DOI QR코드

DOI QR Code

Properties of integral operators in complex variable boundary integral equation in plane elasticity

  • Chen, Y.Z. (Division of Engineering Mechanics, Jiangsu University) ;
  • Wang, Z.X. (Division of Engineering Mechanics, Jiangsu University)
  • 투고 : 2012.02.04
  • 심사 : 2013.01.11
  • 발행 : 2013.02.25

초록

This paper investigates properties of integral operators in complex variable boundary integral equation in plane elasticity, which is derived from the Somigliana identity in the complex variable form. The generalized Sokhotski-Plemelj's formulae are used to obtain the BIE in complex variable. The properties of some integral operators in the interior problem are studied in detail. The Neumann and Dirichlet problems are analyzed. The prior condition for solution is studied. The solvability of the formulated problems is addressed. Similar analysis is carried out for the exterior problem. It is found that the properties of some integral operators in the exterior boundary value problem (BVP) are quite different from their counterparts in the interior BVP.

키워드

참고문헌

  1. Brebbia, C.A., Telles, J.C.F. and Wrobel, L.C. (1984), Boundary Element Techniques - Theory and Applications in Engineering, Springer, Heidelberg
  2. Chen, J.T., Liang, M.T. and Yang, S.S. (1995), "Dual boundary integral equations for exterior problems", Eng. Anal. Bound. Elem., 16, 333-340. https://doi.org/10.1016/0955-7997(95)00078-X
  3. Chen, J.T. and Hong, H.K. (1999), "Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series", Appl. Mech. Rev., 52, 17-33. https://doi.org/10.1115/1.3098922
  4. Chen, J.T. and Chen, Y.W. (2000), "Dual boundary element analysis using complex variable for potential problems with or without a degenerate boundary", Eng. Anal. Bound. Elem., 24, 671-684. https://doi.org/10.1016/S0955-7997(00)00025-4
  5. Chen, J.T. and Chiu, Y.P. (2002a), "On the Pseudo- differential operators in the dual boundary integral equations using degenerate kernels and circulants", Eng. Anal. Bound. Elem., 26, 41-53. https://doi.org/10.1016/S0955-7997(01)00087-X
  6. Chen, J.T., Kuo, S.R. and Lin, J.H. (2002b), "Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two-dimensional elasticity", Int. J. Numer. Meth. Eng., 54, 1669-1681. https://doi.org/10.1002/nme.476
  7. Chen, J.T., Lin, S.R. and Chen, K.H. (2005), "Degenerate Scale problem when solving Laplace's equation by BEM and its treatment", Int. J. Numer. Meh. Eng., 62, 233-261. https://doi.org/10.1002/nme.1184
  8. Chen, J.T. and Wu, A.C. (2007), "Null-field approach for the multi-inclusion problem under antiplane shears", J. Appl. Mech., 74, 469-487. https://doi.org/10.1115/1.2338056
  9. Chen, J.T. and Lee, Y.T. (2009), "Torsional rigidity of a circular bar with multiple circular inclusions using the null-field integral approach", Comput. Mech., 44, 221-232. https://doi.org/10.1007/s00466-009-0365-8
  10. Chen, Y.Z. and Lin X.Y. (2006), "Complex potentials and integral equations for curved crack and curved rigid line problems in plane elasticity", Acta Mech., 182, 211-230. https://doi.org/10.1007/s00707-005-0305-4
  11. Chen, Y.Z., Wang, Z.X. and Lin, X.Y. (2007), "Eigenvalue and eigenfunction analysis arising from degenerate scale problem of BIE in plane elasticity", Eng. Anal. Bound. Elem., 31, 994-1002. https://doi.org/10.1016/j.enganabound.2007.05.003
  12. Chen, Y.Z. and Lin, X.Y. (2008), "Regularity condition and numerical examination for degenerate scale problem of BIE for exterior problem of plane elasticity", Eng. Anal. Bound. Elem., 32, 811-823. https://doi.org/10.1016/j.enganabound.2008.02.004
  13. Chen, Y.Z., Wang, Z.X. and Lin, X.Y. (2009), "A new kernel in BIE and the exterior boundary value problem in plane elasticity", Acta Mech., 206, 207-224. https://doi.org/10.1007/s00707-008-0088-5
  14. Chen, Y.Z., Lin, X.Y. and Wang, Z.X. (2010) "Influence of different integral kernels on the solutions of boundary integral equations in plane elasticity", J. Mech. Mater. Struct., 5, 679-692. https://doi.org/10.2140/jomms.2010.5.679
  15. Chen, Y.Z., Hasebe, N. and Lee, K.Y. (2003), Multiple Crack Problems in Elasticity, WIT Press, Southampton.
  16. Cheng, A.H.D. and Cheng, D.S. (2005) "Heritage and early history of the boundary element method", Eng. Anal. Bound. Elem., 29, 286-302.
  17. Cruse, T.A. (1969), "Numerical solutions in three-dimensional elastostatics", Int. J. Solids Struct., 5, 1259- 1274. https://doi.org/10.1016/0020-7683(69)90071-7
  18. Cruse, T.A. and Suwito, W. (1993), "On the Somigliana stress identity in elasticity", Comput. Mech., 11, 1-10. https://doi.org/10.1007/BF00370069
  19. Davey, K. and Farooq, A. (2011), "Evaluation of free terms in hypersingular boundary integral equations", Eng. Anal. Bound. Elem., 35, 1060-1074. https://doi.org/10.1016/j.enganabound.2011.04.002
  20. Jaswon, M.A. and Symm, G.T. (1967), Integral Equation Methods in Potential Theory and Elastostatics, Academic Press, London.
  21. Kolte, R., Ye, W., Hui, C.Y. and Mukherjee, S. (1996), Complex variable formulations for usual and hypersingular integral equations for potential problems- with applications to corners and cracks, Comput. Mech., 17, 279-286. https://doi.org/10.1007/BF00368550
  22. Linkov, A.M. (2002), Boundary Integral Equations in Elasticity, Kluwer, Dordrecht.
  23. Mogilevskaya, S.G. and Linkov, A.M. (1998), "Complex fundamental solutions and complex variables boundary element method in elasticity", Comput. Mech., 22, 88-92. https://doi.org/10.1007/s004660050342
  24. Mogilevskaya, S.G. (2000), "Complex hypersingular equation for piece-wise homogenous half-plane with cracks", Inter. J. Fract., 102, 177-204. https://doi.org/10.1023/A:1007633814813
  25. Muskhelishvili, N.I. (1953), Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, The Netherlands.
  26. Rizzo, F.J. (1967), "An integral equation approach to boundary value problems in classical elastostatics", Quart. J. Appl. Math., 25, 83-95.
  27. Savruk, M.P. (1981), Two-dimensional problems of elasticity for body with crack, Naukoya Dumka, Kiev. (In Russian)
  28. Vodicka, R. and Mantic, V. (2004), "On invertibility of elastic single-layer potential operator", J. Elastics, 74, 147-173. https://doi.org/10.1023/B:ELAS.0000033861.83767.ce
  29. Vodicka, R. and Mantic, V. (2008), "On solvability of a boundary integral equation of the first kind for Dirichlet boundary value problems in plane elasticity", Comput. Mech., 41, 817-826. https://doi.org/10.1007/s00466-007-0202-x
  30. Whitley, R.J. and Hromadka, II.T.V. (2006), "Theoretical developments in the complex variable boundary element method", Eng. Anal. Bound. Elem., 30, 1020-1024. https://doi.org/10.1016/j.enganabound.2006.08.002
  31. Zhang, X.S. and Zhang X.X. (2008), "Exact solution for the hypersingular boundary integral equation of two-dimensional elastostaticcs", Struct. Eng. Mech., 30, 279-296. https://doi.org/10.12989/sem.2008.30.3.279

피인용 문헌

  1. Properties of integral operators and solutions for complex variable boundary integral equation in plane elasticity for multiply connected regions vol.52, 2015, https://doi.org/10.1016/j.enganabound.2014.11.009
  2. Numerical solution of the t-version complex variable boundary integral equation for the interior region in plane elasticity vol.46, 2014, https://doi.org/10.1016/j.enganabound.2014.05.007
  3. Some general properties in the degenerate scale problem of antiplane elasticity or Laplace equation vol.64, pp.6, 2017, https://doi.org/10.12989/sem.2017.64.6.695
  4. Solution for null field CVBIE in plane elasticity using an accurate shape function vol.6, pp.2, 2021, https://doi.org/10.12989/acd.2021.6.2.077