A C 0 scalar field approach for describing sprain variables in finite element analysis of fracture

A new approach is proposed to account for the impact of sprain variables for computational simulation of fracture in quasi-brittle materials. For an isotropic solid, these variables are equal to the Laplacians of the components of the displacement field, and they were introduced in a recent study by Zhang and Bazant to resolve spurious mesh effects for cases involving softening material response. This is possible by establishing localization-resisting forces which are conjugate to the displacements with respect to a sprain energy functional, defined as a function of the sprain variables. To enable the evaluation of the sprain variables in a conventional, C0 continuum finite element approximation, this paper introduces a set of scalar fields, each being weakly equal to the Laplacian of one of the components of the displacement vector field. This leads to a straightforward implementation with standard continuum (solid) elements, eliminating the need for approximations satisfying higher continuity requirements or for ad-hoc, finite-difference algorithms which were adopted in the original pertinent work for the calculation of the sprain variables.