A one-dimensional model for localized and distributed failure

For an elastic body with limited strength, the equilibrium configurations can be obtained by minimization of an energy functional containing two contributions, bulk and cohesive: the bulk energy is a function of strain and the cohesive energy is a function of the relative displacement on a surface of discontinuity. In the present communication we consider the simplest one-dimensional problem for a bar with this type of energy in a hard device. We assume that the bulk energy is convex, and we vary the concavity properties of the cohesive energy, obtaining thereby three distinct modes of failure. If the cohesive energy is concave for all admissible displacements, failure occurs with the formation of a single crack, and the opening of the crack may be either abrupt or gradual, depending on the length of the bar. If the cohesive energy is concave at large displacements but convex at the origin, the deformation may progress at constant stress (yielding), through formation of an infinite numer of infinitesimal cracks (structured deformation). Finally, when the cohesive energy is characterized by two domains of concavity, (in the vicinity, and far away from the origin), separated by a domain of convexity, fracture precedes through a successive formation of a finite number of cracks of small but finite size. We conjecture that the different modes of fracture, produced by this simple model, may be associated with various experimentally well-documented regimes of localized and distributed damage.