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The generalized finite difference method for in-plane crack problems. (English) Zbl 1404.74194
Summary: This paper documents the first attempt to apply the generalized finite difference method (GFDM), a recently-developed meshless method, for the numerical solution of problems with cracks in general anisotropic materials. To solve the resulting second-order elliptic partial differential equations with mixed boundary conditions, the explicit formulae for the partial derivatives of unknown functions in the equations are derived by using the Taylor series expansions combining with the moving-least squares approximation in this meshless GFD method. To deal with the strong discontinuous crack-faces, some special treatments are applied to modify this GFDM. The node distributions are locally refined in the vicinity of the crack-tips. By dividing the crack domain into two parts, the sub-domain method is also used for comparing with the single-domain method. The direct displacement extrapolation method, the path-independent \(J\)-integral and the interaction integral methods are, respectively, used to compute the stress intensity factors and compared. Finally, some classical crack examples are presented to show the effectiveness and accuracy of the proposed meshless method for crack problems.

74S20 Finite difference methods applied to problems in solid mechanics
65N06 Finite difference methods for boundary value problems involving PDEs
74R10 Brittle fracture
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