Quasi-Static Griffith Fracture Evolution with Boundary Loads

Yousefi, Pooya

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Quasi-Static Griffith Fracture Evolution with Boundary Loads

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In this dissertation, we study the well-posedness of a variational formulation for modeling quasi-static evolution of cracks in elastic materials under boundary loads. Quasi-static evolution of fracture for displacement loads, i.e., Dirichlet boundary conditions, has been studied extensively in the past couple of decades, using models based on global and local minimization. However, boundary loads, i.e., Neumann boundary conditions, had been seen as problematic with the usual variational formulation, due to a straightforward non-existence argument. Recently, a variational formulation, namely dual minimization, was proposed as a method for finding solutions for fracture problem with boundary loads. Adopting this method, we study the existence of quasi-static fracture evolutions under time-varying boundary loads. Global minimizers of the quasi-static Dirichlet problem have always balanced the sum of stored elastic plus crack dissipated surface energies. Nonetheless, even though our formulation for the quasi-static Neumann problem is based on global minimization, we show that evolutions here do not necessarily satisfy this energy balance, and describe how there can be decreases in the energy. Note that decreases in the sum of stored and dissipated energies in time might be expected since the effect of kinetic energy caused by the jumps in the evolution of cracks is not considered in the quasi-static energy equation. We also give estimates on how big energy drops can be. Also, in a separate problem, we prove that a regularized Ambrosio-Tortorelli type energy functional that models fracture in layered structures with interfaces Γ−converges to a sharp interface energy, where the surface energy of a crack at the interface is proportional to an effective toughness, that in a sense averages the toughness of the interface and the bulk materials, whereas away from the interface, it is proportional to the toughness of the bulk.

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