On the solution of plane boundary-value problems of elasticity in a rectangle

An algorithm for solving plane boundary-value problems of elasticity for a rectangular domain is expounded. The algorithm is based on a complex-valued representation of the general solution to the differential equations of the plane problem and on the use of Lagrange polynomials to satisfy the boundary conditions. The algorithm can quite easily be implemented in a computer program. This is probably the simplest way of solving boundary-value problems of this class __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 97–102, January 2006.     

Existence and uniqueness of the solution of the boundary-value problem for a parabolic equation of unsteady filtration

The problem of the motion of a filtration front in a zero background in the case of a power-law dependence of the filtration coefficient on gas density is considered, and the existence and uniqueness theorem for solutions in the class of analytic functions is proved. The solution is constructed in explicit form, recurrence formulas for computing the coefficients in the series are obtained, and the convergence of the series is proved by the majorant method. The filtration front construction procedure is proposed.     

On approximate solution of boundary-value problems by the least squares method

Using the least squares method, we construct a new iterative procedure for finding solutions of a weakly nonlinear boundary-value problem for a system of ordinary differential equations in the critical case in the form of an expansion of a solution in a generalized Fourier polynomial in the neighborhood of the generating solution. We obtain an estimate for the range of values of the small parameter for which this iterative procedure converges to the required solution. Translated from Neliniini Kolyvannya, Vol. 11, No. 4, pp. 554–573, October–December, 2008.     

Numerical solution of finite geometry boundary-value problems in nonlinear magnetoelasticity

This paper provides examples of the numerical solution of boundary-value problems in nonlinear magnetoelasticity involving finite geometry based on the theoretical framework developed by Dorfmann and co-workers. Specifically, using a prototype constitutive model for isotropic magnetoelasticity, we consider two two-dimensional problems for a block with rectangular cross-section and of infinite extent in the third direction. In the first problem the deformation induced in the block by the application of a uniform magnetic field far from the block and normal to its larger faces without mechanical load is examined, while in the second problem the same magnetic field is applied in conjunction with a shearing deformation produced by in-plane shear stresses on its larger faces. For each problem the distribution of the magnetic field throughout the block and the surrounding space is illustrated graphically, along with the corresponding deformation of the block. The rapidly (in space) changing magnitude of the magnetic field in the neighbourhood of the faces of the block is highlighted.     

On the approximate solution of autonomous boundary-value problems by the least-squares method

Using the least-squares method, we construct a new iterative procedure for finding solutions of an autonomous weakly nonlinear boundary-value problem in the critical case in the form of a generalized Fourier polynomial expansion.     

On the solution of three-dimensional boundary-value problems for layered bodies with nonplanar interfaces

The interfaces play an important role in various buildup bodies, and also in the composite materials and structural elements. Special monographs [7, 8] have been devoted to this question, presenting the results of scientific studies of the physical and chemical phenomena on the interfaces, the mechanical behavior, and the role of the interfaces in the damage processes, and also their influence on the basic mechanical properties of the composites. In many cases the interfaces deviate from the ideal geometric shapes: planar (in the layered composites), circular cylindrical (in the fibrous composites), and spherical (in the granular composites). Numerous theoretical and experimental studies confirm this. Thus, in the explosive welding of metals (and nonmetals) there form wavy surfaces, the sections of which may be close to sinusoids, for example in the welding of niobium and copper [9]. If the densities of the materials differ significantly, then the sinusoidal nature of the interface distorts as illustrated in [12] for the example of the welding of lead and steel. In addition, in view of the nature of the technological processes [10] the interfaces may become curved in the layered composite materials and deviate locally or periodically from the ideal coordinate planes. Theoretical and experimental studies have shown that the shape of the interface has a significant influence on the physical and mechanical processes and phenomena (bond strength, stress concentration, wave diffraction, thermal conduction, and so on). Numerous publications that are cited in the survey works [1, 3, 11] confirm this. A second variant of the boundary shape perturbation method was developed in [4, 5] for the solution of the three-dimensional boundary-value problems for nonorthogonal surfaces that are close to the coordinate planes. It was assumed that the equations of the interfaces are linear relative to the small parameter characterizing the degree of deviation from the coordinate planes. This narrowed significantly the class of the examined boundary-value problems and their practical importance. In the present work we examine the three-dimensional boundary-value problems of the mechanics of layered bodies with interfaces that are described by nonlinear equations relative to a small parameter. We construct in general form the recurrence relations and the differential operators of the boundary conditions, making it possible to solve the three-dimensional boundary-value problems with the accuracy that is required for applications. We examine particular cases and present one of the possible criteria for evaluating the accuracy of the approximate solutions that are obtained with the aid of the described variant of the boundary shape perturbation method.S. P. Timoshenko Institute of Mechanics, Ukrainian Academy of Sciences, Kiev. Translated from Prikladnaya Mekhanika, Vol. 30, No. 2, pp. 23–32, February, 1994.     

Axisymmetric boundary-value problems in a piezoelectric medium

Based on the general solution of three-dimensional problems in piezoelectric medium, with the method of Green's functins^[2], axisymmetric boundary-value problems are discussed. The purpose of this research is for analyzing the effective on mechanics and electricity of the piezoelectric ceramics caused by voids and inclusions. The displacement, traction and electric Green's functions corresponding to circular ring loads acting in the interior of a piezoelectric ceramic are obtained. A cylindrical coordinate system is employed and Hankel transform are applied with respect to radial coordinates. Explicit solutions for Green's functions are presented in terms of infinite integrals of Lipshitz-Hankel type. By solving a traction boundary-value problem, the solution scheme is illustrated. Supported by the National Natural Science Foundation of China and the Foundation of the Open Laboratory of Solid Mechanics.     

An analog of the Saint-Venant principle and the uniqueness of a solution of the first boundary-value problem for a third-order equation of combined type in unbounded domains

We consider the first boundary-value problem for a third-order equation of combined type. Using the Saint-Venant principle, we study the uniqueness class for solutions of the problem in an unbounded domain. __________ Translated from Neliniini Kolyvannya, Vol. 9, No. 1, pp. 117–126, January–March, 2006.     

The non-isothermal elastic dumbbell: A model for the thermal conductivity of a polymer solution

In a flowing polymeric liquid, molecular orientation will give rise to anisotropic conduction of heat. In this paper, a theory is presented relating the thermal conductivity tensor to the deformation history of the fluid. The basis of this theory is formed by the Hookean dumbbell. It is shown that the anisotropy of the thermal conductivity is proportional to the polymer contribution to the extra-stress tensor. This stress-thermal law makes it relatively simple to incorporate anisotropic heat conduction into the numerical simulation of a flowing polymeric liquid.