Fundamental solution of a plane problem of elasticity theory for a composite anisotropic medium

The plane problem on the action of an arbitrarily oriented concentrated force, applied at some point of an elastic plane, composed of two different anisotropic half-planes, is considered. By a special choice of a particular solution the problem reduces to a well-known differential equation of the anisotropic theory of elasticity with discontinuous coefficients. The latter reduces, by the method of the integral Fourier transform, to the Riemann boundary value problem. Expressions for the stresses and displacement derivatives at an arbitrary point of the plane are obtained. The application of the obtained results is illustrated on the example of a problem on an elastic linear inclusion (strap).Translated from Dinamicheskie Sistemy, No. 4, pp. 40–45, 1985.     

Complex potentials for the plane problem of elasticity theory for a multiconnected anisotropic body with cracks

Results are given for a study and modernization of basic relationships obtained previously by the author for generalized complex potentials of crack theory generated by solving contact problems in the case of multiconnected anisotropic plates. In contrast to the author's previous work devoted to this question, general presentations of complex potentials, and boundary and additional conditions for finding them, are simplified. Use of them is shown in the case of a crack and concentrated forces, and when a plate has a finite number of cracks along a single line. Independence of stress intensity factor on anisotropy parameters is demonstrated in the last case.Donetsk. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 24–34, 1990.     

A mixed dynamic problem of the theory of elasticity for a piecewise homogeneous plane

We study the process of splitting of a compressed piecewise homogeneous medium under high-speed motion of a wedge with the formation of a crack of unknown extent ahead of the wedge. The motion of the wedge occurs along the interface between physico-mechanical properties of the piecewise homogeneous plane. We carry out numerical studies. We obtain asymptotic representations of the elastic potentials at the limiting velocities of motion of the wedge. Two figures. One table. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 26, 1996, pp. 48–58.     

Asymptotics of solutions to the spectral elasticity problem for a spatial body with a thin coupler

We construct asymptotics for the eigenvalues and vector eigenfunctions of the elasticity problem for an anisotropic body with a thin coupler (of diameter h) attached to its surface. In the spectrum we select two series of eigenvalues with stable asymptotics. The first series is formed by eigenvalues O(h ²) corresponding to the transverse oscillations of the rod with rigidly fixed ends, while the second is generated by the longitudinal oscillations and twisting of the rod, as well as eigenoscillations of the body without the coupler. We check the convergence theorem for the first series and derive the error estimates for both series.     

The Cauchy problem of the moment elasticity theory in R m

In this paper we consider the problem of analytical continuation of the solution to the system of equations of the moment theory of elasticity in spatial many-dimensional domain. We give an explicit formula of restoring of solution inside the domain by values of sought-for solution and values of strains on part of the boundary of this domain.     

A plane problem in elasticity theory for a half-strip with given normal displacement at the base

This paper presents a solution for the elasticity problem for a half-strip with load-free lateral edges and given normal displacement with zero tangential stress at the base. The functions describing the load at the base are not assumed smooth. Translated fromDinamicheskie Sistemy. Vol. 12. pp. 8–14. 1993.     

Numerical solution of a boundary-value problem of elasticity theory for a body with an inclusion

We consider the problem of steady-state oscillations of a plane body with an inclusion. A solution algorithm is constructed. Computational experiments are described, which establish the applicability of the proposed procedure for analyzing the effect of defects inside the body on wave fields excited by ultrasonic sources.Kiev University. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 75, pp. 83–87, 1991.     

Cauchy problem for a system of equations of three-dimensional elasticity theory

Samarkand. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 33, No. 1, pp. 186–190, January–February 1992.     

On Kolosov’s relations in the plane problem of elasticity theory

Three different Kolosov-type expressions for stresses in elastic bodies are given.     

Stability of solutions to the Cauchy problem for a plane hyperbolic system with time-periodic coefficients

Considering a plane hyperbolic system with time-periodic coefficients, we construct a version of the direct Lyapunov method with the condition on the derivative of the Lyapunov functional along the trajectories of the system which is weakened by use of periodicity of the coefficients. We exhibit an illustrative example.     

Variational formulations of the static problem of elasticity theory under specified external forces

The initial-value problem is considered for the equations of elastic equilibrium of moving bodies when stresses are specified on the surface of the body. Two variational problems that may be solved in unique fashion over the entire space are formulated for this problem, which is known to possess a unique solution in a subspace.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 43, No. 2, pp. 158–161, February, 1991.     

Estimates for deviations from exact solutions to plane problems in the cosserat theory of elasticity

A functional a posteriori estimate is obtained for control of the accuracy of approximate solutions to plane problems arising in the Cosserat theory of elasticity. We deal with the case where displacements and independent rotation are given on the boundary of the domain, a continuous medium is isotropic, and there is a linear dependence between stresses and strains. The proposed method is based on the duality theory of Calculus of Variations and can be also applied to the case of anisotropic media.