Regularization of localization due to material softening using a nonlocal hardening variable in an Eulerian formulation of inelasticity

Abstract

It is known that damage or inelastic softening can cause an ill-posed problem leading to localization and mesh-dependence in finite element simulations. In this paper, a nonlocal hardening variable, κ̄, is introduced in a finite deformation Eulerian formulation of inelasticity with a rate-independent smooth elastic–inelastic transition. This nonlocal variable is defined over an Eulerian region of nonlocality, which is a sphere with radius equal to the characteristic length, Lc, defined in the current deformed geometry of the material. Two models of this nonlocal hardening variable are explored. One model where κ̄ is the minimum value of the local hardening κ within the region of nonlocality, and another model where κ̄ is the average of κ in the same region. The influence of the nonlocal hardening variable is studied using an example of a plate that is loaded by a prescribed boundary displacement causing formation of a shear band. Predictions of the applied load vs. displacement curves and contour plots of the total distortional deformation of the plate and the hardening variable κ are studied. The model based on the minimum value of κ in the nonlocal region predicts mesh-independent post-peak response of the load vs. displacement curve. Also, it is shown that the characteristic material length, Lc, controls the structure of the shear band developed in the plate.