Biorthogonal expansions in the first fundamental problem of elasticity theory

The first fundamental boundary-value problem of elasticity theory is considered for a rectangular semi-infinite strip whose long sides are free of stress. Separation of variables is used to reduce the solution to a series expansion of two functions defined in a closed interval (the “end” of the half-strip), in terms of homogeneous solutions. The system of homogeneous solutions over an interval of the real axis is proved to be complete in L₂. Systems of functions biorthogonal to the systems of homogeneous solutions are constructed on a certain contour on the Riemann surface of the logarithm. This biorthogonality concept is a natural generalization of biorthogonality over a closed interval. The biorthogonal systems constructed are used to find explicit expressions for the expansion coefficients.     

Constructing boundary-influence functions of the three-dimensional problem of the theory of elasticity in a parallelepiped

General solutions in the form of boundary-influence functions for the three-dimensional problem of the theory of elasticity in a parallelepiped are constructed. The Il'yushin block method, the Papkovich-Neiber representation, and the straightline method in combination with finite Fourier series are used.Translated from Matematicheskie Metody i Fiziko-mekhanicheskie Polya, No. 26, pp. 68–72, 1987.     

The first and second fundamental boundary-value problems of elasticity theory for a transversally isotropic paraboloid of revolution

We give an exact solution of the interior and exterior problems of elasticity theory for a transversally isotropic paraboloid of revolution in the case when the stresses prescribed on its surface or the displacements along one variable can be represented by a Hankel integral and those along the other variable can be expanded in a trigonometric series. It is assumed that the roots of the characteristic equation are of multiplicity greater than one.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 60–70.     

On a combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped

A combined grid method for solving the Dirichlet problem for the Laplace equation in a rectangular parallelepiped is proposed. At the grid points that are at the distance equal to the grid size from the boundary, the 6-point averaging operator is used. At the other grid points, the 26-point averaging operator is used. It is assumed that the boundary values have the third derivatives satisfying the Lipschitz condition on the faces; on the edges, they are continuous and their second derivatives satisfy the compatibility condition implied by the Laplace equation. The uniform convergence of the grid solution with the fourth order with respect to the grid size is proved     

A two-stage difference method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A novel two-stage difference method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. At the first stage, approximate values of the sum of the pure fourth derivatives of the desired solution are sought on a cubic grid. At the second stage, the system of difference equations approximating the Dirichlet problem is corrected by introducing the quantities determined at the first stage. The difference equations at the first and second stages are formulated using the simplest six-point averaging operator. Under the assumptions that the given boundary values are six times differentiable at the faces of the parallelepiped, those derivatives satisfy the Hölder condition, and the boundary values are continuous at the edges and their second derivatives satisfy a matching condition implied by the Laplace equation, it is proved that the difference solution to the Dirichlet problem converges uniformly as O(h ⁴lnh ^?1), where h is the mesh size.     

Modified combined grid method for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped

A modified combined grid method is proposed for solving the Dirichlet problem for the Laplace equation on a rectangular parallelepiped. The six-point averaging operator is applied at next-to-the-boundary grid points, while the 18-point averaging operator is used instead of the 26-point one at the remaining grid points. Assuming that the boundary values given on the faces have fourth derivatives satisfying the Hölder condition, the boundary values on the edges are continuous, and their second derivatives obey a matching condition implied by the Laplace equation, the grid solution is proved to converge uniformly with the fourth order with respect to the mesh size.     

The second fundamental problem of elasticity applied to a plane circular ring

Zusammenfassung Ein ebener Kreisring mit vorgegebenen Randverschiebungen wird mittels der komplexen Methode nachN. I. Muskhelishvili behandelt. Es wird ein einfaches Beispiel gelöst und die Lösung mit der fehlerhaften vonH. Reissner verglichen.     

Solution of the fundamental problems of elasticity theory for a multiconnected anisotropic half-plane

By analytic continuation of generalized complex potentials to upper half-planes we reduce the boundary conditions on a rectilinear boundary to problems of linear coupling for cuts of a multiconnected extended plane. By solving the latter problems we obtain general representations of the complex potentials in the case of a multiconnected anisotropic half-plane for different types of boundary conditions on intervals of the rectilinear boundary. As particular cases we give expressions for the complex potentials in the cases of action of external forces on the rectilinear boundary and dies both with and without friction. Two figures. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 27, 1997, pp. 44–63.     

Neumann problem for the operator of elasticity theory

Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 41, No. 9, pp. 1279–1281, September, 1989.     

A method for solving the first initial-and-boundary-value problem of the linear theory of elasticity for isotropic bodies

A guadrature of the solution of the first dynamic problem of the linear theory of elasticity in which the deformable body occupies a finite volume and is bounded by a piecewise-smooth surface, is obtained. The material of the body is assumed to be homogeneous and isotropic. It is proved that the quadrature satisfies a system of equations, as well as the initial and boundary conditions at the original problem.     

The contact problem of elasticity theory for bodies with cracks

The plane contact problem of a stamp impressed into an elastic half-plane containing arbitrarily arranged rectilinear subsurface cracks is formulated and investigated by asymptotic methods. Partial or total overlapping of the crack edges is assumed. The problem reduces to a system of linear singular integrodifferential equations with side conditions in the form of equalities and inequalities. An analytic solution of the problem is obtained in the form of asymptotic power series /1/ in the relative dimension of the greatest crack. Dependences of the first terms of the asymptotic expansions of the desired functions on the mutual location of the cracks and the contact domains, the pressure and friction stress distributions, and the crack size and orientation are determined. Numerical results are presented.

Analysis of the influence of the stress-free boundary of the half-plane on the state of stress and strain of the elastic material near the cracks is presented in /2, 3/. The problem of a crack in an elastic plane whose edges overlap partially is also examined in /3/ by numerical methods.     

A problem in elasticity theory

The problem of determining the dimensions of the transverse cross-sections of a beam from the given frequencies of its natural vibrations is examined. Frequency spectra are indicated that determine the dimenions of the transverse cross-sections of the beam uniquely, an effective procedure is presented for solving the inverse problem, and a uniqueness theorem is proved. The method of standard models /1/ is used to solve the inverse problem.     

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.     

Local solvability of a problem of nonlinear elasticity theory

One considers an integral functional depending only on the trace of the metric tensor induced by a mapping of ann-dimensional domain into ⁿ. One seeks a mapping which minimizes this functional. Under a well-defined smallness in the boundary conditions, one proves the existence of an infinite set of critical points of the functional. Under additional restrictions, one discusses an existence theorem and the character of the extremum. The convexity of the functional is not assumed. Such functionals are encountered in elasticity theory.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 110, pp. 163–173, 1981.