A two dimensional problem for a transversely isotropic generalized thermoelastic thick plate with spatially varying heat source

This paper deals with a two dimensional problem for a transversely isotropic thick plate having heat source. The upper surface of the plate is stress free with prescribed surface temperature while the lower surface of the plate rests on a rigid foundation and is thermally insulated. The study is carried out in the context of generalized thermoelasticity proposed by Green and Naghdi. The governing equations for displacement and temperature fields are obtained in Laplace–Fourier transform domain by applying Laplace and Fourier transform techniques. The inversion of double transform has been done numerically. The numerical inversion of Laplace transform is done by using a method based on Fourier Series expansion technique. Numerical computations have been done for magnesium (Mg) and the results are presented graphically. The results for an isotropic material (Cu) have been deduced numerically and presented graphically to compare with those of transversely isotropic material (Mg).     

Generalized thermoelastic functionally graded solid with a periodically varying heat source

This paper deals with the problem of thermoelastic interactions in a functionally graded isotropic unbounded medium due to the presence of periodically varying heat sources in the context of the linear theory of generalized thermoelasticity without energy dissipation (TEWOED). The governing equations of generalized thermoelasticity without energy dissipation (GN model type II) for a functionally graded materials (FGM) (i.e. material with spatially varying material properties)are established. The governing equations are expressed in Laplace–Fourier double transform domain and solved in that domain. Now, the inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand is obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the displacement, temperature, stress and strain are obtained for a hypothetical material. The solution to the analogous problem for homogeneous isotropic material is obtained by taking nonhomogeneity parameter suitably. Finally the results obtained are presented graphically to show the effect of nonhomogeneity on displacement, temperature, stress and strain.     

Unsteady Couette flow of a micropolar fluid with slip

The unsteady Couette flow of an isothermal incompressible micropolar fluid between two infinite parallel plates is investigated. The motion of the fluid is produced by a time-dependent impulsive motion of the lower plate while the upper plate is set at rest. A linear slip, of Basset type, boundary condition on both plates is used. Two particular cases are discussed; in the first case we have assumed that the plate moves with constant speed and in the second case we have supposed that the plate oscillates tangentially. The solution of the problem is obtained in the Laplace transform domain. The inversion of the Laplace transform is carried out numerically using a numerical method based on Fourier series expansion. Numerical results are represented graphically for the velocity, microrotation, and volume flux for various values of the time, slip and micropolar parameters.     

A half-space problem in the theory of generalized thermoelastic diffusion

In this work we consider the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The chemical potential is also assumed to be a known function of time on the bounding plane. Laplace transform techniques are used. The solution is obtained in the Laplace transform domain by using a direct approach. The solution of the problem in the physical domain is obtained numerically using a numerical method for the inversion of the Laplace transform based on Fourier expansion techniques.The temperature, displacement, stress and concentration as well as the chemical potential are obtained. Numerical computations are carried out and represented graphically.     

A two-dimensional thermoelasticity problem for a half space subjected to heat sources

The two-dimensional problem for a half space whose surface is traction free and subjected to the effects of heat sources is considered within the context of the theory of thermoelasticity with two relaxation times. Laplace and Fourier transform techniques are used. The solution in the transformed domain is obtained by using a direct approach. Numerical inversion of both transforms is carried out to obtain the temperature, stress and displacement distributions in the physical domain. Numerical results are represented graphically and discussed.     

Two-dimensional interactions due to moving load in generalized thermoelastic solid with diffusion

The present paper is concerned with the investigation of disturbances in＇a homogeneous, isotropic elastic medium with generalized thermoelastic diffusion, when a moving source is acting along one of the co-ordinate axis on the boundary of the medium. Eigen value approach is applied to study the disturbance in Laplace-Fourier transform domain for a two dimensional problem. The analytical expressions for displacement components, stresses, temperature field, concentration and chemical potential are obtained in the physical domain by using a numerical technique for the inversion of Laplace transform based on Fourier expansion techniques. These expressions are calculated numerically for a copper like material and depicted graphically. As special cases, the results in generalized thermoelastic and elastic media are obtained. Effect of presence of diffusion is analyzed theoretically and numerically.     

Generalized thermoelastic infinite transversely isotropic body with a cylindrical cavity due to moving heat source and harmonically varying heat

In this paper, the induced temperature, displacement, and stress fields in an infinite transversely isotropic unbounded medium with cylindrical cavity due to a moving heat source and harmonically varying heat are investigated. This problem is solved in the context of the linear theory of generalized thermoelasticity with dual phase lag model. The governing equations are expressed in Laplace transform domain. Based on Fourier series expansion technique the inversion of Laplace transform is done numerically. The numerical estimates of the displacement, temperature and stress are obtained and presented graphically. The theories of coupled thermoelasticity, generalized thermoelasticity with one relaxation time, and thermoelasticity without energy dissipation can extracted as special cases. Some comparisons have been shown in figures to present the effect of the heat source, dual phase lags parameters and the angular frequency of thermal vibration on all the studied fields.     

Thermo-elastic interaction with energy dissipation in an unbounded body with a spherical hole

The distribution of stresses due to step input of temperature at the boundary of a spherical hole in a homogeneous isotropic unbounded body is investigated by applying Laplace transform technique in the context of generalized theories of thermo-elasticity. The inverse of the transformed solution is carried out by applying a method of Bellman et al. The stresses are computed numerically and presented graphically in a number of figures for aluminum–epoxy composite material. The comparison among the theories i.e. classical thermo-elasticity (CTE), classical coupled thermo-elasticity (CCTE), temperature-rate-dependent thermo-elasticity (TRDTE(GL)) and thermo-elasticity with energy dissipation (TEWED(GN)) theory is presented graphically.     

热冲击载荷作用下的含球形空腔的广义热弹性功能梯度球形各向同性体

This paper is concerned with the determination of thermoelastic displacement, stress and temperature in a functionally graded spherically isotropic infinite elastic medium having a spherical cavity, in the context of the linear theory of generalized thermoelasticity with two relaxation time parameters （Green and Lindsay theory）. The surface of cavity is stress-free and is subjected to a time-dependent thermal shock. The basic equations have been written in the form of a vector-matrix differential equation in the Laplace transform domain, which is then solved by an eigenvalue approach. Numerical inversion of the transforms is carried out using the Bellman method. Displacement, stress and temperature are computed and presented graphically. It is found that variation in the thermo-physical properties of a material strongly influences the response to loading. A comparative study with a corresponding homogeneous material is also made.     

Dynamic analysis of a cracked magnetoelectroelastic medium under antiplane mechanical and inplane electric and magnetic impacts

The transient analysis of a magnetoelectroelastic medium containing a crack is made under antiplane mechanical and inplane electric and magnetic impacts. The crack is assumed to penetrate through the solid along the poling direction. By using the Fourier and Laplace transforms, the associated mixed boundary value problem is reduced to a Fredholm integral equation of the second kind, which is solved numerically. By means of a numerical inversion of the Laplace transform, dynamic field intensity factors are obtained in the time domain. Numerical results are presented graphically to show the effects of the material properties and applied electric and magnetic impacts on the dynamic intensity factors of COD and stress, and dynamic energy density factors. The results indicate that except for the intensity factors of electric displacement and magnetic induction, other field intensity factors exhibit apparent transient feature. Moreover, they depend strongly on mechanical input as well as electric and magnetic impacts.     

Dynamic Response of a Piezoelectric Material with a Conducting Rigid Inclusion

The problem of a conducting rigid inclusion embedded in an infinite piezoelectric matrix is considered under the action of combined electromechanical impact loads. By using integral transform techniques, the mixed initial-boundary value problem for the case of anti-plane shear load and in-plane electric field is transformed into two systems of dual integral equations, the solutions of which give the singularity coefficients of electroelastic field near the inclusion tips in closed-form in the Laplace transform domain. Numerical results for the stress singularity coefficient in the physical space are presented graphically by numerically solving the resulting Fredholm integral equation and carrying out the numerical inversion of Laplace transform for a PZT-5H material with a conducting rigid line inclusion.     

Effects of three-phase-lag on two-temperature generalized thermoelasticity for infinite medium with spherical cavity

The thermoelastic interaction for the three-phase-lag (TPL) heat equation in an isotropic infinite elastic body with a spherical cavity is studied by two-temperature generalized thermoelasticity theory (2TT). The heat conduction equation in the theory of TPL is a hyperbolic partial differential equation with a fourth-order derivative with respect to time. The medium is assumed to be initially quiescent. By the Laplace transformation, the fundamental equations are expressed in the form of a vector-matrix differential equation, which is solved by a state-space approach. The general solution obtained is applied to a specific problem, when the boundary of the cavity is subjected to the thermal loading (the thermal shock and the ramp-type heating) and the mechanical loading. The inversion of the Laplace transform is carried out by the Fourier series expansion techniques. The numerical values of the physical quantity are computed for the copper like material. Significant dissimilarities between two models (the two-temperature Green-Naghdi theory with energy dissipation (2TGN-III) and two-temperature TPL model (2T3phase)) are shown graphically. The effects of two-temperature and ramping parameters are also studied.     

Deformation due to time harmonic sources in micropolar thermoelastic medium possessing cubic symmetry with two relaxation times

The response of a micropolar thermoelastic medium possessing cubic symmetry with two relaxation times due to time harmonic sources is investigated. Fourier transform is employed and the transform is inverted by using a numerical inversion technique. The components of displacement, stress, microrotation and temperature distribution in the physical domain are obtained numerically. The results of normal displacement, normal force stress, tangential couple stress and temperature distribution are compared for micropolar cubic crystal and micropolar isotropic solid. The numerical results are illustrated graphically for a particular material. Some special cases are also deduced.     

Dynamic fracture behavior of a cracked piezoelectric half space under anti-plane mechanical and in-plane electric impact

Summary  The dynamic response of a cracked piezoelectric half-space under anti-plane mechanical and in-plane electric impacting loads is investigated in the present paper. In the study, the crack is assumed parallel to the free surface of the half-space. Laplace and Fourier transforms are used to reduce the mixed boundary value problems to Cauchy-type singular integral equations in the Laplace transform domain, which are solved numerically. Then, a numerical Laplace inversion is performed and the dynamic stress and electric displacement factors are obtained as functions of time and geometry parameters. The dynamic energy release rate is derived for piezoelectric materials in terms of the electroelastic intensities and is displayed graphically. Received 5 January 2000; accepted for publication 28 June 2000     

Time harmonic sources at micropolar thermoelastic medium possessing cubic symmetry with one relaxation time

The response of a micropolar thermoelastic medium possessing cubic symmetry with one relaxation time due to time harmonic sources has been investigated. Fourier transform has been employed and the transform has been inverted by using a numerical inversion technique. The components of displacement, stress, microrotation and temperature distribution in the physical domain are obtained numerically. The results of normal displacement, normal force stress, tangential couple stress and temperature distribution have been compared for micropolar cubic crystal and isotropic micropolar solid. The numerical results are illustrated graphically for a particular material. Some special cases have also been deduced.     

Moving load response in micropolar thermoelastic medium without energy dissipation possessing cubic symmetry

The steady state response of a micropolar thermoelastic medium without energy dissipation possessing cubic symmetry due to a moving load has been studied. Fourier transform has been employed and the transform has been inverted by using a numerical inversion technique. The components of displacement, stress, microrotation and temperature distribution in the physical domain are obtained numerically. The results of normal displacement, normal force stress, tangential couple stress and temperature distribution have been compared for micropolar cubic crystal and micropolar isotropic solid. The numerical results are illustrated graphically for a particular material. Some special cases have also been deduced.     

Thermoelastic interaction with energy dissipation in a transversely isotropic thin circular disc

The distribution of stresses due to step input of temperature on the boundaries of a homogeneous transversely isotropic circular disc is investigated by applying Laplace transform technique in the context of generalized theories of thermo-elasticity. The inverse of the transformed solution is carried out by applying a method of Bellman et al. The stresses are computed numerically and presented graphically in a number of figures. A comparison of the results for different theories (CTE, CCTE, TRDTE(GL), TEWED(GN)) and the effect of anisotropy on the stresses are also presented. When the material is isotropic and outer radius of the disc tends to infinity, the corresponding result agrees with that of existing literature.     

Generalized thermoviscoelastic interaction due to periodically varying heat source with three-phase-lag effect

The present paper aims at studying the thermo-visco-elastic interaction in a homogeneous, infinite Kelvin-Voigt-type viscoelastic, thermally conducting medium due to the presence of periodically varying heat sources. Three-phase-lag thermoelastic model, GN model II (TEWOED) and GN model III (TEWED) are employed to study the thermomechanical coupling, thermal and mechanical relaxation effects. In the absence of mechanical relaxations (viscous effect), the results for various generalized theories of thermoelasticity may be obtained as particular cases. The governing equations are expressed in Laplace-Fourier double transform domain and are solved in that domain. The inversion of the Fourier transform is carried out by using residual calculus, where poles of the integrand are obtained numerically in complex domain by using Laguerre’s method and the inversion of Laplace transform is done numerically using a method based on Fourier series expansion technique. The numerical estimates of the displacement, temperature and stress are obtained for a hypothetical material. A comparison of the results for different theories (three-phase-lag model, GN model II, GN model III) is presented and the effect of viscosity is also shown. In absence of viscous effect the results corresponding to GN model II and GN model III agree with the results of the existing literature.     

Generalized magneto-thermoelasticity in a perfectly conducting medium

A model of the equations of generalized magneto-thermoelasticity in a perfectly conducting medium is given. The formulation is applied to generalizations, Lord–Shulman theory with one relaxation time and the Green–Lindsay theory with two relaxation times, as well as to the coupled theory.Laplace transforms and Fourier transforms techniques are used to get the solution. The resulting formulation is used to solve a specific two-dimensional problem. The inverses of Fourier transforms are obtained analytically.Laplace transforms are obtained using the complex inversion formula of the transform together with Fourier expansion techniques.Numerical results for the temperature distribution, thermal stress and displacement components are represented graphically. A comparison was made with the results predicted by the three theories.     

Analysis of vibrations in a nonhomogeneous thermoelastic thin annular disk under dynamic pressure

The forced vibration analysis of nonhomogeneous thermoelastic, isotropic, thin annular disk under periodic and exponential types of axisymmetric dynamic pressures applied on its inner boundary has been performed and analytical benchmark solution has been obtained. The material has been assumed to have inhomogeneity according to a simple power law in the radial coordinate. The present analysis has been worked out in the context of generalized theory of thermoelasticity with one relaxation time. The two coupled partial differential equations have been clubbed and solved by employing Laplace transform technique to obtain the solution for radial displacement and temperature change in the space domain. In order to obtain the solution in physical domain, the inversion of the transform has been carried out by using residue calculus. The radial displacement, radial stress, circumferential stress, and temperature change have been computed numerically for copper material annular disk. The numerically computed results have been presented graphically to demonstrate the effect of two different types of dynamic pressure in reference to homogeneous and nonhomogeneous material disk. The results for homogeneous material disk have been deduced and validated with that available in literature. The closed-form solution obtained here is interesting and allows further parametric studies of nonhomogeneous structures.