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.     

A thermoelastic spherical shell with and without energy dissipation

In this work, we apply the Green and Naghdi generalized thermoelasticity theory to a one-dimensional problem of distribution of thermal stresses and temperature in a generalized thermoelastic medium in the form of a spherical shell subjected to sudden change in the temperature of its external boundary. The results are compared to the generalized thermoelasticity theory with one relaxation time. Numerical results are computed and represented graphically for temperature, displacement, and stress distributions.     

Thermoelastic behavior of elastic media with temperature-dependent properties under transient thermal shock

This article is concerned with thermoelastic behavior of an elastic media with temperature-dependent properties. The formulations of anisotropic media with variable material properties are proposed by the Clausius inequality and generalized theory of thermoelasticity with one relaxation time, where the higher-order expansion of the Helmholtz free energy with respect to increment temperature is used to obtain the relations between each parameter and real temperature. The governing equations of isotropic media with temperature-dependent properties are obtained based on these formulations. The problem of a half-space formed of an isotropic media with variable material properties and subjected to a sudden temperature rise in the boundary has been conducted. The propagations of thermoelastic wave and thermal wave, as well as the distributions of displacement, temperature, and stresses in the different cases, including constant properties and variable properties with specific temperature and real temperature, are obtained and plotted to reveal the effect of variable material properties on thermoelastic behavior.     

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.     

Fractional Ultrafast Laser–Induced Thermo-Elastic Behavior In Metal Films

In this work, a new mathematical model of modification heat conduction for an isotropic generalized thermoelasticity is derived using the methodology of fractional calculus. Some theorems of generalized thermoelasticity follow as limit cases. An ultrafast fractional thermoelasticity model utilizing the modified hyperbolic heat conduction model and the generalized fractional thermoelastic theory was formulated to describe the thermoelastic behavior of a thin metal film irradiated by a femtosecond laser pulse. The temporal profile of the ultrafast laser was regarded as being non-Gaussian. An analytical–numerical technique based on the Laplace transform was used to solve the governing equations and the time histories of the temperature, displacement and stress in a gold film were analyzed. Some comparisons have been shown in figures to estimate the effects of the relaxation time and fractional order parameter α on all the studied fields.     

Analysis of thermoelastic response in functionally graded hollow sphere without load

In this article, a model concerning free vibrations of spherically symmetric, thermoelastic, isotropic, and functionally graded sphere has been developed and analyzed in the context of linear theory of generalized thermoelasticity with one relaxation time. Laplace transform has been used to solve the problem which yields natural frequencies of free vibrations without performing inversion of the transform. The analytical results for coupled, uncoupled, and homogeneous spheres have been deduced as special cases of the general case. Natural frequencies of first 10 modes of vibrations have been obtained for different values of grading index of cobalt material regarding coupled thermoelastic, generalized thermoelastic, and elastic functionally graded spheres. The frequency shift and thermoelastic damping for Fourier and non-Fourier processes of heat propagation, temperature change, radial, and hoop stresses have been presented graphically. It has been analyzed here that grading index parameter helps in detecting the strength of signals in such material devices and the thermal relaxation time contributes in improving the quality of signals. The analysis also leads to the fact that grading index parameter is useful from design point of view and it can be tailored to specific applications for controlling the stress.     

EFFECTS OF THERMAL RELAXATIONS ON THERMOVISCOELASTIC INTERACTIONS IN AN UNBOUNDED BODY WITH A SPHERICAL CAVITY SUBJECTED TO A PERIODIC LOADING ON THE BOUNDARY

Thermoviscoelastic interactions in an infinite homogeneous viscoelastic medium with a spherical cavity are studied. The cavity surface is subjected to a periodic loading and zero temperature change. The classical dynamical theory of thermoelasticity as well as the generalized theories of thermoelasticity are applied to consider the thermoelastic coupling. The analytical expressions for the closed-form solutions of displacement, temperature, and stresses are obtained; and the thermal relaxation effects on the interactions are studied to compare the three theories. The numerical values of the physical quantities are computed for a suitable material. The results are presented graphically to illustrate 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.     

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.     

Thermoelastic damping of graphene nanobeams by considering the size effects of nanostructure and heat conduction

Thermoelastic damping of nanobeams by considering the size effects of nanostructure and heat conduction is studied herein. The size effect of nanostructure is investigated based on Euler–Bernoulli beam assumptions in the framework of nonlocal strain gradient elasticity, and the size dependence of heat conduction is taken into account by incorporating phase-lagging and nonlocal effects. Closed-form solutions of thermoelastic damping and quality factor characterized by thermoelastic coupling are derived. Graphene nanoribbon is chosen as a nanobeam. The effects of relaxation time, aspect ratio, elastic modulus, thermal expansion, and thermal conductivity on quality factor of graphene nanobeams are discussed in detail.     

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.     

Thermoelastic Damping in Nanomechanical Resonators with Finite Wave Speeds

ABSTRACT

The operation of micro-/nanobeams vibrating at very high frequencies, such as encountered in micro-/nanoelectromechanical systems (MEMS/NEMS), hinges on the minimization of intrinsic material losses. We study the associated thermoelastic damping in such beams from the standpoint of a generalized theory of thermoelasticity with one relaxation time. Some of our results relate to: (i) the cooling (instead of heating) in the compressed surface of the beam; (ii) the existence of not one damping peak appearing in the classical theory, but many peaks, with a decreasing amplitude as the frequency tends to infinity; (iii) the relevance of thermoelasticity with finite wave speeds for frequencies on the order of 10¹² Hz.     

An investigation on thermoelastic interactions under an exact heat conduction model with a delay term

The present work is concerned with a very recently proposed heat conduction model: an exact heat conduction model with a single delay term. A generalized thermoelasticity theory was proposed by Roy Choudhuri based on the heat conduction law with three-phase-lag effects for the purpose of considering the delayed response in time due to the microstructural interactions in the heat transport mechanism. However, the model defines an ill-posed problem in Hadamard sense. Quintanilla has recently proposed to reformulate this heat conduction model as an alternative heat conduction theory with a single delay term and subsequently, Leseduarte and Quintanilla investigated the spatial behavior of the solutions for this theory and they extended the results to a thermoelasticity theory by considering the Taylor series approximation of the equation of heat conduction with one delay term. In the present work, we consider the thermoelasticity theory based on this newly proposed heat conduction model and investigate a problem of thermoelastic interactions. State-space approach is used to formulate the problem and the formulation is then applied to a problem of an isotropic elastic half-space with its plane boundary subjected to sudden increase in temperature and zero stress. The integral transform method is applied to obtain the solution of the problem. A detailed analysis of analytical results is provided by finding the short-time approximated solutions of different field variables analytically and comparing the results of the present model with the corresponding results reported for other existing theories. An attempt has also been made to illustrate the problem and numerical values of field variables are obtained for a particular material. Results are analyzed with different graphs. To the best of the author\textquoteright s knowledge, this thermoelastic model is not yet investigated by any researcher in this direction.     

Exact solution for thermal damping of functionally graded Timoshenko microbeams

This article deals with the thermoelastic damping problem in a functionally graded (FG) Timoshenko microbeam. Thermal and mechanical properties of the microbeam vary in the thickness direction according to the power law relation. Employing Timoshenko beam theory, the governing dynamic equation coupled with thermal effects of the FG microbeam is developed. Afterwards, Using the Taylor series expansion for material properties, the heat conduction equation is solved analytically for temperature in the form of a power series. The free vibration of the FG microbeam is analyzed to achieve the natural frequencies and thermal damping ratio of the FG microbeam. The effect of FG index on the thermoelastic damping ratio is investigated in different aspect ratios. Also comparison studies are made between the results obtained from the models based on the Euler–Bernoulli and Timoshenko beam theories.     

BOUNDARY ELEMENT ANALYSIS OF GREEN AND LINDSAY THEORY UNDER THERMAL AND MECHANICAL SHOCK IN A FINITE DOMAIN

A boundary element formulation based on the Laplace transform method is developed for transient coupled thermoelasticity problems with relaxation times in a two-dimensional finite domain. The dynamic thermoelastic model of Green and Lindsay is selected for the present study. The Laplace transform method is applied to the time domain, and the resulting equations in the transformed field are discretized using the boundary element method. The nodal dimensionless temperature and displacements in the transformed domain are inverted to obtain the actual physical quantities using the numerical inversion of the Laplace transform method. This work considers the Green and Lindsay theory of thermoelasticity with the thermal and mechanical loading in a finite domain. The creation and propagation of elastic and thermoelastic waves in a finite domain and their effects on each other are investigated. Different relaxation times are chosen to briefly illustrate the events that take place in GL theory. Details of the formulation and numerical implementation are also presented.     

Thermal Fracture of an Interpenetrating Phase Composite: Effects of Local Thermal Nonequilibrium

When subjected to thermal shocks, an interpenetrating phase composite may undergo significant, long range temperature difference between the constituent phases due to the interconnected microstructural networks, which facilitate faster heat transfer in the phase of higher thermal diffusivity. This temperature differential may alter the macroscopic temperature field thereby inducing additional thermal stresses in the composite. This work presents a local thermal nonequilibrium (LTNE) thermoelasticity theory for interpenetrating phase composites. In the LTNE thermoelasticity theory, the temperatures of the constituent phases are governed by the LTNE heat conduction equations based on the continuum theory of mixtures. A weighted average of temperatures for the constituents is employed in the thermoelastic constitutive equations of the homogenized composite. The model is subsequently applied to an infinite composite strip with an edge crack subjected to a thermal shock. Asymptotic solutions of temperature, thermal stress, and thermal stress intensity factor are obtained using the Laplace transform technique. The numerical results for an interpenetrating Al₂O₃/Al composite show that the temperature and thermal stress fields of the LTNE theory deviate from those of the classical theory. More importantly, the thermal stress intensity factor is reduced by considering the LTNE effect, which indicates that interpenetrating networks enhance the thermal fracture resistance of ceramic-metal composites.     

THERMAL RELAXATION TIMES EFFECT ON RAYLEIGH WAVES IN GENERALIZED THERMOELASTIC MEDIA

Rayleigh waves in a half-space exhibiting generalized thermoelastic properties based on Green-Lindsay (G-L), Lord-Shulman (L-S), and classical dynamical coupled (C-D) theories are discussed. The phase velocity of Rayleigh waves in the previous three different theories has been obtained. A comparison is carried out between the phase velocities of Rayleigh waves, displacements, stresses, and temperature as calculated from the different theories of generalized thermoelasticity. The C-D theory is recovered as a special case. It appears, in particular, that the results obtained from G-L theory tend to those of L-S theory as the values of the two relaxation times become closer to each other. The second relaxation time is well pronounced when it becomes larger than the first one. Furthermore, it is found that the thermal relaxation times decrease the speed of the elastic waves and modify the phase velocities of the Rayleigh waves. The results obtained and the conclusions drawn are discussed numerically and illustrated graphically. Relevant results of previous investigations are deduced as special cases.     

A Domain of Influence Theorem for Thermoelasticity with Three-Phase-Lag Model

We establish the theory of domain of influence for a potential-temperature disturbance under the generalized theory of thermoelasticity with three-phase-lag model. For a finite time t > 0, it has been proved that the potential due to the displacement and the temperature fields does not produce any disturbances outside a bounded domain and the domain of influence is dependent on the support of load, the thermoelastic coupling constant and the phase-lag parameters. It has been shown that the domain of influence of the present case reduces to the domain of influence for classical thermoelasticity theory and the thermoelasticity theory with one relaxation parameter.     

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.