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Analysis of 3D cracks in anisotropic multi-material domain with weakly singular SGBEM. (English) Zbl 1195.74254
Summary: A weakly singular, symmetric Galerkin boundary element method (SGBEM) is established for analysis of cracks in a three-dimensional, linearly elastic domain consisting of subdomains which are made from different materials. The integral equations governing each subdomain are obtained by employing a pair of weakly singular, weak-form displacement and traction integral equations. A system of governing equations for the entire domain is subsequently obtained in a symmetric form by properly combining the integral equations for each subdomain along with the use of the continuity of displacements and tractions on the subdomain interface. The technique developed possesses several important features: (1) it is applicable for modeling cracks with arbitrary geometry and under general loading; (2) the governing integral equations contain only weakly singular kernels (of $𝒪\left(1/r\right)$) such that their validity requires only the continuity of the displacement boundary data; (3) the formulation is symmetric and gives rise to a system of linear equations with a symmetric coefficient matrix; and (4) the formulation allows the treatment of a multi-material domain where a material constituting each subdomain can be generally anisotropic. In the numerical implementation, standard ${C}^{o}$ elements are employed everywhere except along the boundary of the crack surface where special crack-tip elements are utilized to model the asymptotic behavior in the vicinity of the crack front. The use of this crack-tip element allows, additionally, the mixed-mode stress intensity factors to be determined directly and accurately in terms of extra degrees of freedom introduced at nodes along the crack front. Several example problems are presented to demonstrate the accuracy and capability of the technique; numerical results indicate that highly accurate stress intensity factors can be obtained with use of relatively coarse meshes.
MSC:
 74S15 Boundary element methods in solid mechanics 74R10 Brittle fracture 74E10 Anisotropy