Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On derivative of energy functional for elastic bodies with cracks and unilateral conditions


Authors: A. M. Khludnev, K. Ohtsuka and J. Sokołowski
Journal: Quart. Appl. Math. 60 (2002), 99-109
MSC: Primary 74G65; Secondary 74P10, 74R99
DOI: https://doi.org/10.1090/qam/1878261
MathSciNet review: MR1878261
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Abstract: In this paper we consider elasticity equations in a domain having a cut (a crack) with unilateral boundary conditions considered at the crack faces. The boundary conditions provide a mutual nonpenetration between the crack faces, and the problem as a whole is nonlinear. Assuming that a general perturbation of the cut is given, we find the derivative of the energy functional with respect to the perturbation parameter. It is known that a calculation of the material derivative for similar problems has the difficulty of finding boundary conditions at the crack faces. We use a variational property of the solution, thus avoiding a direct calculation of the material derivative.


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  • [1] A. M. Khludnev and J. Sokolowski, The Griffith formula and the Rice-Cherepanov integral for crack problems with unilateral conditions in nonsmooth domains, European J. Appl. Math. 10 (1999), no. 4, 379–394. MR 1713077, https://doi.org/10.1017/S0956792599003885
  • [2] A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, Southampton-Boston, WIT Press, 2000
  • [3] V. G. Maz′ya and S. A. Nazarov, Asymptotic behavior of energy integrals under small perturbations of the boundary near corner and conic points, Trudy Moskov. Mat. Obshch. 50 (1987), 79–129, 261 (Russian); English transl., Trans. Moscow Math. Soc. (1988), 77–127. MR 912054
  • [4] Kohji Ohtsuka, Generalized 𝐽-integral and three-dimensional fracture mechanics. I, Hiroshima Math. J. 11 (1981), no. 1, 21–52. MR 606833
  • [5] Kohji Ohtsuka, Generalized 𝐽-integral and its applications. I. Basic theory, Japan J. Appl. Math. 2 (1985), no. 2, 329–350. MR 839334, https://doi.org/10.1007/BF03167081
  • [6] Jan Sokołowski and Jean-Paul Zolésio, Introduction to shape optimization, Springer Series in Computational Mathematics, vol. 16, Springer-Verlag, Berlin, 1992. Shape sensitivity analysis. MR 1215733
  • [7] V. P. Parton and E. M. Morozov, Mechanics of Elastoplastic Fracture, Moscow, Nauka, 1985 (in Russian)
  • [8] A. M. Khludnev and J. Sokołowski, Modelling and Control in Solid Mechanics, Birkhäuser, Basel-Boston-Berlin, 1997
  • [9] P. Destuynder and M. Djaoua, Sur une interprétation mathématique de l’intégrale de Rice en théorie de la rupture fragile, Math. Methods Appl. Sci. 3 (1981), no. 1, 70–87 (French, with English summary). MR 606849, https://doi.org/10.1002/mma.1670030106
  • [10] A. M. Khludnev, On a Signorini problem for inclusions in shells, European J. Appl. Math. 7 (1996), no. 5, 499–510. MR 1419645, https://doi.org/10.1017/S0956792500002515
  • [11] Alexander M. Khludnev, Contact problem for a plate having a crack of minimal opening, Control Cybernet. 25 (1996), no. 3, 605–620. Distributed parameter systems: modelling and control (Warsaw, 1995). MR 1408722
  • [12] P. Grisvard, Singularities in boundary value problems, Recherches en Mathématiques Appliquées [Research in Applied Mathematics], vol. 22, Masson, Paris; Springer-Verlag, Berlin, 1992. MR 1173209
  • [13] G. P. Cherepanov, Mechanics of Brittle Fracture, McGraw-Hill, 1979
  • [14] M. Bonnet, Équations intégrales variationnelles pour le problème en vitesse de propagation de fissures en élasticité linéaire, C. R. Acad. Sci. Paris, série II, vol. 318, 1994, pp. 429-434
  • [15] S. A. Nazarov and B. A. Plamenevskii, Elliptic problems in domains with piecewise smooth boundaries, Moscow, Nauka, 1991 (in Russian)
  • [16] Q. S. Nguyen, C. Stolz, and G. Debruyne, Energy methods in fracture mechanics: stability, bifurcation and second variations, European J. Mech. A Solids 9 (1990), no. 2, 157–173. MR 1093078
  • [17] H. Petryk and Z. Mróz, Time derivatives of integrals and functionals defined on varying volume and surface domains, Arch. Mech. (Arch. Mech. Stos.) 38 (1986), no. 5-6, 697–724 (English, with Russian and Polish summaries). MR 900269
  • [18] Edward J. Haug, Kyung K. Choi, and Vadim Komkov, Design sensitivity analysis of structural systems, Mathematics in Science and Engineering, vol. 177, Academic Press, Inc., Orlando, FL, 1986. MR 860040
  • [19] A. M. Khludnev, On the contact of two plates, one of which contains a crack, Prikl. Mat. Mekh. 61 (1997), no. 5, 882–894 (Russian, with Russian summary); English transl., J. Appl. Math. Mech. 61 (1997), no. 5, 851–862 (1998). MR 1632059, https://doi.org/10.1016/S0021-8928(97)00109-3
  • [20] A. M. Khludnev, A contact problem for a shallow shell with a crack, Prikl. Mat. Mekh. 59 (1995), no. 2, 318–326 (Russian, with Russian summary); English transl., J. Appl. Math. Mech. 59 (1995), no. 2, 299–306. MR 1350047, https://doi.org/10.1016/0021-8928(95)00033-L
  • [21] L. V. Ovsyannikov, \cyr Lektsii po osnovam gazovoĭ dinamiki, “Nauka”, Moscow, 1981 (Russian). MR 665918

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DOI: https://doi.org/10.1090/qam/1878261
Article copyright: © Copyright 2002 American Mathematical Society


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