Action of a variable concentrated load on a viscous half-space with allowance for surface tension

An attempt is made to estimate the effect of surface tension on the distribution of velocities and pressure and the shape of the free surface under the action of a variable concentrated load.Mekhanika Polimerov, Vol. 3, No. 2, pp. 273–278, 1967     

The stress-strain state of shells under the action of concentrated heat sources

We construct a fundamental solution of a static problem of heat conduction for thin isotropic shells of positive curvature under the action of concentrated heat sources. We study the problems involved in simplifying the heat equations. Four figures. Two tables. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 25, 1995, pp. 70–80.     

Determination of three-dimensional temperature fields in multiply connected orthotropic bodies under heating with heat sources and flows

We present an algorithm for calculating temperature fields in orthotropic bodies of complex shape, which is based on the method of integral equations. To develop the algorithm, the heat conduction boundary-value problem for orthotropic bodies is preliminarily reduced to the corresponding heat conduction problems for isotropic bodies with modified boundary conditions and heat sources. An investigation of the influence of anisotropy on temperature fields in a bounded and an infinite body with a cavity that are heated by heat sources and flows is performed.     

Determination of the stress state in a half-space in the vicinity of cylindrical defects appearing on the surface under torsional vibrations

We solve the problem of the determination of the stress state under torsional vibrations of a half-space with a cylindrical defect (a crack or a thin rigid inclusion) that crosses its surface. The method of solution is based on the use of discontinuous solutions of the equations of torsional vibrations and consists in the reduction of the initial boundary-value problems to integral equations for the unknown jumps of an angular displacement or a tangential stress.     

Viscoelastic jet drawn under conditions of heat exchange with the ambient medium

The effect of a constant or variable pull on a heated (cooled) viscoelastic jet is investigated. The problem is resolved into the subproblems of the extension of a viscoelastic rod and the motion determined by the rate of the process. The formation of polystyrene fibers is treated as an example.Mekhanika Polimerov, Vol. 4, No. 3, pp. 540–546, 1968     

The influence of a spherical defect on the temperature field in a half-space under local heating

We study the nonlinear axisymmetric problem of determining the temperature field near a spherical defect irradiated by a heat flux. The solution is constructed by the Fourier method using the Kirchhoff transform in a bipolar coordinate system. We carry out a numerical analysis of two special cases of boundary conditions. Translated fromMatematichni Metodi ta Fiziko-mekhanichni Polya, Vol. 39, No. 1, 1996, pp. 47–50.     

Determination of a time-dependent heat source under nonlocal boundary and integral overdetermination conditions

This paper investigates the inverse problem of finding a time-dependent heat source in a parabolic equation with nonlocal boundary and integral overdetermination conditions. Under some natural regularity and consistency conditions on the input data the existence, uniqueness and continuously dependence upon the data of the solution are shown by using the generalized Fourier method. Numerical tests using the Crank-Nicolson finite difference scheme combined with an iterative method are presented and discussed.     

The Reynolds analogy for the mixed convection over a vertical surface with prescribed heat flux

The steady mixed convection boundary layer flow over a vertical surface with prescribed heat flux is revisited in this Note. The subset of solutions which can be obtained with the aid of the Reynolds analogy is discussed in a close relationship with the dual solutions reported by Merkin and Mahmood [1] for impermeable, and more recently by Ishak et al. [2], for permeable surfaces.     

The Reynolds analogy for the mixed convection over a vertical surface with prescribed heat flux

The steady mixed convection boundary layer flow over a vertical surface with prescribed heat flux is revisited in this Note. The subset of solutions which can be obtained with the aid of the Reynolds analogy is discussed in a close relationship with the dual solutions reported by Merkin and Mahmood [1] for impermeable, and more recently by Ishak et al. [2], for permeable surfaces.      

On the deformation of an elastic half-space with a thin slit for mixed conditions on its boundary

The problem of the state of stress and strain in an elastic half-space with a cutout in the shape of a circular slot is solved by Kelvin's method /1/. The conditions on the slot are satisfied in a suitable manner by selecting the scalar and vector mass force potentials as generalized functions concentrated at the slot. The problem reduces to a system of Fredholm integral equations of the second kind in a semi-infinite interval. The solution for an elastic space with a slot is obtained in final form in the limiting case, which enables an estimate to be made of the magnitude of the settling of the earth's surface as a result of oil or gas deposit development.     

Interface conditions for Stokes equations with a discontinuous viscosity and surface sources

The interface conditions, or jump conditions, for the pressure and the velocity of the solution to the incompressible Stokes equations with a discontinuous viscosity and a singular source along an interface are derived in this work. While parts of the results agree with those in the literature, some of the results are new. These theoretical results are useful for developing accurate numerical methods for the interface problem.     

Temperature distribution in a parallelopiped under the influence of heat source with boundary conditions of second kind

In the present paper, the general expressions for temperature distribution in an orthotropic plate of finite dimensions have been derived under the influence of an arbitrary volume heat source and arbitrary initial temperature distribution. The boundary conditions prescribed on the faces are of second kind. The solution is obtained by constructing an eigen value problem. The expression for unsteady temperature distribution consists of two parts—steady and pseudo-steady. The pseudo-steady parts of the solution have been obtained with the help of finite integral transform technique. The general expressions are further studied under different cases. The whole analysis is presented in the dimensionless form with the help of various dimensionless parameters,viz., F _o , P _o , K _i , etc.     

3D dynamic responses of a multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure

Dynamic Green's function plays an important role in the study of various wave radiation, scattering and soil-structure interaction problems. However, little research has been done on the response of transversely isotropic saturated layered media. In this paper, the 3D dynamic responses of a multi-layered transversely isotropic saturated half-space subjected to concentrated forces and pore pressure are investigated. First, utilizing Fourier expansion in circumferential direction accompanied by Hankel integral transform in radial direction, the wave equations for transversely isotropic saturated medium in cylindrical coordinate system are solved. Next, with the aid of the exact dynamic stiffness matrix for in-plane and out-of-plane motions, the solutions for multi-layered transversely isotropic saturated half-space under concentrated forces and pore pressure are obtained by direct stiffness method. A FORTRAN computer code is developed to achieve numerical evaluation of the proposed method, and its accuracy is validated through comparison with existing solutions that are special cases of the more general problems addressed. In addition, selected numerical results for a homogeneous and a layered material model are performed to illustrate the effects of material anisotropy, load frequency, drainage condition and layering on the dynamic responses. The presented solutions form a complete set of Green's functions for concentrated forces (including horizontal load in x(y)-direction, vertical load in z-direction) as well as pore pressure, which lays the foundation for further exploring wave propagation of complex local site in a layered transversely isotropic saturated half-space by using the BEMs.     

On the conditions for a Hamiltonian matrix to have an eigen-value density with some prescribed characteristics

The following problem is considered: “Find the necessary and sufficient conditions to be fulfilled by the components of a hamiltonian operator to have an eigenvalue density with certain prescribed characteristics”. The solution when the hamiltonian operator is a Jacobi matrix and the prescribed characteristic is the unimodality, is shown. A sufficient condition for a real Jacobi matrix to have a prescribed density is also given.     

Thermoelasticity of orthotropic shells under the action of a moving concentrated heat source

We solve the problem of thermoelasticity for thin orthotropic shells of nonnegative curvature under the action of a concentrated heat source that moves over the surface of the shell. A linear temperature distribution over the thickness of the shell and convective heat exchange from its lateral surfaces by the Newton law are established. Using Fourier and Laplace integral transformations, we obtain a solution in the analytic form. The influences of the thermomechanical properties of the material and parameters of heat exchange with the surrounding medium on the stress-strain state of the shell are investigated.     

Determination of an unknown source term and the temperature distribution for the linear heat equation involving fractional derivative in time

We consider the inverse problem of finding the temperature distribution and the heat source whenever the temperatures at the initial time and the final time are given. The problem considered is one dimensional and the unknown heat source is supposed to be space dependent only. The existence and uniqueness results are proved.     

On a class of problems of determining the temperature and density of heat sources given initial and final temperature

We consider a class of problems modeling the process of determining the temperature and density of heat sources given initial and finite temperature. Their mathematical statements involve inverse problems for the heat equation in which, solving the equation, we have to find the unknown right-hand side depending only on the space variable. We prove the existence and uniqueness of classical solutions to the problem, solving the problem independently of whether the corresponding spectral problem (for the operator of multiple differentiation with not strongly regular boundary conditions) has a basis of generalized eigenfunctions.     

The displacements in an elastic half-space when a load moves along a beam lying on its surface

The motion, with constant velocity, of a normal load along an elastic beam lying on an elastic isotropic homogeneous half-space is considered. A method for the approximate calculation of the normal displacements of the surface of the half-space for subsonic velocities of motion is developed. An estimate is given of the expressions obtained and a comparison is made with existing results for the problem of the motion of a point load along a half-space.