Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites

International audienceWe report on the mathematical analysis of two different, FFT-based, numerical schemes for the homogenization of composite media within the framework of linear elasticity: the basic scheme of Moulinec and Suquet (1994, 1998) [9] and [10], and the energy-based scheme of Brisard and Dormieux (2010) [13]. Casting these two schemes as Galerkin approximations of the same variational problem allows us to assert their well-posedness and convergence. More importantly, we extend in this work their domains of application, by relieving some stringent conditions on the reference material which were previously thought necessary. The origins of the flaws of each scheme are identified, and a third scheme is proposed, which seems to combine the strengths of the basic and energy-based schemes, while leaving out their weaknesses. Finally, a rule is proposed for handling heterogeneous pixels/voxels, a situation frequently met when images of real materials are used as input to these schemes

Corrigendum to “Combining Galerkin approximation techniques with the principle of Hashin and Shtrikman to derive a new FFT-based numerical method for the homogenization of composites” [Comput. Methods Appl. Mech. Engrg. 217–220 (2012) 197–212]

International audienceAssumption 1 in our original paper can be replaced with the following, less stringent assumption

FFT-based methods for the mechanics of composites: A general variational framework

International audienceFor more than a decade, numerical methods for periodic elasticity, based on the fast Fourier transform, have been used successfully as alternatives to more conventional (fem, bem) numerical techniques for composites. These methods are based on the direct, point-wise, discretization of the Lippmann-Schwinger equation, and a subsequent truncation of underlying Fourier series required for the use of the fast Fourier transform. The basic FFT scheme is very attractive, because of its simplicity of implementation and use. However, it cannot handle pores or rigid inclusions, for which a specific (and significantly more involved) treatment is required. In the present paper, we propose a new FFT-based scheme which is as simple as the basic scheme, while remaining valid for infinite contrasts. Since we adopted an energy principle as an alternative to the Lippmann-Schwinger equation, our scheme is derived within a variational framework. As a by-product, it provides an energetically consistent rule for the homogenization of boundary voxels, a question which has been pending since the introduction of Fourier-based methods

Hashin-Shtrikman bounds on the bulk modulus of a nanocomposite with spherical inclusions and interface effects

International audienceNanocomposites are becoming more and more popular and mechanical models are needed to help with their design and optimization. One of the key issues to be addressed by such models is the surface-stresses arising at the inclusion-matrix boundary, due to its high curvature. In this paper, we show that, contrary to what has previously been suggested, polarization techniques can be employed in the context of composites with interface effects. This requires a specific mathematical treatment of the interface, which must be regarded as a thin elastic layer. We then apply the proposed general methods to the specific case of nanocomposites with monodisperse spherical inclusions, for which a lower bound on the bulk modulus is derived. When interface effects are disregarded, this bound coincides with the classical Hashin-Shtrikman bound. In the presence of interface effects, we show that the existing Mori-Tanaka estimate is in fact a lower bound on the effective bulk modulus. Finally, lower bounds on the effective bulk modulus of nanocomposites with polydisperse spherical inclusions are proposed. Although this result can be considered as a by-product of the previous one, it is new, and has no published Mori-Tanaka counterpart

Propriétés macroscopiques de résistance de matériaux nanoporeux : une affaire de taille

On s’intéresse dans cette étude à la résistance macroscopique des milieux nanoporeux ductiles. Pour cela, on rappelle d’abord brièvement le cadre de l’analyse limite appliquée aux matériaux mésoporeux, ainsi que les modèles classiques de type Gurson et ses dérivés qui s’en déduisent par approche cinématique. Une attention particulière est accordée aux effets de forme de cavités. Puis on expose des extensions de ces modèles que nous avons récemment proposés pour des systèmes nanoporeux. À cette fin, l’homogénéisation du milieu est réalisée en considérant des contraintes interfaciales entre la matrice solide et les nano cavités, l’interface obéissant à une loi de plasticité surfacique déduite de façon asymptotique et étant susceptible d’un saut du vecteur contraintes. Le critère macroscopique obtenu, pour le matériau nanoporeux, prédit des caractéristiques inhabituelles telles que (i) une dépendance significative de la résistance macroscopique avec la taille des nano cavités, (ii) la possibilité d’une asymétrie entre la résistance en traction et en compression, (iii) un effet combiné de la taille et de la forme et des cavités, particulièrement marqué pour des nano cavités aplaties. Certains des résultats obtenus ont pu être évalués à l’aide des données récentes issues de simulations atomistiques. [1] L. Dormieux, D. Kondo, An extension of Gurson model incorporating interface stresses effects. Int. J. Eng. Sci., 48: 575-581, 2010. [2] V. Monchiet, D. Kondo, ombined voids size and shape effects on the macroscopic criterion of ductile nanoporous materials. Int. J. Plasticity., Sous Presse, en ligne 2013

Strength of a matrix with elliptic criterion reinforced by rigid inclusions with imperfect interfaces

International audienceElliptic effective strength criteria in the mean-deviatoric stress plane are encountered in porous media for a granular material made of rigid grains with cohesive frictional interfaces or a material with pores in a Drucker-Prager matrix. The macroscopic strength criterion of a heterogeneous material comprising a matrix with elliptic strength criterion reinforced by rigid inclusions with perfect or imperfect interfaces is studied. Considered imperfect interfaces follow either a Tresca or a Mohr-Coulomb strength criterion. Derived macroscopic criteria are shown to be a combination of a larger ellipse, which corresponds to the criterion for perfectly bounded interfaces, conditionally truncated by a smaller ellipse resulting from the activation of interfacial mechanisms. The activation of the interfacial mechanisms depends on the matrix and interfaces strength properties, inclusions concentration, as well as the macroscopic strain triaxiality ratio

New boundary conditions for the computation of the apparent stiffness of statistical volume elements

International audienceWe present a new auxiliary problem for the determination of the apparent stiffness of a Statistical Volume Element (SVE). The SVE is embedded in an infinite, homogeneous reference medium, subjected to a uniform strain at infinity, while tractions are applied to the boundary of the SVE to ensure that the imposed strain at infinity coincides with the average strain over the SVE. The main asset of this new auxiliary problem resides in the fact that the associated Lippmann-Schwinger equation involves without approximation the Green operator for strains of the infinite body, which is translation-invariant and has very simple, closed-form expressions. Besides, an energy principle of the Hashin and Shtrikman type can be derived from this modified Lippmann-Schwinger equation, allowing for the computation of rigorous bounds on the apparent stiffness. The new auxiliary problem requires a cautious mathematical analysis, because it is formulated in an unbounded domain. Observing that the displacement is irrelevant for homogenization purposes, we show that selecting the strain as main unknown greatly eases this analysis. Finally, it is shown that the apparent stiffness defined through these new boundary conditions "interpolates" between the apparent stiffnesses defined through static and kinematic uniform boundary conditions, which casts a new light on these two types of boundary conditions

Self-influence and influence pseudotensors of d-dimensional spheres

This document is a companion to Ref. [1]. It gathers analytical expressions for the self-influence and influence pseudotensors of d-dimensional spheres (d=2, 3)

Ductile sliding between mineral crystals followed by rupture of collagen crosslinks: Experimentally supported micromechanical explanation of bone strength

International audienceThere is an ongoing discussion on how bone strength could be explained from its internal structure and composition. Reviewing recent experimental and molecular dynamics studies, we here propose a new vision on bone material failure: mutual ductile sliding of hydroxyapatite mineral crystals along layered water films is followed by rupture of collagen crosslinks. In order to cast this vision into a mathematical form, a multiscale continuum micromechanics theory for upscaling of elastoplastic properties is developed, based on the concept of concentration and influence tensors for eigenstressed microheterogeneous materials. The model reflects bone's hierarchical organization, in terms of representative volume elements for cortical bone, for extravascular and extracellular bone material, for mineralized fibrils and the extrafibrillar space, and for wet collagen. In order to get access to the stress states at the interfaces between crystals, the extrafibrillar mineral is resolved into an infinite amount of cylindrical material phases oriented in all directions in space. The multiscale micromechanics model is shown to be able to satisfactorily predict the strength characteristics of different bones from different species, on the basis of their mineral/collagen content, their intercrystalline, intermolecular, lacunar, and vascular porosities, and the elastic and strength properties of hydroxyapatite and (molecular) collagen