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This paper develops a modeling approach for the simulation of thermal contacts delimited by curvilinear edges. The study introduces a correction for semi-analytical methods based on the half-space assumption to account for thermal boundary conditions. The approach establishes a correction radius to control the heat flux and adjusts the corrective heat load to handle the thermoelastic normal stress on free surfaces. Compared to finite element analysis (FEA), the displacement calculated with the approach demonstrates a 99% improvement. A comparison of the complete thermal modification and normal stress elimination with FEA for a thermal contact shows more than 73% improvement, leading to an error margin of under 5%. The calculation burden is also drastically reduced as compared to that of an FEA.

This paper addresses the wave propagation problem within the context of the theory of materials with triple porosity under local thermal non-equilibrium. The time-harmonic wave solution is studied and the dispersion relation is explicitly established as an eighth-degree algebraic equation. It is shown that there are two shear waves undamped in time, that are non-dispersive and unaltered by the presence of the pore system or by thermal effects. Also, there are seven longitudinal wave solutions that are dispersive and damped in time: one longitudinal quasi-elastic wave, three longitudinal quasi-pore standing waves and three other longitudinal quasi-thermal standing waves. An illustrative simulation on a material like Berea sandstone shows that the effects of coupling triple porosity with local thermal imbalance and mechanical deformations leads to an increase in the propagation speed of the longitudinal quasi-elastic wave accompanied by an increase in its rate of decrease over time, as well as to a relaxation of the decrease in time for the quasi-pore and quasi-thermal standing waves.

In this work, we present a method to simultaneously detect the mass and the position of an attached microparticle using a bridge-type micro-beam resonant mass sensor under axial pre-tension. Simultaneous detection of the mass and the position is innovatively enabled by deliberately varying the axial pre-tension, utilizing only the fundamental frequency. As an important dissipation mechanism, thermoelastic damping (TED) in micro-/nano-resonators imposes an upper limit on the quality factor and the minimum detectable mass of resonant mass sensors. Accordingly, TED of micro-/nano-beam resonant mass sensors with axial pre-tension is analyzed by the thermal energy method, and heat conduction along both the thickness direction and axial direction is considered. The effects exerted by the pre-tension on the sensitivity and TED of resonant mass sensors are systematically investigated. Numerical results reveal that axial pre-tension not only enhances the mass sensitivity but also reduces the TED of the resonant mass sensor. Furthermore, effects of other factors, such as aspect ratio, vibration mode, mass and position of the attached particle, on TED are examined as well. This work provides a theoretical guidance to the extraction of mass of microparticles, cells and molecules by experiment.

This article describes the impacts of variable thermal conductivity and diffusivity on an infinite thermoelastic diffusion circular plate of finite width with a heat source due to axisymmetric thermal and chemical potential loadings in the light of two-temperature generalized thermoelastic diffusion theory. The thermal conductivity and diffusivity are assumed to be linear functions of thermodynamic temperature and concentration, respectively. The governing equations are transformed into the linear form by applying Kirchhoff’s transform. These equations are solved by using the Laplace–Hankel transform technique. In the transformed region, the closed form expressions for conductive and thermodynamic temperatures, displacement and stress components, concentration, and chemical potential are obtained. To transform the solutions to the original domain, a numerical inversion technique is applied. Numerical results for thermodynamic and conductive temperatures, normal stress component, and chemical potential are depicted graphically to illustrate the impacts of two temperatures, ramping time parameter, variable thermal conductivity and diffusivity. A validation of the obtained results is also presented.

This paper investigates the axisymmetric thermoelastic contact of a functionally graded material (FGM) coated half-space. A rigid insulated punch rotates on the surface of the FGM coating with a constant angular velocity. The frictional heating is generated within the contact region. The thermoelastic properties of the coating vary exponentially along the thickness direction. By using the Hankel integral transform, the problem is reduced to Cauchy singular integral equations, which are solved numerically to obtain the normal contact stress, radial stress and surface temperature. The effects of the friction coefficient, angular velocity, gradient index, coating thickness and punch geometry on the surface thermoelastic contact characteristics are investigated and discussed in detail.

A new model of equations of generalized thermoelasticity for an isotropic medium with mechanical properties that are dependent on temperature is established. The present problem is a generalization of the three-phase-lag model, Lord and Shulman's coupled theory. The elasticity modulus is a reference temperature function which is linear. Analytical expressions of the considered variables are obtained by using the Laplace–Fourier transforms technique. The results are analysed in a deeper manner by comparing them with unique cases of absence of the magnetic field, temperature-dependent properties of the body, and two types of mechanical loads. The most significant points are highlighted.

It is prominent that the Galerkin-type representation plays a dominant role in probing various challenges of mathematical physics, continuum mechanics and occupies an important place in the field of partial differential equations (PDEs). Thus, the contemporary analysis of different boundary value problems (BVPs) in thermoelasticity theory commonly begins by analyzing the Galerkin-type representation of the field equations in terms of elementary functions (harmonic, biharmonic, and metaharmonic, etc). This work is aimed at formulating the representation of a Galerkin-type solution by means of elementary functions for the recently developed Moore–Gibson–Thompson (MGT) thermoelasticity theory. The MGT theory is a generalized form of the Lord–Shulman (LS) model as well as of the Green–Naghdi (GN) thermoelastic model. Here, we establish a theorem and derive the Galerkin-type solution for the basic governing equations under this theory. Later, the Galerkin representation of a system of equations for steady oscillations is derived. Based on this representation, we finally establish the general solution (GS) for the system of homogeneous equations of stable oscillation, neglecting the extrinsic body force and extrinsic heat supply.

The present paper is devoted to the development of the method for searching an approximate solution of non-stationary problems on a variable interval. This method was first proposed by L.I. Slepyan. Here, the field of its applicability has been expanded to systems of equations, including equations of both hyperbolic and parabolic type. We describe the procedure of such approach in detail on a number of classical partial differential equations and compare the obtained results with exact analytical solution. Also, we consider a dynamic non-coupled thermoelastic problem. The method of expansion on a variable interval allows us to estimate the material response to a thermal disturbance at a large distance from the source.

In this article, the propagation of Rayleigh surface waves in a piezothermoelastic transversely isotropic layer lying over a piezothermoelastic transversely isotropic half-space is investigated in the context of the Green–Naghdi model type III of hyperbolic thermoelasticity. The secular equation of Rayleigh surface waves is derived, and different cases are discussed. Phase velocity, attenuation coefficient and specific loss of surface waves are computed and presented graphically with respect to frequency, and a comparison of different wave characteristics for classical and generalized thermoelastic models is presented in the figures.

The present paper is focused on the Moore–Gibson–Thompson (MGT) thermoelasticity theory. The MGT thermoelasticity theory is a generalized form of the Lord–Shulman (LS) thermoelasticity theory as well as the Green-Naghdi thermoelasticity theory with energy dissipation (GN-III). The present work is aimed at establishing the domain of influence results in the context of this new thermoelasticity theory. We consider a mixed boundary-initial value problem representing natural stress–heat-flux disturbance inside an isotropic and homogeneous medium. We establish an identity regarding this present problem. Further, we derive the domain of influence theorem based on this identity under the MGT thermoelasticity theory. From this theorem, we conclude that for prescribed bounded support of thermomechanical loading and for a finite time, the disturbance generated by the pair of stress and heat flux vanishes outside a bounded domain. It is also analyzed that the domain of influence relies on the thermoelastic material parameters. We further compare the present domain of influence results with the corresponding results of LS thermoelasticity theory.

The paper deals with Rayleigh wave propagation in a nonlocal thermoelastic layer, and the layer is lying over a nonlocal thermoelastic half-space. The problem is treated in the context of Eringen’s nonlocal thermoelasticity and Green–Naghdi model type III of hyperbolic thermoelasticity. The frequency equation of Rayleigh waves is derived, and different cases are also discussed. The effect of the nonlocal parameter on phase velocity, attenuation coefficient, specific loss, and penetration depth is presented graphically.

The present paper studies the fundamental solution and plane wave propagation in swelling porous thermoelastic material. It is observed that there exist three longitudinal waves and two transversal waves propagating with different velocities. Numerical computations are performed to obtain the phase velocity and attenuation coefficients of plane waves in the medium and the values so obtained are presented graphically. By means of elementary functions for swelling porous thermoelastic medium in case of steady oscillations, the fundamental solution is obtained. Some basic properties of this solution are established and special cases are also deduced.

Microbeam resonators are widely used due to their scientific and engineering applications. The accurate prediction of thermoelastic damping (TED) is necessary to evaluate the performance of resonators at micro- and nanoscales with less energy dissipation. This article aims to present an analytical method for analyzing TED and dynamic behavior of microbeam resonators based on the Moore–Gibson–Thompson (MGT) generalized thermoelasticity theory. The finite Fourier sine transform and Laplace transform methods are used to solve the coupled thermoelastic equations. The analytical solutions are obtained for deflection and thermal moment of beams. The vibration responses of deflection and thermal moment are established in microbeams with simply supported and isothermal boundary conditions. The responses of deflection and thermal moment in beams are analyzed by comparing the results obtained under the MGT model with the corresponding results under the Lord–Shulman (LS) and Green–Naghdi (GN-III) models. The obtained results show that the amplitudes of deflection and thermal moment are attenuated, and the vibration frequency is increased due to the effect of thermoelastic coupling. It has been observed that the amplitudes of deflection under these three models are approximately the same, while the amplitude of thermal moment under the MGT model is higher than under the GN-III model and agrees with the LS model. It has been further noticed that TED depends on the size of the beams when the thermoelastic coupling effect is considered.

The temperature variation within two elastic bodies in perfect thermoelastic contact may cause the contact area to become convex and hence lead to a reduction in the size of the contacting surface. In receding thermoelastic contact problems, the final size of the contact zone is independent of the applied loads, being only affected by the thermomechanical properties of the solids. However, the final size of the contact zone can also be affected by the conductive and convective boundary conditions at the separation zones of the contacting surfaces. In those regions, the heat flux is a function of the separation between the solids, so the thermal and the thermoelastic problems are highly coupled. For this reason, this work studies the three-dimensional receding thermomechanical contact problem under conductive and convective boundary conditions at the interstitial zones of the contact area. After the validation of the numerical scheme presented to solve this problem, several examples are presented and discussed in detail. The results reveal that conductive and convective interstitial boundary conditions have a significant effect not only on the size of the contact zone, but also on the resulting tractions, temperature and heat flux distributions.

The present article deals with the propagation of Rayleigh surface waves in a homogeneous orthotropic medium based on Eringen’s nonlocal thermoelasticity. This thermoelastic problem is studied under the purview of Green–Naghdi model type III of hyperbolic thermoelasticity in the presence of voids. The normal mode analysis is employed to obtain a vector matrix differential equation which is then solved by an eigenvalue approach. The frequency equations for different cases are derived. The path of surface particles during Rayleigh wave propagation is found to be elliptical. In order to illustrate the analytical developments, the numerical solution is carried out and the computer-simulated results with respect to phase velocity, attenuation coefficient and specific loss are presented graphically.

In this article, a novel mathematical model is theoretically developed by incorporating a memory-dependent derivative (MDD) into the temperature-rate-dependent thermoelasticity theory (Green–Lindsay) on three mathematical aspects in order to study the spatial behavior of the thermal signals in an isotropic, homogeneous, thermoelastic continuum. Firstly, to ensure the thermodynamic consistency of the proposed model, the constitutive equations involving MDD are derived from the principle of thermodynamics or continuum mechanics in conjunction with the corresponding kinematic assumptions. Secondly, to prove the finite propagation speeds of the thermal signals of the proposed model, a domain of influence theorem is established. Finally, to analyze the spatial propagation of the thermal signals inside the domain of influence, a spatial decay theorem is established. As an immediate outcome of this theoretical analysis, a uniqueness theorem of the proposed model is also derived.

The present investigation is concerned with the reflection of plane waves at the free surface of a homogeneous, isotropic, nonlocal, micropolar rotating thermoelastic medium. The entire thermoelastic medium is rotating with a uniform angular velocity. It is observed that there exist four coupled plane waves, which travel through the medium with distinct speeds. Using appropriate boundary conditions, the reflection coefficients and energy ratios of various reflected waves are computed numerically with the help of the software MATLAB. The numerical values of modulus of reflection coefficients are presented graphically to show the effects of nonlocal, rotation and micropolar parameters. It has been verified that during reflection phenomena, the sum of modulus of energy ratios is approximately equal to unity at each angle of incidence. The effect of micropolarity on the phase velocities is also observed and shown graphically.

The present work is devoted to the investigation of the free vibrations of a homogeneous isotropic nonlocal thermoelastic cylinder with void. Time-harmonic variations are used to reduce the governing partial differential equations to a system of ordinary differential equations. The frequency equation for the continuation of vibrations for the mode numbers in the considered cylinder is deduced in closed form for traction-free and isothermal/thermally insulated boundary conditions. To observe the free vibration, the frequency equation is further studied by using the numerical iteration method with the help of MATLAB software. The numerically simulated results from the analytical solutions are shown graphically for the natural frequency, thermoelastic damping and the frequency shift against mode numbers for the nonlocal as well as the local thermoelastic cylinders in the presence and absence of the void.

The present paper is concerned with plane and surface wave solutions of the governing equation for an incompressible, homogeneous, transversely isotropic and rotating thermoelastic medium in the context of the Green–Naghdi theory. The velocity equation for homogeneous plane waves is obtained which indicates the existence of two coupled plane shear waves. The secular equation for Rayleigh waves is derived for thermally insulated as well as isothermal boundaries. The non-dimensional speeds of plane waves are plotted against the angle of incidence, non-dimensional anisotropy parameter and rotation/frequency ratio. The non-dimensional speed of Rayleigh surface waves is also computed against the non-dimensional anisotropy parameter and rotation/frequency ratio.

The vibration of a functionally graded axisymmetric nonlocal thermoelastic hollow sphere with dual-phase-lag effect is addressed in this paper. Surfaces of the sphere are assumed to be thermally insulated or isothermal and stress free. According to a simple power law, the material is assumed to be graded in the radial direction. The linear theory of modified thermoelasticity with a dual phase lag based on Eringen’s nonlocal elasticity is employed to model this problem. The Matrix Frobenius method of continued power series is introduced to derive the analytical solutions. The phase velocity relations for the existence of various modes of vibrations in the designed hollow sphere are derived in compact forms. In order to explore the attributes of vibrations, the fixed-point numerical iteration technique is used to solve the secular equations. The numerical computations for the material crust in respect of the natural frequencies, thermoelastic damping and the frequency shifting are presented graphically using MATLAB software tools.

The large deformation and stresses (normal and shear) of the graded nanotube-reinforced sandwich structure are numerically investigated under the influence of mechanical loading and a uniform temperature field. A higher-order nonlinear finite element model in conjunction with the direct iterative technique has been adopted for the solution purpose. Also, the structural distortion was modeled via the full-scale geometrical nonlinearity (Green–Lagrange strain) in the framework of higher-order displacement functions. Further, to replicate the actual operational conditions, the temperature-dependent properties of the individual material constituents (i.e., carbon nanotube and polymer) have been implemented in the current material modeling steps. The final deflections and stress values are evaluated via an own developed computer code using the currently proposed nonlinear mathematical formulation. The model accuracy and solution stability are checked by comparing the responses (deflection and stress) with available published results. Lastly, a variety of numerical examples is solved for different design parameters and deliberated in detail.

Based upon the Lord and Shulman theory of thermoelasticity, the new governing equations of thermoelasticity with nonlocal heat conduction are formulated. The above model is then employed to study the transient responses of a thermoelastic rod of finite length when subjected to a moving heat source. Both ends of the rod are assumed to be fixed and thermally insulated. The Laplace transform technique is used to obtain the analytical solutions for the field variables such as temperature, displacement, and stress. The inverse Laplace transform based on the Zakian algorithm is numerically implemented to obtain the solutions of the above physical variables in the space–time domain. Specific attention is paid to the study of the effect of the thermal nonlocal parameter on the distributions of the field variables and the speed of the moving heat source.

This work aims to highlight, by a series of simulations, the effect of the presence of pores in the material matrix on the transmission depth of a thermomechanical signal: two linear thermoelastic models are considered, for which the thermal response obeys a time-differential constitutive equation with two relaxation times, derived from the dual-phase lag theory. A porous material matrix, modeled on the basis of the Cowin–Nunziato theory, is compared with its counterpart without voids: this is done under hypotheses of linearity, compatible with the assumption to deal with very small spatial scales, in the order of the micrometer or nanometer. The results of such simulations are graphically presented and are completely unexpected; in the opinion of the authors, they can represent an interesting starting point for further discussions and insights.

This present work is devoted to the investigation of the transient phenomena for a fiber-reinforced medium with a cylindrical cavity in the context of the three-phase-lag model of generalized thermoelasticity with a new form of derivative of the Caputo–Fabrizio (CF) type in the heat transport equation, where the medium is under the action of an induced magnetic field. The Laplace transform is incorporated as a tool for the solution of the problem when the boundary of the cavity is exposed to harmonically varying heat with a constant angular frequency of thermal vibration. The numerical inversion of the Laplace transforms is computed using the Zakian method. Excellent predictive capability is demonstrated due to the presence of reinforcement, the angular frequencies on thermal vibrations, CF fractional parameter and magnetic field.

Ultrafast heating has gained serious attention due to technological advancements in recent years. The thermoelastic process accounting for the finite speed of thermal signals is therefore becoming increasingly important. With the help of extended thermodynamics, Yu et al. (Meccanica 53(10):2543–2554, 2018) have established a thermoelastic model by including the strain rate and temperature rate in the constitutive equations. This model is also an attempt to remove the discontinuity in the displacement field under temperature rate-dependent thermoelasticity theory. The present work seeks to derive the representation of a Galerkin-type solution in the context of this recently proposed model in terms of elementary functions. The representation theorem of the Galerkin-type solution of the system of equations for steady oscillation is also proved. In accordance with this theorem, we finally establish a theorem which expresses the general solution of the system of homogeneous equations of steady oscillation in terms of metaharmonic functions.

The present investigation is concerned with the reflection and transmission phenomena of plane waves between a rotating thermoelastic transversely isotropic solid half space and a fiber-reinforced thermoelastic rotating solid half space under the effect of a magnetic field. Green–Lindsay and Lord–Shulman theories have been employed for addressing the mathematical analysis. Relevant boundary conditions are considered in order to determine the reflection and transmission coefficients. The phase speeds, reflection and refraction coefficients and energy ratios are computed numerically with the help of MATLAB. The numerical values of the moduli of reflection and transmission coefficients are depicted graphically to estimate and highlight the effects of fiber reinforcement, rotation parameter and angular frequency. The effect of anisotropy on the velocities of the reflected and transmitted waves is also observed. One particular case of interest has also been inferred from the present investigation.

Based on the complex variable method, the decoupled thermoelastic problem of an infinite matrix containing an arbitrarily shaped inclusion subjected to plane deformations and uniform remote heat flux is studied in this paper. The shape of the inclusion is defined by a polynomial conformal mapping. Faber series and Fourier expansion techniques are used to solve the corresponding boundary value problems. Several numerical examples are presented to study the influence of the hardness and the heat conductivity of the inclusion on the concentration of the interfacial Von Mises stress and tangential stress in the matrix for a uniform remote uniaxial heat flux. It is shown that for given thermal expansion coefficients and given heat conductivities of the inclusion–matrix system, the Von Mises stress and the tangential stress in the matrix around the inclusion both increase significantly with increasing hardness of the inclusion whether the inclusion is softer or harder than the matrix. On the other hand, it is found that for given thermal expansion coefficients of the inclusion–matrix system, the Von Mises stress in the matrix around the inclusion decreases significantly with increasing heat conductivity of the inclusion, while the tangential stress concentration first decreases and then increases with increasing heat conductivity of the inclusion whether the inclusion is softer or harder than the matrix.

Inspired by the seminal works of Eshelby (Philos Trans R Soc A 244A:87–112, 1951, J Elast 5:321–335, 1975) on configurational forces and of Noll (Arch Ration Mech Anal 27:1–32, 1967) on material uniformity, we study a thermoelastic continuum undergoing volumetric growth and in a dynamical setting, in which we call the divergence of the Eshelby stress the Eshelby force. In the classical statical case, the Eshelby force coincides with the negative of the configurational force. We obtain a differential identity for the modified Eshelby stress, involving the torsion of the connection induced by the material isomorphism of a uniform body, which includes, as a particular case, that found by Epstein and Maugin (Acta Mech 83:127–133, 1990). In this identity, the divergence of the modified Eshelby stress with respect to this connection of the material isomorphism takes the name of modified Eshelby force. Moreover, we show that Eshelby’s variational approach (1975) can be used to formulate not only the balance of material momentum, but also the balance of energy. In this case, we find that what we call Eshelby power is the temporal analogue of the Eshelby force, and we obtain a differential identity for the modified Eshelby power. This leads to concluding that the driving force for the process of growth–remodelling is the Mandel stress. Eventually, we find that the relation between the differential identities for the modified Eshelby force and modified Eshelby power represents the mechanical power expended in a uniform body to make the inhomogeneities evolve.

Recent developments in nanostructured materials have led to the use of graphene sheets as resonators in advanced micro- and nanoelectromechanical systems. An important feature of micro- and nanoresonators is their ability to function with low power dissipation. The main intrinsic mechanism of energy loss in these advanced devices is thermoelastic damping (TED). In this article, we study TED effects in orthotropic graphene sheets of varied lengths operating at different temperatures using nonlocal elasticity theory. For this purpose, the fundamental thermoelastic relations are used to develop a system of coupled partial differential equations to describe the behavior of graphene nanoresonators. The orthotropic mechanical and thermal properties of graphene were taken into account in our model for zigzag and armchair chiralities operating at different temperatures. The free in-plane vibration of the graphene nanoresonator is analyzed using Galerkin method. Decidedly, we show that the developed system of equations is capable of describing the TED behavior of graphene nanoresonators along the two considered chiralities during thermoelastic vibration. Specifically, we examined the influence of size, chirality, and temperature upon thermoelastic damping, as measured by the so-called quality factor, of the graphene nanoresonator. Our results reveal that the nanoresonator experiences higher energy dissipation with increased temperature. They also reveal the dependence of the energy dissipation upon the size and chirality of the graphene sheet.

Enlightened by the Caputo fractional derivative, the present study deals with a novel mathematical model of generalized thermoelasticity to investigate the transient phenomena due to the influence of a non-Gaussian pulsed laser type heat source in a stress free isothermal half-space in the context of Lord–Shulman (LS), dual-phase lag (DPL), and three-phase lag (TPL) theories of thermoelasticity simultaneously. The memory-dependent derivative is defined in an integral form of a common derivative on a slipping interval by incorporating the memory-dependent heat transfer. Employing Laplace transform as a tool, the problem has been transformed to the space-domain, and it is then solved analytically. To get back all the thermophysical quantities as a function of real time, we use two Laplace inversion formulas, viz. Fourier series expansion technique (Honig in J Comput Appl Math10(1):113–132, 1984) and Zakian method (Electron Lett 6(21):677–679, 1970). According to the graphical representations corresponding to the numerical results, a comparison among LS, DPL, and TPL model has been studied in the presence and absence of a memory effect simultaneously. Moreover, the effects of a laser pulse have been studied in all the thermophysical quantities for different kernels (randomly chosen) and different delay times. Then, the results are depicted graphically. Finally, a comparison of results, deriving from the two numerical inversion formulas, has been made.

We study the consequences of thermoelastic coupling on heat and stress pulse propagation along equilibrium and nonequilibrium reference states. We use a generalized heat-transport equation accounting for relaxational and nonlinear effects. We compare the obtained results with those for heat pulses without thermoelastic coupling and with previous results obtained by using the relaxational Maxwell–Cattaneo equation for the heat flux without nonlinear terms. The difference of the speed of heat pulses along and against an imposed average heat flux in nonequilibrium states is also obtained.

Reflection and transmission phenomena at the plane interface between thermoelastic half-space and an initially stressed fiber-reinforced thermoelastic half-space are analyzed. The formulation is applied to generalized thermoelasticity based on the dual-phase-lag theory. Using convenient boundary conditions, the amplitude ratios and energy ratios for the reflected and transmitted waves are derived, and the numerical computations are carried out with the help of MATLAB programming. The numerical values of the modulus of reflection and transmission coefficients are presented graphically to highlight fiber-reinforced and initial stress parameter effects. The expressions of energy ratios obtained in explicit form are shown graphically as functions of angle of incidence. It has been verified that during reflection and transmission phenomena, the sum of energy ratios is equal to unity at each angle of incidence. The effect of anisotropy on velocities of reflected and transmitted waves is also depicted.

In this paper, we analyze, from the numerical point of view, a dynamic problem involving a thermoelastic body with diffusion, whose microelements are assumed to possess microtemperatures and microconcentrations. Using the linear theory, the mechanical problem is written as a coupled system of hyperbolic and parabolic partial differential equations for the displacement, temperature, chemical potential, microconcentrations, and microtemperatures fields. The variational formulation is derived, and it leads to a coupled system of parabolic linear variational equations, for which an existence and uniqueness result is stated. Then, using the finite element method and the implicit Euler scheme, fully discrete approximations are introduced. Stability properties and a priori error estimates are obtained, from which the linear convergence of the algorithm is deduced under suitable additional regularity conditions. Finally, some numerical simulations are presented to show the accuracy of the approximation and the behaviour of the solution.

This paper analyzes a thermal shock problem of an elastic half-space with a penny-shaped crack near the surface based on a fractional thermoelasticity theory. The embedded crack is assumed to be insulated. The Hankel transform and Laplace transform are employed to solve an initial-boundary value problem associated with a fractional partial differential equation. Explicit expressions of temperature and thermal stresses induced by the penny-shaped crack are obtained by solving a system of singular integral equations. Numerical results of the thermoelastic fields in the time domain are given by applying the numerical inverse Laplace transform. The temperature jump between the upper and lower crack surfaces and the thermal stress intensity factors at the crack front are illustrated graphically for various relaxation times and fractional orders, as well as the distance between the crack plane and the half-space surface. A comparison of the temperature, thermal stresses, and their intensity factors is made when adopting the fractional heat conduction model and the classical Fourier heat conduction model. Numerical results show that the temperature overshooting phenomenon may occur for the fractional heat conduction model, whereas it does not occur for the classical Fourier heat conduction model.

Thermoelastic damping (TED) is one of the main energy dissipation mechanisms in structures with small scales. On the other hand, the classical continuum theory is not capable of describing the mechanical behavior of small-scale structures. In this paper, small-scale effects on the thermoelastic damping in microplates are studied. To this end, the coupled governing equations of motion and heat conduction are obtained based on the non-classical continuum theory of the modified couple stress and the dual-phase-lag heat conduction model. By solving these coupled equations, an explicit expression including small-scale effects for calculating TED in microplates is derived. The results are compared with those given by the classical continuum and heat transfer theories. In addition, numerical results are presented to investigate the influences of some parameters on TED and critical thickness, such as microplate thickness, aspect ratio, boundary conditions, and the type of material.

The hot spot caused by frictional heat and thermal contact resistance is one of the most important reasons for hot-cracks and brake failure in brake systems. This paper simulates the brake system as a functionally graded material (FGM) coated half-plane sliding against a homogeneous half-plane. The motivation of using the FGM coating is to improve the coupled thermoelastic instability (TEI) of the brake system due to the thermal contact resistance and frictional heat. The thermoelastic properties of the FGM coating are assumed to vary arbitrarily along the thickness direction. The homogeneous multi-layered model is employed to simulate arbitrary properties of the coating. The perturbation method and transfer matrix method are used to derive the characteristic equation of the coupled TEI problem. The effects of the thermal contact resistance, friction coefficient, heat generation factor and different gradient types of the FGM coating on the stability boundaries are discussed in detail. The results show that the stability boundary is sensitive to the varying thermal parameters of the FGM coating, and an appropriate gradient type can adjust the coupled TEI of the sliding system.

We present a rigorous treatment to the problem of a circular-arc crack in an infinite thermoelectric solid subjected to a combined electrical and thermal loading. Formulating the problem in terms of the complex potentials and reducing it to the Hilbert arc problem, the solutions of quantities in both thermoelectric field and the associated thermoelastic field are presented in a closed form based on the electrically permeable and thermally insulated crack model. The results show that the fields of heat flux, energy flux, and stress exhibit the traditional square-root singularity at tips of arc crack. The applied electric current and energy flux generate both mode I and mode II stress intensity factors (SIFs), which are dependent on the loading direction, electric conductivity, thermal conductivity, central angle of the crack, and thermoelastic isotropy. Electrically induced SIF is a quadratic function of applied electric current density, and thermally induced SIF is a linear function of the imposed total energy flux. In addition, the effects of the loading direction and half crack angle on the thermal SIFs are also presented in a graphic form. This is the first theoretical paper to study the effect of the crack shape on the fracture of fully coupled thermoelectric materials by a rigorous inference of physics and mathematics.

The present paper is aimed to make a detail analysis on the correct simulation of the associated natural thermal boundary conditions and their interesting effects on the thermally nonlinear generalized thermoelastic response of a continuum medium. The paper presents some explanations regarding the literally conventional thermal boundary conditions associated with Cattaneo’s heat conduction law. The importance of modeling the natural thermal boundary conditions is investigated by applying three practical thermal loads to a homogeneous one-dimensional layer. The results of the investigations are illustrated by propagation of thermoelastic waves through the layers. These results clearly show the profound effects of the correct statement of natural thermal boundary conditions on the thermoelastic response of the layer. It is shown that these effects intensify the importance of thermally nonlinear generalized thermoelastic analyses.

The present study aimed to investigate the size-dependent quality factor of thermoelastic damping in a microbeam resonator based on modified strain gradient elasticity theory. The governing differential equation of motion is derived by using the principle of virtual displacements. The generalized thermoelasticity theory of the well-known Lord–Shulman model is also utilized to derive the equation of coupled thermoelasticity. The quality factor of thermoelastic damping in the microbeam is obtained based on both the complex frequency and the entropy generation methods. Further, by applying the generalized thermoelasticity theory and entropy generation method, explicit formulae for the quality factor of thermoelastic damping in the microbeam resonators are developed in the frameworks of both classical and modified couple stress elasticity theories. The effects of several parameters including size effect are investigated on thermoelastic damping in cantilever and clamped–clamped silicon microbeams as case studies. The results are validated by observing an excellent agreement regarding the comparison of the results of the present study and with previously published results.

In this paper, the thermoelastic behavior of a functionally graded nanodisk is studied based on the strain gradient theory. It is assumed that the nanodisk thickness is constant, and a power-law model is adopted to describe the variation of functionally graded material properties. Furthermore, the nanodisk angular acceleration is taken to be zero while it is subjected to an axisymmetric loading. Also, it is assumed that any variation in temperature occurs only in the radial direction. The equilibrium equation and the boundary conditions are deduced from Hamilton’s principle. The obtained results are compared with those of classical theory. These results show that both theories predict the same trend for the variation in radial displacements. The differences between the stresses obtained from classical and strain gradient theories are clearly highlighted. Increasing the value of the material inhomogeneity parameter, n, considerably affects the magnitudes and the corresponding peak values of the high-order stress $$\bar{\tau }_{rrr}$$ . Any rise in temperature at the outside radius has a direct effect on the total stresses and radial displacements in the nanodisk. Also, the effects of external load at the inner and outer radii on radial displacement as well as stress components are fully investigated.

This paper is concerned with the deformation of a chiral elastic bar subjected to a thermal field. The bar is composed of two different materials welded together along the surface of separation. The behavior of chiral materials is of interest for the investigation of bones and auxetic materials. We study the deformation of isotropic chiral materials by using the equilibrium theory of Cosserat thermoelasticity. The temperature variation is assumed to be a polynomial in the axial coordinate. It is shown that a plane thermal field, in contrast with the result predicted by the theory of achiral materials, produces torsional effects. The solution of the problem could be used to investigate the behavior of bone implants and other compound cylinders. The method is applied to the case of a circular bar reinforced by a longitudinal circular rod.

In this work, an investigation of electro–magneto interaction of a two-dimensional problem is presented for an infinitely long solid conducting circular cylinder. The problem is considered in the context of generalized electro–magneto–thermoelasticity with one relaxation time. The lateral surface of the solid is traction free and having a known temperature in the presence of a uniform magnetic field in the direction of the axis. The Laplace transform technique is used to obtain the solution in the transform domain. The solution is obtained by a direct approach without the customary use of potential functions. Numerical inversion of the Laplace transform is carried out to obtain the temperature, displacement, stress and induced magnetic field distributions. Numerical results are represented graphically and discussed.

A mathematical model for taking into consideration the interaction of mechanical, thermal and electrical dynamics in an anisotropic porous piezo-thermoelastic (APPT) medium is presented. Basic equations for APPT are derived. The three-dimensional problem of wave propagation is studied which explains the existence and propagation of six waves in the considered medium. The phase velocities and attenuation factors for each of the six attenuated waves are determined in terms of complex slowness vectors. The particular results are obtained for triclinic, monoclinic and orthotropic types of crystal symmetries. The effects of piezoelectricity, thermoelasticity, frequency, and porosity on the wave propagation phenomenon are analyzed.

Thermoelastic damping is an important issue for micro-/nanoscale mechanical resonators. In this work, an Euler–Bernoulli beam with different types of mechanical boundary conditions at the two ends is adopted. Once it bends, the half compressed (stretched) will be heated (cooled), then a transverse heat conduction appears, and energy dissipation occurs. First, size-dependent thermoelasticity based on generalized thermodynamics is introduced by combining the nonlocal elasticity and hydrodynamic heat-conductive model heat conduction, and the governing equations of the nonlocal thermal Euler–Bernoulli beam are sequently formulated. Second, an analytical solution to the inverse quality factor is obtained by using the complex-frequency approach, and it is observed that the solution is related to the nonlocal parameter of both elastic and thermal fields, as well as material constants. Meanwhile, another numerical method to get the inverse quality factor is proposed. Third, the effects of nonlocal parameters of both thermal and elastic fields, the height of the beam, and the material constants on the quality factor are evaluated. Finally, conclusive remarks are summarized. The predicted results are expected to be beneficial to micro-/nanomechanical resonators design.

Propagation of coupled Rayleigh-type waves in a 2-mm piezoelectric layer over a porous piezo-thermoelastic half-space is investigated in this paper. The governing equations for an orthorhombic porous piezo-thermoelastic half-space and 2-mm piezoelectric layer are discussed. The dispersion relations have been numerically derived. The phase velocities of Rayleigh waves are obtained for different types of electric–thermal boundary conditions at the interface and at the free surface. The variations of mechanical displacements, electric potential and temperature component with the non-dimensional wave number are obtained. The effects of different electric–thermal boundary conditions on the phase velocity have also been discussed. The dispersion curves and attenuation coefficients of wave propagation are presented graphically. Some reduced cases are also discussed.

This article presents new constructive formulas to steady-state thermoelastic Green’s functions for a plane generalized boundary value problem of thermoelasticity for a generalized rectangle. The constructive formulas are expressed in terms of Green’s functions for Poisson’s equation. These results are formulated in a special theorem, which is proved using the author’s developed harmonic integral representations method. On the base of derived constructive formulas, it is possible to obtain many analytical expressions for Green’s functions for thermoelastic displacements and stresses to 28 concrete boundary value problems for: rectangle-16, half-strip-8, strip-4. An example of such kind is presented for a concrete plane boundary value problem for a rectangle, Green’s functions of which are presented in the form of a sum of elementary functions and ordinary series. These results are presented in another theorem, which is proved on the base of derived general constructive formulas. In the particular case for a half-strip and strip, ordinary series vanish and Green’s functions are presented in elementary functions. New analytical expressions for thermal stresses to a particular plane problem for a thermoelastic rectangle subjected to a constant boundary temperature gradient are also derived. Numerical investigation has shown that the infinite series are convergent. All solutions obtained for thermal stresses, caused by a constant temperature gradient and by a unit heat source are validated by checking the respective equilibrium equation and continuity of deformation equations (Beltrami–Michel equations), written in the terms of thermal stresses. The graphics for thermal stresses and their infinite series also are presented.