×

zbMATH — the first resource for mathematics

A space-time variational formulation for the boundary integral equation in a 2D elastic crack problem. (English) Zbl 0817.73067
Summary: This paper investigates the transient elastic wave scattering by a crack in \(\mathbb{R}^ 2\) by means of boundary integral equation (BIE) method. The analysis of the Laplace-Fourier transform (in time) of the integral operator allows to obtain existence, uniqueness and continuity dependence of the solution with respect to the data, in a Sobolev functional framework. A regularisation of the hypersingular \(BIE\) is applied in order to remove the hypersingularity and to write the associated time- space variational formulation in a tractable form. A Galerkin-type approximation is then performed to solve this variational formulation, and finally we present some numerical results.

MSC:
74S15 Boundary element methods applied to problems in solid mechanics
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
74J20 Wave scattering in solid mechanics
74R99 Fracture and damage
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] A. BAMBERGER and T. HA DUONG, 1986, Formulation variationnelle espace-temps pour le calcul par potentiel retardé d’une onde acoustique, Math. Methods Appl. Sci., 8, 405-435. Zbl0618.35069 MR859833 · Zbl 0618.35069
[2] A. BAMBERGER and T. HA DUONG, 1986, Formulation variationnelle espace-temps pour le calcul par potentiel retardé d’une onde acoustique; Problème de Neumann, Math. Methods Appl. Sci., 8, 598-608. Zbl0636.65119 MR870995 · Zbl 0636.65119
[3] A. BAMBERGER, 1983, Approximation de la diffraction d’ondes élastiques, une nouvelle approche (I), (II), (III), Technical report, École Polytechnique, CMAP, Rapports Internes n^\circ 91, 96, 98. Zbl0571.73020 · Zbl 0571.73020
[4] E. BÉCACHE, 1991, Résolution par une méthode d’équations intégrales d’un problème de diffraction d’ondes élastiques transitoires par une fissure. PhD thesis, Université de Paris 6. Thèse.
[5] E. BÉCACHE, 1993, A Variational Boundary Integral Equation Method for an Elastodynamic Antiplane Crack, Int. J. for Numerical Meth. in Eng., 36, 969-984 Zbl0772.73088 MR1208455 · Zbl 0772.73088
[6] E. BÉCACHE, J.-C. NÉDÉLEC, N. NISHIMURA, 1993, Regularization in 3D for Anisotropic Elastodynamic Crack and Obstacle Problems, J. of Elasticity, 31, 25-46. Zbl0773.73029 MR1221204 · Zbl 0773.73029
[7] D. E. BESKOS, 1987, Boundary elements methods in dynamic analysis, Appl. Mech. Rev., 40, 1-23.
[8] M. BONNET, 1986, Méthode des équations intégrales régularisées en élastodynamyque, PhD thesis, ENPC, Thèse. Zbl0612.73083 MR884382 · Zbl 0612.73083
[9] H. D. BUI, 1977, An intgral equations method for sol ving the problems of a plane crack of arbitrary shape, J. Mech. Phys. Solids, 25, 29-39. Zbl0355.73074 MR443528 · Zbl 0355.73074
[10] P. CORTEY-DUMONT, 1984, Simulation Numérique de Problèmes de Diffraction d’Ondes par une Fisure, PhD thesis, Université Paris VI, Thèse d’État.
[11] R. DAUTRAY and J. L. LIONS, 1985, Analyse Mathématique et Calcul Numérique pour les Sciences et les Techniques, vol. 2. Masson. Zbl0642.35001 · Zbl 0642.35001
[12] T. HA DUONG, 1990, On the transient acoustic scattering by a flat object, Japan J. Appl. Math., 7, 489-513. Zbl0719.35063 MR1076300 · Zbl 0719.35063
[13] T. HA DUONG, 1992, On the boundary integral equations for the crack opening displacement of flat cracks, Integr. Equat. Oper. Th., 15, 427-453. Zbl0753.45005 MR1155713 · Zbl 0753.45005
[14] V. A. KONDRAT’EV and O. A. OLEINIK, 1988, Boundary-value problems for the System of elasticity theory in unbounded domains. Korn’s inequalities, Russian Math. Surveys, 43, 65-119. Zbl0669.73005 MR971465 · Zbl 0669.73005
[15] G. KRISHNASAMY, F. J. RIZZO and T. J. RUDOLPHI, 1991, Hypersingular boundary integral equations : Their occurrence interpretation, regularization and computation. In P. K. Banerjee and S. Kobayashi, editors, Developments in Boundary Element Methods, vol. 7 ; Advanced Dynamic Analysis, Elsevier Applied Science Publishers.
[16] J. L. LIONS and E. MAGENES, 1968, Problèmes aux limites non homogènes et Applicaitons, vol. l, Dunod. Zbl0165.10801 · Zbl 0165.10801
[17] Ch. LUBICH, On multistep time discretization of linear initial-boundary value problems and their boundary integral equations, submitted to Numerische Mathematik. Zbl0795.65063 · Zbl 0795.65063
[18] P. A. MARTIN and F. J. RIZZO, 1989, On boundary integral equations for crack problems, Proc. Roy. Soc. London A, 421, 341-355. Zbl0674.73071 MR985268 · Zbl 0674.73071
[19] J. C. NÉDÉLEC, 1982, Intégral Equations with non Integrable Kernels, Intégral Equations and Operator Theory, 5, 562-572. Zbl0479.65060 MR665149 · Zbl 0479.65060
[20] J. C. NÉDÉLEC, 1983, Le Potentiel de Double Couche pour les Ondes Élastique, Internal report n^\circ 99, C.M.A.P., École Polytechnique.
[21] N. NISHIMURA, Q. C. GUO, S. KOBAYASHI, 1987, Boundary Integral Equation Methods in Elastodynamic Crack Problems, In Brebbia, Wendland, and Kuhn, editors, Proc. 9th Int. Conf. BEM, vol. 2 : Stress Analysis Applications, pp. 279-291. Springer-Verlag.
[22] N. NISHIMURA and S. KOBAYASHI, 1989, A regularized boundary integral equation method for elastodynamic crack problems, Computat. Mech., 4, 319-328. Zbl0675.73065 · Zbl 0675.73065
[23] J. A. NITSCHE, 1981, On Korn’s second inequality, RAIRO, Analyse numérique, 15, 237-248. Zbl0467.35019 MR631678 · Zbl 0467.35019
[24] V. SLADEK and J. SLADEK, 1984, Transient elastodynamic three-dimensional problems in cracked bodies, Appl. Math. Model, 8, 2-10. Zbl0525.73110 MR734034 · Zbl 0525.73110
[25] I. N. SNEDDON and M. LOWENGRUB, Crack Problems in the Classical Theory of Elasticity, John Wiley and Sons. Zbl0201.26702 MR258339 · Zbl 0201.26702
[26] E. P. STEPHAN, 1986, A Boundary Integral Equation Method for Three-Dimensional Crack Problem in Elasticity, Math. Meth. in the Appl. Sci., 8, 609-623. Zbl0608.73097 MR870996 · Zbl 0608.73097
[27] E. P. STEPHAN, 1987, Boundary Integral Equation for screen problem in R3 Integral Eq. and Oper. Theory, 10, 263. Zbl0653.35016 · Zbl 0653.35016
[28] TREVES, 1975, Basic Linear Partial Differential Equations, Academic Press. Zbl0305.35001 · Zbl 0305.35001
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.