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.     

EFFECTS OF THERMAL RELAXATIONS ON THERMOVISCOELASTIC INTERACTIONS IN AN UNBOUNDED BODY WITH A SPHERICAL CAVITY SUBJECTED TO A PERIODIC LOADING ON THE BOUNDARY

Thermoviscoelastic interactions in an infinite homogeneous viscoelastic medium with a spherical cavity are studied. The cavity surface is subjected to a periodic loading and zero temperature change. The classical dynamical theory of thermoelasticity as well as the generalized theories of thermoelasticity are applied to consider the thermoelastic coupling. The analytical expressions for the closed-form solutions of displacement, temperature, and stresses are obtained; and the thermal relaxation effects on the interactions are studied to compare the three theories. The numerical values of the physical quantities are computed for a suitable material. The results are presented graphically to illustrate the problem.     

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.     

Study of the effective parameters on stress distribution around triangular hole in metallic plates subjected to uniform heat flux

On the basis of the steady-state two-dimensional theory of thermoelasticity, stress field around a triangular hole in an infinite isotropic plate is discussed. A metallic plate subjected to uniform heat flux and thermal-insulated condition along the hole boundary is assumed. The method used for this study is the expansion of Goodier and Florence's method. They used the complex variable method for stress analysis of infinite isotropic plates with an elliptical or circular hole. The rotation angle of the hole, bluntness, aspect ratio of hole size, and angle of heat flux are important parameters considered in this paper.     

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.     

Eigenfunction expansion method to analyze thermal shock behavior in magneto-thermoelastic orthotropic medium under three theories

This article is concerned with a study of thermal shock response in infinite thermoelastic medium under the purview of Lord–Shulman model, Green–Naghdi theory III, and three-phase-lag model of generalized thermoelasticity. The medium under consideration is assumed to be homogeneous, orthotropic, and thermally conducting. The fundamental equations of the two-dimensional problem of generalized thermoelasticity with three-phase-lag model in an infinite elastic medium under the influence of magnetic field are obtained as a vector–matrix differential equation form using normal mode analysis which is then solved by the Eigenfunction expansion method. Numerical results for the temperature, displacements and thermal stress distribution are presented graphically.     

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.     

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.     

PROPAGATION OF GENERALIZED VISCOTHERMOELASTIC RAYLEIGH–LAMB WAVES IN HOMOGENEOUS ISOTROPIC PLATES

The present paper is aimed at studying the thermoelastic interaction in an infinite Kelvin–Voigt-type viscoelastic, thermally conducting plate. The upper and lower surfaces of the plate are subjected to stress-free, thermally insulated or isothermal conditions. The coupled dynamic thermoelasticity and generalized theories of thermoelasticity, namely, Lord and Shulman's, Green and Lindsay's, and Green and Nagdhi's are employed to understand the thermomechanical coupling and thermal and mechanical relaxation effects. Secular equations for the plate in closed from and isolated mathematical conditions for symmetric and skew-symmetric wave mode propagation in completely separate terms are derived. In the absence of mechanical relaxations (viscous effect), the results for generalized and coupled theories of thermoelasticity have been obtained as particular cases from the derived secular equations. In the absence of thermomechanical coupling, the analysis for a viscoelastic plate can be deduced from the present one. The various forms and regions of Rayleigh–Lamb-type secular equation have been obtained and discussed in addition to Lame modes, decoupled shear horizontal (SH) modes, and thin-plate results. At short-wavelength limits, the secular equations for symmetric and skew-symmetric waves in a stress-free insulated and stress-free isothermal plate reduce to the Rayleigh surface wave frequency equation. The amplitudes of temperature and displacement components during symmetric and skew-symmetric motion of the plate have been computed and discussed. Finally, the numerical solution is carried out for copper material. The dispersion curves, and amplitudes of temperature change and displacements for symmetric and skew-symmetric wave modes are presented to illustrate and compare the theoretical results.     

Generalized Fractional Thermoelasticity Associated with Two Relaxation Times

In this work, a new theory of thermoelasticity associated with two relaxation times is derived using the methodology of fractional calculus. The theories of coupled thermoelasticity and of generalized thermoelasticity (Green–Lindsay Model) with two relaxation times follow as limiting cases. A uniqueness theorem and a reciprocity theorem for this model are derived. A variational principle theorem is obtained.     

Thermoelastic diffusion with memory-dependent derivative

The constitutive equations are derived for the thermoelastic diffusion in anisotropic and isotropic solids, in the context of a new generalized thermoelasticity theory with two time delays and kernel functions. The coupled thermoelastic diffusion and the Lord–Shulman theories result from the given theory as particular cases. For anisotropic solid, the reciprocity theorem is proved; the convolutional variational principle is given and the uniqueness theorem based on the variational principle is proved.     

A Unified Generalized Thermoelasticity Formulation; Application to Thick Functionally Graded Cylinders

In this article a new unified formulation for the generalized coupled thermoelasticity theories is presented. The generalized coupled thermoelasticity theories proposed by Lord–Shulman, Green–Lindsay, and Green–Naghdi are combined into a unified formulation introducing the unified parameters. The formulation is given for the general anisotropic heterogeneous linear thermoelastic materials and then is reduced to the system of equations for the isotropic heterogeneous materials. As an example, a functionally graded cylinder is considered and the thermoelastic waves based on the new unified formulation, using the generalized theories, are obtained and discussed. Utilizing the transfinite element method, the equations for a long thick circular cylinder are solved in the Laplace domain and the results are inverted to the real time domain using a numerical inversion technique of the Laplace transform. The results for the propagation of thermoelastic waves based on the Lord–Shulman, Green–Lindsay, and Green–Naghdi models are derived and compared.     

Propagation of waves in an initially stressed generalized electromicrostretch thermoelastic medium with temperature-dependent properties under the effect of rotation

In the present article, a model of the equations of generalized electromicrostretch thermoelasticity in an initially stressed perfectly conducting elastic medium under the effect of temperature-dependent properties is studied. The entire elastic medium is rotating with a uniform angular velocity. Reflection phenomena of plane waves in electromicrostretch thermoelasticity is investigated under two theories proposed by Lord and Shulman (L–S) and Green and Lindsay (G–L). Amplitude ratios and energy ratios of various reflected waves are presented when an elastic wave is made incident obliquely at the plane boundary of an electromicrostretch thermoelastic solid half-space. It has been verified that there is no dissipation of energy at the boundary surface during reflection. Numerical examples calculate the amplitude ratios to evince the effects of initial stress parameter, rotation, and temperature-dependent properties, and the results obtained are depicted graphically.     

Propagation of Rayleigh Surface Waves in Microstretch Thermoelastic Continua Under Inviscid Fluid Loadings

The present article is aimed at an investigation of the propagation of generalized Rayleigh surface waves in a homogeneous, isotropic, microstretch thermoelastic solid half-space underlying an inviscid liquid half-space or layer of finite thickness, in the context of classical (coupled) and non-classical (generalized) theories of thermoelasticity. The secular equations in close form and isolated mathematical conditions are derived for generalized Rayleigh waves in the considered composite structure after obtaining general wave solutions of the model. The fluid overlying the solid half-space has been successfully modeled as thermal load in addition to normal (hydrostatic pressure) one. Some special cases of dispersion equations have also been deduced and discussed. The analytic expressions for the amplitudes of displacement, microstretch, microrotation and temperature change at the interfacial surface during the Rayleigh wave propagation are also derived. The results have been deduced and compared with the relevant publications available in the literature at the appropriate stages of this work. Finally, the analytical developments have been illustrated numerically for aluminum–epoxy-like material half-space under the action of inviscid liquid (water) half-space or layer of finite thickness. The computer simulated results in respect of phase velocity, attenuation coefficient, specific loss factor of energy dissipation and relative frequency shift due to fluid loadings are presented graphically in normalized form to observe their distinctions from those in the context of the well established theory of coupled thermoelasticity.     

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.     

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.     

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.     

Two temperature fractional order thermoelasticity theory in a spherical domain

This article is an application of fractional thermoelasticity in association with two-temperature theory. The fractional heat conduction model has been proposed to investigate the thermal variations within the bounded spherical region. The corresponding heat conduction equation has been derived in the context of the generalized two-temperature theory of fractional thermoelasticity. The analytical solutions of thermal variations have been obtained in the Laplace domain, which are inverted using the Gaver–Stehfest algorithm in the time domain. Kuznetsov convergence criterion has been discussed for the bounded variations and stability of the problem. The delay time translations used in the heat flux vector and the temperature gradient result in the finite speed of thermal wave propagation. As a special case of time fractional derivative, the classical and generalized thermoelasticity theories have been recovered.     

Unified finite element approach for generalized coupled thermoelastic analysis of 3D beam-type structures,part 1: Equations and formulation

An innovative 1D finite element (FE) approach is developed to analyze the 3D static, transient, and dynamic problems in the coupled and uncoupled thermoelasticity for the nonhomogeneous anisotropic materials. The Galerkin method is directly applied to the governing equations to obtain a weak formulation of the thermoelasticity problems with arbitrary loads and boundary conditions. To surmount the restrictions of the classical beam theories, a 1D FE procedure is proposed in the context of the Carrera Unified Formulation (CUF). Since coupled thermoelastic analyses are computationally demanding, the proposed 1D FE approach can be used as a powerful means to simulate the generalized coupled thermoelastic behavior of structures. This methodology, indeed, reduces the 3D problems to 1D models with 3D-like accuracies and very low computational costs. The Lord-Shulman and the Green-Lindsay models are considered as the generalized theories of thermoelasticity. Furthermore, as simplified cases, the classical coupled, dynamic uncoupled, quasi-static uncoupled and steady-state uncoupled theories of thermoelasticity may be derived from the formulation. Moreover, effects of the structural damping can be taken into account in the present formulation. The accuracy of the formulation has been evaluated through numerical simulations and comparisons, which have been presented in a companion article (Part 2).