Energy Minimizing Brittle Crack Propagation

We propose a minimizing movement model for quasi-static brittle crack evolution. Cracks (fissures) appear and/or grow without any prescription of their shape or location when time-dependent displacements are imposed on the exterior boundary of the body. We use an energetic approach based on Mumford-Shah type functionals. By the discretization of the time variable we obtain a sequence of free discontinuity problems. We find exact solutions and estimations which lead us to the conclusion that in this model crack appearance is allowed but the constant of Griffith G and the critical stress which causes the fracture in an uni-dimensional traction experiment cannot be both constants of material. A weak formulation of the model is given in the frame of special functions with bounded deformation. We prove the existence of weak constrained incremental solutions of the model. A partial existence result for the minimizing movement model is obtained under the assumption of uniformly bounded (in time) power communicated to the body by the rest of the universe. The model is of applicative interest. A numerical approach and examples, using an Ambrosio–Tortorelli variational approximation of the energy functional, are given in the last section.

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