Nonlocal Elastic-Damage Models

A theory of nonlocal isotropic damage for elastic quasi-brittle materials is presented under the assumption of isothermal conditions and small deformations. Key ingredients of this theory are a self-adjoint (regularization) operator which transforms a local field into a related nonlocal one while preserves uniform fields and a free energy which depends on the strain and (linearly) on the nonlocal damage variable, as well as on an (scalar) internal variable accounting for the damage hardening. The relevant thermodynamic restrictions on the constitutive equations are obtained by means of two alternative procedures, one based on the principle of virtual power and the other on the concept of “nonlocality energy residual,” both of which lead to the state equations and the dissipation inequality. The damage evolution laws, based on the normality rule and on associativeness, admit both local and nonlocal forms of the maximum dissipation principle, as long with a minimum principle.