An NTFA-based homogenization framework considering softening effects

In this work, we consider a two-scale mechanical problem of solids, where a microscopic heterogeneity is taken into account. In addition to plasticity, another irreversible process is our focus: softening, which is introduced on the micro scale and homogenized on the macro scale by model reduction with the so-called nonuniform transformation field analysis (NTFA, originally proposed by Michel and Suquet, 2003). Based on dissipative considerations, new NTFA constitutive equations with even model structure, accounting for softening effects, are proposed and validated theoretically and numerically for a homogeneous microstructure. For an accuracy improvement, we propose two new methods: the 'uneven NTFA' method and the 'adaptive NTFA' method, which introduce the additional aspects of parameter identification and adaptive modeling, respectively. The related numerical issues of both new methods are outlined. Two procedures for mode identification are studied for the present case, where the modes are actually basis functions for the reduced homogenization scheme. By means of the finite element method (FEM), numerical examples with regard to a fiber-reinforced composite are presented, where the accuracy and the numerical efficiency of the NTFA methods are investigated by comparison with the FEM solution. The mesh dependence of the different NTFA methods is also studied.