One-dimensional ductile-brittle transition, yielding, and structured deformations

The present communication refers to some research in progress, based upon the use of structured deformations, introduced by D.R. Owen and the present author to study the mechanics of bodies undergoing changes in internal structure, and on some recent work of Choksi and Fonseca on the definition and on the mathematical properties of the energy associated with such deformations. I show that, prescribing only the forms of the bulk energy of the material constituting the bar and of the energy of the cohesive cracks, and characterizing the equilibrium configurations as local minima of the total energy, it is possible to reproduce some phenomena, such as the formation of a crack or a change in the internal structure of the material, and to predict the conditions of their initiation. The form of the bulk energy need not allow for strain softening, and no initial cracks, inhomogeneities, imperfections, or weakening bands are needed. The formation of a crack may cause either ductile or brittle fracture, and the type of fracture is influenced by the total length of the bar, which therefore plays the role of a scale factor