Solver-free classical computational homogenization for nonlinear periodic heterogeneous media

Modeling the behavior of composite materials is an important application of computational homogenization methods. Classical computational homogenization (CCH), based on asymptotic analysis, is such a method. In CCH, equilibrium equations are separated into two length scales and solved numerically. Solving the fine-scale equilibrium equations at every coarse-scale Gauss point, in every iteration of a Newton-Raphson loop, is often too computationally expensive for real engineering applications. In this study, we propose a modified CCH approach that avoids solving the fine-scale equilibrium equations. The proposed method, which we call solver-free CCH, works by pre-computing a set of eigenstrain influence function tensors based on data from a small number of numerical experiments. Then, during the online stage of the computation, these eigenstrain influence tensors are used in the fine-scale problem to evaluate and homogenize strains and stresses. This article begins by formulating the solver-free CCH approach for small-deformation problems involving composites with nonlinear constituent phase material models, including computation of the eigenstrain influence tensors and their application within the online stage. To verify the proposed approach, we consider loading cases outside the training set used to derive the eigenstrain influence tensors. The combinational efficiency and accuracy of the solver-free CCH in comparison to the conventional CCH is studied on a multilayer composite plate in three point bending (3pt-bend) and open hole tension.