Deriving exact Green’s functions and integral formulas for a thermoelastic wedge

In this paper an exact Green’s function and a Poisson-type integral formula for a boundary-value problem (BVP) for a thermoelastic wedge are derived in elementary functions. The thermoelastic displacements are generated by a heat source distributed in the inner points of the wedge. On the boundary half-planes the temperature is given, the mechanical boundary conditions being locally-mixed and homogeneous. Similar results for a quarter-space and for a half-space as particular cases of the wedge also are included. According to the well-known analogy between thermoelasticity and poroelasticity, the obtained results can be extended to poroelasticity problems, by making use of changes in respective constants. This paper introduces an approach for deriving new thermoelastic and poroelastic influence functions and Poisson-type integral formulas in closed form not only for wedge, quarter-space and half-space but also for many other canonical domains.     

A Nonlinear Generalized Thermoelasticity Model of Temperature-Dependent Materials Using Finite Element Method

In this article, a general finite element method (FEM) is proposed to analyze transient phenomena in a thermoelastic model in the context of the theory of generalized thermoelasticity with one relaxation time. The exact solution of the nonlinear model of the thermal shock problem of a generalized thermoelastic half-space of temperature-dependent materials exists only for very special and simple initial- and boundary problems. In view of calculating general problems, a numerical solution technique is to be used. For this reason, the FEM is chosen. The results for the temperature increment, the stress components, and the displacement component are illustrated graphically with some comparisons.     

On the complex potentials of the linear thermoelasticity with microlocal parameters

Summary The linear theory of thermoelasticity with microlocal parameters was proposed lately in [1], [2], [3] as a certain alternative approach to the modelling of microperiodic composites. In this paper we adopt the complex variable method for two-dimensional problems of the periodically layered thermoelastic composites treated within the framework of the linear thermoelasticity with microlocal parameters. The presented complex potentials reduce two-dimensional static problems of the microperiodically layered thermoelastic composites to the boundary values problems for analytical functions. As an example illustrating this method the plane-strain problem of stress and temperature distribution in the microperiodic two-layered half-space subjected to general loading and temperature on the boundary is considered.With 3 Figures     

Two-Temperature Generalized Thermoelastic Interactions in an Infinite Body with a Spherical Cavity

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 in the context of the two-temperature generalized thermoelasticity theory (2TT). The two-temperature Lord-Shulman (2TLS) model and two-temperature Green–Naghdi (2TGN) models of thermoelasticity are combined into a unified formulation introducing the unified parameters. The medium is assumed initially quiescent. 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 (a) the state-space approach and (b) the eigenvalue approach for any set of boundary conditions. The general solution obtained is applied to a specific problem when the boundary of the cavity is subjected to thermomechanical loading. The numerical inversion of the transform is carried out using Fourier-series expansion techniques. The computed results for thermoelastic stresses, conductive temperature, and thermodynamic temperature are shown graphically for the Lord Shulman model and for two models of Green–Naghdi and the effects of two temperatures are discussed. A comparative study of the two methods has also been carried out.     

A three-dimensional thermoelastic problem for a half-space without energy dissipation

A three-dimensional problem for a homogeneous, isotropic and thermoelastic half-space subjected to a time-dependent heat source on the boundary of the space, which is traction free, is considered in the context of Green and Naghdi model II (thermoelasticity without energy dissipation) of thermoelasticity. The normal mode analysis and eigenvalue approach techniques are used to solve the resulting non-dimensional coupled equations. Numerical results for the temperature, thermal stress, strain and displacement distributions are represented graphically and discussed.     

Fractional Order Thermoelasticity Theory for a Half-Space Subjected to an Axisymmetric Heat Distribution

The present work is concerned with a traction-free thermoelastic half-space subjected to a known axisymmetric temperature distribution. The thermoelastic interactions inside the medium are investigated by employing the fractional order theory of thermoelasticity. The problem is solved by using Laplace's and Hankel's transforms. The inverse transforms are computed numerically. The variations of temperature, displacements, and stresses inside the half-space are investigated. The field variables for a particular material are graphically presented. Comparisons are made within the theory in the presence and absence of fractional order parameter.     

定常温度热弹性梁的精化理论

首先给出了定常温度热弹性Biot通解的一种新的简化形式,它看起来与各向同性弹性力学的Papkovich-Neuber通解十分相似。不作预先假设,从热弹性理论出发,利用Biot通解和Lur’e算子方法构造了梁的精化理论,得出了自由表面热弹性梁的三个精确方程:四阶方程、超越方程和温度方程。由一般的各向同性弹性梁推广到热弹性梁,导出了在反对称载荷和介质温度作用下热弹性梁的近似控制微分方程。     

Transient Disturbance in a Half Space Under Thermoelasticity with Two Relaxation Times Due to Moving Internal Heat Source

In this paper transient waves caused by a line heat source moving with a uniform velocity inside isotropic homogeneous thermoelastic half-space are studied under the GL model of generalized thermoelasticity. The problem is reduced to the solution of three differential equations by introducing the elastic vector potential and the thermoelastic scalar potential. Using Laplace and Fourier transforms solutions are obtained in transforms domain. Applying inverse transforms approximate solutions of the displacement at the boundary valid in the small time range are given. Also the approximate region valid for the solutions is given and two special cases, (i) the source is motionless and (ii) the relaxation times vanish, are studied. Numerical evaluations are presented for the medium of copper.     

Analysis of wave motion at the boundary surface of orthotropic thermoelastic material with voids and isotropic elastic half-space

The purpose of this research is to study the effect of voids on the surface wave propagation in a layer of orthotropic thermoelastic material with voids lying over an isotropic elastic half-space. The frequency equation is derived on the basis of the developed mathematical model under the boundary conditions for welded and smooth contacts. The dispersion curves giving the phase velocity and attenuation coefficient versus the wave number enable one to reveal the effects of voids and anisotropy for welded contact boundary conditions. The specific loss and amplitudes of the volume fraction, normal stress, and temperature change for welded contact are obtained and presented graphically for a particular model showing the voids and anisotropy effects. Some special cases are also deduced.     

The coupled magnetothermoelastic problem in elastic half-space with two relaxation times

This paper deals with the magnetothermoelastic disturbances in a perfectly electrically conducting elastic half-space, in contact with a vacuum, due to a thermal shock applied on the plane boundary of the half-space. The theory of thermoelasticity proposed by Green and Lindsay is used to account for the interaction between the elastic and thermal fields. The solution achieved is in analytical form, which reduces to that of Kaliski and Nowacki when the coupling between the temperature and strain fields and the relaxation time of Green and Lindsay theory are neglected.     

Magneto-rotation-fibre-reinforced thermoelastic with gravity and energy dissipation

In this article, two theories of the generalized thermoelasticity Green-Naghdi theory (of type II and III) are applied, as well as the coupled theory to study the effect of magnetic field and rotation under influence of gravity on 2D problem of a fibre-reinforced thermoelastic. The normal mode analysis is used to obtain the expressions for the temperature, displacement components and the thermal stresses distributions. The resulting formulation is applied for two different concrete problems. The first concerns the case of a punch moving across the surface of semi-infinite thermoelastic half-space subjected to appropriate boundary conditions. The second deals with a thick plate subjected to a time-dependent heat source on each face. Numerical results are illustrated graphically for each problem considered. A comparison is made with the results predicted obtained by the two theories in the presence and absence of magnetic field, rotation and gravity field.     

Generalized Magneto-thermoelasticity in a Fiber-Reinforced Anisotropic Half-Space

The propagation of plane waves in a fiber-reinforced, anisotropic thermoelastic half-space proposed by Lord–Shulman under the effect of a magnetic field is discussed. The problem has been solved numerically using a finite element method. Numerical results for the temperature distribution, the displacement components, and the thermal stress are given and illustrated graphically. Comparisons are made with the results predicted by the theory of generalized thermoelasticity with one relaxation time for different values of time. It is found that the reinforcement has a great effect on the distribution of field quantities.     

Solution for the 3D Temperature Problem of the Thermoelasticity Theory in Displacements

A new method of solution for the 3D problem of the thermoelasticity theory in displacements is proposed. The resultant general solution can be used in solving thermoelastic problems both for half-space and an elastic layer. The second boundary-value problem for half-space is detailed and numerical calculation is given.__________Translated from Problemy Prochnosti, No. 3, pp. 86 – 95, May – June, 2005.     

Investigation on a Two-dimensional Generalized Thermal Shock Problem with Temperature-dependent Properties

The dynamic response of a two-dimensional generalized thermoelastic problem with temperature-dependent properties is investigated in the context of generalized thermoelasticity proposed by Lord and Shulman. The governing equations are formulated, and due to the nonlinearity and complexity of the governing equations resulted from the temperature-dependent properties, a numerical method, i.e., finite element method is adopted to solve such problem. By means of virtual displacement principle, the nonlinear finite element equations are derived. To demonstrate the solution process, a thermoelastic half-space subjected to a thermal shock on its bounding surface is considered in detail. The nonlinear finite element equations for this problem are solved directly in time domain. The variations of the considered variables are obtained and illustrated graphically. The results show that the effect of the temperature-dependent properties on the considered variables is to reduce their magnitudes, and taking the temperature-dependence of material properties into consideration in the investigation of generalized thermoelastic problem has practical meaning in predicting the thermoelastic behaviors accurately. It can also be deduced that directly solving the nonlinear finite element equations in time domain is a powerful method to deal with the thermoelastic problems with temperature-dependent properties.     

Influence functions, integral formulas, and explicit solutions for thermoelastic spherical wedges

In this study, new exact Green’s functions and a new exact Green-type integral formula for a boundary value problem (BVP) in thermoelasticity for some spherical wedges with mixed homogeneous mechanical boundary conditions are derived. The thermoelastic displacements are subjected to a heat source applied in the inner points of the spherical wedges and to a mixed non-homogeneous boundary heat conditions. When the thermoelastic Green’s function is derived, the thermoelastic displacements are generated by an inner unit point heat source, described by Dirac’s δ-function. All results are obtained in elementary functions that are formulated in a special theorem. Exact solutions in elementary functions for two particular BVPs of thermoelasticity for spherical wedges also are included. In these particular BVPs, the thermoelastic displacements are subjected to a constant temperature (in the first particular BVP) or to a constant heat source (in the second particular BVP). In both BVPs, the constant temperature or the constant heat source is given on the segment of the radius of the quarter-space. On the boundary half-planes of the quarter-space zero temperature and zero heat flux are prescribed.     

Effect of Gravitational Field and Temperature Dependent Properties on Two-Temperature Thermoelastic Medium with Voids under G-N Theory

This investigation is aimed to study the two dimensional problem of thermoelastic medium with voids under the effect of the gravity. The modulus of elasticity is taken as a linear function of the reference temperature and employing the two-temperature generalized thermoelasticity. The problem is studied in the context of Green-Naghdi (G-N) theory of types II and III. The normal mode analysis method is used to obtain the exact expressions for the physical quantities which have been shown graphically by comparison between two types of the (G-N) theory in the presence and the absence of the gravity, the temperature dependent properties and the two-temperature effect.     

A Generalized Thermoelasticity Problem for a Half-Space with Heat Sources and Body Forces

In this work, a two-dimensional problem of distribution of thermal stresses and temperature in a linear theory of a generalized thermoelastic half-space under the action of a body force and subjected to a thermal shock on the bounding plane is considered. Heat sources permeate the medium. The problem is in the context of the theory of generalized thermoelasticity with one relaxation time. Laplace and exponential Fourier transform techniques are used. The solution in the transformed domain is obtained by a direct approach. The inverse double transform is evaluated numerically. Numerical results are obtained and represented graphically.     

A two-dimensional problem for a fibre-reinforced anisotropic thermoelastic half-space with energy dissipation

The theory of thermoelasticity with energy dissipation is employed to study plane waves in a fibre-reinforced anisotropic thermoelastic half-space. We apply a thermal shock on the surface of the half-space which is taken to be traction free. The problem is solved numerically using a finite element method. Moreover, the numerical solutions of the non-dimensional governing partial differential equations of the problem are shown graphically. Comparisons are made with the results predicted by Green–Naghdi theory of the two types (GNII without energy dissipation) and (GNIII with energy dissipation). We found that the reinforcement has great effect on the distribution of field quantities. Results carried out in this paper can be used to design various fibre-reinforced anisotropic thermoelastic elements under thermal load to meet special engineering requirements.     

Two-dimensional thermal shock problem of fractional order generalized thermoelasticity

In this study, 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 fractional order generalized thermoelasticity theory. 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 is applied to a specific problem of a half-space subjected to thermal shock. The inverse Fourier transforms are obtained analytically, while the inverse Laplace transforms are computed numerically. Some comparisons have been shown in figures to estimate the effect of the fractional order on all the studied fields.