A new method for determining the overall behavior of composite materials comprised of nonlinear inelastic constituents is presented. Upon use of an implicit timediscretization 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 Castañeda 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.

A new method for determining the overall behavior of composite materials comprised of linear viscoelastic constituents is presented. Unlike classical methods which are based on the Laplace transform, the present method operates directly in the time-domain. 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 linear thermoelastic problem with a transformation strain which is a nonuniform field (not even uniform within the phases). The variational technique of Ponte Castañ eda is used to approximate the nonuniform eigenstrains by piecewise uniform eigenstrains. The latter 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 obtained either by full finite element simulations in time-domain or by available analytical results obtained by the Laplace transform.

International audienceThe aim of this study is to estimate as accurately as possible the eff ective response, as well as the statistics of the fields (first and second moments), in elasto-(visco)plastic heterogeneous materials with isotropic and/or kinematic hardening. After time-discretization, a new incremental variational principle for the increments in strain and internal variables in materials governed by two potentials is derived. This variational principle, together with the variational method of Ponte Casta~neda (1992) is used to introduce a linear comparison composite (LCC) at each time step, approximating in a variational sense the original problem. The e ffective response of the LCC, as well as the fi rst and second moments of the stress and strain fields in each phase of the LCC are shown to provide estimates for the same quantities in the actual nonlinear elasto-(visco)plastic composite. The accuracy of the model is assessed by comparison with full-fi eld simulations. The agreement is found to be quite satisfactory, in particular the asymmetry between tension and compression observed in elasto-plastic composites (Bauschinger eff ect) is well reproduced, unlike in other mean- field models. The statistics of the stress and strain fields, and to a certain extent, that of the back-stress field, are also in good agreement with full- field simulations