

1) The supra-transus thermo-mechanical loads occurring near the contact interface caused the formation of a hard joint characterized by three distinct zones: the WL made of highly textured ultra-fine martensitic α' laths precipitated within fine recrystallized prior-β grains, the TMAZ consisting of highly deformed αp nodules and αs fragments with scattered retained β phase, and the HAZ indistinguishable from the base material but displaying a slight increase in hardness which might be due to residual stresses.

2) The high strain and strain rates combined with supra-transus temperatures activated

Figure 2 :

 2 Figure 2 : Fully-equiaxed microstructure of the as-received Ti6242 base material. SEM

Figure 3 :

 3 Figure 3 : Preliminary inspection of the weld by OM after etching revealing an offset in

Figure 4 :

 4 Figure 4 : Temperature measurements across the joint showing a) the thermocouple data

Figure 5 :

 5 Figure 5 : Local temperature maxima measured by the thermocouples (black dots) in

Figure 6 :

 6 Figure 6 : Combined data from OM observations with polarized-light and corresponding

Figure 7 :

 7 Figure 7 : Microscale observations by SEM in Back-Scatter Electron (BSE) mode of the

Figure 8 :

 8 Figure 8 : Microscale observations by SEM in Back-Scatter Electron (BSE) mode of the

Figure 9 :

 9 Figure 9 : EBSD orientation map of the HCP phase (step size=0.3 μm) within the joint

Figure 10 :

 10 Figure 10 : EBSD analysis in the FTMAZ highlighting of the Burgers Orientation

Figure 12 :

 12 Figure 12 : XRD diffractograms of restricted ranges of the diffraction angle of the core

  confirmed the microstructural refinement in the TMAZ visible by OM with polarized light. At the HAZ/TMAZ border a sharp local drop in the grain diameter occurs. Those changes in the grain features are accompanied by a gradual decrease in the volume fraction of the β phase throughout the FTMAZ; the absence of β phase in the WL and the NTMAZ is confirmed corroborating the XRD measurements.

Figure 13 :

 13 Figure 13 : Evolution of the α/α' grain diameter (longest distance between any two

Figure 14 :

 14 Figure 14 : Cumulated correlated misorientation angle distributions across the

  1.1 tool box were tested and validated on clusters of entangled α lamellae within large prior-β equiaxed grains (D=1 mm) of a Ti17 Widmanstätten α+β microstructure.

Figure 15 :

 15 Figure 15 : Orientation map of the BCC β phase obtained by parent grain reconstruction

Figure 17 :

 17 Figure 17 : a) Correlated misorientation angle distributions across the boundaries of the

Figure 18 :

 18 Figure 18 : Local analysis of the BCC {hkl}<111> texture development among the

Figure 19 :

 19 Figure 19 : Spatial distribution of the three main {hkl}<111> texture components of the

Figure 20 :

 20 Figure 20 : 110 and 111 pole figures of the mean orientation of all recrystallized nuclei

Figure 21 :

 21 Figure 21 : Microstructure observations of the weld after PWHT with a) OM

Figure 22 :

 22 Figure 22 : Comparison of the microstructural features between the base material in the

Figure 23 :

 23 Figure 23 : XRD diffractograms of restricted ranges of the diffraction angle of both heat

Figure 24 :

 24 Figure 24 : a) 0001 and 112 0 pole figures of the α phase b) 110 and 111 pole figures of

Figure 1 :

 1 Figure 1 : a) Sketch of the geometry of the welded parts and the sampling method for

Figure 2 : 3 Figure 3 :Figure 4 :

 2334 Figure 2 : Fully-equiaxed microstructure of the as-received Ti6242 base material. SEM

Figure 5 :

 5 Figure 5 : Local temperature maxima measured by the thermocouples (black dots) in

Figure 6 :Figure 7 :Figure 8 :Figure 9 :Figure 10 :Figure 11 :

 67891011 Figure 6 : Combined data from OM observations with polarized-light and corresponding

Figure 12 :

 12 Figure 12 : XRD diffractograms of restricted ranges of the diffraction angle of the core

Figure 13 :

 13 Figure 13 : Evolution of the α/α' grain diameter (longest distance between any two pixels

Figure 14 :

 14 Figure 14 : Cumulated correlated misorientation angle distributions across the boundaries

Figure 15 :

 15 Figure 15 : Orientation map of the BCC β phase obtained by parent grain reconstruction

Figure 16 :Figure 17 :Figure 18 :

 161718 Figure 16 : Cumulated correlated misorientation angle distributions excluding the

Figure 19 :Figure 20 :Figure 21 : 22 Figure 22 :

 1920212222 Figure 19 : Spatial distribution of the three main {hkl}<111> texture components of the

Figure 23 :Figure 24 1

 23241 Figure 23 : XRD diffractograms of restricted ranges of the diffraction angle of both heat

DATA AVAILABILITY

The raw/processed data required to reproduce these findings can be shared on request.

(FTMAZ) transformations upon cooling resulted in a strong texture in the core of the joint with the {0001} plane being parallel with the friction plane {xy} and the <112 0> direction being aligned with the transversal direction <x>.

3) The material consumption and the temperature fields revealed a rather symmetrical behavior between the oscillating and the forging block.

ACKNOWLEDGEMENT

The authors acknowledge the financial support from the French National Research Agency (ANR) through the OPTIMUM ANR-14-CE27-0017 project as well as the Spatial and Aeronautic Research Foundation, the Hauts-de-France Region and the European Regional Development Fund (ERDF) 2014/2020 for the funding of this work.

The authors are also grateful to ACB for providing LFW welded samples and Airbus for both their technical support.


M Chabchoub

B Vieille
email: benoit.vieille@insa-rouen.fr

R Moutou Pitti

Numerical estimation of the mode I strain energy release rate in woven-ply thermoplastic-based composites at high temperature based on Gθ method

Keywords: Thermoplastic-based composites, high temperature, translaminar fracture, strain energy release, Gθ method, numerical modelling

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. Introduction

The generalization of composite materials (50% mass ratio in recent aircrafts) in aeronautical structures raises issues at various levels: design, manufacture, assembly and control in service.

Thus, the increasing use of composites has highlighted the limits of TS resins predominantly used in aeronautics. In addition, manual draping of TS prepregs is a well-controlled, but long and expensive process. Several manufacturing processes offer an alternative to manual draping of TS prepregs: Resin Transfer Molding, infusion, compression molding of thermoplastics. The constraints associated with TS have thus promoted the emergence of new materials Carbon fiber composites and high performance thermoplastic resin (TP) with a better compromise. If this type of polymer has been put aside for a long time due to poor mechanical properties at high temperature (i.e. about 120°-200°C for polymer matrix composites), the development of high performance resins (PEEK, PES, PI, PPS) in the 1990s, opened the door to structural composites combining carbon fibers and high-performance TP matrices. However, the use of polymer matrix composites (PMC) raises several problems concerning their large-scale exploitation in the aeronautics sector in terms of manufacturing (Stamping, RTM, Sheet Molding Compound, interpenetration welding of surface layers, etc.), behavior (under monotonic or cyclic loading), properties under critical conditions and more particularly their fracture behavior at high temperature.

I.1 Fracture mechanics in composite materials

There are different ways to approach fracture mechanics: locally through stress and strain fields around the crack tip, and more globally with the energy release rate (associated with material toughness). Depending on the material behaviour (e.g. linear elastic or elasticplastic), different fracture mechanics frameworks can be considered. Thus, the determination of G-R (brittle fracture) and J-R curves (ductile fracture) is the key point to quantify fracture in terms of energy required to grow a crack in laminates with different lay-ups. In elastic brittle composite materials, fracture is usually analyzed by using the Linear Elastic Fracture Mechanics (LEFM) parameters: R-curve analysis, critical strain energy release rate GIc, stress intensity factor KIc. In elastic-ductile composite materials, fracture analysis is conducted by means of Elastic Plastic Fracture Mechanics (EPFM): J-integral, crack opening displacement or J-R curves. The knowledge of the R-curves is therefore of the utmost importance in composite materials design as it represents the energy brought to the material to extend a crack in heterogeneous and anisotropic media. This energy is directly associated with damage mechanisms resulting from composite materials' fracture toughness. In fibers reinforced polymer composites, the energy dissipated during controlled crack propagation is usually evaluated in terms of different energy-absorbing mechanisms: fiber breakage, local plastic deformation, matrix-cracking, fiber/matrix debonding and fiber pull-out [START_REF] Jk | Crack-tip energy absorption processes in fibre composites[END_REF][START_REF] Jk | The prediction of R-curves and notched tensile strength for composite laminates[END_REF]. In composite laminates, failure is determined by the combination of loading, location of defect and material heterogeneity (presence of matrix-rich regions at the crimp area in woven-ply laminates).

When failure is initiated from an existing notch, a sequence of energy absorbing events occurs in a region surrounding the notch tip. The knowledge of energy-absorbing processes is therefore important since they are responsible for the toughness of the composite. Depending on laminates' stacking sequence, the contribution of matrix behavior to strain energy release rate can be evaluated during damage in both brittle and ductile composite laminates subjected to high temperature conditions (T > Tg) when matrix ductility and toughness are enhanced [START_REF] Vieille | Influence of matrix ductility and toughness on strain energy release rate and failure behavior of woven-ply reinforced thermoplastic structures at high temperature[END_REF][START_REF] Vieille | A fracture mechanics approach using Acoustic Emission technique to investigate damage evolution in woven-ply thermoplastic structures at temperatures higher than glass transition temperature[END_REF]. In composite materials with a brittle matrix, the energies dissipated during crack initiation and propagation in the composite with a weak fiber/matrix interface are derived principally from the work of fiber pull-out (which is due to the frictional forces that hold a broken fiber in its matrix socket). In ductile matrix composite systems, the local plastic deformation of the matrix also must be taken into account to determine the total fracture toughness [START_REF] Lauke | Fracture toughness of short-fibre reinforced thermoplastics[END_REF][START_REF] Kim | Interlaminar fracture toughness of CF/PEI and GF/PEI composites at elevated temperatures[END_REF]. The fracture behavior (and therefore the fracture toughness) of TP polymers exhibits temperature dependence (viscoelastic and viscoplastic behaviors) not only above Tg but also below Tg.

I. 2 Fracture toughness of TP-based composites

When it comes to damage tolerance, high-performance TP matrix laminates are characterized by high values of fracture toughness compared to thermosetting-based composites. In addition, potential advantages of using woven-ply laminates as opposed to UD-ply laminates are discussed in the literature as woven-ply composites have higher values of interlaminar critical strain energy release rates . Most of the studies available in the literature deal with the mode I interlaminar failure of TP based composites by means of double cantilever beam tests. In orthotropic or quasi-isotropic composite laminates, transverse matrix cracking and fibers breakage (also known as translaminar failure modes) are usually the primary damage mechanisms occurring in the early phase of mechanical loading [START_REF] Vieille | Influence of matrix ductility and toughness on strain energy release rate and failure behavior of woven-ply reinforced thermoplastic structures at high temperature[END_REF]. To the author's best knowledge, there is no study in the literature quantifying the influence of time-dependent behaviours on translaminar fracture of woven-ply composites. In the present study, 5-Harness Satin weave carbon fabrics reinforced PolyPhenylene Sulfide (5HS C/PPS) laminates have been studied with two different stacking sequences: a Quasi-Isotropic (QI) sequence displaying a fibre-dominated behaviour and an Angle-Ply (AP) sequence with a matrixdominated behavior [START_REF] Chabchoub | Determination of J-R curves by load separation criterion in highly ductile TP-based composites under high temperature conditions[END_REF]. It is expected that QI specimens show weak time-dependency in contrast to AP specimens. Tensile tests have been carried out on Single-Edge-Notch (SEN) specimens subjected to quasi-static loadings to investigate the effect of time-dependent behaviors on translaminar failure and strain energy release rate [START_REF] Vieille | A fracture mechanics approach using Acoustic Emission technique to investigate damage evolution in woven-ply thermoplastic structures at temperatures higher than glass transition temperature[END_REF]. In other words, the fracture behavior and toughness depend on the contribution of both fiber and matrix nature to damage mechanisms. In addition, the translaminar crack growth resulting from an initial transverse notch is usually self-similar in quasi-isotropic brittle laminates [START_REF] Vieille | Influence of matrix ductility and toughness on strain energy release rate and failure behavior of woven-ply reinforced thermoplastic structures at high temperature[END_REF][START_REF] Vieille | A fracture mechanics approach using Acoustic Emission technique to investigate damage evolution in woven-ply thermoplastic structures at temperatures higher than glass transition temperature[END_REF]. In angle-ply ductile laminates, a self-similar crack growth may be expected in materials that have very strong matrix materials such as the matrix and the fiber/matrix interface are sufficiently strong to enable matrix cracking which results in the breakage of fibers due to stress concentration [START_REF] Vieille | A fracture mechanics approach using Acoustic Emission technique to investigate damage evolution in woven-ply thermoplastic structures at temperatures higher than glass transition temperature[END_REF]. Thus, this idealization of a damage zone, which extends in a self-similar manner, allows a global analysis based on the evaluation of the total strain energy release-rate [START_REF] Pwr | Physical modelling of damage development in structural composite materials under stress[END_REF][START_REF] Whitney | Stress fracture criteria for laminated composites containing stress concentrations[END_REF]. The experimental data used in the present work for comparison purposes with the numerical modelling results are drawn from previous works [START_REF] Vieille | A fracture mechanics approach using Acoustic Emission Technique to investigate damage evolution in woven-ply thermoplastic structures at temperatures higher than glass transition temperature[END_REF][START_REF] Vieille | Influence of matrix ductility and toughness on strain energy release rate and failure behavior of woven-ply reinforced thermoplastic structures at high temperature[END_REF].

I.3 Modeling background

I.3.1 Numerical modelling of the strain energy release rate

The study of fracture mechanics in composite materials is not recent [START_REF] Valentini | Crack propagation in a Brittle elastic material with defects[END_REF][START_REF] Belmonte | A physically-based model for the notched strength of woven quasi-isotropic CFRP laminates[END_REF][START_REF] Pinho | Fracture toughness of the tensile and compressive fibre failure modes in laminated composites[END_REF][START_REF] Shameli | Fracture criterion of woven Glass-Epoxy Composite using a new modified mixed-mode loading fixture[END_REF][START_REF] Fan | Elastic-plastic fracture behavior analysis on a Griffith crack in the cylindrical three-phase composites with generalized Irwin model[END_REF]. When it comes to investigate numerically the crack propagation in TP-based composites at high temperature as matrix toughness and viscous behavior are significantly enhanced, there is virtually no reference in the literature. It is known that the comprehension and the prediction of both initiation and propagation of primary damage in TP composites is highly related to the presence of singular stress and strain fields [START_REF] Pupurs | Micro-crack Initiation and Propagation in Fiber Reinforced Composites[END_REF] responsible for the accommodation of overstresses in high gradients zones (e.g. notches or holes). According to Vieille et al [START_REF] Vieille | Overstress accommodation in notched woven-ply thermoplastic laminates at high-temperature: numerical modeling and validation by digital image correlation[END_REF][START_REF] Vieille | About the influence of temperature and matrix ductility on the behaviour of carbon woven-ply PPS or epoxy laminates: notched and unnotched laminates[END_REF],

the overstress accommodation is highly temperature-dependent in high-performance TP-based composites mainly under off axis loading. Hence, in order to account for the non-linear behavior (time-dependent and plastic behaviors) of TP-based laminates at high temperature, there is a need to develop a reliable model to capture their visco-elastic-plastic behaviors. The present work is based on a viscoelastic and viscoplastic model developed by Albouy et al. [START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF]. In the literature, many studies have been conducted to describe the time dependent behavior of Polymer Matrix Composites (PMCs). A comprehensive review is proposed in [START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF]. Most of the viscoelastic models presented in the literature such as Schapery's model, were initially elaborated for polymer materials and was later extended to composites [START_REF] Bonnet | Matériaux composites soumis à de grandes vitesses de sollicitation. Analyse expérimentale et modélisation[END_REF]. At higher loading levels, the viscoplastic behavior becomes prominent and not surprisingly, the viscoelastic models lose their accuracy. Finally, in the vicinity of , it turns out that most of the viscoelastic models are unable to precisely predict the viscous behaviors [START_REF] Al-Haik | Prediction of nonlinear viscoelastic behavior of polymeric composites using an artificial neural network[END_REF]. That's the reason why, a viscoelastic viscoplastic model has been developed for TP-based composites [START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF] and implemented into a Finite Element code Cast3m®.

The study and the comprehension of fracture mechanics in high performance TP composite require the development of numerical models able to evaluate the energy release rate and predict the initiation and crack growth process. The main purpose of the present work is then to determine the strain energy release rate at crack initiation in high performance TP composite at > . Many numerical tools have been developed to provide the mechanical field state in the vicinity of the crack tip. Among them, the energy methods, based on unvarying integrals, appear to be efficient to evaluate the energy release rate in accordance with a thermodynamic approach [START_REF] Al-Haik | Prediction of nonlinear viscoelastic behavior of polymeric composites using an artificial neural network[END_REF][START_REF] Pitti | Finite element analysis and Fracture in viscoelastic materials by Mv integral-Part I: crack initiation[END_REF]. Using such methods, it is possible to evaluate the fracture parameters far from the crack tip where the mechanical fields are largely disturbed by a strong singularity. However, some of these tools are mathematically limited to simple or global fracture modes for isotropic or orthotropic media [START_REF] Pitti | Finite element analysis and Fracture in viscoelastic materials by Mv integral-Part I: crack initiation[END_REF][START_REF] Pitti | Mixed mode fracture separation in viscoelastic orthotropic media: numerical and analytical approach by the 4 9 -integral[END_REF].

Using a contour integral related to energy in the vicinity of a crack, Rice proposed a method to solve 2D notch and crack problems in elastic-plastic materials [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF]. The -integral can be seen as the amplitude of the stress and strain singularity; the latter is often referred to as the HRR (Hutchinson-Rice-Rosengren) singularity [START_REF] Rice | A path independent integral and the approximate analysis of strain concentration by notches and cracks[END_REF][START_REF] Rice | Plane strain deformation near a crack tip in a power-law hardening material[END_REF][START_REF] Hutchinson | Plastic stress and strain fields at a crack tip[END_REF] which is formally defined by the pathindependent line integral:

= . -. . ! "# (1)
Where # is an arbitrary curve encircling the crack tip oriented by its normal , $ designates the displacement component and is the strain energy density defined as:

= % "ε ( ) (2)
where and ε are the components of stress and strain, respectively.

However, the J-integral leads to difficulties in integrating fields at Gaussian points [START_REF] Dubois | Modelling of the crack growth initiation in viscoelastic media by the -integral[END_REF][START_REF] Dubois | A finite element analysis of creep-crack growth in viscoelastic media[END_REF][START_REF] Bouchard | Contribution à la modélisation numérique en mécanique de la rupture et structures multi matériaux[END_REF].

Alternatively, the Gθ-integral method, applicable to materials whose behavior can be elastic, viscoelastic and viscoplastic [START_REF] Dubois | A finite element analysis of creep-crack growth in viscoelastic media[END_REF], provides a way to determine the strain energy release rate by using a surface integral. Initially developed by Destuynder et al. [START_REF] Destuynder | Sur une interprétation mathématique de l'intégrale de Rice en théorie de la rupture fragile[END_REF], the Gθ-integral is deduced from the J-Integral by operating, on the contour integral domain, a Gauss-Ostrogradsky transformation which changes the integration domain from a contour to a surface V. It is given by:

= % -. * +,+ + . $ ,+ . * +, . "/ (3)
where V is a surface bounded by two contours.

* 0 1 * * 2 3 is the virtual displacement field. It represents the virtual kinematics of the crack.

If and integrals provide to determine an invariant leading at the mechanical state in the crack vicinity, they provide an overall computation regardless the modes of fracture (mode I or II, mixed modes). For this reason, these integrals are used only for pure opening or pure shear fracture modes [START_REF] Pitti | Finite element analysis and Fracture in viscoelastic materials by Mv integral-Part I: crack initiation[END_REF][START_REF] Pitti | Mixed mode fracture separation in viscoelastic orthotropic media: numerical and analytical approach by the 4 9 -integral[END_REF]. In order to take into account the mixed mode separation, Noether proposed the conservative law method, based on an independent path integral [START_REF] Noether | Invariant variations problem[END_REF].

This method is characterized by an expensive finite element discretization.

4 = 1 2 • % , (9) • $ - ( ) • ; , • "# ! (4)
where ( ( ) , $) and ( (9) , ;) are the real and virtual stresses and displacement field respectively.

Then, this integral was modified from a curvilinear integral to a surface integral .

Thus, for plane problems, 4 -integral takes the following form: 9) • $ • * +, "/ .

4 = 1 2 • % ( ) • ; , -,+ (
(5)

In the case of orthotropic materials and for each fracture mode, the virtual displacements fields are expressed as follows [START_REF] Bui | Découplage des modes mixtes de rupture en thermo-élasticité par des intégrales indépendantes du contour[END_REF]:

; = 2 • < (=) • > ? 2 • @ • ℜB C 1 D -D 2 • E 2 • D • FG 2 -E • D 2 • FG H +2 • < 2 (=) • > I 2•J • ℜB K L M NL O • E 2 • FG 2 -E • FG P ; 2 = 2 • < (=) • > ? 2 • @ • ℜB C 1 D -D 2 • Q 2 • D • FG 2 -Q • D 2 • FG H +2 • < 2 (=) • > I 2•J • ℜB K L M NL O • Q 2 • FG 2 -Q • FG P (6)
with

G = cos(*) + U • D • sin (*) avec X ∈ Z1; 2\ and E = ] • D 2 + ] 2 et Q = ^OO L _ + ] 2
D represent the roots of the following equation:

] • D `a + (2 • ] 2 + ] bb ) • D `2 + ] 22 = 0 (7)
] , ] 2 , ] 22 and ] bb are the components of the compliance tensor in an orthotropic body.

More recently, Moutou Pitti et al. have generalized this approach into viscoelastic behavior by combining the generalized Kelvin Voigt model and 4 -integral in the case of an elastic body [START_REF] Pitti | Finite element analysis and Fracture in viscoelastic materials by Mv integral-Part I: crack initiation[END_REF][START_REF] Pitti | Mixed mode fracture separation in viscoelastic orthotropic media: numerical and analytical approach by the 4 9 -integral[END_REF]. Eq. ( 5) becomes:

4* 9 (+) = 1 2 • % (+) ($) • $ (+) -,+ (9)
(;) • ; ,+ (+) • * +, "/ . ( 8
)
with k = (0,1, … . , N), where N is the number of springs.

(+) ($), $ (+) and ,+ (9) (;), ; (+) are the real and virtual stresses and displacements in the k gh spring.

This method is only applicable in the case of 2D configurations. In addition, although the viscoelastic behavior can be implicitly taken into account in the model through 4* 9 -integral, the mechanical fields are determined by considering an elastic behavior. Destunyder et al.

have extended the Gθ method by introducing a vector field * 0 allowing a virtual crown definition [31][36]. This crown should be far enough from the crack tip in order to obtain good approximation of the solution. The vector field * 0 should verify the following properties:

-* is parallel to the crack plan;

-* is normal to the crack front;

the support of * is concentrated in the vicinity of the crack;

-‖*‖ is constant in a defined area around the crack tip

In 2D, two paths j and j 2 can be considered around the crack tip. These paths divide the structure into three domains. For a given angle of inclination k, the vector field * 0 can be expressed as follow [START_REF] Dubois | A finite element analysis of creep-crack growth in viscoelastic media[END_REF]:

l m n* = 11 - o4 o 3 cos (k) * 2 = 11 - o4 o 3 sin (k)
where 4 is an integration point, o and are the intersections between p4 and j and j 2 respectively.

Thus, * is continuously differentiable and takes the following values in the three domains (see Figure 1):

in j qr , ‖*‖ = 1; in j s r , ‖*‖ = 0;

in j tu I , ‖*‖ varies continuously from 1 to 0.

I.3.2 Numerical modelling of time-dependent behaviors

The behavior of woven-ply thermoplastic laminates at temperature higher than their was simulated using a time-dependent model. The purpose of this section was not to detail the model, but it is necessary to briefly recall the viscoelastic viscoplastic model constitutive laws. For more details, the reader can refer to [START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF]. In order to predict accurately the time dependent behavior carbon fibers woven-ply reinforced PolyPhenylene Sulphide (PPS) laminates at temperature higher than their , a linear spectral viscoelastic model and a generalized Norton-type viscoplastic formulation developed for TS-based composites have been considered [20][37]. To a first approximation, the small strain formulation is used. Thus, under an incremental formalism, the total strain can be divided into three parts as follows:

∆ = ∆ s + ∆ 9s + ∆ 9w (9)
where s is the elastic strain, 9s is the viscoelastic one and 9w is the viscoplastic one.

Viscoelastic spectral linear model

In order to simulate the viscoelastic behavior of Polymer Matrix Composites (PMCs) at high temperature, a viscoelastic spectral linear model was chosen [START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF]. It consists in decomposing the viscoelastic strain rate x 9s into elementary mechanisms ξ z which are associated with relaxation time spectrum such as:

x 9s = { | q } ~ ( 10
)
where • is the total number of mechanisms, and the elementary viscous mechanisms | are defined by:

| x = € (• S ƒ" σ -| ) (12)
where is the Cauchy stress tensor and ] 9s is the viscoelastic compliances tensor.

The viscoelastic formulation assumes a normal distribution of the relaxation mechanisms weights • according to the relaxation time of the i th mechanism (U) = log (ˆ ) where ˆ is the relaxation time:

• = 1 ) √@ × B‹E OE-• (U) -t ) Ž 2 • (11)
with = t + ) + (U -1)∆ is the ith relaxation mechanism and ∆= 2 ) / • -1 is the time interval separating two relaxation times. The spectrum is only described by two parameters: its standard deviation ) and its average value t .

In the case of woven ply, under the assumption of a purely elastic behavior in fibers directions (direction 1 and 2 in the case of a woven-ply lamina) and a plane stress state (thin laminated plate), S ƒ" can be defined by:

S ƒ" = • • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 k aa / 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 • • ( 13
)
where 2 is the shear modulus and k aa is a material viscosity parameter.

Generalized Norton-type viscoplastic model

In order to develop a satisfactory modelling at high stress levels, a generalized Norton-type viscoplastic model [START_REF] Boubakar | A meso-macro finite element modelling of laminate structures: part II: time-dependent behaviour[END_REF] is added to the viscoelastic model:

' 9w -' = -' -ˆ"( ) (14)
with -' = > -' 4 -' " 4 describes the viscoplastic flow anisotropy associated with the PPS matrix in shear loading:

M = • • 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 • • (15)
' is the thermodynamic force. In the case of a linear kinematics hardening, it is associated withsuch as ' = --.

is a material parameter associated with the linear kinematic hardening.

The constitutive laws for the viscoplastic strain rate x 9w and the kinematic hardening rate -x are derived from the thermodynamical potential ' 9w such as:

x 9w = <〈' 9w 〉 š › oe• = and αx = -<〈' 9w 〉 š › oe• Ÿ ( 16
)
where represents the material rate sensitivity and < is a penalty coefficient. When < → ∞, the classical time-independent plasticity expression is recovered. Therefore, the evolution laws can be expressed as follows:

x 9w = £ x 9w ¤ =NŸ =NŸ and -x = £ x 9w ¤ =NŸ =NŸ = x 9w (17)
where £ x 9w = > " x 9w 4 N x 9w is the Lagrange viscoplastic multiplier homogenous to a strain rate. Using a backward Euler method, the previous constitutive laws have been time-discretized, and the resulting incremental laws have been implemented into the finite element code Cast3m® [START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF].

Parameters identification

It is noteworthy that the main advantage of the present modelling compared to other models found in the literature is the reduced number of parameters to identify. The parameters of the viscoelastic and viscoplastic constitutive laws are summarized in Table 1. The mechanical tests carried out to identify the parameters of the models are detailed in [START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF].

I.4 Objectives of the work

The Gθ-integral can be computed by means of the procedure G_THETA implemented in the finite element code Cast3m®. This procedure can deal with elastic, elasto-plastic and viscoplastic problems based on 2D, 3D or axisymmetric geometries, and can account for the behavior of homogenous, non-homogeneous or composite materials. It can also be used to separate the mixed modes for homogenous and isotropic materials. However, this procedure is complicated and it's very difficult to adapt it to external constitutive laws. In addition, by analyzing the analytical form of Gθ-integral, one can notice that it is expressed in term of displacement, strain and stress mechanical fields. In this context, the idea was to apply directly the Gθ integral form based on the CH_THETA procedure implemented into the finite element code Cast3m® in order to create a * field. In order to accurately evaluate the Gθintegral using Gθ method, it is required to predict accurate local mechanical fields in the vicinity of the crack tip by means of optimized meshing. This can be achieved by using the viscoelastic viscoplastic model constitutive laws implemented in Cast3M [START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF].

Thus, the mode I strain energy release rate was evaluated by using Gθ-integral method based on the mechanical fields at the crowns surrounding the crack tip. The independence path integral was examined by representing the evolution of the strain energy release rate versus each crown in laminates characterized by viscoelastic viscoplastic behaviours at high temperature. Then, the strain energy release rate was plotted as a function of the applied load and crack growth. The obtained results were compared to experimental ones. It's worth precising that the purpose of the present work is not to simulate the crack growth under mechanical loading but to evaluate the capability of this method to determine the mode I strain energy release rate for different loading levels at crack initiation.

II. Results and discussion

II. 1 Meshing strategies

In order to test the capability of the numerical model to predict the response (overstress and overstrain profiles) of high-gradient structures [START_REF] Vieille | Viscoelastic viscoplastic model for aeronautical thermoplastic laminates at high temperature: Validation on high stress gradient structures[END_REF], simulations have been carried out on Single Edge Notched (SEN) specimens, characterized by a ratio a/w=0.3. For symmetry reasons, only a half of the plate is modelled. Although reduction in the mesh size leads to more accurate FE solution, the finer mesh refinement requires more computation time. That is the reason why, it is necessary to find a compromise between accuracy requirements and computation time [START_REF] Chabchoub | Investigations on crack propagation and strain energy release rate in notched woven-ply thermoplastic laminates at high-temperature[END_REF]. To this end, three-node triangular elements (Discrete Kirchhoff Triangles -DKT) have been used to mesh the crack tip with higher mesh density and two different types of meshing (radiant and concentric -see Figure 2) around the crack tip were tested, allowing us to generate crowns around the crack tip. Indeed, crowns, whose boundaries coincide with elements sides, have a crucial role when computing the strain energy release rate by means of the method. In addition, the numerical integration is performed at Gauss integration points of elements belonging to the crown, giving therefore more accurate results.

The numerical overstress and overstrain profiles was validated through the comparison with theoretical stress fields and longitudinal overstrain fields obtained from experimental fullfield measurements (by Digital Image Correlation technique). The selected meshing type was then used to simulate the numerical overstrain and overstress profiles for AP laminates [START_REF] Chabchoub | Investigations on crack propagation and strain energy release rate in notched woven-ply thermoplastic laminates at high-temperature[END_REF].

One of the most important criteria for selecting an efficient mesh is its ability to generate accurate overstress and overstrain fields in high gradient zones. Indeed, the issue of stress concentration in composite materials design has always been of great interest to researchers [START_REF] Tan | Finite-width correction factors for anisotropic plate containing a central opening[END_REF][START_REF] Tan | Stress concentration in laminated composites[END_REF][START_REF] Huang | Stress concentration factor in matrix of a composite subjected to transverse compression[END_REF], and many studies have been conducted to investigate the effect of stress concentration zones (notches, holes…) on the mechanical properties of composites [START_REF] Liu | A new ultimate strength model of notched composite laminates including the effects of matrix[END_REF].

II. 2 Numerical modelling of the strain energy release rate

The strain energy release rate was computed by means of the numerical model in the case of laminates with two different stacking sequences: a Quasi-Isotropic (QI) sequence characterized by a fibre-dominated behaviour and an Angle-Ply (AP) sequence whose behaviour is matrix-dominated. It is expected that the PPS matrix viscoelastic-viscoplastic behavior has significant influence on the values of the strain energy release rate.

First of all, from the macroscopic standpoint, the proposed model predicts suitably the tensile mechanical response of notched specimens consisting of angle-ply and quasi-isotropic stacking sequences (see Figure 3). In QI specimens, a radiant meshing appears to be slightly more accurate than the concentric meshing. From the ¥ ¦ method introduced in section I.3, it is possible to estimate the strain energy release rate in AP and QI laminates. For the two stacking sequences, two ratios a/w=0.2 and a/w=0.3 were studied. These ratios were chosen to obtain a strain energy release rate which is stabilized far from both the crack tip and the specimen edge. In fact, it is necessary to choose an integration crown in such a way as to validate the path independence domain. Figures 4567show the evolution of the strain energy release rate according to the i th integration crown in QI and AP laminates. It appears that ¥ ¦ highly depends on the integration crown. In both cases, ¥ ¦ increases gradually until a constant value is reached. It's the high singularity of the mechanical fields which disturbs the field integration around the crack tip. Based on Eq. ( 5) and -method detailed in section I.3, the strain energy release rate was determined for different testing conditions.

The behavior of QI laminates is almost elastic brittle and it's characterized by rapid crack extension until brittle failure. However, the material exhibits a stable crack growth before crack initiation. Thus, it is important to consider the gradual propagation of the crack in the model. Figure 8 represents a comparison between numerical and experimental evolution of the strain energy release rate as a function of the applied load by considering computations with gradual crack growth (referred to as artificial propagation in Fig. 8). For a/w=0.2, the numerical results are satisfactory by considering the artificial propagation, though the simulation with a/w=0.3 shows some discrepancy between experimental and numerical results. Similarly for G-R curves in the case of QI laminates (see Figure 9).

In AP specimens with small ratios a/w (e.g. a/w=0.2), crack tip blunting is observed before reaching the critical strain energy release rate without important crack propagation [START_REF] Chabchoub | Determination of J-R curves by load separation criterion in highly ductile TP-based composites under high temperature conditions[END_REF], hence justifying that only crack opening is taken into account. Figure 10a represents the evolution of the strain energy release rate as a function of the applied load before the critical load. For a/w=0.3, the numerical result seems to be in accordance with the experimental results.

However, a discrepancy is observed in the case of a/w=0.2, as the model doesn't take into account crack tip blunting coming along with large fiber rotation and displacement near the crack tip [START_REF] Chabchoub | Investigations on crack propagation and strain energy release rate in notched woven-ply thermoplastic laminates at high-temperature[END_REF] to accurately simulate the J-R curves (see Figure 10b). In addition, along with the instability of the crack propagation, it appears essential to use an automatic remesher which takes into account the crack propagation at each time step as well as its direction.

Finally, -method provides an overall computation of the strain energy release rate in mode I; and cannot give an accurate approximation of the strain energy release rate in mode II as AP laminates are characterized by a mixed failure mode [START_REF] Vieille | Influence of matrix ductility and toughness on strain energy release rate and failure behavior of woven-ply reinforced thermoplastic structures at high temperature[END_REF].

III. Conclusion

Within the framework of fracture mechanics, the determination of G-R (brittle fracture) and J-R curves (ductile fracture) is utmost important to quantify fracture in terms of energy required to grow a crack in laminates with different lay-ups. Thus, the purpose of the present work was to apply the -integral in the case of C/PPS laminates characterized by a locally more or less significant time-dependent behavior at T>Tg in order to determine the mode I strain energy release rate (associated with the fracture toughness) at crack initiation. Depending on the stacking sequence, the highly viscous behavior of TP-based laminates at high temperature contributes to the increase in the fracture energy. By means of a time-dependent model implemented in the finite element code Cast3m and a meshing optimization, accurate mechanical fields have been simulated in the vicinity of the crack tip. A radiant mesh was selected to compute the mode I strain energy release rate by applying the method to AP and QI laminates with different ratios a/w. The R-curves have been derived from the computation of the strain energy release rate and the corresponding crack length. Ultimately, it is possible to build R-curves based on these numerical evaluations of fracture toughness and to compare these curves to the ones obtained from experiments.

Due to different phenomena (such as damage and crack instability) which are not taken into account by the model, a few discrepancies between experimental and numerical can be observed after crack initiation. The obtained results are highly sensitive to crack stability as, once the crack is initiated, it starts propagating in instable manner. To control the stability of the crack, compact tension specimens could prove to be relevant as the crack growth is inherently stable in such specimens. In addition, a remeshing technique will be considered to account for translaminar crack propagation, and to allow the model to improve its predictive capability.

Figure captions

  Kim et al. have studied the influence of temperature on the interlaminar failure of composites with TP coated carbon fibers [4]. Vieille et al. have studied the evolution of mode I translaminar failure along with the strain energy release rate at high temperature in carbon fibers reinforced PolyPhenylene Sulfide (PPS) composites [3-4].

Figure 1 -Figure 2 -Figure 3 -Figure 4 -Figure 5 -Figure 6 -Figure 7 -Figure 8 - 3 Figure 9 - 2 Figure 10 -

 1234567839210 Figure 1 -Integration domain used for Gθ method

Table 1 .

 1 2 Parameters identification -mechanical properties, viscoelastic and viscoplastic parameters[START_REF] Albouy | Experimental and numerical investigations on the timedependent behavior of woven-ply PPS thermoplastic laminates at temperatures higher than glass transition temperature[END_REF]

		Mechanical properties		Viscoelastic		Viscoplastic
						parameters		parameters
	E1(GPa) E2(GPa) G12(GPa) υ12 τy(MPa)	nc	n0	β44	δ(MPa)	K	N
	56.5	56.6	1.35	0.04	10	4.05	6.9 0.6	400	8.4e -12 9.5