Nonlocal Damage Theory Based on Micromechanics of Crack Interactions

Abstract

A nonlocal continuum model for strain‐softening damage is derived by micromechanics analysis of a macroscopically nonhomogeneous (nonuniform) system of interacting and growing microcracks, using Kachanov's simplified version of the superposition method. The homogenization is obtained by seeking a continuum field equation whose possible discrete approximation coincides with the matrix equation governing a system of interacting microcracks. The result is a Fredholm integral equation for the unknown nonlocal inelastic stress increments, which involves two spatial integrals. One integral, which ensues from the fact that crack interactions are governed by the average stress over the crack length rather than the crack center stress, represents short‐range averaging of inelastic macro‐stresses. The kernel of the second integral is the long‐range crack influence function which is a second‐rank tensor and varies with directional angle (i.e., is anisotropic), exhibiting sectors of shielding and amplification. For long distances r, the weight function decays as r-2 in two dimensions and as r-3 in three dimensions. Application of the Gauss‐Seidel iteration method, which can conveniently be combined with iterations in each loading step of a nonlinear finite element code, simplifies the handling of the nonlocality by allowing the nonlocal inelastic stress increments to be calculated from the local ones explicitly. This involves evaluation of an integral involving the crack influence function, for which closed‐form expressions are derived. Because the constitutive law is strictly local, no difficulties arise with the unloading criterion or continuity condition of plasticity.