Green Naghdi Thermoelasticity Research Articles

Application of Chebyshev collocation element method to Green-Naghdi thermoelasticity of a bi-layered media consists of FGM and viscoelastic domains

In this study, the generalized coupled thermoelastic wave propagation and temporal evolution in a bi-layered system are analyzed. The main layer is made of functionally graded materials (FGM), while the holder layer is viscoelastic. The constituent volume fractions in the FGM layer are calculated by a power law function. Moreover, the Voigt rule of mixtures is employed to obtain the homogenized material properties. On the other hand, the constitutive law for the viscoelastic environment is considered based on the Kelvin-Voigt model. The heat conduction equation for each layer is represented based on the general form of the Green-Naghdi theory. First, for each layer, two motion and heat conduction coupled equations are derived, as well as the boundary and appropriate continuity conditions. The Chebyshev collocation element (CCE) method is used to solve the spatial-dependent parts of equations and also assemble the governing matrices into a global system. Finally, the Newmark numerical integration method is implemented to solve the ODEs and calculate the system response. The achieved response is verified with limited articles in the literature. Some parametric results are shown to evaluate the performance of FG and viscoelastic characteristics on the FGM layer response which is subjected to a thermal shock.

Magneto-thermoelasticity Green–Naghdi theory with memory-dependent derivative in the presence of a moving heat source

In the present work, a mathematical model of the Green-Naghdi thermoelasticity theory of type III (GN-III) with memory-dependent derivative (MDD) heat transfer for a perfectly conducting isotropic media has been constructed. The state-space and Laplace transform techniques are adopted for the solution of a half-space problem in the presence of a moving heat source with constant velocity. The inversion of the Laplace transforms is carried out using a numerical approach. Numerical results for all fields are given and illustrated graphically. Comparison is made with the results predicted by coupled thermoelasticity (DCT). The influences of MDD parameters and heat source speed on all fields are examined.

Some results on stability and continuous dependence in Green-Naghdi thermoelasticity of Cosserat bodies

In our paper, we consider the mixed problem in the context of the Green-Naghdi theory of thermoelastic Cosserat media. Using very accessible mathematical calculations, we prove two qualitative results on the solutions of the formulated mixed problem. Thus, in the first of these approaches, we obtain a result of stability, of the Hölder type, with regards to the loads. In the second main result, we prove a continuous dependence result regarding the initial values from the mixed problem. It should be noted that we obtain these results without imposing very restrictive conditions on the thermoelastic tensors in the constitutive equations. In fact, imposed restrictions are commonly used in Mechanics of Continuous Media.

Size-dependent thermoelastic damping analysis in nanobeam resonators based on Eringen’s nonlocal elasticity and modified couple stress theories

Thermoelastic damping has emerged as a critical issue in the modeling and design of micro and nanomechanical systems. Therefore, an extensive research interest is being devoted towards the reduction in thermoelastic damping for micro and nanomechanical systems. The present study intends to examine thermoelastic damping in nanobeam resonators using the modified couple stress theory and Eringen’s nonlocal elasticity theory, thus so-called modified nonlocal couple stress theory within the context of the recently proposed Moore–Gibson–Thompson thermoelasticity theory. In order to observe size effects, the size-dependent coupled thermoelastic equations are derived by combining modified couple stress theory and nonlocal elasticity theory in the frame of Moore–Gibson–Thompson thermoelasticity. The coupled governing equations for thermoelastic damping of nanobeam resonators are solved analytically in terms of the inverse quality factor. The size-dependent thermoelastic damping of nanobeams is presented graphically with the help of numerical results predicted by modified nonlocal couple stress theory. The results obtained under the combined effects of modified couple stress theory and nonlocal theory in the present context are further compared to those of the classical, modified couple stress, and nonlocal elasticity theories as special cases of the modified nonlocal couple stress theory. It is observed that thermoelastic damping weakens at the submicron scale under Eringen’s nonlocal elasticity theory, whereas it becomes greater at the submicron scale under modified couple stress theory. However, at the submicron scale, the modeling of nanobeam resonators using modified nonlocal couple stress theory reveals a smaller amount of thermoelastic damping than modified couple stress, nonlocal, and classical theories. In addition, as compared to the Green–Naghdi thermoelasticity theory of type III (GN-III), the Moore–Gibson–Thompson model predicts a higher thermoelastic damping value, while agreeing with the results predicted by the Lord–Shulman (LS) model under modified nonlocal couple stress theory. Some important points highlighted here are believed to be useful in the design of mechanical resonators at micron and submicron scales.

A domain of influence theorem for a natural stress–heat-flux problem in the Moore–Gibson–Thompson thermoelasticity theory

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.

Types of flow in Green-Naghdi Thermoelasticity Theory

In this paper we will show a review of the Green-Naghdi thermoelasticity theory.Such model have beautiful foundations which contains since the law of heat conduction proposed byFourier, fundamentals variables such as Helmholtz free energy, entropy, flux of heat and etc. and the constitutive hypothesis. The nature of partial differential equation of parabolic (PDE) type derived from classical thermoelasticity lead us to a mathematical inconsistency well-know as thermal signal speed paradox. We will show how Green-Naghdi model solved those mathematical inconsistency introducing different types of flow and giving us a hyperbolic PDE.

Thermoelastic interactions in a hollow cylinder due to a continuous heat source without energy dissipation

The linear generalized Green–Naghdi thermoelasticity theory without energy dissipation is employed. The study of thermoelastic interactions in a hollow cylinder under a continuous heat source is carried out. Firstly, Laplace and Hankel transforms are employed to solve the problem without the time domain. Then, the state space approach is employed to get the exact solution of the problem in the space domain. Once again, the inverse Laplace transforms is used to get the solutions in the time domain. Accurate terminologies for the temperature, thermoelastic potential, axial displacement, dilatation, and stresses are derived. Numerical outcomes for field variables are presented with the view of illustrating the theoretical results.

Theoretical analysis of thermoelastic damping of silver nanobeam resonators based on Green–Naghdi via nonlocal elasticity with surface energy effects

The vibration of a silver nanobeam resonator, considering the surface effect as well as the thermal effect has been investigated in this paper. The emerged bending in nanobeam resonator causes the surface effects to appear in nanobeam. The governing equations for nanobeam are obtained considering the surface and thermal effects and using the nonlocal elasticity theory. The temperature effects based on the Green–Naghdi thermoelasticity theory, and considering the thermoelastic damping, are taken into account. The vibration governing equations are derived by the coupled Green–Naghdi thermoelastic, nonlocal elasticity theory, and surface effect for Euler–Bernoulli beam model. The dynamic and temperature responses of the nanobeam are obtained in the Laplace domain using the Laplace method. The technique of inverse Laplace, called a Talbot method, is utilized to calculate the dynamic and thermal responses of the nanobeam in the time domain. To investigate the results, the effects of the various parameters, such as the surface effects, nonlocal parameter, and the initial temperature conditions, on the dynamic and temperature responses of the microbeam are scrutinized.

An exact solution on gold microbeam with thermoelastic damping via generalized Green-Naghdi and modified couple stress theories

The vibration of the gold microbeam resonator considering the effects of the material length scale under the temperature effect is investigated in this article. The modified couple stress theory is utilized to show the material length scale effects. The effects of temperature considering the thermoelastic damping are taken into account based on Green-Naghdi thermoelasticity theory. The Green-Naghdi coupled thermoelasticity and modified couple stress governing equations for Euler-Bernoulli beam are derived based on Hamilton’s principle. The Laplace method is utilized to solve the obtained equations. The inverse Laplace method as Talbot method is used to obtain the lateral deflection and temperature response of the microbeam in the time domain. The effects of the various parameters such as the nondimensional material length scale parameter, the thickness of the microbeam, and the initial temperature conditions, on the lateral deflection as well as temperature responses of the microbeam are studied to investigate the results. Raising the material length scale parameter and width to length ratio makes the lateral deflection of the beam decrease in the time domain. Increase in temperature shock and Poisson’s ratio makes the lateral deflection increase, and both have a direct effect on the lateral deflection in the time domain, except that this increase is linear for temperature shock. The temperature of the microbeam decreases by an increase in Poisson’s ratio and progressing in the axial direction. For the final observation, it should be noted that the couple stress has an inverse effect on the lateral deflection.

On vibrations in Green-Naghdi thermoelasticity of dipolar bodies

Abstract This study is concerned with the theory of thermoelasticity of type III proposed by Green and Naghdi, which is extended to cover the bodies with dipolar structure. In this context we construct a boundary value problem for a prismatic bar which is subjected to some harmonic in time vibrations. For the oscillations whose amplitudes have the frequency lower than a critical value, we deduce some estimates for describing the spatial behavior.

The effect of a dipolar structure on the Hölder stability in Green–Naghdi thermoelasticity

This study is concerned with the mixed initial boundary value problem for a dipolar body in the context of the thermoelastic theory proposed by Green and Naghdi. For the solutions of this problem we prove a result of Holder’s-type stability on the supply terms. We impose middle restrictions on the thermoelastic coefficients, which are common in continuum mechanics. For the same conditions we propose a continuous dependence result with regard to the initial data.

Fractional phase-lag Green–Naghdi thermoelasticity theories

ABSTRACTThe constitutive equations are given with a fractional Maxwell–Cattaneo heat conduction law using the Caputo fractional derivative and the fractional order heat transport equation is given. The uniqueness theorem is proved, the reciprocity relation is deduced and the variational characterization of solution is given. Seven thermoelasticity theories result from the given problem as special cases (Coupled, Lord–Shulman, phase-lag Green–Naghdi theories, Green–Naghdi theory without energy dissipation and the Green–Naghdi theory of type III).

Two-temperature theory in Green–Naghdi thermoelasticity with fractional phase-lag heat transfer

A mathematical model of two-temperature phase-lag Green–Naghdi thermoelasticty theories based on fractional derivative heat transfer is given. The GN theories as well as the theories of coupled and of generalized thermoelasticity with thermal relaxation follow as limit cases. The resulting non dimensional coupled equations together with the Laplace transforms techniques are applied to a specific problem of a half space subjected to arbitrary heating which is taken as a function of time and is traction free. The inverse transforms are obtained by using a numerical method based on Fourier expansion techniques. The predictions of the theory are discussed and compared with those for the generalized theory of thermoelasticity with one relaxation time. The effects of temperature discrepancy and fractional order parameters on copper-like material are discussed in different types of GN theories.

Energy decay in Green–Naghdi thermoelasticity with diffusion and dissipative boundary controls

ABSTRACTWe study the asymptotic behavior of the solutions of a three-dimensional hyperbolic system arising in the Green–Naghdi models of thermoelasticity with diffusion of type-II and III, in the presence of dissipative boundary controls and prove that the energy exponentially decays in time.

Free vibration of a microbeam resting on Pasternak’s foundation via the Green–Naghdi thermoelasticity theory without energy dissipation

This article investigates the effect of length-to-thickness ratio and elastic foundation parameters on the natural frequencies of a thermoelastic microbeam resonator. The generalized thermoelasticity theory of Green and Naghdi without energy dissipation is used. The governing frequency equation is given for a simply supported microbeam resting on Winkler–Pasternak elastic foundations. The influences of different parameters are all demonstrated. Natural vibration frequencies are graphically illustrated and some tabulated results are presented for future comparisons.

A domain of influence theorem for a natural stress–heat-flux disturbance in thermoelasticity of type-II

ABSTRACTThe aim of the present work is to establish the domain of influence theorem for a stress–heat-flux disturbance under Green and Naghdi thermoelasticity theory of type-II. We consider a mixed problem of natural type represented as stress–heat-flux disturbance in the context of Green–Naghdi thermoelasticity theory of general type. We establish a general energy identity for the problem. Then we establish the domain of influence theorem for natural stress–heat-flux disturbance in the context of Green–Naghdi model of thermoelasticity oftype-II. We prove that for a finite time t > 0, the pair of stress and the heat-flux field generates no disturbance outside a bounded domain. The domain of influence is shown to be dependent on the thermoelastic parameters.

On the phase- lag Green–Naghdi thermoelasticity theories

In this work, we proposed three models of generalized thermoelasticity: a single –phase - lag Green–Naghdi theory of type III, a dual-phase–lag Green–Naghdi theory of type II and of type III. The solid is assumed to be linear anisotropic inhomogeneous. A unified heat conduction law and heat transport equation are given, which consolidate the three theories and also the Lord–Shulman theory and the Green–Naghdi theory of type II. Dissipative inequality is derived, constitutive equations and thermodynamic restrictions are given. The speed of thermal wave propagation is given, for each of the three theories in case of isotropic solid. A uniqueness theorem is proved for the considered theories. Variational characterization of solution is established. An application is given for isotropic thermoelastic solid and the results are presented graphically.

A Green-Naghdi approach for thermo-electroelasticity

The constitutive relations of piezoelectric ceramics are essentially nonlinear since the so-called piezoelectric moduli depend on the induced strains. Pioneering papers in these topics dealt mainly with the isothermal case. In view of applications, however, thermal effects have to be taken into account in connection with thermo-electric behaviors. Here we briefly compare continuum theories for nonlinear thermoelettroelasticity. In particular we describe an extension of Green-Naghdi thermoelasticity theory for an electrically polarizable and finitely deformable heat conducting elastic continuumn, which interacts with the electric field. In this theory, unlike other, thermal waves propagate at a finite speed.

On the Two-Temperature Green–Naghdi Thermoelasticity Theories

The constitutive laws for two-temperature Green–Naghdi theories are given. It is proved that the two-temperature thermoelasticity theory admits dissipation of energy and the theory of elasticity without energy dissipation is valid only when the two-temperatures coincide. The uniqueness and reciprocal theorems are proved for a linear anisotropic and inhomogeneous thermoelastic centro-symmetric solid in the frame of two-temperature Green–Naghdi theories. The convolutional variational principle is established for the two-temperature Green–Naghdi theory of type III. A continuous dependence result is given for an isotropic solid.

Material Forces in Non-Classical Thermo-Hyperelasticity

The contribution's main objective is the treatment of non-classical nonlinear thermoelasticity based on the approach of Green and Naghdi within the context of the material force method. Material forces have proven to be well suited for the examination of defect mechanics and have become a very powerful and active research topic over the last decades. The numerical realization is based on finite elements in space as well as in time. Examples of investigating material forces within Green–Naghdi thermoelasticity are provided.