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A variational boundary integral equation method for an elastodynamic antiplane crack. (English) Zbl 0772.73088
Summary: This paper investigates the transient wave scattering by a crack by means of the boundary integral equation method (BIEM). The author has developed a new formulation to solve the BIE for the crack opening displacement (COD). The resolution is done directly in the time domain. The solution is represented by means of a retarded double layer potential, and the resulting BIE, with the COD as unknown, has a hypersingular kernel. The corresponding difficulty is overcome by using a variational method. We present the application of this method to an antiplane crack, describe the approximate problem and finally give 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 74R99 Fracture and damage
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##### References:
  ’Variational integral method in the time domain for elastic scattering problems’, in and (eds.), Boundary Elements in Mechanical, and Electrical Engineering, Proc. Int. Boundary Element Symposium, Nice, Computational Mechanics Publications, and Springer-Verlag, Berlin, 1990.  ’Résolution par une méthode d’équations intégrales d’un probléme de diffraction d’ondes élastiques transitoires par une fissure’, Thesis, University Paris 6, 1991.  Bécache, J. Elasticity  Beskos, Appl. Mech. Rev. 40 pp 1– (1987)  ’Méthode des équations intégrales régularisées en élastodynamique’, Thesis, ENPC, 1986.  Bui, J. Mech. Phys. Solids 25 pp 29– (1977)  ’Boundary integral operators for the heat equation’, Technische Hochschule Darmstadt, Fachbereich Mathematik, Preprint Nr. 1269, 1989.  Costabel, J. Reine Angew. Math. 372 pp 39– (1986)  Ha Duong, Japan J. Appl. Math. 7 pp 489– (1990)  Hamdi, CRAS, Série II 292 pp 17– (1981)  Hirose, Int. j. numer. methods eng. 28 pp 629– (1989)  Hsiao, J. Comp. Math. 7 pp 121– (1989)  and , ’Coercivity of the single layer heat operator’, Technical Report 89-2, Department of Mathematical Sciences, University of Delaware, 1989.  and , ’Hypersingular boundary integral equations: Their occurrence, interpretation, regularization, and computation’, Chapter 7 in and (eds.), Developments in Boundary Element Methods, Vol. 7: Advanced Dynamic Analysis, Elsevier Applied Science Publishers, Barking, U.K., 1991.  Martin, Proc. Roy. Soc. London A 421 pp 341– (1989)  ’Approximation des équations intégrales en mécanique et en physique’, Lecture Notes, Centre de Mathematiques Appliquées, Ecole Polytechnique, Palaiseau, 1977.  Nédélec, Integral Equations, and Operators Theory 5 pp 562– (1982)  , and , ’Boundary integral equation methods in elastodynamic crack problems’, in et al. (eds.) Boundary Elements IX, Proc. 9th Int. Conf. BEM. Vol. 2: Stress Analysis Applications, Springer-Verlag, Berlin, 1987, pp. 279-33.  Nishimura, Computat. Mech. 4 pp 319– (1989)  Polch, Computat. Mech. 2 pp 253– (1987)  Sladek, Appl. Math. Model. 8 pp 2– (1984)  and , Crack Problems in the Classical Theory of Elasticity, SIAM Series in Applied Mathematics, Wiley, New York.
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