Reflection of three-dimensional plane waves at the free surface of a rotating triclinic half-space under the context of generalized thermoelasticity

The reflection of three-dimensional(3D) plane waves in a highly anisotropic(triclinic) medium under the context of generalized thermoelasticity is studied. The thermoelastic nature of the 3D plane waves in an anisotropic medium is investigated in the perspective of the three-phase-lag(TPL), dual-phase-lag(DPL), Green-Naghdi-III(GNIII), Lord-Shulman(LS), and classical coupled(CL) theories. The reflection coefficients and energy ratios for all the reflected waves are obtained in a mathematical form. The rotational effects on the reflection characteristics of the 3D waves are discussed under the context of generalized thermoelasticity. Comparative analyses for the reflection coefficients of the waves among these generalized thermoelastic theories are performed. The energy ratios for each of the reflected waves establish the energy conservation law in the reflection phenomena of the plane waves. The highly anisotropic materials along with the rotation may have a significant role in the phenomenon of the reflection behavior of the 3D waves. Numerical computations are performed for the graphical representation of the study.     

A reciprocal theorem in generalized thermoelasticity

A reciprocal theorem for initial mixed boundary value problems is obtained in the context of the linearized anisotropic thermoelasticity theory of Green and Lindsay.     

On the propagation waves in the theory of thermoelasticity with microtemperatures

The present paper studies the propagation of plane time harmonic waves in an infinite space filled by a thermoelastic material with microtemperatures. It is found that there are seven basic waves traveling with distinct speeds: (a) two transverse elastic waves uncoupled, undamped in time and traveling independently with the speed that is unaffected by the thermal effects; (b) two transverse thermal standing waves decaying exponentially to zero when time tends to infinity and they are unaffected by the elastic deformations; (c) three dilatational waves that are coupled due to the presence of thermal properties of the material. The set of dilatational waves consists of a quasi-elastic longitudinal wave and two quasi-thermal standing waves. The two transverse elastic waves are not subjected to the dispersion, while the other two transverse thermal standing waves and the dilatational waves present the dispersive character. Explicit expressions for all these seven waves are presented. The Rayleigh surface wave propagation problem is addressed and the secular equation is obtained in an explicit form. Numerical computations are performed for a specific model, and the results obtained are depicted graphically.     

A generalized variational principle in micromorphic thermoelasticity

Recently Nappa obtains a Gurtin-type variational principle for micromorphic thermoelasticity. However, the use of convolutions is of course restricted to the linear case, which sets a limit to the rang of applicability. This paper establishes a classic variational principles for the discussed problem by the semi-inverse method.     

Some theorems on two-temperature generalized thermoelasticity

The aim of the present work is to establish a reciprocal principle of Betti type in the context of linear theory of two-temperature generalized thermoelasticity (Youssef in IMA J Appl Math 71:383–390, 2006; Arch Appl Mech 75:553–565, 2006) for homogeneous and isotropic body. Generalizations of the theorems of Somigliana and Green to two-temperature generalized thermoelasticity are also established on the basis of our reciprocal principle.     

Well-posedness of the equations of generalized thermoelasticity

In this paper the local existence, uniqueness and continuous dependence for smooth solutions to the initial value problem for a class of generalized (dependent on the time derivative of temperature) thermoelastic materials is proved. The field equations are written as a quasilinear hyperbolic system and the known results by Hughes, Kato and Marsden are applied.     

A generalized theory of linear micropolar thermoelasticity

Summary The linear theory of thermoelasticity established by Green and Lindsay with the help of an entropy production inequality proposed by Green and Laws is generalized to the case of a homogeneous micropolar continuum. The basic equations are derived using invariance conditions under superposed rigid body motions.

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Sommario La teoria lineare della termoelasticità stabilita da Green e Lindsay sulla base della disuguaglianza della produzione dell'entropia proposta da Green e Laws è generalizzata al caso di un continuo micropolare omogeneo. Le equazioni fondamentali vengono derivate usando condizioni di invarianza rispetto a moti rigidi sovrapposti.

     

Energy and dissipation of inhomogeneous plane waves in thermoelasticity

Inhomogeneous small-amplitude plane waves of (complex) frequency ω are propagated through a linear dissipative material. For thermoelasticity we derive an energy-dissipation equation that contains all the quadratic dependence on the field quantities, see Eq. (10). In addition, we derive a new energy-dissipation equation (Eq. (22)) involving the total energy density which contains terms linear in the field quantities as well as the usual quadratic terms. The terms quadratic in the small quantities in the energy density, energy flux and dissipation give rise to inhomogeneous plane waves of frequency 2ω and to (attenuated) constant terms. Usually these quadratic quantities are time-averaged and only the attenuated constant terms remain. We derive a new result in thermoelasticity for these terms, see Eq. (54). The present innovation is to retain the terms of frequency 2ω, since they are comparable in magnitude to the attenuated constant terms, and a new result, see Eq. (44), is derived for a general energy-dissipation equation that connects the amplitudes of the terms of the energy density, energy flux and dissipation that have frequency 2ω. Furthermore, for dissipative waves or inhomogeneous conservative waves the (complex) group velocity is related to these amplitudes rather than to the attenuated constant terms as it is for homogeneous waves in conservative materials.     

Propagation of discontinuities in electromagneto generalized thermoelasticity in cylindrical regions

In this work we study wave propagation for a problem of an infinitely long solid conducting circular cylinder whose lateral surface is traction free and subjected to known surrounding temperatures in the presence of a uniform magnetic field in the direction of the axis. The problem is in the context of generalized magneto-thermo-elasticity theory with one relaxation time. Laplace transform techniques are used to derive the solution in the Laplace transform domain. The inversion process is carried out using a numerical method based on Fourier series expansions. Wave propagation in the elastic medium and in the free space, bounding it, is investigated.     

Classical and generalized coupled thermoelasticity analysis in one-dimensional layered media

The behavior of thermoelastic waves at the interface of layered medium and distributions of these waves through the domain are examined by applying the direct finite element method to obtain the field variables directly within the spatial and temporal domains. The analysis is performed in a one-dimensional domain with two different layers to provide a means to follow the behavior of the reflected thermoelastic waves at the interface. It appears that the distributions of thermoelastic waves in an isotropic slab with one layer are significantly different from those in multilayered slabs. For instance, the negative displacement waves, several stresses with positive or negative signs and temperature distributions produced in the multilayered domains, are quite different from those in a single layer. This method may be generalized to simulate the propagation of thermoelastic waves in various multilayered regions and analyze the behavior of the layered composite structures under the mechanical or thermal impact loads.     

State-space approach to thermoelastic interactions in generalized thermoelasticity type III

The present work is attempted to formulate the state-space approach to the problems of thermoelastic interactions with energy dissipation on the basis of the theory of generalized thermoelasticity type-III, recently developed by Green and Naghdi. The formulation is then applied to solve a boundary value problem of an isotropic elastic half space with its plane boundary subjected to two different types of boundary conditions: (1) sudden increase in temperature and zero stress and (2) sudden increase in load and zero temperature change. Integral transform method is applied to obtain the solution of the problem. The short time approximated solutions for the field variables have been constructed analytically for both the cases. The problem is illustrated with the help of different graphs of numerical values of the field variables.     

State-space approach of two-temperature generalized thermoelasticity of one-dimensional problem

In this paper, we will consider a half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory [Youssef, H., 2005a. The dependence of the modulus of elasticity and the thermal conductivity on the reference temperature in generalized thermoelasticity for an infinite material with a spherical cavity, J. Appl. Math. Mech., 26(4), 4827; Youssef, H., 2005b. Theory of two-temperature generalized thermoelasticity, IMA J. Appl. Math., 1–8]. The medium is assumed initially quiescent. Laplace transform and state space techniques are used to obtain the general solution for any set of boundary conditions. The general solution obtained is applied to a specific problem of a half-space subjected to thermal shock and traction free. The inverse Laplace transforms are computed numerically using a method based on Fourier expansion techniques. Some comparisons have been shown in figures to estimate the effect of the two-temperature parameter.     

On the propagation of generalized transverse waves in heat-conducting elastic materials

A generalized transverse wave is a propagating acceleration discontinuity on which the temperature and the entropy, together with their gradients, are continuous. In a heat-conducting elastic material the propagation and growth of such waves are uninfluenced by thermomechanical interaction. It is shown in this paper that in any given plane there is at least one direction in which a generalized transverse wave may propagate, and the existence is also proved of at least one direction in which a pair of generalized transverse waves may travel. Necessary and sufficient conditions are established for the speeds of propagation of these waves to be real. Relationships between transverse and generalized transverse waves are also studied, and in the last two sections of the paper the directions in which generalized transverse waves may propagate in an isotropic heat-conducting elastic material are systematically worked out and classified.     

On thermodynamic potentials in linear thermoelasticity

The four thermodynamic potentials, the internal energy u=u(ε_ij,s), the Helmholtz free energy f=f(ε_ij,T), the Gibbs energy g=g(σ_ij,T) and the enthalpy h=h(σ_ij,s) are derived, independently of each other, by using the Duhamel–Neumann extension of Hooke's law and an assumed linear dependence of the specific heat on temperature. A systematic procedure is then presented to express all thermodynamic potentials in terms of four possible pairs of independent state variables. This procedure circumvents a tedious transition from one potential to another, based on the formal change of variables, and inversions of the stress–strain and entropy–temperature relations. The general results are applied to uniaxial loading paths under isothermal, adiabatic, constant stress, and constant strain conditions. An interplay of adiabatic and isothermal elastic constants in the expressions for exchanged heat along certain thermodynamic paths is indicated.     

On the hamiltonian theory of surface waves

It is shown that the classical theory of gravity driven waves on the surface of a non-viscous liquid can be derived from a set of canonical equations. Various approximate equations then can be found by introducing suitable approximations to the kinetic and potential energy functionals. The stability of these approximate equations then can be insured beforehand by using positive definite approximate energy functionals. For fairly long, fairly low waves a stable equation of Boussinesq type is derived in this way. This equation is also valid for waves which are not approximately simple.     

On hypo-elastic transverse surface waves in an internal stratum

Transverse surface waves in a stratum of uniform thickness, bounded on both sides by very deep layers of different materials are investigated in the context of hypo-elasticity of grade zero. It is found that the wave motion is possible and that the phase-velocity equation is similar to the one in the corresponding pure elastic problem. But unlike in the pure elastic problem, the waves in the present problem produce non-zero normal stresses.     

Two-temperature generalized thermoelasticity under ramp-type heating by finite element method

In this work, a general finite element model is proposed to analyze transient phenomena in thermoelastic half-space filled with an elastic material, which has constant elastic parameters. The governing equations are taken in the context of the two-temperature generalized thermoelasticity theory (Youssef in IMA J. Appl. Math. 71(3):383–390, 2006). A linear temperature ramping function is used to more realistically model thermal loading of the half-space surface. The medium is assumed initially quiescent. A finite element scheme is presented for the high accuracy numerical purpose. The numerical solutions of the non-dimensional governing partial differential equations of the problem have been shown graphically and some comparisons have been shown in figures to estimate the effect of the ramping parameter of heating and the parameter of two-temperature.     

On the dual-phase-lag thermoelasticity theory

The uniqueness and reciprocal theorems are proved without the use of Laplace Transforms for the Dual-Phase-Lag thermoelasticity theory. Variational principle is established for a linear anisotropic and inhomogeneous thermoelastic solid. The dissipative inequality is used to obtain a continuous dependence result for isotropic solid.     

On a generalized Broer-Kaup-Kupershmidt system for the long waves in shallow water

Describing the long waves in shallow water, a generalized Broer-Kaup-Kupershmidt system is investigated in this paper. With respect to the horizontal velocity of the water wave and the height of the water surface, we use symbolic computation to build up (A) a scaling transformation, (B) a set of the hetero-Bäcklund transformations, from that generalized system to a known linear partial differential equation, as well as (C) two sets of the similarity reductions, each of which from that generalized system to a known ordinary differential equation. Our results depend on all the shallow-water coefficients for that generalized system.