Thermally nonlinear generalized thermoelasticity of a layer

This article addresses a clarification study on the thermally nonlinear Fourier/ non-Fourier dynamic coupled (generalized) thermoelasticity. Based on the Maxwell-Cattaneo’s heat conduction law and the infinitesimal theory of thermoelasticity, governing equations for the thermally nonlinear small deformation type of generalized thermoelasticity are derived. The Bubnov–Galerkin scheme is implemented for spatial discretization. The spatially discretized equations are directly discretized in time domain using the fully damped Newmark method. The Newton–Raphson procedure is used to linearize the finite element equations. The layers are exposed to a thermal shock, so that the displacement, temperature, and stress waves propagate in layers. The effects of the time evolution, thermoelastic coupling, and thermal relaxation time on the response of the layers are investigated. Results reveal the significance of the thermally nonlinear analysis of generalized thermoelasticity for the conditions where large temperature elevations exist.     

Numerical analysis of coupled effects of pulsatile blood flow and thermal relaxation time during thermal therapy

The main purpose of this study is to investigate the coupled effects of the pulsatile blood flow in thermally significant blood vessels and the thermal relaxation time in living tissues on temperature distributions during thermal treatments. Considering the fact that propagation speed of heat transfer in solid tissues is actually finite according to experiments, the traditional Pennes bioheat transfer equation (PBTE) was modified to a wave bioheat transfer equation (WBTE) that contains both wave transportation and diffusion competing with each other and characterized by the thermal relaxation time. The wave behavior will be more dominant when the relaxation time is large. WBTE together with a coupled energy transport equation for blood vessel flow was used to describe the temperature evolution of our current tumor–blood vessel system, and the equations were numerically solved by the highly accurate multi-block Chebyshev pseudospectral method. Numerical results showed that temperature evolution from WBTE was quite different from their counterparts from PBTE due to the dominant wave feature under large relaxation time. For example, larger relaxation time would preserve high temperature longer and this effect is even more pronounced when heating is fast. It further implies that heat is drained more slowly when relaxation time is large, and would make thermal lesion region cover the tumor tissue, the heating target, better. This phenomenon would therefore hint that the traditional PBTE simulations might under-estimate the thermal dose exerted on tumor. As to the pulsation frequency of blood flow from heart beat which was originally predicted to be important here, it turned out that the thermal behavior is quite insensitive to pulsation frequency in the current study.     

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.     

Thermoelastic interaction in functionally graded thick hollow cylinder with temperature-dependent properties

The prediction of thermoelastic behavior induced by transient thermal shock is important to evaluate the durability of functionally graded materials. The purpose of this article is to study the axisymmetric thermoelastic interaction in a functionally graded thick hollow cylinder by an asymptotic approach. The governing equations with variable material properties, which are spatially graded and temperature dependent, are proposed based on the generalized theory of thermoelasticity with one relaxation time (L–S theory). The Laplace transform technique is used to derive the general solutions with the cylinder divided into thin cylinders and material properties assumed constant in each thin cylinder. The inverse Laplace transform is then conducted analytically by some approximations in the time domain, and the short-time solution of the problem with its interior boundary subjected to a sudden temperature rise and the outer surface maintained at constant temperature are obtained. Utilizing these asymptotic solutions, the propagation of thermal and thermoelastic waves are studied, which display dependence of each wave’s propagation upon the relaxation time, volume fraction parameter and temperature. The distributions of the radial displacement, temperature and stresses are also plotted and discussed. These results reveal effects of these variable material properties with spatial position and temperature on thermoelastic behavior.     

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.     

Analysis of Thermoelastic Waves in Functionally Graded Hollow Spheres Based on the Green-Lindsay Theory

In this article, the Green–Lindsay theory of thermoelasticity is employed to study the thermoelastic response of functionally graded hollow spheres. This generalized coupled thermoelasticity theory admits the second sound phenomena and depicts a finite speed for temperature wave propagation. The materials of the hollow sphere are assumed to be graded through its thickness in the radial direction while a symmetric thermal shock load is applied to its boundary. The Galerkin finite element method via the Laplace transformation is used to solve the coupled form of governing equations. A numerical inversion of the Laplace transform is employed to obtain the results in time domain. Using the obtained solution, the temperature, displacement, radial stress, and hoop stress waves propagation are studied. Also the material distribution effects on temperature, displacement and stresses are investigated. Finally, the obtained results for the Green–Lindsay theory are compared with the results of classical thermoelasticity theory.     

Thermodynamically consistent description of heat conduction with finite speed of heat propagation

In this work, governing equations for heat conduction with finite speed of heat propagation are derived directly from classical thermodynamics. For a one-dimensional flow of heat, the developed governing equation is linear and of parabolic type. In a three dimensional case, the system of nonlinear equations is formulated.Analytical solutions of the equations for one-dimensional flow of heat are obtained, and their analysis shows characteristic features of heat propagation with finite speed, being fully consistent with classical thermodynamics.     

Generalized thermoelasticity of a piezoelectric layer

In the present research, the response of a one-dimensional piezoelectric layer is investigated using the generalized thermoelasticity theory of Lord and Shulman. The layer is subjected to thermal shock on one surface. Three coupled equations, namely, motion equation, energy equation and Maxwell equation in terms of displacement, temperature, and electric potential are established. Using the proper transformation, the mentioned equations are given in a dimensionless form. These equations are discretized by means of the generalized differential quadrature method and traced in time by means of the Newmark time marching scheme. Numerical examples are provided to show the propagation and reflection of thermal, mechanical and electrical waves in a layer. It is shown that under the Lord and Shulman theory, temperature propagates with a finite speed, similar to mechanical displacement wave. However, the electric displacement and potential propagate with infinite speed.     

Magnetothermoelastic creep analysis of functionally graded cylinders

This paper describes time-dependent creep stress redistribution analysis of a thick-walled FGM cylinder placed in uniform magnetic and temperature fields and subjected to an internal pressure. The material creep, magnetic and mechanical properties through the radial graded direction are assumed to obey the simple power law variation. Total strains are assumed to be the sum of elastic, thermal and creep strains. Creep strains are time, temperature and stress dependent. Using equations of equilibrium, stress–strain and strain–displacement a differential equation, containing creep strains, for displacement is obtained. Ignoring creep strains in this differential equation a closed form solution for the displacement and initial magnetothermoelastic stresses at zero time is presented. Initial magnetothermoelastic stresses are illustrated for different material properties. Using Prandtl–Reuss relation in conjunction with the above differential equation and the Norton’s law for the material uniaxial creep constitutive model, the radial displacement rate is obtained and then the radial and circumferential creep stress rates are calculated. Creep stress rates are plotted against dimensionless radius for different material properties. Using creep stress rates, stress redistributions are calculated iteratively using magnetothermoelastic stresses as initial values for stress redistributions. It has been found that radial stress redistributions are not significant for different material properties, however major redistributions occur for circumferential and effective stresses.     

Stochastic Assessment of Thermo-Elastic Wave Propagation in Functionally Graded Materials (FGMs) with Gaussian Uncertainty in Constitutive Mechanical Properties

The main objective of this article is focused on stochastic analysis of wave propagation and effects of uncertainty in mechanical properties on transient behaviors of displacement and temperature fields in functionally graded materials under thermo-mechanical shock loading. The problem is studied in a cylindrical domain and the governing equations of a functionally graded thick hollow cylinder are solved. To assess the wave propagation, the generalized coupled thermoelasticty equations based on Green-Naghdi theory (without energy dissipation) are analyzed in a FG thick hollow cylinder. The FG cylinder is considered to have infinite length and axisymmetry conditions. The constitutive mechanical properties of FGM are assumed as random variables with Gaussian distribution and also the mechanical properties are considered to vary across thickness of FG cylinder as a nonlinear power function of radius. The FG cylinder is divided into many elements across thickness of cylinder and hybrid numerical method (Galerkin finite element and Newmark finite difference methods) along with the Monte Carlo simulation are employed to solve the statistical coupled equations. The effects of uncertainty in functionally graded materials on thermal and elastic waves, transient behaviors of radial displacement and temperature fields and variance and maximum values of displacement and temperature are discussed in details for various grading patterns in FGMs and various points on thickness at several times.     

On thermodynamics of a class of temperature-rate-dependent viscoelastic solids

Thermodynamics of a class of temperature-rate viscoelastic materials is studied using the reduced energy equation and the Green–Naghdi dissipation inequality embodying certain aspects of the second law of thermodynamics. Constitutive response function for entropy is determined explicitly in terms of the Helmholtz free energy function. The restrictions imposed by the Green–Naghdi dissipation inequality are obtained for other thermomechanical response functions. For the linear theory, a coupled system of differential equations is obtained which under special circumstances allow propagation of thermal waves without energy dissipation. In addition, the method presented here is applied to the temperature-rate-dependent thermoelastic materials.     

Three-dimensional thermoviscoelastic analysis of a FGM cylindrical panel using state space differential quadrature method

Mechanical behavior of a viscoelastic cylindrical panel with various edge boundary conditions, made up of functionally graded material (FGM) and subjected to thermal or mechanical load is investigated. For cases of simply supported boundary conditions, analytical solution is presented through Fourier series expansion along the axial and circumferential coordinates as well as state space method along the radial coordinate. For nonsimply supported conditions, semi-analytical solution is performed using differential quadrature method instead of Fourier series solution. Governing differential equations are transformed to Laplace transform domain and using inverse Laplace transform, obtained solutions are converted to time domain. In the present work, relaxation modulus of FGM is supposed to be according to the Prony series with power law variations along the radial direction. Numerical comparison was made with the available published results to assess the validity of present technique. Effect of relaxation time constant, thickness to mid-radius ratio, edges boundary condition and outer surface temperature on stress and displacement fields are discussed. Besides, time history of stresses for different relaxation time constant and for various boundary conditions is presented.     

Generalized differential quadrature analysis of MHD mixed convective motion of Casson fluids past an isothermal irregular slender geometry via Cattaneo–Christov's approach

Classical Fourier's theory is well-known in continuum physics and thermal sciences. However, the primary drawback of this law is that it contradicts the principle of causality. To explore the thermal relaxation time characteristic, Cattaneo–Christov's theory is adopted thermally. In this regard, the features of magnetohydrodynamic (MHD) mixed convective flows of Casson fluids over an impermeable irregular sheet are revealed numerically. In addition, the resulting system of partial differential equations is altered via practical transformations into nonlinear ordinary differential equations. An advanced numerical algorithm is developed in this respect to get higher approximations for temperature and velocity fields, as well as their corresponding wall gradients. For validating our numerical code, the current outcomes are compared with the available literature results. Moreover, it is revealed that the velocity field is more prominent in the suction flow situation as compared with the injection flow case. It is also found that the Casson fluid is hastened in the case of lower yield stress. Larger values of thermal relaxation parameters create a lessening trend in the temperature distribution and its related boundary layer breadth.     

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.     

Thermomechanical coupling and second sound in superfluid flows

The present paper is concerned with the modelling of the influence of the thermomechanical coupling on the propagation of temperature waves (second sound) in superfluids. For an adequate heat transfer analysis in superfluids, finite thermal wave speed must be considered. Besides hyperbolic heat transfer, turbulent flows are generally observed despite the lost of internal friction. Due to the thermo-mechanical coupling, density waves may induce moving heat sources or sinks while temperature waves propagate at a different and independent speed. In particular, a rotational flow in this kind of fluid can strongly affect heat propagation. A general procedure, developed within the framework of thermodynamics of irreversible processes, is proposed to obtain constitutive relations that verify automatically the second law of thermodynamics and the principle of material objectivity. Such a phenomenological continuum approach allows a rational identification of the terms responsible for the thermomechanical coupling in the heat equation, which is a first step to better understand its influence on the superfluid flow.     

ON UNDAMPED HEAT WAVES IN AN ELASTIC SOLID

This paper is concerned with thermoelastic material behavior whose constitutive response functions possess thermal features that are more general than in the usual classical thermoelasticity. After a general development of the constitutive equations in the context of both nonlinear and linear theories, attention is focused on the latter. In particular, the one-dimensional version of the equation for the determination of temperature in the linearized theory provides an easy comparative basis of its predictive capability: In one special case where the Fourier conductivity is dominant, the temperature equation reduces to the classical Fourier law of heat conduction, which does not permit the possibility of undamped thermal waves; however,'in another special case in which the effect of conductivity is negligible, the equation has undamped thermal wave solutions without energy dissipation.     

THERMOELASTICITY SOLUTION OF A LAYER USING THE GREEN–NAGHDI MODEL

The Green–Naghdi (GN) linear theory of thermoelasticity of types II (without energy dissipation) and III (with energy dissipation) for homogeneous and isotropic materials is employed to study thermal and mechanical waves in a layer. The disturbances are generated by sudden application of temperature to the boundary. The dimensionless forms of the governing equations are solved utilizing the Laplace transform method. Closed-form solutions are obtained for a layer in the Laplace domain, and a numerical inversion of the Laplace transform method is used to obtain the temperature, displacement, and stress fields in the physical time domain. Thermomechanical wave propagation and reflection from the boundary layer are investigated and the influence of the damping parameter on the temperature, displacement, and stress fields in the GN type III theory is discussed.     

A comparison of hyperbolic and parabolic models of phase change of a pure metal

The solidification history of individual thermal spray particles has been the subject of many experimental and theoretical studies. Yet it is customary to assume that solidification occurs at the equilibrium temperature, and that heat propagates according to Fourier’s Law. To account for a finite thermal diffusion speed, a hyperbolic heat conduction equation is usually adopted to analyze heat transfer. However, under certain circumstances, this equation can violate the second law of thermodynamics, and so others have modified the original hyperbolic equation via theories of extended irreversible thermodynamics. In this work, we study non-equilibrium effects of rapid solidification of a pure metal particle, and compare the so-called parabolic, hyperbolic and modified hyperbolic equations for heat transfer, to predict the interface undercooling due to thermal effects and velocity as a function of time, for different relaxation times. Results indicate that differences are limited to the early part of the solidification process, when undercooling is most significant, the interface velocity is highest, and non-equilibrium effects are most evident. As solidification progresses, the non-equilibrium effects wane and solidification can then be properly modeled as an equilibrium process.     

Wave Propagation in an Inhomogeneous Anisotropic Generalized Thermoelastic Solid

The plane wave propagation in an inhomogeneous anisotropic thermally conducting elastic solid is studied with two thermal relaxation times. Three types of plane waves—quasi-P, quasi-S, and thermal waves—are shown to exist. The analytical expressions for their velocities of propagation are obtained. The inhomogeneity drastically affects the velocity of these waves and also depends on the angle of propagation and frequency. The effects of these parameters are shown graphically.     

Axisymmetric nonlinear rapid heating of FGM cylindrical shells

Large amplitude thermally induced vibrations of cylindrical shells made of a through-the-thickness functionally graded material (FGM) are investigated in the current research. All of the thermo-mechanical properties of the FGM shell are assumed to be functions of temperature and thickness coordinate. Shell is subjected to rapid surface heating on the ceramic-rich surface while the other surface of the shell is kept at reference temperature. One dimensional heat conduction equation is constructed and solved by means of a hybrid finite difference-Crank–Nicolson algorithm. The constructed heat conduction equation is nonlinear since the thermal conductivity is temperature dependent. With the aid of first-order shear deformation shell theory under the axisymmetric Donnell kinematic assumptions and von Kármán type of strain-displacement relations, the total energy of the shell is established. Implementing the conventional Ritz method, a set of nonlinear coupled algebraic equations are obtained which govern the dynamics of the shell under thermal shock. These equations are solved in time domain using the Newmark time marching scheme and the simple Picard successive method. Parametric studies are given to explore the dynamics of an FGM cylindrical shell under thermal shock.