On the nature of the relaxation time,the Maxwell–Cattaneo and Fourier law in the thermodynamics of a continuous medium,and the scale effects in thermal conductivity

We present the theory of space–time elasticity and demonstrate that it is the extended reversible thermodynamics and gives the coupled model of thermoelasticity and heat conductivity and involves traditional thermoelasticity. We formulate the generally covariant variational model’s dynamic thermoelasticity and heat conductivity in which the basic kinematic and static variables are unified tensor objects (subject, matter). Variation statement defines the whole set of the initial-boundary problems for the 4D vector governing equation (Euler equation), the spatial projections of which define motion equations and the time projection gives the heat conductivity equation. We show that space–time elasticity directly implies the Fourier and the Maxwell–Cattaneo laws of heat conduction. However, space–time elasticity is richer than classical thermoelasticity, and it advocates its own equations of motion for coupled thermoelasticity. Moreover, we establish that the Maxwell–Cattaneo law and Fourier law can be defined for the reversible processes as compatibility equations without introducing dissipation. We argue that the present framework of space–time elasticity should prove adequate to describe the thermoelastic phenomena at low temperatures for interpreting the results of molecular simulations of heat conduction in solids and for the optimal heat and stress management in the microelectronic components and the thermoelectric devices.     

Theory of 4D-media with stationary dislocations

In earlier studies, the authors showed that an application of classical methods of mechanics of deformable media to the study of properties of 4D-space-time continuum permit stating consistent models of nonholonomic media mechanics consistent with the first and second laws of thermodynamics. In the present paper, we show that the classical methods of continuum mechanics are also promising when modeling physical processes. It is shown that, just as in the three-dimensional theory of stationary dislocations, there exist dislocations of three types for a generalized 4D-medium. They correspond to the decomposition of the free distortion tensor into a spherical tensor, a deviator tensor, and a pseudotensor of rotations. We interpret several particular models, thus showing that the proposed model describes the spectrum of known physical interactions: electromagnetic, strong, weak, and gravitational. We show that the resolving equations include the Maxwell equations of electrodynamics and the Yukawa equations for strong interactions as subsystems.     

Continuum model for a composition of shells of revolution with thermosensitive thickness

An integral variational principle is used to derive the equations of uncoupled thermoelasticity for a composition of thin-walled shells of revolution—a cone, a torus or sphere, and a cylinder smoothly connected along the junction lines. The study is based on a Reissner-typemodel under the assumption that the thickness is sensitive to heating. The generalized position vector of any point on the middle surface is constructed, which permits standardly determining the principal curvatures and components of the basis metric tensor of the middle surface. Equations for temperature functions are derived under the assumption that the temperature field is linear across the thickness of the composition and there are no internal heat sources. The equations of static thermostability and axially symmetric thermoelasticity are written.     

Fundamentals of viscoelastoplasticity and hardening theory revisited

Thermodynamic and statistical methods for setting up the constitutive equations describing the viscoelastoplastic deformation and hardening of materials are proposed. The thermodynamic method is based on the law of conservation of energy, the equations of entropy balance and entropy production in the presence of self-balanced internal microstresses characterized by conjugate hardening parameters. The general constitutive equations include the relationships between the thermodynamic flows and forces, which follow from nonnegative entropy production and satisfy the generalized Onsager’s principle, and the thermoelastic relations and the expression for entropy, which follow from the law of conservation of energy. Specific constitutive equations are derived by representing the dissipation rate as a sum of two terms responsible for kinematic and isotropic hardening and approximated by power and hyperbolic-sinus functions. The constitutive equations describing viscoelastoplastic deformation and hardening are derived based on stochastic microstructural concepts and on the linear thermoelasticity model and nonlinear Maxwell model for the spherical and deviatoric components of microstresses and microstrains, respectively. The problem of determining the effective properties and stress-strain state of a three-component material found using the Voigt-Reuss scheme leads to constitutive equations similar in form to those produced by the thermodynamic method __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 3–18, February 2008.     

THERMOELASTIC DAMPING IN A MICRO-BEAM RESONATOR TUNABLE WITH PIEZOELECTRIC LAYERS

In this paper,thermoelastic damping (TED) in a micro-beam resonator with a pair of piezoelectric layers bonded on its upper and lower surfaces is investigated.Equation of motion is derived and the ther...     

RESTUDY OF COUPLED FIELD THEORIES FOR MICROPOLAR CONTINUA (Ⅰ)──MICROPOLAR THERMOELASTICITY

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A microstructure continuum approach to electromagneto-elastic conductors

A micromorphic continuum model of a deformable electromagnetic conductor is established introducing microdensities of bound and free charges. The conductive part of electric current consists of contributions due to free charges and microdeformation. Beside the conservation of charge, we derive suitable evolution equations for electric multipoles which are exploited to obtain the macroscopic form of Maxwell’s equations. A constitutive model for electromagneto-elastic conductors is considered which allows for a natural characterization of perfect conductors independently on the form of the constitutive equation for the conduction current. A generalized Ohm’s law is also derived for not ideal conductors which accounts for relaxation effects. The consequences of the linearized Ohm’s law on the classic magnetic transport equation are shown.     

Generalized thermoelastic infinite transversely isotropic body with a cylindrical cavity due to moving heat source and harmonically varying heat

In this paper, the induced temperature, displacement, and stress fields in an infinite transversely isotropic unbounded medium with cylindrical cavity due to a moving heat source and harmonically varying heat are investigated. This problem is solved in the context of the linear theory of generalized thermoelasticity with dual phase lag model. The governing equations are expressed in Laplace transform domain. Based on Fourier series expansion technique the inversion of Laplace transform is done numerically. The numerical estimates of the displacement, temperature and stress are obtained and presented graphically. The theories of coupled thermoelasticity, generalized thermoelasticity with one relaxation time, and thermoelasticity without energy dissipation can extracted as special cases. Some comparisons have been shown in figures to present the effect of the heat source, dual phase lags parameters and the angular frequency of thermal vibration on all the studied fields.     

EFFECT OF TEMPERATURE-DEPENDENT PROPERTIES ON THERMOELASTIC PROBLEMS WITH THERMAL RELAXATIONS

Based on the generalized thermoelasticity proposed by Green and Lindsay, the dynamic response of generalized thermoelastic problems with temperature-dependent material properties is investigated. The governing equations are formulated and found to be nonlinear because of the temperature-dependence of properties. Owing to the nonlinearity of the governing equations, the finite element method is resorted to for solution. The results obtained show that the temperature-dependent properties influence the variables considered by reducing their magnitudes. This indicates that taking the temperature-dependence of properties into consideration in the investigation of generalized thermoelastic problems is necessary and practical for accurately predicting the thermoelastic behavior.     

Closed solutions of boundary-value problems of coupled thermoelasticity

Coupled equations of thermoelasticity take into account the effect of nonuniform heating on the medium deformation and that of the dilatation rate on the temperature distribution. As a rule, the coupling coefficients are small and it is assumed, sometimes without proper justification, that the effect of the dilatation rate on the heat conduction process can be neglected. The aim of the present paper is to construct analytical solutions of some model boundary-value problems for a thermoelastic bounded body and to determine the body characteristic dimensions and the medium thermomechanical moduli forwhich it is necessary to take into account that the temperature and displacement fields are coupled. We consider some models constructed on the basis of the Fourier heat conduction law and the generalized Cattaneo-Jeffreys law in which the heat flux inertia is taken into account. The solution is constructed as an expansion in a biorthogonal system of eigenfunctions of the nonself-adjoint operator pencil generated by the coupled equations of motion and heat conduction. For the model problem, we choose a special class of boundary conditions that allows us to exactly determine the pencil eigenvalues.     

Thermoelasticity without energy dissipation

This paper deals with thermoelastic material behavior without energy dissipation; it deals with both nonlinear and linear theories, although emphasis is placed on the latter. In particular, the linearized theory of thermoelasticity discussed possesses the following properties: (a) the heat flow, in contrast to that in classical thermoelasticity characterized by the Fourier law, does not involve energy dissipation; (b) a constitutive equation for an entropy flux vector is determined by the same potential function which also determines the stress; and (c) it permits the transmission of heat as thermal waves at finite speed. Also, a general uniqueness theorem is proved which is appropriate for linear thermoelasticity without energy dissipation.     

ELECTRO-MAGNETO-THERMOELASTIC PLANE WAVES IN MICROPOLAR SOLID INVOLVING TWO TEMPERATURES

<正>The model of equations of micropolar generalized magneto-thermoelasticity is introduced within the context of the theory of two temperatures generalized thermoelasticity and we consider a problem of an isotropic homogeneous micropolar medium taking into account the heat effects and allowing the magnetic field effects.A plane wave analysis is employed to obtain the exact formulas of the two temperatures(conductive and mechanical),displacement components, micro-rotation components,stresses,couple stresses,induced electric current,electric field and magnetic field.Arbitrary application is chosen to enable us to get the complete solution.The considered variables are presented graphically and discussions are made for the results.     

属性与温度相关材料的广义热弹性问题有限元分析

计及材料特性与温度的相关性,基于Lord和Shulman(L-S)广义热弹性理论,建立了此类问题的有限元控制方程. 由于材料属性的温度相关性,温度控制方程具有非线性,积分变换求解方法难以采用,因而将有限元方程直接在时间域求解. 利用所建立方法研究了材料特性与温度相关、带有孔洞的无限大体在热冲击和机械冲击作用下的广义热弹性问题. 分析表明,在时间域直接求解材料属性与温度相关的广义热弹性问题是可行的,所得结果具有很高的精度,热的波动性得到充分的展现. 同时发现,热冲击载荷作用时,材料属性与温度的相关性对结构的机械响应影响显著,对温度响应影响很小;机械载荷作用时,材料参数与温度的相关性对所有响应影响都很小. 因此,研究热冲击载荷作用的机械响应时,必须考虑材料属性的温度相关性,而研究温度响应时,无论何种冲击载荷,都可以不考虑材料属性的温度相关性.      

A continuum theory for viscoelastic composite beams

Summary A dynamical continuum theory is developed for laminated composite beams. Starting with an assumed displacement- and temperature field, the one-dimensional approximate theory is consistently constructed within the frame of the three-dimensional theory of linear, nonisothermal, anisotropic, coupled viscoelasticity. Each constituent of the beam may possess different constant thickness and mechanical properties. All dynamic interactions between the adjacent constituents are included. Further, the effects of transverse shear and normal strains and rotatory inertia as well as those of cross-sectional distortion are all taken into account. The resulting equations consist of the macroscopic beam equations of motion and heat conduction, the kinematical relations, the initial and boundary conditions and the constitutive equations, and they govern the extensional, flexural and torsional motions of laminated composite beams. The special cases of constituents which made of either isotropic thermoviscoelastic or anisotropic thermoelastic materials are discussed briefly.Supported by the Office of Naval Research.With 1 figure     

On the modeling of a prestressed thermoelectroelastic half-space with a coating

The constitutive equations of nonlinear mechanics of a prestressed electrothermoelastic continuum are linearized in the framework of the theory of small strains imposed on finite strains. Simple and convenient-to-operate formulas of linearized constitutive equations and equations of motion of the mediumare obtained. A model of electrothermoelastic half-space with inhomogeneous coating, which is a structure of functionally graded layers, is proposed. It is assumed that each of the medium components is under the action of initial mechanical strains and initial temperature, and the materials of the medium components are orthotropic pyroelectric materials of hexagonal crystal system of class 6 mm. The integral representation of the mediumwave field is constructed by a hybrid numerical-analytical method based on a combination of analytical solutions and numerical schemes used to reconstruct the Green function for the inhomogeneous components of the coating and the matrix approach used to satisfy the boundary conditions.     

Two-temperature generalized thermoelastic infinite medium with cylindrical cavity subjected to moving heat source

In this work, a problem of thermoelastic interactions in an elastic infinite medium with cylindrical cavity thermally shocked at its bounding surface and subjected to moving heat source with constant velocity has been solved. The governing equations are taken in the context of two-temperature generalized thermoelasticity theory (Youssef model). The analytical solution with direct approach in the Laplace transforms domain has been obtained. The derived analytical expressions have been computed for specific situations. Numerical results for the dynamical and conductive temperatures, stress, strain, and displacement are represented graphically with comparisons by one-temperature generalized thermoelasticity (Lord–Shulman model).     

Convection and diffusion of polymer network strands

A review of several important constitutive equations is herein conducted with an eye towards determining those most suitable for use in modelling polymer melt processing. General principles are invoked for a priori screening of the equations without needing detailed comparison of the model predictions with experimental data. These principles, which are derived from continuum mechanics, thermodynamics and molecular kinetic theory, and dela with convection and diffusion of entangled polymer strands during flow, are: (1) During sudden deformations, the stress is a unique function of the total strain. (2) The second law of thermodynamics holds for all deformations. (3) The constitutive equation can be derived from a plausible molecular model which describes the convection and diffusion. (4) The model parameters can be determined by a reasonable number of rheometric experiments. Based on these principles, it is concluded that separable free energy models are the most promising. These are either BKZ integral models with a kernel factorable into a time-dependent and a strain-dependent part. or sets of Maxwell-type differential equations that employ a generalized convected derivative, and that are linear in stress in the absence of flow.     

On canonical equations of continuum thermomechanics

It is shown that the canonical balance of momentum of continuum mechanics can be formulated in a general way, but not independently of the usual balance of linear momentum, even in the absence of specified constitutive equations. A parallel construct is made of necessity for the accompanying time-like canonical energy equation. On specifying the energy, previous particular cases can be deduced including pure elasticity, inhomogeneous thermoelasticity of conductors, and the case of dissipative solid-like materials described by means of a diffusive internal variable (such as in damage or weakly non-local plasticity). A redefinition of the entropy flux is necessarily accompanied by a redefinition of the Eshelby stress tensor.     

On the equations of motion for accelerated frames

In this communication we present the equations of Euler generalized for the motion of a body in an accelerated reference frame using the generalized work-energy principle. The equivalence among the generalized Euler equation, the generalized Lagrange equation, and the generalized Kane equation are shown when applied to the motion of a body of a holonomic system that depend onn generalized coordinates. Therefore when the generalized coordinates can be reduced to two sets of independent coordinates, the generalized Euler equation can be split into two uncoupled equations that are not independent of each other.Universidade da Beira Interior, Covilhã, Portugal. Published in Prikladnaya Mekhanika, Vol. 31, No. 9, pp. 79–89, September, 1995.     

Restudy of coupled field theories for micropolar continua (I)—Micropolar thermoelasticity

Problems of micropolar thermoelasticity have been presented and discussed by some authors in the traditional framework of micropolar continuum field theory. In this paper the theory of micropolar thermoelasticity is restudied. The reason why it was restricted to a linear one is analyzed. The rather general principle of virtual work and the new formulation for the virtual work of internal forces as well as the rather complete Hamilton principle in micropolar thermoelasticity are established. From this new Hamilton principle not only the equations of motion, the balance equation of entropy, the boundary conditions of stress, couple stress and heat, but also the boundary conditions of displacement, microrotation and temperature are simultaneously derived. Contributed by DAI Tian-min Foundation item: the National Natural Science Foundation of China (10072024); the International Cooperation Project of the NSFC (10011130235) and the DFG (51520001); the Research Foundation of Liaoning Education Committee (990111001) Biography: DAI Tian-min (1931-)