On the boundary integral-equations for the crack opening displacement of flat cracks. (English) Zbl 0753.45005

The author considers the first kind boundary integral equations \(Du=f\) in a bounded smooth domain in the plane, where \(D\) is the hypersingular operator associated with the Helmholtz equation and the Lamé system, respectively. Regarding \(D\) is a pseudodifferential operator and computing its complete symbol, he proves a coerciveness estimate and a solvability result.


45F15 Systems of singular linear integral equations
35C15 Integral representations of solutions to PDEs
47G30 Pseudodifferential operators
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
74R99 Fracture and damage
74J20 Wave scattering in solid mechanics
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[1] ACHENBACH J.D.: Wave Propagation in Elastic Solids. North Holland, Amsterdam. (1975). · Zbl 0313.73083
[2] BAMBERGER A.: Approximation de la diffraction d’ondes Elastiques (I). in:Nonliner Partial Differential Equations and their Applications; (eds: H. Brezis and J.L.Lions). Pitman, London 1984, 48-95.
[3] BENDALI A.: Numerical Analysis of the Exterior Boundary Value problem for the Time Harmonic Maxwell Equations by a Boundary Finite Element Method.Math. of Comp. v.43 (1984),p. 29-46 &47-68. · Zbl 0555.65082
[4] BENDALI A., DEVYS C.: Calcul numérique du rayonnement électromagnétiques.Onde Electrique, v.66 (1986),no1,p. 77-81.
[5] Benjelloun-Touimi Z.: Diffraction par un réseau 1-périodique dans R3. Thèse, Université Paris VI, (1988).
[6] BUDRECK D.E., ACHENBACH J.D.,: Scattering from Three-Dimensional Planar Cracks by the Boundary Integral Equation Method.Journ. Appl. Mech., v.55 (1988),p.405-412 · Zbl 0663.73073 · doi:10.1115/1.3173690
[7] Bui H.D., Loret B., Bonnet M., Régularisation des équations intégrales de l’élastodynamique et de l’élastostatique.CRAS Paris, série II, t. 300 (1985).
[8] CHAZARAIN J., PIRIOU A., Introduction à la théorie des équations aux Dérivées Partielles. Gauthiers Villars, Paris (1981). · Zbl 0446.35001
[9] COLTON D., KRESS R., Integral Equation Method in Scattering Theory John Wiley, N.Y. (1983). · Zbl 0522.35001
[10] Cortey-Dumont P.: Thèse Docteur ès Sciences, Université PARIS VI, (1985).
[11] Costabel M., Stephan E.P.: A Boundary Element Method for 3-d crack Problems. (Invited Lecture at the4th International Symposium on Numerical Methods in Engineering, Atlanta, Georgia 1986). · Zbl 0625.73117
[12] DING Y., FORESTIER A., HA DUONG T., A Galerkin Scheme for the Time Domain Integral Equation of Acoustic Scattering from a Hard Surface.Journ. Acoust. Soc. Am., v.86 (4 (1989),p. 1566-1572. · doi:10.1121/1.398777
[13] Giroire J.: Thèse Docteur ès Sciences, Université PARIS VI, (1987).
[14] Ha Duong T.: On the Transient Acoustic Scattering by a Flat Object.Japan J. of Appl. Math (to appear). (1990). · Zbl 0719.35063
[15] Ha Duong T.: Thèse Docteur ès Sciences, Université Paris VI, (1987).
[16] HAMDI M.: Une Formulation variationnelle par Equations Intégrales pour la Résolution de l’Equation de Helmholtz avec des Conditions aux limites mixtes.CRAS, série II, T292 (1981),p. 17-20.
[17] Hirose S., Niwa Y.: Scattering of Elastic waves by a Three-Dimensional Crack.Proc. of the VIII Conference on BEM, Tokyo, Springer Verlag (1986). · Zbl 0613.73103
[18] Hsiao G.C., Kopp P., Wendland W.L.: Some Applications of a Galerkin Collocation Method for Boundary Integral Equations of the First Kind.Preprint no 768 (1983)Fach. Mathematik, Technische Hochschule Darmstadt. · Zbl 0546.65091
[19] JONES D.S.: A New Method for Calculating Scattering with Particular Reference to the Circular Disc.C.P.A.M., v.9 (1956),p.713-746. · Zbl 0073.43202
[20] KUPRADZE S. et al.: Three Dimensional Problems of the Mathematical Theory of Elasticity and Thermoelacticity. North Holland, Amsterdam (1979). · Zbl 0421.73009
[21] LIONS J.L., MAGENES E.: Non Homogeneous Boundary Value Problems and Applications. Springer Verlag, Berlin (1972). · Zbl 0227.35001
[22] MARTIN P.A., WICKHAM G.R.: Diffraction of Elastic Waves by a Penny-shaped Crack; Analytical and Numerical Results.Proc. R. Soc. London A. 390 (1983),p. 91-129. · Zbl 0537.73017 · doi:10.1098/rspa.1983.0124
[23] MARTIN P.A., RIZZO F.J.: On Boundary Integral Equations for Crack Problems.Proc. R. Soc. London A. 421 (1989),p. 341-355 · Zbl 0674.73071 · doi:10.1098/rspa.1989.0014
[24] MEISTER E., SECK F.O.: The Explicit Solution of Elastodynamical Diffraction Problems by Symbol Factorization.Zeitschrift. fur Analysis und ihre Anwendungen, Bd8 (1989),p.307-328. · Zbl 0697.73022
[25] NEDELEC J.C.: Résolution par Potentiel de Double Couche du Problème de Neumann extérieur.CRAS, T286 (1978),serie A, p. 103-106. · Zbl 0375.65047
[26] NEDELEC J.C.: Curved Finite Element Method for the Solution of Integral Singular Equations on Surfaces inR 3.Comp. Meth. Appl. Mech. Eng. 8 (1976),p.61-80. · Zbl 0333.45015 · doi:10.1016/0045-7825(76)90053-0
[27] NEDELEC J.C., PLANCHARD J.: Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dansR 3.RAIRO, série rouge, v.7 (1973),p.105-129.
[28] NISHIMURA N., KOBAYASHI S. A.: Regularised Boundary Integral Method for Elastodynamic Crack Problem.Computational Mechanics v.4 (1989),p.319-328. · Zbl 0675.73065 · doi:10.1007/BF00301390
[29] STEPHAN E.P.: A Boundary Integral Equation Method for 3-d Crack Problems in Elasticity.Math. Meth. in the Appl. Sci. v.8 (1986),p.609-623. · Zbl 0608.73097 · doi:10.1002/mma.1670080140
[30] STEPHAN E.P., WENDLAND W.L.: An augmented Galerkin Procedure for the Boundary Integral Method applied to Two-dimensional Screen and Crack Problems.Applicable Analysis 18 (1986),p.183-219. · doi:10.1080/00036818408839520
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