FFT-based homogenization of hypoelastic plasticity at finite strains

Abstract

Fast Fourier Transform (FFT) based methods are becoming increasingly popular for modeling the texture evolution and local mechanical response of polycrystalline materials within the representative volume element (RVE). Originally, the FFT-based method was formulated through the Lippman–Schwinger (L–S) equation in terms of a homogeneous elastic reference medium. More recently, the Fourier–Galerkin method was proposed to discretize the RVE weak form independently of the reference medium using trigonometric polynomials. The Fourier–Galerkin method, albeit efficient and robust, has not been extended to solve crystal plasticity (CP) problems with spatial heterogeneity and fine resolution. Also, its algorithmic homogenization has not yet been investigated, which is essential for concurrent multiscale modeling. In this paper, a general framework is proposed that connects the FFT-based method and objective rate constitutive models, in particular a generalized implementation of crystal plasticity. Consistent linearization is achieved by pulling back the tangent stiffness from unrotated configuration to reference configuration, which improves the convergence behavior of the Newton–Krylov solver. Applying the inexact Newton method further improves the numerical efficiency. Also, the algorithmic homogenized tangent stiffness for the Fourier–Galerkin method is derived to enable mixed boundary conditions and concurrent multiscale modeling. Lastly, the Fourier–Galerkin method’s accuracy and efficiency for solving CP problems are studied in a parallelized environment, and significant speedup is observed versus the finite element method and L–S based FFT method.