Equations Of Thermoelasticity Research Articles

Numerical study of the non-linear vibrations of electrically actuated curved micro-beams considering thermoelastic damping

Thermoelastic damping is a major source of intrinsic damping in vacuum operated Micro Electro Mechanical Systems (MEMS). In this article, the non-linear vibration of electrically actuated curved micro-beams is investigated, considering the effect of thermoelastic damping. The coupled thermoelasticity equations are derived on the basis of non-classical formulation, which is the modified couple stress theory and generalized heat conduction thermoelasticity equation. Afterwards, the order of the governing equations is reduced using the Generalized Differential Quadrature Method (GQDM). The validation of the derived model is achieved through comparison with existing published results. Then the effects of design parameters on snap-through buckling and pull-in instability have been investigated. Furthermore, it is observed that the GDQM with small number of nonuniform grid spacing performs accurate results without dealing with complicated shape functions and integrations. Finally the coupled thermo-mechanical modal analysis is carried out by FE simulation.

Modified G-L thermoelasticity theory for nonlinear longitudinal wave in a porous thermoelastic medium

This paper presents a nonlinear analysis of the transient response of a porous medium affected by external traction. The formulation is presented based on a new modified G-L theory of thermoelasticity while temperature and strain rate parameters are included in the governing equations. Based on the finite strain theory (FST), the Lagrangian strain–displacement equation and Second Piola-Kirchhoff stress are applied to express the generalized form of thermoelasticity equations including large deformation. Both mechanical and thermal shocks are applied and finally, the effects of loading rate on transient response are discussed. An updated finite element method and Newmark’s numerical time integration method are employed to solve the nonlinear and time-dependent equations. It is determined that the modified G-L model is more powerful in predicting the wave propagation phenomenon. Furthermore, the findings indicate that the external surface shock induces compressive stresses, temperature raise, and volume fraction field increase in a porous medium. It is found compared to the classic model, the modified G-L model is superior in capturing the perceived wave propagation phenomenon.

High-temperature effect on the material constants and elastic moduli for solid rocks

Abstract Thermally coupled constitutive relations are generally used to determine material constants and elastic moduli (Young's modulus and shear modulus) of solid media. Conventional studies on this issue are mainly based on the linear temperature dependence of elastic moduli, whereas analytical difficulties are often encountered in theoretical studies on nonlinear temperature dependence, particularly at high temperatures. This study investigates the thermally coupled constitutive relations for elastic moduli and material constants using the assumption of axisymmetric fields, with applications to geologic materials (marble, limestone and granite). The Taylor power series of the Helmholtz free energy function within dimensionless temperatures could be used to develop the thermally coupled constitutive relations. The thermoelastic equivalent constitutive equations were formulated under the generalized Hooke's law. The material constants of solid rocks were determined by fitting experimental data using axisymmetric stress and strain fields at different temperatures, based on their thermomechanical properties. For these geologic materials, the resultant equivalent elastic moduli and deformations were in good agreement with those from the experimental measurements. Thermal stresses, internal moisture evaporation and internal rock compositions significantly affected the experimental results. This study provides a profound understanding of the thermally coupled constitutive relations that are associated with the thermomechanical properties of solid rocks exposed to high temperatures.

DG-GMsFEM for Problems in Perforated Domains with Non-Homogeneous Boundary Conditions

Problems in perforated media are complex and require high resolution grid construction to capture complex irregular perforation boundaries leading to the large discrete system of equations. In this paper, we develop a multiscale model reduction technique based on the Discontinuous Galerkin Generalized Multiscale Finite Element Method (DG-GMsFEM) for problems in perforated domains with non-homogeneous boundary conditions on perforations. This method implies division of the perforated domain into several non-overlapping subdomains constructing local multiscale basis functions for each. We use two types of multiscale basis functions, which are constructed by imposing suitable non-homogeneous boundary conditions on subdomain boundary and perforation boundary. The construction of these basis functions contains two steps: (1) snapshot space construction and (2) solution of local spectral problems for dimension reduction in the snapshot space. The presented method is used to solve different model problems: elliptic, parabolic, elastic, and thermoelastic equations with non-homogeneous boundary conditions on perforations. The concepts for coarse grid construction and definition of the local domains are presented and investigated numerically. Numerical results for two test cases with homogeneous and non-homogeneous boundary conditions are included, as well. For the case with homogeneous boundary conditions on perforations, results are shown using only local basis functions with non-homogeneous boundary condition on subdomain boundary and homogeneous boundary condition on perforation boundary. Both types of basis functions are needed in order to obtain accurate solutions, and they are shown for problems with non-homogeneous boundary conditions on perforations. The numerical results show that the proposed method provides good results with a significant reduction of the system size.

Influence of gravity and initial stress on Rayleigh wave propagation in magneto-thermoelastic medium

The main aim of present study is to show the effect of gravity, initial stress and magnetic field on the Rayleigh waves propagation in homogeneous orthotropic magneto-thermoelastic medium in the context of Three Phase Lag (TPL) model at two temperature. The governing equations of thermoelasticity have been solved by normal mode technique to deduce the frequency equation for Rayleigh wave with relevant boundary conditions. Special cases have been derived for isothermal and thermally insulated surfaces. Computer simulation is used for numerical discussion to show the effects of various parameters on phase velocity of Rayleigh waves. The variation in phase velocity corresponding to wave number has been demonstrated graphically in the presence of gravity, initial stress and magnetic field.

Thermal postbuckling analysis of variable angle tow composite cylindrical panels

The advanced Automated Fiber Placement (AFP) technologies allow the fiber trajectories to be curvilinearly steered over the panel platform, which brings out a novel type of cylindrical panel structure, termed as Variable Angle Tow (VAT) composite cylindrical panel. So far, the thermal postbuckling problem of VAT composite cylindrical panels under temperature loadings remain unexplored. The aim of this article is to cover this lack by developing a methodology based on a generalized Rayleigh–Ritz formulation in the framework of the principle of thermoelastic complementary energy. The proposed panel model is based on Donnell’s shallow shell theory and accounts for von Kármán’s geometrical nonlinearity. With the aid of the Lagrange multiplier method and the introduction of Airy’s stress function, a mixed variational formula that governs the nonlinear thermoelastic problem of VAT composite cylindrical panels is derived from the principle of thermoelastic potential energy. The Rayleigh-Ritz formulation combined with the mixed variational formula is then applied to derive the simultaneous nonlinear thermoelastic equations, and the Crisfield’s arc-length method, which can cope well with snap-through or snap-back problems, is employed to trace the equilibrium path that characterizes the thermal postbuckling features. Two different modeling strategies, namely, semi-inverse method and Lagrangian multiplier method, are adopted to formulate the nonlinear thermoelastic problem of VAT composite cylindrical panels undergoing temperature change. The application of Lagrangian multiplier method overcomes some limitations inherent in semi-inverse method in terms of constructing Rayleigh-Ritz formulation, and thus provides generality to model general in-plane boundary constraint. The accuracy and robustness of the proposed Rayleigh-Ritz model are validated against finite element solutions and previously published results. Effects of geometric imperfection, curvature radius, aspect ratio and fiber orientation angle on thermal postbuckling responses of VAT cylindrical plates are discussed by several examples. The study presented here may be helpful for the preliminary design of VAT curved panels in thermal environments.

LS-based and GL-based thermoelasticity in two dimensional bounded media: A Chebyshev collocation analysis

In this paper, the Chebyshev collocation numerical method is developed for solving generalized thermoelasticity problems of the isotropic homogeneous two dimensional media. The coupled thermoelastic equations are derived based on Lord-Shulman (LS) and Green-Lindsay (GL) theories. The temperature and displacement fields are approximated in space domain by linear combinations of Chebyshev polynomials. Also, the direct collocation method is applied to governing differential equations to generate the system of differential equations with respect to time. The resulted set of differential equations are solved in time domain by Wilson method. Both temperature and traction loadings are considered to be applied at the left side of the media. The obtained results from the present paper for the classical coupled thermoelasticity of two dimensional finite domains are compared with the same results extracted analytically in the literature and a very close agreement is observed.

A strain rate dependent thermo-elasto-plastic constitutive model for crystalline metallic materials

The strain rate and temperature effects on the deformation behavior of crystalline metal materials have always been a research hotspot. In this paper, a strain rate dependent thermo-elasto-plastic constitutive model was established to investigate the deformation behavior of crystalline metal materials. Firstly, the deformation gradient was re-decomposed into three parts: thermal part, elastic part and plastic part. Then, the thermal strain was introduced into the total strain and the thermo-elastic constitutive equation was established. For the plastic behavior, a new relation between stress and plastic strain was proposed to describe the strain rate and temperature effects on the flow stress and work-hardening. The stress–strain curves were calculated over wide ranges of strain rates (10–6–6000 s−1) and temperatures (233–730 K) for three kinds of crystalline metal materials with different crystal structure: oxygen free high conductivity copper for face centered cubic metals, Tantalum for body centered cubic metals and Ti–6Al–4V alloy for two phase crystal metals. The comparisons between the calculation and experimental results reveal that the present model describes the deformation behavior of crystalline metal materials well. Also, it is concise and efficient for the practical application.

Fully coupled meshfree numerical approach based on the finite pointset method for static linear thermoelasticity problems

In this work, a promising fully coupled meshfree numerical approach is extended and implemented for the first time in the field of linear static thermoelasticity. A real meshfree method, the so-called finite pointset method (FPM), is applied and implemented in order to solve the strong/classical form of the governing partial differential equations for static linear thermoelasticity. Several benchmark problems are numerically solved in order to show the proposed coupled FPM numerical performance. The presented FPM meshfree approach shows excellent behavior for 2D linear static thermoelasticity problems even for complex geometries.

Thermoelastic medium in the context of four theories subjected to gravity field and laser pulse

The paper aims at studying the impact of gravity and laser pulse on the overall structure of the generalized thermoelasticity equations concerning a standardized elastic isotropic half-space. Using four generalized thermoelasticity theories, i.e. the Green-Naghdi (GN) theory, Green-Lindsay (GL) theory, Lord-Schulman (LS) theory, and Coupled Theory (CT), the authors applied the formulation. In the physical sphere, they attained the analytical expressions for the (Mechanical and Maxwell’s) stresses distribution, displacement components, and temperature through utilizing the normal mode analysis. Moreover, they calculated these expressions numerically and plotted the conforming graphs to demonstrate and make comparisons between theoretical findings. The paper concludes that the impact of gravity and laser pulse field is highly significant.

Simulation of thermoelastic waves based on the Lord-Shulman theory

Thermoelasticity is important in seismic propagation due to the effects related to wave attenuation and velocity dispersion. We have applied a novel finite-difference (FD) solver of the Lord-Shulman thermoelasticity equations to compute synthetic seismograms that include the effects of the thermal properties (expansion coefficient, thermal conductivity, and specific heat) compared with the classic forward-modeling codes. We use a time splitting method because the presence of a slow quasistatic mode (the thermal mode) makes the differential equations stiff and unstable for explicit time-stepping methods. The spatial derivatives are computed with a rotated staggered-grid FD method, and an unsplit convolutional perfectly matched layer is used to absorb the waves at the boundaries, with an optimal performance at the grazing incidence. The stability condition of the modeling algorithm is examined. The numerical experiments illustrate the effects of the thermoelasticity properties on the attenuation of the fast P-wave (or E-wave) and the slow thermal P-wave (or T-wave). These propagation modes have characteristics similar to the fast and slow P-waves of poroelasticity, respectively. The thermal expansion coefficient has a significant effect on the velocity dispersion and attenuation of the elastic waves, and the thermal conductivity affects the relaxation time of the thermal diffusion process, with the T mode becoming wave-like at high thermal conductivities and high frequencies.

Dual-phase-lag thermoelastic damping in nonlocal rectangular nanoplates

The present article pertains to the assessment of thermoelastic damping (TED) in rectangular nanoplates by incorporation of size effect within the constitutive and heat conduction relations. With the purpose of establishing size-dependent coupled thermoelastic equations, nonlocal elasticity theory and dual-phase-lag heat conduction model are utilized. By considering time-harmonic vibrations, nonclassical frequency equation affected by thermoelastic coupling is derived. By solving this equation and extracting the real and imaginary parts of damped natural frequency via some mathematical operations, an explicit expression for description of TED in circular nanoplates is obtained. With the goal of better understanding the small-scale effect on TED, a comparison is made between the results predicted by the size-dependent formulation and those determined according to classical continuum mechanics and heat conduction theories for two kinds of boundary constraints. Considerable difference between classical and nonclassical results, especially at smaller sizes, implies the necessity of incorporating small-scale effect into both structural and thermal areas.

The meso-scale behavior of anisotropic particle-reinforced thermo-elastic composites

Linear thermo-elasticity equations are derived in this paper for anisotropic, particle-reinforced composite structures at mesoscale. The full set of equations is first derived at microscale where the structure is represented as a three-dimensional body with spherical inclusions of known spatial distribution. Limiting the consideration by infinitesimal strains and assuming that the inclusions are embedded into the body firmly and that there are no any defects at the interface of inclusions, the mesoscale limit of the set of equations is derived as the scale parameter decreases to 0. Convergence of the corresponding solutions is established. Material characteristics at mesoscale are expressed in terms of Dirac distribution corresponding to a structure with point inhomogeneities. It is shown that the effective or macroscale material characteristics evaluated in terms of the mesoscale ones, match the well-known rule of mixtures. A particular case of Ambarcumyan plate with random distribution of spherical inclusions is studied numerically.

On thermoelastic damping in axisymmetric vibrations of circular nanoplates: incorporation of size effect into structural and thermal areas

The paper at hand intends to evaluate thermoelastic damping (TED) in circular plates by incorporating nonlocal effect within the constitutive and heat conduction frameworks. To attain this purpose, nonclassical coupled thermoelastic equations are established on the basis of nonlocal elasticity theory and Guyer–Krumhansl (GK) heat conduction model. By considering symmetric time–harmonic vibrations, the size-dependent thermoelastic frequency equation is derived. By solving this nonclassical eigenvalue problem, real and imaginary parts of damped natural frequency are separated. According to the definition of TED in the framework of complex frequency approach, a closed-form expression characterizing TED in circular nanoplates is introduced. With the aim of surveying the nonlocal effect on TED, a comparison study is performed between the size-dependent outcomes and those extracted by way of classical continuum mechanics and heat conduction theories for simply supported and clamped circular nanoplates. When the dimensions of nanoplate become smaller, an obvious discrepancy between classical and nonclassical results is observed, which is irrefutable evidence of size effect on mechanical behavior of nanostructures.

Application of Chebyshev collocation method to unified generalized thermoelasticity of a finite domain

In this article, the Chebyshev collocation numerical method is developed for solving generalized thermoelasticity problems of the isotropic layer. The coupled thermoelastic equations are derived based on Lord-Shulman (LS), Green-Lindsay (GL) and Green-Naghdi (GN) theories. Two kinds of shock loading are considered. In the first model, a temperature shock is applied at the left side of the layer, and in the second model a heat flux shock is applied at the mentioned side. The displacement and temperature fields are approximated in the layer by Chebyshev polynomials and then the collocation is imposed on some collocation points to derive the system of ordinary differential equations from the coupled partial differential equations. The derived system of differential equations is then solved by the Wilson method to achieve the displacement and temperature and also stress at any location and time. The obtained results from the present article are compared with the same results in the open literature and a very close agreement is observed. Finally, it is concluded that the Chebyshev collocation numerical method can be utilized as a very strong method for solving the transient wave propagation problems.

Decay properties for evolution-parabolic coupled systems related to thermoelastic plate equations

<abstract><p>In this paper, we consider the Cauchy problem for a family of evolution-parabolic coupled systems, which are related to the classical thermoelastic plate equations containing non-local operators. By using diagonalization procedure and WKB analysis, we derive representation of solutions in the phase space. Then, sharp decay properties in a framework of $ L^p-L^q $ are investigated via these representations. Particularly, some thresholds for the regularity-loss type decay properties are found.</p></abstract>

Effect of convection boundary condition on hyperbolic heat conduction in thermoelastic medium

This paper aims to study the effect of heat transfer by convection on thermoelastic rectangular solid medium in the context of hyperbolic heat conduction. The studied geometry is a two dimensional finite thin rectangular plate without internal heat generation, which is initially at a uniform temperature and subjected to convection heat transfer from the extreme edge(y=a) while the opposite side is kept insulated and remaining two sides are at a constant temperature. The material properties are assumed to be constant. The differential transform method is applied to solve the hyperbolic heat conduction equation to obtain temperature distribution in the spatial and temporal domain. Then by applying the obtained temperature in the thermoelastic equation, the displacement component, and the stress field are calculated. Also, the effect of convection boundary conditions on temperature distribution and thermoelastic field are illustrated numerically and graphically for a copper plate. It is observed that the compressive and tensile stress occurs along y-direction due to heat transfer through convection at one end.

Influence of gravity and initial stress on Rayleigh wave propagation in magneto-thermoelastic medium

The main aim of present study is to show the effect of gravity, initial stress and magnetic field on the Rayleigh waves propagation in homogeneous orthotropic magneto-thermoelastic medium in the context of Three Phase Lag (TPL) model at two temperature. The governing equations of thermoelasticity have been solved by normal mode technique to deduce the frequency equation for Rayleigh wave with relevant boundary conditions. Special cases have been derived for isothermal and thermally insulated surfaces. Computer simulation is used for numerical discussion to show the effects of various parameters on phase velocity of Rayleigh waves. The variation in phase velocity corresponding to wave number has been demonstrated graphically in the presence of gravity, initial stress and magnetic field.

Thermomechanical deformation of the orthotropic shell taking into account the deformation anisotropy

A variant of the rotation shell in the particular form of a closed circular cylindrical shell, which is often used in the design practice of civil, power and other industrial structures, is considered. The specificity of the considered shell lies in the features of its material, which has a manifestation of dual anisotropy. In particular, this material is orthotropic in structure, and the nature of deformation shows the dependence of stiffness and strength on the type of stress state. The loading of the shell is assumed to be axisymmetric, taking into account the influence of a medium with variable thermal parameters. The temperature difference between the shell surfaces is taken into account here. The statement of the general thermomechanical problem is carried out in an unrelated form, taking into account a certain independence of the problems of thermodynamics and mechanics. Taking into account the limitations of the classical thermomechanical theories of shells made of materials with dual anisotropy and the fact that the known models for such materials have significant drawbacks, the authors used a variant of the normalized stress space. Differential equations of thermoelasticity for a cylindrical shell are obtained, taking into account the complicated thermomechanical properties of its material. Particular solutions with the features of the results of calculating the shell states are illustrated, and their analysis is carried out.

Ⅲ型热弹性方程的空间二择性及稳定性

The spatial alternative of type Ⅲ thermoelastic equations in a semi-infinite region was considered. The equations were assumed to satisfy certain initial and boundary conditions, and the solution was proved to grow exponentially or alternatively decay exponentially respectively, with spatial variables on a semi-infinite cylinder with the energy analysis method. The continuous dependence of solutions on the coefficients of equations was established.