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In this work we derive a general model for N −phase isotropic, incompressible, rateindependent elasto-plastic materials at finite strains. The model is based on the nonlinear homogenization variational (or modified secant) method which makes use of a linear comparison composite (LCC) material to estimate the effective flow stress of the nonlinear composite material. The homogenization approach leads to an optimization problem which needs to be solved numerically for the general case of a N −phase composite. In the special case of a two-phase composite an analytical result is obtained for the effective flow stress of the elasto-plastic composite material. Next, the model is validated by periodic three-dimensional unit cell calculations comprising a large number of spherical inclusions (of various sizes and of two different types) distributed randomly 1 in a matrix phase. We find that the use of the lower Hashin-Shtrikman bound for the LCC gives the best predictions by comparison with the unit cell calculations for both the macroscopic stress-strain response as well as for the average strains in each of the phases. The formulation is subsequently extended to include hardening of the different phases. Interestingly, the model is found to be in excellent agreement even in the case where each of the phases follows a rather different hardening response.Keywords: Homogenization; Elasto-plasticity; Composite materials; Finite strains * Corresponding author. Fax: +30 2421 074009. E-mail addresses: aravas@uth.gr (N. Aravas), kdanas@lms.polytechnique.fr (K. Danas). 1 IntroductionThe present work deals with the analytical and numerical estimation of the effective as well as the phase average response of N−phase incompressible isotropic elasto-plastic metallic composites. Special attention is given to particulate microstructures, i.e., composite materials which can be considered to comprise a distinct matrix phase and an isotropic distribution of spherical particles [46] (or in a more general setting an isotropic distribution of phases [45]). In the present study, the particles are considered to be stiffer than the matrix phase, which is the case in most metallic materials of interest, such as TRIP steels, dual phase steels, aluminum alloys and others. Such materials, usually contain second-phase particles (e.g., intermetallics, carbon particles) or just second and third phase variants (e.g., retained austenite, bainite, martensitic phases). In addition, these phases/particles tend to reinforce the yield strength of the composite while they usually have different strength and hardening behavior than the host matrix phase.In the literature of nonlinear homogenization there exists a large number of studies for twophase composite materials. The reader is referred to Ponte Castañeda and Suquet [33], Ponte Castañeda [32], Idiart et al. [15], and Idiart [16] for a review of the nonlinear homogenization schemes such as the ones used in the present work and relevant estimates. Nonetheless, very few studies exist in the conte...

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A multiscale investigation of the microstructure and the mechanical behavior of TRIP steels is presented. A multi-phase field model is employed to predict the microstructure of a low-alloy TRIP700 steel during a two-stage heat treatment. The resulting stability of retained austenite is examined through the M s σ temperature. The phase field results are experimentally validated and implemented into a model for the kinetics of retained austenite during strain-induced transformation. The kinetics model is calibrated by using experimental data for the evolution of the martensite volume fraction in uniaxial tension. The transformation kinetics model is used together with homogenization methods for non-linear composites to develop a constitutive model for the mechanical behavior of the TRIP steel. A methodology for the numerical integration of the constitutive equations is developed and the model is implemented in a general-purpose finite element program (ABAQUS). Necking of a bar in uniaxial tension is simulated and “forming limit diagrams” (FLDs) for sheets made of TRIP steels are calculated. The models developed provide an integrated simulation toolkit for the computer-assisted design of TRIP steels and can be used to translate mechanical property requirements into optimised microstructural characteristics and to identify the appropriate processing routes.