SHORT TIME SOLUTION FOR A PROBLEM IN MAGNETOTHERMOELASTICITY WITH THERMAL RELAXATION

In this work, we study a problem in electromagnetothermoelasticity with thermal relaxation for a half-space whose surface is subjected to a thermal shock and is laid on a rigid foundation. Laplace transform techniques are used to obtain the solution by a direct approach. Wave propagation in the elastic medium and in the free space, bounding it, is investigated. The solution of the problem is obtained analytically using asymptotic expansions valid for short times. The temperature, displacement, stress, and the induced magnetic and electric field distributions are obtained analytically. The numerical values of these functions are represented graphically.     

A MODE-I CRACK PROBLEM FOR AN INFINITE SPACE IN GENERALIZED THERMOELASTICITY

ABSTRACT

In this work, we solve a dynamical problem of an infinite space with a finite linear crack inside the medium. The Fourier and Laplace transform techniques are used. The problem is reduced to the solution of a system of four dual integral equations. The solution of these equations is shown to be equivalent to the solution of a Fredholm integral equation of the first kind. This integral equation is solved numerically using the method of regularization. The inverse Laplace transforms are obtained numerically using a method based on Fourier expansion techniques. Numerical values for the temperature, stress, displacement, and the stress intensity factor are obtained and represented graphically.     

THERMAL STRESSES FOR AN INFINITE BODY WITH A SPHERICAL CAVITY WITH TWO RELAXATION TIMES

Thermoelastic interactions caused in a homogeneous and isotropic infinite body with a spherical cavity are considered for the two different theories of generalized thermoelasticity, that is, Lord, and Shulman's theory and Green and Lindsay's theory. Analytical expressions for the temperature, displacement, and thermal stress fields are obtained; and the results are compared with the classical dynamical coupled theory.     

GENERALIZED THERMOELASTICITY OF AN INFINITE BODY WITH A CYLINDRICAL CAVITY AND VARIABLE MATERIAL PROPERTIES

ABSTRACT

The equations of generalized thermoelasticity with one relaxation time in an isotropic elastic medium with temperature-dependent mechanical and thermal properties are established. The modulus of elasticity and the thermal conductivity are taken as linear function of temperature. A problem of an infinite body with a cylindrical cavity has been solved by using Laplace transform techniques. The interior surface of the cavity is subjected to thermal and mechanical shocks. The inverse of the Laplace transform is done numerically using a method based on Fourier expansion techniques. The temperature, the displacement, and the stress distributions are represented graphically. A comparison was made with the results obtained in the case of temperature-independent mechanical and thermal properties.     

GENERALIZED THERMOELASTICITY IN AN INFINITE SOLID WITH A HOLE

This paper deals with one-dimensional generalized thermoelasticity based on the theories of Lord and Shulman and of Green and Lindsay. A formulation of generalized thermoelasticity that combines both generalized theories is derived. The generalized thermoelastic problems for an infinite solid with a cylindrical hole and an infinite solid with a spherical hole are analyzed by means of the Laplace transform technique. Numerical calculations for temperature, displacement, and stresses under the generalized formulation are carried out and compared with those of classical dynamic coupled theory.     

EXISTENCE OF A GENERALIZED SOLUTION IN THERMOELASTICITY WITH ONE RELAXATION TIME

The existence of a generalized solution to a natural stress-flux initial boundary-value problem of thermoelasticity with one relaxation time is established. It is shown that the solution is the limit of a sequence of classical solutions belonging to a functional space of the Sobolev type.     

THERMALLY INDUCED VIBRATIONS IN AN ELASTIC BODY WITH A SPHERICAL CAVITY

An elastic heat-conducting body with a cavity can be set in vibration if the temperature of the cavity varies cyclically with time. A simple closed solution can be derived if the geometry is spherical and if the temperature oscillates harmonically. The amplitude of the stresses increases with increasing inertia. If the velocities of temperature waves and elastic waves coincide, no resonancelike amplification of the stresses occurs.     

THERMAL STRESS-FOCUSING EFFECT IN A TRANSVERSELY ISOTROPIC SPHERICAL INCLUSION EMBEDDED IN AN ISOTROPIC INFINITE ELASTIC MEDIUM

When an isotropic infinite elastic medium with a transversely isotropic spherical inclusion is suddenly subjected to a uniform temperature rise, stress waves occur at the interface of spherical inclusion at the moment thermal impact is applied. The stress wave in a transversely isotropic inclusion proceeds radially inward to the center of the inclusion. The wave may accumulate at the center and cause very large stress magnitudes even though the initial thermal stress should be relatively small. This phenomenon is called the stress-focusing effect. The stress wave in an infinite elastic medium proceeds radially to infinity. This paper analyzes, in an exact manner, the effects of these waves using the ray integrals. The results give a clear indication of the mechanism of the thermal stress-focusing effect in a transversely isotropic inclusion embedded in the isotropic infinite elastic medium.     

RELAXATION EFFECTS ON THERMALLY INDUCED VIBRATIONS IN A GENERALIZED THERMOVISCOELASTIC MEDIUM WITH A SPHERICAL CAVITY

Thermally induced vibrations of an infinite isotropic viscoelastic solid containing a spherical cavity are investigated. The stress-free boundary of the cavity is subjected to a temperature that varies harmonically with time. The classical dynamical coupled theory of thermoelasticity and the generalized theories of thermoelasticity proposed by Lord and Shulman and Green and Lindsay are applied to consider the thermoelastic coupling. The analytical expressions for the solutions of displacement, temperature, and stresses are determined. The relaxation effects on the vibrations are studied to compare the three theories. Numerical values of displacement, temperature, and stresses are computed for a particular material; and the results are presented graphically to illustrate the solution.     

GREEN'S FUNCTION FOR A THERMOMECHANICAL MIXED BOUNDARY VALUE PROBLEM OF AN INFINITE PLANE WITH AN ELLIPTIC HOLE

This article derives the Green's function for a thermomechanical mixed boundary value problem of an infinite plane with an elliptic hole under a pair of heat source and sink. To derive the Green's function in closed form, the Cauchy integral method and a basic Green's function for an external force boundary value problem with a pair of heat source and sink are employed. Illustrative numerical results for temperature, heat flux, and stress along the hole edge and stress intensity factors when the hole collapses into a crack are presented graphically.     

THERMOELASTIC INTERACTIONS WITHOUT ENERGY DISSIPATION IN AN UNBOUNDED MEDIUM WITH A SPHERICAL CAVITY DUE TO A THERMAL SHOCK AT THE BOUNDARY

Thermoelastic interactions without energy dissipation in an unbounded elastic medium with a spherical cavity have been investigated. The cavity surface is assumed to be stress free and is subjected to a thermal shock. The solutions for displacement, temperature, and stresses are obtained using the Laplace transform procedure. The discontinuities of the distributions of the physical quantities are determined and compared with earlier findings. The inversions are also carried out with a numerical method based on Fourier series expansions of functions. The results are compared with the corresponding results obtained in cases of conventional thermoelasticity theory and the generalized theories of thermoelasticity with thermal relaxation time parameters.     

DOMAIN OF INFLUENCE IN THERMOELASTICITY WITH ONE RELAXATION TIME

A domain-of-influence theorem for an initial-boundary-value problem of thermoelasticity with one relaxation time, proposed by Lord and Schulman, is established. The theorem asserts that for a finite time t > 0 a pair (<φ, ), in which φ and are the thermoelastic potential and temperature, respectively, generates no thermoelastic disturbances outside a bounded domain D_t.     

A PROBLEM IN GENERALIZED THERMOELASTICITY FOR AN INFINITELY LONG ANNULAR CYLINDER COMPOSED OF TWO DIFFERENT MATERIALS

The problem of heat conduction of a thermoelaslic cylindrical medium composed of two different materials is considered. The problem is solved in the framework of the generalized thermoelasticity theory with one relaxation time. The solution is obtained in the Laplace transform domain by using the potential function approach. Numerical inversion formula is used to obtain the corresponding solutions in the physical domain     

A NONIDEAL CONTACT PROBLEM OF THERMOELASTICITY FOR TWO SOLIDS WITH A HEAT SOURCE

A generalized Hertz problem of thermoelastic solids in pressure, contacting over convex surfaces, has been discussed. At a certain moment of time a concentrated source of heat starts acting. Therefore the heat flux flowing through the region of contact is nonstationary. The problem considered is axially symmetric. The purpose is twofold: first the problem of thermoelasticity with time variation of temperature is taken into account; second the “paradox of a cooled sphere” has been investigated under time-dependent conditions. There is a possibility that the character of the boundary conditions can change in time. To obtain the solution we have applied the Laplace and Hankel integral transforms. The main point is to discuss the cases when the boundary conditions are such that the problem can be considered in terms of classical thermoelasticity and when the Barber-type boundary conditions have to be used. The solution has been obtained by means of a devised numerical algorithm such that the procedure is simplified. The results have been presented in diagram form suitable for discussion.     

AN INVERSE SOURCE PROBLEM IN RADIATIVE TRANSFER FOR SPHERICAL MEDIA

Abstract

A method is presented to identify the source term in a one-dimensional, absorbing, emitting, scattering spherical medium from the knowledge of the exit radiation intensities. The inverse radiation problem is formulated as an optimization problem. The sensitivity problem and the gradient equation are derived. The conjugate gradient method is used for its solution. Although the source term is a function of the space variable, only radiation intensities exiting the outer boundary are required. Both data with and without measurement errors are used as input to identify the source term. The study shows that the estimation of the source term is more sensitive to increases in measurement errors as the optical thickness increases.     

THERMAL STRESS-FOCUSING EFFECT IN A TRANSVERSELY ISOTROPIC SPHERICAL ZIRCONIA INCLUSION IN AN INFINITE ELASTIC MEDIUM CAUSED BY IMPACT COOLING

When an isotropic infinite elastic medium with a transversely isotropic spherical zirconia inclusion is suddenly subjected to a uniform temperature fall, stresswavesoccur at the interface of the spherical inclusion the moment thermal impact is applied. The stress wave in a transversely isotropic spherical inclusion proceeds radially inward to the center of the inclusion. The wave may accumulate at the center and cause very large stress magnitudes, even though the initial thermal stress should be relatively small. This phenomenon is called the stress-focusing effect. The stress wave in an inclusion may be affected by phase transformation during the tempering process. The stress-focusing effects are analyzed in an exact manner, and a clear indication of the mechanism of thermal and phase transformational stress-focusing effects in a transversely isotropic spherical Zirconia inclusion in the isotropic infinite elastic domain is given.     

SOLUTION OF AN ELLIPTICAL RIGID INCLUSION WITH DEBONDINGS IN AN INFINITE PLANE UNDER THE UNIFORM HEAT FLUX

A problem of an elliptical rigid inclusion in an infinite plane subjected to uniform heat flux in an arbitrary direction and with a number of debondings on the interface of the elliptical rigid inclusion and the elastic matrix is solved. The rotation of the inclusion under the uniform heat flux is considered. The complex variable method is used, and the closed-form solution is obtained. The stress intensity of debonding at the debonding tips is calculated and the extension of debonding is investigated. Examples of stress distributions and resultant moments on the inclusion are shown for the inclusion with two debondings.     

A NOTE ON UNIQUENESS IN THERMOELASTICITY WITH ONE RELAXATION TIME

Abstract

Uniqueness for a general initial boundary-value problem of linear dynamic thermoelasticity with one relaxation time is established using the associated conservation law involving higher-order time derivatives.     

TWO-DIMENSIONAL PROBLEM IN GENERALIZED THERMOELASTICITY WITH HEAT SOURCES

In this work, a two-dimensional problem for a half-space is solved. The problem is in the context of the theory of generalized thermoelasticity with one relaxation time. The surface of the half-space is taken to be traction free and the temperature on it is specified. Heat sources permeate the medium. Laplace and exponential Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. The inverse double transform is evaluated numerically. Numerical results are obtained and represented graphically.