Mechanics of Solids (about journal) Mechanics of Solids
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IssuesArchive of Issues2008-1pp.31-33

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N. V. Minaeva, "Application of the perturbation method in mechanics of deformable solids," Mech. Solids. 43 (1), 31-33 (2008)
Year 2008 Volume 43 Number 1 Pages 31-33
DOI 10.3103/S0025654408010044
Title Application of the perturbation method in mechanics of deformable solids
Author(s) N. V. Minaeva (Voronezh State Technological Academy, pr-t Revolyutsii 19, Voronezh, 394000, Russia)
Abstract Although the solutions of the classical problems of continuum mechanics have been studied sufficiently well, the smallest deviations, for example, of the body boundary or of the material characteristics from the traditional values prevent one from obtaining exact solutions of these problems. In this case, one has to use approximate methods, the most common of which is the perturbation method. The problems studied in [1-6] belong to classical problems in which the perturbation method is used to study the behavior of deformable bodies. A wide survey of studies analyzing the perturbations of the body boundary shape caused by variations in its stress-strain state is given in [5, 6].

In numerous studies, it was noted that the problem on the convergence of approximate solutions and hence the studies of the continuous dependence of the solution of the original problem on the characteristics of perturbations ("imperfections") play an important role. In the present paper, we analyze the forms of mathematical models of deformable bodies by studying whether the solution of the original problem continuously depends on the characteristics of the perturbed shape of the body boundary on which the boundary conditions are posed in terms of stresses and on the characteristics of the material properties. We use the results of this analysis to conclude that, when using the perturbation method, one should state the boundary conditions in terms of stresses on the boundary of the real body in stressed state.
1.  D. D. Ivlev and L. V. Ershov, Perturbation Method in the Theory of Elastoplastic Body (Nauka, Moscow, 1978) [in Russian].
2.  D. D. Ivlev, Mechanics of Plastic Media, Vol. 2 (Fizmatlit, Moscow, 2002) [in Russian].
3.  D. D. Ivlev, "Approximate Solution of Problems of Theory of Small Elastoplastic Strains," Dokl. Akad. Nauk SSSR 113(3), 527-528 (1957) [Soviet Math. Dokl.].
4.  D. D. Ivlev, "Approximate Solution of Elastoplastic Problems of Theory of Perfect Plasticity," Dokl. Akad. Nauk SSSR 113(2), 294-296 (1957) [Soviet Math. Dokl.].
5.  A. N. Guz and Yu. N. Nemish, "Perturbation of Boundary Shape in Continuum Mechanics (Review)," Prikl. Mekh. 23(9), 3-29 (1987) [Int. Appl. Mech. (Engl. Transl.) 23 (9), 799-822 (1987)].
6.  A. N. Guz and Yu. N. Nemish, Method of Boundary Shape Perturbations in Mechanics of Solids (Vyshcha Shkola, Kiev, 1989) [in Russian].
7.  A. N. Guz and Yu. N. Nemish, Perturbation Method in Spatial Problems of Elasticity (Vyshcha Shkola, Kiev, 1982) [in Russian].
Received 03 March 2005
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