Pierre Suquet

Effective properties of composite materials with periodic microstructure: a computational approach

Abstract This study reviews several problems which are specific of composites with periodic microstructure composed of linear or nonlinear constituents. The theoretical background of the method is recalled first. Two different families of numerical methods are considered to solve the problem. The first is based on the Finite Element Method. The concept of ‘macroscopic degrees of freedom’ is presented. The implementation of periodicity conditions is discussed. A general framework permitting either a strain or stress control is proposed. The second numerical method is based on Fast Fourier Transforms. It considers first the problem of a homogeneous linear reference material undergoing a nonhomogeneous periodic eigenstrain. The solution of this problem is based on the explicit form of the periodic Greens function of the reference medium. The relative merits of the two methods are compared and several examples are discussed. Both methods give very comparable results on test examples and their domains of applications appear to be complementary.

An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals

Variational approaches for nonlinear elasticity show that Hill’s incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too stiff and may even violate rigorous bounds. This paper aims at proposing an alternative ‘affine’ formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is first described for nonlinear elasticity and specified by making use of Hashin–Shtrikman estimates for the linear comparison composite; the associated affine self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure (Ponte Castaneda, P., 1996. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827–862) and some limitations of the affine method are pointed out; explicit comparisons between different procedures are performed for isotropic, two-phase materials. Finally, the affine formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it offers an improved alternative to Hill’s incremental formulation.

Nonuniform transformation field analysis

The exact description of the overall behavior of composites with nonlinear dissipative phases requires an infinity of internal variables. Approximate models involving only a finite number of those can be obtained by considering a decomposition of the microscopic anelastic strain field on a finite set of transformation fields. The Transformation Field Analysis of Dvorak [Proc. R. Soc. Lond. A 437 (1992) 311] corresponds to piecewise uniform transformation fields. The present theory considers nonuniform transformation fields. Comparison with numerical simulations shows the accuracy of the proposed model.

Exact results and approximate models for porous viscoplastic solids

Abstract The aim of this paper is to present new approximate macroscopic models for porous viscoplastic materials, based on partial but exact results applicable to such media. Available results are first supplemented by providing a new inequality (which, in addition to its intrinsic interest, allows one to rederive in a simpler way some previous bounds of Ponte-Castaneda and Talbot and Willis), and by exhibiting the exact form of the overall potential of a typical porous viscoplastic volume element, namely a hollow cylinder loaded in generalized plane strain. Approximate expressions for the macroscopic viscoplastic potentials of materials containing cavities of cylindrical or spherical shape are then proposed, based on these and other results; these expressions satisfy, in particular, the three following natural requirements: (i) reproduce the exact solution of a hollow cylinder or sphere loaded in hydrostatic tension or compression; (ii) be a quadratic form of the overall stress tensor in the extreme case of a Newtonian (linear) behaviour; and (iii) yield the currently accepted Gurson criterion in the other extreme case of an ideal-plastic behavior.

The constitutive law of nonlinear viscous and porous materials

Abstract This Paper examines with the help of micromechanics the overall behaviour of nonlinear viscous materials containing voids. In complement to variational bounds derived here and to several models proposed in the recent literature we derive a simple model which meets exactly a closed-form solution of a hollow sphere under hydrostatic tension. More generally we consider the problem of a hollow sphere under hydrostatic tension when the constitutive material of the sphere is already porous. The solution is used in a self-consistent scheme and a differential scheme to derive a constitutive law for a porous material containing different populations of micro-voids with distributed sizes. These schemes predict a higher damage effect for the same porosity than the models based on a single size of voids. All the models considered in this paper make use of a strain-rate potential and most of them assume a simple “quadratic” form for it.

On the effective behavior of nonlinear inelastic composites: I. Incremental variational principles

A new method for determining the overall behavior of composite materials comprised of nonlinear inelastic constituents is presented. Upon use of an implicit time-discretization scheme, the evolution equations describing the constitutive behavior of the phases can be reduced to the minimization of an incremental energy function. This minimization problem is rigorously equivalent to a nonlinear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). In this first part of the study the variational technique of Ponte Castaneda is used to approximate the nonuniform eigenstrains by piecewise uniform eigenstrains and to linearize the nonlinear thermoelastic problem. The resulting problem is amenable to simpler calculations and analytical results for appropriate microstructures can be obtained. The accuracy of the proposed scheme is assessed by comparison of the method with exact results.

Intraphase strain heterogeneity in nonlinear composites: a computational approach

Recent progress in the analysis of nonlinear composites or polycrystals suggests that higher-order moments of the local fields in individual phases could play a significant role on the effective properties of such composites. The present study is devoted to a detailed computational investigation of the first and second moment of the strain fields. It shows that the predictive schemes belonging to the class of secant methods underestimate significantly the second moments of the strain field. As a consequence, their predictions are too stiff in comparison with numerical simulations.

Comparison of FFT-based methods for computing the response of composites with highly contrasted mechanical properties

Abstract Numerical simulations of the behavior of composite materials have to take into account the complexity of their microstructural geometry. Realistic simulations involve a large number of degrees of freedom. FFT-based methods have proved their efficiency for mechanical problems (composites with linear elastic or elastic-plastic phases), and can often be much more efficient than classical finite element methods. However the iterative method initially proposed in Moulinec and Suquet (C.R. Acad. Sci. Paris II 318 (1994) 1417) requires a number of iterations roughly proportional to the contrast between the moduli of the individual phases. The improved method proposed in Eyre and Milton (J. Phys. III 6 (1999) 41) has a rate of convergence proportional to the square root of the contrast, but still cannot be applied to composites with infinite contrast (like porous materials). An alternate scheme based on augmented Lagrangians and Fourier Transforms has been proposed in Michel et al. (Comput. Modelling Eng. Sci. (2000) accepted) for highly contrasted or even infinitely contrasted materials. The efficiency of these three methods are compared as a function of the contrast between the phases.

Analysis of Inhomogeneous Materials at Large Strains using Fast Fourier Transforms

A computational technique based on fast Fourier transforms has been recently developed to analyze the response of inhomogeneous materials with complex microstructure, both at the global level (effective properties) and local level (stress and strain fields). Initially proposed for elastic constituents and subsequently extended to elastoplastic constituents in the approximation of infinitesimal strains, the method is extended here to large strains, using either a Lagrangian formulation in the reference configuration, or an Eulerian formulation in the deformed configuration. In the latter case, the algorithm inspired by “particle-in-cell” method, makes use of two grids, a computational grid and a material grid. Applications to hyperelastic materials reinforced by stiff inclusions (Lagrangian formulation) and to the motion of a rigid particle in a viscous fluid (Eulerian formulation) are given.

Homogenization, Plasticity and Yield Design

We consider an epi-convergence problem arising from the theory of yield design. The functional under consideration has a linear growth with respect to the deformation tensor of the displacement field, and the problem is naturally posed in a space of displacement fields with bounded deformation. The problem includes a linear constraint which can be closed or not closed, depending on the type of boundary conditions considered. In the case where the constraint is not closed (applied forces on a part of the boundary) a relaxation term appears. Physically the strength of the loaded boundary turns out to be smaller than the natural guess deduced from the well known Average Variational Principle.