Reduction of elasticity theory boundary problems for inhomogeneous media to sets of integral equations

The boundary problem of elasticity theory in stresses or displacements for materials which are continuously inhomogeneous along one coordinate is reduced by means of Laplace and Helmholtz equations to a set of four integro-differential equations, two of which are singular. Each of the equations contains integrals for the contour of the transverse section of a body which is assumed to be piecewise-smooth, and integrals for a region coincident with the section of the body.Sumy. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 21, pp. 20–23, 1990.     

Some basic boundary value problems for plane theory of elasticity of porous Cosserat media with triple-porosity

The static equilibrium of porous elastic materials with triple-porosity is considered in the case of an elastic Cosserat medium. A two-dimensional system of equations of plane deformation is written in the complex form and its general solution is represented by means of three analytic functions of a complex variable and three solutions of Helmholtz equations. The constructed general solution enables one to solve analytically a sufficiently wide class of plane boundary value problems of the elastic equilibrium of porous Cosserat media with triple-porosity. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

External boundary value problems in the quasi static theory of elasticity for triple porosity materials

In this paper the quasi static linear theory of elasticity for materials with triple porosity is considered. Basic external boundary value problems (BVPs) of steady vibrations are formulated. The uniqueness and existence theorems for regular (classical) solutions of these BVPs are established. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Singular boundary integral equations of boundary value problems of the elasticity theory under supersonic transport loads

We consider a transport boundary value problem for an isotropic elastic medium bounded by a cylindrical surface of arbitrary cross-section and subjected to supersonic transport loads. We pose the corresponding hyperbolic boundary value problem and prove the uniqueness of the solution with regard to shock waves. To solve the problem, we use the method of generalized functions. In the space of generalized functions, we obtain the solution, perform its regularization, and construct a dynamic analog of the Somigliana formula and singular boundary equations solving the boundary value problem.     

Integral equations for the mixed boundary value problem in plane elastostatics

Six formulations of the mixed boundary value problem of plane elastostatics integral equations are presented. All equations are of purely second kind and are characterized by a uniform structure of the kernels with respect to geometrical and statical boundary values. The kernels of two formulations are regular, the remaining formulations contain Cauchy principal value integrals.     

Integral equations for the Dirichlet and Neumann boundary value problems in a plane domain with a cusp on the boundary

The boundary equations of the logarithmic potential theory corresponding to the interior Dirichlet problem and the exterior Neumann problem for a plane domain with a cusp on the boundary are studied. Solvability theorems are proved for these integral equations in the spacesL ^p. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 881–892, June, 1996.     

Weak solutions of the interior boundary value problems of plane Cosserat elasticity

In this paper we formulate interior Dirichlet and Neumann boundary value problems of plane Cosserat elasticity in Sobolev spaces, show that these problems are well-posed and find the corresponding weak solutions in the form of integral potentials.     

Weak solutions of the interior boundary value problems of plane Cosserat elasticity

In this paper we formulate interior Dirichlet and Neumann boundary value problems of plane Cosserat elasticity in Sobolev spaces, show that these problems are well-posed and find the corresponding weak solutions in the form of integral potentials. Received: April 7, 2005     

Equivalent boundary integral equations for plane elasticity

Indirect and direct boundary integral equations equivalent to the original boundary value problem of differential equation of plane elasticity are established rigorously. The unnecessity or deficiency of some customary boundary integral equations is indicated by examples and numerical comparison.     

Integral boundary value problems for nonlinear non-instantaneous impulsive differential equations

In this paper, we study integral boundary value problems for two classes of nonlinear non-instantaneous impulsive ordinary differential equations. More precisely, one is integer order impulsive model, the other is fractional order impulsive model. By using standard fixed point approach, a series of existence results are presented under different conditions.     

An outer layer method for solving boundary value problems of elasticity theory

In this paper, an algorithm for solving boundary value problems of elasticity theory suitable for solving contact problems and those whose deformation domain contains thin layers is presented. The solution is represented as a linear combination of auxiliary and fundamental solutions to the Lame equations. The singular points of the fundamental solutions are located in an outer layer of the deformation domain near the boundary. The linear combination coefficients are determined by minimizing deviations of the linear combination from the boundary conditions. To minimize the deviations, a conjugate gradient method is used. Examples of calculations for mixed boundary conditions are presented.     

Integral boundary value problems for first order integro-differential equations with deviating arguments

This paper deals with first order integro-differential equations of mixed type with deviating arguments. We investigate the existence of solutions of such problems with integral boundary conditions by establishing a comparison result and applying the monotone iterative technique. To obtain corresponding results, we also discuss first order differential inequalities with deviating arguments. Two examples demonstrate the application of our results.     

Integral equations in the linear theory of elasticity in semiinfinite domains

We investigate linear integral equations on a semiaxis that appear in the course of construction of solutions of boundary-value problems in the theory of elasticity in such domains as a semiinfinite strip or a cylinder. By using the Mellin transformation and the theory of perturbations of linear operators, we establish general results concerning the solvability and asymptotic properties of solutions of the equations considered. We give examples of application of the general statements obtained to specific integral equations in the theory of elasticity. Translated from Ukrainskii Matematicheskii Zhumal, Vol. 50, No. 5, pp. 613–622, May, 1998.     

Integral boundary value problems for first order impulsive integro-differential equations of mixed type

This paper considers existence of solutions for a class of first order impulsive differential equation with integral boundary value conditions. We present a new comparison theorem and show that the monotone iterative technique coupled with lower and upper solutions is still valid. The results we obtained generalize and improve many recent results.     

Solvability in Lebesgue spaces of nonlinear boundary equations of one-sided contact problems of elasticity theory

We investigate the solvability of one-sided contact problems with nonlinear boundary conditions. The results generalize the author's previous results to the case of equations in Lebesgue spaces.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 77–83, 1988.     

Integral boundary value problems for first order nonlinear impulsive functional integro-differential differential equations

In this paper, we consider the existence of solutions for first order nonlinear impulsive functional integro-differential equations with integral boundary value conditions. We firstly build a new comparison theorem. Then by utilizing the monotone iterative technique and the method of lower and upper solutions, we obtain the existence of extremal solutions or quasi-solutions.     

Reduction of three-dimensional dynamical elasticity theory problems with arbitrarily located plane slits to integral equations

By generalizing a method described earlier /1/ for reducing three-dimensional dynamical problems of elasticity theory for a body with a slit to integral equations, integral equations are obtained for an infinite body with arbitrarily located plane slits. The interaction of disc-shaped slits located in one plane is investigated when normal external forces that vary sinusoidally with time (steady vibrations) are given on their surfaces.

Problems of the reduction of dynamical three-dimensional elasticity theory problems to integral equations for an infinite body weakened by a plane slit were examined in /1, 2/. The solution of the initial problem is obtained in /1/ by applying a Laplace integral transform in time to the appropriate equations and constructing the solution in the form of Helmholtz potentials with densities characterizing the opening of the slit during deformation of the body. The problem under consideration is solved in /2/ by using the fundamental Stokes solution /3/ with subsequent construction of the solution in the form of an analogue of the elastic potential of a double layer.