# Calculation of T-stress for cracks in two-dimensional anisotropic elastic media by boundary integral equation method

- 33 Downloads

## Abstract

A numerical procedure is proposed to compute the T-stress for two-dimensional cracks in general anisotropic elastic media. T-stress is determined from the sum of crack-face displacements which are computed via an integral equation of the boundary data. To smooth out the data in order to perform accurately numerical differentiation, the sum of crack-face displacement is established in a weak-form integral equation in which the integration domain is simply the crack-tip element. This weak-form integral equation is then solved numerically using standard Galerkin approximation to obtain the nodal values of the sum of crack-face displacements. The procedure is incorporated in a weakly-singular symmetric Galerkin boundary element method in which all integral equations for the traction and displacement on the boundary of the domain and on the crack faces include (at most) weakly-singular kernels. To examine the accuracy and efficiency of the developed method, various numerical examples for cracks in infinite and finite domains are treated. It is shown that highly accurate results are obtained using relatively coarse meshes.

## Keywords

T-stress SGBEM Crack Anisotropic Weakly singular## References

- Al-Ani AM, Hancock JW (1991) J-dominance of short cracks in tension and bending. J Mech Phys Solids 39:23–43CrossRefGoogle Scholar
- Ayatollahi MR, Pavier MJ, Smith DJ (1998) Determination of t-stress from finite element analysis for mode I and mixed mode I/II loading. Int J Fract 91:283–298CrossRefGoogle Scholar
- Cardew GE, Goldthorpe MR, Howard IC, Kfouri AP (1985) On the elastic T-Term, fundamentals of deformation and fracture. In: Bilby BA, Miller KJ, Willis JR (eds) Proceedings of the Eshelby memorial symposium, sheffield, 2–5 April 1984, pp 465–476Google Scholar
- Carka D, Mear ME, Landis CM (2011) The Dirichlet-to-Neumann map for two-dimensional crack problems. Comput Methods Appl Mech Eng 200:1263–1271CrossRefGoogle Scholar
- Chen CS, Krause R, Pettit RG, Banks-Sills L, Ingraffea AR (2001) Numerical assessment of T-stress computation using a p-version finite element method. Int J Fract 107:177–199CrossRefGoogle Scholar
- Cotterell B, Ricet JR (1980) Slightly curved or kinked cracks. Int J Fract 16:155–169CrossRefGoogle Scholar
- Gao H, Chiu C-H (1992) Slightly curved or kinded cracks in anisotropic elastic solids. Int J Solids Struct 29:947–972CrossRefGoogle Scholar
- Henry BS, Luxmoore AR (1995) Three-dimensional evaluation of the T-stress in centre cracked plates. Int J Fract 70:35–50CrossRefGoogle Scholar
- Hoenigh A (1982) Near-tip behavior of a crack in a plane anisotropic elastic body. Eng Fract Mech 16:393–403CrossRefGoogle Scholar
- Kfouri AP (1986) Some evaluations of the elastic T-term using Eshelby’s method. Int J Fract 30:301–315CrossRefGoogle Scholar
- Kim J, Paulino GH (2003) T-stress, mixed-mode stress intensity factors, and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method. Comput Methods Appl Mech Eng 192:1463–1494CrossRefGoogle Scholar
- Kim J, Paulino GH (2004) T-stress in orthotropic functionally graded materials: Lekhnitskii and Stroh formalisms. Int J Fract 126:345–384CrossRefGoogle Scholar
- Larsson SG, Carlsson AJ (1973) Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack tips in elastic-plastic materials. J Mech Phys Solids 21:263–277CrossRefGoogle Scholar
- Leevers PS, Radon JC (1982) Inherent stress biaxiality in various fracture specimen geometries. Int J Fract 19:311–325CrossRefGoogle Scholar
- Melin S (2002) The influence of the T-stress on the directional stability of cracks. Int J Fract 114:259–265CrossRefGoogle Scholar
- Nakamura T, Parks DM (1992) Determination of T-stress along three-dimensional crack fronts using an interaction integral method. Int J Solids Struct 29:1597–1611CrossRefGoogle Scholar
- Phan A-V (2011) A non-singular boundary integral formula for determining the T-stress for cracks of arbitrary geometry. Eng Fract Mech 78:2273–2285CrossRefGoogle Scholar
- Richardson JD, Cruse TA (1999) Weakly singular stress-BEM for 2D elastostatics. Int J Numer Methods Eng 45:13–35CrossRefGoogle Scholar
- Rungamornrat J, Mear ME (2008) A weakly-singular SGBEM for analysis of cracks in 3D anisotropic media. Comput Methods Appl Mech Eng 197:4319–4332CrossRefGoogle Scholar
- Shah PD, Tan CL, Wang X (2005) T-stress solutions for two-dimensional crack problems in anisotropic elasticity using boundary element method. Fatigue Fract Eng Mater Struct 29:342–356Google Scholar
- Sladek J, Sladek V, Fedelinski P (1997) Contour integrals for mixed-mode crack analysis: effect of nonsingular terms. Theor Appl Fract Mech 29:115–127CrossRefGoogle Scholar
- Su RKL, Sun HY (2003) Numerical solutions of two-dimensional anisotropic crack problems. Int J Solids Struct 40:4615–4635CrossRefGoogle Scholar
- Su RKL, Sun HY (2005) A brief note on elastic T-stress for centred crack in anisotropic plate. Int J Fract 131:53–58CrossRefGoogle Scholar
- Sutradhar A, Paulino GH (2004) Symmetric Galerkin boundary element computation of T-stress and stress intensity factors for mixed-mode cracks by the interaction integral method. Eng Anal Bound Elem 28:1335–1350CrossRefGoogle Scholar
- Tran HD, Mear ME (2013) Regularized boundary integral equations for two-dimensional crack problems in multi-field media. Int J Fract 181:99–113CrossRefGoogle Scholar
- Tran HD, Mear ME (2014) A weakly singular SGBEM for analysis of two-dimensional crack problems in multi-field media. Eng Anal Bound Elem 41:60–73CrossRefGoogle Scholar
- Ueda Y, Ikeda K, Yao T, Aoki M (1983) Characteristics of brittle failure under general combined modes including those under bi-axial tensile loads. Eng Fract Mech 18:1131–1158CrossRefGoogle Scholar
- Williams ML (1957) On the stress distribution at the base of a stationary crack. J Appl Mech 24:109–114Google Scholar
- Williams JG, Ewing PD (1972) Fracture under complex stress—the angled crack problem. Int J Fract 8:416–441Google Scholar
- Yang B, Ravi-Chandar K (1999) Evaluation of elastic T-stress by the stress difference method. Eng Fract Mech 64:589–605CrossRefGoogle Scholar
- Yang S, Yuan F-G (2000a) Determination and representation of the stress coefficient term by path-independent integrals in anisotropic cracked solids. Int J Fract 101:291–319CrossRefGoogle Scholar
- Yang S, Yuan F-G (2000b) Kinked crack in anisotropic bodies. Int J Solids Struct 37:6635–6682CrossRefGoogle Scholar