Nonlocal damage theory

In the usual local finite element analysis, strain softening causes spurious mesh sensitivity and incorrect convergence when the element is refined to vanishing size. In a previous continuum formulation, these incorrect features were overcome by the imbricate nonlocal continuum, which, however, introduced some unnecessary computational complications due to the fact that all response was treated as nonlocal. The key idea of the present nonlocal damage theory is to subject to nonlocal treatment only those variables that control strain softening, and to treat the elastic part of the strain as local. The continuum damage mechanics formulation, convenient for separating the nonlocal treatment of damage from the local treatment of elastic behavior, is adopted in the present work. The only required modification is to replace the usual local damage energy release rate with its spatial average over the representative volume of the material whose size is a characteristic of the material. Avoidance of spurious mesh sensitivity and proper convergence are demonstrated by numerical examples, including static strain softening in a bar, longitudinal wave propagation in strain-softening material, and static layered finite element analysis of a beam. In the last case, the size of the representative volume serving in one dimension as the averaging length for damage must not be less than the beam depth, due to the hypothesis of plane cross sections. It is also shown that averaging of the fracturing strain leads to an equivalent formulation, which could be extended to anisotropic damage due to highly oriented cracking.