A common practice in multiscale problems for heterogeneous materials with well separated scales, is to look for homogenized, or effective, constitutive relations. In linear elasticity the structure of the homogenized constitutive relations is strictly preserved in the change of scales. The linear effective properties can be computed once for all by solving a finite number of unit-cell problems. Unfortunately there is no exact scale-decoupling in multiscale nonlinear problems which would allow one to solve only a few unit-cell problems and then use them subsequently at a larger scale. Computational approaches developed to investigate the response of representative volume elements along specific loading paths, do not provide constitutive relations. Most of the huge body of information generated in the course of these costly computations is often lost.

Model reduction techniques, such as the Non Uniform Transformation Field Analysis ([1]), may be used to exploit the information generated along such computations and, at the same time, to account for the commonly observed patterning of the local plastic strain field. A new version of the model was proposed in [2] (NTFA-TSO: Non Uniform Transformation Field Analysis using Tangent Second Order expansion), with the aim of preserving the underlying variational structure of the constitutive relations (similar objective in [3]), while using approximations which are common in nonlinear homogenization.

This study presents a micromechanical modeling by the new NTFA-TSO model [2] on heterogeneous material: a three-phase particulate composite material with two inclusion phases dispersed in a contiguous matrix. In the study realized in [4], the phases had an aging linear (one dissipation potential) viscoelastic behavior with swelling. The NTFA [1] was applied in a three-dimensional setting and extended to account for inhomogeneous eigenstrains in the individual phases. In the study presented in this paper, the phases have an aging viscoelastic behavior with swelling, defined by the sum of two dissipation potentials, a quadratic one corresponding to linear viscoelastic behavior and a power-law second one corresponding to nonlinear viscoelastic behavior. Due to the nonlinear dissipation potential within phases, the procedure developed in [2] is required. First, the model-reduction approach is introduced by the authors in this paper. The local fields of internal variables are decomposed on a reduced basis of modes and the dissipation potentials of the phases are replaced by its TSO expansion [2]. The reduced evolution equations of the model can be entirely expressed in terms of quantities which are pre-computed once for all. These pre-computed quantities depend only on the average and fluctuations per phase of the modes and of the associated stress fields. Second, the accuracy of the NTFA-TSO model is assessed by comparison with full-field simulations (results for the overall as well as for the local response of the composites) on different tests (e.g. creep test).

References:

1 J.C. Michel, P. Suquet, Int. J. Solids Structures 40, 6937-6955 (2003)

2 J.C. Michel, P. Suquet, J. Mech. Phys. Solids 90, 254–285 (2016)

3 F. Fritzen, M. Leuschner, Comput. Meth. Appl. Mech. Eng. 260, 143–154 (2013)

4 R. Largenton, J.C. Michel, P. Suquet, Mech. Mater. 73, 76–100 (2014)