A computational procedure for analysis of fractures in three dimensional anisotropic media



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A symmetric Galerkin boundary element method (SGBEM) is developed for analysis of fractures in three dimensional anisotropic, linearly elastic media, and the method is coupled with standard finite element procedures. Important features of the technique are that the formulation is applicable to general anisotropy, the kernels in the governing integral equations are only weakly-singular (of order 1/r) hence allowing the application of standard Co elements in the numerical treatment, and a special crack tip element is utilized which allows general mixed-mode fracture data (viz. the stress intensity factors) to be efficiently determined as a function of position along the crack front. The weakly-singular, weak-form displacement and traction integral equations which constitute a basis for the SGBEM are obtained via a regularization technique. The technique utilizes a particular decomposition for the stress fundamental solution and for the strongly-singular kernel in order to facilitate an integration by parts via Stokes’ theorem. The final integral equations contain only weakly-singular kernels (given explicitly in terms of a line integral) which are applicable to gen- eral anisotropic materials. These weakly-singular kernels are obtained by solving a system of partial differential equations via the Radon transform. A symmetric formulation is developed by a suitable use of the weakly-singular displacement and traction integral equations. As part of the numerical implemen- tation, a Galerkin approximation strategy is utilized to discretize the governing integral equations. Standard isoparametric Co elements are employed everywhere except along the crack front where a special crack-tip elements is used. To demon- strate the accuracy and versatility of the method, various examples for cracks in both unbounded and finite domains are considered. Finally, a symmetric coupling of the SGBEM and the standard finite element method is established. The coupling strategy exploits the versatility and capabil- ity of the finite element method to treat structures with complex geometry and loading, while employing the SGBEM to efficiently and accurately treat a (local) region containing the crack. In the numerical implementation, both conforming and nonconforming discretization along the interface of the two regions are treated. In addition, the coupling of the SGBEM with a commercial finite element code is ex- plored and successfully implemented. Several examples are presented to illustrate the capability and accuracy of the method.