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.     

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.     

3D thermoelastic analysis of rotating disks having arbitrary profile based on a variable kinematic 1D finite element method

A variable kinematic 1D finite element (FE) method is presented for 3D thermoelastic analysis of rotating disks with variable thickness. The principle of minimum potential energy is used to derive general governing equations of the disks subjected to body forces, surface forces, concentrated forces, and thermal loads. To solve the equations, the 1D Carrera unified formulation (CUF), which enables to go beyond the kinematic assumptions of classical beam theories, is employed. Based on the 1D CUF, the disk is considered as a beam, which can be discretized into a finite number of 1D elements along its axis. The displacement field over the beam’s cross section is approximated by Lagrange expansions. This methodology leads to an FE formulation that is invariant with respect to the order of expansions used over the cross sections, and thus the 3D problem reduces to a 1D problem. The effect of the cross section discretization on displacement and stress fields is investigated. Results obtained from this method are in good agreement with the reference analytical and finite difference solutions. The proposed innovative method can be very effective in the thermoelastic analysis of rotating disks.     

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.     

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.     

Eigenvalue Approach to Linear Micropolar Thermoelasticity Under Distributed Loading

The eigenvalue approach is developed for the two-dimensional problem in a micropolar thermoelastic medium for a half-space subjected to distributed loading and zero temperature change. The formulation is applied to the coupled theory as well as to two generalizations, the Lord–Shulman and the Green–Lindsay theories. The Fourier transforms are inverted analytically. The inversion of the Laplace transforms are carried out using the inversion formula of the transform together with Fourier expansion techniques. Numerical methods are used to accelerate the convergence of the resulting series and to evaluate the improper integrals involved to obtain temperature, displacement, force and couple stress in the physical domain. The results of micropolar elasticity and generalized thermoelasticity are deduced as special cases from the present formulation. Numerical results are represented graphically and discussed.     

NUMERICAL SOLUTIONS OF THERMOELASTIC WAVE PROBLEMS BY THE METHOD OF CHARACTERISTICS

This article is concerned with the numerical treatment of thermal and thermal stress waves in thermoelastic solids. To keep the numerical treatment general, the development of the formulation is based on the generalized theory of thermoelasticity. A number of thermoelastic wave problems, which involve one or two space variables, are treated, in a uniform manner, by a system of first-order partial differential equations with stress, velocity, heat flow, and temperature as dependent variables. This system of equations is analyzed by the method of characteristics, yielding the characteristics and the characteristic equations. Procedures of numerical integration along the characteristics are established and carried out for several generalized and classical thermoelastic wave problems in homogeneous materials, composite materials, nonhomogeneous materials, and nonlinear elastic solids.     

On the plane waves of generalized thermo-microstretch elastic half-space under three theories

A general model of the equations of generalized thermo-microstretch for a homogeneous isotropic elastic half space is given. The formulation is applied to generalized thermoelasticity theories, the Lord-?hulman and Green-Lindsay theories, as well as the classical dynamical coupled theory. The normal mode analysis is used to obtain the exact expressions for the displacement components, force stresses, temperature, couple stresses and microstress distribution. The variations of the considered variables through the horizontal distance are illustrated graphically. Comparisons are made with the results in the presence and absence of microstretch constants and between the three theories for two different times.     

Analysis of thermoelastic response in functionally graded hollow sphere without load

In this article, a model concerning free vibrations of spherically symmetric, thermoelastic, isotropic, and functionally graded sphere has been developed and analyzed in the context of linear theory of generalized thermoelasticity with one relaxation time. Laplace transform has been used to solve the problem which yields natural frequencies of free vibrations without performing inversion of the transform. The analytical results for coupled, uncoupled, and homogeneous spheres have been deduced as special cases of the general case. Natural frequencies of first 10 modes of vibrations have been obtained for different values of grading index of cobalt material regarding coupled thermoelastic, generalized thermoelastic, and elastic functionally graded spheres. The frequency shift and thermoelastic damping for Fourier and non-Fourier processes of heat propagation, temperature change, radial, and hoop stresses have been presented graphically. It has been analyzed here that grading index parameter helps in detecting the strength of signals in such material devices and the thermal relaxation time contributes in improving the quality of signals. The analysis also leads to the fact that grading index parameter is useful from design point of view and it can be tailored to specific applications for controlling the stress.     

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.     

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.     

Generalized Thermoelastic Models for Linear Elastic Materials with Micro-Structure Part I: Enhanced Green–Lindsay Model

In this article, we derive constitutive thermoelastic models for linear elastic materials with micro-structure. The elastic behavior is assumed to be consistent with Mindlins’ Form II gradient elasticity theory, whereas for the thermal behavior the generalization of Clausius-Duhem inequality, proposed by Green and Laws, is adopted. The resulting model is actually a generalization of the thermoelastic theory of Green and Lindsay for linear elastic materials with micro-structure, taking into account micro-inertia effects, as well. It is demonstrated that classical thermoelasticity models are retrieved from the present general formulation, when some of the model constants are set to zero. Finally, the uniqueness of solution for the general case of anisotropic materials is proved.     

Wave propagation in a generalized thermoelastic plate using eigenvalue approach

In the present work, we obtain a dispersion relation for Rayleigh–Lamb wave propagation in a plate of thermoelastic material. For this aim, we consider the theory of generalized thermoelasticity with one relaxation time. The thickness of the plate is taken to be finite and the faces of the plate are assumed to be isothermal and free from stresses. We obtain the analytical solution for the temperature, displacement components, and stresses using an eigenvalue approach. Finally, we derive a dispersion relation for the plate in closed form taking into account isothermal boundary conditions for wave mode propagation. To obtain the phase velocity and attenuation coefficients of propagating wave mode, we use the function iteration numerical scheme to solve the complex dispersion relation. The phase velocity and attenuation coefficients for the first five modes of waves are represented graphically for Lord–Shulman and classical coupled dynamical theories.     

Wave scattering through an orthotropic thermoelastic slab sandwiched between two isotropic elastic half-spaces

Abstract

Present study deals with the scattering of a plane wave through an orthotropic thermoelastic slab sandwiched between two elastic half-spaces. Unlike the classical theory of thermoelasticity, we have employed non-classical thermoelastic theories (LS-theory and GL-theory) to analyze the scattering of plane waves. The amplitude ratios for different waves have been computed numerically for the considered generalized theories of thermoelasticity. The effect of the slab thickness on the amplitude ratios has been shown graphically. Moreover, the amplitude ratios of different waves (i.e., reflected, transmitted, forward and backward waves) are compared for different values of slab thickness under both the LS-theory and GL-theory.     

THE UNIQUENESS AND RECIPROCITY THEOREMS FOR GENERALIZED THERMOVISCOELASTICITY FOR ANISOTROPIC MEDIA

A new model of the equations of generalized thermoviscoelasticity for anisotropic media is given. The formulation is applied to both generalizations, the Lord-Shulman theory with one relaxation time and the Green-Lindsay theory with two relaxation times, as well as to the coupled theory. Using Laplace-Carson transforms, a uniqueness theorem for this model is proved, the dynamic reciprocity theorem is derived, and some applications are given. The cases of isotropic thermoviscoelasticity (with or without the volume rheological properties), anisotropic thermoelasticity, and isotropic thermoelasticity can be obtained from the given general model for the coupled and generalized theories.     

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.     

Thermoelastic transient response of multilayered conical shells subjected to vapor field

The problems of the three-dimensional axisymmetric quasi-static coupled thermoelastic for the laminated circular conical shells subjected to vapor field. The water vapor temperature and pressure relation assumed for the inner boundary. The water vapor temperature and pressure data were obtained from a thermodynamic steam table. The formulation begins with the basic equations of thermoelasticity in curvilinear circular conical coordinates. Laplace transform and finite difference methods are used to analyze problems. The solution is obtained by using the matrix similarity transformation and inverse Laplace transform. We obtain solutions for the temperature and thermal deformation distributions in a transient and steady state. Moreover, the computational procedures established in this thesis, can solve the generalized thermoelasticity problem for multilayered conical shells with nonhomogeneous materials.     

An investigation on thermoelastic interactions under an exact heat conduction model with a delay term

The present work is concerned with a very recently proposed heat conduction model: an exact heat conduction model with a single delay term. A generalized thermoelasticity theory was proposed by Roy Choudhuri based on the heat conduction law with three-phase-lag effects for the purpose of considering the delayed response in time due to the microstructural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in Hadamard sense. Quintanilla has recently proposed to reformulate this heat conduction model as an alternative heat conduction theory with a single delay term and subsequently, Leseduarte and Quintanilla investigated the spatial behavior of the solutions for this theory and they extended the results to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction with one delay term. In the present work, we consider the thermoelasticity theory based on this newly proposed heat conduction model and investigate a problem of thermoelastic interactions. State-space approach is used to formulate the problem and the formulation is then applied to a problem of an isotropic elastic half-space with its plane boundary subjected to sudden increase in temperature and zero stress. The integral transform method is applied to obtain the solution of the problem. A detailed analysis of analytical results is provided by finding the short-time approximated solutions of different field variables analytically and comparing the results of the present model with the corresponding results reported for other existing theories. An attempt has also been made to illustrate the problem and numerical values of field variables are obtained for a particular material. Results are analyzed with different graphs. To the best of the author\textquoteright s knowledge, this thermoelastic model is not yet investigated by any researcher in this direction.     

Thermomechanical Interactions in a Generalized Thermo-Microstretch Elastic Half Space

Interactions caused by thermal and mechanical sources in a generalized thermo-microstretch elastic medium are studied by the use of Laplace-Fourier transform techniques. The formulation is applied to the coupled theory as well as to two generalizations, the Lord-Shulman and the Green-Lindsay theories. The integral transforms are inverted using a numerical technique to obtain the solutions field in the physical domain. Stretch effects lead to the existence of a new wave that is called longitudinal microstretch wave. The results of micropolar generalized thermoelasticity and generalized thermoelasticity are deduced as special cases from the present formulation.