< . < - - - - - - - - - - - - - - - - - - - - - - - - c : 2 3 l Nonlocal Continuum Damage , Localization Instability and Convergence

Center for Concrete and Geomaterials, Northwestern University, Tech-241 0, Evanston, I L 60208 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 the usual classical form, and no zeroenergy spurious modes of instability are encountered. However, the field :;perator 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 regio.'1s of vanishing volume. The static strain-localization instability, whose solution is reduced to an integral equation, is found to be controllr!d by the characteristic length of the material introduced in the averaging rule. The :alculated static stability limits are close to those obtained in the previous non local studies, as well as to those obtained by the crack band model in which the continuum is treated as local bu: the minimum size of the strain-softening region (localization region) is prescribed as a localization iimiter. Furthermore, the rate of convergence of static finite-element solutions with nonlocal damage is studied and is found to be of a power type, almost quadratric. A smooth weighting function in the averaging operator is found to lead to a much better convergence than unsmootr. functions.