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The Representative Volume Element (RVE) plays a central role in the homogenization of random heterogeneous microstructures, especially for composite and porous materials, with a view to predicting their effective properties. A quantitative evaluation of its size is proposed in this work in linear elasticity and linear thermal conductivity of random heterogeneous materials. A RVE can be associated with different physical and statistical properties of microstructures. The methodology is applied to specific two–phase microstructure–based random sets. Statistical parameters are introduced to study the variation in the RVE size versus volume fractions of components and the contrast in their properties. The key notion of the integral range is introduced to determine these variations. For a given desired precision, we can provide a minimal volume size for the computation of effective mechanical and thermal properties. Numerical simulations are performed to demonstrate that a volume exists which is statistically representative of random microstructures. This finding is an important component for homogenization–based multiscale modeling of materials.

In this paper, a formula for estimating the effective thermal conductivity of lotus-type porous materials (LTPM) combining an analytical formula with a mean-field homogenisation technique is developed and validated. LTPMs are considered multi-phase materials in which each phase is defined by the pore shape. To estimate the effective thermal conductivities of multi-phase LTPMs, a two-step mean-field homogenisation method is implemented and validated. The validation is applied by comparing the results obtained with those of the representative volume element based finite element homogenisation method, which has been taken as a reference. The proposed formula is applied by replacing the first step of the two-step mean field homogenisation method using a formula for estimating the effective thermal conductivity of two-phase lotus-type materials. The good agreement between the results of the proposed formula and the reference results indicates that the proposed formula ensures an accurate estimation of the effective macroscopic thermal conductivities of multi-phase lotus-type porous materials.

This paper deals with the evaluation of several homogenization methods for short fiber-reinforced composites (SFRC) in isothermal linear elasticity. In order to have an unbiased assessment, only full-field finite element (FE) versions of the methods are developed and studied, no analytical mean-field models are considered. Firstly, the FE homogenization of unidirectional (UD) SFRC is performed using two computational models: representative volume elements (RVE) and unit cells, and comparing their predictions. Secondly, for the homogenization of misaligned SFRC, a model RVE seen as an aggregate of UD pseudo-grains (PG) is homogenized in two steps. In the first step, the effective response of PGs is obtained by FE analysis on UD unit cells. The second step is performed with one of three homogenization schemes: Voigt, Reuss and a FE-based version of the Mori-Tanaka (MT) model. The latter is equivalent to a direct MT homogenization of the RVE. The actual SFRC microstructure is 3D, but because of limited computer resources, the current article is restricted to 2D analyses, which nevertheless provide some guidance. A parametric study is conducted using several fiber volume fractions and orientation tensor components. Both the effective properties of RVEs and the mean strains inside individual fibers are assessed against reference results obtained by direct FE homogenization of the actual RVEs.

The purpose of this study was to establish a method based on an iterative scheme to approximate the numerical solution obtained from finite elements analysis for an RVE in two and three dimensions based on the homogenization concept for the assessment of the effective properties. The bounds of Hashin–Shtrikman and Voigt–Reuss were considered in the iterative process based on an updating of the constitutive relations of these models respectively. In this study, by assumption, we took the particular case of the heterogeneous materials with several elastic isotopic phases. The output variables considered using the iterative process are the bulk, shear modulus and the thermal conductivity. We have found a fast convergence of the iterative solution to the numerical result with a suitable concordance between the two solutions at the final step.

The present paper concerns a computational study of a three-dimensional (3D) unit cells with identical spherical voids in a von Mises matrix. The objective is to estimate the effective plastic flow surface of 3D microstructures. The work originality is to deal with identical spherical overlapping voids, covering a wide range of stress triaxiality ratios. The effective plastic flow surface is computed for nine distinct loadings on four distinct microstructures. The result indicates that the classical Gurson–Tvergaard–Needleman (GTN) model obtained using the Fritzen et al. [40] parameters, matches with our numerical simulations. Keywords: Porous media, Random overlapping spherical voids, Computational homogenization, Macroscopic yield surface, Representativeness.

International audience; In the context of polycrystalline metals, the damage process analysis is generally restricted to surface of a sample containing some hundreds of grains, due to the inherent difficulties microstructural analysis. Determination of the grain size influence, misorientation and neighboring grains effect remains difficult with experimental studies. In this work, the influence of microstructural characteristics of polycrystalline metals on the evolution of the damage process under cyclic loading is investigated. On the basis of the quaternion theory, the orientation of each (crystal) grain is represented by a single angle (theta). The relative misorientation of each grain is set by the ratio of its misorientation over the average value of polycrystalline aggregate. The relative volume of grain is used as a parameter representing the grain size. The damage evolution under cyclic loading is identified using the coupling of the crystal plasticity model with the Continuum Damage Mechanics (CDM) model. The damage evolution and its distribution on polycrystalline aggregate are calculated by means of numerical homogenization over each grain. The heterogeneity distribution and damage evolution for polycrystalline metals have been analyzed. The obtained results show that neighboring grains effects are larger and may change the tendency that larger grains with great misorientation are the most damaged.

International audience; The main goal of this paper is to predict the elastic modulus of partially intercalated and exfoliated polymer-clay nano-composites using numerical homogenization techniques based on the finite element method. The representative volume element was employed here to capture nano-composites microstructure, where both intercalated exfoliated and clay platelets coexisted together. The effective macroscopic properties of the studied microstructure are obtained with two boundary conditions: periodic boundary conditions and kinematic uniform boundary conditions. The effect of particle volume fractions, aspect ratio, number and distribution of particles and the type of boundary conditions are numerically studied for different configurations. This paper investigate also the performance of several classical analytical models as Mori and Tanaka model, Halpin and Tsai model, generalized self consistent model through their ability to estimate the mechanical properties of nano-composites. A comparison between simulation results of polypropylene clay nano-composites, analytical methods and experimental data has confirmed the validity of the set results.

This paper aims at developing 3D numerical simulations of the mechanical properties of polymer composites based on the aggregates of argan shell. The microstructure is made up of two phases, corresponding to the Poly–Propylene PP matrix and the natural aggregates. Due to the grinding operation of agran shell during the manufacturing process, the aggregates shapes, crushed aggregates, greatly vary with their surfaces. The majority of published works used the spherical or ellipsoidal forms in order to study numerically the mechanical behavior of such composites, because of the inherent simplification in algorithm formulations. However, real microstructures of these composites based simple microscopic, show that the shape of aggregates does not takes a spherical form, but a polyhedral form. In this study, the microstructure is described and modeled as a combination of Poisson polyhedral and PP matrix. In the numerical model, the obtained aggregate morphology is similar to that of many argan shell powders often used as reinforcement of the PP matrix. This technique allows generating models with different reinforcement volume fractions up to 28%. The numerical homogenization technique is used to estimate the mechanical properties of these composites. The set obtained numerical results with 3D polyhedral model are compared with the pertinent experimental data and analytical models reported in the literature. The performance of the most used micromechanical schemes used in predicting the effective elastic properties of biocomposites was discussed.

The aim of this work is to efficiently select samples of non-overlapping parallel fiber reinforced composites with regard to their elasticity and their fiber distribution in the composite cross-section. The samples were built with the help of the simulated annealing technique according to chosen Radial Distribution Functions. For each sample the fields of local stresses were simulated by finite element method, then homogenized by volume averaging in order to investigate their elastic properties. The effect of RDF shape on elastic properties was quantified. The more the fiber distributions deviate from Poisson’s Law the higher the effective elastic moduli are. A method to select samples of real fiber reinforced composites according to their elasticity is proposed.

In this study, we have explored the numerical modelling of the thermoforming of high-density polyethylene wood particle composites (WPC) into a formwork intended for construction applications. By using the parameters identified from oscillatory shear test, the Lodge model is used to describe the viscoelastic behavior of the WPC sheet. The forming stage of the WPC sheet is performed under the action of gas flow governed by the van der Waals equation of state. The thermoformability is firstly determined by simulating a free blowing of HDPE and highly charged WPC. Then, the numerical simulation of the formwork thermoforming is carried out on four types of WPC. The results have shown that the structure is subjected to increasing stress as the wood particle content increases, which consequently increases the risk of failure. Nevertheless, thermoforming is possible for WPC sheets with less than 40 % wood content. Simulations of the formwork under loadings have also shown good mechanical resistance at low thicknesses.

International audience; Wood polymer composites (WPC) have well proven their applicability in several fields of the plasturgy sector, due to their aesthetics and low maintenance costs. However, for plasturgy applications, the characterization of viscoelastic behavior and thermomechanical and thermophysical properties of WPC with the temperature and wood filler contents is essential. Therefore, the processability of polymer composites made up with different percentage of wood particles needs a better understanding of materials behaviors in accordance with temperature and wood particles contents. To this end, a numerical analysis of the viscoelastic, mechanical, and thermophysical properties of composite composed of high density polyethylene (HDPE) reinforced with soft wood particles is evaluated.

The determination of the size of the Representative Volume Element “RVE” is extremely important to predict mechanical and physical effective properties of heterogeneous materials. In the present work, the optimal size of RVE for polycrystal materials, which is made–up of grains having a polyhedron Voronoi shape, is investigated for the prediction of an elastoplastic behavior using the crystal plasticity model. Ten realizations with a different number of grains ranging from 10 to 250 have been used in the numerical simulation. The homogenization technique combined with the Finite Element Method (FEM) is used to obtain the macroscopic strain and stress. Several new criteria are introduced as indicators for estimating the optimal RVE's size. Their formulation is based on the standard deviation with respect to the average value of different mechanical quantities. These criteria ensure that, for an optimal RVE size, the response of the crystalline aggregate is not sensitive to the orientations evolution and grains' dimensions. It turns out that, for the cyclic behavior, the influence of the loading mode is very important, in particular the asymmetrical stress mode. The influence of the value of strain or stress used to perform different simulation tests on the RVE size has been analyzed. The simulation results are in good agreement with the experimental results from the literature.

This paper presents an efficient method to automatically generate and mesh random non–periodic three dimensional (3D) microstructures for three classes of complex heterogeneous media having a wide range of important engineering applications, porous media, composites with interfacial debonding and composites with high density particles. The resulting 3D microstructure is intentionally constructed to be easily and efficiently implemented in standard finite element computational codes. Several examples of 3D representative volume elements are shown. The performance of the proposal in finite element analysis is demonstrated in numerical implementation to predict the effective non–linear elastic–plastic response of two–phase particulate composites reinforced with spherical particles. The main result achieved is the estimation of the effective plastic tangent modulus by a simple linear regression equation for different volume fractions.

International audience; A numerical homogenization technique and morphological analysis based on the finite element method are used to compute mechanical properties of porous materials. This is achieved by considering two–dimensional matrix containing random distribution of identical non–overlapping circular or elliptical voids. Several microstructure configurations are obtained by varying the voids morphology and the porosity of the matrix. The notion of the representative volume element is used for numerical simulations in order to estimate the morphology effects of the voids on the effective ultimate tensile strength of the called Lotus–Type Porous Metals. A confrontation of the obtained numerical results of the representative microstructures for different morphologies of voids and different porosities to an analytical model and experimental data is performed. Finally, a formula improving the Boccaccini model is proposed to estimate effective tensile strength of porous metals taking into account the voids morphology.

International audience; It is well known that in high cycle fatigue (HCF), macroscopically, structures undergo elastic shakedown and the stress level commonly determines the lifetime. In this domain, the fatigue phenomena is due to local plasticity at the grain scale. Therefore, some multiscale HCF multiaxial fatigue criteria were proposed, among them the well-known Dang Van criterion. This criterion supposes that in a polycrystal, some misoriented grains can undergo plastic shakedown which conducts to crack initiation. The objective of this work is to validate this assumption by conducting numerical simulations on polycrystalline aggregates. As it is necessary to estimate the stabilized state in each grain of the polycrystal, classical incremental simulations are not the best way as it will be highly time-consuming because of the size of the aggregate. In the recent years, Pommier proposed a method called Direct Cyclic Algorithm to obtain the stabilized response of a structure under cyclic periodic loading, which it is shown to be more efficient compared to an incremental analysis in such situation. However, errors can be obtained in certain case with respect to the incremental solution. In this work, a Crystal Plasticity FEM model, based on dislocation densities, was used. As a first step, an aggregate of 20 grains of AISI 316L stainless steel under strain controlled cyclic loading was studied. Precise comparisons were conducted with incremental analysis and the results show that DCA seems to be an efficient solution in order to estimate the shakedown state of polycrystalline aggregates.

International audience; The microstructure studied in this paper is a Composite made up of Randomly oriented Short Fibers (RSFC) and a computational homogenization is performed to investigate its effective properties. The area fraction is also varied to study its effect on the size of the Deterministic Representative Volume Element (DRVE). It appeared that at certain area fractions, the RSFC does not respect the convergence of the apparent properties calculated under different boundary conditions. This indicates that it does not adhere to the definition of the DRVE in the studied range of scale. The concerned area fraction is found to be around the percolation threshold. The present work consists primarily in investigating the causes of this problem by studying an extreme example of a percolating medium which is a Composite made up of Randomly oriented Long Fibers (RLFC). By identifying the contributing factors, the calculability of the DRVE of a random composite can be predicted by simple verification of the microstructure morphology.

During the ballistic impact process of fabrics, yarns are locally subjected to both longitudinal tensile and transverse compressive loading. This paper presents a microscopic numerical investigation of the transverse compressive behavior of a single and bulk aramid fibers. Interactions with friction between single fibers in a yarn have been taken into account. Numerical calculations were performed in the case of a transverse compression using the assumption of plane strain and validated by experimental data. Effects of friction and representative volume size were also discussed. It can be highlighted that using a numerical homogenization technique, the variation of the transverse mechanical effective property versus fiber volume fraction has been determined based on the microscopic transverse behavior of bulk aramid fibers. Finally, power relationships describing effective transverse Young's modulus E ¯ and apparent strain e ¯ versus the normalized fiber volume fraction have been proposed and discussed. This effective homogenized material allows the development of homogenized Kevlar yarns and fabrics in the further works of homogenized approaches.

The focus of this paper is on aligned fiber-reinforced composites, where fiber centers were randomly distributed in their cross-sections. The volume fractions of fibers were [Formula: see text]% and [Formula: see text]%. Samples were built with the help of the simulated annealing technique according to the chosen Radial Distribution Functions (RDFs). For each sample, the fields of local stresses and heat fluxes were simulated by finite element method. Then, homogenization by volume averaging was performed in order to investigate both the effective mechanical and thermal properties. The effect of RDF shape on elastic and thermal properties was quantified along with the influence of the probability of near neighbors of fibers on the physical properties. The more the fiber distributions deviate from Poisson’s Law, the higher the results compared to the lower bound of Hashin–Shtrikman.

International audience; The purpose of this work was to study the influence of microstructure on effective transverse elastic behavior of fiber reinforced composites. Two microstructures were taken into account, hexagonal periodic and random arrangements of fibers. Unlike classical results at low fiber volume fractions and low Young’s modulus contrast between fibers and matrices, results provided by finite elements simulations have shown that microstructure strongly affect the effective properties of composite for both high volume fractions and Young’s modulus contrast. Results were compared to most common analytical models for composites elasticity.

International audience; Homogenization of effective elastic properties of fiber reinforced composites frequently supposed periodic arrangement of fibers because such microstructures allow analytical or low computational cost integrations. Hexagonal frame was often preferred than square one which is strongly anisotropic. In practical situations those periodical microstructures are not realistic. Real microstructures are often random or if they are periodic their boundaries don't fit with the periodic scheme. We studied with the help of finite elements samples that exhibit hexagonal arrangement of fibers embedded in a random distribution. Characteristic length scales of hexagonal area were extracted from observation of stress maps. Principal results are a short scale in which bulk and shear stresses become structured. On the other hand we nether reached a size large enough to observe local stress maps similar to those produced by a periodic model.

International audience; Uniformly stretched thin plates do not buckle unless they are in special boundary conditions. However, buckling commonly occurs around discontinuities, such as cracks, cuts, narrow slits, holes, and different openings, of such plates. This study aims to show that buckling can also occur in thin plates that contain no defect or singularity when the stretching is local. This specific stability problem is analyzed with the finite element method. A brief literature review on stretched plates is presented. Linear and nonlinear buckling stress analyses are conducted for a partially stretched rectangular plate, and various load cases are considered to investigate the influence of the partial loading expanse on the critical tensile buckling load. Results are summarized in iso-stress areas, tables and graphs. Local stretching on one end of the plate induces buckling in the thin plate even without geometrical imperfection.

International audience; This paper focuses on validating a numerical modelling of woven materials subjected to ballistic impact based on experimental data. Studied impact system is a 3D warp interlock fabric (4 layers of angle interlock – through the thickness) and FSP projectile (Fragment Simulating Projectile). Numerical prediction is compared with ballistic tests on data obtained from ultra-rapid camera in two typical impact cases: no perforation and perforation. Effect of deformation rate is investigated by using both mechanical properties of yarn in static and dynamic states for numerical modelling. The numerical model using dynamic parameters shows a better prediction of impact behaviour of the 3D woven fabric than the one using static parameters.

A numerical model of ballistic impact on a two-dimensional Kevlar KM2® plain-woven fabric has been validated by experiment. This paper shows that it is necessary to experimentally measure material constants of yarns for having good input parameters of the model. Effects of yarn Poisson’s ratio, transverse and shear modulus on impact behaviors of a simple crimped yarn and a complete fabric have been carried out. The effect of the Poisson’s ratio can be negligible in both impact cases: on a single crimped yarn and a complete fabric. The same conclusion has been proven for the effect of the transversal modulus except the cases of its so low values that can cause yarn early damage. The shear modulus of a yarn appears to be an important material parameter that mainly influences the ballistic performance of a two-dimensional plain-woven fabric. When using a very high value of a shear modulus of yarn, a crimped single yarn is broken immediately after contact with projectile in pure shearing mode.

International audience; 3D interlock woven fabrics are promising materials to replace the 2D structures in the field of ballistic protection. The structural complexity of this material caused many difficulties in numerical modeling. This paper presents a new tool that permits to generate a geometry model of any woven fabric, then, mesh this model in shell or solid elements, and apply the mechanical properties of yarns to them. The tool shows many advantages over existing software. It is very handy in use with an organization of the functions in menu and using a graphic interface. It can describe correctly the geometry of all textile woven fabrics. With this tool, the orientation of the local axes of finite elements following the yarn direction facilitates defining the yarn mechanical properties in a numerical model. This tool can be largely applied because it is compatible with popular finite element codes such as Abaqus, Ansys, Radioss etc. Thanks to this tool, a finite element model was carried out to describe a ballistic impact on a 3D warp interlock Kevlar KM2® fabric. This work focuses on studying the effect of friction onto the ballistic impact behavior of this textile interlock structure. Results showed that the friction among yarns affects considerably on the impact behavior of this fabric. The effect of the friction between projectile and yarn is less important. The friction plays an important role in keeping the fabric structural stability during the impact event. This phenomenon explained why the projectile is easier to penetrate this 3D warp interlock fabric in the no-friction case. This result also indicates that the ballistic performance of the interlock woven fabrics can be improved by using fibers with great friction coefficients.

International audience; In this paper, a numerical homogenization technique is used to estimate the effective thermal conductivity of random two-dimensional two-phase heterogeneous materials. The thermal computational leads essentially bring out the effect of the voids/inclusions morphology on the effective physical properties. This is achieved using two different heterogeneous materials: microstructure 1 with non-overlapping spherical pores and microstructure 2 with non overlapping spherical rigid inclusions taking into account five different volume fractions from each case. The notion of the representative volume element is introduced for numerical simulations using periodic boundary conditions and uniform gradient of temperature conditions. The obtained effective material properties on the representative microstructures are compared with different analytical models as: series model, parallel model, effective medium theory and Maxwell models, for different morphologies of rigid inclusions and voids. This paper compares the performance of several classical effective medium approximations. Finally, an analytical expression developing the Maxwell model is proposed to estimate the effective thermal conductivity of heterogeneous materials taking into account the inclusion morphology.

International audience; A computational homogenization study of three-dimensional cubic cells is presented to estimate the overall yield surface of random porous media covering a wide range of stress triaxiality ratios. The representativity of the overall yield surface estimates is examined using cubic cells containing randomly distributed and oriented voids with different volume fractions and shapes (spherical and oblate/prolate). The computational results are compared with some existing Gurson-type yield criteria. The extension of the Gurson-Tvergaard model recently proposed by Fritzen et al. (2012) for spherical cavities is shown in good agreement with the computational results for all investigated porosities.

International audience; In this paper, damage mechanisms of a 3D interlock woven fabric subjected to ballistic impact were analyzed using a numerical model. Two impact configurations were carried out in order to validate the numerical model with experimental observations: perforation (900 m/s) and no-perforation (90 m/s). Global deformation of the fabric during impact is determined continuously to detail fabric impact behavior. Also, in this study, the effects of boundary conditions on failuremechanisms have been investigated. Boundary conditions are divided into two cases: (1) only warp yarns fixed and (2) only weft yarns fixed. Basing on continuous evolutions of global deformation, projectile velocity, different energies and reaction force onto projectile, the influence of both these fixation conditions is investigated.

International audience; In this study, a FEM analysis has been carried out to find out pertinent multi scale model for an investigation of a ballistic impact on 2D KM2® plain woven fabrics. Multi scale models are a combination between macroscopic and mesoscopic models. This study aims at testing a multi scalec model in order to minimize the computing time. Three configurations were analyzed by varying the ratio of macroscopic and mesoscopic areas: 75.3-24.7%, 65.5-34.5%, 56.3-43.7% with two impact velocities 60 m/s and 245 m/s. In these multi scale models, the continuity in macroscopic-mesoscopic interfaces is ensured by checking the evolution of global displacements of the fabric during impact. The effect of the macroscopic area of multi scale models on the ballistic performance of the fabric is also investigated. The optimal multi scale model was validated by comparison with results obtained from a mesoscopic model in terms of the evolutions of the projectile velocity, energy forms, the overall behavior of the fabric during impact and the force applied on the projectile. The failure criterion Forming Limited Diagram (FLD) is suggested for bundle failure. The observed damage mechanisms of the fabric during penetration time of the projectile are discussed and compared among numerical models.