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A weakly-singular SGBEM for analysis of cracks in 3D anisotropic media. (English) Zbl 1194.74501
Summary: A weakly-singular, symmetric Galerkin boundary element method (SGBEM) is developed for analysis of fractures in three-dimensional, anisotropic linearly elastic media. The method constitutes a generalization of that by S. Li, M.E. Mear and L. Xiao [Comput. Methods Appl. Mech. Eng. 151, No. 3-4, 435–459 (1998; Zbl 0906.73074)] for isotropic media, and is based upon a pair of weak-form displacement and traction integral equations recently established by J. Rungamornrat and M. E. Mear [Int. J. Solids Struct. 45, No. 5, 1283–1301 (2008; Zbl 1169.74549)]. The formulation involves only weakly-singular kernels and, as a consequence, standard ${C}^{0}$ elements can be employed in the numerical discretization. The kernels possess a relatively simple form (for general anisotropy) which involves an equatorial line integral like that associated with the displacement fundamental solution. Despite their simple form, it is still necessary to evaluate these kernels efficiently in the numerical implementation; to do so, certain symmetry properties associated with the specific material type under consideration are exploited, and a simple yet accurate interpolation strategy is introduced. Another important aspect of the numerical treatment is the use of a special crack-tip element which allows highly accurate mixed-mode stress intensity factor data to be extracted as a function of position along the crack front. This crack-tip element was originally introduced by S. Li, M.E. Mear and L. Xiao [Comput. Methods Appl. Mech. Eng. 151, No. 3-4, 435–459 (1998; Zbl 0906.73074)] in their treatment of isotropic media, and here it is extended to allow evaluation of the stress intensity factors for generally anisotropic media. The element involves “extra” degrees of freedom associated with the nodes along the crack front (with these quantities being directly related to the stress intensity factors), and it is capable of representing the asymptotic behavior in the vicinity of the crack front to a sufficiently high order that relatively large crack-tip elements can be utilized. Various examples are treated for cracks in an unbounded domain and for both embedded and surface-breaking cracks in a finite domain, and it is demonstrated that very accurate results can be obtained even with relatively coarse meshes.
##### MSC:
 74S15 Boundary element methods in solid mechanics 74R10 Brittle fracture 74E10 Anisotropy