On bifurcation in local and nonlocal materials with tension and compression damage

In the present work an isotropic damage model for concrete accounting separately for progressive degradation of the elastic properties under prevailing tension and compression stress states is considered in both local and nonlocal forms. The possible occurrence of material instabilities in the form of strain localization is analyzed. For the local model, the bifurcation conditions at the onset of strain localization are first solved in terms of the critical values of the damage variables. Next, in order to avoid ill-posedness of the initial-boundary value problem, two nonlocal enhancements are provided through integral averaging or gradient dependency. The regularizing properties of both nonlocal formulations are explored. The expressions of the characteristic lengths implicitly introduced by the nonlocal enhancements are derived through a one-dimensional wave propagation analysis. Such results are confirmed by preliminary numerical FE simulations of the quasi-static response of a tensile bar analyzed with the nonlocal damage model.