Why continuum damage is nonlocal: Justification by quasiperiodic microcrack array

Strain-softening damage due to distributed cracking is modeled by an elastic continuum with a quasiperiodic array of cracks of regular spacing but varying sizes. As a model for the initial stage, the cracks are penny-shaped and small compared to their spacing, and as a model for the terminal stage the uncracked ligaments between the cracks are circular and small compared to their spacing. The strain due to cracks and the compliance per crack are calculated. The cracked material is homogenized in such a manner that the macroscopic continuum strains satisfy exactly the condition of compatibility with the actual strains due to cracks, and the macroscopic continuum stress satisfies exactly the condition of work equivalence with the actual stresses in the cracked material. The results show that, contrary to the existing theories, the damage variable used in continuum damage mechanics should be nonlocal, while the elastic part of the response should be local. In particular, the nonlocal continuum damage should be considered as a function of the spatial average of the cracking strain rather than its local value. The size of the averaging region is determined by the crack spacing.