Study of various estimates of the macroscopic tangent operator in the incremental homogenization of elastoplastic composites

This paper contains a theoretical and numerical investigation of the incremental formulation for the mean-field homogenization of two-phase elastoplastic composites. We study several variants of the formulation and try to understand why some of them give good predictions while others do not. We define six instantaneous operators for a fictitious homogeneous reference matrix (two anisotropic, two isotropic, and two transversely isotropic) and various estimates of the macroscopic tangent operator. Theoretically, we present mathematical results that prove that some estimates are stiffer or softer than others. Numerically, we carry out a wide range of validated simulations with different types of inclusions under various loads. The findings confirm the theoretical results and shed new light on a rather complicated problem.