A second Piola-Kirchhoff stress-driven homogenization scheme for nonlinear elasticity

Abstract

This paper presents a novel stress-driven computational homogenization framework for imposing the second Piola-Kirchhoff (IIPK) stress tensor in the context of finite deformation. Unlike the commonly used first Piola-Kirchhoff stress (IPK), which is asymmetric and lacks frame and rotation invariance, the IIPK stress is frame-invariant and independent of rigid body rotations. In this study, we develop a variational framework integrated with finite element method to impose the IIPK stress on the representative volume element (RVE). For a prescribed IIPK stress tensor, the framework yields the corresponding equivalent Green-Lagrange strain tensor and the associated linearized elasticity tensor. The proposed formulation is analytically and numerically shown to satisfy the Hill-Mandel condition ensuring consistent micro-macro transitions. To facilitate the finite element implementation, an easy-to-implement linear constraint is derived to enforce the periodic boundary condition while eliminating rigid body modes and preventing the singularity of the global stiffness matrix. The effectiveness of this approach is validated through various numerical examples involving material and geometric nonlinearities, showcasing the framework’s robustness and accuracy.