×

An improved singular curved boundary integral evaluation method and its application in dual BEM analysis of two- and three-dimensional crack problems. (English) Zbl 1477.74112

Summary: In this paper, an improvement is made to an efficient direct method for numerical evaluation of high order singular curved boundary integrals. Then this improved singular integral evaluation method is employed to solve the strongly and hypersingular integrals involved in the dual boundary element method (Dual BEM), which combine the use of displacement and traction boundary integral equations to solve crack problems in a single domain formulation. The singular integral evaluation method is carried out based on a parameter plane expansion and radial integral approach, this paper proposed a new strategy for treating the singular radial integral, which plays a vital role in this method. In isoparametric coordinate system, the singular curved boundary integral is mapped into a singular square plane integral in intrinsic coordinates, then the radial integration method (RIM) is employed to transform the singular square plane integral into a regular line integral over the contour of intrinsic square plane and a singular radial integral over the path from source point to the contour of intrinsic square plane. A singularity isolation technique is utilized to divide the singular radial integral into two parts, the regular radial integral can be evaluated normally using Gauss quadrature and the singular radial that can be evaluated analytically by expanding the non-singular part of the integrand function into a power series. Compared with conventional local interpolation approach to deal with the singular radial integral, the newly proposed method has a more rigorous mathematical derivation, and can achieve more stable and precise results. Based on the successful implementation of direct evaluation of singular boundary integrals, Dual BEM is successfully applied to solve two- and three-dimensional elastic crack problems including straight and curved crack paths with continuous or discontinuous elements. Two different approaches, geometrical extrapolation method and \(J\)-integral method are used in the evaluation of stress intensity factors. Several numerical examples are given to validate effectiveness of the presented method.

MSC:

74S15 Boundary element methods applied to problems in solid mechanics
74R10 Brittle fracture
74G70 Stress concentrations, singularities in solid mechanics

Software:

BEMECH
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aliabadi, M. H., Boundary element formulations in fracture mechanics, Appl. Mech. Rev., 50, 83-96 (1997)
[2] Aliabadi, M. H., A new generation of boundary element methods in fracture mechanics, Int. J. Fract., 86, 91-125 (1997)
[3] Aliabadi, F. M.H.; Rooke, D. P., Numerical Fracture Mechanics (1991), Kluwer Academic Publishers, Computational Mechanics Publications: Kluwer Academic Publishers, Computational Mechanics Publications Dordrecht, Southampton · Zbl 0769.73001
[4] Blandford, G. E.; Ingraffea, A. R.; Liggett, J. A., Two-dimensional stress intensity factor computations using the boundary element method, Int. J. Numer. Methods Eng., 17, 387-404 (1981) · Zbl 0463.73082
[5] Brebbia, C. A.; Dominguez, J., Boundary Elements: an Introductory Course (1996), Computational Mechanics Publicatons: Computational Mechanics Publicatons Southampton · Zbl 0691.73033
[6] Bueckner, H. F., Field singularities and related integral representations, (Sih, G. C., Mechanics of Fracture (1973), Leyden: Leyden the Netherlands: Nordhoff) · Zbl 0319.73055
[7] Chen, J. T., Recent development of dual BEM in acoustic problems, Comput. Methods Appl. Mech. Eng., 188, 833-845 (2000) · Zbl 0980.76055
[8] Chen, J. T.; Chen, Y. W., Dual boundary element analysis using complex variables for potential problems with or without a degenerate boundary, Eng. Anal. Bound. Elem., 24, 671-684 (2000) · Zbl 1001.74115
[9] Chen, J. T.; Hong, H. K., Review of dual boundary element methods with emphasis on hypersingular integrals and divergent series, Appl. Mech. Rev., 52, 17-33 (1999)
[10] Chen, J. T.; Kuo, S. R.; Lin, J. H., Analytical study and numerical experiments for degenerate scale problems in the boundary element method for two‐dimensional elasticity, Int. J. Numer. Methods Eng., 54, 1669-1681 (2002) · Zbl 1098.74725
[11] Cruse, T. A., Recent advances in boundary element analysis methods, Comput. Methods Appl. Mech. Eng., 62, 227-244 (1987) · Zbl 0614.73087
[12] Cruse, T. A., BIE fracture mechanics analysis: 25 years of developments, Comput. Mech., 18, 1-11 (1996) · Zbl 0946.74073
[13] Cruse, T. A.; Novati, G., Traction Boundary Integral Equation (BIE) Formulations and Applications to Nonplanar and Multiple Cracks (1992), Fracture Mechanics; National Symposium
[14] Dell’Erba, D. N.; Aliabadi, M. H., BEM analysis of fracture problems in three-dimensional thermoelasticity using J-integral, Int. J. Solid Struct., 38, 4609-4630 (2001) · Zbl 0989.74074
[15] Dell’Erba, D. N.; Aliabadi, M. H.; Rooke, D. P., Dual boundary element method for three-dimensional thermoelastic crack problems, Int. J. Fract., 94, 89-101 (1998)
[16] Feng, W. Z.; Gao, X. W., An interface integral equation method for solving transient heat conduction in multi-medium materials with variable thermal properties, Int. J. Heat Mass Tran., 98, 227-239 (2016)
[17] Feng, W. Z.; Liu, J.; Gao, X. W., An improved direct method for evaluating hypersingular stress boundary integral equations in BEM, Eng. Anal. Bound. Elem., 61, 274-281 (2015) · Zbl 1403.74166
[18] Feng, W. Z.; Gao, X. W.; Liu, J.; Yang, K., A new BEM for solving 2D and 3D elastoplastic problems without initial stresses/strains, Eng. Anal. Bound. Elem., 61, 134-144 (2015) · Zbl 1403.74165
[19] Feng, W. Z.; Yang, K.; Cui, M.; Gao, X. W., Analytically-integrated radial integration BEM for solving three-dimensional transient heat conduction problems, Int. Commun. Heat Mass Tran., 79, 21-30 (2016)
[20] Feng, W. Z.; Gao, L. F.; Du, J. M.; Qian, W.; Gao, X. W., A meshless interface integral BEM for solving heat conduction in multi-non-homogeneous media with multiple heat sources, Int. Commun. Heat Mass Tran., 104, 70-82 (2019)
[21] Feng, W. Z.; Li, H. Y.; Gao, L. F.; Qian, W.; Yang, K., Hypersingular flux interface integral equation for multi-medium heat transfer analysis, Int. J. Heat Mass Tran., 138, 852-865 (2019)
[22] Frangi, A.; Guiggiani, M., Boundary element analysis of Kirchhoff plates with direct evaluation of hypersingular integrals, Int. J. Numer. Methods Eng., 46, 1845-1863 (1999) · Zbl 0965.74075
[23] Frangi, A.; Guiggiani, M., A direct approach for boundary integral equations with high‐order singularities, Int. J. Numer. Methods Eng., 49, 871-898 (2000) · Zbl 1013.74075
[24] Frangi, A.; Guiggiani, M., Free terms and compatibility conditions for 3D hypersingular boundary integral equations, Z. Angew. Math. Mech., 81, 651 (2001) · Zbl 0991.65128
[25] Gao, X. W., The radial integration method for evaluation of domain integrals with boundary-onl54discretization, Eng. Anal. Bound. Elem., 26, 905-916 (2002) · Zbl 1130.74461
[26] Gao, X. W., An effective method for numerical evaluation of general 2D and 3D high order singular boundary integrals, Comput. Methods Appl. Mech. Eng., 199, 2856-2864 (2010) · Zbl 1231.65236
[27] Gao, X. W.; Davies, T. G., Boundary Element Programming in Mechanics (2002), Cambridge University Press: Cambridge University Press Cambridge, UK
[28] Gao, X. W.; Zhang, C.; Sladek, J.; Sladek, V., Fracture analysis of functionally graded materials by a BEM, Compos. Sci. Technol., 68, 1209-1215 (2008)
[29] Gao, X. W.; Feng, W. Z.; Yang, K.; Cui, M., Projection plane method for evaluation of arbitrary high order singular boundary integrals, Eng. Anal. Bound. Elem., 50, 265-274 (2015) · Zbl 1403.65197
[30] Gao, X. W.; Feng, W. Z.; Zheng, B. J.; Yang, K., An interface integral equation method for solving general multi-medium mechanics problems, Int. J. Numer. Methods Eng., 107, 696-720 (2016) · Zbl 1352.74031
[31] Gray, L. J.; Martha, L. F.; Ingraffea, A. R., Hypersingular integrals in boundary element fracture analysis, Int. J. Numer. Methods Eng., 29, 1135-1158 (1990) · Zbl 0717.73081
[32] Guiggiani, M., Hypersingular boundary integral equations have an additional free term, Comput. Mech., 16, 245-248 (1995) · Zbl 0840.65117
[33] Guiggiani, M.; Krishnasamy, G.; Rudolphi, T. J.; Rizzo, F. J., A general algorithm for the numerical solution of hypersingular boundary integral equations, Journal of Applied Mechanics-Transactions of the ASME, 59, 604-614 (1992) · Zbl 0765.73072
[34] Hong, H. K.; Chen, J. T., Derivations of integral equations of elasticity, Journal Of Engineering Mechanics-ASCE, 114, 1028-1044 (1988)
[35] Karami, G.; Derakhshan, D., An efficient method to evaluate hypersingular and supersingular integrals in boundary integral equations analysis, Eng. Anal. Bound. Elem., 23, 317-326 (1999) · Zbl 0940.65139
[36] Krishnasamy, G.; Schmerr, L.; Rudolphi, T.; Rizzo, F., Hypersingular boundary integral equations: some applications in acoustic and elastic wave scattering, Journal of Applied Mechanics-Transactions of the ASME, 57, 404 (1990) · Zbl 0729.73251
[37] Lachat, J.; Watson, J., Effective numerical treatment of boundary integral equations: a formulation for three‐dimensional elastostatics, Int. J. Numer. Methods Eng., 10, 991-1005 (1976) · Zbl 0332.73022
[38] Lee, K. Y.; Choi, H. J., Boundary element analysis of stress intensity factors for dimaterial interface cracks, Eng. Fract. Mech., 29, 461-472 (1988)
[39] Lee, K. Y.; Kwak, S. G., Determination of stress intensity factors for bimaterial interface stationary rigid line inclusions by boundary element method, Int. J. Fract., 113, 285-294 (2002)
[40] Li, J.; Feng, W. Z.; Gao, X. W., A dual boundary integral equation method based on direct evaluation of higer order singular integral for crack problems, Chin. J. Theor. Appl. Mech., 48, 387-398 (2016)
[41] Liu, Y. J., On the simple-solution method and non-singular nature of the BIE/BEM - a review and some new results, Eng. Anal. Bound. Elem., 24, 789-795 (2000) · Zbl 0974.65110
[42] Liu, Y. J.; Rizzo, F., A weakly singular form of the hypersingular boundary integral equation applied to 3-D acoustic wave problems, Comput. Methods Appl. Mech. Eng., 96, 271-287 (1992) · Zbl 0754.76072
[43] Liu, Y. J.; Rudolphi, T. J., Some identities for fundamental solutions and their applications to weakly-singular boundary element formulations, Eng. Anal. Bound. Elem., 8, 301-311 (1991)
[44] Liu, Y. J.; Rudolphi, T. J., New identities for fundamental solutions and their applications to non-singular boundary element formulations, Comput. Mech., 24, 286-292 (1999) · Zbl 0969.74073
[45] Liu, Y. J.; Shen, L., A dual BIE approach for large-scale modelling of 3-D electrostatic problems with the fast multipole boundary element method, Int. J. Numer. Methods Eng., 71, 837-855 (2007) · Zbl 1194.78058
[46] Liu, Y. J.; Mukherjee, S.; Nishimura, N.; Schanz, M.; Ye, W.; Sutradhar, A.; Pan, E.; Dumont, N. A.; Frangi, A.; Saez, A., Recent advances and emerging applications of the boundary element method, Appl. Mech. Rev., 64, Article 030802 pp. (2011)
[47] Liu, Y. J.; Li, Y. X.; Xie, W., Modeling of multiple crack propagation in 2-D elastic solids by the fast multipole boundary element method, Eng. Fract. Mech., 172, 1-16 (2017)
[48] Lv, J. H.; Feng, X. T.; Chen, B. R.; Jiang, Q.; Guo, H. S., The CPCT based CBIE and HBIE for potential problems in three dimensions, Eng. Anal. Bound. Elem., 67, 53-62 (2016) · Zbl 1403.65214
[49] Lv, J. H.; Feng, X. T.; Yan, F.; Chen, B. R., Implementation of CPCT based BIEs for 3D elasticity and its application in fracture mechanics, Eng. Anal. Bound. Elem., 71, 1-10 (2016) · Zbl 1403.74200
[50] Mi, Y.; Aliabadi, M. H., Dual boundary element method for three-dimensional fracture mechanics analysis, Eng. Anal. Bound. Elem., 10, 161-171 (1992)
[51] Mi, Y.; Aliabadi, M. H., Three-dimensional crack growth simulation using BEM, Comput. Struct., 52, 871-878 (1994) · Zbl 0900.73900
[52] Murakami, Y., Stress Intensity Factors Handbook (1987), Pergamon Press: Pergamon Press Oxford
[53] Portela, A.; Aliabadi, M. H.; Rooke, D. P., The dual boundary element method - effective implementation for crack problems, Int. J. Numer. Methods Eng., 33, 1269-1287 (1992) · Zbl 0825.73908
[54] Portela, A.; Aliabadi, M. H.; Rooke, D. P., Dual boundary element incremental analysis of crack propagation, Comput. Struct., 46, 237-247 (1993) · Zbl 0825.73888
[55] Prasad, N. N.V.; Aliabadi, M. H.; Rooke, D. P., The dual boundary element method for thermoelastic crack problems, Int. J. Fract., 66, 255-272 (1994)
[56] Prasad, N. N.V.; Aliabadi, M. H.; Rooke, D. P., The dual boundary element method for transient thermoelastic crack problems, Int. J. Solid Struct., 33, 2695-2718 (1996) · Zbl 0903.73076
[57] Qu, W. Z.; Zhang, Y. M.; Liu, C. S., Boundary stress analysis using a new regularized boundary integral equation for three-dimensional elasticity problems, Arch. Appl. Mech., 87, 1213-1226 (2017)
[58] Rice, J. R., A path independent integral and the approximate analysis of strain concentration by notches and cracks, Journal of Applied Mechanics-Transactions of the ASME, 35, 379-386 (1968)
[59] Shih, C. F.; Moran, B.; Nakamura, T., Energy release rate along a three-dimensional crack front in a thermally stressed body, Int. J. Fract., 30, 79-102 (1986)
[60] Sladek, J.; Sladek, V., Boundary-element analysis for interface crack between dissimilar elastoplastic materials, Comput. Mech., 16, 396-405 (1995) · Zbl 0848.73076
[61] Sladek, J.; Sladek, V.; Wünsche, M.; Zhang, C., Analysis of an interface crack between two dissimilar piezoelectric solids, Eng. Fract. Mech., 89, 114-127 (2012)
[62] Tan, C. L.; Gao, Y. L., Stress intensity factors for cracks at spherical inclusions by the boundary integral equation method, J. Strain Anal. Eng. Des., 25, 197-206 (1990)
[63] Tan, C. L.; Gao, Y. L., Treatment of bimaterial interface crack problems using the boundary element method, Eng. Fract. Mech., 36, 919-932 (1990)
[64] Watson, J. O., Hermitian cubic and singualr elements for plane strain, (Banerjee, P. K.; Watson, J. O., Developments in Boundary Element Methods 4 (1986), Elsvier Applied Science Publishers: Elsvier Applied Science Publishers Barking, UK) · Zbl 0586.73169
[65] Xie, G.; Zhang, J.; Cheng, H.; Lu, C.; Li, G., A direct traction boundary integral equation method for three-dimension crack problems in infinite and finite domains, Comput. Mech., 53, 575-586 (2014) · Zbl 1398.74445
[66] Zhang, C.; Gao, X. W.; Sladek, J.; Sladek, V., Fracture mechanics analysis of 2-D FGMs by a meshless BEM, Key Eng. Mater., 324-325, 1165-1172 (2006)
[67] Zhang, C.; Cui, M.; Wang, J.; Gao, X. W.; Sladek, J.; Sladek, V., 3D crack analysis in functionally graded materials, Eng. Fract. Mech., 78, 585-604 (2011)
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.