A Mixed Finite Element Implementation of a Gradient-enhanced Coupled Damage—Plasticity Model

The numerical implementation of a gradient-enhanced continuum coupled damage—plasticity model as a constitutive framework to model the nonlocal response of materials is presented. By the introduction of “nonlocal, ” gradient-enhanced measures in the plasticity potential function and yield criterion and in the damage potential function and damage criterion, the proposed model introduces microstructural characteristic material length scales, which allow us to predict the size of localized zones based on material constants, as opposed to local models where the loss of ellipticity causes localized zones to be mesh dependent. As numerical methods are used to compute the gradients of the hardening terms, no additional gradient internal state variables are introduced into the Helmholtz free energy. As opposed to previous theories that only incorporate linear hardening, the gradient model proposed here uses coupled nonassociative plasticity — nonassociative damage and can account for a wide range of material models because of the consistently expanded Laplacian evolution equations. However, due to the nonassociative material model, anisotropic hardening, and the corresponding gradients in plasticity, the model becomes highly complex as additional equations have to be introduced for the evolution of the gradients of plastic normal, plastic strains, damage normal, and damage tensor. The numerical implementation uses a small deformation finite element formulation and includes the displacements, plastic multiplier, and damage multiplier as nodal degrees of freedom, thus allowing the three fields to have different interpolation functions. Higher-order elements are used for the plastic multiplier and damage multiplier to enforce continuity of the second gradients, which requires the introduction of additional boundary conditions.