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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A. M. Khludnev and J. Sokołowski, 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)
A. M. Khludnev and V. A. Kovtunenko, Analysis of Cracks in Solids, Southampton-Boston, WIT Press, 2000
V. G. Mazya and S. A. Nazarov, Asymptotic behavior of energy integrals under small perturbations of the boundary near corner and conic points, Proceeding of Moscow Math. Society, Moscow State Univ., 1987, pp. 79–129 (in Russian)
K. Ohtsuka, Generalized J-integral and three-dimensional fracture mechanics I, Hiroshima Math. J. 11, 21–52 (1981)
K. Ohtsuka, Generalized J-integral and its applications. I. - Basic theory, Japan Journal of Applied Mathematics 2, 329–350 (1985)
J. Sokołowski and J. P. Zolesio, Introduction to Shape Optimization—Shape Sensitivity Analysis, Springer-Verlag, New York, 1992
V. P. Parton and E. M. Morozov, Mechanics of Elastoplastic Fracture, Moscow, Nauka, 1985 (in Russian)
A. M. Khludnev and J. Sokołowski, Modelling and Control in Solid Mechanics, Birkhäuser, Basel-Boston-Berlin, 1997
P. Destuynder and M. Jaoua, Sur une Interprétation Mathématique de l’Intégrale de Rice en Théorie de la Rupture Fragile, Math. Mech. in the Appl. Sci. 3, 70–87 (1981)
A. M. Khludnev, On a Signorini problem for inclusions in shells, European J. Appl. Math. 7, 499–510 (1996)
A. M. Khludnev, Contact problem for a plate having a crack of minimal opening, Control and Cybernetics 25, 605–620 (1996)
P. Grisvard, Singularities in Boundary Value Problems, Masson, Paris; Springer, Berlin, 1992
G. P. Cherepanov, Mechanics of Brittle Fracture, McGraw-Hill, 1979
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
S. A. Nazarov and B. A. Plamenevskii, Elliptic problems in domains with piecewise smooth boundaries, Moscow, Nauka, 1991 (in Russian)
Q. S. Nguen, C. Stolz, and G. Debruyne, Energy methods in fracture mechanics: Stability, bifurcation and second variations, European J. of Mech., A/Solids 9, 157–173 (1990)
H. Petryk and Z. Mróz, Time derivatives of integrals and functionals defined on varying volume and surface domains, Arch. Mech. 38, 697–724 (1986)
J. H. Edward, K. C. Kyung, and V. Komkov, Design Sensitivity Analysis of Structural Systems, Academic Press, 1986
A. M. Khludnev, The contact between two plates, one of which contains a crack, J. Appl. Math. Mech. 61, 851–862 (1997)
A. M. Khludnev, The contact problem for a shallow shell with a crack, J. Appl. Math. Mech. 59, 299–306 (1995)
L. V. Ovsyannikov, Lections on the foundation of gas dynamics, Moscow, Nauka, 1981 (in Russian)
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© Copyright 2002
American Mathematical Society
