Thermoelastic Damping in Nanomechanical Resonators Based on Two-Temperature Generalized Thermoelasticity Theory

This work is concerned with the study of the thermoelastic damping of nanobeam resonators in the context of the two-temperature generalized thermoelasticity theory. An explicit formula of thermoelastic damping has been derived. Influences of the beam height, the relaxation time parameter, the two-temperature parameter and the isothermal value of frequency have been studied with some comparisons between the Biot model and Lord–Shulman model (L–S). Numerical results show that the values of thermal relaxation parameter and the two-temperature parameter have a strong influence on thermoelastic damping in nanoscales.     

On the Theory of Two-Temperature Thermoelasticity with Two Phase-Lags

The present paper is concerned with the theory of two temperature thermoelasticity with two phase-lags in which the theory of heat conduction in deformable bodies depends on two distinct temperatures – the conductive temperature and the thermodynamic temperature. A generalized heat conduction law with dual-phase-lag effects was proposed by Tzou (1995) for the purpose of considering the delayed response in times due to the microstructural interactions in the heat transport mechanism. Recently, Quintanilla (2008) has proposed to combine this constitutive equation with a two temperature heat conduction theory and has proved that a dual-phase-lag theory with two temperatures is a well-posed problem. In the present work we consider the basic equations concerning this dual-phase lag theory of two temperature thermoelasticity and make an attempt to establish some important theorems in this context. A uniqueness theorem has been established for a homogeneous and isotropic body. An alternative characterization of mixed boundary initial value problem is formulated and a variational principle as well as reciprocal principle have been established.     

Variational and Reciprocal Principles in Two-Temperature Generalized Thermoelasticity

The aim of the present work is to establish a variational principle of convolutional type and a reciprocal principle in the context of linear theory of two-temperature generalized thermoelasticity, for a homogeneous and isotropic body.     

Two-Temperature Generalized Thermoelasticity with Variable Thermal Conductivity

In this work, the consideration of variable thermal conductivity as a linear function of temperature has been taken into account in the context of two-temperature generalized thermoelasticity (Youssef's model). The governing equations have been derived and used to solve the one-dimensional problems of an elastic half-space. The governing equations have been cast into a matrix form by using Bahar–Hetnarski method, and Laplace transform is used to get the general solution for any set of boundary conditions. The solution has been applied for a thermally shocked medium that has no strain on its bounding plane. The numerical inversion of the Laplace transform has been calculated by using the Riemann-sum approximation method. The distribution of the conductive temperature, the thermo-dynamical temperature, the strain, the displacement, and the stress have been shown graphically with some comparisons.     

Normal Mode Method for Two-Temperature Generalized Thermoelasticity Under Thermal Shock Problem

The theory of two-temperature generalized thermoelasticity, based on Youssef's theory, was used to solve boundary value problems of one-dimensional generalized thermoelasticity half-space by heating its boundary with different types of heating. The governing equations are solved using new mathematical methods within the purview of the Lord-?hulman (L-S) theory and the classical dynamical coupled theory (CD). The general solution obtained is applied to a specific problem of a half-space subjected to one type of heating—thermal shock type. The separation of variables method is used to get the exact expressions for distributions of displacement, the stresses, and temperature distribution. Variations of the considered functions through the horizontal distance are illustrated graphically. Comparisons are made with results between the two theories. Numerical work is also performed for a suitable material and results are discussed, specifically the conductive temperature, the dynamical temperature, and the stress and strain distributions are shown graphically when discussed.     

On the Harmonic Vibrations in Linear Thermoelasticity Without Energy Dissipation

In the present paper we consider a prismatic cylinder occupied by an anisotropic homogeneous compressible linear thermoelastic material that is subject to zero body force and heat supply and zero displacement and thermal displacement on the lateral boundary. The motion is induced by a harmonic time–dependent displacement and thermal displacement specified pointwise over the base. We establish some spatial estimates for an appropriate cross–sectional measure associated with the amplitude of the corresponding harmonic vibration in the high frequency range that describe how the amplitude evolves with respect to the axial distance to the excited base. In fact, in the high frequency range we establish certain algebraical estimates that predict decay or growth slower than that for the low frequency range.     

Effects of Rotation on Harmonic Plane Waves Under Two-Temperature Thermoelasticity

The present article is aimed at a detailed analysis of the effects of rotation on the propagation of harmonic plane waves under two-temperature thermoelasticity theory. We consider a homogeneous and isotropic elastic medium that is rotating with uniform angular velocity. After formulating the problem, we obtain the dispersion relations for the longitudinal and transverse plane waves propagating in the medium and the solutions of dispersion relations are obtained analytically for high-frequency as well as for low-frequency values. On the basis of these solutions, the asymptotic expressions of the important characterizations of the wave fields for high and low frequency values are derived. To observe the behavior of the wave characterizations for the intermediate values of frequency and to examine the effects of rotation on them, computational work is carried out and numerical values of different wave fields for various values of frequency and for various angle of rotation are computed. The results are shown graphically. An in-depth analysis of the effects of rotation on plane wave is presented on the basis of our analytical and numerical results and the several significant points are highlighted. A comparative analysis of our results with the corresponding results in absence of rotation of the body is also presented.     

Convolutional Variational Principle,Reciprocal and Uniqueness Theorems in Linear Fractional Two-Temperature Thermoelasticity

Two general models of fractional heat conduction law for non-homogeneous anisotropic elastic solid is introduced and the constitutive equations for the two-temperature fractional thermoelasticity theory are obtained, uniqueness and reciprocal theorems are proved and the convolutional variational principle is established and used to prove a uniqueness theorem with no restrictions imposed on the elasticity or thermal conductivity tensors except symmetry conditions. The two-temperature dynamic coupled, Lord-Shulman and fractional coupled thermoelasticity theories result as limit cases. The reciprocity relation in case of quiescent initial state is found to be independent of the order of differintegration.     

On the Spatial Behavior in Thermoelasticity Without Energy Dissipation

In the present paper we consider a prismatic cylinder occupied by an anisotropic homogeneous compressible linear thermoelastic material that is subject to zero body force and heat supply and zero displacement and thermal displacement on the lateral boundary. The motion is induced by a harmonic time-dependent displacement-thermal displacement specified pointwise over the base. We establish some spatial estimates for some appropriate cross-sectional measures associated with the transient and steady state solutions that describe how they evolves with respect to the axial distance to the excited base. For the transient solutions a domain of influence is observed inside which a measure of the solution decays exponentially with respect to the axial variable, while outside of the domain the solution vanishes. For the harmonic in time vibrations we establish spatial estimates describing how the amplitude of vibration evolves with respect to the axial variable, provided the frequency is lower than a critical value. All results are established for the thermoelastic materials whose constitutive coefficients satisfy strong ellipticity conditions.     

A Problem on Thermoelastic Interactions in an Infinite Medium with a Cylindrical Hole in Generalized Thermoelasticity III

The aim of the present article is to study the thermoelastic interactions in an infinite elastic medium with a cylindrical hole in the context of generalized thermoelasticity III, recently developed by Green and Nagdhi [1 A. E. Green and P. M. Nagdhi , On Undamped Heat Waves in an Elastic Solid , J. Thermal Stresses , vol. 15 , pp. 253 – 264 , 1992 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]. The boundary of the hole is assumed to be stress free and is subjected to a ramp type heating. In order to make a comparison between this thermoelastic model with other thermoelastic models, the problem is formulated on the basis of three different theories of thermoelasticity, namely: the extended thermoelasticity proposed by Lord and Shulman [2 H. W. Lord and Y. Shulman , A Generalized Dynamical Theory of Thermoelasticity , J. Mech. Phys. Solids , vol. 15 , pp. 299 – 309 , 1967 .[Crossref], [Web of Science ®] , [Google Scholar]], the thermoelasticity without energy dissipation (Green and Nagdhi [3 A. E. Green and P. M. Nagdhi , Thermoelasticity Without Energy Dissipation , J. Elasticity , vol. 31 , pp. 189 – 209 , 1993 .[Crossref], [Web of Science ®] , [Google Scholar]]) and thermoelasticity with energy dissipation (thermoelasticity type III [1 A. E. Green and P. M. Nagdhi , On Undamped Heat Waves in an Elastic Solid , J. Thermal Stresses , vol. 15 , pp. 253 – 264 , 1992 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]]) in a unified way. The solutions for displacement, temperature and stresses are obtained with the help of Laplace transform procedure. Firstly the short time approximated solutions for three different theories have been obtained analytically. Then following a numerical method for the inversion of Laplace transforms, the numerical values of the physical quantities are also computed for the copper material and results are displayed in graphical forms to compare the results obtained from different models of generalized thermoelasticity.     

Thermoelastic Wave with Energy Dissipation in an Unbounded Medium with a Spherical Cavity

The generalized linear theory of thermoelasticity has been used to study waves emanating from the boundary of a spherical cavity in a homogeneous and isotropic infinite thermoelastic body. The basic equations are written in the form of a vector-matrix differential equation and solved by an eigenvalue approach. Solutions in closed form have been achieved in the Laplace transform domain, and the inversion in the space-time domain for the field variables are done numerically. Finally, the results are analyzed and represented graphically.     

Dissipation Function in Hyperbolic Thermoelasticity

The equations governing two hyperbolic thermoelasticity theories (with one or with two relaxation times) are examined from the standpoint of thermodynamics with internal variables, i.e., employing the free energy and the dissipation function explicitly. In the case of thermoelasticity with one relaxation time (the Lord–Shulman model), the derivation relies on an extended state space and a representation theory of Edelen. In the case of thermoelasticity with two relaxation times (the Green–Lindsay model), the derivation process by simply using the thermodynamic orthogonality of Ziegler.     

The Effect of Gravity Field on the Plane Waves in a Fiber-Reinforced Two-Temperature Magneto-Thermoelastic Medium Under Lord-Shulman Theory

The aim of the present work is to investigate the influence of the gravity field on the plane waves in a linearly fiber-reinforced magneto-thermoelastic isotropic medium in the context of the two-temperature theory of generalized thermoelasticity under Lord-Shulman model. The problem has been solved numerically using the normal mode analysis together with the eigenvalue approach. Numerical results for the conductive temperature, thermodynamic temperature, displacement components and the stresses are represented graphically and the results are analyzed. The graphical results indicate that the effect of the two-temperature parameter, gravity field and the magnetic field on the plane waves in the fiber-reinforced thermoelastic medium are very pronounced. Comparisons are made with the results in the presence and absence of the two-temperature parameter, gravity field and the magnetic field. Such problems are very important in many dynamical systems.     

Thermoelastic Material Response Due to Laser Pulse Heating in Context of Four Theorems of Thermoelasticity

This article describes the study of induced temperature and stress fields in an elastic half-space in the context of classical coupled thermoelasticity (Biot) and generalized thermoelasticity (Lord–Shulman, Green–Lindsay and Green–Naghdi) in a unified system of equations. The medium is considered to be made of an isotropic homogeneous thermoelastic half-space. The bounding plane of the surface is heated by a non-Gaussian laser beam with pulse duration of 2 ps. An exact solution of the problem is first obtained in Laplace transform space. Because the response is of more interest in the transient state, the inversion of Laplace transforms were carried out numerically. The derived expressions were computed numerically for copper, and the results are presented in graphical form.     

The Effect of Initial Stress on Thermoelastic Rotating Medium with Voids Due to Laser Pulse Heating with Energy Dissipation

This article reports studies on the rotation of initially stressed thermoelastic medium with voids subjected to thermal loading due to laser pulse. The bounding plane surface is heated by a non-Gaussian laser beam. The medium rotated with a uniform angular velocity. The problem was studied in the context of Green-Naghdi (G-N) theory of types II and III using the normal mode analysis method. The comparisons between the both types II and III of (G-N) theory in the presence and the absence of the rotation, the initial stress and for two values of time for considered medium are shown graphically.     

Type II Thermoelasticity. A New Aspect

In this paper we investigate the thermomechanical theory of thermoelasticity without energy dissipation. We consider the case where the assumptions on the constitutive tensor are different from the usual ones in the current literature. These assumptions are suitable because they correspond to situations where the material is prestressed and has initial entropy flux. When we restrict our attention to the one-dimensional problem, we first establish a condition on the coefficients to guarantee the hyperbolicity of the system. Then we prove that the problem is well posed and stable. When the hyperbolicity condition is not satisfied we prove that the problem is ill-posed. We also prove a similar result when the dimension is greater than one whenever the domain satisfies a certain condition. Though we cannot expect in general the stability of solutions in dimension greater than one, we prove the uniqueness and stability of radial solutions in the general case. To obtain the results, we need the use of a conservation law which has not previously been considered in the literature. Similar results are not known for other thermoelastic theories.     

Two-Temperature Theory in Linear Micropolar Thermoviscoelastic Anisotropic Solid

The reciprocal theorem is proved and the convolutional variational principle is established for the two-temperature linear anisotropic and inhomogeneous micropolar thermoviscoelastic solid. A proof, based on variational principle of a uniqueness theorem with no restrictions imposed on the relaxation or thermal conductivity tensors except symmetry conditions, is given. The theorems for classical dynamic coupled micropolar-thermoviscoelasticity theory result as special case when the two temperatures coincides.     

On the Fundamental Solutions of Generalized Thermoelasticity with Three Phase-Lags

The aim of the present work is to derive the fundamental solutions in the linear theory of generalized thermoelasticity with three phase-lags, recently introduced by Roychoudhuri (2007). We consider the case of homogeneous and isotropic bodies and present a solution of the Galerkin-type of field equations. The solution is then used to determine the effects of concentrated loads and heat sources in an unbounded medium. The fundamental solutions of the field equations in case of steady vibrations are also derived.     

On the Spatial Behavior of Constrained Motion in Type III Thermoelasticity

Green and Naghdi proposed three theories of thermoelasticity labelled as type I, II and III, respectively. Here we investigate the spatial behavior of solutions of a problem determined by the type III thermoelasticity. We study the growth/decay alternative in the spatial variable for the linear theory in the case that the thermoelastic deformations at the instant T are constrained to be proportional to the ones at the instant zero. In the case that the solutions decay, an upper bound for the amplitude term by means of the data of the problem can be obtained. We sketch how to obtain it. We finish by noting (through examples) that for several families of cases the solutions can be periodic in the spatial variable and the Phragmen–Lindelof alternative is not satisfied.     

掺气分流墩设施消能效果的试验研究

本文研究了在消力塘上游溢流面增设掺气分流墩设施后的消能效果,研究结果证明,掺气分流墩设施具有非常显著的消能效率,可以大幅度的缩减消力塘的容积,因而可以大大地节省工程造价。