A Rogani

P Navarro

S Marguet

J-F Ferrero

C Lanouette

Tensile post-impact behaviour of thin carbon/epoxy and glass/epoxy hybrid woven laminates -Part I : experimental study

Keywords: Woven composites, thin laminates, post-impact behaviour

This article concerns the study of post-impact damage propagation in two thin hybrid woven composite laminates loaded in tension. Low velocity normal drop weight impact tests are first performed. Post-impact quasi-static tensile tests are then carried out and monitored by Digital Image Correlation. The influence of the impact energy variation on the fracture surfaces and the residual strength is studied. In order to have a better understanding of the mechanisms involved into the hybrid laminates, four monomaterial sample configurations are also tested. When the damage after impact is barely visible, the sample behaves as a non-impacted sample. When the damage is visible, it propagates whatever the impact energy. The damage grows in the direction perpendicular to the loading for all the monomaterial laminates while two different damage scenarios are observed for the hybrid laminates. The residual strength evolves in three steps depending on the impact energy : first a very low decrease, then an important drop and finally a plateau.

Introduction

This two parts article deals with the post-impact behaviour of thin hybrid woven composite laminates under tensile loading. In this paper, an experimental study is carried out on hybrid and monomaterial woven composite samples submitted to a tensile loading after having been impacted with a low velocity in a drop tower device.

Composite structures are widely used, specifically in the field of aeronautics. These structures are known to be very sensitive to impact loadings. Indeed, impacted composites usually present matrix cracking, delamination and fibre breakage. These damages, that can be barely detectable, induce a decrease of the mechanical properties and a loss of the strength. Thus, applying a load on an impacted composite structure can lead to the final failure of the structure and have dramatic consequences. For all these reasons, understanding and characterizing the post-impact behaviour of composite structures, that is to say the growth of the damages induced by impacts, is important.

Post-impact behaviour of unidirectional composite has been widely studied. In particular, the compressive and tensile post-impact behaviour has been investigated [START_REF] Caprino | Residual strength prediction of impacted cfrp laminates[END_REF][START_REF] Cantwell | An assessment of the residual strength of an impact-damaged carbon fibre reinforced epoxy[END_REF][START_REF] Richardson | Review of low-velocity impact properties of composite materials[END_REF][START_REF] Mosallam | Assessment of residual tensile strength of carbon/epoxy composites subjected to low-energy impact[END_REF][START_REF] Wang | Low-velocity impact and residual tensile strength analysis to carbon fiber composite laminates[END_REF][START_REF] Balasubramani | Prediction of residual tensile strength of laminated composite plates after low velocity impact[END_REF]. The residual tensile strength versus impact energy evolves in three steps. First, no damage occurs as long as the impact energy is below the damage threshold. Then the residual strength decreases rapidly with the impact energy. Finally, when the impact perforates the laminate, the residual strength remains constant. The critical damage mode is fibre breakage. For the compressive post-impact loadings, the main failure mode is local buckling due to the delamination induced by impact.

Concerning post-impact behaviour of woven laminates, tensile and compressive behaviours have been studied. The compressive behaviour after impact is the most studied because it is the most critical failure mode [START_REF] Kim | Impact and delamination failure of woven-fabric composites[END_REF][START_REF] Davies | Impact damage and residual strengths of woven fabric glass/polyester laminates[END_REF][START_REF] Shim | Characterization of the residual mechanical properties of woven fabric reinforced composites after low-velocity impact[END_REF][START_REF] Schubel | Impact and post impact behavior of composite sandwich panels[END_REF][START_REF] Aktas | The effect of stacking sequence on the impact and post-impact behavior of woven/knit fabric glass/epoxy hybrid composites[END_REF]. Indeed, the impact generates local delamination that induces micro-buckling of the bundles of fibres that can lead to a total failure. Studies on post-impact tensile behaviour on woven fabric laminates are more recent [START_REF] Koo | Prediction of residual strength of cfrp after impact[END_REF][START_REF]Investigation of impact residual strength of woven and non-woven composites[END_REF][START_REF] Kounain | Low velocity impact and post impact tensile properties of plain weave woven gfrp composite laminates[END_REF][START_REF] Mouhoubi | Residual properties of composite plates subjected to impact fatigue[END_REF]. Most of them are conducted on carbon with layers oriented at 0 • /90 • . The residual tensile strength decreases with the impact energy, while the global stiffness does not seem to be significantly affected. In these studies, the post-impact strength can be directly connected to the failure of the fibres.

However, all the studies are made on composites made up with only one material. During the recent years, authors started to take an interest in hybrid composites, more specifically in composites composed of two materials or more. The objective is an improvement of the properties and a reduction of the costs. Indeed, the in-plane mechanical properties of hybrid laminates composites can be better than single-material laminates composites. For example, adding glass plies to a carbon laminate improves the tensile behaviour as the glass fibres have a greater strain to failure and are more ductile in comparison to carbon fibres [START_REF] Bunsell | Hybrid carbon and glass fibre composites[END_REF][START_REF] Aveston | Synergistic fibre strengthening in hybrid composites[END_REF][START_REF] Summerscales | Carbon fibre and glass fibre hybrid reinforced plastics[END_REF][START_REF] Marom | Hybrid effects in composites: conditions for positive or negative effects versus rule-of-mixtures behaviour[END_REF][START_REF] Manders | The strength of hybrid glass/carbon fibre composites[END_REF][START_REF] Kretsis | A review of the tensile, compressive, flexural and shear properties of hybrid fibre-reinforced plastics[END_REF]. On the contrary, adding carbon plies to a glass laminate increases the tensile stiffness and strength to failure. Concerning the strain to failure of the hybrid laminates, some authors use the term of "hybrid effect" or "synergistic effect" : the value experimentally obtained is greater than that calculated with the rule of mixtures. Furthermore, this hybridization can prevent the catastrophic failure behaviour of carbon fibres. Indeed, some studies show that the final break of hybrid laminates is not sudden but progressive while the failure of the different materials is not simultaneous [START_REF] Bunsell | Hybrid carbon and glass fibre composites[END_REF][START_REF] Aveston | Synergistic fibre strengthening in hybrid composites[END_REF][START_REF]Tensile properties of bidirectional glass-carbon hybrid composites: Part i[END_REF]. It leads also to an improvement of the impact response. As for the tensile behaviour, glass fibres can be added to carbon laminates [START_REF] Harris | Impact properties of glass fibre/carbon fibre hybrid composites[END_REF][START_REF] Summerscales | Carbon fibre and glass fibre hybrid reinforced plastics[END_REF][START_REF] Saka | A simple laminate theory approach to the prediction of the tensile impact strength of woven hybrid composites[END_REF][START_REF] Enfedaque | Effect of glass fiber hybridization on the behavior under impact of woven carbon fiber/epoxy laminates[END_REF], their strain-rate and strength-rate response being much better. However, in this case, the "hybrid effect" depends on the stacking sequence and the position of the glass plies [START_REF] Hosur | Studies on the low-velocity impact response of woven hybrid composites[END_REF]. Other combinations are also tested as carbon/aramid [START_REF] Imielińska | Air-coupled ultrasonic c-scan technique in impact response testing of carbon fibre and hybrid: glass, carbon and kevlar/epoxy composites[END_REF][START_REF] Gustin | Low velocity impact of combination kevlar/carbon fiber sandwich composites[END_REF][START_REF] Sayer | An experimental investigation on the impact behavior of hybrid composite plates[END_REF] or carbon/polyethylene fibres [START_REF] Jang | Impact resistance and energy absorption mechanisms in hybrid composites[END_REF]. Adding these materials to carbon plies increases the energy absorbed during the impact. Combinations without carbon are also investigated like graphite/glass [START_REF] Wang | Impact behavior of hybridfiber and hybrid-matrix composites[END_REF][START_REF] Kowsika | Static and low-velocity impact response characteristics of pultruded hybrid glass-graphite/epoxy composite beams[END_REF] or graphite/nylon [START_REF] Jang | Impact resistance and energy absorption mechanisms in hybrid composites[END_REF] and during the last decade, the use of synthetic and natural fibres (as basalt fibres) to replace carbon or glass fibres has been also studied [START_REF] Rosa | Post-impact mechanical characterisation of glass and basalt woven fabric laminates[END_REF][START_REF] Nunna | A review on mechanical behavior of natural fiber based hybrid composites[END_REF][START_REF] Sarasini | Hybrid composites based on aramid and basalt woven fabrics: Impact damage modes and residual flexural properties[END_REF][START_REF] Petrucci | Impact and post-impact damage characterisation of hybrid composite laminates based on basalt fibres in combination with flax, hemp and glass fibres manufactured by vacuum infusion[END_REF]. Recently, an other type of hybridization has been studied : the same fiber and matrix materials are used but with different layers architectures, for example a combination of woven fabrics and unidirectionnal plies (UD [START_REF] Liu | Experimental and numerical studies on the impact response of damage-tolerant hybrid unidirectional/woven carbon-fibre reinforced composite laminates[END_REF][START_REF] Mahmoud | Experimental and numerical study of the damage mechanisms in hybrid unidirectional/woven composites under impact loading[END_REF]. The impact behaviour in terms of damage extent is improved when UD plies are placed between woven fabrics plies.

The post-impact behaviour of hybrid laminates has been less studied. Concerning the unidectionnal composites, Naik et al. studied the post-impact compressive behaviour of glass-carbon/epoxy laminates [START_REF] Naik | Impact response and damage tolerance characteristics of glass-carbon/epoxy hybrid composite plates[END_REF] and Petrucci et al. worked on the flexural behaviour of glass-basalt/epoxy hybrid laminates [START_REF] Petrucci | Impact and post-impact damage characterisation of hybrid composite laminates based on basalt fibres in combination with flax, hemp and glass fibres manufactured by vacuum infusion[END_REF]. For woven composites, Dehkordi et al. studied the post-impact compressive behaviour of basalt-nylon laminates [START_REF] Dehkordi | The influence of hybridization on impact damage behavior and residual compression strength of intraply basalt/nylon hybrid composites[END_REF] while other authors investigated the flexural behaviour of glass-basalt/epoxy and aramid-basalt hybrid laminates [START_REF] Rosa | Post-impact mechanical characterisation of glass and basalt woven fabric laminates[END_REF][START_REF] Sarasini | Hybrid composites based on aramid and basalt woven fabrics: Impact damage modes and residual flexural properties[END_REF]. Different stacking sequences were compared through these studies. In the end, the authors showed that the hybridization of the material allows a better residual buckling strength for the laminate in compression and a better residual flexural strength for the laminate in bending.

But, neither study deals with the tensile behaviour of an impacted hy-brid carbon/glass woven laminate while this material combination has been quite studied in the litterature [START_REF] Bunsell | Hybrid carbon and glass fibre composites[END_REF][START_REF] Aveston | Synergistic fibre strengthening in hybrid composites[END_REF][START_REF] Summerscales | Carbon fibre and glass fibre hybrid reinforced plastics[END_REF][START_REF] Marom | Hybrid effects in composites: conditions for positive or negative effects versus rule-of-mixtures behaviour[END_REF][START_REF] Manders | The strength of hybrid glass/carbon fibre composites[END_REF][START_REF] Kretsis | A review of the tensile, compressive, flexural and shear properties of hybrid fibre-reinforced plastics[END_REF][START_REF]Tensile properties of bidirectional glass-carbon hybrid composites: Part i[END_REF][START_REF] Harris | Impact properties of glass fibre/carbon fibre hybrid composites[END_REF][START_REF] Saka | A simple laminate theory approach to the prediction of the tensile impact strength of woven hybrid composites[END_REF][START_REF] Enfedaque | Effect of glass fiber hybridization on the behavior under impact of woven carbon fiber/epoxy laminates[END_REF]. Thus, this paper focuses on the post-impact damage propagation in two thin hybrid carbon-glass/epoxy woven composite laminates under tensile loading. In parallel, monomaterial carbon/epoxy and glass/epoxy laminates with different ply orientations are also studied. Low velocity impacts are first performed with a drop weight device. Then quasi-static tensile tests are carried out on the impacted samples. Impact energies are varied. The influence of the hybridization, the initial damage and the stacking sequence are investigated. An analysis of the damage scenario based on measures from Digital Image Correlation is provided. The post-impact damage propagation in the hybrid laminate happens to be highly influenced by the presence of a glass ply and by the carbon plies orientations.

Material and samples

In this study, post-impact tensile behaviour of thin hybrid woven composite laminates is investigated. Two hybrid sample configurations commonly used as skin of helicopter blades, presented in Table 1, are studied. They are made with one ply of glass/epoxy and two plies of carbon/epoxy woven fabrics : G 0 C 45 C 45 is made up of one ply of glass woven oriented at 0 • and two woven carbon plies oriented at ±45 • , G 0 C 45 C 0 is made up of one ply of glass woven at 0 • , one ply of carbon woven at ±45 • and one ply of carbon woven at 0 • .

In addition, in order to have a better understanding of the mechanisms involved in the post-impact tensile behaviour, four monomaterial sample configurations, that represent parts of the studied hybrid configurations, are also tested. These four configurations, presented in Table 2, are made with two plies of the same material : C 0 C 0 is made up of two plies of carbon woven oriented at 0 • , C 45 C 45 is made up of two plies of carbon woven at ±45 • , C 45 C 0 is made up of one ply of carbon woven at ±45 • and one ply of carbon woven at 0 • and G 0 G 0 is made up of two plies of glass woven oriented at 0 • .

The samples have been designed to be able to perform low velocity impact tests and tensile tests on the same specimen. The geometry is presented in Figure 1. The dimensions are 100 mm x 300 mm with a thickness of 0.62 mm

(G 0 G 0 ), 0.71 mm (C 0 C 0 , C 45 C 45 , C 45 C 0 ) or 1.02 mm (G 0 C 45 C 45 , G 0 C 45 C 0 ).
The tabs are made with three plies of glass/epoxy woven fabrics oriented at ±45 • .

Low velocity impact tests

The experiments are conducted in two steps. First an impact damage is produced in the samples, then tensile tests are carried out.

The low velocity impact tests are performed with a drop weight device. The impactor has a mass of 2 kg and a 16 mm diameter hemispherical steel head. The samples are clamped on a 75 mm x 125 mm rectangular frame. The reaction force and the impactor displacement are recorded during the impact. Impact velocities and corresponding impact energies are varied. The values are given in Table 3.

Figures 234567show the fracture surfaces of the impacted samples for all the configurations. The values in yellow represent the size of the damaged zone on the upper ply and the values in white represent the size of the fibres breaks observed in the top and bottom plies.

For the C 0 C 0 configuration, no visible damage is observed for the laminate impacted with an energy of 1 J. For higher impact energies, cross-shaped failures oriented in the direction of the fibres are observed. The sizes of the damage increase with the energy of impact until an energy of 4 J. Indeed, for the impacts higher than 4J the impactor totally perforates the sample. For the C 45 C 45 configuration, there is practically no visible damage for an impact with an energy of 2.25 J. The sizes of the damage increase with the energy of impact. Until 4 J, the impactor rebounds and from 6.25 J, it completely perforates the sample. For the C 45 C 0 configuration, there is no visible damage for an energy of impact of 1J. The size of the damage increases with the energy of impact until an energy of 4 J, which corresponds to the perforation of the sample by the impactor with no rebound. The directions of the breaks of fibres are between ±45 • and 0 • /90 • in the upper ply and are 0 • /90 • in the lower ply. The damage mainly follows the directions of the lower ply in which the damage initiates during the impact. For the G 0 G 0 configuration, no visible damage is observed until an energy of 4 J. Then, the sizes of the damage increases with the energy of impact until 6.25 J, which corresponds to the energy from which the impactor totally perforates the sample. The difference with the carbon woven configurations in terms of impact energies are explained by a lower stiffness and a strong dynamic effect for the glass woven, which leads to an increase of the breaking limit with the strain rate [START_REF] Navarro | Etude de l'impact oblique à haute vitesse sur des structures en sandwich composite : application aux pales d'hélicoptère[END_REF][START_REF] Aubry | Etude expérimentale et numérique d'impact sur structures : application aux pales d'hélicoptères[END_REF].

For the G 0 C 45 C 45 configuration, the damage is barely visible for an energy of 1 J. Then, the damage increases until 6.25 J and a rebound of the impactor is observed. Carbon fibre failures are noticed in the lower ply and the resin of the upper ply is damaged. For the impact energy of 6.25 J, slight fibre breaks are noticed in the upper ply. Finally for the 9 J and 16 J tests, the impactor perforates the three plies so that the damage size is the same for the two impacts. These observations are the same for the G 0 C 45 C 0 configuration : the fibres breaks in the glass ply and the carbon ply and the resin damaging in the glass ply increases with the energy of impact. The damage starts to be really visible for 4 J. The fibres breaks appear in the lower carbon ply at 2.25 J and in the glass ply at 6.25 J. For 9 J and 16 J, the impactor completely perforates the three plies and the damages are similar. Finally, for the two hybrid laminates, the add of the glass ply G 0 leads to a decrease of the damage size in the carbon plies.

Post-impact tensile tests

In this section, the behavior of the impacted samples under tensile loading is described and analysed. Quasi-static tensile tests on damaged samples are conducted with an INSTRON machine equipped with a load cell of 250 kN at 2 mm/min (Figure 8). The reaction load and the displacement of the crosshead are measured during the test. The strain fields in the samples are measured by the use of Digital Image Correlation (DIC). The tests are recorded with two cameras, placed in front and behind the sample.

The residual normalized tensile strength versus impact energy is given in Figure 9. For all the configurations, the curves can be divided in three parts. First, for very small impact energies, no loss of strength is observed. In fact, practically no damage is visible after the impact, so the residual strength seems to be unaffected. Then, an important loss of strength is noticed when the impact energy increases. Finally, the value of the residual strength remains constant with the impact energy. Indeed, after a given impact energy, the impact damage size does not grow, so that the residual strength is not altered. Figure 9 also shows that the relative loss of strength is higher when the plies are oriented at 0 • /90 • , especially for the glass configuration, which the post-impact tensile behaviour seems to be more sensitive to the impact damage.

In the next paragraphs, the mechanisms leading to the failure, deduced from DIC measurements and fracture shape observations, are analyzed, first for the monomaterial configurations, then for the hybrid configurations. Figure 10 shows the fracture surfaces obtained after the quasi-static tensile tests for each energy of impact and Figure 11 regroups the corresponding load/displacement curves. Except for 1 J impacted sample, for all the other impacted sample, the fracture surfaces are the same whatever the energy of impact used : the final break of the sample is horizontal and the two plies break at the same time. The 1 J impacted sample has the same behaviour as the clean sample. In that case, the damage is not important enough to propagate (Figure 2) and it breaks for a displacement similar to the clean sample (Figure 11).

An analyze of the damage propagation, based on DIC measures, is made. The strains fields are identical in the two plies (Figure 12) and the first damage propagation appears simultaneously in these two plies just before the final break. The strain field in the load direction (ε yy ) in the area around the post-impact damage is around four times higher than far from the impact location (Figure 12a). The principal strain in these areas is oriented in the direction y : near the damaged zone, the fibres are loaded in tension. More, in plane shearing ε xy is also noticed under and above the impact location (Figure 12b). In these areas, the resin is mainly loaded. However, as the damage propagates in the direction where ε yy is high, the failure of the laminate is governed by the fibres breakage.

C 45 C 45 samples

Figure 13 shows the fracture surfaces obtained after the quasi-static tensile tests for each energy of impact and Figure 14 regroups the corresponding curves. As for configuration C 0 C 0 , all the fracture surfaces are the same whatever the energy of impact used, except for the sample impacted with an energy of 2.25 J for which the damage generated by impact is barely visible (Figure 3). The final break of the sample is mainly horizontal and the two plies break at the same time (Figure 13). For the 2.25 J impacted sample, the fracture surface and the load-displacement curve are similar to those of a clean (non-impacted) sample. The initial damage is to small to propagate and the final failure is oriented at 45 • . For the other impacted samples (from 3 J to 9 J), the final failure is almost instantaneous, contrary to what observed with a clean sample which presents a progressive failure. In comparison to the laminate C 0 C 0 , the curves for the C 45 C 45 are first linear and then non-linear. This is due to the pseudo-plastic behaviour of the carbon woven oriented at ±45 • . Indeed, for in plane-shear, the laminate behaviour is governed by the resin. For epoxy matrix based laminates, the non-linear stage corresponds to a stage of pseudo-hardening, for which anelastic strains emerge and grow, and matrix damaging with the occurrence of resin crackings.

This behaviour explains why the residual normalized tensile strength is lower in comparison to C 0 C 0 (Figure 9). In fact, in the non-linear part, the laod increases slower than in the linear part.

For the laminate C 45 C 45 , the results provided by the digital image processing show that the post-impact behaviour concerning the damage propagation is different compared to the laminate C 0 C 0 . The first damage propagation starts for a crosshead displacement of 1.6 mm (Figure 15b). The propagation initiates at each tip of the initial fibres breaks and follows a direction of ±45 • (Figure 15c), which corresponds to the edge of the zone damaged after impact, until they merge each other (Figure 15d). Then the behaviour is similar to C 0 C 0 . An area around the post-impact damage where the strain field ε yy is high is identified (Figure 15d). The final rupture quickly occurs.

C 45 C 0 samples

For the laminate C 45 C 0 , the same observation than C 0 C 0 can be done for the 1 J case : it is the same behaviour as a clean sample (Figure 16). The post-impact damage is to small to propagate. For the other impact energies, the propagation direction of the damage is between 0 • and 45 • (Figure 16). It shows that the orientation of the plies has an influence on the propagation mechanisms. Nevertheless, the curves (presented on the Figure 17) show that the stiffness of the laminate is predominantly due to the carbon ply C 0 . In fact, the curves are linear and there is no indication of pseudo-plasticity. Furthermore, for example on the 6.25 J curve, the first break of fibre, which appears for a crosshead displacement of 1.55mm and which is noticeable through a slight decrease of the load (just before the final break), occurs in the ply C 0 . So the final break of the laminate is handled by the breakage of the fibres oriented at 0 • . Moreover, during the tensile test, the strain field yy in the upper ply C 45 (Figure 18a) is more similar to what observed for C 0 C 0 (Figure 18b) than for C 45 C 45 (Figure 18c) at the same displacement. The elongation of the ply C 45 is managed by the elongation of the ply C 0 .

G 0 G 0 samples

The Figure 19 shows the fracture surfaces obtained after the quasi-static tensile tests for each energy of impact and the Figures 20 regroups the corresponding curves. Except for the sample impacted with an energy of 4 J, all the fracture surfaces are the same whatever the energy of impact used : the final break of the sample is horizontal and the two plies break at the same time (Figure 19). Concerning the 4 J case, for which the is no visible damage after impact (Figure 5), the fracture surface is similar to those of a clean (non-impacted) sample (Figure 19). Nevertheless, Figure 20 shows that the failure occurs for a load 25% lower than for the clean sample.

The analysis of the damage propagation is identical as for C 0 C 0 . Indeed, there is an area around the post-impact damage where the strain field ε yy is high (Figure 21a). It correspond to the loading of the fibres. Areas where the strain ε xy is not negligible can also be noticed (Figure 21b). They are areas where the resin is loaded. However, the break of the laminate is governed by the fibres breaks.

Comparison

Whatever the monomaterial configuration studied, the damage propagation during the post-impact tensile simulation is always governed by fibres breakages and the impacted sample behaves as a non-impacted one when as long as the impact does not produce fibres breakages in the two plies (1 J for C 0 C 0 & C 45 C 0 , 2.25 J for C 45 C 45 and 4 J for G 0 G 0 ). When the two plies of the laminate are identical, the post-impact damage propagation is mainly horizontal (C 0 C 0 , C 45 C 45 and G 0 G 0 ), even if the final break of the laminate occurs much later for C 45 C 45 due to its pseudo-plastic behaviour (visible on the load curves with a non-linear part). For C 45 C 0 , the behaviour is governed by the ply C 0 , the curves being linear and the first fibres breakages initiating in this ply, but the post-impact damage propagation is influenced by each single ply with a propagation between 0 • and 45 • .

4.2. Hybrid samples 4.2.1. G 0 C 45 C 45 samples
Figure 22 shows the fracture surfaces obtained after the quasi-static tensile test for each energy of impact and the Figure 23 represents the corresponding curves. The fracture surfaces are identical for all the energies tested. The post-impact damage propagates and leads to a final break of sample oriented at ±45 • . The curves are not linear. It is due to the pseudoplastic behaviour of the two plies of carbon woven at ±45 • . The damage propagates in three steps. First the break at ±45 • of the glass ply and the resin cracking at ±45 • of the carbon plies is noticed. It corresponds to the first drop of load. Then the stress is recovered by the two carbon plies oriented at ±45 • . It corresponds to the part for which the load increases slowly. Finally, the sample breaks with the failure at ±45 • of the two carbon plies which corresponds to the last drop of load.

For the sample impacted at an energy of 1 J, its post-impact behaviour is similar to those of a non-impacted sample. The post-impact damage is too small and does not propagate.

Digital Image Correlation results are analyzed for the sample impacted at 4 J in order to identify the mechanisms involved in the damage propagation. A comparison between the strain field in the tensile direction ε yy in the glass ply of the configuration G 0 C 45 C 45 (Figure 24a) and the strain field ε yy in the upper ply of C 45 C 45 (Figure 24b) and G 0 G 0 (Figure 24c) shows that the strain field in the ply G 0 is similar to those of the upper ply of C 45 C 45 and different of the upper ply of G 0 G 0 . Thus, in this laminate, the two carbon plies at ±45 • manage the deformation of the upper ply G 0 .

When the load increases and when the break of fibres initiates, areas of compressive strains transversal to the direction of the load are observed (Figure 25). The fibres of the glass ply are compressed due to the Poisson's effect of the two plies of carbon at ±45 • . As a consequence, the glass fibres break in compression along a line oriented at ±45 • (Figure 26a). Finally, a concentration of strains appear in the carbon plies in the zone of breakage of the glass ply (Figure 26b). It leads to a progressive failure of the carbon plies on a strip oriented at ±45 • . The final rupture is due to the breakage of the carbon fibres.

4.2.2. G 0 C 45 C 0 samples
Figure 27 shows the fracture surfaces obtained after the quasi-static tensile test for each energy of impact and the Figure 28 represents the corresponding curves. Two propagations of the post-impact damage are visible on the Figure 27. For the 1 J and the 2.25 J impacted samples, the propagation is horizontal alhtought for the 4 J to the 16 J impacted samples, the propagation starts at ±45 • and then continues horizontally.

For the 1 J and 2.25 J cases, low damage is observed in the glass ply after impact. As a consequence, the strain field ε yy in the glass ply near the impact zone is smaller than what observed for the G 0 C 45 C 45 sample ((Figure 29). The glass fibres are not loaded enough to propagate the glass ply damage. Figure 30 compares the strain field ε yy in the lower carbon ply of G 0 C 45 C 0 samples and C 45 C 0 samples just before the final break. The shapes and values are similar. The fibres oriented at 0 • are mainly loaded and their breakage manage the horizontal final rupture of the sample.

For the samples impacted above 4 J, fibre breakages in the glass ply are observed after impact. Thus, during the post-impact tensile tests, the damage starts to propagate in the glass ply. In the first stage of the propagation the strain field in the glass ply is similar to that observed in the G 0 C 45 C 45 samples : the damage starts to propagate at ±45 • because it is managed by the carbon ply oriented at ±45 • (Figure 31). However, when the failure of the glass ply grows, the loads are carried by the 0 • carbon layer. The mechanisms leading to the failure of the sample change : the carbon fibres oriented at 0 • are mainly loaded. Their breakage leads to a final horizontal global failure of the specimen. Indeed, Figure 32 shows that the strain field in the bottom plies of G 0 C 45 C 0 and C 45 C 0 samples are the same just before the final break.

Conclusion

The post-impact damage propagation in thin hybrid carbon-glass/epoxy woven composite laminates under tensile loading has been investigated in this article. Besides the study of the initial damage through the variation of the impact energy, a precise analyze of the damage propagation scenario has been carried out. This analyze is based on the study of the behaviour of monomaterial laminates and on the observation of the strain fields given by Digital Image Correlation. From this study, two main significant results can be highlighted.

First, the hybridization of carbon woven laminates with a ply of glass woven fabric changes significantly the damage propagation scenario. Indeed, if the impact generates fibres breakages, for C 0 C 0 , C 45 C 45 and G 0 G 0 , the sample breaks horizontally althought for C 45 C 0 , the sample breaks between 0 • and ±45 • . But, for G 0 C 45 C 45 , the laminate breaks at ±45 • in three steps and for G 0 C 45 C 0 , the damage propagation is mainly horizontal. In the G 0 C 45 C 0 case, the hybridization has a beneficial effect. The final tensile break of the impacted hybrid laminate occurs for higher load and elongation than that of C 45 C 0 . For G 0 C 45 C 45 , the conclusion depends on the desired effect.

The addition of the glass ply also leads to an increase of the maximum load during the tensile test, but the break elongation decreases. This consequence of the hybridization is visible for the sample of which the damage does not propagate. The addition of the glass ply G 0 reduces the influence of the pseudo-plastic behaviour of the two carbon plies C 45 C 45 , with a non-linearity of the load curves less pronounced in comparison to C 45 C 45 .

The second significant results is that the damage propagation scenario is the result of an interaction between the post-impact damage size, the stacking sequence and the contribution of each ply of the laminate. In fact, two different behaviour are observed depending on the hybrid laminate. The damage propagation scenarios are completely different. For G 0 C 45 C 0 , the propagation depends also on the energy of impact. For lower impact energies, the fracture surface is horizontal while for higher impact energies, the propagation begins at ±45 • and finishes horizontally.

This study has also confirmed some litterature obervations for monomaterial laminates and has shown that they can be applied to hybrid laminates : the propagation of the damage is always governed by fibres breakages and the residual strength always decreases in three steps with the impact energy. First the decrease is very low, then it strongly grows and finally it becames constant. The start of the second and the third steps seems to correspond to the energy respectively required to create fibre breakage and completely perforate the laminate.

The second part of this article will be focused on the modelling of the tests studied in this part. First, numerical developements, based on the experimental observations made in this part, will be presented. Then, the modelling will be validated through a numerical/experimental comparison. Results will be used in order to explain the influence of the impact energy on the damage propagation scenario in a laminate of configuration G 0 C 45 C 0 .

4. 1 .

 1 Monomaterial samples 4.1.1. C 0 C 0 samples

Figure 1 :

 1 Figure 1: Sample geometry

Figure 2 :Figure 4 :Figure 6 :Figure 8 :Figure 9 :

 24689 Figure 2: Post-impact fracture surfaces on C 0 C 0 samples

Figure 10 :Figure 12 :

 1012 Figure 10: Fracture surfaces after tensile tests on post-impacted C 0 C 0 samples

Figure 15 :

 15 Figure 15: Strain field yy from DIC for C 45 C 45 for a 6.25 J impacted sample (identical in the two plies)

Figure 16 :Figure 17 :

 1617 Figure 16: Fracture surfaces for post-impact quasi-static tensile tests of C 45 C 0

Figure 18 :

 18 Figure 18: Strain field yy from digital image processing in the upper ply of C 45 C 0 for a crosshead displacement of 1mm -Comparison with C 0 C 0 and C 45 C 45

Figure 19 :

 19 Figure 19: Fracture surfaces for post-impact quasi-static tensile tests of G 0 G 0 (identical in the two plies)

Figure 21 :Figure 26 :

 2126 Figure 21: Strain fields from digital image processing for G 0 G 0 (identical in the two plies)

Figure 27 :

 27 Figure 27: Fracture surfaces for post-impact quasi-static tensile tests of G 0 C 45 C 0

Figure 28 :Figure 29 :Figure 30 :Figure 31 :Figure 32 :

 2829303132 Figure 28: Load -Crosshead displacement curves for post-impact quasi-static tensile tests of G 0 C 45 C 0

Figure 13: Fracture surfaces for post-impact quasi-static tensile tests of C 45 C 45

Figure 22: Fracture surfaces for post-impact quasi-static tensile tests of G 0 C 45 C 45

Configuration

Lower ply

Middle ply Upper ply


A Rogani

P Navarro

S Marguet

J-F Ferrero

C Lanouette

Tensile post-impact behaviour of thin carbon/epoxy and glass/epoxy hybrid woven laminates -Part II : numerical study

Keywords: Woven composites, thin laminates, hybrid laminates, post-impact behaviour, explicit F.E modeling, damage

This article concerns the modelling of post-impact damage propagation in thin woven composite laminates. Simulations of low velocity impacts and post-impact quasi-static tension are performed on single-material and hybrid laminates. The modelling is based on the semi-continuous approach implemented into the explicit finite element code RADIOSS. The bundles are modelled with rod elements and a specific damageable shell element is used to stabilize this truss structure. Improvements are brought with the introduction of a compressive failure criterion for the rod elements and the development of a pseudo-plastic law with damaging for in in-plane shear. The results provided by the modelling well correlates the experimental observations in terms of damage propagation and load-displacement curves for all the configurations studied.

Introduction

This article deals with the modelling of post-impact damage propagation in thin hybrid woven laminates (multiple materials and orientations) subjected to quasi-static tensile test. This paper is the second part of a two-parts article : the phenomenons which occur during the tensile test have been investigated through several experimental tests on single-material and hybrid laminates, and are described in part I.

Concerning the modelling of the post-impact behaviour of composite laminates, most of the studies are realized on unidirectionnal carbon laminates. Moreover, as the buckling due to compression is a critic aspect in order to evaluate the toughness of a composite laminate, only the compression after impact (CAI) is studied. First, simple models, based on analytical approaches [START_REF] Husman | Residual strength characterization of laminated composites subjected to impact loading[END_REF][START_REF] Xiong | A prediction method for the compressive strength of impact damaged composite laminates[END_REF][START_REF] Naik | Estimation of compressive strength of delaminated composites[END_REF], semi-empirical approaches [START_REF] Ostré | Experimental analysis of cfrp laminates subjected to compression after edge impact[END_REF][START_REF] Qi | An engineering approach for predicting residual strength of carbon/epoxy laminates after impact and hygrothermal cycling[END_REF][START_REF] Caprino | The significance of indentation in the inspection of carbon fibre-reinforced plastic panels damaged by low-velocity impact[END_REF] or finite element (FE) calculations [START_REF] Soutis | Prediction of the post-impact compressive strength of cfrp laminated composites[END_REF][START_REF] De Moura | Modeling compression failure after low velocity impact on laminated composites using interface elements[END_REF][START_REF] Habib | A new method for evaluating the residual compression strength of composites after impact[END_REF][START_REF] Hawyes | Effect of impact damage on the compressive response of composite laminates[END_REF][START_REF] Suemasu | A numerical study on compressive behavior of composite plates with multiple circular delaminations considering delamination propagation[END_REF] have been developped to estimate the residual compressive strength of the impacted laminate. They mainly use criterions based on delamination growth or buckling instability in order to evaluate the residual strength of the laminate, while [START_REF] Ostré | Experimental analysis of cfrp laminates subjected to compression after edge impact[END_REF] and [START_REF] Qi | An engineering approach for predicting residual strength of carbon/epoxy laminates after impact and hygrothermal cycling[END_REF] draw a parallel with notched laminates using the point stress failure criterion (PSFC) to predict the compressive strength of various CFRP laminates. However, these studies are not sufficient because the damage used for the CAI calculation is only implemented in the model using experimental observations (such as permanent indentation shape or stiffness degradation) and non-destructive testings. Thus, the most recent studies use models capable of predict the damage due to impact and then realize the CAI simulation to take into account the permanent indentation after impact [START_REF] Davies | Compression after impact strength of composite sandwich panels[END_REF][START_REF] Wang | Low-velocity impact and residual tensile strength analysis to carbon fiber composite laminates[END_REF][START_REF] Gonzàlez | Simulation of drop-weight impact and compression after impact tests on composite laminates[END_REF][START_REF] Dang | A numerical study on impact and compression after impact behaviour of variable angle tow laminates[END_REF][START_REF] Rivallant | Failure analysis of cfrp laminates subjected to compression after impact: Fe simulation using discrete interface elements[END_REF][START_REF] Tan | Predicting low velocity impact damage and compression-after-impact (cai) behaviour of composite laminates[END_REF][START_REF] Panettieri | Delaminations growth in compression after impact test simulations: Influence of cohesive elements parameters on numerical results[END_REF][START_REF] Montero | Multi-scale impact and compression-after-impact modeling of reinforced benzoxazine/epoxy composites using micromechanics approach[END_REF][START_REF] Dubary | Damage tolerance of an impacted composite laminate[END_REF]. Also in this case, most of the models use criterions based on delamination growth (taken into account through cohesive elements) and matrix cracking. But, some authors, who have observed during CAI tests that the presence of fiber breakage around the impact area are the reason of the final break, introduce fiber failure criterion.

Very few studies on the post-impact behaviour are conducted on woven composites laminates. The compressive post-impact behaviour is the most studied [START_REF] Yan | Compression-after-impact response of woven fiber-reinforced composites[END_REF][START_REF] Mendes | Numerical prediction of compression after impact behavior of woven composite laminates[END_REF] and the developed models try to predict the compressive strength of the impacted laminate. Yan et al. use a complex FE simulation in which the interlaminar debonding, the matrix and fibers crackings, the buckling and the interactions between these mechanisms are represented. The matrix and the fibers are modelized with an homogenized volumic element and the delamination is calculated through cohesive elements. However, the damage due to the impact is initially introduced in the model as a result of experimental observations [START_REF] Yan | Compression-after-impact response of woven fiber-reinforced composites[END_REF]. Althought Mendes et al. develop a FE model able to simulate the impact and the CAI test. The matrix and the fibers are represented by homogenized shell elements and the interface between the plys by cohesive elements. The intralaminar and interlaminar damage propagations are calculated through criterions based on energy re-lease rates [START_REF] Mendes | Numerical prediction of compression after impact behavior of woven composite laminates[END_REF]. The simulations, respectively performed on glass/vinyl-ester and carbon/epoxy woven composite laminates are capable of reproducing experimental results with good accuracy for the impact tests and CAI tests. The flexural post-impact behaviour is studied by Papanicolaou et al. [START_REF] Papanicolaou | Low energy impact and post impact behavior of epoxy matrix-woven flax fabric composites[END_REF] who compare two different analytical model to predict the residual flexural strength after impact. Colombo et al. develop a simple model to investigate the post-impact tensile behavior of a carbon/epoxy laminate, but only to evaluate the post-impact residual stiffness [START_REF] Colombo | A micromechanical approach to evaluate the post-impact residual stiffness of woven composites[END_REF]. A Unit Cell (UC) is developed at the microscale in which warp and weft bundles, resin between the bundles and resin in the bundles are modelled. The idea is to define a simple numerical procedure to simulate the degradationof the impacted woven by changing the mechanical properties of this UC. To do that, a range of experimental impact tests are realized to evaluate the size and type of damage and the residual stiffness corresponding. Then, to simulate the damage due to the impact, the elastic modulus of the matrix in UC, corresponding to the impacted region, is degraded by a coefficient of reduction. Finally, a numerical analysis is ran to determine the laminate residual stiffness. It is compared with the experimental data and give good results. However, this model can not represent the damage propagation and is very depending on the experimental observations.

The objective of this study is to model quasi-static tensile tests on impacted woven laminates with the semi-continuous approach [START_REF] Navarro | Semi-continuous approach for the study of impacts on woven composite laminates: Modeling interlaminar behavior with a specific interface element[END_REF][START_REF] Pascal | On the modelling of low to medium velocity impact onto woven composite materials with a 2d semi-continuous approach[END_REF][START_REF] Pascal | Impact damage prediction in thin woven composite laminates -part i: Modeling strategy and validation[END_REF][START_REF] Pascal | Impact damage prediction in thin woven composite laminates -part ii: Application to normal and oblique impacts on sandwich structure[END_REF]. For this purpose, the existing semi-continuous strategy has been improved. Indeed, some add-ons, based on experimental observations made in Part I, are implemented through the development of a damaging pseudo-plastic law in plane-shear and the introduction of a new criterion for the break of fibres bundles in compression. Finally, the model is validated by several simulations on the same single-material laminates (Table 1) and hybrid laminates (Table 2) used in Part I. The results are finally compared to the experiments.

Extension of the semi-continuous model to tensile simulations

Semi-continuous model for impact simulations

The semi-continuous model, developped in the research team for woven laminates subjected to impact loadings, has been already presented in the articles [START_REF] Navarro | Semi-continuous approach for the study of impacts on woven composite laminates: Modeling interlaminar behavior with a specific interface element[END_REF][START_REF] Pascal | On the modelling of low to medium velocity impact onto woven composite materials with a 2d semi-continuous approach[END_REF][START_REF] Pascal | Impact damage prediction in thin woven composite laminates -part i: Modeling strategy and validation[END_REF][START_REF] Pascal | Impact damage prediction in thin woven composite laminates -part ii: Application to normal and oblique impacts on sandwich structure[END_REF]. The modelling of the woven ply decouples the resin and tows behaviour to represent simultaneously the damaged resin and the undamaged fibers tows. A specific finite element has been developped (Figure 1) with a truss structure of rod elements, modelizing the woven fabric bundles and following the woven pattern geometry, and a shell element, placed at the neutral axis of the ply, which represent the epoxy matrix. The mesh size respects the woven fabric pattern and the nodes of the rods are offset from the shell to a distance of h/4 in the thickness direction where h is the thickness of the ply. The connection between the rods and the shell elements is realized at the nodes through rigid links. Finally, an interface element, with an elasting damaging behaviour and a bilinear cohesive law in order to represent the delamination, is added in order to connect two plies.

The rod element has an elastic and brittle behaviour with a maximal tensile strain criterion for its break :

If ε > ε tension max then break of the rod element (1)
The resin is modelized by a specific elastic and isotrop shell element for which an anisotropic damaging has been implemented for each direction of the woven throught two damage variables d 1 and d 2 , based on thermodynamical functions Y i , which came degrade shell and bending matrices :

C shell = C bend =         (1 -d 1 ) E 11 1 -(1 -d 1 ) -(1 -d 2 ) ν 2 (1 -d 1 ) (1 -d 2 ) νE 11 1 -(1 -d 1 ) -(1 -d 2 ) ν 2 0 (1 -d 1 ) (1 -d 2 ) νE 22 1 -(1 -d 1 ) -(1 -d 2 ) ν 2 (1 -d 2 ) E 22 1 -(1 -d 1 ) -(1 -d 2 ) ν 2 0 0 0 G 12         (2)
Concerning impact simulations, in comparison to the experimental results, the semi-continous model gives accurate results in terms of sizes and shape of damages and load curves for diverse laminates configurations (singlematerial or hybrid laminates). But, it requires some adds-on in order to correctly represent the post-impact quasi-static tensile behaviour of these laminates.

Adds-on to the semi-continuous model

Compressive strain criterion for rod element

The experimental testings on the hybrid laminate G0C45C45 have shown in the Part I that the elongation of the upper glass ply G0 is piloted by the two carbon plies at ±45 • , what generates areas of compressive strains in the glass plys due to the Poisson effect of these two carbon plies. Thus, the propagation of the post-impact damage in the glass ply is made at ±45 • due to compressive break of fibres. So, in the modelling, it is necessary to add a compressive strain criterion for the rod element in addition to the tensile strain criterion :

If ε(x) < ε comp max then break of the bar element (3)
The value of ε comp max for glass and carbon woven are chosen such as :

|ε comp max
Furthermore, in a woven ply, the fibres bundles located at a cross-point are in local bending. So, they need a more important tensile strain in order to spread out and reach the tensile strain criterion and on the contrary, they need a less important compressive strain to reach the compressive strain criterion. So, at a cross-point, ε tension max is increased and |ε comp max | is reduced. For this purpose, ε comp max cross and ε tension max cross are introduced in the model such as :

|ε comp max cross | |ε comp max | = ε tension max ε tension max cross = 0.7 (6)

Damaging pseudo-plastic law

Introduction.

The experimental testings of the Part I have shown, for the hybrid configuration G0C45C45, that the two carbon plies at ±45 • was piloting the elongation of the upper ply G0, but also the existence of areas of strain field ε xy with vectors of principal strain directed at ±45 • (which are areas where the resin is loaded) for the single-material laminates. So, in order to represent the loading and the damaging of the resin, it is necessary to intruce in the model a damaging law in plane shear for the carbon woven.

Experimental observations.

A cyclic tensile test is performed on two plies of carbon woven oriented at ±45 • with the same characteristics of the sample and the tensile as in the Part I. A classic load -displacement curve is obtained and presented on the Figure 2. First, from a certain value of shear stres, after the unload, the crosshead displacement does not come back to zero. Which is specific of the emergence of plastic shear strains (irreversible strains) in the laminate and means that the carbon woven has a pseudo-plastic behaviour in plane shear. The curve shows also a decrease of the shear stiffness G 12 during the test. It is driven by the emergence and the growth of damage in the laminate. In fact, during quasi-static tensile tests on carbon/epoxy woven laminate oriented at ±45 • , [START_REF] Karahan | Investigation of damage initiation and propagation in 2x2 twill woven carbon/epoxy multi-layer composites[END_REF][START_REF] Bogdanovich | Quasi-static tensile behavior and damage of carbon/epoxy composite reinforced with 3d non-crimp orthogonal woven fabric[END_REF] observed first the initiation of cracks in areas full of resin (fibres/resin splitting) for a strain of 0.88 and then the initiation of cracks in bundles for a strain of 1.2 % which propagate thereafter in metadelamination (delamination between warp and weft bundles).

The splittings quicky evolve at the beginning of the test then they stabilize athought the cracks in the bundles and the meta-delaminations slowly grow at the beginning and suddenly accelerate once the splittings no longer evolve. Finally, the meta-delaminations come dissociate the warp and the weft, what leads to their rotation and generates break of bundles at a high strain rate. The emergence of each type of damage is represented on the Figure 2 by Damage A Initiation in red and Damage B Initiation in purple.

The Figure 2 presents also the progress of the damage at a macroscopic scale. The strain fields ε yy and ε xx , obtained in the top ply for a crosshead displacement of 12 mm, have a rhombus shape (Point A) which is typical of a carbon woven ±45 • subject to a tensile load. Then, for a crosshead displacement of 17.5 mm (Point B), two bands of cracking, following the edges of the rhombus area, emerge in the laminate. But, the laminate does not break directly. A rotation and a distortion of the bundles is noticeable (Point C) until a crosshead displacement of 21.1 mm for which breaks of fibers finally propagate along one of the two cracked bands (Point D).

Pseudo-plastic behaviour in plane shear.

The emergence of irreversible strains in the carbon woven is taken into account in the model through the introduction of a pseudo-plasticity criterion on the in-plane shear stress based on an isotropic hardening using a classic elastic prediction and a plastic correction. An elastic field is defined by :

f = |τ 12 | -K plas p β -τ 0 (7)
where τ 12 is the in-plane shear stress, τ 0 is the plastic strength, (K plas , β) are parameters of the plastic hardening law and p the cumulative plastic strain. Thus, if f > 0, the plastic correction is performed throught the update of the plastic variables γ p 12 and p, based on a Newton-Raphson iterative scheme, with γ p 12 the plastic shear strain defined by :

γ 12 = γ e 12 + γ p 12 ( 8
)
where γ 12 is the total shear strain and γ e 12 is the elastic shear strain. Shear damaging.

In order to modelize the two types of damage mentioned above, two shear damage variables are added in the modelling and came degrade the shear modulus :

G 12 = 1 -d a 12 -d b 12 G 0 12 (9)
They depend on thermodynamical function, as it is typically used [START_REF] Hochard | A progressive first ply failure model for woven ply cfrp laminates under static and fatigue loads[END_REF][START_REF] Hochard | A generalized damage model for woven ply laminates under static and fatigue loading conditions[END_REF] :

           d a 12 = d a max 12 × 1 -exp - √ Y 12 - √ Y 0 ≥0 √ Y c d b 12 = max exp √ Y 12 - √ Y 0 bis ≥0 √ Y c bis , 1 (10)
where Y 0 (respectively Y 0 bis ) and Y c (respectively Y c bis ) are parameters of damage threshold and damage speed of d 12 a (respectively d 12 b ). d max 12 a is the maximal limit of d 12 a . This limit has been introduced for the damage A because it is not responsible for the final break of the laminate.

Here, the shear strain energy is only due to shear stress and the coupling with the stresses in the other directions is neglected. Thus, the thermodynamical function Y 12 is defined by :

Y 12 = ∂W 12 ∂d 12 (11)
where d 12 = d a 12 + d b 12 and W 12 is the shear strain energy defined by :

W 12 = τ 2 12 2G 0 12 (1 -d 12 ) (12)
Hence :

Y 12 = 1 2 G 0 12 γ e 12 2 (13)
Furthermmore, during a tensile test on a ±45 • woven carbon laminate, when the resin is totally damaged, that amounts to saying that G 12 = 0. However, in the model, there already are two damage variables d 1 and d 2 in order to modelize the resin damaging in the warp and weft directions and which come to degrade stifnesses E 11 and E 22 by

E 11 = (1 -d 1 )E 0
11 and

E 22 = (1 -d 2 )E 0 22 .
Even if G 12 = 0, the load is going to be recovered because E 11 and E 22 are not equal to zero. So, the two shear damage variables are also used to degrade E 11 and E 22 , and the shell and the bending matrix are expressed by :

            1 -(1 -d 1 )(1 -d 2 ) 1 -d a 12 -d b 12 ν 12 ν 21 0 0 0 1 -d a 12 -d b 12 G 0 12             Angle criterion.
Experimentally, rotation and distorsion of the bundles appear once the resin is totally cracked and meta-delamination emerges between warp and weft [START_REF] Bogdanovich | Quasi-static tensile behavior and damage of carbon/epoxy composite reinforced with 3d non-crimp orthogonal woven fabric[END_REF]. However, in the modelling, this meta-delamination is taken into account by a damage variable which degrades the stiffnesses of the shell element. When the resin is totally cracked, that is to say that the shell element is broken, the finite element is suddenly distorted (Figure 3). But, a link between the rod elements of the warp and weft still remains. Moreover, the rod elements don't recover a lot of load, thus they can't elongate and the break criterion on tensile strain can't be reached for these rods. So, a criterion on the minimum or maximum angle between the rods has been added in order to represent this distortion : if φ i ≤ 90 • -φ max or φ i ≥ 90 • + φ max then break of the bar elements constituting the angle.

Identification and validation.

Before realizing simulations on the impacted laminates experimentally tested in the Part I, the damaging pseudo-plastic law has to be validated trought the identification of the several parameters presented in the Table 3.

For this purpose, a simulation of tensile test on two plies of woven carbon oriented at ±45 • is made. The sizes of numerical sample are the same as the experimental sample (100mm×200mm) and numerical tabs (100mm×20mm) are added in order to avoid edge effects throughout the tensile simulation. They are modelized by an elastic Johnson-Cook law with an infinite break limit and a Young's modulus corresponding to the lamination plus six plies of glass woven at ±45 • . In the strategy of the semi-continuous model, the mesh size respects the woven fabric pattern. So, the mesh size of the modelling is 1.4mm×1.4mm×0.36mm. The computation is performed with the explicit finit element code RADIOSS on 60 cores from HPC resources.

The Figure 4 shows a good correspondance between the numerical and experimental curves, with a good stiffness of the laminate and a good progress of the load, for a cycled and a non-cycled quasi-static tensile tests on two plies of carbon woven at ±45 • , but also a numerical fracture surface close to those obtained experimentally.

The Figure 5 shows the damage development during the simulation. As for the experimental testing, cracking bands appear first in the laminate (Point A), which results in deletions of the shell elements in the model. Then, the resin being totally cracked, these elements endure a distorsion (Point B-B') until the criterion on the angle between the rod elements is reached (Point B"). From this time, the rod elements are deleted and the finite element is totally removed. Finally, fibres breakings propagate along the cracked bands, what leads to the final break of the laminate (Point C).

Post-impact quasi-static tensile simulations

Model

Each configuration experimentally tested in the Part 1 (Tables 12) is now modelized. The numerical sample is the same as the simulations carried out for the validation of the damaging pseudo-plastic law. Each ply of the sample are modelized separately with a difference on the mesh size between carbon and glass woven, the measured woven fabric pattern being differend according to the material. So, the mesh size of the modelling is 1.4mm×1.4mm×0.36mm for the carbon woven and 0.5mm×0.5mm×0.31mm for the glass woven. Thus, the highest required number of developed finite elements among all the tested configurations is 160 000. Finally, between two plies of different nature and/or orientation (with different meshings), a kinematic condition is added in order to connect the nodes of these two plies.

The conditions of the impact simulation are the same as the ones used for the experimental test presented in Part I. The impactor is modelized by a halph-sphere with an elastic law (with the steel characteristics : ρ steel = 7900 kg/m 3 , E steel = 210 GP a and ν steel = 0.3), a contact condition is used between the sample and the impactor and the double frame is modelized by two rigid wall in contact with the bottom and the top of the sample. The different energies of impact tested are gathered in the Table 4.

For the tensile test, on side of the tabs is embedded and the other is subject to an imposed velocity.

The computations are performed with the explicit finit element code RA-DIOSS on 180 cores from HPC resources.

The method to simulate the post-impact quasi-static tensile simulation is first to realize the impact simulation on the sample and then to initiate the tensile simulation with the damaged state obtained at the end of the impact simulation. A Python script has been developped for purposes of doing that.

Single-material laminates

Before testing the semi-continuous modelling on the hybrid laminates, a validation is first realized on the monomaterial laminates C0C0, C45C45, C45C0 and V0V0 for one energy of impact. The propagation of the postimpact damage and the load -crosshead displacement curves numerically obtained are compared to the experimental one for each configuration.

Drop weight impact simulations

In a first time, impact simulations are made for each monomaterial laminates. The Figure 6 shows a comparaison between experimental and numerical fracture shapes after impact. On the numerical results, the red represents the fibres breaks and the rest matrix cracks which occurs around these breaks. The model is able to represent the good shape of damage after impact : the fibres breaks are well oriented and follow the direction of the plies. The damages sizes are correct with maximal relative error lower than 20%. In this case, the initial damage states used for the next tensile simulations are similar to the experimental ones what confirms that the two adds-on implemented in the semi-continuous model don't modify the modelling of impact behaviour.

Post-impact quasi-static tensile simulations

The fracture surfaces (identical for the upper and lower plies) and the corresponding load -crosshead displacement curves obtained for the postimpact tensile simulations on the single-material laminates are presented in Figure 7. For the configuration C0C0, C45C45 and V0V0, the propagation of the post-impact damage is tranversal and for the configuration C45C0, the direction of the propagation is mainly transversal and slightly guided by the carbon ply at ±45 • . Furthermore, for C45C0, the model manages to represent the emergence of little vertical break of fibres thanks to the introduction of the compressive criterion on the rod elements. In any case, the final break of the laminate is due to fibres breaks propagation.

Concerning the numerical load -crosshead displacement curves, they are identical to the experimental ones for all the single-material laminates. The stiffness of the damaged sample is correct and the final break occurs for the good load and crosshead displacement (the relative error on the displacement is only 8% for C0C0, 10% for C45C45, 5% for C45C0 and 3% for G0G0). Thus, the pseudo-plastic damage law, developed and presented previously, provide thus the good damage propagation in the impacted samples C45C45 and C45C0.

So, the semi-continuous modelling is validated for the monomaterial laminates and can be now tested for the hybrid laminates.

Hybrid laminates

In this section, the semi-continuous modelling is first validated on the hybrid laminates for one energy of impact. As for the monomaterial laminates, the post-impact damage propagation and the corresponding load -crosshead displacement curves are compared to the experimental ones. Then, the simulations are used to explain the particular behaviour of the configuration G0C45C45 and the differences observed for the configuration G0C45C0 depending on the energy of impact.

Drop weight impact simulations

As for the single-material laminates, the Figure 8 shows a comparaison between experimental and numerical fracture shapes after impact for the hybrid laminates. The fibres breaks in the lower ply follow the direction of this ply and the cracked area in the upper ply has the same shape than those experimentally obtained, with a maximal relative error on the damages sizes lower than 15%. Moreover, the numerical and experimental loaddisplacement curves are identical, what indicate that the damaging scenario during the drop weight impact is correctly taken into account with the model.

Post-impact quasi-static tensile simulations

The fracture surfaces (identical for the upper and lower plies) and the corresponding curves obtained for hybrid laminates are presented on the Figure 9. As for the monomaterial laminates, the fracture surfaces numerically obtained for the hybrid laminates are identical to the experimental ones. The propagation of the post-impact damage for G0C45C45 is at ±45 • while for G0C45C0, it starts at ±45 • and finishes horizontally. Moreover, the curves show that the stiffness of the damaged sample is correct. The influence of the pseudo-plastic behaviour of the carbon ply ±45 • is noticeable for the laminate G0C45C45 with a non-linear curve althought for the laminate G0G45C0, the influence is not visible, the curve being linear.

The strain fields numerically observed in each ply of the configuration G0C45C45 give the damage propagation scenario during the tensile simulation (Figure 10). First, the strain field ε yy in the upper glass ply is very similar to those observed in the upper ply of the configuration C45C45 and very different to those observed in the upper ply of the configuration V0V0 at the same crosshead displacement (Point A). It indicates that the elongation of the glass ply is piloted by the elongation of the carbon plies. Then, just before the first fall of load (Point B), areas of compressive strains are visible around the damaged zone in the glass ply. The new criterion on compressive strain of rods is reached here by some elements what initiate the propagation in this ply by compressive fibres breaks at ±45 • . After that, compressive and tensile fibres breaks propagate what leads to the breaks at ±45 • of the glass ply and the cracking at ±45 • of the carbon plies on the corresponding band and corresponds to the first fall of load on the curve (Point C). However, the carbon plies don't break what localize the strains on the cracked band. They are able to recover the load what it is visible on the curve by a plateau betwenn the Points C and D. Finally, the two carbon plis break at ±45 • along the cracked band what coincide which the final fall of the load on the curve (Point E).

In summary, the break of the impacted laminate G0C45C45 is in three steps (first the glass ply breaks at ±45 • , then the two carbon plies recover the load and finally these carbon plies break at ±45 • ) and this scenario is connected to the curve evolution. Well, the numerical curve evolution is close to the experimental one (Figure 9) : the relative error on the displacement is 8% for the first fall of load, 3% for the final fall of load and the relative error on the load is 13% for the plateau. So, it indicates that, during the simulation, each break occurs at the same time as for the experimental test. The results of the numerical simulations are also used to explain the propagation of the post-impact damage in the laminate G0C45C0. The Figure 11 presents the evolution of the damage in each ply of a 4J impacted laminate during the tensile simulation. The damage starts to propagate in the glass ply from the fibres breaks generated by the impact. After that, fibres breaks initiate in the lower carbon ply but not in the middle ply. Then, propagation of compressive fibres breaks at ±45 • is noticeable in the glass ply. Thus, the beginning of the damage in the glass ply is similar to the the propagation in the laminate G0C45C45. Finally, fibres breaks appear in the middle carbon ply, what causes the final horizontal and simultaneous propagation in the three plies. The middle carbon ply is thus the critical ply of the laminate.

The final break of the laminate corresponds to the unique fall of load on the load -crosshead displacement curve (Figure 9). The numerical and experimental curves being identical (the relative error on the break displacement is only 5%), it means that the break of the numerical sample occurs at the same time in comparison to the experimental one.

The modelling is also used to explain the difference of fractured surfaces observed in the Part I for the laminate G0C45C0 depending on the energy of impact. In fact, for 1 J and 2.25 J impacted samples, the propagation of the damage during the tensile test is horizontal althought, from 4 J, the damage starts to propagate at ±45 • and finishes horizontally. For an energy of 1 J (Figure 12), the initial damage is almost absent in the upper glass ply and the fibres breaks start to propagate in the lower carbon ply. And after that, they start to propagate quasi-simultaneously in the glass ply and the middle carbon plies. The compressive fibres breaks do not have the time to appear in the glass ply and the three plies break horizontally and simultaneously.

For an energy of 2.25 J (Figure 13), even if there is no fibres breaks in the glass ply after the impact, the resin damaging is quite extanded, what initiates the propagation of fibres breaks in this ply during the tensile simulation. But, quasi-simultaneously, the propagation of fibres breaks is also initiated in the lower and middle carbon plies. So, the final break of the three plies is, in this case too, horizontal and simultaneous.

Finally, the Figure 14 links the fracture surface obtained after the tensile simulation with the post-impact damage and the ply in which the damage initiates :

• When there is no fibre break and almost no resin damaging in the glass ply, the damage propagation initiates in the lower carbon ply. Then, fibres breaks initiate simultaneously in the glass ply and in the middle carbon ply what lead to a transversal fracture surface of the laminate, the middle carbon ply being the critical ply of the laminate.

• When there is no fibres breaks and resin damaging quite developed in the glass ply, the damage propagation initiates quasi-simultaneously in the glass ply and the middle and lower carbon plies, what leads also to a transversal fracture surface.

• When there are fibres breaks in the glass ply, the damage propagation initiates in this ply and then in the lower carbon ply. Fibres breaks starts to propagate at ±45 • as long as fibres breaks propagation are not initiated in the middle carbon ply. Once they are initiated, the propagation finishes horizontally in the three plies.

Conclusion

In this article, post-impact quasi-static tensile tests have been modeled with the semi-continuous strategy. The modelling manages to represent on the one hand the propagation of resin crackings and on the other hand the propagation of bundles breakages during the tensile simulation while taking into account the laminate hybridization, especially through the implementation of a compressive strain criterion for the bundle breakage and of an original damaging pseudo-plastic law in plane-shear able to represent different types of shear damage visible experimentally.

The damaging pseudo-plastic law has been identified and validated with cycled and non-cycled quasi-static tensile tests on two undamaged carbon plies ±45 • . Then, the model has been validated through post-impact quasistatic tensile simulations on all the configurations experimentally tested in Part 1 (monomaterial and hybrid laminates). The model well represents the propagation of the post-impact damage. The fracture surfaces and the load -crosshead displacement curves well correlate the experimental results.

The first major contribution of this study is to provide a modelling that simulates as well impact as post-impact tensile tests. The initial damage state of the tensile simulation is the damage state obtained after the impact simulation and is not introduced through experimental observations.

The second major contribution of this article is to provide a more accurate analysis of the post-impact damage scenario based on the analysis of numerical results. For instance, concerning the hybrid laminate G0C45C0, the analyzis of the results provided by the modelling helps us to understand that the two post-impact damage propagations observed for low and higher impact energies are directly related to the fibre breakage in the glass ply. Moreover, in any case, it can be observed from the result that the middle carbon ply is the critical ply of the laminate.

Figure 1 :

 1 Figure 1: Modelling strategy of the woven ply

Figure 2 :

 2 Figure 2: Emergence and progress of shear damage during a quasi-static tensile test on two plies of carbon/epoxy woven ±45 •

Figure 3 :

 3 Figure 3: Distorsion of an element at the break of the resin

Figure 4 :Figure 5 :

 45 Figure 4: Comparison of numerical and experimental results after (a) normal (b) cycled quasi-static tensile tests on two plies of carbon woven at ±45 •

Figure 6 :Figure 7 :

 67 Figure 6: Drop weight impact results on single-material laminates -Experimental and numerical comparaison

Figure 8 :Figure 9 :Figure 10 :

 8910 Figure 8: Drop weight impact results on hybrid laminates -Experimental and numerical comparaison

Figure 11 :

 11 Figure 11: Scenario of damage propagation during a quasi-static tensile simulation on a 4 J impacted sample of G0C45C0

Figure 12 :

 12 Figure 12: Scenario of damage propagation during a quasi-static tensile simulation on a 1 J impacted sample of G0C45C0

Figure 13 :

 13 Figure 13: Scenario damage propagation during a quasi-static tensile simulation on a 2.25 J impacted sample of G0C45C0