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

Abstract

ABSTRACT: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,' gradientenhanced 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.

On thermoelastostatics of composites with nonlocal properties of constituents I. General representations for effective material and field parameters

“…The mentioned representation for U was generalized to U ¼ 1 2 ½eLe þ aK 0 ðeÞLK 0 ðeÞ (see Polizzotto et al, 2006), where the argument K 0 (2.8) acts in a way similar to the strain gradient for strain gradient-dependent materials; in particular, like the strain gradient, K 0 vanishes identically for any uniform strain field in w. However, the differential form of the nonlocal operator (2.19) and (2.20) has obvious advantages due to the more readily realized finite element analysis (see, e.g., Dorgan and Voyiadjis, 2006;Voyiadjis and Song, 2006) with respect to the integral form (see, e.g., Polizzotto, 2001).…”

On thermoelastostatics of composites with nonlocal properties of constituents II. Estimation of effective material and field parameters

“…The differential operator can be obtained by truncation of the corresponding Tailor approximation of the integral one (see, e.g., I), and, because of this, the form of the differential operator is less general. However, the differential form of the nonlocal operator (8.11) and (8.12) has obvious advantage due to the more readily realized FEA (see, e.g., Dorgan and Voyiadjis, 2006;Voyiadjis and Song, 2006) with respect to the integral form (see, e.g., Polizzotto, 2001). Therefore, for estimations of the tensors A e i (x) and B e i (x) (rather than the operators (8.11)) by the use of the method considered by Buryachenko (2007) for the local problems, this approach can be more effective than the volume integral equation method (8.1).…”

An anisotropic gradient damage model based on microplane theory

“…In this situation, deformation in the damaged region is localized in a narrow zone, called the shear band (Dorgan and Voyiadjis, 2006). This induces severe mesh sensitivity in the process of numerical solving using the finite element method.…”