vincent monchiet | Université Paris-Est Marne-la-Vallée (original) (raw)

Papers by vincent monchiet

Research paper thumbnail of Numerical Implementation of Higher-Order Homogenization Problems and Computation of Gradient Elasticity Coefficients

Volume 3: Advanced Composite Materials and Processing; Robotics; Information Management and PLM; Design Engineering, 2012

ABSTRACT A micromechanics-based approach for the derivation of the effective properties of period... more ABSTRACT A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of tridimensional elasticity, but, since the assumption of strict separation of scale is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which, higher-order terms (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. All these terms are then obtained by solving a hierarchy of higher-order elasticity problems with prescribed body forces and eigen-strains whose expression depends on the solution at the lower-order. An energy based micro-macro transition is then proposed for the change of scale and constitutes, in fact, a generalization of the Hill-Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. It is rigorously proved that the macroscopic quantities derived from this homogenization procedure comply with the equations of strain gradient elasticity. As an illustration, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed whose efficiency is shown in the particular case of composites reinforced by long fibers.

Research paper thumbnail of A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media

International Journal of Solids and Structures, 2012

ABSTRACT A micromechanics-based approach for the derivation of the effective properties of period... more ABSTRACT A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of three-dimensional elasticity, but, since the assumption of strict separation of scales is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which higher-order terms, (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. An energy based micro–macro transition is then proposed for upscaling and constitutes, in fact, a generalization of the Hill–Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. As an illustration purpose, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed in the last part of the paper. Its efficiency is shown in the particular case of composites reinforced by long fibers.

Research paper thumbnail of Microstructural effects in non linear stratified composites

International Journal of Solids and Structures, 2014

ABSTRACT In this paper, we analyze the microstructural effects on non linear elastic and periodic... more ABSTRACT In this paper, we analyze the microstructural effects on non linear elastic and periodic composites within the framework of asymptotic homogenization. We assume that the constitutive laws of the individual constituents derive from strain potentials. The microstructural effects are incorporated by considering the higher order terms, which come from the asymptotic series expansion. The complete solution at any order requires the resolution of a chain of cell problems in which the source terms depend on the solution at the lower order. The influence of these terms on the macroscopic response of the non linear composite is evaluated in the particular case of a stratified microstructure. The analytic solutions of the cell problems at the first and second order are provided for arbitrary local strain–stress laws which derive from potentials. As classically, the non-linear dependence on the applied macroscopic strain is retrieved for the solution at the first order. It is proved that the second order term in the expansion series also exhibits a non linear dependence with the macroscopic strain but linearly depends on the gradient of macroscopic strain. As a consequence, the macroscopic potential obtained by homogenization is a quadratic function of the macroscopic strain gradient when the expansion series is truncated at the second order. This model generalizes the well known first strain gradient elasticity theory to the case of non linear elastic material. The influence of the non local correctors on the macroscopic potential is investigated in the case of power law elasticity under macroscopic plane strain or antiplane conditions.

Research paper thumbnail of Numerical Implementation of Higher-Order Homogenization Problems and Computation of Gradient Elasticity Coefficients

A micromechanics-based approach for the derivation of the effective properties of periodic linear... more A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of tridimensional elasticity, but, since the assumption of strict separation of scale is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which, higher-order terms (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. All these terms are then obtained by solving a hierarchy of higher-order elasticity problems with prescribed body forces and eigen-strains whose expression depends on the solution at the lower-order. An energy based micro-macro transition is then proposed for the change of scale and constitutes, in fact, a ...

Research paper thumbnail of Low frequency locally resonant metamaterials containing composite inclusions

The Journal of the Acoustical Society of America, 2015

One main feature of metamaterials is the occurrence of a negative dynamic mass density that is pr... more One main feature of metamaterials is the occurrence of a negative dynamic mass density that is produced when an inner local resonance is present. The inner resonance can be obtained in composite materials containing composite inclusions. For suitable ratios of the physical properties of the constituting materials, the composite inclusions act as spring-mass systems. The scaling of physical properties leading to such an inner resonance and the associated effective dynamic properties of materials containing composite inclusions are briefly recalled. The resonance frequencies and dynamic mass densities are obtained in a closed form for materials containing cylindrical composite fibers or spherical composite inclusions, after solving the related boundary value elasticity problems.

Research paper thumbnail of A Micromechanical Model for Crack Initiation in High Cycle Fatigue of Metallic Materials

Fracture of Nano and Engineering Materials and Structures, 2006

Most of structural components resisting to high cycle fatigue are subjected to a multiaxial state... more Most of structural components resisting to high cycle fatigue are subjected to a multiaxial state of stress. Since fatigue cracks generally initiate and propagate in a plane of maximal shear stress (stage I), the first approaches of Crossland and Sines considered the octahedral plane and their criteria are based on the amplitude of the second invariant of the deviatoric stress

Research paper thumbnail of Un modèle micromécanique pour la nucléation de fissures en fatigue polycyclique

Mécanique & Industries, 2007

ABSTRACT Une approche multiéchelle de la fatigue multiaxiale en endurance illimitée est proposée.... more ABSTRACT Une approche multiéchelle de la fatigue multiaxiale en endurance illimitée est proposée. Elle vise à rendre compte des mécanismes couplés de plasticité et d'endommagement qui surviennent le long des Bandes de Glissement Persistantes (BGP). L'endommagement, qui est couplé avec la plasticité, est modélisé comme la conséquence de la croissance de microcavités le long des BGP. Le critère de fatigue macroscopique proposé correspond à une condition de nucléation d'une fissure à l'interface BGP/matrice, et est obtenu pour le schéma d'homogénéisation de Kröner. On montre le rôle de la contrainte moyenne et de la pression hydrostatique en fatigue à grand nombre de cycles. Cette dépendance est ici liée à la prise en compte des micromécanismes d'endommagement. Enfin, on présente quelques illustrations dans le cas particulier des trajets de chargement affines. Des comparaisons du critère obtenu avec des résultats expérimentaux montrent la pertinence de l'approche

Research paper thumbnail of A HOMOGENIZATION APPROACH FOR THE DETERMINATION OF THE EFFECTIVE PROPERTIES OF GRADIENT ELASTIC MEDIA

The aim of this paper is to provide the macroscopic elastic properties of a gradient elastic medi... more The aim of this paper is to provide the macroscopic elastic properties of a gradient elastic media from a homogenization framework. To reach this objective, the clas-sical conditions at the boundary of the representative volume element (RVE) are replaced by nonlinear boundary conditions. The macroscopic measures associated to polynomial boundary conditions are obtained along the lines of Rodin (2007). The RVE is constituted of an elastic matrix containing voids randomly distributed within the RVE. The elastic matrix comply with Cauchy equations at the microscopic scale. Nonlinear boundary con-ditions are taken under the form of a polynomial function which depends explicitly of the strain gradient or the hyperstress. The generalization of the homogenization approach in this context consists in replacing the RVE by an equivalent gradient elastic medium at the macroscopic scale. The macroscopic model which is thus obtained takes into account the effect of the strain gradient and a char...

Research paper thumbnail of A polarization-based fast numerical method for computing the effective conductivity of composites

International Journal of Numerical Methods for Heat & Fluid Flow, 2013

ABSTRACT Purpose ‐ The paper deals with the development of an improved fast Fourier transform (FF... more ABSTRACT Purpose ‐ The paper deals with the development of an improved fast Fourier transform (FFT)-based numerical method for computing the effective properties of composite conductors. The convergence of the basic FFT-based methods is recognized to depend drastically on the contrast between the phases. For instance, the primal formulation is not suited for solving the problems with high conductivity whereas the dual formulation is computationally costly for problems with high resistivity. Consequently, it raises the problem of computing the properties of composites containing both highly conductive and resistive inclusions. Design/methodology/approach ‐ In the present work, the authors' propose a new iterative scheme for solving that kind of problems which is formulated in term of the polarization. Findings ‐ The capability and relevance of this iterative scheme is illustrated through numerical implementation in the case of composites containing squared inclusions. It is shown that the rate of convergence is increased and thus, particularly when the case of high contrasts is considered. The predominance of the polarization based iterative scheme (PBIS) over existing ones is also illustrated in the case of a composite containing both highly conductive and highly resistive inclusions. Originality/value ‐ The method is easy to implement and uses the same ingredients as the basic schemes: the FFT and the exact expression of the Green tensor in the Fourier space. Moreover, its convergence conditions do not depend on the conductivity properties of the constituents, which then constitutes the main difference with other existing iterative schemes. The method can then be applied for computing the effective properties of composites conductors with arbitrary contrasts.

Research paper thumbnail of Crack opening displacements under remote stress gradient: Derivation with a canonical basis of sixth order tensors

International Journal of Engineering Science, 2015

ABSTRACT In this paper, we derive the crack opening displacement of a penny-shaped crack embedded... more ABSTRACT In this paper, we derive the crack opening displacement of a penny-shaped crack embedded in an infinite isotropic elastic medium and subjected to a remote constant stress gradient. The solution is derived by taking advantage of the solution of the equivalent ellipsoidal inclusion problem subjected to a linear polarization. The case of the penny-shaped crack is thereafter investigated by considering the case of a spheroidal cavity which has one principal axis infinitesimally small compared to both others. The derivation of the explicit solution for the inhomogeneity subjected to a remote stress gradient raises the problem of the inversion of a sixth order tensor. For the problem having a symmetry axis (this including the case of penny shaped crack), this problem can be tackled by using a decomposition on the canonical basis for transversely isotropic sixth order tensors.

Research paper thumbnail of A Gurson-type model accounting for void size effects

International Journal of Solids and Structures, 2013

ABSTRACT In this paper we present an extension of the Gurson model of cavity growth which include... more ABSTRACT In this paper we present an extension of the Gurson model of cavity growth which includes the void size effect. To this end, we perform the limit analysis of a hollow sphere made up of a Fleck and Hutchinson’s strain gradient plasticity material. Based on the trial velocity field of Gurson, we derive an approximate closed form expression of the macroscopic criterion. The latter incorporates the void size dependency through a non dimensional parameter defined as the ratio of the cavity radius and the intrinsic length of the plastic solid. The accuracy of this approximate criterion is demonstrated by its comparison with numerical data. In the last part of the paper we present a complete plasticity model involving the damage rate and a power-law strain hardening of the matrix. It is shown that the cavity size effect has a strong dependency on damage growth as well as on the stress strain response.

Research paper thumbnail of Micromechanical Modelling of Fracture-Induced Anisotropy and Damage in Orthotropic Materials

Matrix cracking is commonly recognised as one of the main inelastic deformation mechanisms of Bri... more Matrix cracking is commonly recognised as one of the main inelastic deformation mechanisms of Brittle Matrix Composites. The modelling of such phenomenon still presents some difficulties which are mainly related to the description of the interaction between the initial anisotropy and the cracks-induced anisotropy. The present study concerns a new micro-macro approach of the non linear behavior and damage propagation

Research paper thumbnail of A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media

International Journal of Solids and Structures, 2012

ABSTRACT A micromechanics-based approach for the derivation of the effective properties of period... more ABSTRACT A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of three-dimensional elasticity, but, since the assumption of strict separation of scales is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which higher-order terms, (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. An energy based micro–macro transition is then proposed for upscaling and constitutes, in fact, a generalization of the Hill–Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. As an illustration purpose, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed in the last part of the paper. Its efficiency is shown in the particular case of composites reinforced by long fibers.

Research paper thumbnail of Microstructural effects in non linear stratified composites

International Journal of Solids and Structures, 2014

ABSTRACT In this paper, we analyze the microstructural effects on non linear elastic and periodic... more ABSTRACT In this paper, we analyze the microstructural effects on non linear elastic and periodic composites within the framework of asymptotic homogenization. We assume that the constitutive laws of the individual constituents derive from strain potentials. The microstructural effects are incorporated by considering the higher order terms, which come from the asymptotic series expansion. The complete solution at any order requires the resolution of a chain of cell problems in which the source terms depend on the solution at the lower order. The influence of these terms on the macroscopic response of the non linear composite is evaluated in the particular case of a stratified microstructure. The analytic solutions of the cell problems at the first and second order are provided for arbitrary local strain–stress laws which derive from potentials. As classically, the non-linear dependence on the applied macroscopic strain is retrieved for the solution at the first order. It is proved that the second order term in the expansion series also exhibits a non linear dependence with the macroscopic strain but linearly depends on the gradient of macroscopic strain. As a consequence, the macroscopic potential obtained by homogenization is a quadratic function of the macroscopic strain gradient when the expansion series is truncated at the second order. This model generalizes the well known first strain gradient elasticity theory to the case of non linear elastic material. The influence of the non local correctors on the macroscopic potential is investigated in the case of power law elasticity under macroscopic plane strain or antiplane conditions.

Research paper thumbnail of Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011

In this paper, the derivation of irreducible bases for a class of isotropic (2n) thorder tensors ... more In this paper, the derivation of irreducible bases for a class of isotropic (2n) thorder tensors having particular "minor symmetries" is presented. The methodology used for obtaining these bases consists in extending the concept of deviatoric and spherical parts, commonly used for 2 nd -order tensors, to the case of a n th -order tensor. It is shown that those bases are useful for effecting the classical tensorial operations and specially the inversion of a 2n th -order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

Research paper thumbnail of On the inversion of non symmetric sixth-order isotropic tensors and conditions of positiveness of third-order tensor valued quadratic functions

Mechanics Research Communications, 2011

In the present paper we propose new results concerning linear tensorial algebra for third-order a... more In the present paper we propose new results concerning linear tensorial algebra for third-order and non symmetric isotropic sixth-order tensors in the most general case (i.e. having not the major and minor symmetries). Such tensors are used, for instance, in the theory of microstructured elastic media. A formalism based on an irreducible basis for isotropic sixth-order tensors is introduced, which is useful for performing the classical tensorial operations. Specially, a condensed expression for the product between two isotropic sixth-order tensors is provided, which allows the obtaining of a condition on these tensors for being invertible and a closed form expression of the inverse of such a tensor. Finally, the condition of positiveness of third-order tensor-valued quadratic functions is derived. For instance, such conditions are required for computing the elastic energy of microstructured media.

Research paper thumbnail of Algebra of Transversely Isotropic Sixth Order Tensors and Solution to Higher Order Inhomogeneity Problems

Journal of Elasticity, 2013

ABSTRACT In this paper we provide a complete and irreducible representation for transversely isot... more ABSTRACT In this paper we provide a complete and irreducible representation for transversely isotropic sixth order tensors having minor symmetries. Such tensors appear in some practical problems of elasticity for which their inversion is required. For this kind of tensors, we provide an irreducible basis which possesses some remarkable properties, allowing us to provide a representation in a compact form which uses two scalars and three matrices of dimension 2, 3 and 4. It is shown that the calculation of sum, product and inverse of transversely isotropic sixth order tensors is greatly simplified by using this new formalism and appears to be appropriate for deriving new various solutions to some practical problems in mechanics which use such kinds of higher order tensors. For instance, we derive the fields within a cylindrical inhomogeneity submitted to remote gradient of strain. The method of resolution uses the Eshelby equivalent inclusion method extended to the case of a polynomial type eigenstrain. It is shown that the approach leads to a linear system involving a sixth order tensor whose closed form solution is derived by means of the tensorial formalism introduced in the first part of the paper.

Research paper thumbnail of Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids

International Journal of Plasticity, 2008

The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile p... more The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile porous solids is investigated. The Gologanu-Leblond-Devaux's (GLD) analysis of an rigid-ideal plastic (von Mises) spheroidal volume containing a confocal spheroidal cavity loaded axisymmetrically is extended to the case when the matrix is anisotropic (obeying Hill's (1948) anisotropic yield criterion) and the representative volume element (RVE) is subjected to arbitrary deformation. To derive the overall anisotropic yield criterion, a limit analysis approach is used. Conditions of homogeneous boundary strain rate are imposed on every ellipsoidal confocal with the cavity. A two-field trial velocity satisfying these boundary conditions are considered. It is shown that for cylindrical and spherical void geometries, the proposed criterion reduces to existing anisotropic Gurson-like yield criteria. Furthermore, it is shown that for the case when the matrix is considered isotropic, the new results provide a rigorous generalization to the GLD model. Finally, the accuracy of the proposed approximate yield criterion for plastic anisotropic media containing non-spherical voids is assessed through comparison with numerical results.

Research paper thumbnail of Exact solution of a plastic hollow sphere with a Mises–Schleicher matrix

International Journal of Engineering Science, 2012

In this paper we establish the exact solution for a hollow sphere with a rigid-plastic pressure-s... more In this paper we establish the exact solution for a hollow sphere with a rigid-plastic pressure-sensitive matrix and subjected to hydrostatic tension or compression. The matrix is assumed to obey to a parabolic Mises-Schleicher criterion. The closed-form expressions of the velocity field and of the stress field are provided. These exact solutions, expressed by means of the Lambert W function, allow to assess and discuss existing results.

Research paper thumbnail of Interfacial models in viscoplastic composites materials

International Journal of Engineering Science, 2010

The aim of the present work is to extend the concept of interphase and equivalent imperfect inter... more The aim of the present work is to extend the concept of interphase and equivalent imperfect interface in the context of viscoplasticity. The interphase takes the form of a thin curved layer of constant thickness, made up of a rigid viscoplastic material located between two other surrounding materials. We aim at representing this interphase by an interface with appropriately devised interface conditions. To reach this objective, a Taylor expansion of the relevant physical fields in the thin region is used. It is shown that, depending of the degree of stiffness of the layer with respect to the neighboring media, this interphase can be replaced by an idealized imperfect interface involving the jump of the velocity field or the traction vector. The first kind of interface model, applicable to soft interphases, is the "spring-type" interface across which the traction are continuous but the velocity field exhibits a discontinuity which is given in term of the traction by a power-law type relation. Moreover, it is shown that the constant of the model can be expressed in terms of the material parameters of the interphase. When the interphase is stiffer than the two surrounding media, one obtain a "stress-type" interface across which the velocity is continuous and a jump condition for the traction is given by a generalization of the so-called Young Laplace model to viscoplastic solids.

Research paper thumbnail of Numerical Implementation of Higher-Order Homogenization Problems and Computation of Gradient Elasticity Coefficients

Volume 3: Advanced Composite Materials and Processing; Robotics; Information Management and PLM; Design Engineering, 2012

ABSTRACT A micromechanics-based approach for the derivation of the effective properties of period... more ABSTRACT A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of tridimensional elasticity, but, since the assumption of strict separation of scale is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which, higher-order terms (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. All these terms are then obtained by solving a hierarchy of higher-order elasticity problems with prescribed body forces and eigen-strains whose expression depends on the solution at the lower-order. An energy based micro-macro transition is then proposed for the change of scale and constitutes, in fact, a generalization of the Hill-Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. It is rigorously proved that the macroscopic quantities derived from this homogenization procedure comply with the equations of strain gradient elasticity. As an illustration, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed whose efficiency is shown in the particular case of composites reinforced by long fibers.

Research paper thumbnail of A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media

International Journal of Solids and Structures, 2012

ABSTRACT A micromechanics-based approach for the derivation of the effective properties of period... more ABSTRACT A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of three-dimensional elasticity, but, since the assumption of strict separation of scales is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which higher-order terms, (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. An energy based micro–macro transition is then proposed for upscaling and constitutes, in fact, a generalization of the Hill–Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. As an illustration purpose, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed in the last part of the paper. Its efficiency is shown in the particular case of composites reinforced by long fibers.

Research paper thumbnail of Microstructural effects in non linear stratified composites

International Journal of Solids and Structures, 2014

ABSTRACT In this paper, we analyze the microstructural effects on non linear elastic and periodic... more ABSTRACT In this paper, we analyze the microstructural effects on non linear elastic and periodic composites within the framework of asymptotic homogenization. We assume that the constitutive laws of the individual constituents derive from strain potentials. The microstructural effects are incorporated by considering the higher order terms, which come from the asymptotic series expansion. The complete solution at any order requires the resolution of a chain of cell problems in which the source terms depend on the solution at the lower order. The influence of these terms on the macroscopic response of the non linear composite is evaluated in the particular case of a stratified microstructure. The analytic solutions of the cell problems at the first and second order are provided for arbitrary local strain–stress laws which derive from potentials. As classically, the non-linear dependence on the applied macroscopic strain is retrieved for the solution at the first order. It is proved that the second order term in the expansion series also exhibits a non linear dependence with the macroscopic strain but linearly depends on the gradient of macroscopic strain. As a consequence, the macroscopic potential obtained by homogenization is a quadratic function of the macroscopic strain gradient when the expansion series is truncated at the second order. This model generalizes the well known first strain gradient elasticity theory to the case of non linear elastic material. The influence of the non local correctors on the macroscopic potential is investigated in the case of power law elasticity under macroscopic plane strain or antiplane conditions.

Research paper thumbnail of Numerical Implementation of Higher-Order Homogenization Problems and Computation of Gradient Elasticity Coefficients

A micromechanics-based approach for the derivation of the effective properties of periodic linear... more A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of tridimensional elasticity, but, since the assumption of strict separation of scale is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which, higher-order terms (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. All these terms are then obtained by solving a hierarchy of higher-order elasticity problems with prescribed body forces and eigen-strains whose expression depends on the solution at the lower-order. An energy based micro-macro transition is then proposed for the change of scale and constitutes, in fact, a ...

Research paper thumbnail of Low frequency locally resonant metamaterials containing composite inclusions

The Journal of the Acoustical Society of America, 2015

One main feature of metamaterials is the occurrence of a negative dynamic mass density that is pr... more One main feature of metamaterials is the occurrence of a negative dynamic mass density that is produced when an inner local resonance is present. The inner resonance can be obtained in composite materials containing composite inclusions. For suitable ratios of the physical properties of the constituting materials, the composite inclusions act as spring-mass systems. The scaling of physical properties leading to such an inner resonance and the associated effective dynamic properties of materials containing composite inclusions are briefly recalled. The resonance frequencies and dynamic mass densities are obtained in a closed form for materials containing cylindrical composite fibers or spherical composite inclusions, after solving the related boundary value elasticity problems.

Research paper thumbnail of A Micromechanical Model for Crack Initiation in High Cycle Fatigue of Metallic Materials

Fracture of Nano and Engineering Materials and Structures, 2006

Most of structural components resisting to high cycle fatigue are subjected to a multiaxial state... more Most of structural components resisting to high cycle fatigue are subjected to a multiaxial state of stress. Since fatigue cracks generally initiate and propagate in a plane of maximal shear stress (stage I), the first approaches of Crossland and Sines considered the octahedral plane and their criteria are based on the amplitude of the second invariant of the deviatoric stress

Research paper thumbnail of Un modèle micromécanique pour la nucléation de fissures en fatigue polycyclique

Mécanique & Industries, 2007

ABSTRACT Une approche multiéchelle de la fatigue multiaxiale en endurance illimitée est proposée.... more ABSTRACT Une approche multiéchelle de la fatigue multiaxiale en endurance illimitée est proposée. Elle vise à rendre compte des mécanismes couplés de plasticité et d'endommagement qui surviennent le long des Bandes de Glissement Persistantes (BGP). L'endommagement, qui est couplé avec la plasticité, est modélisé comme la conséquence de la croissance de microcavités le long des BGP. Le critère de fatigue macroscopique proposé correspond à une condition de nucléation d'une fissure à l'interface BGP/matrice, et est obtenu pour le schéma d'homogénéisation de Kröner. On montre le rôle de la contrainte moyenne et de la pression hydrostatique en fatigue à grand nombre de cycles. Cette dépendance est ici liée à la prise en compte des micromécanismes d'endommagement. Enfin, on présente quelques illustrations dans le cas particulier des trajets de chargement affines. Des comparaisons du critère obtenu avec des résultats expérimentaux montrent la pertinence de l'approche

Research paper thumbnail of A HOMOGENIZATION APPROACH FOR THE DETERMINATION OF THE EFFECTIVE PROPERTIES OF GRADIENT ELASTIC MEDIA

The aim of this paper is to provide the macroscopic elastic properties of a gradient elastic medi... more The aim of this paper is to provide the macroscopic elastic properties of a gradient elastic media from a homogenization framework. To reach this objective, the clas-sical conditions at the boundary of the representative volume element (RVE) are replaced by nonlinear boundary conditions. The macroscopic measures associated to polynomial boundary conditions are obtained along the lines of Rodin (2007). The RVE is constituted of an elastic matrix containing voids randomly distributed within the RVE. The elastic matrix comply with Cauchy equations at the microscopic scale. Nonlinear boundary con-ditions are taken under the form of a polynomial function which depends explicitly of the strain gradient or the hyperstress. The generalization of the homogenization approach in this context consists in replacing the RVE by an equivalent gradient elastic medium at the macroscopic scale. The macroscopic model which is thus obtained takes into account the effect of the strain gradient and a char...

Research paper thumbnail of A polarization-based fast numerical method for computing the effective conductivity of composites

International Journal of Numerical Methods for Heat & Fluid Flow, 2013

ABSTRACT Purpose ‐ The paper deals with the development of an improved fast Fourier transform (FF... more ABSTRACT Purpose ‐ The paper deals with the development of an improved fast Fourier transform (FFT)-based numerical method for computing the effective properties of composite conductors. The convergence of the basic FFT-based methods is recognized to depend drastically on the contrast between the phases. For instance, the primal formulation is not suited for solving the problems with high conductivity whereas the dual formulation is computationally costly for problems with high resistivity. Consequently, it raises the problem of computing the properties of composites containing both highly conductive and resistive inclusions. Design/methodology/approach ‐ In the present work, the authors' propose a new iterative scheme for solving that kind of problems which is formulated in term of the polarization. Findings ‐ The capability and relevance of this iterative scheme is illustrated through numerical implementation in the case of composites containing squared inclusions. It is shown that the rate of convergence is increased and thus, particularly when the case of high contrasts is considered. The predominance of the polarization based iterative scheme (PBIS) over existing ones is also illustrated in the case of a composite containing both highly conductive and highly resistive inclusions. Originality/value ‐ The method is easy to implement and uses the same ingredients as the basic schemes: the FFT and the exact expression of the Green tensor in the Fourier space. Moreover, its convergence conditions do not depend on the conductivity properties of the constituents, which then constitutes the main difference with other existing iterative schemes. The method can then be applied for computing the effective properties of composites conductors with arbitrary contrasts.

Research paper thumbnail of Crack opening displacements under remote stress gradient: Derivation with a canonical basis of sixth order tensors

International Journal of Engineering Science, 2015

ABSTRACT In this paper, we derive the crack opening displacement of a penny-shaped crack embedded... more ABSTRACT In this paper, we derive the crack opening displacement of a penny-shaped crack embedded in an infinite isotropic elastic medium and subjected to a remote constant stress gradient. The solution is derived by taking advantage of the solution of the equivalent ellipsoidal inclusion problem subjected to a linear polarization. The case of the penny-shaped crack is thereafter investigated by considering the case of a spheroidal cavity which has one principal axis infinitesimally small compared to both others. The derivation of the explicit solution for the inhomogeneity subjected to a remote stress gradient raises the problem of the inversion of a sixth order tensor. For the problem having a symmetry axis (this including the case of penny shaped crack), this problem can be tackled by using a decomposition on the canonical basis for transversely isotropic sixth order tensors.

Research paper thumbnail of A Gurson-type model accounting for void size effects

International Journal of Solids and Structures, 2013

ABSTRACT In this paper we present an extension of the Gurson model of cavity growth which include... more ABSTRACT In this paper we present an extension of the Gurson model of cavity growth which includes the void size effect. To this end, we perform the limit analysis of a hollow sphere made up of a Fleck and Hutchinson’s strain gradient plasticity material. Based on the trial velocity field of Gurson, we derive an approximate closed form expression of the macroscopic criterion. The latter incorporates the void size dependency through a non dimensional parameter defined as the ratio of the cavity radius and the intrinsic length of the plastic solid. The accuracy of this approximate criterion is demonstrated by its comparison with numerical data. In the last part of the paper we present a complete plasticity model involving the damage rate and a power-law strain hardening of the matrix. It is shown that the cavity size effect has a strong dependency on damage growth as well as on the stress strain response.

Research paper thumbnail of Micromechanical Modelling of Fracture-Induced Anisotropy and Damage in Orthotropic Materials

Matrix cracking is commonly recognised as one of the main inelastic deformation mechanisms of Bri... more Matrix cracking is commonly recognised as one of the main inelastic deformation mechanisms of Brittle Matrix Composites. The modelling of such phenomenon still presents some difficulties which are mainly related to the description of the interaction between the initial anisotropy and the cracks-induced anisotropy. The present study concerns a new micro-macro approach of the non linear behavior and damage propagation

Research paper thumbnail of A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media

International Journal of Solids and Structures, 2012

ABSTRACT A micromechanics-based approach for the derivation of the effective properties of period... more ABSTRACT A micromechanics-based approach for the derivation of the effective properties of periodic linear elastic composites which exhibit strain gradient effects at the macroscopic level is presented. At the local scale, all phases of the composite obey the classic equations of three-dimensional elasticity, but, since the assumption of strict separation of scales is not verified, the macroscopic behavior is described by the equations of strain gradient elasticity. The methodology uses the series expansions at the local scale, for which higher-order terms, (which are generally neglected in standard homogenization framework) are kept, in order to take into account the microstructural effects. An energy based micro–macro transition is then proposed for upscaling and constitutes, in fact, a generalization of the Hill–Mandel lemma to the case of higher-order homogenization problems. The constitutive relations and the definitions for higher-order elasticity tensors are retrieved by means of the “state law” associated to the derived macroscopic potential. As an illustration purpose, we derive the closed-form expressions for the components of the gradient elasticity tensors in the particular case of a stratified periodic composite. For handling the problems with an arbitrary microstructure, a FFT-based computational iterative scheme is proposed in the last part of the paper. Its efficiency is shown in the particular case of composites reinforced by long fibers.

Research paper thumbnail of Microstructural effects in non linear stratified composites

International Journal of Solids and Structures, 2014

ABSTRACT In this paper, we analyze the microstructural effects on non linear elastic and periodic... more ABSTRACT In this paper, we analyze the microstructural effects on non linear elastic and periodic composites within the framework of asymptotic homogenization. We assume that the constitutive laws of the individual constituents derive from strain potentials. The microstructural effects are incorporated by considering the higher order terms, which come from the asymptotic series expansion. The complete solution at any order requires the resolution of a chain of cell problems in which the source terms depend on the solution at the lower order. The influence of these terms on the macroscopic response of the non linear composite is evaluated in the particular case of a stratified microstructure. The analytic solutions of the cell problems at the first and second order are provided for arbitrary local strain–stress laws which derive from potentials. As classically, the non-linear dependence on the applied macroscopic strain is retrieved for the solution at the first order. It is proved that the second order term in the expansion series also exhibits a non linear dependence with the macroscopic strain but linearly depends on the gradient of macroscopic strain. As a consequence, the macroscopic potential obtained by homogenization is a quadratic function of the macroscopic strain gradient when the expansion series is truncated at the second order. This model generalizes the well known first strain gradient elasticity theory to the case of non linear elastic material. The influence of the non local correctors on the macroscopic potential is investigated in the case of power law elasticity under macroscopic plane strain or antiplane conditions.

Research paper thumbnail of Inversion of higher order isotropic tensors with minor symmetries and solution of higher order heterogeneity problems

Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2011

In this paper, the derivation of irreducible bases for a class of isotropic (2n) thorder tensors ... more In this paper, the derivation of irreducible bases for a class of isotropic (2n) thorder tensors having particular "minor symmetries" is presented. The methodology used for obtaining these bases consists in extending the concept of deviatoric and spherical parts, commonly used for 2 nd -order tensors, to the case of a n th -order tensor. It is shown that those bases are useful for effecting the classical tensorial operations and specially the inversion of a 2n th -order tensor. Finally, the formalism introduced in this study is applied for obtaining the closed form expression of the strain field within a spherical inclusion embedded in an infinite elastic matrix and subjected to linear or quadratic polynomial remote strain fields.

Research paper thumbnail of On the inversion of non symmetric sixth-order isotropic tensors and conditions of positiveness of third-order tensor valued quadratic functions

Mechanics Research Communications, 2011

In the present paper we propose new results concerning linear tensorial algebra for third-order a... more In the present paper we propose new results concerning linear tensorial algebra for third-order and non symmetric isotropic sixth-order tensors in the most general case (i.e. having not the major and minor symmetries). Such tensors are used, for instance, in the theory of microstructured elastic media. A formalism based on an irreducible basis for isotropic sixth-order tensors is introduced, which is useful for performing the classical tensorial operations. Specially, a condensed expression for the product between two isotropic sixth-order tensors is provided, which allows the obtaining of a condition on these tensors for being invertible and a closed form expression of the inverse of such a tensor. Finally, the condition of positiveness of third-order tensor-valued quadratic functions is derived. For instance, such conditions are required for computing the elastic energy of microstructured media.

Research paper thumbnail of Algebra of Transversely Isotropic Sixth Order Tensors and Solution to Higher Order Inhomogeneity Problems

Journal of Elasticity, 2013

ABSTRACT In this paper we provide a complete and irreducible representation for transversely isot... more ABSTRACT In this paper we provide a complete and irreducible representation for transversely isotropic sixth order tensors having minor symmetries. Such tensors appear in some practical problems of elasticity for which their inversion is required. For this kind of tensors, we provide an irreducible basis which possesses some remarkable properties, allowing us to provide a representation in a compact form which uses two scalars and three matrices of dimension 2, 3 and 4. It is shown that the calculation of sum, product and inverse of transversely isotropic sixth order tensors is greatly simplified by using this new formalism and appears to be appropriate for deriving new various solutions to some practical problems in mechanics which use such kinds of higher order tensors. For instance, we derive the fields within a cylindrical inhomogeneity submitted to remote gradient of strain. The method of resolution uses the Eshelby equivalent inclusion method extended to the case of a polynomial type eigenstrain. It is shown that the approach leads to a linear system involving a sixth order tensor whose closed form solution is derived by means of the tensorial formalism introduced in the first part of the paper.

Research paper thumbnail of Macroscopic yield criteria for plastic anisotropic materials containing spheroidal voids

International Journal of Plasticity, 2008

The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile p... more The combined effects of void shape and matrix anisotropy on the macroscopic response of ductile porous solids is investigated. The Gologanu-Leblond-Devaux's (GLD) analysis of an rigid-ideal plastic (von Mises) spheroidal volume containing a confocal spheroidal cavity loaded axisymmetrically is extended to the case when the matrix is anisotropic (obeying Hill's (1948) anisotropic yield criterion) and the representative volume element (RVE) is subjected to arbitrary deformation. To derive the overall anisotropic yield criterion, a limit analysis approach is used. Conditions of homogeneous boundary strain rate are imposed on every ellipsoidal confocal with the cavity. A two-field trial velocity satisfying these boundary conditions are considered. It is shown that for cylindrical and spherical void geometries, the proposed criterion reduces to existing anisotropic Gurson-like yield criteria. Furthermore, it is shown that for the case when the matrix is considered isotropic, the new results provide a rigorous generalization to the GLD model. Finally, the accuracy of the proposed approximate yield criterion for plastic anisotropic media containing non-spherical voids is assessed through comparison with numerical results.

Research paper thumbnail of Exact solution of a plastic hollow sphere with a Mises–Schleicher matrix

International Journal of Engineering Science, 2012

In this paper we establish the exact solution for a hollow sphere with a rigid-plastic pressure-s... more In this paper we establish the exact solution for a hollow sphere with a rigid-plastic pressure-sensitive matrix and subjected to hydrostatic tension or compression. The matrix is assumed to obey to a parabolic Mises-Schleicher criterion. The closed-form expressions of the velocity field and of the stress field are provided. These exact solutions, expressed by means of the Lambert W function, allow to assess and discuss existing results.

Research paper thumbnail of Interfacial models in viscoplastic composites materials

International Journal of Engineering Science, 2010

The aim of the present work is to extend the concept of interphase and equivalent imperfect inter... more The aim of the present work is to extend the concept of interphase and equivalent imperfect interface in the context of viscoplasticity. The interphase takes the form of a thin curved layer of constant thickness, made up of a rigid viscoplastic material located between two other surrounding materials. We aim at representing this interphase by an interface with appropriately devised interface conditions. To reach this objective, a Taylor expansion of the relevant physical fields in the thin region is used. It is shown that, depending of the degree of stiffness of the layer with respect to the neighboring media, this interphase can be replaced by an idealized imperfect interface involving the jump of the velocity field or the traction vector. The first kind of interface model, applicable to soft interphases, is the "spring-type" interface across which the traction are continuous but the velocity field exhibits a discontinuity which is given in term of the traction by a power-law type relation. Moreover, it is shown that the constant of the model can be expressed in terms of the material parameters of the interphase. When the interphase is stiffer than the two surrounding media, one obtain a "stress-type" interface across which the velocity is continuous and a jump condition for the traction is given by a generalization of the so-called Young Laplace model to viscoplastic solids.