Variable electrical and thermal conductivity in the theory of generalized thermoelastic diffusion

This paper deals with the problem of a thermoelastic half-space with a permeating substance in contact with the bounding plane in the context of the theory of generalized thermoelastic diffusion with one relaxation time and with variable electrical and thermal conductivity. The bounding surface of the half-space is taken to be traction free and subjected to a time dependent thermal shock. The solution is obtained in the Laplace transform domain by a direct approach. A numerical technique is employed to obtain the solution in the physical domain. It is found that there exist two coupled waves, one of which is elastic and the other is thermal, and a third wave affects diffusion mainly. A comparison is made with the results obtained in a thermoelastic medium with and without diffusion in the following cases : (a) the electrical and thermal conductivities have constant values, (b) the presence of magnetic field and (c) the generalized theory in thermoelasticity. Received: June 1, 2005     

A stochastic half-space problem in the theory of generalized thermoelastic diffusion including heat source

A stochastic half-space problem, driven by an additive Gaussian white noise, is considered within the context of the theory of generalized thermoelastic diffusion with one relaxation time. The bounding surface is traction free and subjected to a time dependent thermal shock. A permeating substance is considered in contact with the bounding surface. Laplace transform technique is used to obtain the solution in the transformed domain by using a direct approach. The mean and variance are derived and analyzed for temperature, displacement, stress, strain, concentration and chemical potential. The asymptotic behavior for the solution is discussed. Numerical results are carried out and represented graphically. The second sound effect is observed in the simulation.     

Boundary integral equation formulation for generalized thermoelastic diffusion – Analytical aspects

The present work deals with the formulation of the boundary integral equations for the solution of equations under linear theory of generalized thermoelastic diffusion in a three-dimensional Euclidean space. A mixed initial-boundary value problem is considered in the present context and the fundamental solutions of the corresponding coupled differential equations are obtained in the Laplace transform domain by employing the treatment of scalar and vector potential theory. A reciprocal relation of Betti type is established. Then we formulate the boundary integral equations for generalized thermoelastic diffusion on the basis of these fundamental solutions and the reciprocal relation.     

State-space approach for an infinite medium with a spherical cavity based upon two-temperature generalized thermoelasticity theory and fractional heat conduction

This paper is concerned with the determination of the thermoelastic displacement, stress, conductive temperature, and thermodynamic temperature in an infinite isotropic elastic body with a spherical cavity. A general solution to the problem based on the two-temperature generalized thermoelasticity theory (2TT) is introduced. The theory of thermal stresses based on the heat conduction equation with Caputo’s time-fractional derivative of order α is used. Some special cases of coupled thermoelasticity and generalized thermoelasticity with one relaxation time are obtained. The general solution is provided by using Laplace’s transform and state-space techniques. It is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading (thermal shock). Some numerical analyses are carried out using Fourier’s series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically and the effects of two-temperature and fractional-order parameters are discussed.     

带扩散的广义热弹性固体中移动荷载引起的二维相互作用

当一个移动荷载沿着一个坐标轴作用在介质边界上时,研究了该具有广义热弹性扩散的均匀各向同性介质中的扰动.应用特征值逼近方法,研究了Laplace-Fourier变换域中的二维扰动问题.在Fourier扩展技术的基础上,利用Laplace数值逆变换技术,求解了位移分量、应力、温度场、浓度和化学势的解析表达式.数值计算了铜类材料的这些表达式,并给出有关图形.作为特殊情况,给出了广义热弹性介质和弹性介质中,扩散和热效应的理论结果和数值结果.     

Wave propagation in a micropolar generalized thermoelastic body with stretch

In the present investigation, we discuss two different problems namely

1. Rayleigh-Lamb problem in micropolar generalized thermoelastic layer with stretch and
2. Rayleigh wave in a micropolar generalized thermoelastic half-space with stretch. The frequency and wave velocity equations for symmetric and anti-symmetric vibrations are obtained for the first problem. The frequency equation has also been derived for the second problem. The special cases of the above said problems of micropolar generalized thermoelasticity with stretch for Green-Lindsay and Lord-Shulman theory have been discussed in detail. Results of these analysis reduce to those without thermal and stretch effects

     

Steady-state response of a micropolar generalized thermoelastic half-space to the moving mechanical/thermal loads

Microrotation effect of a load applied normal to the boundary and moving at a constant velocity along one of the co-ordinate axis in a generalized thermoelastic half-space is studied. The analytical expressions of the displacement component, force stress, couple stress and temperature field for two different theories i.e. Lord-Shulman (L-S) and Green-Lindsay (G-L) for supersonic, subsonic and transonic velocities in case of mechanical and thermal sources applied, are obtained by the use of Fourier transform technique. The integral transforms have been inverted by using a numerical technique and the numerical results are illustrated graphically for magnesium crystal-like material.     

Two-dimensional problem of a two-temperature generalized thermoelastic half-space subjected to ramp-type heating

In this work, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the theory of two-temperature generalized thermoelasticity. A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. Laplace and Fourier transform techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to ramp-type heating. The inverse Fourier transforms are obtained analytically while the inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating.     

On dynamic effects in an elastic half-space under “thermal impact” : PMM vol. 38, n 6, 1974, pp. 1105–1113

The general uncoupled dynamical problem of thermoelasticity for a half-space under the condition of a thermal impact with a finite rate of change in temperature on its boundary is solved by the method of principal (fundamental) functions within the framework of a generalized theory of heat conduction.An elastic steel half-space is analyzed as an illustration. The problem on thermal stresses originating in an elastic half-space due to thermal impact produced by a jump change in temperature on the boundary was first analyzed in [1]. Since the temperature change on the boundary occurs at a finite rate, it is generally impossible to realize the thermal impact considered in [1] physically. The dynamic effects in an elastic half-space under a thermal impact with finite rate of change in the temperature on the boundary have been studied in [2]. For high rates of change of the heat flux we obtain a generalized wave equation of heat conduction [3] taking into account the finite velocity of heat propagation. Hence, the solution of the ordinary parabolic heat conduction equation used in [1, 2] does not correspond to the true temperature field. The problems of [1, 2] have been examined in [4, 5], respectively, within the framework of a generalized theory of heat conduction.     

An idea of half-space thermal mirror

The paper describes a theoretical discussion of the optical reflecting properties of an isotropic thermoelastic half-space subjected to an instantaneous heat pulse at a point of its surface (such a system is called the half-space thermal mirror). The fundamental optical properties of the mirror so obtained (i.e.—its aberration characteristic, including optical power and focal length), and their time evolution are derived in the paper. The half-space thermal mirror represents an optical device suffering rather high aberrations; the paraxial optics approximation, and criteria of its applicability for such a mirror are given. Possibilities of an application of the thermal mirror for determining the temperature conductivity of a material are considered.     

Generalized thermoelastic infinite medium with spherical cavity subjected to moving heat source

This paper deals with a problem of thermoelastic interactions in an isotropic unbounded medium with spherical cavity due to the presence of moving heat sources in the context of the linear theory of generalized thermoelasticity with one relaxation time. The governing equations are expressed in the Laplace transform domain and solved in that domain. The inversion of the Laplace transform is done numerically using the Riemann-sum approximation method. The numerical estimates of the displacement, temperature, stress, and strain are obtained for a hypothetical material. The results obtained are presented graphically to show the effect of the heat source velocity and the relaxation time parameters on displacement, temperature, stress, and strain.     

Effect of rotation on plane waves of generalized electro-magneto-thermoviscoelasticity with two relaxation times

A new model of the equations of generalized thermovisco-elasticity for a thermally, isotropic and electrically conducting half-space solid whose surface is subjected to a thermal shock is given. The formulation is applied to the generalized thermoelasticity based on the Green and Lindsay (GL) theory under the effect of rotation, where there is an initial magnetic field parallel to the plane boundary of the half-space. The medium is deformed because of thermal shock and due to the application of the magnetic field, it results an induced magnetic and electric fields in the medium. The normal mode analysis is used to obtain the expressions for the variables considered. The distributions of temperature, displacement, stress, induced magnetic and electric fields are represented graphically. Comparisons are made with the results predicted by the coupled theory (CD) in the presence and absence of rotation.     

具有两个驰豫时间的热弹性立方晶体材料中平面波的传播

研究具有两个驰豫时间的、两个不同弹性和热性质的、广义传热立方晶体固体半空间的有缺陷结合面上,热弹性平面波的反射和折射问题.具有两个驰豫时间的广义热弹性理论,是1972年由Green和Lindsay提出并应用于问题的研究.对有缺陷边界,给出了反射系数和折射系数(即反射波和折射波振幅与入射波振幅之比)的表达式,并推演了法向刚性边界、横向刚性边界、接触传热边界、滑动边界和结合面边界时的表达式.给出了在不同边界条件及出射角时,不同的反射波和折射波的振幅比,在不同的入射波时的变化图.发现反射波和折射波振幅比受到介质刚性和热性质的影响.     

3D dynamic coupled thermoelastic solution for constant thickness disks using refined 1D finite element models

This paper deals with the generalized coupled thermoelastic solution for disks with constant thickness. It is a sequel to the authors’s previous work in which refined 1D Galerkin finite element models with 3D-like accuracies are developed for theories of coupled thermoelasticity. Use of the reduced models with low computational costs may be of interest in a laborious time history analysis of the dynamic problems. In this paper, the developed models are applied and evaluated for a 3D solution of the dynamic generalized coupled thermoelasticity problem in the disk subjected to thermal shock loads. Comparison of the obtained result with the results available in the literature verified the proposed finite element models are quite efficient with very high rate of convergence and able to provide results with analytical accuracy. In addition, propagation of the thermoelastic waves, the wave reflection from the boundaries and the Poisson effect in an axisymmetric and asymmetric disk problem are represented as contour plots to demonstrate 3D capabilities of the models.     

A theory of thermoelastic diffusion materials with voids

In the first part of this paper, we derive the equations of the linear theory of thermoelastic diffusion in porous media based on the concept of volume fraction. Then, we establish a reciprocal relation which leads to reciprocity, uniqueness and continuous dependence theorems for anisotropic materials. Finally, we prove the existence of a generalized solution by means of the semigroup of linear operators theory.     

三相滞后对有球形空腔无限介质双温广义热弹性的影响

就各向同性的无限弹性体,具有一个球形空腔时,从双温广义热弹性理论（2TT）角度,研究三相滞后热方程的热弹性相互作用问题．在三相滞后理论中,热传导方程是一个含时间四阶导数的、双曲型的偏微分方程．假设无限介质初始时静止,通过Laplace变换,将基本方程用向量矩阵微分方程的形式表示,然后通过状态空间法求解．将得到的通解应用于特殊问题:空腔边界上承受着热荷载（热冲击和坡型加热）和力学荷载．使用Fourier级数展开技术,实现Laplace变换的求逆．计算了铜类材料物理量的数值解．图形显示,两种模型:带能量耗散的双温Green-Naghdi理论（2TGNIII）和双温3相滞后模型（2T3相）明显不同．还对双温和坡型参数的影响进行了研究．     

变温度荷载作用下半无限成层饱和介质的热固结分析

对半无限成层饱和多孔介质作用随时间变化的温度荷载的热固结问题进行解析求解．其中,热-水-力耦合线性弹性控制方程考虑了热渗效应和等温热流效应的影响．先采用Laplace变换求其在变换域上的解,然后用数值方法求逆变换．对半无限体表面作用呈指数衰减热荷载的双层体系进行研究,分析了两层介质热固结系数、弹性模量等的差异性对热固结特征的影响．研究表明:位移场和应力场对温度场的耦合作用可以忽略,而热渗效应对温度和孔压有显著影响．     

Transient thermoelastic response in a cracked strip of functionally graded materials via generalized fractional heat conduction

This work is devoted to analyzing a thermal shock problem of an elastic strip made of functionally graded materials containing a crack parallel to the free surface based on a generalized fractional heat conduction theory. The embedded crack is assumed to be insulated. The Fourier transform and the Laplace transform are employed to solve a mixed initial-boundary value problem associated with a time-fractional partial differential equation. Temperature and thermal stresses in the Laplace transform domain are evaluated by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying a numerical inversion of the Laplace transform. The temperature jump between the upper and lower crack faces and the thermal stress intensity factors at the crack tips are illustrated graphically, and phase lags of heat flux, fractional orders, and gradient index play different roles in controlling heat transfer process. A comparison of the temperature jump and thermal stress intensity factors between the non-Fourier model and the classical Fourier model is made. Numerical results show that wave-like behavior and memory effects are two significant features of the fractional Cattaneo heat conduction, which does not occur for the classical Fourier heat conduction.     

A generalized Cauchy problem for the linear differential equations of coupled physical - mechanical fields

A generalized Cauchy problem for a partial differential equation with constant coefficients, which is encountered in the study of physical processes in continuous media with widened physical - mathematical fields (see /1/) (generalized coupled thermoeleasticity /2/, coupled thermoeleasticity, porous media saturated with a viscous fluid /5/, mass and heat transfer /6/, linearized magnetoelasticity /7/, etc.) is considered. The characteristic properties of the solution of the problem, under certain constraints imposed on an equation by the stability condition, are studied. The presence of waves of higher and lower order is characteristic for the solution; in the course of time the lower-order waves are maintained and take a characteristic form. In the general case, the solution is represented in the form of integrals over the segments which link the singular points of Fourier - Laplace transforms with respect to time of the solution under consideration. The methods proposed enable an exact investigation to be made of the processes described by the equation for any time constants, and they also enable one to isolate the singularities at the fronts of propagating perturbations. As an application, the dynamic processes taking place in a thermoelastic subsapce (2) as a result of applying a mechanical and a thermal input at the boundary is studied. It is shown that in the case of unit perturbation of the boundary, the stress and temperature waves in the course of time assume a bell-shaped form and propagate with adiabatic velocity. A numerical analysis of the process which occurs due to sudden application of the force and of the thermal shock at the boundary is given.     

Propagation of nonlinear thermoelastic waves in porous media within the theory of heat conduction with memory: physical derivation and exact solutions

In this paper, we study a mathematical model of nonlinear thermoelastic wave propagation in fluid‐saturated porous media, considering memory effect in the heat propagation. In particular, we derive the governing equations in one dimension by using the Gurtin–Pipkin theory of heat flux history model and specializing the relaxation function in such a way to obtain a fractional Erdélyi–Kober integral. In this way, we obtain a nonlinear model in the framework of time‐fractional thermoelasticity, and we find an explicit analytical solution by means of the invariant subspace method. A second memory effect that can play a significant role in this class of models is parametrized by a generalized time‐fractional Darcy law. We study the equations obtained also in this case and find an explicit traveling wave type solution. Copyright © 2016 John Wiley & Sons, Ltd.