Coupling continuous damage and debris fragmentation for energy absorption prediction by cfrp structures during crushing
 878 Downloads
 1 Citations
Abstract
Energy absorption during crushing is evaluated using a thermodynamic based continuum damage model inspired from the Matzenmiller–Lubliner–Taylors model. It was found that for crashworthiness applications, it is necessary to couple the progressive ruin of the material to a representation of the matter openings and debris generation. Element kill technique (erosion) and/or cohesive elements are efficient but not predictive. A technique switching finite elements into discrete particles at rupture is used to create debris and accumulated mater during the crushing of the structure. Switching criteria are evaluated using the contribution of the different ruin modes in the damage evolution, energy absorption, and reaction force generation.
Keywords
Composite Crushing Damage mechanics Computational modeling Discrete particles Energy absorption1 Introduction
Nowadays crashoriented design of vehicles or aeronautical structures or structural subcomponents is realized through comparison of experimental and numerical testing on elementary geometries [2, 7, 8, 9, 10, 11]. Efforts have been made to represent material properties degradations, debris wedge generation, and residual shape of structures. To experimentally study the effect of the initiator for typical aeronautical components, Guillon [7] realized different tests and tried to link the observed phenomena with the registered load in time, and with macroscopic properties that we can extract of it.
Regarding numerical modelling, if it is desired to model both the material degradation and the structural efficiency in term of energy absorption, it is necessary to use methods that are able to represent both scales of behaviors and to calculate local as well as global quantities and fields.
As far as is known by the authors, numerical studies devoted to the analysis of the structural behavior and design of tubes or plates during crushing found in the literature are based upon the hypothesis of a 2D or multilayer shell behavior for the structure. Under this hypothesis, behavior laws for the plies material can be reduced to a plane stress or plane strain hypothesis and shell or 3D finite elements are used [9, 11]. To numerically represent fractures creation and propagation, essentially two techniques are used: decohesion (loss of links between elements) and element kill (suppression of elements). Erosion by the element kill technique consists in eliminating from the computation the elements that suffers too large strains or stresses, simulating the local loss of mechanical strength [2, 10]. The criterion is often based on a mean state of damage for composites which is directly linked to the mechanical strength. To represent the structure pealing [12] between sublamina (often of different orientation only) or to represent wedge debris or fracture [7, 9], a cohesive element is suppressed when the maximum stress is reached in the corresponding opening mode. Cohesive criteria can also be coupled with damage in sublamina [10, 11, 13, 14]. These studies are focused on the global properties of the structure and are able to create debris if the crack path is known a priori.
Other numerical methods exist to represent the generation of fragments, chips or debris from structures subjected to impulsive or high transient loadings, in particular the particlebased methods. The advantage of these methods is that they do not suffer from localized large deformations as do the meshbased methods. The Lagrangian SPH particle method for example [15] allows to use classical continuum mechanics material models, so that it is appropriate to selfgenerate a natural separation of matter and fragments as soon as the material reaches a certain state criteria defined in the frame of the classical continuum mechanics [16]. The SPH method is in particular well suited and robust enough to represent matter decohesion, holes, or matter accumulation due to shear and compression in ductile materials, since it is a continuous particle method (through the kernel approximation). But SPH becomes instable in tension and needs regularization if the ruptures/cracks that generate the fragments are due to a brittle (instantaneous stress relaxation after rupture) or semi brittle behavior (delayed stress relaxation process after process) [17, 18]. Furthermore, SPH simulations require huge models and big computation efforts (time and space). To keep advantage of the continuous particle method approximation to represent large to very large and severe deformations up to rupture while limiting numerical instabilities, and allowing linking the particles to classical finite elements, Johnson and his woworkers have developed the generalized particle algorithm (GPA) [19, 20]. The method is able to deal with unequal spacing of particles and recent developments allow switching finite elements into SPHlike particles when a failure (or a nonadmissible state) criterion is reached. Matter is then kept during fragmentation. The latter method is called CPEM for Combined Particleelement Method [21]. The method is indeed very efficient in modelling a perforation process and seems to capture very well the distribution of matter ejected from the impacted or from the bottom faces of a target impacted at a high velocity. As can be seen in [22], it is nevertheless still difficult to determine the residual size (mass) and velocity of the generated fragments, since fragments are generated by brittle or semi brittle (cohesive like) phenomena. To represent the size of fragments and distribution of matter, another particle based method seems to be useful, the materialpoint method (MPM). The method has been introduced by D. Sulsky et al. [23]. The MPM method is inspired from the particleincell method used for fluid mechanics [24, 25, 26]. It uses an Eulerian grid that computes the matter state variables (velocity, pressure, and internal energy) without any mesh distortion. The particles are Lagrangian material points which velocities and displacements are tracked through the grid and used to compute the particles state. Information exchanges on state variables between materialpoint particles are thus done in an indirect way using the grid as a support. This presents the advantage of stabilizing the computation (in particular the time step [27]) and allows the computation of noncohesive material behaviors [28] such as that of snow [29], seeds in a silo [30], or soils [31, 32]. Comparisons of simulations results obtained with a SPH model and a MPM model for a case of hyper velocity impact are presented by Zhang [27]. The method is more stable especially for large deformations due to high tensions, but needs a special treatment of cracks and contacts. As an example, Ambati et al. used the MPM method to simulate the orthogonal cutting process [33]. The chip formation was well recovered. As in classical FE models, it was nevertheless necessary to set friction coefficients to get the right cutting force and the right shear bands formation, whereas the friction coefficient is an output in SPH simulations [34]. The only need on SPH simulations is to set enough particles to catch strain localization, which again gives huge models and is not efficient in computation time because the time step becomes very small. For crack initiation and propagation, the MPM is attractive as it is possible to take into account the effect of a local failure on farfiled stress [35] thus avoiding localization effects that can be observed for example when using cohesive elements in classical finite element simulations of impacts on composite structures. But the treatment of explicit cracks needs to set discontinuous velocities in the support grid [36, 37]. New developments have been done to compute stress concentrations in 3D structures containing a crack placed between the particles [38], or multiple cracks supported by the particles for brittle materials [39]. Keeping in mind the idea that local stresses must be correctly computed to create and propagate cracks, and that the rupture can be supported by particles instead of being supported by surfaces between the particles, recent research activities have been devoted to the quality of numerical integration of meshess methods [40]. These works show the necessity to link the computation of the large deformations due to plasticity or damage (up to rupture) in the continuum description of the material, and its representation through Lagrangian material points.
To model both the local material behavior and the structural fragmentation process a very interesting application similar to what can happen in the crushing process of our composite plates, is the case of fragmentation of structures under impulsive high energy loading, typically either hyper velocity impacts or explosions. Two strategies are retained here as the reference. The approach of Banerjee [41, 42] consists in modelling the structure with discrete materialpoint. As a consequence, the structure is allowed to fragment following free paths from particles to particles that are set broken when their state variables reach continuum thermodynamically based criteria. The Eulerian grid that is used as a support for the material points is also used to compute the loading from the surrounding gas of explosion. On the opposite, Borvik and Wadley [43, 44] use discrete particles for the gas and the sand clouds generated by the explosion and isogeometric finite element with a nodal splitting technique to fragment the structure. The high order isogeometric computation in the structure allows large deformations and the nodal splitting technique allows the structure to create fragments that have a minimum size under interest. This approach is more appropriate to our objectives and is easier to implement in the industrial finite element codes that are expected to be used.
Regarding the crushing process initiation, it is still a challenge to find the method or the methods that represent the fragment sizes, masses, velocities, and compressibility after it has been created because the debris itself becomes part of the crushing process. The computational strategy presented in this paper is a new methodology capable to simulate the different ruin and fragmentation modes during the crushing process of composite plates that selfinitiates and propagates debris generation from local material degradations, and predict the global structural efficiency of the structure. A corpuscular method that switches finite elements into discrete particles is used for small debris. The analysis presented in this paper is focused on the capabilities of the methods to reproduce the strong interactions between the physical ruin modes and to self both initiate and propagate fractures in a full 3D simulation. This is why the first part of this paper is devoted to the analysis of the experimental analysis of the crushing process itself. The second part presents the particle method. The third one presents the strategy that has been adopted here to represent the continuous and discontinuous behaviors of a composite sample plate during crushing, and their ability to predict energy absorption and the fragmentation processes. The final part of this paper presents the simulation results and examines the added value of using discrete particles to adequately represent debris accumulation and their effect on the crushing process and energy consumption computation. It is concluded that discrete particle methods in conjunction with classical finite elements methods are the best suited combination to represent both the global structural behavior and the local fragmentation of composites while keeping a certain computational efficiency. efficiency.
2 Technical discussion
2.1 Experimental observations

\(0\)—contact: at the first beginning of the contact, the force grows very rapidly and the extremity of the plate is destroyed

\(1\)—first damage: the force has reached about half the maximum value and the outer 0\(^{\circ }\) ply delaminates; transverse shearing appears in lamina neighbors to the external 0\(^{\circ }\) ply and some fragments are created

\(2\)—peak: the force reaches the maximum value and about one half thickness of the wedge has been eroded, the plate is slightly flexed and touches the right horizontal guide; a long crack (delamination) appears at an interface situated near the middle thickness; the outer left lamina bend

\(3\)—full contact: the force is maintained when the external right ply touches the rigid base; the plate is slightly bend and touches the left horizontal guide; fragments of +45 and \(\)45 lamina are created by transverse shearing; the middle 0\(^{\circ }\) lamina starts a local buckling; a lot of small debris are generated

\(4\)—two halves: the plate is separated into two half parts; the left part lamina bend and debris are generated in the confined region under the bending zones; the right half lamina starts bending on a long dimension rod between the rigid base and a damaged zone near the horizontal guide (fracture in the outer right 0 degree lamina); the force just dropped down from a value (point 3.5) and crossing the displacement measures on the pictures with the force curve, one can suggest that this is due to the fracture in the outer 0 degree lamina that has been created by the contact with the horizontal guide

\(5\)—stationary crush; one can see the big fracture in the outer right half plate sliding on the rigid base; fragments are generated in the two confined regions under the bending lamina with a bigger size than debris at stage 0 to 3.
2.2 A basis of five ruin modes
The risk of rupture of a composite structure is evaluated using structural criteria based on stresses (as the famous criteria proposed by Hashin [45]), and rely on the hypothesis that the matter is continuous to compute the loading path up to the rupture. These criteria are discussed by different authors among whom the work of Davila et al. [46] or Brewer [47]. These criteria have in common the will to link the material elementary ruin modes and the structural rupture prediction. The question that is addressed in the present work is to find five basis ruin criteria of the structure that could drive coupling between inside plies continuous damage (in the matter volume) and discontinuous fragmentation of a CFRP structure during crushing. We propose here five independent ruin modes: three sublamina modes corresponding to damage in fibers, matrix and fiber matrix interface, one delamination mode (cohesive elements at the interface between lamina) and one interaction mode. These modes must be considered as an elementary basis from which more sophisticated modes can be derived.

\(< >\) defines the Macaulay brackets (positive part or ramp function)

\(X_T\) and \(X_C\) are respectively tensile/compressive failure stresses in fiber direction

\(Y_T\) and \(Y_C\) are respectively tensile/compressive failure stresses in transverse direction

\(Z_T\) and \(Z_C\) are respectively tensile/compressive failure stresses in outofplane direction

\(S_{fs}\) is the debonding strength of fiber matrix interface

\(S_{12}\), \(S_{23}\) and \(S_{13}\) are respectively shear damage threshold stresses

\(r_j \in [1,+\infty ]\) is called the limit load ratio

The coefficient \(\phi \) is given in order, first, to increase the failure shear stress when normal compressive stress occurs and secondly to represent friction
Effects of localized temperature are not taken into account in the behavior model during the process, neither on the evolution of material parameters such as the Young’s modulus, nor on the stressstrain relations or ruin criteria. Indeed, the effect of temperature on the apparent Young’s moduli of the unidirectional CFRP ply has been proven to be low (a few percent) because temperature affects essentially the resin behavior while the composite ply Young’s moduli are essentially due to the carbon fibers resistance. Carbon fibers moduli are not affected by local temperature increase because the loading condition make them brake in compression or shear in a brittle way before the local temperature has reached a significant value for carbon [52]. As a consequence, as for machining in aluminum work pieces, event though the local temperature could reach high values due to strain localization, it is dissipated in the fragmentation process and debris are evacuated rapidly so that the whole crushing process can be considered as adiabatic. In some cases of quasistatic loading such as a composite corner unfolding (LShape structures), it has been proven that it is necessary to take into account for the residual curing stresses in the identification of the rupture yield stresses \(Z_T\) and \(Y_T\) which appear to have different values [53]. For the dynamic crushing process, these considerations were not taken into account.
3 Orthotropic damage driven fracture modeling
The key issue in fragmentation is to be able to transform a material behavior into a series of domains while keeping each domain a realistic matter resistance and kinematics state properties while releasing the tensile strengths at domains connections. A special attention is paid here at choosing appropriate numerical methods to disconnect domains and create smaller matter domains that correspond to physical debris or fragments. For computational efficiency, it is aimed to keep as long as possible in the simulation large 3D finite element domains to represent a continuous behavior for the composite structure.
As other commercial codes Impetus Afea Solver (http://www.impetusafea.com) offers different ways to create fractures. Decohesion between domains is generated following different criteria: isotropic or anisotropic CockcroftLatham failure criterion, plastic strain failure criterion, JohnsonCook failure criterion, geometric failure strain criterion. Elements can be eroded following the criteria, or they can be split and new nodes are created to allow matter opening. Among other features of Impetus Afea Solver, GPU computing allows much more rapid computation durations and makes it possible to use higher degree isogeometric interpolation finite elements. It has been chosen here to try different combinations of these methods with the previous described CDM material model to generate the different debris.
3.1 From the continuous volumes to particlebased small debris
3.2 Cohesive failure
3.3 Proposed strategy
 Criterion I: the material inplane shear modulus \(G_{12}\) has dropped down to zero, which means that the damage variable \(d_4\) has reached 1:$$\begin{aligned} d_4=1 \end{aligned}$$(17)
 Criterion II: the inplane \(E_{11}\) fiber or \(E_{22}\) matrix moduli have dropped down to their minimal values (not zero in axial fiber direction to keep a residual compressive strength) :$$\begin{aligned} d_1=d_1^{max} \; or \; d_2=1 \end{aligned}$$(18)
 Criterion III: the limit load ratios for matrix cracking and delamination have reached their limit value (see [50] for details)$$\begin{aligned} \phi _4=1 \; and \; \phi _5=1 \end{aligned}$$(19)

A progressive finite element to particle switching is controlled by the damage state computed with the 3D continuous damage material model only (CDM)

Delamination is created using a cohesive failure contact between the damageable plies while damage in plies will not create fracture

Both cohesive delamination and finite element damage driven erosion are simultaneous. Two kinds of soft coupling are evaluated, each using a different criterion for FE to particles switching.
Numerical models presented
Model identification number  Continuous damage mechanics (CDM)  CDM erosion criterion EF to particles switching  Cohesive failure 

1  X  Criterion I  
2  X  X  
3  X  Criterion II  X 
4  X  Criterion III  X 
4 Numerical models and input data
4.1 Wedged plate crush FE model
The numerical model presented here is set in the framework of plane strain hypothesis, defining symmetry constraints on the lateral boundaries of the 3D one element in depth finite element model. In the simulation presented here, a value of 72 kg was chosen upon the possible values. In order to reduce the model size, the length of the plate in the model was shorter than in the experiment that is 100 mm instead of 160 mm. The free height is fixed to 20 mm in the model.
The composite structure is the only deformable part of the model. A flexural modulus is introduced in place of the Young axial modulus. The reader is invited to read reference [50] for further details. The heavy carriage is perfectly tied at the right end of the composite structure. This system has an initial velocity of 5.4 m/s. The mass of the carriage has been scaled down to the appropriate value for a 0.27 mm depth model, and represent a physical mass of 72 kg. Contacts are defined between the composite structure and the rigid guides, and between the structure and the rigid base. A static friction coefficient of \(10~\%\) is introduced on the rigid base, and \(8~\%\) on the rigid lateral guides. Faces of elements perpendicular to the y direction are constrained with symmetry conditions to obtain a 2D plane strain model.
4.2 Inputs for the CDM material model and the cohesive failure

Static tests to get failure stresses

Cycled static tests for damage evolution parameters, irreversibility or saturation effects, m parameters, damage thresholds

Dynamic tests (Hopkinson tests) for strain rate effects, and validate \(m\) parameters
T800S/M21e input data for the CDM model
\(E_{11}^0=165\) GPa  \(E_{22}^0=E_{33}^0 =7.64\) GPa  \(X_T=2.2\) GPa  \(X_C=1.2\) GPa  \(E_f=112\) GPa 
\(\nu _{32}=0.4\)  \(\nu _{21}=\nu _{31} =0.0162\)  \(Y_T=45\) MPa  \(Y_C=280\) MPa  \(m_i=10\) 
\(G_{23}=2.75\) GPa  \(G_{12}=G_{13} =5.61\) GPa  \(Z_T = 45\) MPa  \(Z_C=0.7\) GPa  \(d_4^{max}=0.87\) 
\(S_{23}=0.05\) GPa  \(S_{12}=S_{31} =0.05\) GPa  \(S_{fs}=1.5\) GPa  \(S_{ffc}=0.5\) GPa  \(\phi =10\) 
\(\rho =1,550\) kg/m\(^{3}\)  \(\sigma _{strength_{stat}}=120\) MPa  \(\dot{\epsilon }_{vis}=400\) s\(^{1}\)  \(\dot{\epsilon }_{ref}=750\) s\(^{1}\)  \(C=4.7\) 
Input data chosen for the cohesive failure
\(\sigma _{fail}=50\) MPa  \(\tau {fail}=50\) MPa  \(G_{Ic}=750\) J/m\(^{2}\)  \(G_{IIc}=1{,}200\) J/m\(^{2}\)  \(\Delta _{ref} = 0.25\) mm 
5 Comparison of experimental and numerical failure modes
Model 2 shows global and local buckling from the first beginning of crushing on the rigid base. The global flexural behavior of the plate that was observed in the experiment is also reproduced by the cohesive failure model 2. The problem of this model is that fully damaged finite elements stay in the simulation, and create too much instability in the crushing process. In order to get both the stable fragmentation process of the pure CDM model 1, and the local and global flexural behaviors of the plate obtained with the pure cohesive model 2, the two coupling models 3 and 4 are tested.
Model 4 is considered to be the most representative of what was observed by Guillon. 0 degree lamina is subjected to lamina bending or local buckling while other directions lamina is essentially subjected to transverse shearing. Debris of 45/90/45 lamina are half the plate thickness long most of the time, which is characteristic of this ruin mode, and delamination about 0 degree lamina are stopped by the circumferential directions lamina. The simulation however did not exhibit the bending of outer right half part lamina as was observed in the experiment. Apart from the form of the failure criteria or limit values of the material, many reasons could be invoked to explain this difference: free space between the guides and the plate could be different in the model than in the experiment, the mass of the heavy carriage could be different as well, the rigid base could be not completely rigid during the experiment, and all conditions that are not perfectly controlled during tests and that must be rigorously defined as inputs in a computational model. Damage \(d_1\) is almost not visible meaning that the elements that reach the limit value are eroded. Damages \(d_2\) and \(d_4\) reach slightly the same values at the same locations, meaning the strong interaction between ruin modes in the plies and at the interfaces, and a predominance of \(\phi _4\) and \(\phi _5\) on the loss of mechanical strength potential. Damage zones are visible in the outer lamina where the plate impacted the guides, as in the experiment.
6 Energy absorption and crushing forces prediction
One can isolate on the energy curve of model 4 three time periods each composed of a small unstable part and then a quite stable part between stages 1 and 4.5, stages 4.5 to 6 and stages 6 to 8. Oscillations are quite always 0.08 ms long, while stable parts are about 0.12 ms long. It can be seen that model 1 reaches quite instantaneously a value of about 4J at stage 1 of the crushing process, and remains at this plateau all the crushing long. This is coherent with the localization of fragmentation at the tip of the sample in the vicinity of the rigid support. The depth of fragmentation and the debris are so small that no oscillations are visible on the energy curve. Model 2 follows almost the same dissipation tendency than model 4 in the first time period and becomes unstable at stage 4. It is concluded that stage 4 is the point where matter erosion is necessary in the simulation, and the point at which coupling between in plane damage due to out of plane delamination becomes the controlling phenomenon. Model 3 also follows model 4 in the first time period, remains stable in the second time period, but absorbs less energy staying at a plateau level characteristic of localized fragmentation at the sample tip. The energy curves decreases when the process becomes unstable at stage 6. It is concluded that in plane damage that was observed to be too brittle in model 3 interacts well with out of plane delamination in model 4. It is also concluded that the controlling phenomenon is the effect of damage on delamination and out of plane fractures at this stage.
7 Discussion
CDM manages distributed defects in volumes of materials by decreasing the mechanical elastic moduli. If used alone, the proposed CDM has proved its limitation to represent the global behavior of wedge plates during crush if it is expected to represent the three types of ruin that are involved in experiments: transverse shearing, lamina bending and local buckling. The same limitation was found by [12]. The solution that was found by Israr was to artificially stop the lateral deformations of 90\(^{\circ }\) plies in order to allow an axial crushing up to \(90\) or \(95~\%\) enabling the initiation of delamination in the neighboring cohesive elements. In our case, no numerical supplementary yields for strains or stresses where introduced. Using only cohesive failure is not adequate either since the real process also creates debris that are partially ejected and partially confined under the bending lamina. In the case of crushing plates, it is necessary to take into account for the redistribution of loads between the laminas coming from decohesions and debris generation and accumulation, in a closed loop interaction. The proposed strategy that switches finite elements into particle allows to keep the low cost finite elements in the model as long as possible and to create matter separations and continuous debris which lengths are coherent with experimental observations of Farley and Jones [3]. Regarding the debris dimensions and accumulation during crushing, the proposed strategy is considered to be more adequate than the CPEM method proposed by Johnson [21] that gives either large clouds of small debris or even single particles, or large unbroken shrapnel. Results are comparable in quality with those obtained with the MPM method by Banerjee regarding the number of fragments [41]. The difference seems essentially to state in the necessity for the MPM method to model the entire surrounding environment in order to catch the right velocity fields whereas in our model only the structure is modelled. Macroscopic discontinuities that are clearly visible on pictures of experiments are strongly interacting with micro cracking. Here, a soft coupling model has been proposed that uses in plane damage to release the thermodynamic potential of the matter of the plies, and a mixedmode cohesive failure for the interfaces. The inplane damage model takes into account the load path redistribution due to through ply cracking and delamination rates, through a coupling matrix between five bases ruin modes and intrinsic damage variables, but in a passive way. If we compare with the very interesting work of Li [39] who uses the MPM method for brittle materials, it can be noticed that our model does not need any distribution default neither to initiate nor to propagate cracks, even if the coupling is weak. Indeed the transient dynamics introduces enough (and probably too much) differences in the integration points states of stresses and strains.
It is not really surprising that the computed force signal is not perfectly comparable with the one of Guillons experiments. We did not try to fit both of them in this study. The first reason of the discrepancies is that the materials are different and it is known that both materials do not have the same damage resistance under dynamic loading [50]. In this context, it is possible to compare qualitatively the behaviors but not quantitatively until a complete characterization of the T700GC/M21e has been done. Furthermore, the finite element model itself will create lower contact forces because even though finite element are switched into particles after erosion, the bulk resistance of the particle is not really the one of the real matter and higher values for the contact stiffness slow down the time step size. The purpose of this work was to analyze the energy absorption phenomena and the effect of coupling between fragmentations and debonding of the plies on the absorption. It has been shown that the global damage shapes are well qualitatively represented. The degree one for the finite element interpolation could not be enough to capture the stresses and strains gradients. Local bending is not well computed, and it is not possible for example to capture kink band phenomena which seem to be replaced by FE to particle switching instead. The 2D plane strain hypothesis is also very restrictive. It is then not necessary at this level of study, to fit exact failure criteria, especially when they are only weakly coupled as it was the case here.
8 Summary and conclusions
In this work we have used a continuous damage model coupled in a weak form with a cohesive failure model available in Impetus Afea Solver to compute the damage induced fracturing of a composite plate during edge crushing. The CDM model has been implemented as a user defined material model in different commercial computation codes. LSDYNA was used to identify the material and damage parameters and with success to predict damage after low velocity impact or compression after impact residual strength. Samcef was used to strongly couple damage in plies and interface cohesive failure or low velocity impacts. Impetus Afea Solver has been used in this study to determine the combination of criteria that could be used to reproduce fragmentation and energy absorption of a composite plate during crushing. The best combination that has been found is to use continuous damage to decrease inplane elastic moduli of plies and use the cohesive failure to open interfaces using criteria of model 4. In model 4, finite element to particles switching is achieved when in plane damage is maximum. The best found criterion for switching was driven by the loss of thermodynamic internal potential of the matter due to ruin rates of through ply cracking and delamination. Using model 4, the three kinds of ruin modes and the stable evolution of the crushing force(displacement) and energy(time) are qualitatively reproduced. Next work will be oriented on strong coupling, and localization of load path to quantify the effect of local hinges or bending on the stability of the crushing process.
Notes
Acknowledgments
This work has been partially funded by the Région MidiPyrénée which is thanked here for support to the EPICEA project MODCOMP.
References
 1.Wiggenraad JFM, Michielsen ALPJ, Santoro D, Lepage F, Kindervater C, Beltran F, AlKhalil M (2001) Finite element methodologies development to simulate the behaviour of composite fuselage structure and correlation with drop test. Air Space Eur 3(3–4):228–233CrossRefGoogle Scholar
 2.McCarthy MA, Wiggenraad JFM (2001) Numerical investigation of a crash test of a composite helicopter subfloor structure, Compos Struct 51:345–359256 PII: S 0 2 6 3–8 2 2 3 (0 0) 0 0 1 5 0–1Google Scholar
 3.Farley GL, Jones R (1992) Crushing characteristics of continuous fiberreinforced composite tubes. J Compos Mater 26(1):37–50CrossRefGoogle Scholar
 4.Hull D (1991) A unified approach to progressive crushing of fiberreinforced composite tubes. Compos Sci Technol 40:377–421CrossRefGoogle Scholar
 5.Qiao P, Davalos J, Barbero EJ (1998) Design optimization of fiber reinforced plastic composite shapes. J Compos Mater 32(2):177–196CrossRefGoogle Scholar
 6.Feraboli P (2008) Development of a corrugated test specimen for composite materials energy absorption. J Compos Mater 42(3):229–256. doi: 10.1177/0021998307086202 CrossRefGoogle Scholar
 7.Guillon D (2008) Etude des mécanismes dabsorption dénergie lors de lécrasement progressif de structures composites base de fibre de carbone (Ph. D. Thesis), 271 (in French), Université de ToulouseICAISAE, FranceGoogle Scholar
 8.Davies GAO, Olsson R (2004) Impact on composite structures. Aeronaut J 108:541–563Google Scholar
 9.Joosten MW, Dutton S, Kelly D, Thomson R (2011) Experimental and numerical investigation of the crushing response of an open section composite energy absorbing element. Compos Struct 93:682–689. doi: 10.1016/j.compstruct.2010.08.011 CrossRefGoogle Scholar
 10.Xiao X (2009) Modeling energy absorption with a damage mechanics based composite material model. J Compos Mater 43(5):427–444. doi: 10.1177/0021998308097686 CrossRefGoogle Scholar
 11.Mc Gregor CJ, Vaziri R, Poursartip A, Xiao X (2007) Simulation of progressive damage development in braided composite tubes under axial compression. Composites A 38:2247–2259. doi: 10.1016/j.compositesa.2006.10.007 CrossRefGoogle Scholar
 12.Israr HA, Rivallant S, Bouvet C, Barrau JJ (2014) Finite element simulation of 0/90 CFRP laminated plates subjected to crushing using a freefacecrushing concept. Composites A 62:16–25. doi: 10.1016/j.compositesa.2014.03.014 CrossRefGoogle Scholar
 13.Espinosa HD, Zavattieri PD, Dwivedi SK (1998) A finite deformation continuum/discrete model for the description of fragmentation and damage in brittle materials. J Mech Phys Solids 46(10):1909–1942CrossRefMATHMathSciNetGoogle Scholar
 14.Guinard S, Allix O, GuedraDegeorges D, Vinet A (2002) A 3D damage analysis of lowvelocity impacts on laminated composites. Compos Sci Technol 62:585–589. doi: 10.1016/S02663538(01)001531 CrossRefGoogle Scholar
 15.Liu GR, Liu MB (2003) Smoothed Particle hydrodynamics: a meshfree particle method. World Scientific Publishing Co Pte Ltd, Singapore ISBN 10:9812384561CrossRefGoogle Scholar
 16.Limido J, Espinosa C, Salan M, Lacome JL (2006) A new approach of high speed cutting modelling: SPH method. Journal de Physique IV 134:1195–1200. doi: 10.1051/jp4:2006134182 CrossRefGoogle Scholar
 17.Lacome JL, Limido J, Espinosa C (2009) SPH formulation with Lagrangian Eulerian adaptive kernel. In: 4th SPHERIC Workshop, NantesGoogle Scholar
 18.Michel Y, Chevalier JM, Durin C, Espinosa C, Malaise F, Barrau JJ (2006) Hypervelocity impacts on thin brittle targets: experimental data and SPH simulations. Int J Impact Eng 33:441–451. doi: 10.1016/j.ijimpeng.2006.09.081 CrossRefGoogle Scholar
 19.Johnson GR, Beissel SR, Stryk RA (2000) A generalized particle algorithm for high velocity impact computations. Comput Mech 25(2–3):245–256. doi: 10.1007/s004660050473 CrossRefMATHGoogle Scholar
 20.Johnson GR (2011) Numerical algorithms and material models for highvelocity impact computations. Int J Impact Eng 38:456–472. doi: 10.1016/j.ijimpeng.2010.10.017 CrossRefGoogle Scholar
 21.Johnson GR, Beissel SR, Gerlach CA (2013) A combined particleelement method for highvelocity impact computations. In: Proceedings of the 12th hypervelocity impact symposium on procedia engineering, vol 58. Balimore, pp 269–278. doi: 10.1016/j.proeng.2013.05.031
 22.Johnson GR (2011) Another approach to a hybrid particlefinite element algorithm for highvelocity impact. Int J Impact Eng 38:456–472. doi: 10.1016/j.ijimpeng.2011.01.002 CrossRefGoogle Scholar
 23.Sulsky D, Chen Z, Schreyer L (1994) A particle method for historydependent materials. Comput Methods Appl Mech Eng 18:179–196CrossRefMathSciNetGoogle Scholar
 24.Sulsky D, Zhou SJ, Schreyer HL (1995) Application of a particleincell method to solid mechanics. Comput Phys Commun 87:236–252CrossRefMATHGoogle Scholar
 25.Sulsky D, Schreyer L (2004) MPM simulation of dynamic material failure with a decohesion constitutive model. Eur J Mech A 23:423–445. doi: 10.1016/j.euromechsol.2004.02.007
 26.Chen Z, Brannon R (2002) An evaluation of the Material Point Method. Report of the Sandia National Laboratory, SAND20020482Google Scholar
 27.Ma S, Zhang X, Qiu XM (2009) Comparison study of MPM and SPH in modeling hypervelocity impact problems. Int J Impact Eng 36:272–282. doi: 10.1016/j.ijimpeng.2008.07.001 CrossRefGoogle Scholar
 28.Bardenhagen SG, Brackbill JU, Sulsky D (2000) The materialpoint method for granular materials. Comput Methods Appl Mech Eng 187:529–541CrossRefMATHGoogle Scholar
 29.Stomakhin A, Schroeder C, Chai L, Teran J, Selle A (2013) A material point method for snow simulation. ACM Trans Graph 32(4):102. doi: 10.1145/2461912.2461948 CrossRefGoogle Scholar
 30.Wieckowski Z (2004) The material point method in large strain engineering problems. Comput Methods Appl Mech Eng 193:4417–4438. doi: 10.1016/j.cma.2004.01.035 CrossRefMATHGoogle Scholar
 31.Andersen SM (2009) MaterialPoint Analysis of LargeStrain Problems: Modelling of Landsildes, Ph.D. Thesis, Aalborg University, Denmark, ISSN 1901–7294 DCE Thesis No. 20Google Scholar
 32.AlKafaji IKJ (2013) Formulation of a Dynamic Material Point Method (MPM) for Geomechanical Problems, D93 Dr.Ing. Dissertation, Universitt Stuttgart, Germany, ISBN 9789053357057Google Scholar
 33.Ambati R, Pan X, Yuan H, Zhang X (2012) Application of material point methods for cutting. Comput Mater Sci 57:102–110. doi: 10.1016/j.commatsci.2011.06.018 CrossRefGoogle Scholar
 34.Limido J, Espinosa C, Salan M, Mabru C, Chieragatti R, Lacome JL (2011) Metal cutting modelling SPH approach. Int J Mach Mach Mater 9(3–4):177–196. doi: 10.1504/IJMMM.2011.039645 Google Scholar
 35.Schreyer HL, Sulsky DL, Zhou SJ (2002) Modeling delamination a s a strong discontinuity with the material point method. Comput Methods Appl Mech Eng 191:2483–2507CrossRefMATHGoogle Scholar
 36.Nairn JA (2003) Material point method calculations with explicit cracks. Comput Model Eng Sci 4(6):649–663MATHGoogle Scholar
 37.Nairn JA (2009) Analytical and numerical modeling of R curves for cracks with bridging zones. Int J Fract 5(2):167–181CrossRefMathSciNetGoogle Scholar
 38.Guo YJ, Nairn JA (2006) Threedimensional dynamic fracture analysis using the material point method. Comput Model Eng Sci 1(1):11–25Google Scholar
 39.Li F, Pan J, Sinka C (2011) Modelling brittle impact failure of disc particles using material point method. Int J Impact Eng 38:653–660. doi: 10.1016/j.ijimpeng.2011.02.004 CrossRefGoogle Scholar
 40.Babuska I, Banerjee U, Osborn JE, Zhang Q (2009) Effect of numerical integration on meshless methods. Comput Methods Appl Mech Eng 198:2886–2897. doi: 10.1016/j.cma.2009.04.008 CrossRefMATHMathSciNetGoogle Scholar
 41.Banerjee B (2004) Material point method simulations of fragmenting cylinders. In: 17th ASCE engineering mechanics conference, University of Delaware, NewarkGoogle Scholar
 42.Banerjee B, Guilkey JE, Harman TB, Schmidt JA, McMurtry PA (2009) Simulation of impact and fragmentation with the material point method, computational physics. In: 11th international conference on fracture 2005, Torino. arXiv:1201.2452 [physics.compph].
 43.Borvik T, Olovsson L, Hanssen AG, Dharmasena KP, Hansson H, Wadley HNG (2011) A discrete particle approach to simulate the combined effect of blast and sand impact loading of steel plates. J Mech Phys Solids 59:940–958. doi: 10.1016/j.jmps.2011.03.004 CrossRefGoogle Scholar
 44.Wadley HNG, Borvik T, Olovsson L, Wetzel JJ, Dharmasena KP, Hopperstad OS, Deshpande VS, Hutchinson JW (2013) Deformation and fracture of impulsively loaded sandwich panels. J Mech Phys Solids 61:674–699. doi: 10.1016/j.jmps.2012.07.007 CrossRefGoogle Scholar
 45.Hashin Z (1980) Failure criteria for unidirectional fiber composites. J Appl Mech 47(2):329–334. doi: 10.1115/1.3153664 CrossRefGoogle Scholar
 46.Davila CG, Camanho PP, Rose CA (2005) Failure criteria for FRP laminates. J Compos Mater 39(4):323–345. doi: 10.1177/0021998305046452 CrossRefGoogle Scholar
 47.Brewer JC (1988) Failure of graphite/epoxy induced by delamination, PhD Massachusetts Institute of Technology, CambridgeGoogle Scholar
 48.Matzenmiller A, Lubliner J, Taylor TL (1995) A constitutive model for anisotropic damage in fibercomposites. Mech Mater 20(2):125–152Google Scholar
 49.Xiao JR, Gama BA, Gillespie JW Jr (2007) Progressive damage and delamination in plain weave S2 glass/SC15 composites under quasistatic punchshear loading. Compos Struct 78(2):182–196. doi: 10.1016/j.compstruct.2005.09.001
 50.Ilyas M (2010) Damage modeling of carbon epoxy laminated composites submitted to impact loading (Ph. D. Thesis), 260. Université de ToulouseICAISAE, FranceGoogle Scholar
 51.Ilyas M, Espinosa C, Lachaud F, Michel L, Salan M (2011) Modeling aeronautical composite laminates behavior under impact using a saturation damage and delamination continuous material model. Key Eng Mater 452–453:639–672. doi: 10.4028/www.scientific.net/KEM.452453.369 Google Scholar
 52.Didierjean S (2004) tude du comportement des matriaux composites carbone/poxy en environnement hygrothermique, (Ph. D. Thesis). Université de ToulouseICAISAE, FranceGoogle Scholar
 53.Michel L, Garcia S, Chen Y, Espinosa C, Lachaud F (2013) Experimental and numerical investigation of delamination in curvedbeam multidirectional laminated composite specimen. Key Eng Mater 389:577–578. doi: 10.4028/www.scientific.net/kem.577578.389 Google Scholar
 54.Olovsson L, Hanssen AG, Borvik T, Langseth M (2010) A particlebased approach to closerange blast loading. Eur J Mech A 29:1–6. doi: 10.1016/j.euromechsol.2009.06.003 CrossRefGoogle Scholar
 55.Cundall PA, Strack ODL (1979) A discrete numerical model for granular assemblies. Geotechnique 29(1):47–65CrossRefGoogle Scholar
 56.Salot C, Gotteland P, Villard P (2009) Influence of relative density on granular materials behavior: DEM simulations of triaxial tests. Granular Matter 11(4):221–236CrossRefMATHGoogle Scholar
 57.Chen C, Espinosa Ch, Michel L, Lachaud F (2012) A numerical approach for analyzing postimpact behavior of composite laminate plate under in plane compression. In: 15th European conference on composite materials, VeniceGoogle Scholar
 58.Espinosa Ch, Michel L, Lachaud F (2012) Modélisation numérique de linitiation et de la propagation des dommages lors du dépliage de cornières composites aéronautiques Comparaison des prévisions théoriques. NAFEMS Congres, ParisGoogle Scholar
 59.Lachaud F, Espinosa C, Michel L, Salan M (2011) Impacts on fuselages: a trial in using numerical simulation to predict residual strength. Workshop Dynamic failure of composites and sandwich structures Toulouse, ParisGoogle Scholar
 60.Briche F (2009) Etude du comportement mécanique des matériaux composites stratifiés sous impact basse vitesse Application au fuselage de lA350, Masters Thesis, Ecole des Mines, FranceGoogle Scholar
 61.Prombut P (2007) Caractérisation de la propagation de délaminage des stratifiés composites multidirectionnels (Ph. D. Thesis), Université de ToulouseICAISAE, 335, FranceGoogle Scholar
 62.Pinho ST, Iannucci L, Robinson P (2006) Formulation and implementation of decohesion elements in an explicit finite element code. Composites A 37:778–789. doi: 10.1016/j.compositesa.2005.06.007 CrossRefGoogle Scholar
 63.Ilyas M, Espinosa Ch, Lachaud F, Salan M (2009) Dynamic delamination using cohesive finite elements, In: 9th international DYMAT conference. BrusselsGoogle Scholar