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.     

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.     

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.     

THERMOELASTIC INTERACTIONS IN A TRANSVERSELY ISOTROPIC ELASTIC MEDIUM WITH A CYLINDRICAL HOLE SUBJECTED TO RAMP-TYPE INCREASE IN BOUNDARY TEMPERATURE OR LOAD

This paper deals with the study of thermoelastic interactions on the basis of the generalized theory of thermoelasticity and with a comparison of the generalized theory of thermoelasticity with the coupled dynamical theory of thermoelasticity for the case of a transversely isotropic elastic medium with a cylindrical cavity when the surface of the cavity is subjected to a ramp-type increase in temperature or load. The solution for the distributions of temperature and stresses are obtained with the help of the Laplace transform technique. Inversions of Laplace transforms are carried out analytically. Discontinuities of the field variables are analyzed. Numerical results for a particular material are also determined in order to extend the problem.     

Dynamic analysis of cracks under thermal shock considering thermoelasticity without energy dissipation

This article reports a study of a cracked finite isotropic medium under nonclassic thermal shock based on thermoelasticity without energy dissipation. The time history of stress intensity factors as well as the temperature distribution around the crack tip is analyzed thoroughly. The fully coupled governing equations are discretized in the space by employing the extended finite-element method. The Newmark method is used as the time integration scheme to solve discretized equations. The stress intensity factors, which are extracted using the interaction integral method, are compared with other theories of thermoelasticity. The results of a cracked plate under temperature shock demonstrate that the stress intensity factors based on thermoelasticity without energy dissipation are significantly greater than those based on classic and Lord–Shulman models, whereas the peaks of stress intensity factors under heat flux shock are nearly equal for various theories of thermoelasticity. Furthermore, a mobile cold region is created along slanted crack in the temperature distribution, in which the temperature is less than the applied thermal boundary condition.     

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.     

TIME-HARMONIC SOURCES IN A GENERALIZED THERMOELASTIC CONTINUUM

The disturbance due to a time-harmonic normal point load and thermal source in a homogeneous isotropic thermoelastic, half-space is investigated by applying the Hankel transform technique in the context of generalized theories of thermoelasticity. The inverse transform integrals are evaluated using Romberg integration with adaptive stepwise after using the results from successive refinements of the extended trapezoidal rule followed by extrapolation of the results to the limit when the step-size tends to zero. The displacement, temperature, and stresses so obtained in the physical domain are computed numerically and presented graphically in Figures 1-12 in different situations for Aluminium-epoxy composite material. A comparison of the results for different theories of generalized thermoelasticity is also presented.     

Generalized Thermoelastic Infinite Layer Subjected to Ramp-Type Thermal and Mechanical Loading under Three Theories—State Space Approach

A problem of thermoelastic interactions in an elastic infinite layer 0 ≤ x ≤ h with an elevated temperature field arising from ramp-type heating and loading has been constructed. The governing equations are written in a unified system from which the field equations for coupled thermoelasticity as well as for generalized thermoelasticity can be easily obtained as particular cases. Due attention has been paid to the finite rise times of temperature and stress. The problem has been solved analytically by using a state-space approach. Solutions from the derived analytical expressions have been computed for a specific situation. The solution for the half-space when (h → ∞) has been found also. Numerical results for the temperature distribution and thermal stress are represented graphically. A comparison was made among the results predicted by the theories.     

A SHORT TIME SOLUTION FOR A PROBLEM IN THERMOELASTICITY OF AN INFINITE MEDIUM WITH A SPHERICAL CAVITY

The problem of a thermoelastic infinite medium with a spherical cavity is considered within the context of the theory of thermoelasticity with two relaxation times. The surface of the cavity is stress free and suddenly subjected to a time-dependent thermal shock. Laplace transform techniques are used. The inverse Laplace transforms are obtained analytically using asymptotic expansions valid for small values of time. Numerical computations for the temperature, the displacement, and stress distribution are carried out and represented graphically.     

MECHANICAL AND THERMAL SOURCES IN A GENERALIZED THERMOELASTIC HALF-SPACE

The disturbance due to mechanical point loads and thermal sources acting on the boundary of a homogeneous isotropic thermoelastic half-space has been investigated upon applying the Laplace and Hankel transforms in the context of generalized theories of thermoelasticity. The integral transforms have been inverted using a numerical technique to obtain the displacements, temperature, and stresses in the physical domain. The numerical technique expresses the integrand as a Fourier series representation with respect to the Laplace transform parameter and evaluates the inverse Hankel transform integral via Romberg integration with an adaptive stepsize after using the results from successive refinements of the extended trapezoidal rule followed by extrapolating the results to the limit when the stepsize tends to zero. The results for various physical quantities are computed and presented graphically. A comparison of the results for different generalized theories of thermoelasticity are also presented.     

PROPAGATION OF MICROPOLAR THERMOELASTIC WAVES IN AN INFINITE ELASTIC SPACE CONTAINING A CYLINDRICAL BORE

The propagation of waves in an infinite micropolar elastic solid containing a cylindrical cavity and under the influence of temperature is investigated. Waves with axial symmetry with respect to the axis of the cavity are discussed, using the linear theory of micropolar thermoelasticity.     

Reflection of magneto-thermoelasticity waves with temperature dependent properties in generalized thermoelasticity

The model of the equations of generalized magneto-thermoelasticity in an isotropic perfectly conducting elastic medium under the effect of temperature dependent properties is established. The modulus of elasticity is taken as a linear function on reference temperature. Reflection of plane harmonic waves in magneto generalized thermoelasticity theories is investigated. The formulation is applied under four theories of the generalized thermoelasticity: Lord-Shulman (LS) with one relaxation time, Green-Naghdi theory GN-II, without energy dissipation and Chandrasekharaiah-Tzou (CT) theory with dual-phase-lag, as well as the coupled theory (CD). The expressions for the reflection coefficients, which are the relations of the amplitudes of the reflected waves to the amplitude of the incident waves are obtained. Effects of dependence of modulus of elasticity on the amplitude ratios have been depicted graphically for a (LS), (GN-II) and (CT) theories.     

An investigation on strain and temperature rate-dependent thermoelasticity and its infinite speed behavior

Abstract

The present work is aimed at a mathematical analysis of the newly proposed strain and temperature rate-dependent thermoelasticity theory, also called a modified Green–Lindsay model (MGL) theory, given by Yu et al. (2018). This model is also an attempt to remove the discontinuity in the displacement field observed under temperature rate-dependent thermoelasticity theory proposed by Green and Lindsay. We study thermoelastic interactions in an infinite homogeneous, isotropic elastic medium with a cylindrical cavity based on this model when the surface of the cavity is subjected to thermal shock. The solutions for the distribution of displacement, temperature, and stress components are obtained by using the Laplace transform technique. The inversion of the Laplace transform is carried out by short-time approximation. A detailed comparison of the analytical results predicted by the MGL model with the corresponding predictions by the Lord–Shulman model and the Green–Lindsay model is performed. It is observed that strain rate terms in the constitutive equation avoid the prediction of discontinuity in the displacement field and other significant effects are noted. However, the new theory predicts the infinite speed of disturbance like the classical theory. Variations of field variables at different time are graphically displayed for different models and compared by using a numerical method.     

Fractional Order Generalized Thermo-Piezoelectric Semi-Infinite Medium with Temperature-Dependent Properties Subjected to a Ramp-Type Heating

The generalized thermoelasticity theory that, based on a fractional order model, is used to solve a one-dimensional boundary value problem of a semi-infinite piezoelectric medium. The resulting formulation is applied to a half-space subjected to ramp-type heating and traction free. The generalized thermo-piezoelectricity model in an isotropic elastic medium with temperature-dependent mechanical properties is established. The modulus of elasticity is taken as a linear function of the reference temperature. The Laplace transform technique is used to obtain the general solution for any set of boundary conditions. The inverse Laplace transforms are numerically computed using the Fourier expansion techniques. The effects of fractional order and the ramping of heating parameters are studied and comparison with different theories of thermoelasticity are considered. The results are also compared to results obtained in the case of a temperature-independent modulus of elasticity.     

REFLECTION OF GENERALIZED THERMOELASTIC WAVES FROM THE BOUNDARY OF A HALF-SPACE

We investigated the problem of thermoelastic wave reflection from the insulated and isothermal stress-free as well as rigidly fixed boundaries of homogeneous isotropic solid half-spaces in the context of various linear theories of thermoelasticity, namely, Lord-Shulman, Green-Lindsay, Green-Nagdhi, coupled thermoelasticity, and uncoupled thermoelasticity. The ratios of reflection coefficients to that of incident coefficients are obtained for P- and SV-wave incidence cases. The results for partition of the energy for various values of the angle of incidence are computed numerically and presented graphically for aluminum-epoxy composite material in case of incident P- and SV-waves from the stress-free and rigidly fixed thermally insulated boundaries. The results obtained are discussed and compared in various models of thermoelasticity.     

Thermoelastic interactions without energy dissipation in an unbounded body with a cylindrical cavity

In this work, we apply Green and Naghdi's generalized thermoelasticity theory to a one-dimensional problem of an infinitely long cylindrical cavity. Laplace transform techniques are used to solve the problem. A comparison of generalized thermoelasticity theory with one relaxation time is made. Numerical results are computed and represented graphically for the temperature,displacement, and stress distributions.     

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.     

A PROBLEM OF GENERALIZED MAGNETOTHERMOELASTICITY FOR AN INFINITELY LONG,PERFECTLY CONDUCTING CYLINDER

This article concerns the investigation of the stress, temperature, and magnetic field in a transversely isotropic, elastic cylinder of infinite length and perfectly conducting material placed in a primary constant magnetic field when the curved surface of the cylinder is subjected to periodic loading. The analysis encompasses Lord and Shulman and Green and Lindsay theories of generalized thermoelasticity to account for the finite velocity of heat equation. The analysis of the numerical results for stress, temperature, and numerical values of the perturbed magnetic field in the free space is carried out at various points of the cylindrical medium. It is found that the effect of the applied magnetic field is an increase in the elastic wave velocity or, in other words, the increase of the solidity of the body. Furthermore, it has been shown graphically that the stress and perturbed magnetic field are modified due to the thermal relaxation time effect. In the absence of the magnetic field or relaxation times, our results reduce to those of generalized thermoelasticity and classical dynamical thermoelasticity, respectively.     

ON THE AXISYMMETRIC PROBLEMS OF GENERALIZED ANISOTROPIC THERMOELASTICITY

In this article a two-dimensional axisymmetric problem in a homogeneous transversely isotropic medium has been studied by employing the eigenvalue approach after applying the technique of Laplace and Hankel transforms. An example of infinite space with concentrated force at the origin has been presented to illustrate the application of the approach. The results for coupled thermoelasticity and in the case of a homogeneous isotropic medium have also been deduced. The results obtained can be used for a broad class of problems in generalized thermoelasticity. The integral transforms have been inverted by using a, numerical technique to obtain the displacements, temperature, and stresses in the physical domain. The results for these quantities are given and illustrated graphically.     

THERMAL RELAXATION TIMES EFFECT ON RAYLEIGH WAVES IN GENERALIZED THERMOELASTIC MEDIA

Rayleigh waves in a half-space exhibiting generalized thermoelastic properties based on Green-Lindsay (G-L), Lord-Shulman (L-S), and classical dynamical coupled (C-D) theories are discussed. The phase velocity of Rayleigh waves in the previous three different theories has been obtained. A comparison is carried out between the phase velocities of Rayleigh waves, displacements, stresses, and temperature as calculated from the different theories of generalized thermoelasticity. The C-D theory is recovered as a special case. It appears, in particular, that the results obtained from G-L theory tend to those of L-S theory as the values of the two relaxation times become closer to each other. The second relaxation time is well pronounced when it becomes larger than the first one. Furthermore, it is found that the thermal relaxation times decrease the speed of the elastic waves and modify the phase velocities of the Rayleigh waves. The results obtained and the conclusions drawn are discussed numerically and illustrated graphically. Relevant results of previous investigations are deduced as special cases.