Finite element simulation of strain localization with large deformation: capturing strong discontinuity using a Petrov–Galerkin multiscale formulation

Abstract

The finite element model for strain localization analysis developed in a previous work is generalized to the finite deformation regime. Strain enhancements via jumps in the displacement field are captured and condensed on the material level, leading to a formulation that does not require static condensation to be performed on the element level. A general evolution condition is first formulated laying out conditions for the continued activation of a strong discontinuity. Then, a multiscale finite element model is formulated to describe the post-bifurcation behavior, highlighting the key roles played by the continuous and conforming deformation maps on the characterization of the finite deformation kinematics of a localized element traced by a strong discontinuity. The resulting finite element equation exhibits the features of a Petrov–Galerkin formulation in which the gradient of the weighting function is evaluated over the continuous part of the deformation map, whereas the gradient of the trial function is evaluated over the conforming part of the same map. In the limit of infinitesimal deformations the formulation reduces to the standard Galerkin approximation described in a previous work. Numerical examples are presented demonstrating absolute objectivity with respect to mesh refinement and insensitivity to mesh alignment of finite element solutions.