A diffuse cohesive energy approach to fracture and plasticity: the one-dimensional case

In the fracture model presented in this paper, the basic assumption is that the energy is the sum of two terms, one elastic and one cohesive, depending on the elastic and inelastic part of the deformation, respectively. Two variants are examined: a local model, and a nonlocal model obtained by adding a gradient term to the cohesive energy. While the local model only applies to materials which obey Drucker's postulate and only predicts catastrophic failure, the nonlocal model describes the softening regime and predicts two collapse mechanisms, one for brittle fracture and one for ductile fracture. In its nonlocal version, the model has two main advantages over the models existing in the literature. The first is that the basic elements of the theory (the yield function, hardening rule, and evolution laws) are not assumed, but are determined as necessary conditions for the existence of solutions in incremental energy minimization. This reduces to a minimum the number of independent assumptions required to construct the model. The second advantage is that, with appropriate choices of the analytical shape of the cohesive energy, it becomes possible to reproduce, with surprising accuracy, a large variety of observed experimental responses. In all cases, the model provides a description of the entire evolution, from the initial elastic regime to final rupture.