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1 Oen Archive Toulouse Archive Ouverte (OATAO) OATAO is an oen access reository that collects the work of Toulouse researchers and makes it freely available over the web where ossible. This is an author-deosited version ublished in: htt://oatao.univ-toulouse.fr/ Erints ID: 9 To link to this article: DOI:./j.comstruct..7. URL: htt://dx.doi.org/./j.comstruct..7. To cite this version: Hongkarnjanakul, Natthawat and Bouvet, Christohe and Rivallant, Samuel Validation of low velocity imact modelling on different stacking sequences of CFRP laminates and influence of fibre failure. () Comosite Structures, vol ISSN - Any corresondence concerning this service should be sent to the reository administrator: staff-oatao@in-toulouse.fr

2 Validation of low velocity imact modelling on different stacking sequences of CFRP laminates and influence of fibre failure N. Hongkarnjanakul, C. Bouvet, S. Rivallant Université de Toulouse, ISAE, INSA, UPS, Emac, ICA (Institut Clément Ader), ISAE (Institut Suérieur de l Aéronautique et de l Esace), Avenue Edouard Belin, BP, Toulouse Cedex, France a r t i c l e i n f o a b s t r a c t Keywords: Damage tolerance Imact behaviour Finite element analysis (FEA) Damage mechanics Fibre failure This aer resents a validation of low-velocity imact Finite Element (FE) modelling. Based on switching ly location of reference layu [,,9, ] s T7GC/M laminated lates from Bouvet et al. () [], twelve ossible layus under a constraint of double-ly, mirror-symmetric, balanced, and quasi- isotroic are allowed. However only seven layus are chosen for the study and one of them reveals the imortance of longitudinal fibre comressive failure during imact events. Therefore, the second asect of this work is the introduction of a fibre comressive failure law associated with fracture damage develoment. This makes it ossible to imrove the simulation for all seven different layus. Good corresondence is achieved between simulation and exeriment for asects such as delamination areas/ shaes and force dislacement resonses. The influence of the addition of fibre comressive failure according to fracture toughness in mode I is discussed.. Introduction Low-velocity imact in comosite structures has been studying since 97s. In aeronautics, many researchers have attemted to design otimum structures mainly with resect to their weight. The structures may lose u to % of their strength when facing low-velocity imact roblems due to accidents during manufacturing or maintenance rocesses []. To coe with this roblem, many comosite structures are over-designed with a high safety factor to comensate for their low damage tolerance [], therefore otimised design with Finite-Element Analysis (FEA) becomes necessary and increasingly used in lace of exerimental tests. This resent work is an extension of Bouvet et al. []. An imact FE modelling of UD carbon/eoxy lates has been develoed. Using their own model, this team has studied the imact damage in three conventional failure modes, namely fibre failure, matrix cracking, and delamination. The results of imact damage from the model were accurately catured, e.g. force dislacement history, delamination surface, ermanent indentation. However, the model was only validated on a single reference layu at a given imact energy. To the authors knowledge, the literature contains no validation of different imact conditions. Also, in order to enlarge the validity of the reference model, this work will therefore focus on model validation of imact on different stacking sequences, including the Corresonding author. Address: ISAE/DMSM, Avenue Edouard Belin, BP, Toulouse Cedex, France. Tel.: + () 9; fax: + (). address: christohe.bouvet@isae.fr (C. Bouvet). resentation of a variety of sulementary ost-rocessing results, e.g. high quality C-scan, and microscoic observation. The reference stacking sequence is an -double-ly, mirrorsymmetric, quasi-isotroic laminated late of T7GC/M (carbon/eoxy): [,,9, ] s, which contains equal numbers of lies in each direction. Note that double-ly is used to facilitate imact damage observation, esecially delamination, as well as to reduce calculation time for the simulation. Seven different layus derived from the reference layu were chosen to be tested (cf. Table ). Thanks to the different imact behaviours of different stacking sequences, redominant fibre comressive failure was observed in a articular case. Thus, the second oint of this work is to imrove fibre failure simulation by combining the effect of fibre comressive failure which is often considered as a comlex mechanism in the failure of comosite structures [ ]. Some research resents fibre failure models for comosites subjected to low-velocity imact. Recently, the energy method seems to be an effective aroach to simulate fibre failure or intralaminar ly failure based on the crack band theory from Baz ant and Oh [7]. This method uses fracture toughness to dissiate the fracture energy and a characteristic length to avoid the mesh deendent solution. For examle, Falson and Aruzzese [,9] resented the fibre failure which was included in the intralaminar failure mechanism. In their constitutive damage model, the material roerties would degrade according to a linear strain-softening law with seven defined damage variables, i.e. fibre failure in tension/comression, matrix failure in tension/comression and three other for the three

3 Table Total ossible stacking sequences with seven exerimental tested cases. Layu name Stacking sequences Remarks A [,,9, ] s Reference case [,] A [9,,, ] s Rotation of A B [,,,9 ] s B [9,,, ] s Rotation of B C [,9,, ] s C [9,,, ] s Rotation of C D [,,,9 ] s D [,9,, ] s Rotation of D E [,,9, ] s E [,,,9 ] s Rotation of E F [,,9, ] s F [,9,, ] s Rotation of F shear directions. This law was assigned as a smeared formulation which assumed constant energy dissiation er unit area in the volume element to generate a mesh-size-indeendent solution. It was alied and tested for examle on an oen-hole laminate couon under comressive loading. Fibre comressive behaviour from their rogressive damage law accurately redicted the results of the exeriment. Iannucci and Ankersen [] reviously used a similar smeared formulation to develo their low-velocity imact model. They secifically tested this formulation with bi-linear stress strain damage. They showed that to have a mesh-size-indeendent FE model, with a given intralaminar fracture energy as an inut arameter, the strain at final failure and the evolution of the damage variable may vary with the element size. Shi et al. [] added non-linear irreversible shear behaviour to their D continuum model of intralaminar failure to simulate ermanent indentation for low-velocity imact modelling. Thanks to the energy based criterion and the shear model under rogressive damage, ermanent indentation was well established. Also, Loes et al. [] used energy-based criteria for matrix cracking and fibre failure in D lies in their low-velocity imact damage model, with an exonential damage evolution law for the damage roagation. Matrix cracking and fibre failure were accurately redicted. Comression after imact (CAI) modelling was consequently studied by González et al. []. According to the same aroach for the fibre failure law, they reorted that the collase of the late was initially triggered by fibre failure, and the CAI strengths for their two tested cases were accurately redicted. Nevertheless, this model was still limited by the calculation time rocess, e.g. imact and CAI analysis lasted twelve days for the laminate [,,,9 ] s on CPUs. As resented above, the modelling of rogressive damage with the energy based method could well reresent intralaminar damage as a multi-urose law, e.g. fibre failure, matrix cracking and ermanent indentation. In addition to the revious work, Bouvet et al. [] also used fracture mechanics for their fibre failure modelling. Moreover, they have roosed a new methodology to distribute unequal fracture energy at each integration oint (eight integration oints in a volume element) deending on the strain magnitude (cf. Section.). Only longitudinal tensile damage for fibre failure was addressed in revious work, whereas the comressive effect would be erformed in this aer in the following sections. However, the value of intralaminar fracture energy, which should be a material roerty as a model s inut, still becomes a sensitive issue since there is no valid standard yet [,,]. Many studies have roosed aroaches to measure this value. For examle, Soutis and Curtis [] measured the comressive fracture toughness of T/9C carbon/eoxy [,9,] s laminates associated with fibre micro-buckling to be. N/mm. Also, Pinho et al. [] used comact comression tests to evaluate the comressive kinking fracture toughness of T/9 carbon/eoxy laminates. A 79.9 N/mm of fracture toughness was reorted. However this value was contested because it was considered only for damage initiation, whereas the roagation was not reliable. Therefore, this current aer will also consider the imortance of comressive fracture toughness for damage roagation in order to make the low-velocity imact modelling match the exerimental tests according to data available from the literature [,].. al study and secimen configurations Imact tests were erformed using a dro tower system with a mm diameter, kg imactor, according to the Airbus Industries Test Method (AITM -). Before imacting the secimen, an otical laser measures the initial velocity. A iezoelectric force sensor is laced inside the imactor to measure contact force during imact. All data are recorded in an oscilloscoe. The rectangular secimen measures mm simly suorted on a 7 mm frame, as shown in Fig.. According to the reference case, -double-ly layus (. mm nominal ly thickness) of UD carbon- eoxy T7GC/M were manufactured based on,, and ly directions. Considering only balanced and mirror-symmetric laminates, switching ly locations makes it ossible to have u to configurations. Half of these configurations are symmetric to each other, along the longitudinal axis, i.e. layus [,,9, ] s and [,,9, ] s. Therefore, the otential cases can be reduced to cases studied. However, only seven cases (including the reference case), summarised in Table, were exerimentally tested, chosen according to the late s behaviour from analytical calculations and the first-trial simulation. Half of the secimens are derived from a orientation in relation to another secimen. Therefore, to number secimen configurations, the same letter will be used for a laminate and its associated rotation laminate. The quasi-isotroic reference layu A contains lies stacked at a constant interface angle of. However, when ly orientation is changed, the interface angle of becomes unavoidable in layus B, B, C, C, E and E.. Numerical modelling In the revious study, [] resented a discrete D imact model which was simulated with the Abaqus v.9 exlicit solver and a user-defined Vumat subroutine. In their model, three major failure modes observed in comosite imact tests were considered: (i) fibre failure in intra-ly, (ii) matrix cracking in intra-ly, and (iii) delamination in inter-ly. The mesh construction from the revious work was maintained. Positions of nodes are uniformly stacked in row and column for all oriented lies. However, the shaes of mesh are different: and lies are meshed in square shae, while and lies are meshed in arallelogram shae in order to follow the fibre Fig.. Imact test setu with the boundary condition.

4 total number of elements (volume elements and interface elements) is aroximately, elements for each layu configuration. Then, the laminated late is laced on analytical rigid body suorts, and is imacted by a semi-sherical analytical rigid body... ling fibre failure (a) (b) Fig.. (a) ling of imact damage and element tyes and (b) mesh shae in each oriented ly []. direction and to have coincident nodes in adjacent lies (Fig. (b)). The fibre failure was assigned in volume elements CD, where non-thickness cohesive elements COHD of delamination are horizontally inserted in-between. Also, vertical non-thickness cohesive elements COHD are laced between volume element stris in the fibre direction to imose the region of matrix cracking, as shown in Fig. (a). Meshing of these three damage tyes is generated only on half the late, due to the symmetry consideration. The Since the fibres rimarily break in the fibre direction, the fibre failure mode in this current model means urely longitudinal failure. Shear failure s is assumed to have no effect on intralaminar damage, whereas s is instead assigned to matrix cracking criterion (cf. Section.). In the exeriment, the fibres are locally broken crossing the fibre direction, as illustrated in Fig. and the micrograh in Fig. (a). For FE modelling, an alternative method is to dissiate the energy release rate of fibre failure sreading over the volume finite element such as the modelling of [,,,,]. Based on crack band theory from [7], a simlified formulation to dissiate the constant energy release rate er unit area in the D continuum element can be written as: Z Z e r de dv ¼ S G fibre ðþ V where G fibre ; e; e are the fracture toughness for the oening mode (I), the strain in fibre direction, and the strain in fibre direction at final failure, resectively. As seen in Fig. (a), these arameters are alicable either in tension (G fibre;t and e T ), or in comression (G fibre;c and e C ). V and S are element s volume and element s cross section normal to fibre direction, resectively. Then, V and S can be reduced in terms of an internal element length l, which is comarable to the FE characteristic length [7,,] to make FE model mesh-size indeendent. Note that the subscrits and denote damage initiation and final failure, resectively. In addition to distributing the fracture energy over the whole volume element, [] have roosed a new aroach to dissiate the fracture energy defined in terms of eight integration oints of each volume element, shown in Fig. (b).... Damage initiation... Damage initiation of fibre tensile failure. Before reaching the damage initiation state, linear elastic evolution of stress according to longitudinal strain e at each integration oint is defined. As the volume elements can be subjected to bending, the value of strain calculated at nodes can reach the criterion before the strain ε i : Longitudinal strain at integration oint (ii) (i) σ Tension Suerscrits T : tension C : comression i : value at integration oint node : value at node Subscrits : damage initiation : final failure i σ t = t at max( ε node= node ) node ε : Longitudinal strain at node re ε : Reresentative strain (max. at integration oint) t t t : Time increment at undergoing damage : Time increment at damage initiation : Time increment at final failure crush X C ε C ε Comression (a) T ε (i) (ii) T ε (i) Damage initiation (ii) Damage roagation ε H Damage initiation T,i ε T ε t = t ( d f ) H re i ε = max( ε ) i= (b) t = t Final failure T ε i ε Fig.. Constitutive laws of fibre failure in the longitudinal direction: (a) overall fibre failure law with damage initiation and damage roagation under tension and comression and (b) detail of the law when alied to the eight integration oints of a D element under tension.

5 Table Mechanical roerties of T7GC/M unidirectional ly as inut in simulation. Density kg/m Orthotroic elastic roerties [ ] E T Tensile Young s modulus in fibre direction GPa E C Comressive Young s modulus in fibre direction GPa E Transverse Young s modulus 7.7 GPa G Shear modulus. GPa t Poisson s ratio. Matrix cracking [,9] Y T Transverse tensile strength MPa S L In-lane shear strength MPa Fibre failure [,,,] e T Tensile strain in fibre direction at damage initiation. e C Comressive strain in fibre direction at damage initiation. X crush Longitudinal comressive mean crushing stress 7 MPa G fibre;t Fracture toughness for mode I in traction N/mm a G fibre;c Fracture toughness for mode I in comression N/mm b Delamination [,] G del G del I a Material: T/9 []. b Predicted value in this study. Interface fracture toughness for oening mode (I) Interface fracture toughness for shear mode (II and III). N/mm. N/mm calculated from integration oints. Thus, in this model the strain values obtained from eight integration oints are comuted with the shae function and extraolated to nodes in order to take into account bending behaviour. Then, the extraolated strain at each node drives the maximum strain failure criterion defined as: max node¼ e node < e T ðþ where the suerscrit node means the extraolation to node and e T is the tensile strain in the fibre direction at damage initiation, given in Table. When any one of the eight strains calculated at nodes reaches the tensile strain at damage initiation (beyond the limit in Eq. ()), all stresses at the eight integration oints are simultaneously established in the damage initiation state at t = t, as illustrated in Fig. (b).... Damage initiation of fibre comressive failure. Similarly to fibre tensile failure, damage initiation of fibre comressive failure can be defined as: min node¼ e node > e C ðþ where e C is the comressive strain in fibre direction at damage initiation, given in Table.... Damage roagation... Damage roagation of fibre tensile failure. The reresentative strain is the maximum strain of eight integration oints which is comuted at each time increment. This reresentative strain will be used for the linear degradation of damage, defined as: e re ¼ maxðe i Þ i¼ where the suerscrit i means the value at integration oint (fundamentally comuted by FE method), and e re is the reresentative strain. Due to fracture toughness for the oening mode (I) in traction G fibre;t ðþ i.e. the material inut roerty for the calculation, the fracture energy in volume elements can be dissiated. The tensile final failure strain (e T ) can then be determined by solving Eq. (). At each time increment, e ;T and e re will be udated during the undergoing damage roagation state, and the linear degradation of strain-softening can be assigned in terms of the damage variable: e T e re e T;i d f ¼ e re ðþ e T et;i where e T;i is tensile strain at damage initiation which is translated to the integration oint in order to take into account d f at the integration oints instead of nodes. Note that the damage variable d f, comuted from the reresentative strain, will be the same for eight integration oints, and it will govern the linear degradation behaviour, as illustrated in Fig. (b).... Damage roagation of fibre comressive failure. Due to the comlexity of damage roagation state in comression, arametric study of the longitudinal comressive fracture toughness G fibre;c is erformed based on available information from the literature [,]. The comressive strain at final failure (e C ) is then determined by solving Eq. ()similar to the tensile damage. And the reresentative strain in comression can be defined as: e re ¼ min ðe i Þ i¼ with the same fibre failure damage variable, d f as in tension. e C e re e C;i d f ¼ ð7þ e re e C ec;i Additionally, the fibre comressive failure behaviour is slightly more comlicated than in tension. Crack initiation in comression is due to the kink band, but when one continues to aly comression, the two sides of the crack come into contact and lead to crushing of acks of fibres. Therefore, the comressive mean crushing stress of T7/M, X crush roosed by [] is then alied as a ðþ

6 lateau to comlete the law. Moreover, the lasticity is also taken into account to revent comressive strain from returning to zero during the unloaded state, as illustrated in Fig. (a). Then, the stress tensor of orthotroic elasticity in terms of the elastic stiffness matrix and the single fibre damage variable, d f (both for tension and comression) can be secified as: 9 9 r ð d f ÞH ð d f ÞH ð d f ÞH e r H H e >< r >= H >< e >= ¼ r ð d f ÞH e 7 r sym ð d f ÞH e >: >; >: >; r ð d f ÞH e ðþ where the lamina s stiffness matrix [H] is comuted by using the orthotroic elastic roerties in Table... ling of matrix cracking A articular model of matrix cracking is introduced using interface elements (different rincial to delamination), between two neighbouring volume elements, which are generated in lanes, as illustrated in Fig.. That means the occurrence of matrix cracking is imosed by the mesh density. The authors assume that it is not necessary to reresent the comlex matrix microcracks network but only stries of lies that enable to simulate the changes in load transfers between arts of lies when the matrix is damaged and thus to drive delamination and fibre failures. Therefore a very fine mesh is not necessary. For the same reason, the energy dissiated in matrix cracking is not taken into account in the interface model (brittle failure), but it is nevertheless included in the energy dissiated in the delamination interfaces to kee the energy balance. Matrix tensile failure (r > ) based according to Hashin s failure criterion is alied to the volume elements. As soon as this criterion (on either one or both neighbouring volume elements) is met, the out-of-lane stresses in the interface elements are set to zero. The two neighbouring volume elements will therefore be indeendent, meaning that the matrix is broken [] hr i þ þ ðs Þ þ ðs Þ Y T ð9þ S L where the subscrit, and denote the direction of stress according to the volume element. Y T is the transverse tensile strength and S L is the in-lane shear strength. In addition to matrix cracking, these interface elements can also model ermanent indentation (see more details in [])... ling of delamination The formation of delamination generally relates to matrix cracking [,,]. For this resent discrete model, even if there is no couling arameter of delamination and matrix cracking, the discontinuity still enables this interaction to be reresented. Delamination normally occurs between different ly directions. It is therefore simulated in interface elements by joining nodes of uer and lower volume ly elements. Thanks to energy dissiation of fracture mechanics, the delamination criterion is simulated as linear couling in three modes based on the ower law criterion of mixed-mode delamination roagation with the energy release rate: mode I is in the thickness direction normal to delamination lane, whilst mode II and mode III are in the in-lane direction, as exlained in []. G I G del þ G II G del I þ G III ¼ G del II ðþ where G I, G II, G III reresent the energy release rate of delamination in mode I, II and III, resectively. G del ; Gdel I ; Gdel II reresent the critical energy release rate of delamination in mode I, II and III, resectively. At the end of the calculation, all layer interface delamination areas are dislayed to create a numerical C-scan. The same colours are used as for the exerimental C-scan in order to comare results from simulation with exeriments.. al validation of the model In exerimental tests, imact energy was exected at J, but in reality the tests were carried out between. J and. J for the seven cases studied. To validate the imact damage from the exeriments, all seven cases were simulated and comared with the exerimental results. Each calculation of imact modelling ( ms of actual time) lasts aroximately h on CPUs. According to the revious study [], imact damage in terms of energy dissiation is mainly searated into two arts, i.e. delamination and fibre failure, as layu D shows in Fig. 9(a). Hence, comarisons between exeriments and numerical simulations are resented as follows:.. Delamination In Fig., delamination is shown. Projected delamination areas are exerimentally obtained by ultrasonic C-scan. Thanks to the double-ly stacking configuration, each inter-ly delamination can clearly be distinguished by different colours. Delamination is visibly obvious in the first inter-ly on the non-imacted side (red colour surface in the figures), and the shae at each interly is often oriented in the fibre direction of the lower ly. The seven exerimental tested configurations are comared. The difference between the biggest delamination area (case D) and the smallest delamination area (case A) is u to % as shown in Fig. (a). This difference cannot be attributed only to the effect of stacking sequence, due to arameters couling when changing the stacking sequence: Stiffness of the anel: switching the order of stacking sequences will not affect the membrane stiffness, while the bending stiffness is changed. For examle, the laminates C is stiffer than C and, as a consequence the elastic resonse during imact is different. The stresses due to bending of laminate C are higher than in any other configurations, thus a comression fibre failure is exected. (cf.section..). Boundary condition: [] showed the imortance of boundary condition in comosite subjected to imact loading when designing real laminated structures near stiffeners of aircraft structures. In this study, excet the laminates A and C, the delamination ti is very close to the boundary conditions, which makes it difficult to conclude on the study of influence on extent of damage. Numerical simulation results show excellent delamination shae comarisons to the exeriments for all seven cases. The margin of error between exerimental determination of the delamination area and the simulation is within %. This margin of error is accetable since there are many factors affecting the delamination results, for examle the quality of exerimental C-scans, or uncertainty concerning the values of the arameters of the inuts in modelling, e.g. G del ; Gdel I. Another advantage of numerical simulation on delamination is the ossibility of searating delamination in each interface and being able to determine the sum of the delamination areas.

7 Fig. (a) shows that the rojected delamination and the total delamination (sum of each interface) follow the same trend... Fibre failure Fibre failure is normally difficult to study exerimentally, esecially during the imact event. However, microscoic observation, X-ray techniques after imact, or de-ly techniques [7] are robably also effective methods to study it. In this work, we simly observe fibre failure using methods:... Major load dro (studied using either the force dislacement curve or the force time curve) Fibre failure can be determined from the major load dro from history curves as resented in Fig.. At this energy level ( J), the major load dro obviously aears in case D and D in exeriment (see Fig. 9(a) for case D), as well as in numerical simulation. This henomenon is robably due to fibre tensile failure rather than fibre comressive failure which will be detailed in Section..... Fibre failure crack on imacted surface (visual insection) An aarent long crack on the uer surface (imacted side) across the fibre direction is visible only in case C, as shown in Fig.. In order to investigate the cause of this crack, the secimen C was cut for microscoic observation. As resented in Fig. 7, disconnected longitudinal fibres at the imacted side, as well as some small debris from the kink band between the disconnected fibres are observed. The formation of these fibres is similar to fibre micro-buckling/kinking in Fig. (b), which could confirm the aearance of the fibre comressive failure. However, ruture modes mm mm A - - mm [,,9,- ] s mm C - - [,9,,- ] s mm 99 mm C - - [9,,-, ] s 77 mm 79 mm D - - [,,-,9 ] s 9 mm Fig.. Comarison between exerimental and numerical C-scan for delamination areas, force dislacement curves, and force time curves of J imact tests.

8 99 mm mm mm D - - [-,9,, ] s E - - [,-,9, ] s E - - [-,,,9 ] s mm 9 mm 7 mm Fig. (continued) Delamination area (mm ) A C C D D E E [,,9,-]s [,9,,-]s [9,,-,]s [,,-,9]s [-,9,,]s [,-,9,]s [-,,,9]s Layus (a) (Projection) (Projection) (Summation) Dissiated energy (J) A C C D D E E [,,9,-]s [,9,,-]s [9,,-,]s [,,-,9]s [-,9,,]s [,-,9,]s [-,,,9]s Layus (b) Fig.. -model comarison of seven different layus: (a) delamination area and (b) energy dissiation.

9 A C C D D E E MODEL EXPERIMENT mm [,, 9, - ] s [, 9,, - ] s [9,, -, ] s [,, -, 9 ] s [-, 9,, ] s [, -, 9, ] s [-,,, 9]s Fibre failure damage variable Fig.. Fibre comressive failure on the uer surface of different layus in both exeriment and simulation. Fibre failure Delamination cut A-A - - cut A-A [9,, -, ] s from the two images might be slightly different because the fibre comressive failure in Fig. (b) involved internal confined lies, whereas in Fig. 7 fibre comressive failure from case C was exterior. To validate the numerical simulation, as seen in Fig., case C shows fibre failure in terms of the fibre failure damage variable similar to the noticeable crack in the exerimental observation. The detail of fibre comressive failure in this simulation is discussed in Section.... Force dislacement history and global imact resonse mm Kink band debris Fig. 7. Microscoic observation of fibre comressive failure of secimen C after imact. This section again demonstrates that this resent model is robust and alicable even if the stacking sequence is changed. The force dislacement and force time history in Fig. show good agreement in terms of maximum force, maximum dislacement and global evolution of imact resonse. The simulation accurately reresents the load dro due to fibre failure as mentioned before, as well as the energy dissiation. The error between the exeriment and the simulation of these seven tested cases is less than %, as shown in Fig. (b).. Discussion of failure mechanisms in simulation.. Fibre tensile failure mechanism According to the revious numerical studies [], although only fibre tensile failure behaviour was studied in the fibre direction, force resonse during the imact event was fairly well corroborated with the exerimental results. This may therefore imly that fibre tensile failure is redominant for longitudinal failure. Consequently, the same fibre tensile failure law is maintained in this case. In fact, fibre tensile failure aeared at the beginning of the imact event but it was not aarent until a dramatic load dro was recorded. This can be an indicator to identify fibre failure which is confirmed by the simulation, resented in case D in Fig. 9. The dissiation of energy in fibre failure mode, esecially on the non-imacted side: ly nd, rd and th lies, suddenly increased. This was simultaneous with the major load dro, recisely in Fig. 9(a a). In addition, the number of elements for fibre failure visibly increased after the major load dro. The number of elements with fibre tensile failure is about % more than in fibre comressive failure, which may be induced by the tension from the late s deflection, as resented in Fig. 9b b... Fibre comressive failure mechanism Some fibre damage was visible after the secimen was imacted, but it does not aear in all cases, as shown in Fig.. C Fig.. Micrograh of: (a) fibre tensile failure in the secimen after imact and (b) fibre micro-buckling/kinking formation in CFRP laminate [].

10 a b a b a b b (a) (b) Fig. 9. Simulation on layu D for: (a) synchronisation of force and energy dissiation; (b and b) fibre failure states before and after the major load dro; (b and b) ly slitting. is an interesting layu, having lies on the exterior, lying in the small edge direction of the boundary condition. This observation led to imrovement of the model by adding the effect of fibre comressive failure which is a key objective of this aer. As mentioned reviously, fibre failure in this simulation is based on fracture energy, therefore an accurate value of fracture toughness is essential. Before obtaining accurate corroboration between the simulation and the exeriment as resented in receding sections, a arametric study of fibre comressive law and fracture toughness was erformed, according to values from the literature. The first value tested for fracture toughness in comression was G fibre;c ¼ N=mm. The result clearly shows the damage variable of fibre comressive failure on the uer surface in case C, but in the force dislacement curve, the load dro is still overestimated, as seen in Fig. (a). In addition to this inaccuracy, other cases (A, C, D, D, E and E) have an overestimation of fibre comressive failure as well, for examle layu E as seen in Fig. (b). Then, G fibre;c ¼ 79:9 N=mm of T/9 [] was tried. This shows a better delamination area according to exerimental results with a bigger delamination interface on the imacted side, but fibre comressive failure is underestimated for all seven configurations. Therefore, the value of G fibre;c between and 79.9 N/ mm should be considered. G fibre;c ¼ N=mm was taken and it could give very accurate results according to exeriments in terms of both fibre comressive failure on the uer surface, load dro in force dislacement and delamination shae as shown in Figs., (a), and (b). Certainly changing G fibre;c directly affects fibre comressive failure. It should be noted that to obtain these accurate results, the value of G fibre;c ¼ N=mm is assigned in the model using the tradeoff method. This value is in accordance with the fracture toughness of T/9C from [] which is in the same material family as the T7GC/M material studied in this work. Nevertheless, further work to determine the fibre comressive failure fracture toughness value of T7GC/M is currently in rogress. The following stage to determine the residual strength after imact, known as CAI, will be erformed based on the same law of imact test fibre comressive failure in this study... Couling between failure modes The validity of the model is broadened. Thanks to the discontinuity of the articular meshing and indeendent material law among three failure modes (without couling arameter), this resent model is able to show very good interactions. Couling among fibre failure, matrix cracking, and delamination: As seen in the view of the non-imacted side in Fig. 9(b), slitting of the lowest ly, induced by matrix cracking and delamination, rather than fibre failure mode, is found. This shows accordance with the exerimental observation, as shown in Fig. 9(b). Couling between fibre failure and delamination: Fibre comressive failure is accomanied by delamination as reorted in [,]. Microscoic section of secimen C in Fig. 7 confirms this henomenon. In addition, numerical modelling of layu C also reveals this interaction, which is in agreement with the exeriment, as shown in Fig..

11 MODEL : layu C [9,, -, ] s MODEL : layu E [-,,, 9 ] s G f = N / mm G fibre, C = N / mm G fibre, C = 7.9 N / mm ibre, C 9 G f = N / mm G fibre, C = N / mm G fibre, C = 7.9 N / mm ibre, C 9 DELAMINATION mm - - DELAMINATION mm - - FIBRE FAILURE ON UPPER SURFACE underestimate Fibre failure damage variable FIBRE FAILURE ON UPPER SURFACE overestimate Fibre failure damage variable fibre,c, G = N/mm, G fibre,c = N/mm, G fibre,c = 79.9 N/mm, G fibre,c = N/mm, G fibre,c = N/mm, G fibre,c = 79.9 N/mm (a) (b) Fig.. Validation of G fibre;c of delamination/fibre comressive failure on the uer surface/force dislacement curve on layus: (a) C and (b) E. Fibre failure Fibre failure mm Delamination Delamination Layu C [9,, -, ] s Fig.. Demonstration of the couling between fibre failure and delamination on the uer ly (non-imacted side) of layu C. Moreover, global behaviour of the imact resonse, e.g. couling of matrix cracking and delamination, force dislacement resonse, overall delamination area, etc., also needs the interaction between each failure mode. Therefore, a careful trade-off must be made to balance the selected arameters of all imact damage tyes, in order to obtain an accurate imact simulation.. Conclusions Fibre failure mechanisms have an imortant role in the resonse of comosite structures subjected to imact. However, in exeriments, fibre failures are difficult to observe because of their location inside the laminate, and the fact that they are not as visible as delamination from ultrasonic C-scans. In this work, we can easily determine the resence of fibre rutures with the noticeable load dro of force dislacement or force time curves. This load dro redominates in fibre tensile failure which can occur when the late is submitted to adequate imact energy. Fibre comressive failure is less well known and is rarely observed during imact tests. In this study, seven layus were exerimentally studied, including an interesting case (C) with a ly on the uer surface of the laminate. Fibre comressive failure was visible in this uer ly after the laminate was imacted. The fact that comressive fibre failure was observed led to imrove the reference model []. A new comressive law was imlemented. This law is similar to the one already used for fibre ruture under tension, with a dissiation of fracture energy in the volume elements, driven by a damage variable calculated at the eight integration oints. Moreover, the effect of crushing in the cracks induced by fibre failure is taken into account with a lateau and a lastic-like law. An identification of the fracture toughness value for comression fibre failure was made by means

12 of a arametric study. The N/mm value is found to be the most aroriate to best describe the observed damage in the C laminate. This new comressive law, alied to the six other laminates, also imroves the results of imact simulations on these lates in terms of force dislacement history, force time history, and delamination. The current model roves to be quite reliable, and resents a certain number of advantages such as the calculation time ( h/calculation), a relatively limited number of material arameters required, and without any couling arameters between failure modes. To widen the validity of this model, other validations such as effect of imact velocity/energy, effect of boundary condition, lydros configuration, and sub-laminate/ly grouing are still in rogress in order to aroach the situations of real aeronautical structures. Thanks to the roosed fibre comressive failure law, modelling of comression after imact (CAI) will soon be continued using the same law. Acknowledgements This work was granted access to the HPC resources of CALMIP under the allocation -P. The authors also gratefully acknowledge the financial suort through the THEOS Oeration Training Program (TOTP). References [] Bouvet C, Rivallant S, Barrau JJ. Low velocity imact modeling in comosite laminates caturing ermanent indentation. Comos Sci Technol ;7():977. [] Richardson MOW, Wilsheart MJ. Review of low-velocity imact roerties of comosite materials. Comos Part A: Al Sci Manuf 99;7A:. [] Iannucci L, Ankersen J. An energy based damage model for thin laminated comosites. Comos Sci Technol ;(7 ):9. [] Laffan MJ, Pinho ST, Robinson P, Iannucci L, McMillan AJ. Measurement of the fracture toughness associated with the longitudinal fibre comressive failure mode of laminated comosites. Comos Part A: Al Sci Manuf ;():9. [] Pinho ST, Robinson P, Iannucci L. Fracture toughness of the tensile and comressive fibre failure modes in laminated comosites. Comos Sci Technol ;():9 79. [] Soutis C, Curtis PT. A method for redicting the fracture toughness of CFRP laminates failing by fibre microbuckling. Comos Part A: Al Sci Manuf ;:7. [7] Bažant ZP, Oh BH. Crack band theory for fracture of concrete. Mater Struct 9;(): 77. [] Falzon BG, Aruzzese P. Numerical analysis of intralaminar failure mechanisms in comosite structures Part I: FE imlementation. Comos Struct ;9():9. [9] Falzon BG, Aruzzese P. Numerical analysis of intralaminar failure mechanisms in comosite structures art II: alications. Comos Struct ;9():7. [] Shi Y, Swait T, Soutis C. ling damage evolution in comosite laminates subjected to low velocity imact. Comos Struct ;9(9):9. [] Loes CS, Camanho PP, Gürdal Z, Maimí P, González EV. Low-velocity imact damage on disersed stacking sequence laminates art II: numerical simulations. Comos Sci Technol 9;9(7 ):97 7. [] González EV, Maimí P, Camanho PP, Turon A, Mayugo JA. Simulation of droweight imact and comression after imact tests on comosite laminates. Comos Struct ;9: 7. [] Donadon MV, Iannucci L, Falzon BG, Hodgkinson JM, de Almeida SFM. A rogressive failure model for comosite laminates subjected to low velocity imact damage. Comos Struct ;( ):. [] Israr HA, Rivallant S, Barrau JJ. al investigation on mean crushing stress characterization of carbon-eoxy lies under comressive crushing mode. Comos Struct ;9:7. [] Bouvet C, Castanié B, Bizeul M, Barrau JJ. Low velocity imact modelling in laminate comosite anels with discrete interface elements. Int J Solids Struct 9;( ):9. [] Faggiani A, Falzon BG. Predicting low-velocity imact damage on a stiffened comosite anel. Comos Part A: Al Sci Manuf ;():77 9. [7] Sztefek P, Olsson R. Tensile stiffness distribution in imacted comosite laminates determined by an inverse method. Comos Part A: Al Sci Manuf ;9(): 9. [] Prombut P. Caractrisation de la roagation de dlaminage des stratifis comosites multidirectionnels. Ph.D. thesis; Universit de Toulouse; 7 [Tye = Phdthesis]. [9] Guillon D. Etude des mécanismes dabsortion dénergie lors de lécrasement rogressif de structures comosites à base de fibre de carbone. Ph.D. thesis; Universit de Toulouse; [Tye = Techreort]. [] Tomblin J, Sherraden J, Seneviratne W, Raju KS. A basis and B basis design allowables for eoxy based rereg TORAY T7GC-K-E/# unidirectional tae. Tech. Re. November; National Institute for Aviation Research. Wichita, KS;. [] Vandellos T, Hautier M, Huchette C. A strategy to identify a cohesive zone model couled with intralaminar damage. In: th Euroean conference on comosite materials (ECCM/). Budaest;.