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The structure theorem for the Eulerian derivative of shape functionals defined on domains with cracks. (English. Russian original) Zbl 0967.74051

Sib. Math. J. 41, No. 5, 889-899 (2000); translation from Sib. Mat. Zh. 41, No. 5, 1183-1202 (2000).
Let \(D\subset {\mathbb R}^2\) be a bounded domain with smooth boundary, let \(\Sigma\) be an arc of a smooth curve such that \(\Omega=D \setminus \overline{\Sigma}\) is the domain with a crack, and let \(J=J(\Omega)\) be a domain functional differentiable at small deformations of the crack with respect to a parameter of deformation. The main result of the article consists in describing the structure of the derivative of this functional. This result is applied to the analysis of an energy functional of a nonlinear boundary value problem in a domain with crack, and to the analysis of a functional generated by a boundary value problem with one-sided (Signorini) conditions.

MSC:

74R10 Brittle fracture
74B99 Elastic materials
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References:

[1] Bui H. D. andEhrlacher A., ”Developments of fracture mechanics in France in the last decades,” in: Fracture Research in Retrospect, Rotterdam, 1997, pp. 369–387.
[2] Destyunder Ph. andJaoua M., ”Sur une Interprétation Mathématique de l’Intégrale de Rice en Théorie de la Rupture Fragile,” Math. Meth. in the Appl. Sci.,3, 70–87 (1981). · Zbl 0493.73087 · doi:10.1002/mma.1670030106
[3] Grisvard P., Singularities in Boundary Value Problems Recherches en Mathématiques Appliquées, Springer-Verlag, Berlin (1992).
[4] Blat J. andMorel J. M., ”Elliptic problems in image segmentation and their relation to fracture theory,” in: Recent Advances in Nonlinear Elliptic and Parabolic Problems. Proceedings of an International Conference, Longman Scientific and Technical, 1988,10, pp. 379–394.
[5] Destyunder Ph., ”Calcul de forces d’avancement d’une fissure en tenant compte du contact unilatéral entre les lèvres de la fissure,” CRAS, Sér. 2,296, 745–748 (1983). · Zbl 0572.73106
[6] Sokolowski J. andZolésio J. P., Introduction to Shape Optimization: Shape Sensitivity Analysis. Springer Series in Computational Mathematics, Vol. 16, Springer-Verlag, Berlin etc. (1992).
[7] Khludnev A. M. andSokolowski J., ”The Griffith formula and the Rice-Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains,” European J. Appl. Math.,10, 379–394 (1999). · Zbl 0945.74058 · doi:10.1017/S0956792599003885
[8] Khludnev A. M. andSokolowski J. ”Griffith formula for elasticity system with unilateral conditions in domains with cracks,” European J. Mech., A/Solids,19, No. 1, 105–120 (2000). · Zbl 0966.74061 · doi:10.1016/S0997-7538(00)00138-8
[9] Aubin J. P., Initiation à l’analyse appliquée (Foundation of applied analysis), Masson, Paris (1994). · Zbl 0809.90002
[10] Delfour M. C., ”Shape optimization and free boundaries,” in: Proceedings of the NATO Advanced Study Institute and Séminaire de Mathématiques Supérieures, Held Montreal, Canada, June 25–July 13, 1990. NATO ASI Series. Series C. Mathematical and Physical Sciences. 380, Kluwer Academic Publishers, Dordrecht, 1992.
[11] Delfour M. C. andZolésio J. P., ”Structure of shape derivatives for nonsmooth domains,” J. Funct. Anal.,104, No. 1, 1–33 (1992). · Zbl 0777.49030 · doi:10.1016/0022-1236(92)90087-Y
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