Nonlocal continuum damage, localization instability and convergence

A recent nonlocal damage formulation, in which the spatially averaged quantity was the energy dissipated due to strain-softening, is extended to a more general form in which the strain remains local while any variable that controls strain-softening is nonlocal. In contrast to the original imbricate nonlocal model for strain-softening, the stresses which figure in the constitutive relation satisfy the differential equations of equilibrium and boundary conditions of usual classical form, and no zero-energy spurious modes of instability are encountered. However, the field operator for the present formulation is in general nonsymmetric, although not for the elastic part of response. It is shown that the energy dissipation and damage cannot localize into regions of vanishing volume. The static strain-localization instability, whose solution is reduced to an integral equation, is controlled by the characteristic length of the material introduced in the averaging rule. The calculated static stability limits are close to those obtained in previous nonlocal studies, as well as those obtained by the crack band model in which the continuum is treated as local but the minimum size of the strain-softening region (localization region) is prescribed as a localization limiter. The rate of convergence of static finite element solutions with nonlocal damage is of a power type, almost quadratic. A smooth weighting function in the averaging operator leads to a better convergence than unsmooth functions.