Generalized thermo-electro-elasticity of a piezoelectric disk using Lord-Shulman theory

Abstract

Current investigation deals with the generalized thermoelastic response of a finite hollow disk made of a piezoelectric material. The constitutive equations of the piezoelectric media are reduced to a two dimensional plane-stress state. To capture the finite speed of temperature wave, the single relaxation time theory of Lord and Shulman is used. Three coupled differential equations in terms of radial displacement, electric potential, and temperature change are obtained. These equations are written in a dimensionless presentation. With the aid of the differential quadrature method (DQM) a time-dependent algebraic system of equations is extracted. The Newmark time marching scheme is applied to trace the temporal evolution of temperature change, electric potential, radial displacement, stresses, and electric displacement. Numerical results demonstrate that radial displacement and temperature waves propagate with finite speed while the electric potential propagates with infinite speed.     

A GDQ approach to thermally nonlinear generalized thermoelasticity of disks

Generalized thermoelasticity response of an annular disk subjected to thermal shock on its inner surface is analyzed in this research. The Lord–Shulman theory, which accounts for one relaxation time in the conventional Fourier law, is used to avoid the infinite speed of thermal wave propagation. Unlike the other available works in which the first law of thermodynamics is linearized, the nonlinearity arising from the temperature change is taken into consideration. The first law of thermodynamics in this case becomes nonlinear and the analysis under such formulation is called thermally nonlinear. Two coupled equations, i.e., the radial displacement wave equation and temperature wave propagation equation are obtained. These equations and the associated boundary conditions are discreted through the generalized differential quadrature method. Solution of the time-dependent system of equations is obtained using the Newmark time marching scheme and the successive Picard method. Numerical results are provided for both thermally linear and thermally nonlinear temperature and radial displacement wave propagations. Parametric studies reveal that at higher temperature levels, thermally nonlinear first law of thermodynamics should be considered instead of thermally linear one. Furthermore, the higher the coupling parameter and/or relaxation time, the higher the divergence of thermally nonlinear-/linear-based results.     

GENERALIZED COUPLED THERMOELASTICITY OF DISKS BASED ON THE LORD–SHULMAN MODEL

A one-dimensional generalized thermoelasticity model of a disk based on the Lord–Shulman theory is presented. The dynamic thermoelastic response of the disk under axisymmetric thermal shock loading is studied. The effects of the relaxation time and coupling coefficient are studied. The Laplace transform method is used to transform the coupled governing equations into the space domain, where the Galerkin finite element method is employed to solve the resulting equations in the transformed domain. The dimensionless temperature, displacement, and stresses in the transformed domain are inverted to obtain the actual physical quantities using the numerical inversion of the Laplace transform method.     

Analytical solution of generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads

In this article, a fully analytical solution of the generalized coupled thermoelasticity problem in a rotating disk subjected to thermal and mechanical shock loads, based on Lord–Shulman model, is presented. The general forms of axisymmetric thermal and mechanical boundary conditions as arbitrary time-dependent heat transfer and traction, respectively, are considered at the inner and outer radii of the disk. The governing equations are solved analytically using the principle of superposition and the Fourier–Bessel transform. The general closed form solutions are presented for temperature and displacement fields. To validate the solutions, the results of this study are compared with the numerical results available in the literature, which show good agreement. For the temperature, displacement and stresses, radial distributions, and time histories are plotted and discussed. The propagation of thermoelastic waves and their reflection from the boundary of the disk are clearly shown. Moreover, effects of relaxation time and angular velocity on temperature, displacement, and stress fields are investigated.     

Propagation of the photothermal waves in a semiconducting medium under L-S theory

The model of the equations of generalized thermoelasticity based on the Lord–Shulman theory with one relaxation time is used to study the photothermal waves in a semiconducting medium. The exact expressions for the displacement components, temperature, carrier density, and stress components are obtained using normal mode analysis. Numerically simulated results are obtained and presented graphically for silicon to depict the effect of time parameter on the different physical quantities.     

Second Sound in a Cracked Layer Based on Lord–Shulman Theory

In this study, a cracked layer is considered under thermal shock and is analyzed once by the Lord–Shulman theory and once by classical theory. In this way the effect of second sound of Lord–Shulman theory on a cracked layer is investigated. The Galerkin method is invoked to obtain finite element modeling of the cracked layer. The eight node rectangular element is used and the nodes near the crack tip are replaced to introduce the crack tip singularity (Barsoum element). The discretized form of Navier and energy equations are solved simultaneously in time domain by Newmark integration algorithm. The J integral formulation in dynamical thermal form is implemented to obtain stress intensity factors from the finite element solution of the problem.     

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.     

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.     

Generalized Theory of Thermoelastic Diffusion for Anisotropic Media

The equations of generalized thermoelastic diffusion, based on the theory of Lord and Shulman with one relaxation time, are given in anisotropic media. A variational principle for the governing equations is obtained. Then we show that the variational principle can be used to obtain a uniqueness theorem under suitable conditions. A reciprocity theorem for these equations is given. The obtained results are valid for some special cases that can be deduced from our generalized model.     

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.     

THERMOELASTICITY AND HEAT CONDUCTION WITH MEMORY EFFECTS

The thermodynamical foundations of generalized thermoelasticity formulated by Lord and Shulman are reviewed and memory effects on the heat flux are generalized. An existence and uniqueness theorem for the evolution integro-differential equations is obtained using semi-group theory.     

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.     

On A Thermoelastic Three-Phase-Lag Model

A three-phase-lag model of the linearized theory of coupled thermoelasticity is formulated by considering the heat condition law that includes temperature gradient and the thermal displacement gradient among the constitutive variables. The Fourier law is replaced by an approximation to a modification of the Fourier law with three different translations for the heat flux vector, the temperature gradient and also for the thermal displacement gradient. The model formulated is an extension of the thermoelastic models proposed by Lord–Shulman, Green–Naghdi and Tzou.     

Reflection of plane wave in a micropolar thermoelastic solid half-space with diffusion

In this article, the governing equations of micropolar thermoelasticity with diffusion are formulated in the context of Lord–Shulman theory of generalized thermoelasticity. The plane wave solutions of these equations indicate the existence of six plane waves, namely, coupled longitudinal displacement (CLD) wave, coupled thermal wave, coupled mass diffusion wave, coupled transverse microrotational wave, coupled transverse displacement wave, and longidudinal microrotational wave. Reflection of CLD wave from a stress-free thermally insulated/isothermal surface is considered. The appropriate potentials of incident and reflected waves satisfy the required boundary conditions at a stress-free thermally insulated/isothermal surface to obtain the reflection coe?cients of various reflected waves for an incident CLD wave and to obtain an extension of Snell’s law. The expressions for energy ratios of various reflected waves are also obtained. A particular material aluminum–epoxy composite is chosen to compute the values of reflection coe?cients and energy ratios of reflected waves. The effects of diffusion and thermal parameters are observed on the reflection coe?cients and energy ratios.     

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.     

Transient Thermal Stresses in an Orthotropic Cylinder under the Hyperbolic Heat Conduction Model

An important consideration in design involving high temperature variation is the determination of the thermal stresses developed. The numerical solution for thermoelastic transient response of orthotropic cylinder subjected to a constant temperature at the surface is presented. The thermoelastic equations with one relaxation time developed by Lord and Shulman with uncoupled thermoelasticity assumption are used in the present work. The hyperbolic heat conduction model is used for the prediction of the temperature history. Thermally induced displacement and stresses are determined. A numerical method based on implicit finite difference scheme is used to calculate the temperature, displacement, and stress distributions within the cylinder. Numerical examples for orthotropic, transverse isotropic, and isotropic cylinders were carried out for the stresses. Furthermore, the results of the numerical solution and the exact solution at the steady state condition are compared.     

Theory of Generalized Micropolar Thermoelastic Diffusion Under Lord–Shulman Model

The general equations of motion and constitutive equations, based on the theory of Lord–Shulman with one relaxation time, are derived for a general homogeneous anisotropic medium with a microstructure, taking into account the effects of heat and diffusion. A variational principle for the governing equations is obtained. Then we show that the variational principle can be used to obtain a uniqueness theorem under suitable conditions. A reciprocity theorem for these equations is given. The obtained results are valid for some special cases which can be deduced from our generalized model.     

Memory response for thermal distributions moving over a magneto-thermoelastic rod under Eringen’s nonlocal theory

Abstract

Enlightened by the Caputo fractional derivative, this study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of magnetic field and moving heat source in a rod in the context of Lord–Shulman (LS) theory of thermoelasticity based on Eringen’s nonlocal elasticity. Both ends of the rod are fixed and heat insulated. Employing Laplace transform as a tool, the problem has been transformed into the space domain and solved analytically. Finally, solutions in the real-time domain are obtained by applying the inverse Laplace transform. Numerical calculation for stress, displacement, and temperature within the rod is carried out and displayed graphically. The effects of moving heat source speed, time instance, memory-dependent derivative, magnetic field and nonlocality on temperature, stress, and temperature are studied.     

Short-time Analysis of Magnetothermoelastic Wave Under Fractional Order Heat Conduction Law

This paper deals with the problem of magnetothermoelastic interactions in a perfectly conducting elastic medium in which the boundary is stress free and subjected to thermal loading in the context of the fractional-order, two-temperature generalized thermoelasticity theory (2TT). The two-temperature, three-phase-lag (2T3P) model and two-temperature Lord–Shulman (2TLS) model of thermoelasticity are combined into a unified formulation introducing unified parameters. The governing equations of generalized thermoelasticity of these models under the influence of a magnetic field are established. 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 the state-space approach. The numerical inversion of the transform is carried out by a method based on Fourier-series expansion techniques. Because of the short duration of the second sound effects, small time approximations of the solutions are studied. The numerical estimates of the quantities of physical interest are obtained and depicted graphically. The effect of the fractional-order parameter and the two-temperature and magnetic field parameters on the solutions has been studied and the comparisons among different thermoelastic models are made.