# zbMATH — the first resource for mathematics

A new formulation of boundary element method for cracked anisotropic bodies under anti-plane shear. (English) Zbl 1050.74052
Summary: By applying integration by parts and other techniques to the traditional boundary integral formulation, a new boundary integral equation is derived to analyze cracked anisotropic bodies under anti-plane shear. The new boundary formulation uses dislocation density as unknown on the crack surface, from which the stress intensity factor $$K_{\text{III}}$$ is determined near the crack tip. The boundary element method based on the new equation is established, in which the singular interpolation functions are introduced to model the singularity of dislocation density (on the order of $$1/{\sqrt r}$$) appropriately. The present traction boundary integral equation involves only singularity of order $$1/r$$ (no hyper-singular terms appear), and all the singular integrals can be evaluated exactly. The equation and method can be directly used for anti-plane problems with cracks of any geometric shapes. Applications to isotropic and anisotropic cases are provided, and numerical results demonstrate the accuracy of our boundary element method.

##### MSC:
 74S15 Boundary element methods applied to problems in solid mechanics 74R10 Brittle fracture 74E10 Anisotropy in solid mechanics 74G70 Stress concentrations, singularities in solid mechanics
##### Keywords:
stress intensity factor; crack tip; singularity
Full Text:
##### References:
 [1] Snyder, M.D.; Cruse, T.A., Boundary-integral equation analysis of cracked anisotropic plates, Int. J. fract., 11, 315-328, (1975) [2] Sollero, P.; Aliabadi, M.H., Fracture mechanics analysis of anisotropic plates by the boundary element method, Int. J. fract., 64, 269-284, (1993) [3] Wang, Y.H.; Cheung, Y.K.; Woo, C.W., Anti-plane shear problem for an edge crack in a finite orthotropic plate, Engrg. fract. mech., 42, 971-976, (1992) [4] Crouch, S.L., Solution of plane elasticity problems by the displacement discontinuity method, Int. J. numer. methods engrg., 10, 301-343, (1976) · Zbl 0322.73016 [5] Portela, A.; Aliabadi, M.H.; Rooke, D.P., The dual boundary element method: effective implementation for crack problems, Int. J. numer. methods engrg., 33, 1269-1287, (1992) · Zbl 0825.73908 [6] Chau, K.T.; Wang, Y.B., A new boundary integral formulation for plane elastic bodies containing cracks and holes, Int. J. solids struct., 36, 2041-2074, (1999) · Zbl 0963.74070 [7] Wang, Y.B.; Chau, K.T., A new boundary element for plane elastic problems involving cracks and holes, Int. J. fract., 87, 1-20, (1997) · Zbl 0900.73895 [8] Sih, G.C.; Chen, E.P., Cracks in materials possessing homogeneous anisotropic, () · Zbl 0488.73105 [9] Vinson, J.R.; Sierakowski, The behavior structures composed of composite materials, (1987), Martinus Nijhoff Dordrecht · Zbl 0624.73068 [10] Brebbia, C.A.; Telle, J.C.F.; Wrobel, L.C., Boundary element techniques, (1984), Springer-Verlag Publishing Berlin · Zbl 0556.73086 [11] Sih, G.C., Handbook of stress intensity factors, (1973), Lehigh University Bethlehem Pennsylvania · Zbl 0263.73057 [12] Zhang, X.S., A central crack at the interface between two different media in a rectangular sheet under anti-plane shear, Engrg. fract. mech., 19, 709-715, (1984) [13] Sun, Y.Z.; Wang, Y.B., Boundary element method for anti-plane crack problem, J. Lanzhou univ. (nature sci.), 37, 4, 25-31, (2001), (in Chinese) [14] Muskhelishvili, N.I., Singular integral equation, (1977), Noordhoof International Publishing Leyden · Zbl 0108.29203 [15] Muskhelishvili, N.I., Some basic problems of mathematical theory of elasticity, (1975), Noordhoff International Publishing Leiden · Zbl 0297.73008
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.