Influence of boundary conditions on the solution of a hyperbolic thermoelasticity problem

We consider a series of problems with a short laser impact on a thin metal layer accounting various boundary conditions of the first and second kind. The behavior of the material is modeled by the hyperbolic thermoelasticity of Lord–Shulman type. We obtain analytical solutions of the problems in the semi-coupled formulation and numerical solutions in the coupled formulation. Numerical solutions are compared with the analytical ones. The analytical solutions of the semi-coupled problems and numerical solutions of the coupled problems show qualitative match. The solutions of hyperbolic thermoelasticity problems are compared with those obtained in the frame of the classical thermoelasticity. It was determined that the most prominent difference between the classical and hyperbolic solutions arises in the problem with fixed boundaries and constant temperature on them. The smallest differences were observed in the problem with unconstrained, thermally insulated edges. It was shown that a cooling zone is observed if the boundary conditions of the first kind are given for the temperature. Analytical expressions for the velocities of the quasiacoustic and quasithermal fronts as well as the critical value for the attenuation coefficient of the excitation impulse are verified numerically.     

Dependence of modulus of elasticity and thermal conductivity on reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity

Introduction Thetheoryofgeneralizedthermoelasticitywithonerelaxationtimebasedonamodified Fourier’slawofheatconductionwasdevelopedbyLordandShulman[1].Thistheoryallowsfor theso_calledsecond_soundeffectsinsolids,hencethermaldisturbancespropagatewithfinite wavespeeds. Themathematicalmodelofthegeneralizedthermoelasticitytheoryisofacomplicatednature thathindersthepossibilityofderivingananalyticalsolution.Mostattemptsdealingwiththese equationsarebasedoneithershort_timesolution[2-4]. Modernstructur…     

Dispersion relations for SH wave in magneto-electro-elastic heterostructures

In the present work the dispersion relations of stationary SH waves in a heteroestructure with magneto-electro-elastic properties have been obtained. The calculations were done taking into consideration the symmetry of the system and separating the solutions in symmetric and anti-symmetric parts. Different limit cases are presented. The dispersion curves and amplitudes of vibration are shown for different configurations.     

Effect of the thermal relaxation and magnetic field on generalized micropolar thermoelasticity

The model of generalized micropolar magneto-thermoelasticity for a thermally and perfectly conducting half-space is studied. The initial magnetic field is parallel to the boundary of the half-space. The formulation is applied to the generalized thermo-elasticity theories of Lord and Shulman, Green and Lindsay, as well as to the coupled dynamic theory. The normal mode analysis is used to obtain expressions for the temperature increment, the displacement, and the stress components of the model at the interface. By using potential functions, the governing equations are reduced to two fourth-order differential equations. By numerical calculation, the variation of the considered variables is given and illustrated graphically for a magnesium crystal micropolar elastic material. Comparisons are performed with the results predicted by the three theories in the presence of a magnetic field.     

A generalized thermal boundary condition for the hyperbolic heat conduction model

 A generalized thermal boundary condition is derived for the hyperbolic heat conduction equation to include all thermal effects of a thin layer, whether solid-skin or fluid film, moving or stationary, in perfect or imperfect thermal contact with an adjacent domain. The thin layer thermal effects include, among others, thermal capacity of the layer, thermal diffusion, enthalpy flow, viscous dissipation within the layer and convective losses from the layer. Six different kinds of thermal boundary conditions can be obtained as special cases of the generalized boundary condition. The importance of the generalized boundary condition is demonstrated comprehensively in an example. The effects of different geometrical and thermophysical properties on the validity of the generalized thermal boundary condition are investigated. Received on 23 May 2001 / Published online: 29 November 2001     

Dispersion relation in the limit of high frequency for a hyperbolic system with multiple eigenvalues

The results of a previous paper (Muracchini et al., 1992) are generalized by considering a hyperbolic system in one space dimension with multiple eigenvalues. The dispersion relation for linear plane waves in the high-frequency limit is analyzed and the recurrence formulas for the phase velocity and the attenuation factor are derived in terms of the coefficients of a formal series expansion in powers of the reciprocal of frequency. In the case of multiple eigenvalues, it is also verified that linear stability implies λ$\lambda$-stability for the waves of weak discontinuity. Moreover, for the linearized system, the relationship between entropy and stability is studied. When the nonzero eigenvalue is simple, the results of the paper mentioned above are recovered. In order to illustrate the procedure, an example of the linear hyperbolic system is presented in which, depending on the values of parameters, the multiplicity of nonzero eigenvalues is either one or two. This example describes the dynamics of a mixture of two interacting phonon gases.     

Dispersion relations for electroelastic shear waves in a periodically layered medium

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 26, No. 11, pp. 84–93, November, 1990.     

Nonlinear problems of the thermal conductivity equation

The asymptotic behavior of solutions of parabolic equations at infinite times has been investigated for various cases [1–6]. Two initial boundary-value problems are considered in this paper. The solution of the thermal conductivity equation with a nonlinear right-hand side is found, including also nonlinear boundary conditions. It is shown that the solution of the corresponding problem tends either to a stable, steady-state solution, or to a periodic solution, depending on the initial values of the functions and constants appearing in the conditions of the problem. Other papers [7, 8] are devoted to finding the periodic solutions of these two problems encountered in hydrodynamics (diffusion, underground hydrodynamics), and to studying the asymptotic behavior of the corresponding initial boundary problems.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 3, pp. 123–128, May–June, 1972.     

Measurements of the thermal conductivity of gases and gas mixtures,methods and results

Two methods for measuring the thermal conductivity have been employed:

1. a cylindrical cell type of apparatus for absolute measurements, with an accuracy of 0.5%.
2. a thermistor katharometer bridge for differential measurements, allowing an accuracy of 0.1%. The thermistor bridge is described in some detail and its performance is analysed.

Results are presented for a number of pure gases and binary mixtures. A special study was made of mixtures with water vapour or heavy water vapour as a component. Measurements were performed at room temperature and 333 K. In general our results compare well with those of other investigators, although differences are noted that surpass the degree of accuracy we believe to hold for our values. For a number of the binary mixtures investigated no results have -to our knowledge-been reported previously. The results for six of the binary mixtures containing water as a component are represented by a Wassiljewa-type formula. No agreement was obtained with values calculated from four different theories based on this formula.     

Laboratory techniques to evaluate thermal conductivity for some soils

The thermal conductivity of two soils was investigated through laboratory studies. These laboratory experiments used the single probe and dual probe methods to measure and compare thermal conductivities. The soils used were classified as sand and loam. Thermal conductivity measured using single probe method ranged from 0.95 to 2.11 for sand and from 0.49 to 0.76 W/m K for loam. Thermal conductivity measured using dual probe method ranged from 0.98 to 2.17 for sand and from 0.51 to 0.78 W/m K for loam. Finally, it was found that sand had higher values of thermal conductivity than loam for all soil conditions studied.     

On thermodynamic potentials in thermoelasticity under small strain and finite thermal perturbation assumptions

Expressions for thermodynamic potentials (internal energy, Helmholtz energy, Gibbs energy and enthalpy) of a thermoelastic material are developed under the assumption of small strains and finite changes in the thermal variable (temperature or entropy). The literature provides expressions for the Helmholtz energy in terms of strain and temperature, most often as expansions to the second order in strain and to a higher order in temperature changes, which ensures an affine stress–strain relation and a certain temperature dependence of the moduli of the material. Expressions are here developed for the four potentials in terms of all four possible pairs of independent variables. First, an expression is obtained for each potential as a quadratic function of its natural mechanical variable with coefficients depending on its natural thermal variable that are identified in terms of the moduli of the material. The form of the coefficients’ dependence on the thermal variable is not specified beforehand so as to obtain the most general expressions compatible with an affine stress–strain relation. Then, from each potential expressed in terms of its natural variables, expressions are derived for the other three potentials in terms of these same variables using the Gibbs–Helmholtz equations. The paper provides a thermodynamic framework for the constitutive modeling of thermoelastic materials undergoing small strains but finite changes in the thermal variables, the properties of which are liable to depend on the thermal variables.     

Dispersion relations and mode shapes for waves in laminated viscoelastic composites by variational methods

The propagation of oscillatory waves through periodic elastic composites has been analysed on the basis of the Floquet theory. This leads to self-adjoint differential equation systems which it was proved convenient to solve by variational methods. Many composites, such as the light-weight high-strength boron-epoxy material, consist of strong reinforcing components in a plastic matrix. The latter can exhibit viscoelastic properties which can have a significant influence on wave propagation characteristics. Replacement of the elastic constant by the viscoelastic complex modulus changes the mathematical structure so that the differential equation system is no longer self-adjoint. However, a modification of the variational principles is suggested which retains formal self-adjointness, and yields variational principles which contain additional boundary terms. These are applied to the determination of wave speeds and mode shapes for a laminated composite made of homogeneous elastic reinforcing plates in a homogeneous viscoelastic matrix for plane waves propagating normally to the reinforcing plates. These results agree well with the exact solution which can be evaluated in this simple case. The variational principles permit solutions for periodic, but otherwise arbitrary variation of material properties.     

Numerical prediction of the thermal conductivity of fibers

The need to determine the thermal conductivity of fibers for design purposes of new composite materials and the inherent difficulties in the direct measurement of the thermal conductivity of fibers motivated the present work due to its importance for energy conservation purposes. In this work, a correlation formula is developed to predict the thermal conductivities of fiber as function of the effective thermal conductivity of a fiber-reinforced composite laminates and their constituents which are easy to measure. The parallel and series thermal models of composite walls have been utilized in developing this correlation equation. The coefficients of this formula can be given as functions of the voids volume fraction for each fiber to resin volume ratio considered. The validity of the models is verified through finite element analysis. This model also shows excellent agreement with the available experimental values.     

Model for predicting effective thermal conductivity of composites with aligned continuous fibers of graded conductivity

Governing differential equations in both transverse and longitudinal directions for predicting the effective thermal conductivities of composites with aligned, graded continuous fibers are derived. It is shown that the effective conductivities of composites with graded fibers are predicted by solving the equations. The results by the present approach are applicable to both dilute and non-dilute cases without additional procedures unlike other approaches. The results are compared with those in the literature, and the applicability of the present approach is justified. A solution by the present approach is obtained analytically or numerically as long as thermal conductivity profile of fibers is given.     

A network theory for the thermal conductivity of an amorphous polymeric material

A model to relate the thermal conductivity tensor to the deformation of an amorphous polymeric material above the glass transition temperature is presented. The basis of the model is formed by the transient network theory for polymer melts. With this theory it is possible to calculate the average orientation of the macromolecular segments as a function of the history of the deformation. Combined with an expression which relates the thermal conductivity to the orientation of the molecules, this provides us with the information needed to calculate the heat conduction tensor. Despite the fact that the simplest possible network model is chosen, there is good agreement with the sparse, experimental results.     

Effect of thermal losses on the microscopic hyperbolic heat conduction model

The effects of radiative losses on the thermal behavior of thin metal films, as described by the microscopic two-step hyperbolic heat conduction model, are investigated. Different criteria, which determine the ranges within which thermal radiative losses are significant, are derived. It is found that radiative losses from the electron gas are significant in thin films having [(C_R e_e^4/₃ T_￥ ⁴ )/(k_e^1/₃ L^2/₃ G)] 3 4.6 ×10⁷{{C_R \epsilon _e^{{4 \over 3}} T_\infty ^4 } \over {k_e^{{1 \over 3}} L^{{2 \over 3}} G}}\geq 4.6 \times 10^7 for /_o > 4 and F_F < 1 and [(C_R e_e^3/₂ T_￥ ^9/₂)/(k_e^1/₂ L^1/₂ G)] 3 7.4 ×10¹⁰{{C_R \epsilon _e^{{3 \over 2}} T_\infty ^{{9 \over 2}}} \over {k_e^{{1 \over 2}} L^{{1 \over 2}} G}}\geq 7.4 \times 10^{10} for /_o < 4 and F_F > 1.