Classical Thermoelasticity Research Articles

Ideal nonsymmetric 4D-medium as a model of invertible dynamic thermoelasticity

The space-time continuum (4D-medium) is considered, and a generalized model of reversible dynamic thermoelasticity is constructed as a theory of elasticity of an ideal (defect-free) nonsymmetric 4D-medium that is transversally-isotropic with respect to the time coordinate. The definitions of stresses and strains for the space-time continuum are introduced. The constitutive equations of the medium model relating the components of nonsymmetric stress and distortion 4D-tensors are stated. Physical interpretations of all tensor components of the thermomechanical properties are given. The Lagrangian of the generalized model of coupled dynamic thermoelasticity is presented, and the Euler equations are analyzed. It is shown that the three Euler equations are generalized equations of motion of the dynamic classical thermoelasticity, and the last, fourth, equation is a generalized heat equation which allows one to predict the wave properties of heat. An energy-consistent version of thermoelasticity is constructed where the Duhamel-Neumann and Maxwell-Cattaneo laws (a nonclassical generalization of the Fourier law for the heat flow) are direct consequences of the constitutive equations.

Thermoelasticity and interdiffusion in CuNi multilayers

X-ray scattering experiments on coherent CuNi multilayers are performed at various annealing temperatures. First, we show that the classical thermoelasticity theory can be applied in such nanosamples to link composition and strain fields at intermediate temperature. Second, when interdiffusion takes place at higher temperature, the satellite peaks measured at different annealing times indicate the presence of a layer-by-layer interdiffusion mode controlled by the asymmetry of the atomic mobilities in this system.

THEORY OF SPACETIME ELASTICITY

We present the theory of spacetime elasticity and demonstrate that it involves traditional thermoelasticity. Assuming linear-elastic constitutive behavior and using spacetime transversely-isotropic elastic constants, we derive all principal thermodynamic relations of classical thermoelasticity. We introduce the spacetime principle of virtual work, and use it to derive the equations of motion for both reversible and dissipative thermoelastic dynamics. We show that spacetime elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, spacetime elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity, complemented by the spectrum of boundary and interface conditions. We argue that the presented framework of spacetime elasticity should prove adequate for describing the thermoelastic phenomena occurring at low temperatures, for interpreting the results of molecular simulations of heat conduction in solids, and also for the optimal heat and stress management in the microelectronic components and the thermoelectric devices.

Theories of thermal stresses based on space–time-fractional telegraph equations

The generalized telegraph equations with time- and space-fractional derivatives are considered. The corresponding theories of thermal stresses are formulated. The proposed theories interpolate the classical thermoelasticity, the theory of Lord and Shulman, thermoelasticity without energy dissipation of Green and Naghdi, and theories of fractional thermoelasticity proposed earlier. The fundamental solution to the nonhomogeneous space–time-fractional telegraph equation as well as the corresponding thermal stresses are obtained in the axisymmetric case.

Strong stabilization of models incorporating the thermoelastic Reissner–Mindlin plate equations with second sound

In this article we are concerned with the strong stabilization of models for the Reissner–Mindlin plate equations with second sound, that is, models that include thermal effects described according to Cattaneo's law of heat conduction instead of Fourier's law in classical thermoelasticity. Two models will be considered which are distinct with respect to the property of compactness or non-compactness of the resolvent of the generator of the underlying semigroup. In accordance with the compactness or non-compactness of the resolvent operator, a different criterion for strong stability is implemented to achieve the strong stabilization of each model. In the compact resolvent case we avail ourselves of a result given by Benchimol [C.D. Benchimol, A note on weak stabilizability of contraction semigroups, SIAM J. Control Optim. 16 (1978), pp. 373–379] and in the non-compact case we resort to a stability criterion Tomilov [Y. Tomilov, A resolvent approach to stability of operator semigroups, J. Operator Theory 46 (2001), pp. 63–98].

Numerical Study of Characteristic Equations of Thermoelastic Waves Propagating in a Uniaxial Prestressed Isotropic Plate

Thermoelastic waves propagating in an isotropic thin plate exerted by a uniaxial tensile stress are represented in this work. Characteristic equation of guided thermoelastic waves is formulated based on the theory of acoustoelasticity and classical thermoelasticity. Curve-tracing method for complex root finding is used to determine the attenuation, which is the imaginary part of the complex-value wavenumber. It is found that each plate mode of thermoelastic wave propagating in an isotropic plate with or without prestress has a minimum attenuation at a specific frequency except the A0 mode. These modes are called the Lamé modes, which are the volume resonances in the thickness direction and propagate along the plate with the least energy dissipation. Frequency spectra of the phase velocity dispersion and attenuation of thermoelastic waves propagating along various orientations in the uniaxial prestressed thin plate have further been discussed.

An Exact Solution for Classic Coupled Thermoelasticity in Cylindrical Coordinates

In this paper, the classic coupled thermoelasticity model of hollow and solid cylinders under radial-symmetric loading condition (r,t) is considered. A full analytical method is used, and an exact unique solution of the classic coupled equations is presented. The thermal and mechanical boundary conditions, the body force, and the heat source are considered in the most general forms, where no limiting assumption is used.

Thermal Stresses in Chiral Elastic Beams

The chiral effects cannot be described by means of the classical thermoelasticity. In the context of the linear theory of Cosserat thermoelasticity we study the deformation of a chiral beam subjected to a prescribed thermal field. This paper points out the importance of the generalized plane strain problem in the analysis of thermal stresses in chiral elastic beams. First, we investigate the effects of a thermal field which is linear in the axial coordinate. It is shown that this temperature variation produces extension, bending, torsion, flexure and a plane deformation. Then, we study the deformation of the beam when the thermal field is a polynomial of degree m (m > 1) in the axial coordinate. The solution is reduced to the solving of some two-dimensional problems. The method is used to solve the problem of a circular cylinder subjected to a temperature that is independent of the axial coordinate.

THERMOPIEZOELECTRIC ANALYSIS OF A FUNCTIONALLY GRADED PIEZOELECTRIC MEDIUM

The thermopiezoelectrical behavior of a functionally graded piezoelectric medium (FGPM) is investigated in the present work. For the special case, the dynamic response of an FGPM rod excited by a moving heat source is studied. The material properties of the FGPM rod are assumed to vary exponentially through the length, except for specific heat and thermal relaxation time which are held constant for simplicity. The governing differential equations in terms of displacement, temperature, and electric potential are obtained in a general form that includes coupled and uncoupled thermoelasticity. The coupled formulation considers classical thermoelasticity as well as generalized thermoelasticity. Employing the Laplace transform and successive decoupling method, unknowns are given in the Laplace domain. Employing a numerical Laplace inversion method, the solutions are gained in the time domain. Numerical examples for the transient response of the FGPM rod are displayed to clarify the differences among the results of the generalized, coupled, and uncoupled theories for various nonhomogeneity indices. The results are verified with those reported in the literature.

Прохождение термоупругого гармонического сигнала через волновод с теплопроницаемой стенкой

The paper is devoted to a study of coupled harmonic thermoelastic impulse guided propagation through an infinite circular cylinder. Heat interchanging is supposed to take place between sidewall of the waveguide and environment. The analysis is carried out according to the principles of coupled generalized thermoelasticity theory of type III (GNIII-thermoelasticity). This theory combines thermodiffusion and wave mechanisms of heat transfer in solids including as limiting cases both the theories: classical thermoelasticity (GNI/CTE) and the theory of hyperbolic thermoelasticity (GNII ). The latter permits field-theoretic formulation and leads to the field equations of hyperbolic analytical type. Closed solution of the coupled GNIII-thermoelasticity equations satisfying the boundary conditions on the surface of waveguide is obtained by separation of variables. The analysis of frequency equation is given and wave numbers and modes of coupled thermoelastic waves of arbitrary order are obtained. The problems of coupled thermal and dynamic impulse propagation in the form of plane and normal waves in a free from tractions thermoisolated waveguide have been studied in our previous papers.

Волновые числа термоупругих волн в волноводе с теплообменом на боковой стенке

The paper presents a study of wavenumbers of coupled harmonic thermoelastic waves propagating via an infinite circular cylinder for higher azimuthal order. Heat interchanging between sidewall of the waveguide and environment is assumed. The analysis is carried in the frameworks of coupled generalized thermoelasticity theory of type III (GNIII-thermoelasticity). This theory synthesizes thermodiffusion and wave mechanisms of heat transfer in solids including as limiting cases both the theories: classical thermoelasticity (GNI/CTE) and the theory of hyperbolic thermoelasticity (GNII). The latter permits field-theoretic formulation and leads to the field equations of hyperbolic analytical type. The two principal problems: separation of 32 independent single-valued square radicals involved in the frequency equation and more accurate computation of the real-valued wavenumbers are resolved and discussed.

An Exact Solution for Classic Coupled Thermoelasticity in Spherical Coordinates

Abstract In this paper, the classic coupled thermoelasticity model of hollow and solid spheres under radial-symmetric loading condition (r,t) is considered. A full analytical method is used and an exact unique solution of the classic coupled equations is presented. The thermal and mechanical boundary conditions, the body force, and the heat source are considered in the most general forms, where no limiting assumption is used. This generality allows to simulate a variety of applicable problems.

Implementation of the extended finite element method for dynamic thermoelastic fracture initiation

The eXtended Finite Element Method (XFEM) is implemented to model the effect of the mechanical and thermal shocks on a body with a stationary crack. The classical thermoelasticity equations are considered. The Newmark and the Crank–Nicolson time integration schemes are used to solve the dynamical system of matrix equations obtained from the spatial discretization of the elastic and thermal parts respectively. The variation of dynamical stress intensity factors versus time are computed by the interaction integral method. A C++code is developed to do the different stages of the calculations from mesh generation to the calculation of interaction integrals. Several elastic and thermoelastic numerical examples are implemented, where the dynamical behaviour of the face and tip enrichments of XFEM are investigated and the results are discussed in detail.

Theory of thermoelasticity based on the space-time-fractional heat conduction equation

The space-time-nonlocal generalization of the Fourier law and the space-time-fractional heat conduction equation are discussed. A theory of thermoelasticity based on such an equation is considered. The proposed theory interpolates classical thermoelasticity and a thermoelasticity without energy dissipation introduced by Green and Naghdi.

Effects of thermal relaxation time on plane wave propagation under two-temperature thermoelasticity

The present paper is concerned with an in-depth investigation of the propagation of harmonic plane waves in elastic media in the context of the linear theory of two-temperature generalized thermoelasticity. The exact dispersion relation solutions for the longitudinal plane wave are determined analytically. Asymptotic expansions of several characterizations of the wave field, like phase velocity, specific loss and penetration depth of the dilatational waves, are obtained for both the high frequency as well as low frequency values. The effects of thermal relaxation parameter on the plane wave is analyzed in details by comparing the theoretical as well as numerical results of the present work with the corresponding results in the context of classical two-temperature thermoelasticity theory as reported earlier.

Differential operators associated with the equations of motion and nondissipative heat conduction in the Green-Naghdi theory of thermoelasticity

We consider a wave (hyperbolic) heat conduction theory of the Green-Naghdi type. In the framework of the continual approach, such a theory permits describing low-temperature phenomena of quantum nature (for example, the second sound effect) that are not within reach of the classical parabolic thermoelasticity. At low temperatures, the medium heat conductivity experiences a substantial increase, which should be reckoned with in the design and analysis of cryogenic devices. This change in heat conduction properties results from the wave character of heat propagation, which cannot be taken into account in classical heat conductivity models of the diffusion type but can be described by hyperbolic models of the Green-Naghdi type. That is why considerable attention has been paid to the development of hyperbolic heat conduction models. The present paper deals with analytic solution methods for the corresponding nonself-adjoint initial-boundary value problems.

Generalized magneto-thermoelasticity in a nonhomogeneous isotropic hollow cylinder using the finite element method

In this paper, we constructed the equations of generalized magneto-thermoelasticity in a perfectly conducting medium. The formulation is applied to generalizations, the Lord–Shulman theory with one relaxation time, and the Green–Lindsay theory with two relaxation times, as well as to the coupled theory. The material of the cylinder is supposed to be nonhomogeneous isotropic both mechanically and thermally. The problem has been solved numerically using a finite element method. Numerical results for the temperature distribution, displacement, radial stress, and hoop stress are represented graphically. The results indicate that the effects of nonhomogeneity, magnetic field, and thermal relaxation times are very pronounced. In the absence of the magnetic field or relaxation times, our results reduce to those of generalized thermoelasticity and/or classical dynamical thermoelasticity, respectively. Results carried out in this paper can be used to design various nonhomogeneous magneto-thermoelastic elements under magnetothermal load to meet special engineering requirements.

Uniqueness and reciprocal theorems in linear micropolar electro-magnetic thermoelasticity with two relaxation times

A general model for the linear micropolar electro-magnetic thermoelastic continuum based on the hyperbolic heat equation, which is physically more relevant than the classical thermoelasticity theory in analyzing problems involving very short intervals of time and/or very high heat fluxes, is introduced. An integral identity that involves two admissible processes at different instants is established. Uniqueness theorem is proved, with no definiteness assumption on the elastic constitutive coefficients and no restrictions on the electro-elastic coupling moduli, magneto-elastic coupling moduli, and thermal coupling coefficients other than symmetry conditions. The reciprocity theorem is derived, without the use of Laplace transforms. The integral representation formula is obtained in case instantaneous concentrated, time-continuous or time-harmonic loads are applied. The Maysel’s, Somigliana’s and Green’s formulas are derived. The mixed boundary value problem is considered and a system of five singular Fredholm integral equations is obtained. The results for dynamic classical coupled theory can be easy deduced from the given general model formulated for the temperature-rate dependent thermoelasticity.

Asymptotic behavior of discontinuous solutions in 3-D thermoelasticity with second sound

This paper is devoted to the study of the Cauchy problem for linear and semilinear thermoelastic systems with second sound in three space dimensions with discontinuous initial data. Due to Cattaneo’s law, replacing Fourier’s law for heat conduction, the thermoelastic system with second sound is hyperbolic. We investigate the behavior of discontinuous solutions as the relaxation parameter tends to zero, which corresponds to a formal convergence of the system to the hyperbolic-parabolic type of classical thermoelasticity. By studying expansions with respect to the relaxation parameter of the jumps of the potential part of the system on the evolving characteristic surfaces, we obtain that the jump of the temperature goes to zero while the jumps of the heat flux and the gradient of the potential part of the elastic wave are propagated along the characteristic curves of the elastic fields when the relaxation parameter goes to zero. An interesting phenomenon is when time goes to infinity: the behavior will depend on the mean curvature of the initial surface of discontinuity. These jumps decay exponentially when time goes to infinity, more rapidly for a smaller heat conductive coefficient in linear problems and in nonlinear problems when certain growth conditions are imposed on the nonlinear functions.

Fractional heat conduction equation and associated thermal stresses in an infinite solid with spherical cavity

In this work, the temperature distribution and thermal stresses in an infinite medium with a spherical cavity are studied in the framework of a quasi-static uncoupled theory of thermoelasticity based on heat conduction equation with a time fractional derivative of order 0 < α≤ 2. The Caputo fractional derivative is used. As the fractional heat conduction equation in the case 1 ≤ α ≤ 2 interpolates the standard heat conduction equation (a = 1) and the wave equation (a = 2), the proposed theory interpolates the classical thermoelasticity and the thermoelasticity without energy dissipation introduced by Green and Naghdi. The solution is obtained using the integral transform technique. Numerical results are illustrated graphically.