On the Stability of Damped Timoshenko Systems: Cattaneo Versus Fourier Law

We consider hyperbolic Timoshenko-type vibrating systems that are coupled to a heat equation modeling an expectedly dissipative effect through heat conduction. While exponential stability under the Fourier law of heat conduction holds, it turns out that the coupling via the Cattaneo law does not yield an exponentially stable system. This seems to be the first example that a removal of the paradox of infinite propagation speed inherent in Fourier’s law by changing to the Cattaneo law causes a loss of the exponential stability property. Actually, for systems with history, the Fourier law keeps the exponential stability known for the pure Timoshenko system without heat conduction, but introducing the Cattaneo coupling even destroys this property. This work was supported by the DFG-project “Hyperbolic Thermoelasticity” (RA 504/3-1).     

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.     

Uniqueness theorems in the theory of thermoelasticity for solids with double porosity

In this paper the coupled linear theory of thermoelasticity for solids with double porosity is considered. The governing system of field equations of this theory is based on motion equations, conservation of fluid mass, constitutive equations, extended Darcy’s law for materials with double porosity and Fourier’s law for heat conduction. A wide class of the basic internal and external boundary value problems (BVPs) of steady vibrations is formulated and uniqueness theorems for regular (classical) solutions of these BVPs are proved.     

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.     

Fractional order theory in thermoelastic solid with three-phase lag heat transfer

In this work, the field equations of the linear theory of thermoelasticity have been constructed in the context of a new consideration of Fourier law of heat conduction with time-fractional order and three-phase lag. A uniqueness and reciprocity theorems are proved. One-dimensional application for a half-space of elastic material in the presence of heat sources has been solved using Laplace transform and state space techniques Ezzat (Canad J Phys Rev 86:1241–1250, 2008). According to the numerical results and its graphs, conclusion about the new theory has been established.     

Heat propagation in a nonuniform rod of variable cross section

An integral formula is used to average a coupled problem of thermoelasticity for a nonuniform rod of variable cross section. Effective characteristics are found. It is shown that, in addition to the expected effective coefficients, there appear five independent coefficients characterizing the temperature change rate effect on the stresses in the rod, on the longitudinal heat flux, and on the entropy distribution along the length of the rod. A feature of these new coefficients is that they become equal to zero in the case of a uniform rod. The homogenization of the thermoelasticity equations for nonuniform rods allows one to propose a new theory of heat conduction in rods. This new theory differs from the classical one by the fact that some new terms are added to the Duhamel–Neumann law, to the Fourier heat conduction law, and to the entropy expression. These new terms are proportional to the temperature change rate with time. It is also shown that, in the new theory of heat conduction, the propagation velocity of harmonic heat perturbations is dependent on the oscillation frequency and is finite when the frequency tends to infinity.     

Modeling of multilayer thin bodies

In this work, we considered the new parametrization of a multilayer thin domain. In particular, in contrast to classic approaches, we used several base surfaces and an analytic method with the application of orthogonal polynomial systems. We gave the vector parametric equation of each layer and the system of vector parametric equations of a multilayer thin domain and introduced the geometric characteristics for the proposed parametrization. We also derived the expressions for the transfer components of the second-rank identity tensor and the relations connecting the various families of bases and presented some differential operators, the system of equations of motion, the heat flow equation, the constitutive relations of the theory of the micropolar elasticity, and the Fourier heat conduction law under this parametrization of the thin-body domain. Finally, we gave the classification and statements of boundary value problems in the theory of thin bodies.     

Evolutionary model of finite-strain thermoelasticity

The equation of state of finite-strain thermoelasticity is obtained using a formalized approach to constructing constitutive relations for complex media under the assumption of closeness of intermediate and current configurations. A variational formulation of the coupled thermoelastic problem is proposed. The constitutive equation, the heat-conduction equation, the relations for internal energy, free energy, and entropy, and the variational formulation of the coupled problem of finite-strain thermoelasticity are tested on the problem of uniaxial extension of a bar. The model adequately describes experimental data for elastomers, such as entropic elasticity, temperature inversion, and temperature variation during an adiabatic process. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 3, pp. 184–196, May–June, 2008.     

Thermoconductivity and thermoelasticity of thin isotropic shells heated by a moving heat pulse

The paper presents a solution to the problem of thermal conduction and thermoelasticity for a thin shallow spherical shell heated by a concentrated or local impulsive heat source moving over the shell surface. It is assumed that temperature is linearly distributed across the shell thickness and that the shell, on its sides, exchanges heat with the environment in accordance with Newton’s law of cooling. The Fourier and Laplace transforms are used to find an analytic solution. The dependence of the temperature field and stress/strain components on the type of heating and the form of heat source is studied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 11, pp. 85–92, November 2006.     

Dispersion relations for the hyperbolic thermal conductivity,thermoelasticity and thermoviscoelasticity

The Maxwell–Cattaneo heat conduction theory, the Lord–Shulman theory of thermoelasticity and a hyperbolic theory of thermoviscoelasticity are studied. The dispersion relations are analyzed in the case when a solution is represented in the form of an exponential function decreasing in time. Simple formulas that quite accurately approximate the dispersion curves are obtained. Based on the results of analysis of the dispersion relations, an experimental method of determination of the heat flux relaxation time is suggested.     

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...     

The Cattaneo type space-time fractional heat conduction equation

The classical heat conduction equation is generalized using a generalized heat conduction law. In particular, we use the space-time Cattaneo heat conduction law that contains the Caputo symmetrized fractional derivative instead of gradient ${{\partial_x}}$ and fractional time derivative instead of the first order partial time derivative ${{\partial_t}}$ . The existence of the unique solution to the initial-boundary value problem corresponding to the generalized model is established in the space of distributions. We also obtain explicit form of the solution and compare it numerically with some limiting cases.     

On some diffusive waves in non-linear heat conduction

We address the non-linear heat conduction in the presence of absorption for the case of spherical symmetry geometry. The non-linear model is based on both a temperature-dependent thermal conductivity and a non-linear generalization of the Fourier law. The governing equation belongs to a class of degenerate parabolic equations. We obtain similarity solutions in closed form for the Cauchy problem corresponding to an instantaneous point source problem. We investigate the non-linear effects on the propagation of the temperature distrubances. We find that in certain cases the temperature distribution displays travelling wave characteristics. The solution for the Cauchy problem is recovered by considering a suitable first boundary value problem.     

Stress solutions to the three-dimensional problem of elasticity

New representations of the stress tensor in the linear theory of elasticity and thermoelasticity are proposed. These representations satisfy the equilibrium equations and the strain compatibility equation. The stress tensor is expressed in terms of a harmonic tensor or a harmonic vector. The second boundary-value problem for an elastic half-space and an elastic layer is solved as an example __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 8, pp. 3–35, August 2006.     

Analytical solution for the time-fractional heat conduction equation in spherical coordinate system by the method of variable separation

In this paper,using the fractional Fourier law,we obtain the fractional heat conduction equation with a time-fractional derivative in the spherical coordinate system.The method of variable separation is used to solve the timefractional heat conduction equation.The Caputo fractional derivative of the order 0 < α≤ 1 is used.The solution is presented in terms of the Mittag-Leffler functions.Numerical results are illustrated graphically for various values of fractional derivative.     

Transient radiative and conductive heat transfer in non-gray semitransparent two-dimensional media with mixed boundary conditions

This paper is devoted to transient heat transfer involving radiation and conduction. Considering a non-gray purely absorbing media, the radiative heat transfer equation (RTE) is solved iteratively with the Discrete Ordinates Method (DOM) using an exponential differencing scheme. The energy balance equation is used to compute temperature at each time step with the Crank–Nicholson technique. Energy equation is coupled to the RTE through the radiative source term. Both equations are discretized with finite differencing schemes. The energy conservation leads to the sparse system of linear equations A× T=B which is solved with a bi-conjugate stabilized gradient technique (BCSG). Validation of the model with different test cases is achieved and application to transient heating of glass is also studied.     

The influence of thermal relaxation on the oscillatory properties of two-gradient convection in a vertical slot

We study the effects of the Maxwell–Cattaneo (MC) law of heat conduction on the flow of a Newtonian fluid in a vertical slot subject to both vertical and horizontal temperature gradients. Working in one spatial dimension (1D), we employ a spectral expansion involving Rayleigh’s beam functions as the basis set, which are especially well-suited to the fourth order boundary value problem (b.v.p.) considered here, and the stability of the resulting dynamical system for the Galerkin coefficients is investigated. It is shown that the absolute value of the (negative) real parts of the eigenvalues are reduced, while the absolute values of the imaginary parts are somewhat increased, under the MC law. This means that the presence of the time derivative of the heat flux increases the order of the system, thus leading to more oscillatory regimes in comparison with the usual Fourier case. Moreover, no eigenvalues with positive real parts were found, which means that in this particular situation, the inclusion of thermal relaxation does not lead to destabilization of the motion.     

A Generalized Piezoelectric Bernoulli–Navier Anisotropic Rod Model

We apply the asymptotic analysis procedure to the three-dimensional static equations of piezoelectricity, for a linear nonhomogeneous anisotropic thin rod. We prove the weak convergence of the rod mechanical displacement vectors and the rod electric potentials, when the diameter of the rod cross-section tends to zero. This weak limit is the solution of a new piezoelectric anisotropic nonhomogeneous rod model, which is a system of coupled equations, with generalized Bernoulli–Navier equilibrium equations and reduced Maxwell–Gauss equations.     

Wave propagation model of heat conduction and group speed

In view of the finite relaxation model of non-Fourier’s law, the Cattaneo and Vernotte (CV) model and Fourier’s law are presented in this work for comparing wave propagation modes. Independent variable translation is applied to solve the partial differential equation. Results show that the general form of the time spatial distribution of temperature for the three media comprises two solutions: those corresponding to the positive and negative logarithmic heating rates. The former shows that a group of heat waves whose spatial distribution follows the exponential function law propagates at a group speed; the speed of propagation is related to the logarithmic heating rate. The total speed of all the possible heat waves can be combined to form the group speed of the wave propagation. The latter indicates that the spatial distribution of temperature, which follows the exponential function law, decays with time. These features show that propagation accelerates when heated and decelerates when cooled. For the model media that follow Fourier’s law and correspond to the positive heat rate of heat conduction, the propagation mode is also considered the propagation of a group of heat waves because the group speed has no upper bound. For the finite relaxation model with non-Fourier media, the interval of group speed is bounded and the maximum speed can be obtained when the logarithmic heating rate is exactly the reciprocal of relaxation time. And for the CV model with a non-Fourier medium, the interval of group speed is also bounded and the maximum value can be obtained when the logarithmic heating rate is infinite.     

Darcy-Forchheimer flow of variable conductivity Jeffrey liquid with Cattaneo-Christov heat flux theory

The role of the Cattaneo-Christov heat flux theory in the two-dimensional laminar flow of the Jeffrey liquid is discussed with a vertical sheet. The salient feature in the energy equation is accounted due to the implementation of the Cattaneo-Christov heat flux. A liquid with variable thermal conductivity is considered in the Darcy-Forchheimer porous space. The mathematical expressions of momentum and energy are coupled due to the presence of mixed convection. A highly nonlinear coupled system of equations is tackled with the homotopic algorithm. The convergence of the homotopy expressions is calculated graphically and numerically. The solutions of the velocity and temperature are expressed for various values of the Deborah number, the ratio of the relaxation time to the retardation time, the porosity parameter, the mixed convective parameter, the Darcy-Forchheimer parameter, and the conductivity parameter. The results show that the velocity and temperature are higher in Fourier's law of heat conduction cases in comparison with the Cattaneo-Christov heat flux model.