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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 propertiess 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 procedes 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. A standard model in fracture mechanics is based on the assumption that the total energy of a body is a sum of a bulk term, representing the strain energy, and of a surface term, representing the energy associated with the displacement discontinuity. This assumption was introduced by Griffith, and modified later by Barenblatt to account for the cohesive forces which oppose fracture opening. A one-dimensional model of this type with convex bulk energy and concave cohesive energy is capable of reproducing a phenomenon of localized fracture: for a bar subject to a prescribed elongation, the minimum of the total energy corresponds to configurations with a single crack 12, 4, 51. The overall response in this case may be either discontinous, with an abrupt drop of stress, or gradual, with a continuous decrease of stress, depending on the length of the bar. These two regimes, which one can loosely associate with brittle and ductile fracture, can also be obtained from a model of a chain made of nonlinear springs with a Lennard-Jones potential: for a sufficiently large number of springs, this discrete system can be approximated by a continuum model with a convex bulk and a concave cohesive energy (8). This type of behavior changes drastically if the cohesive energy is convex at the origin and concave away from the origin (5). In this case, for elongations belonging to a certain interval, there are no piecewise continuous minimizers, and the energy minimum is attained at a configuration with an infinite number of infinitesimal cracks. This situation can be described in the context of the theory of structured defomzations (6). The resulting expression of the energy minimum in the one-dimensional model agrees with the three- dimensional relaxation in the class of functions with this level of regularity (3). In the present communication we report some preliminary results for a one-dimensional model with a cohesive energy which is convex on a finite segment separated from the origin and concave outside. As we show, this model predicts the formation of a finite number of cracks, one after another, as the total elongation increases (7). This regime of distributed cracking can be viewed as a "quantized propagation of damage. When the concave region near the origin shrinks to zero the model recovers structured deformations, and when the convex region disappears a localized fracture appears as another limiting case.