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In this paper, we present numerical computational methods for solving the fracture problem in brittle and ductile materials with no prior knowledge of the topology of crack path. Moreover, these methods are capable of modeling the crack initiation. We perform numerical simulations of pieces of brittle material based on global approach and taken into account the thermal effect in crack propagation. On the other hand, we propose also a numerical method for solving the fracture problem in a ductile material based on elements deletion method and also using thermo-mechanical behavior and damage laws. In order to achieve the last purpose, we simulate the orthogonal cutting process.

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