A Variational Approach to Fracture and Other Inelastic Phenomena

The aim of the present article is a description of the many phenomenological aspects of fracture and of their relations with other inelastic phenomena, such as the formation of microstructure and the changes in the material's strength induced by plasticity and damage. The whole analysis is carried on using a single mathematical tool, the incremental energy minimization. The energy functional is assumed to be made of two parts, an elastic energy and a cohesive energy. The structure assumed for the cohesive energy term determines different modes of inelastic response and fracture. To avoid the heavy technical difficulties met in higher dimension, the whole analysis is one-dimensional. This article reflects the contents of six lectures delivered in a couple of occasions. The object of the first lecture is Griffith's theory of brittle fracture. In the second lecture, the Barenblatt-Dugdale regularization is discussed. In Lecture 3 it is shown that the concept of cohesive energy can be successfully used to describe the formation of microstructure. A description of the phenomenon of elastic unloading, based on the assumption of dissipativity of the cohesive energy, is given in Lecture 4. The last two lectures deal with the diffuse cohesive energy model. In it, the cohesive energy, instead of being defined on surfaces as usual, is supposed to be diffused over the volume. In particular, the non-local model discussed in Lecture 6 provides a comprehensive description of the strain-softening phenomenon.