PSI - Issue 33

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Derouiche Sami et al. / Procedia Structural Integrity 33 (2021) 996–1006 Author name / Structural Integrity Procedia 00 (2019) 000–000

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Fig. 2. (Left) Crack geometry for a kinking (extension δa). (Right) Reorganization of the mesh around the new tip crack A rearrangement of the mesh is necessary around the crack tip affecting four elements; two elements containing the crack extension on its new direction and two belonging to the surrounding of the crack tip; the rest of the mesh remain unchanged, as presented in Fig. 2 (Right). 2.1. Stiffness derivative procedure The method to compute the energy release rates G is developed by Bouzerd (1992). It is evaluated by taking the parameters in the « a �  a » configuration with kinking. The concerned elements will store the elementary matrices of the “a” setup with “ δa ” cancelled. Resuming all the work already done, and taking into account the linearly elastic behavior of small displacements, the analytical solution found in the structure with a crack length "a" and in the same structure with a crack length a � δa are quite similar especially when the variation δa is smaller in contrast with the dimensions of the crack tip. Then it is possible to say that        v a v a a (12) Previous results affirm the Eq(10) that however theoretically coherent and physical satisfying; as long as the conditions are taken into account and respected. Identical process in the analysis of cracked structures is used previously Petit(1990). If we consider that the external loading does not change during the extension δa , then the energy release rate G is estimated as follows: ( ) ( ) a a a G a         (13) where ��a � δa� and ��a� characterize respectively the deformation energy of the cracked structure in the study cases «a � δa » and « a » . In its discretised form, the deformation energy is:         1 1 { } 2 ne T i i i i v K v (14) With ne is total number of elements in discretized structure, ��� i is vertical vector containing the nodal values of element i, and ��� i is elementary matrix of element i, and the exponent t indicates the transposed vector.