Full discontinuous Galerkin formulation of shells in large deformations with parallel and fracture mechanics applications Becker, Gauthier ; Noels, Ludovic Conference (2012, July 11) Fracture mechanical problems can be solved by coupling the finite elements with a cohesive approach. Unfortunately, the classical cohesive methods suffer from severe limitations. Indeed, on one hand, the ... [more ▼] Fracture mechanical problems can be solved by coupling the finite elements with a cohesive approach. Unfortunately, the classical cohesive methods suffer from severe limitations. Indeed, on one hand, the intrinsic approach, which inserts the cohesive elements at the beginning, has to model the prefracture stage. This requires an initial slope in the traction separation law that should tend toward infinity to avoid lack of consistency leading to obvious numerical problems. On the other hand, the extrinsic cohesive method inserts the cohesive elements during the simulation when a fracture criterion is reached. This insertion requires topological mesh modifications and therefore a very complicated implementation, especially in a parallel code. To overcome these limitations, new methods were developed and in particular, an approach based on discontinuous Galerkin formulation (DG) has been pioneered by R. Radovitzky (Radovitzky cmame2011). The use of the DG principle allows to formulate the problem with discontinuous elements and the continuity between them is ensured weakly by terms integrated on the elements interface . These interface elements can be easily replaced by a cohesive element during the simulation. We have recently developed this approach for shells (Becker cmame2011) to obtain a full DG method. Moreover, a new cohesive law based on the reduced stresses of the thin bodies formulation is developed to propagate a fracture through the thickness. This cohesive model dissipates the right amount of energy during crack phenomena. These developments are implemented in parallel and validated by the study the blast of a notched cylinder, for which experimental and numerical (by XFEM method) data are reported in the literature by R. Larsson (Larsson ijnme2011). Finally, as thin structures are often made of ductile materials, which show large deformations before fracture, the formulation is extended to the non linear case with hyperelastic material law. This one can take into account the damage and a criterion based on the work of Huespe (Huespe plasticity2009) is developed to localize the damage leading to the apparition and propagation of cracks. [less ▲] Detailed reference viewed: 60 (8 ULg)Multiscale Simulations of Composites with Non-Local Damage-Enhanced Mean-Field Homogenization Wu, Ling ; Noels, Ludovic ; et al Conference (2012, July) The mean-field homogenization (MFH) approach is an attractive framework for multiscale methods, as it provides predictions of the macroscopic behavior of particle or fiber reinforced composites at a ... [more ▼] The mean-field homogenization (MFH) approach is an attractive framework for multiscale methods, as it provides predictions of the macroscopic behavior of particle or fiber reinforced composites at a reasonable computational cost. Efficient MFH methods have been available for a long time for linear elastic problems, using for example the Mori-Tanaka scheme [2], but they can also be extended in the non-linear regime after linearization of the constitutive behavior at the current strain state, as for the incremental approach, e.g. [1]. In this work, the application of ductile-damage theories to a multiscale analysis of continuous fiber reinforced composites is considered. Toward this end, the incremental MFH approach is extended to account for the damage behavior happening in the matrix material at the microscale and to derive the effective properties of particle or fiber reinforced composites. However, capturing the degradation, damage or failure of material happening at the microscopic scale could lead to loss of uniqueness in the solution as the governing partial differential equations may lose ellipticity at a given level of loading corresponding to the strain-softening onset. Thus, in order to avoid the strain/damage localization caused by matrix material softening, the gradient-enhanced formulation [3] is adopted to describe the material behavior of the matrix during the homogenization process, as we have recently proposed [4]. As illustration, the behavior of a fiber re-enforced elasto-plastic matrix is considered. The properties of the matrix correspond to an elasto-plastic material experiencing damage, with a non-local form of Lemaitre Chaboche model. The fibers are assumed linear elastic, see [4] for details. A loading-unloading cycle is applied in the direction transverse to the fibers. A maximal deformation of 10 % is reached before the unloading proceeds to zero-transverse deformation. The effective behavior predicted by the MFH models is compared to the solutions obtained by finite element computations performed on a unit periodic cell and on RVE where the micro-structure is fully meshed. The results for three fiber volume ratios are presented in Fig. 1. For the three fiber volume ratios, the homogenized property is dominated by the properties of the matrix, with an obvious elasto-plastic behavior exhibiting softening. For vI = 15% and 30%, rather good predictions are given by the MFH model, with, as expected, higher macroscopic stress and damage predicted by the MFH due to the incremental formulation. However for vI = 50%, the MFH model overestimates the macroscopic stress considerably. This error comes from the assumption of Mori - Tanaka based MFH. As it is shown to be an efficient multi-scale approach, the developed gradient enhanced MFH formulation presented can now be used to model the behavior of composite laminates experiencing damage. [less ▲] Detailed reference viewed: 54 (5 ULg)A one-field formulation of elasto-plastic shells with fracture applications ; Noels, Ludovic Conference (2012, July) The main feature of Discontinuous Galerkin (DG) methods is their ability to take into account discontinuities of the unknown field in the interior domain of studied problems. In such formulations the ... [more ▼] The main feature of Discontinuous Galerkin (DG) methods is their ability to take into account discontinuities of the unknown field in the interior domain of studied problems. In such formulations the integration by parts is realized on the elements leading to boundary integral terms These terms between elements ensure the continuity and compatibility of the solution in a stable manner, which makes DG attractive to ensure weakly the C0 continuity and/or the C1 continuity. For non-linear Kirchhoff-Love shells, the C0/DG formulation - the C0 continuity is ensured as usual by considering continuous shape functions and the C1 continuity is weakly ensured by DG interface terms in a stable and consistent way - leads to a one-field formulation, where the displacements are the only unknowns of the problem [5]. DG methods have also an advantage when it comes to simulate fracture with a cohesive approach. Indeed, the classical cohesive methods suffer from severe limitations. The intrinsic approach, which inserts the cohesive elements at the beginning, requires an initial slope in the traction separation law and suffers from lack of consistency, while the extrinsic cohesive method, which inserts the cohesive elements during the simulation when a fracture criterion is reached, suffers from difficult parallel implementation due to the topological mesh modifications happening during the simulations. To overcome these limitations, new methods were developed and in particular, an approach based on a C0-discontinuous Galerkin formulation where continuity between elements is ensured weakly by the interface elements. These interface elements can be easily replaced by a cohesive element during the simulation, leading to an efficient fracture framework [4, 6]. We have recently extended this approach for linear shells [1] to obtain a full DG formulation of thin bodies, where discontinuities in the C0 and C1 fields are weakly enforced using consistent interface elements, and where the interface elements can be used to integrate the cohesive law when a fracture criterion is met. Moreover, a new cohesive law based on the reduced stresses of the thin bodies formulation is developed to propagate a fracture through the thickness. This cohesive model dissipates the right amount of energy during crack initiations and/or propagations. These developments are extended to non-linear elasto-plastic shells and are implemented in parallel, which allows to simulate complex fracture problems. As a validation example, the study of a blasted notched-cylinder is performed, for which experimental and numerical (by XFEM method) data are reported in the literature by [3]. Figure 1 illustrates the results obtained at different times. It can be seen that the simulation allows to model crack propagation, as well as the bifurcation happening when the crack reaches the top of the cylinder. Also, it appears that with the elasto-plastic framework, the crack speed numerically predicted matches the experimental measures [2]. [less ▲] Detailed reference viewed: 51 (2 ULg)Multi-scale computational homogenization analysis of foams with micro-buckling Nguyen, Van Dung ; Noels, Ludovic Conference (2012, July) When studying the behavior of foams by multi-scale computational homogenization procedure, the micro-buckling may occur at the cell walls and edges and reduces the effective stiffness of the structures at ... [more ▼] When studying the behavior of foams by multi-scale computational homogenization procedure, the micro-buckling may occur at the cell walls and edges and reduces the effective stiffness of the structures at macro-scale. This instability can be enhanced by plastic deformation at micro-scale. At sufficiently large value of macro-strain, even if the micro-tangent moduli of micro-material is still elliptic, the homogenized tangent moduli at macro-scale can lose its ellipticity that implies the localization occurs at macro-scale. When localization occurs, the characteristic size of macro- scopic deformation is the same order of the microscopic size. The assumption of material action in standard multi-scale computational homogenization approach where the stress only depends on the strain at this point is no-longer suitable. And the material behavior at given point depends also on the neighborhood of this point. To cover this problem, the second-order multi-scale computational homogenization is suitably used. At macroscopic problem, the high-order stress and the high-order strain are enhanced to the standard formulation by using the Discontinuous-Galerkin formulation while at the micro-scale, the standard continuum formulation is still used. By this procedure, the influence of micro-buckling of foams on structural behaviour is studied. [less ▲] Detailed reference viewed: 101 (25 ULg)Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation Nguyen, Van Dung ; Béchet, Eric ; Geuzaine, Christophe et al in Computational Materials Science (2012), 55 In order to predict the effective properties of heterogeneous materials using the finite element approach, a boundary value problem (BVP) may be defined on a representative volume element (RVE) with ... [more ▼] In order to predict the effective properties of heterogeneous materials using the finite element approach, a boundary value problem (BVP) may be defined on a representative volume element (RVE) with appropriate boundary conditions, among which periodic boundary condition is the most efficient in terms of convergence rate. The classical method to impose the periodic boundary condition requires the identical meshes on opposite RVE boundaries. This condition is not always easy to satisfy for arbitrary meshes. This work develops a new method based on polynomial interpolation that avoids the need of matching mesh condition on opposite RVE boundaries. [less ▲] Detailed reference viewed: 665 (316 ULg)Non-linear mechanical solvers for GMSH Noels, Ludovic ; ; Nguyen, Van Dung et al Scientific conference (2012, March) Detailed reference viewed: 43 (7 ULg)Serial FEM/XFEM-Based Update of Preoperative Brain Images Using Intraoperative MRI ; Noels, Ludovic ; et al in International Journal of Biomedical Imaging (2012), 2012 Current neuronavigation systems cannot adapt to changing intraoperative conditions over time. To overcome this limitation, we present an experimental end-to-end system capable of updating 3D preoperative ... [more ▼] Current neuronavigation systems cannot adapt to changing intraoperative conditions over time. To overcome this limitation, we present an experimental end-to-end system capable of updating 3D preoperative images in the presence of brain shift and successive resections. The heart of our system is a nonrigid registration technique using a biomechanical model, driven by the deformations of key surfaces tracked in successive intraoperative images. The biomechanical model is deformed using FEM or XFEM, depending on the type of deformation under consideration, namely brain shift or resection. We describe the operation of our system on two patient cases, each comprising ¯ve intraoperative MR images, and demonstrate that our approach significantly improves the alignment of nonrigidly registered images. [less ▲] Detailed reference viewed: 89 (23 ULg)Non-local damage-enhanced MFH for multiscale simulations of composites Wu, Ling ; Noels, Ludovic ; et al in Proceedings of the XII SEM International Conference & Exposition on Experimental and Applied Mechanics (2012) In this work, a gradient-enhanced mean-field homogenization (MFH) procedure is proposed for fiber reinforced materials. In this approach, the fibers are assumed to remain linear elastic while the matrix ... [more ▼] In this work, a gradient-enhanced mean-field homogenization (MFH) procedure is proposed for fiber reinforced materials. In this approach, the fibers are assumed to remain linear elastic while the matrix material obeys an elasto-plastic behavior enhanced by a damage model. As classical finite element simulations face the problems of losing uniqueness and strain localization when strain softening of materials is involved, we develop the mean-field homogenization in a non-local way. Toward this end we use the so-called non-local implicit approach, reformulated in an anisotropic way to describe the damage in the matrix. As a result we have a multi-scale model that can be used to study the damage process at the meso-scale, and in particular the damaging of plies in a composite stack, in an efficient computational way. As a demonstration a stack with a hole is studied and it is shown that the model predicts the damaging process in bands oriented with the fiber directions. [less ▲] Detailed reference viewed: 43 (7 ULg)Stiction failure in microswitches due to elasto-plastic adhesive contact Wu, Ling ; Golinval, Jean-Claude ; Noels, Ludovic in Proceedings of the XII SEM International Conference & Exposition on Experimental and Applied Mechanics (2012) Undesirable stiction, which results from contact between surfaces, is a major failure mode in micro-switches. Indeed the adhesive forces can become so important that the two surfaces remain permanently ... [more ▼] Undesirable stiction, which results from contact between surfaces, is a major failure mode in micro-switches. Indeed the adhesive forces can become so important that the two surfaces remain permanently glued, limiting the life-time of the MEMS. This is especially true when contact happens between surfaces where elasto-plastic asperities deform permanently until the surfaces reach plastic accommodation, increasing the surface forces. To predict this behavior, a micro adhesive-contact model is developed, which accounts for the surfaces topography evolutions during elasto-plastic contacts. This model can be used at a higher scale to study the MEMS behavior, and thus its life-time. The MEMS devices studied here are assumed to work in a dry environment. In these operating conditions only the Van der Waals forces have to be considered for adhesion. For illustration purpose, an electrostatic-structural analysis is performed on a micro-switch. To determine the degree of plasticity involved, the impact energy of the movable electrode at pull-in is estimated. Thus the maximal adhesive force is predicted using the developed model. [less ▲] Detailed reference viewed: 58 (9 ULg)Homogenization of fibre reinforced composite with gradient enhanced damage model Wu, Ling ; Noels, Ludovic ; et al in Hogge, Michel; Van Keer, Roger; Dick, Erik (Eds.) et al Proceedings of the 5th International Conference on Advanded COmputational Methods in Engineering (ACOMEN2011) (2011, November) Classical finite element simulations face the problems of losing uniqueness and strain localization when the strain softening of materials is involved. Thus, when using continuum damage model or ... [more ▼] Classical finite element simulations face the problems of losing uniqueness and strain localization when the strain softening of materials is involved. Thus, when using continuum damage model or plasticity softening model, numerical convergence will not be obtained with the refinement of the finite element discretization when strain localization occurs. Gradient-enhanced softening and non-local continua models have been proposed by several researchers in order to solve this problem. In such approaches, high-order spatial gradients of state variables are incorporated in the macroscopic constitutive equations. However, when dealing with complex heterogeneous materials, a direct simulation of the macroscopic structures is unreachable, motivating the development of non-local homogenization schemes. In this work, a non-local homogenization procedure is proposed for fiber reinforced materials. In this approach, the fiber is assumed to remain linear elastic while the matrix material is modeled as elasto-plastic coupled with a damage law described by a non-local constitutive model. Toward this end, the mean-field homogenization is based on the knowledge of the macroscopic deformation tensors, internal variables and their gradients, which are applied to a micro- structural representative volume element (RVE). Macro-stress is then obtained from a homogenization process. [less ▲] Detailed reference viewed: 70 (9 ULg)Influence of the elasto-plastic adhesive contact on Micro-Switches Wu, Ling ; Golinval, Jean-Claude ; Noels, Ludovic in Hogge, Michel; Van Keer, Roger; Dick, Erik (Eds.) et al Proceedings of the 5th International Conference on Advanded COmputational Methods in Engineering (ACOMEN2011) (2011, November) Undesirable stiction, which results from contact between surfaces, is a major failure mode in micro,electro-mechanical systems (MEMS). In previous works, a statistical rough surfaces interaction,model ... [more ▼] Undesirable stiction, which results from contact between surfaces, is a major failure mode in micro,electro-mechanical systems (MEMS). In previous works, a statistical rough surfaces interaction,model, based on Maugis and Kim formulations has been presented to estimate the adhesive forces in MEMS switches. In this model, only elastic adhesive contact has been considered. However, during the impact between rough surfaces, at pull-in process for example, plastic deformations of the rough surfaces cannot be always neglected especially for the MEMS with metallic contact surfaces. In the present work, a new micro-model predicting the adhesive-contact force on a single elasticplastic asperity interacting with a rigid plane is presented. This model will be used later on for the interaction between two elastic-plastic rough surfaces. The MEMS devices studied here are assumed to work in a dry environment. In these operating conditions only the Van der Waals forces have to be considered for adhesion. [less ▲] Detailed reference viewed: 55 (5 ULg)A two-scale model predicting the mechanical sliding and opening behavior of grain boundaries in nanocrystalline solids Péron-Lührs, Vincent ; ; et al in Hogge, Michel; Van Keer, Roger; Dick, Erik (Eds.) et al Proceedings of the 5th International Conference on Advanded COmputational Methods in Engineering (ACOMEN2011) (2011, November) In polycrystalline materials with nanosized grains smaller than 100 nm, the deformation mechanisms taking place at grain boundaries (GBs) become dominant compared to intragranular crystal plasticity ... [more ▼] In polycrystalline materials with nanosized grains smaller than 100 nm, the deformation mechanisms taking place at grain boundaries (GBs) become dominant compared to intragranular crystal plasticity. Recent studies have revealed that more accurate mechanical properties can be obtained by choosing the relevant GB character distribution (GBCD). We use here a numerical multiscale approach (an extension of a previous work [1]) to predict the mechanical behavior of nanostructured metals according to their GBCD composed of either high angle GBs (HAB) or low angle GBs (LAB). The quasicontinuum method (QC) is used to obtain the GB mechanical response at the nanoscale under simple shear (sliding part) and tensile load (opening part). These QC results are then used in a finite element code (direct numerical simulation-DNS) as GB constitutive models. This two-scale framework does not suffer from length scales limitations conventionally encountered when considering the two scales separately. [less ▲] Detailed reference viewed: 124 (30 ULg)Full discontinuous Galerkin formulation of shell in large deformations with fracture mechanic applications Becker, Gauthier ; Noels, Ludovic Different methods have been developed to model tearing prediction, as e.g., the combination between the cohesive principle and the finite element method. Unfortunately, this method has some well known ... [more ▼] Different methods have been developed to model tearing prediction, as e.g., the combination between the cohesive principle and the finite element method. Unfortunately, this method has some well known issues that can be fixed by recourse to discontinuous Galerkin formulation. Such a formulation allows to insert very easily an extrinsic cohesive element at onset of fracture without any mesh modification. This promising technique has been recently developed by the authors for linear shell. Although promising numerical results were obtained, it is difficult to compare the method with experiments due to the large plastic deformation present in material before the fracture apparition. Thus, the method is extent herein to elasto-plastic finite deformations. The simulations of some benchmarks prove the ability of this new framework to model accurately the continuum part of the deformation and the crack propagation. [less ▲] Detailed reference viewed: 78 (29 ULg)Vectorial Incremental Nonconservative Consistent Hysteresis model François-Lavet, Vincent ; ; Stainier, Laurent et al in Hogge, Michel; Van Keer, Roger; Malengier, Benny (Eds.) et al Proceedings of the 5th International Conference on Advanded COmputational Methods in Engineering (ACOMEN2011) (2011, November) This paper proposes a macroscopic model for ferromagnetic hysteresis that is well-suited for finite element implementation. The model is readily vectorial and relies on a consistent thermodynamic ... [more ▼] This paper proposes a macroscopic model for ferromagnetic hysteresis that is well-suited for finite element implementation. The model is readily vectorial and relies on a consistent thermodynamic formulation. In particular, the stored magnetic energy and the dissipated energy are known at all times, and not solely after the completion of closed hysteresis loops as is usually the case. The obtained incremental formulation is variationally consistent, i.e., all internal variables follow from the minimization of a thermodynamic potential. [less ▲] Detailed reference viewed: 98 (14 ULg)Proceedings of the 5th International Conference on Advanded COmputational Methods in Engineering (ACOMEN2011) Hogge, Michel ; ; et al Book published by Université de Liège - Dépôt légal: D/2011/0480/31 (2011) Detailed reference viewed: 53 (7 ULg)Imposing periodic boundary condition on arbitrary meshes by polynomial interpolation Nguyen, Van Dung ; Béchet, Eric ; Geuzaine, Christophe et al In order to predict the effective properties of heterogeneous materials using the finite element approach, a boundary value problem (BVP) may be defined on a representative volume element (RVE) with ... [more ▼] In order to predict the effective properties of heterogeneous materials using the finite element approach, a boundary value problem (BVP) may be defined on a representative volume element (RVE) with appropriate boundary conditions, among which periodic boundary condition is the most efficient in terms of convergence rate. The classical method to impose the periodic boundary condition requires identical meshes on opposite RVE boundaries. This condition is not always easy to satisfy for arbitrary meshes. This work develops a new method based on polynomial interpolation that avoids the need of the identical mesh condition on opposite RVE boundaries. [less ▲] Detailed reference viewed: 129 (62 ULg)A one Field Full Discontinuous Galerkin Method for Kirchhoff-Love Shells Applied to Fracture Mechanics Becker, Gauthier ; Geuzaine, Christophe ; Noels, Ludovic in Computer Methods in Applied Mechanics & Engineering (2011), 200(45-46), 3223-3241 In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the Finite Element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive ... [more ▼] In order to model fracture, the cohesive zone method can be coupled in a very efficient way with the Finite Element method. Nevertheless, there are some drawbacks with the classical insertion of cohesive elements. It is well known that, on one the hand, if these elements are present before fracture there is a modification of the structure stiffness, and that, on the other hand, their insertion during the simulation requires very complex implementation, especially with parallel codes. These drawbacks can be avoided by combining the cohesive method with the use of a discontinuous Galerkin formulation. In such a formulation, all the elements are discontinuous and the continuity is weakly ensured in a stable and consistent way by inserting extra terms on the boundary of elements. The recourse to interface elements allows to substitute them by cohesive elements at the onset of fracture. The purpose of this paper is to develop this formulation for Kirchhoff-Love plates and shells. It is achieved by the establishment of a full DG formulation of shell combined with a cohesive model, which is adapted to the special thickness discretization of shell formulation. In fact, this cohesive model is applied on resulting reduced stresses which are the basis of thin structures formulations. Finally, numerical examples demonstrate the efficiency of the method. [less ▲] Detailed reference viewed: 181 (66 ULg)Projects in Fracture Simulations Noels, Ludovic ; ; Wu, Ling et al Scientific conference (2011, September) Detailed reference viewed: 12 (7 ULg)A Micro-Macroapproach to Predict Stiction due to Surface Contact in Microelectromechanical Systems Wu, Ling ; Noels, Ludovic ; et al in IEEE/ASME Journal of Microelectromechanical Systems (2011), 20(4), 976-990 Stiction, which results from contact between surfaces, is a major failure mode in micro electro-mechanical systems (MEMS). Indeed microscopic structures tend to adhere to each other when their surfaces ... [more ▼] Stiction, which results from contact between surfaces, is a major failure mode in micro electro-mechanical systems (MEMS). Indeed microscopic structures tend to adhere to each other when their surfaces enter into contact and when the restoring forces are unable to overcome the interfacial forces. Since incidental contacts cannot be completely excluded and since contacts between moving parts can be part of the normal operation of some types of MEMS, stiction prediction is an important consideration when designing micro and nano-devices. In this paper a micro-macro multi-scale approach is developed in order to predict possible stiction. At the lower scale, the unloading adhesive contact-distance curves of two interacting rough surfaces are established based on a previously presented model [L. Wu et al., J. Appl. Phys. 106, 113502, 2009]. In this model, dry conditions are assumed and only the van der Waals forces as adhesion source are accounted for. The resulting unloading adhesive contact-distance curves are dependant on the material and on surface properties, such as, elastic modulus, surface energy and on the rough surfaces topography parameters; the standard deviation of asperity heights and the asperities density. At the higher scale, a finite element analysis is considered to determine the residual cantilever beam configuration due to the adhesive forces once contact happened. Toward this end, the adhesive contact-distance curve computed previously is integrated on the surface of the finite elements as a contact law. Effects of design parameters can then be studied for given material and surface properties. [less ▲] Detailed reference viewed: 210 (94 ULg)Multi‐scale modelling of fibre reinforced composite with non‐local damage variable Wu, Ling ; Noels, Ludovic ; et al Conference (2011, July) Classical finite element simulations face the problems of losing uniqueness and strain localization when the strain softening of materials is involved. Thus, when using continuum damage model or ... [more ▼] Classical finite element simulations face the problems of losing uniqueness and strain localization when the strain softening of materials is involved. Thus, when using continuum damage model or plasticity softening model, numerical convergence will not be obtained with the refinement of the finite element discretization when strain localization occurs. Gradient-enhanced softening and non-local continua models have been proposed by several researchers in order to solve this problem. In such approaches, the spatial gradients of state variables are incorporated in the macroscopic constitutive equation [1, 2]. However, when dealing with complex heterogeneous materials, a direct simulation of the macroscopic structures is unreachable, motivating the development of non-local homogenization schemes [3]. In our work, a gradient-enhanced homogenization procedure is proposed for fiber reinforced materials. In the approach, the fiber is assumed to remain linear elastic while the matrix material is modeled as elasto-plastic [4] coupled with damage and is described by a non-local constitutive model [5]. Toward this end, the mean-field homogenization is based on the knowledge of the macroscopic deformation tensors, internal variables and their gradients, which are applied to a micro-structural representative volume element (RVE). Macro-stress is then obtained from a homogenization process. This procedure is applied to simulate damage process occurring in unidirectional carbon-fiber reinforced epoxy composites submitted to different loading histories. [less ▲] Detailed reference viewed: 58 (4 ULg) |
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