A reformulation of nonlocal strain-driven continuum damage mechanics models using peridynamic differential operators

This paper presents some analytical solutions for the static response of a nonlocal continuum damage mechanics (CDM) softening rod in tension. Two families of nonlocal CDM models valid for quasi-brittle materials are explored in this study: a ‘classical’ nonlocal strain-driven CDM model and a peridynamic CDM model. For both nonlocal CDM models, the elasto-damage constitutive law and the damage loading function can be affected through two independent scale-dependence kernels. It is chosen in this paper, to focus on the introduction of the nonlocality, for both CDM models, in the damage loading function. It is shown that the response of a strain-driven nonlocal damage rod may coincide with the one of a peridynamic nonlocal damage rod, for specific calibrated exponential kernels. Both nonlocal damage models, the strain-driven and the peridynamic damage models, governed by integral equations or integro-differential equations, are reformulated in a nonlocal differential framework. The propagation of the localization zone during the softening process is analytically investigated, by solving a moving damage boundary value problem. The size of the damage localization zone is shown to be loading-dependent. The strain profiles for both models are obtained and confirm the strain localization in the finite damage structural area of the homogeneous nonlocal rod in tension. The paper concludes by an analysis of the scale-dependence response of such nonlocal, strain-driven and peridynamic damage rods.