Asymptotic solutions of coupled dynamic problems of thermoelasticity for thin bodies of anisotropic inhomogeneous-in-plan materials

Two-dimensional recurrence resolvents for an inhomogeneous thin body (plates of variable thickness and shells) are derived by an asymptotic method based on the three-dimensional equations of the coupled dynamic problem of the thermoelasticity of an anisotropic body, which are solved in the case of anisotropy, having, at each point, one plane of symmetry perpendicular to the transverse axis. Recurrence formulae are derived in a general formulation for determining the components of the stress tensor, the strain vector and the function of the change in the temperature field, when different boundary conditions of dynamic problems of the theory of coupled thermoelasticity and thermal conductivity are given on the end surfaces of a thin body. An algorithm for determining the analytical and numerical (necessary) solutions of these boundary-value problems with an arbitrarily specified accuracy is developed.     

An analytic method of studying the nonlinearly elastic equilibrium of a rectangular plate of variable thickness under bending

We propose an approximate analytic method of solving three-dimensional boundary-value problems of the physically nonlinear theory of elasticity for thick rectangular plates of variable thickness subject to a transverse load. The method is used to seek a solution of this problem in the form of double power series in small dimensionless parameters. In arbitrary approximation the original problem is reduced to a sequence of linear inhomogeneous boundary-value problems for plates of constant thickness. Bibliography: 6 titles. Translated fromTeoreticheskaya i Prikladnaya Mekhanika, No. 28, 1998, pp. 36–40     

Asymptotic solutions of coupled dynamic problems of thermoelasticity for isotropic plates

The asymptotic method of solving boundary-value problems of the theory of elasticity for anisotropic strips and plates is used to solve coupled dynamic problems of thermoelasticity for plates, on the faces of which the values of the temperature function and the values of the components of the displacement vector or the conditions of the mixed problem of the theory of elasticity are specified. Recurrence formulae are derived for determining the components of the displacement vector, the stress tensor and for the temperature field variation function of the plate.     

Analytic method for solving a class of three-dimensional thermoelasticity problems for layered transversely isotropic plates with variable thicknesses

This paper examines three-dimensional boundary value problems in the theory of heat conduction and thermoelasticity for layered transversely isotropic rectangular plates with variable thicknesses acted on by a nonuniform temperature field. It is assumed that known temperature and heat flux at the surfaces of the plate or temperature of the surrounding medium allow a representation of the solution in terms of double trigonometric series. An approximate analytic method has been developed for solving this class of problems which makes it possible to reduce the initial boundary value problem for a plate of variable thickness to a recurrence sequence of the corresponding problems for plates with constant thicknesses. Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Teoreticheskaya i Prikladnaya Mekhanika, No. 30, pp. 26–36, 1999.     

Thermoelasticity of a regularly inhomogeneous curved layer with wavy surfaces

A curved inhomogeneous anisotropic layer of variable thickness is considered that has wavy surfaces. It is assumed that the elastic, thermo-physical characteristics of the layer material and the shape of its upper and lower surfaces are periodic in structure with a single periodicity cell (PC). The period of the structure is here comparable in magnitude with the layer thickness, which is assumed to be much less than the other linear dimensions of the body and the radius of curvature of its middle surface.On the basis of a general scheme for taking the average of processes in periodic media /1, 2/, a method is developed which enables a transition to be made from a spatial quasistatic thermoelasticity problem to a system of thermoelasticity equations for an average shell whose effective and thermophysical coefficients are determined from the solution of local problems in a PC. Results obtained for the static theory of elasticity in /3/ are used. The heat conduction problem is averaged to determine the temperature components occurring in the equation of motion.The model constructed enables thermoelastic strains, stresses and the temperature distribution to be obtained in shells and plates of composite or porous materials with a different kind of reinforcement of the periodic structure (waffle, ribbed, corrugated) in reinforced and grid-like shells and plates. In the limiting case of “smooth” surfaces and a homogeneous material, the thermoelasticity equations are obtained for shallow anisotropic shells and the heat conduction equations of anisotropic shells assuming a linear temperature distribution law over the thickness.     

Static thermoelasticity problems for layered thermosensitive plates with cubic dependence of the coefficients of heat conductivity on temperature

We describe an analytic-numerical method of solution of one-dimensional static thermoelasticity problems for layered plates, heated in different ways. We take into account the cubic dependence of the coefficients of heat conductivity and arbitrary nature of the dependence of other physicomechanical parameters on temperature. Here, using the constructed exact solution of an auxiliary problem, we have reduced the heat conduction problems, irrespective of the number of layers, to the solution of one or a system of two nonlinear algebraic equations. We have also studied the temperature fields and stresses in four-layer plates under conditions of complex heat exchange.     

Green functions of a quasi-static problem

In this paper Green functions are constructed in analytic form for a deformable half-plane of a quasi-static problem of thermoelasticity when the heat flow on the boundary x₂=0 of the half-plane is zero. To construct the Green functions, certain integral representations are used whose kernels are known Green functions of the corresponding problems of elasticity theory. The functions constructed make it possible to obtain a wide class of new solutions of boundary-value problems of thermoelasticity, in particular solutions for a piecewise homogeneous half-plane. Bibliography: 6 titles. Translated fromObchyslyuwval’na ta Pryklandna Matematyka, No. 77, 1993, pp. 97–104.     

Solution of the stationary problem of heat conduction of laminated anisotropic inhomogeneous plates by the method of initial functions

The problem of stationary heat conduction of laminated plates of constant and variable thickness is formulated in the three-dimensional statement. We reduce the three-dimensional problem to a twodimensional one by the method of initial functions. For plates with layers of variable thickness, a system of resolving equations with variable coefficients is obtained. The obtained two-dimensional boundary-value problems are analyzed. For plates with homogeneous layers of constant thickness, we construct a solution in an analytic form. It is shown that this solution coincides with a solution obtained by the method of separation of variables.     

On the classes of problems for deformable one-layer and multilayer thin bodies solvable by the asymptotic method

A survey of studies by the author and his disciples on the solution of some classes of problems for deformable thin bodies (strip-beams, plates, and shells) is presented. Classical and nonclassical boundary-value problems of the statics and dynamics of anisotropic and layered bodies are considered. Free and forced vibrations of one-layer and multilayer thin bodies are investigated. The coupled problems of thermoelasticity are solved.     

Cylindrical Tank Filled With a Liquid in a Three-Dimensional Temperature Field

In the cylindrical coordinate system, we construct an exact solution of the threedimensional thermoelasticity problem for a tank filled with a liquid. After determining the temperature field from the heat conduction equation, we solve the equations of the asymmetrical problem of the theory of elasticity. In doing so, the system of resolving equations is reduced to four separate equations with respect to the displacements of the construction. Several exact solutions of boundary-value problems are found. The results are presented in the form of rather simple formulas.     

Mixed boundary-value problems in the theory of elasticity of thin laminated bodies of variable thickness consisting of anisotropic inhomogeneous materials

Starting from the three-dimensional equations of the theory of thermoelasticity, two-dimensional equations for thin laminated bodies are derived in a general formulation and solved by an asymptotic method. The bodies and layers, consisting of anisotropic and inhomogeneous materials (with respect to two longitudinal coordinates), bounded by arbitrary smooth non-intersecting surfaces, also have variable thicknesses. Recursion formulae are derived for determining the components of the stress tensor and the displacement vector when the kinematic or mixed boundary conditions of the static boundary-value problem of the theory of thermoelasticity are specified on the faces of the body, assuming that the corresponding heat conduction problem is solved. An algorithm for constructing of the analytical solutions of the boundary-value problems formulated is developed using modern computational facilities.     

The plane dynamic problem of coupled thermoelasticity

A method of solving two-dimensional inner and outer boundary-value problems of coupled thermoelasticity, taking into account the finite propagation velocity of heat pulses, is proposed, based on constructed fundamental solutions of the corresponding equations. An estimate is given of the coupling of thermomechanical fields in these problems, and the hyperbolic and parabolic models of thermal conductivity are compared. It is shown that the effect of the finite propagation velocity of heat is unimportant even for very short periods of the duration of the processes (comparable with the relaxation time of the heat flux).     

Determination of the stress-strain state of anisotropic thermosensitive cylinders

In the three-dimensional formulation we study a class of problems involving the stressed state of an axisymmetrically heated anisotropic cylinder arbitrarily inhomogeneous over the thickness taking account of the dependence of mechanical characteristics on the temperature. The solution of the boundary-value problems is carried out numerically. We study the temperature and mechanical fields in composite cylinders. Translated fromMatematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 41, No. 1, 1998, pp. 73–77.     

The method of boundary integral equations in unsteady boundary-value problems of uncoupled thermoelasticity

A development of the method of boundary integral equations for solving unsteady boundary-value problems of uncoupled thermoelasticity is presented. In the case of plane deformation, an algorithm for the numerical implementation of the method is presented and the results of calculations of a thermally stressed plane with apertures of circular (the test problem) and arched forms are given for the case when there is a specified unsteady heat flux on the boundary.     

Unsteady thermal stresses near a curvilinear hole in a plate with heat transfer under its warming by a heat flow

We describe an algorithm of determining quasistatic thermal stresses in multiply connected plates with heat transfer, induced by the disturbance of heat flow near holes. Our approach is based on the Laplace transformation and a modified relation of its numerical conversion. The boundary-value problems for the Helmholtz equation, from which the Laplace transform is determined, are solved using the method of boundary integral equations. We solve the integral equations by the method of mechanical quadratures. The results of calculation of nonstationary temperature fields and stresses induced by them in a strip with small holes of different shape are also presented. Translated from Matematychni Metody ta Fizyko-Mekhanichni Polya, Vol. 51, No. 1, pp. 105–111, January–March, 2008.     

Method for calculating nonstationary temperature stresses in a thermosensitive half-space

On the basis of a model of a thermosensitive body, we propose an analytic-numerical method for the construction of the solution of an axisymmetric quasistationary problem of thermoelasticity for a half-space heated by an instantaneous linear heat source and exchanging heat through a bounding surface by convective heat transfer with the environment. Using the perturbation method, we reduce the problem to the solution of a sequence of boundary-value problems for the Poisson equations, whose solutions are constructed in the form of rapidly convergent series for each approximation by using expansions in multiple probability integrals.     

Solution of a class of three-dimensional boundary value problems of the theory of heat conduction

We consider a class of connected three-dimensional boundary-value problems for the quasi-linear heat equation that model the temperature fields in molds under a continuous cyclic molding of flat castings. We propose a method of approximate solution. The results of the computations are shown as the dependence of the dimensionless temperature on the time throughout ten cycles.Translated fromVychislitel'naya i Prikladnaya Matematika, Issue 71, 1990, pp. 84–90.     

On the solvability of physically nonlinear problems of Thermoelasticity

We study the problem of existence and uniqueness of generalized solutions of nonlinear vector boundary-value problems arising in the physically nonlinear theory of thermoelasticity. We prove the convergence of iteration processes in the space W ¹ ₂.     

On a method of solving a problem of the theory of thermoelasticity for solids of revolution

We propose a method of solving problems of the theory of thermoelasticity for solids of revolution of variable thickness. The method is based on successive approximations. We analyze numerical examples for two types of temperature field in a disk with surfaces that are nonsymmetric with respect to the Or axis.     

On the solvability of one boundary-value problem using a refined theory of Vekua

In the present paper, a boundary-value problem of plates for St. Venant-Kirchhoff materials is considered. This three-dimensional problem is reduced to a two-dimensional problem by Vekua's method for the geometrical nonlinear theory. In the case of N = 0 approximation, using the small-parameter method of Signorini and the theory of complex-variable functions, we solve boundary-value problems either for infinity plates that include a circle or concentrated circular rings. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 51, Differential Equations and Their Applications, 2008.