zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Weakly-singular, weak-form integral equations for cracks in three-dimensional anisotropic media. (English) Zbl 1169.74549
Summary: Singularity-reduced integral relations are developed for displacement discontinuities in three-dimensional, anisotropic linearly elastic media. An isolated displacement discontinuity is considered first, and a systematic procedure is followed to develop relations for the displacement and stress fields induced by the discontinuity. The singularity-reduced relation for the stress is particularly important since it is in a form which allows a weakly-singular, weak-form traction integral equation to be readily established. The integral relations obtained for a general displacement discontinuity are then specialized to an isolated crack and to dislocations; the relations for dislocations are introduced to emphasize their direct connection to corresponding results for cracks and to allow earlier independent findings for these two types of discontinuities to be put into proper context. Next, the singularity-reduced integral equations obtained for an isolated crack are extended to allow treatment of cracks in a finite domain, and a pair of weakly-singular, weak-form displacement and traction integral equations is established. These integral equations can be combined to obtain a final formulation which is in a symmetric form, and in this way they serve as the basis for a weakly-singular, symmetric Galerkin boundary element method suitable for analysis of cracks in anisotropic media.

74R10Brittle fracture
74B05Classical linear elasticity
[1]Bacon, D. J.; Barnett, D. M.; Scattergood, R. O.: Anisotropic continuum theory of lattice defects, Progress in materials science 23, 51-262 (1978)
[2]Becache, E.; Nedelec, J. C.; Nishimura, N.: Regularization in 3D for anisotropic elastodynamic crack and obstacle problems, Journal of elasticity 31, 25-46 (1993) · Zbl 0773.73029 · doi:10.1007/BF00041622
[3]Blin, J.: Energie mutuelle de deux dislocations, Acta metallurgica 3, 199-200 (1955)
[4]Bonnet, M.: Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity, Engineering analysis with boundary elements 15, 93-102 (1995)
[5]Bui, H. D.: An integral equations method for solving the problem of a plane crack of arbitrary shape, Journal of the mechanics and physics of solids 25, No. 1, 29-39 (1977) · Zbl 0355.73074 · doi:10.1016/0022-5096(77)90018-7
[6]Burgers, J.M. (1939). Some considerations on the fields of stress connected with dislocations in a regular crystal lattice. In: Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen 42, pp. 293 – 325 (part I); pp. 378 – 399 (part II).
[7]Deans, S. R.: The Radon transform and some of its applications, (1983) · Zbl 0561.44001
[8]Frangi, A.; Novati, G.; Springhetti, R.: 3D fracture analysis by the symmetric Galerkin BEM, Computational mechanics 28, No. 3-4, 220-232 (2002) · Zbl 1010.74078 · doi:10.1007/s00466-001-0283-x
[9]Frangi, A.: Fracture propagation in 3D by the symmetric Galerkin boundary element method, International journal of fracture 116, No. 4, 313-330 (2002)
[10]Gel’fand, I. M.; Graev, M. I.; Vilenkin, N. Y.: Generalized functions: integral geometry and representation theory, Generalized functions: integral geometry and representation theory 5 (1966) · Zbl 0144.17202
[11]Gray, L. J.; Martha, L. F.; Ingraffea, A. R.: Hypersingular integrals in boundary element fracture analysis, International journal for numerical methods in engineering 29, 1135-1158 (1990) · Zbl 0717.73081 · doi:10.1002/nme.1620290603
[12]Gu, H.; Yew, C. H.: Finite element solution of a boundary integral equation for mode I embedded three-dimensional fractures, International journal for numerical methods in engineering 26, 1525-1540 (1988) · Zbl 0636.73087 · doi:10.1002/nme.1620260705
[13]Helgason, S.: The Radon transform, (1999)
[14]Hirth, J. P.; Lothe, J.: Theory of dislocations, (1982)
[15]Indenbom, V.L., Orlov, S.S. (1968). In: Proceedings of the Kharkov Conference on Dislocation Dynamics. Physical-Technical Institute of Low Temperatures, Academy of Sciences of the USSR, Moscow, 406.
[16]Li, S.; Mear, M. E.: Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media, International journal of fracture 93, 87-114 (1998)
[17]Li, S.; Mear, M. E.; Xiao, L.: Symmetric weak-form integral equation method for three-dimensional fracture analysis, Computer methods in applied mechanics and engineering 151, 435-459 (1998) · Zbl 0906.73074 · doi:10.1016/S0045-7825(97)00199-0
[18]Lothe, J.: Dislocations in anisotropic media: the interaction energy, Philosophical magazine A 46, No. 1, 177-180 (1982)
[19]Lothe, J. (1983). Dislocations in anisotropic media. Report No. 83-06, Institute of Physics, University of Oslo, Oslo.
[20]Martha, L. F.; Gray, L. J.; Ingraffea, A. R.: Three-dimensional fracture simulation with a single-domain, direct boundary element formulation, International journal for numerical methods in engineering 35, 1907-1921 (1992) · Zbl 0775.73322 · doi:10.1002/nme.1620350911
[21]Martin, P. A.; Rizzo, F. J.: Hypersingular integrals: how smooth must the density be?, International journal for numerical methods in engineering 39, 687-704 (1996) · Zbl 0846.65070 · doi:10.1002/(SICI)1097-0207(19960229)39:4<687::AID-NME876>3.0.CO;2-S
[22]Martin, P. A.; Rizzo, F. J.; Cruse, T. A.: Smoothness-relaxation strategies for singular and hypersingular integral equations, International journal for numerical methods in engineering 42, 885-906 (1998) · Zbl 0913.65105 · doi:10.1002/(SICI)1097-0207(19980715)42:5<885::AID-NME392>3.0.CO;2-X · doi:http://www.interscience.wiley.com
[23]Nedelec, J. C.: Integral equations with non integrable kernels, Integral equations and operator theory 5, 562-572 (1982) · Zbl 0479.65060 · doi:10.1007/BF01694054
[24]Nishimura, N.; Kobayashi, S.: A regularized boundary integral equation method for elastodynamic crack problems, Computational mechanics 4, 319-328 (1989) · Zbl 0675.73065 · doi:10.1007/BF00301390
[25]Sládek, V.; Sládek, J.: Three dimensional crack analysis for an anisotropic body, Applied mathematical modeling 6, No. 5, 374-380 (1982) · Zbl 0492.73102 · doi:10.1016/S0307-904X(82)80101-7
[26]Sládek, V.; Sládek, J.; Le Van, A.: Completely regularized integral representations and integral equations for anisotropic bodies with initial strains, Zeitschrift für angewandte Mathematik und mechanik 78, No. 11, 771-780 (1998) · Zbl 0916.73011 · doi:10.1002/(SICI)1521-4001(199811)78:11<771::AID-ZAMM771>3.0.CO;2-E
[27]Ting, T. C. T.; Lee, V. G.: The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids, The quarterly journal of mechanics and applied mathematics 50, 407-426 (1997) · Zbl 0892.73006 · doi:10.1093/qjmam/50.3.407
[28]Weaver, J.: Three-dimensional crack analysis, International journal of solids and structures 13, 321-330 (1977) · Zbl 0373.73093 · doi:10.1016/0020-7683(77)90016-6
[29]Wang, C. -Y.: Elastic fields produced by a point source in solids of general anisotropy, Journal of engineering mathematics 32, 41-52 (1997) · Zbl 0907.73008 · doi:10.1023/A:1004289831587
[30]Xu, G.; Ortiz, M.: A variational boundary integral method for the analysis of 3-D cracks of arbitrary geometry modelled as continuous distributions of dislocation loops, International journal for numerical methods in engineering 36, 3675-3701 (1993) · Zbl 0796.73067 · doi:10.1002/nme.1620362107
[31]Xu, G.: A variational boundary integral method for the analysis of three-dimensional cracks of arbitrary geometry in anisotropic elastic solids, Journal of applied mechanics 67, 403-408 (2000) · Zbl 1110.74768 · doi:10.1115/1.1305276