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Mode III fracture analysis by Trefftz boundary element method. (English) Zbl 1202.74193

Summary: This paper presents a hybrid Trefftz (HT) boundary element method (BEM) by using two indirect techniques for mode III fracture problems. Two Trefftz complete functions of Laplace equation for normal elements and a special purpose Trefftz function for crack elements are proposed in deriving the Galerkin and the collocation techniques of HT BEM. Then two auxiliary functions are introduced to improve the accuracy of the displacement field near the crack tips, and stress intensity factor (SIF) is evaluated by local crack elements as well. Furthermore, numerical examples are given, including comparisons of the present results with the analytical solution and the other numerical methods, to demonstrate the efficiency for different boundary conditions and to illustrate the convergence influenced by several parameters. It shows that HT BEM by using the Galerkin and the collocation techniques is effective for mode III fracture problems.

MSC:

74S15 Boundary element methods applied to problems in solid mechanics
74R10 Brittle fracture
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[1] Trefftz, T.E.: Ein Gegenstück zum Ritzschen Verfahren. In: Proceedings 2nd International Congress of Applied mechanics, Zurich, pp. 131–137 (1926)
[2] Huang S.C. and Shaw R.P. (1995). The Trefftz method as an internal equation. Adv. Eng. Softw. 24: 57–63 · Zbl 0984.65507 · doi:10.1016/0965-9978(95)00058-5
[3] Mikhlin S.G. (1964). Variational Methods in Mathematical Physics. Pergamon Press, Oxford · Zbl 0119.19002
[4] Kupradze, V.D.: Three Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelasticity. North Holland, Amsterdam (1979) · Zbl 0421.73009
[5] Piltner R. (1995). Rencent developments in the Trefftz method for finite element and boundary element applications. Adv. Eng. Softw. 24: 107–115 · Zbl 0984.65504 · doi:10.1016/0965-9978(95)00063-1
[6] Zienkiewicz, O.C.: Trefftz type of approximation and the finite element method–history and development. In: First International Workshop on Trefftz Method–Recent Development and Perspectives. Cracow University of Technology, Poland, pp. 103 (1996)
[7] Jirousek, J., Zielinski, A.P.: Survey of Trefftz-type element formulations. In: First International Workshop on Trefftz Method–Recent Development and Perspectives. Cracow University of Technology, Poland, pp. 103 (1996)
[8] Qin Q.H. (2000) The Trefftz Finite and Boundary Element Method. WIT Press, Southampton · Zbl 0982.74003
[9] Jirousek, J., Venkatesh, A.: Hybrid-Trefftz plane elasticity elements with p-method capabilities. Int. J. Numer. Meth. Eng. 35, 1443–1472 (1992) · Zbl 0775.73259 · doi:10.1002/nme.1620350705
[10] Piltner R. (1985). Special finite elements with holes and internal cracks. Int. J. Numer. Meth. Eng. 21: 1471–1485 · Zbl 0572.73109 · doi:10.1002/nme.1620210809
[11] Venkatesh A. and Jirousek J. (1995). Accurate representation of local effect due to concentrated and discontinuous loads in hybrid-Trefftz plate bending elements. Comput. Struct. 57: 863–870 · Zbl 0900.73796 · doi:10.1016/0045-7949(95)00082-R
[12] Jirousek J. and Teodorescu P. (1982). Large finite elements method for the solution of problems in the theory of elasticity. Comput. Struct. 15: 575–587 · Zbl 0489.73077 · doi:10.1016/0045-7949(82)90009-8
[13] Portela A., Aliabadi M.H. and Rooke D.P. (1992). The dual boundary element method: effective implementation for crack problems. Int. J. Numer. Methods Eng. 33: 1269–1287 · Zbl 0825.73908 · doi:10.1002/nme.1620330611
[14] Freitas J.A.T. and Ji Z.Y. (1996). Hybrid-Trefftz equilibrium model for crack problems. Int. J. Numer. Meth. Eng. 39: 569–584 · Zbl 0845.73073 · doi:10.1002/(SICI)1097-0207(19960229)39:4<569::AID-NME870>3.0.CO;2-8
[15] Sabino J., Portela A. and Castro P.M.S.T. (1999). Trefftz boundary element method applied to fracture mechanics. Eng. Frac. Mech. 64: 67–86 · doi:10.1016/S0013-7944(99)00062-4
[16] Gray L.J. (1987). Boundary Element Method for Regions with Thin Interal Cavities. IBM Bergen, Norway
[17] Cui Y.H. and Qin Q.H. (2006). Discussion on anti-plane crack problems by hybrid Trefftz finite element approach. Acta Mech. Solida Sinica, 27: 167–174
[18] Cui Y.H., Qin Q.H. and Wang J.S. (2006). Application of HT finite element approach on mode I, II and III complex problems. Eng. Mech. 23: 104–110
[19] Portela A. and Charafi A. (1997). Porgramming Trefftz boundary elements. Adv. Eng. Softw. 28: 509–523 · Zbl 05470136 · doi:10.1016/S0965-9978(97)00035-5
[20] Leitão V.M.A. (1998). Applications of multi-region Trefftz-collocation to fracture mechanics. Eng. Anal. Bound. Elem. 22: 251–256 · Zbl 1122.74547 · doi:10.1016/S0955-7997(98)00049-6
[21] Domingues J.S., Portela A. and Castro P.M.S.T. (1999). Trefftz boundary element method applied to fracture mechanics. Eng. Fract. Mech. 64: 67–86 · doi:10.1016/S0013-7944(99)00062-4
[22] Zielinski A.P. and Zienkiewicz O.C. (1985). Generalized finite element analysis with T-complete boundary solution functions. Int. J. Numer. Meth. Eng. 21: 509–528 · Zbl 0594.65081 · doi:10.1002/nme.1620210310
[23] Rooke D.P. and Cartwright D.S. (1976). Compendium of Stress Intensity Factors. HerMajestys Stationery Office, London
[24] Isida M. (1970). Analysis of stress intensity factors for plates containing a random array of cracks. Bull JSME 13: 635–642
[25] Sih C.G. (1973). Handbook of Stress Intensity Factors. HerMajestys Stationery Office, London
[26] Silling, S.A.: Singularities and phase transitions in elastic solids: numerical studies and stability analysis. (Ph. D. thesis), California Institute of Technology (1986)
[27] Horgan C.O. and Silling S.A. (1987). Stress concentration factors in finite anti-plane shear numerical calculations and analytical estimates. J. Elast. 18: 83–91 · Zbl 0611.73051 · doi:10.1007/BF00155438
[28] Zhang X.S. (1988). Thegeneral solution of an edge crack off the center line of a rectangular sheet for Mode III. Eng. Fract. Mech. 31: 847–855 · doi:10.1016/0013-7944(88)90240-8
[29] Sun Y.Z., Yang S.S. and Wang Y.B. (2003). A new formulation of boundary element method for cracked anisotropic bodies under anti-plane shear. Comput. Methods Appl. Mech. Eng. 192: 2633–2648 · Zbl 1050.74052 · doi:10.1016/S0045-7825(03)00297-4
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