A Galerkin approach to FFT-based homogenization methods

International audienceSince their introduction by Moulinec and Suquet [1, 2], FFT-based full-field simulations of the mechanical properties of composites have become increasingly popular, with applications ranging from the linear elastic behaviour of cementitious materials [3], to the plasticity of polycrystals [4]. Recently, the authors have proposed [5] a new formulation of these numerical schemes, based on the energy principle of Hashin and Shtrikman [6]. While similar in principle to the original scheme of Moulinec and Suquet, the new scheme was shown to be much better-behaved. Indeed, convergence of the scheme is guaranteed for any contrast, without having to resort to augmented Lagrangian approaches [7]. Besides, convergence of the new scheme is generally much faster. However, the new scheme has two drawbacks. First, the reference material must be stiffer (or softer) than all constituants of the composite; this is not always possible, for example when the composite contains both pores and rigid inclusions. Second, the scheme requires the preliminary computation of the so-called consistent Green operator, which turned out to be a difficult task in three dimensions. In order to relax these requirements, an in-depth mathematical analysis of these schemes was carried out by the authors [8]. In this paper, the Lippmann-Schwinger equation and its variational form will briefly be recalled. The Galerkin approach will then be adopted for the discretization of this equation, and it will be shown that the basic scheme of [1] as well as the energy scheme proposed in [5] can both be viewed as well-posed Galerkin approximations of the Lippmann-Schwinger equation. Contrary to what was previously believed [7, 5] these approximations are convergent, regardless of the reference material (provided that its stiffness is positive definite). Comparison of these two approximations leads to the derivation of the so-called filtered, non-consistent approach, which combines the assets of the two former methods. Finally, some applications will be shown. In particular, the important problem of heterogeneous voxels will be addressed

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

A micro-mechanical model for the plasticity of porous granular media and link with the Cam clay model

International audienceA micro-mechanical constitutive model for the plastic behavior of cohesive granular materials with hardening due to porosity changes is proposed. The plasticity model is based on a re-interpretation of a micro-mechanical strength model for cohesive frictional granular media. The hardening law by porosity changes explicitly stems from the homogenization process. The micro-macro plasticity model, analytical and fully explicit, depends only on two constant material parameters with a clear physical signification at the microscopic scale: the friction angle and the tensile strength of the grain to grain interfaces. The seminal ideas of critical state soil mechanics are retrieved: critical state line, state boundary surface in the stress/porosity space, hardening or softening due to change in porosity and ability to describe both dilatancy and contrac-tancy. The established micro-mechanical model is very similar to the phenomenological modified Cam clay model, providing to the latter a microstructural based interpretation

A variational form of the equivalent inclusion method for numerical homogenization

International audienceDue to its relatively low computational cost, the equivalent inclusion method is an attractive alternative to traditional full-field computations of heterogeneous materials formed of simple inhomogeneities (spherical, ellipsoidal) embedded in a homogeneous matrix. The method can be seen as the discretization of the Lippmann--Schwinger equation with piecewise polynomials. Contrary to the original approach of Moschovidis and Mura, who discretized the strong form of the Lippmann--Schwinger equation through Taylor expansions, we propose in the present paper a Galerkin discretization of the weak form of this equation. Combined with the new, mixed boundary conditions recently introduced by the authors, the resulting method is particularly well-suited to homogenization. It is shown that this new, variational approach has a number of benefits: (i) the resulting linear system is well-posed, (ii) the numerical solution converges to the exact solution as the maximum degree of the polynomials tends to infinity and (iii) the method can provide rigorous bounds on the apparent properties of the statistical volume element, provided that the matrix is stiffer (or softer) than all inhomogeneities. This paper presents the formulation and implementation of the new, variational form of the equivalent inclusion method. Its efficiency is investigated through numerical applications in 2D and 3D elasticity

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