Abstract
The static boundary value problems of plane elasticity for an inhomogeneous anisotropic medium in a simply connected domain are reduced to the Riemann–Hilbert problem for a quasianalytic vector. Singular integral equations over the domain are obtained, and their solvability is proved for a sufficiently wide anisotropy class. In the case of a homogeneous anisotropic body, the solutions of the first and second boundary value problems are obtained in closed form.
For compound elastic media with anisotropy varying over a domain (of a sufficiently wide class), uniquely solvable integral equations of boundary value problems of static elasticity for an inhomogeneous anisotropic medium are obtained, which readily permits finding generalized solutions that satisfy the matching conditions on the interfaces between the subdomains.
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References
L. Bers, “Partial Differential Equations and GeneralizedAnalytic Functions,” Proc. Nat. Ac. Se. USA 37 (1), 42–47 (1951).
I. N. Vekua, Generalized Analytic Functions (Nauka, Moscow, 1988) [in Russian].
B. B. Bojarski, “Theory of Generalized Analytic Vector,” Annales PoloniciMathematicy 17, 281–320 (1966).
N. I. Monakhov, Boundary Value Problems with Free Boundaries for Elliptic Systems of Equations (Nauka, Moscow, 1977) [in Russian].
N. I. Monakhov, “Nonlinear Diffusion Processes,” Sibirsk. Mat. Zh. 44 (5), 1082–1097 (2003) [Siberian Math. J. (Engl. Transl.) 44 (5), 845–856 (2003)].
N. I. Muskhelishvili, Some Fundamental Problems of Mathematical Elasticity Theor (Nauka, Moscow, 1966) [in Russian].
N. I. Martynov, “Boundary Value Problems of Elasticity of Inhomogeneous Medium Treated as Boundary Value Problems of Generalized Analytic Vector,” Mat. Zh., No. 3 (25), 69–77 (2007).
N. I. Martynov, “Reduction of Boundary Value Problems of Elasticity to boundary Value Problems of Generalized Analytic Vector,” in Theses of Intern. Sci. Conf. “Differential Equations, Theory of Functions, and Applications Dedicated to Academician I. N. Vekua on the Occasion of His 100th Birthday (2007), pp. 518–519 [in Russian].
L. A. Alekseeva, N. I. Martynov, and I. Yu. Fedorov, “Application of QuasiconformalMapping in Problems of Torsion of Inhomogeneous Anisotropic Bodies,” Mat. Zh. 9 (3(33)), 14–18 (2009).
K. F. Chernykh, Nonlinear Elasticity in Engineering (Mashinostroenie, Leningrad, 1986) [in Russian].
S. G. Lekhnitskii, Theory of Elasticity of Anisotropic Bodies (Nauka, Moscow, 1977) [in Russian].
K. F. Chernykh, Introduction to Anisotropic Elasticity (Nauka, Moscow, 1988) [in Russian].
N. I. Ostrosablin, “Canonical Moduli and General Solution of Equations of a Two-Dimensional Static Problem of Anisotropic Elasticity,” Zh. Prikl. Mekh. Tekhn. Fiz. 51 (3), 94–106 (2010) [J. Appl. Mech. Tech. Phys. (Engl. Transl.) 51 (3), 377–388 (2010)].
I. G. Petrovskii, Lectures on the Theory of Partial Differential Equations (Gos. Tekh. Izd. Tekh. -Teor. Lit., Moscow, 1953) [in Russian].
N. I. Muskhelishvili, Singular Integral Equations (Boundary Value Problems of Theory of Functions and Some of Their Applications in Mathematical Physics (Fiz. -Mat. Lit., Moscow, 1962) [in Russian].
N. P. Vekua, Systems of Singular Integral Equations and Some Boundary Value Problems (Nauka, Moscow, 1970) [in Russian].
F. D. Gakhov, Boundary Value Problems (Nauka, Moscow, 1977) [in Russian].
S. N. Antontsev and N. I. Monakhov, “Boundary Value Problems with Discontinuous Boundary Conditions for Quasilinear Elliptic Systems of 2m (m = 1) First-Order Equations,” Izv. SO AN SSSR. Ser. Tekhn. Nauk 8 (2), 65–73 (1967).
Ch. Ashyraliev and N. I. Monakhov, “Iteration Algorithm for Solving Two-Dimensional Singular Integral Equations,” Din. Sploshnoi Sredy, No. 101, 21–29 (1991).
G. N. Savin, Stress Distribution near Holes (Naukova Dumka, Kiev, 1968) [in Russian].
G. S. Litvinchuk, Boundary Value Problems and Singular Integral Equations with Translation (Nauka, Moscow, 1977) [in Russian].
E. A. Raenko, “Boundary Value Problems for a Quasiholomorphic Vector,” Din. Sploshnoi Sredy, No. 118, 65–68 (2001).
O. A. Ladyzhenskaya and N. I. Ural’tseva, Linear and Quasilinear Equations of Elliptic Type (Nauka, Moscow, 1964) [in Russian].
V. A. Lomakin, Theory of Elasticity of Inhomogeneous Bodies (Izdat. MGU, Moscow, 1976) [in Russian].
N. I. Martynov, “Quasiconformal Mappings in Plane Elasticity of Inhomogeneous Anisotropic Medium,” Vestnik NAN RK, No. 5, 11–19 (2012).
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Original Russian Text © N.I. Martynov, 2016, published in Izvestiya Akademii Nauk, Mekhanika Tverdogo Tela, 2016, No. 4, pp. 94–117.
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Martynov, N.I. Integral equations of plane static boundary value problems of the elasticity theory for an inhomogeneous anisotropic medium. Mech. Solids 51, 451–471 (2016). https://doi.org/10.3103/S0025654416040087
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DOI: https://doi.org/10.3103/S0025654416040087