It seems relevant from the above discussion that little attention has been given to the study of propagation of thermoelastic plane waves in a rotating medium in presence of an external magnetic field based on the generalized **thermoelasticity**. In view of the fact that most large bodies like the earth, the moon, and other planets have an angular veloc- ity, it is important to consider the propagation of magneto-thermoelastic plane waves in an electrically conducting, rotating viscoelastic medium under the action of an external magnetic field. In this connection, Choudhuri and Debnath [9, 10, 11, 12, 13] have stud- ied propagation of magneto-thermoelastic plane waves in rotating thermoelastic media permeated by a primary uniform magnetic field using the generalized heat conduction equation of Lord and Shulman. In the present problem, we have studied the propagation of time-harmonic coupled electromagneto-viscoelastic dilatational thermal shear waves using the **thermoelasticity** **theory** of type II [14, Green-Naghdi model]. This thermoe- lastic model possesses several significant characteristics that di ﬀ er from the traditional classical development in thermoelastic material behaviors: (i) it does not involve thermal energy dissipation, (ii) the entropy flux vector (or equivalently, the heat-flow vector) in the **theory** is determined in terms of the same potential that also determines the stresses, (iii) it permits transmission of heat flow as thermal waves at finite speed. Several problems in **thermoelasticity** of type II (without thermal energy dissipation) have been studied by several authors [4, 5, 6, 7, 8, 23]. In this paper, GN model of **thermoelasticity** of type II is used to obtain a more general dispersion equation to ascertain the eﬀects of rotation, finite thermal waves speed c T of GN **theory**, thermoelastic coupling constant and the

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Green and Nagdhi [18-20] have formulated a new model of **thermoelasticity**. This model predicts that the internal rate of production of entropy is identically zero, i.e., there is no dissipation of thermal energy. This **theory** (GN **theory**) is known as **thermoelasticity** without energy dissipation **theory**. In the development of this **theory** the thermal displacement gradient is considered as a constitutive variable, whereas in the conventional development of a **thermoelasticity** **theory**, the temperature gradient is taken as a constitutive variable [12]. A couple of uniqueness theorems have been proved in [21-22], and one-dimensional waves in a half-space and in an unbounded body have been studied in [23-25].

It seems relevant from the above discussion that little attention has been given to the study of propagation of thermoelastic plane waves in a rotating medium in the presence of external magnetic ﬁeld based on the generalized **thermoelasticity**. In view of the fact that most large bodies, like the earth, the moon, and other planets, have an angular ve- locity, it is important to consider the propagation of magnetothermoelastic plane waves in an electrically conducting, rotating elastic medium under the action of the external magnetic ﬁeld with or without thermal relaxation. In this connection, Roychoudhuri and Debnath [17, 18, 19, 21, 20] have studied propagation of magnetothermoelastic plane waves in a rotating thermoelastic medium permeated by a primary uniform magnetic ﬁeld by using the generalized heat conduction equation of Lord and Shulman. In the present problem, we have studied the propagation of time-harmonic coupled electro- magnetoelastic dilatational thermal shear waves using the **thermoelasticity** **theory** of type II [9] (Green-Naghdi (G-N) model 1993). This thermoelastic model possesses several signiﬁcant characteristics that diﬀer from the traditional classical development in ther- moelastic material behaviors: (i) it does not sustain energy dissipation, (ii) the entropy ﬂux vector (or equivalently heat ﬂow vector) in the **theory** is determined in terms of the same potential that also determines the stress, (iii) it permits transmission of heat ﬂow as thermal waves at ﬁnite speed. Several problems in **thermoelasticity** relating to this Green-Naghdi **theory** of **thermoelasticity** of type II (without thermal energy dissipation) have been studied by several authors [4, 5, 6, 7, 8, 23]. In this paper, G-N model of **thermoelasticity** of type II is used to obtain a more general dispersion equation to as- certain the eﬀects of rotation, thermal parameter c T , the nondimensional thermal wave

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The absence of any elasticity term in the heat conduc- tion equation for uncoupled **thermoelasticity** appears to be unrealistic, since due to the mechanical loading of an elastic body, the strain so produced causes variation in the temperature field. Moreover, the parabolic type of the heat conduction equation results in an infinite velocity of thermal wave propagation, which also contradicts the actual physical phenomena. Introducing the strain-rate term in the uncoupled heat conduction equation, Biot extended the analysis to incorporate coupled thermoelas- ticity [1]. In this way, although the first shortcoming was over, there remained the parabolic type partial differen- tial equation of heat conduction, which leads to the pa- radox of infinite velocity of the thermal wave. To elimi- nate this paradox generalized **thermoelasticity** **theory** was developed subsequently. Due to the advance-

Thermoelastic problems are used to study the thermal stresses in an elastic body under high temperature gradients. The problems of **thermoelasticity** are broadly classified into two categories, namely static and dynamic problems. The problems dealing with dynamic thermal stresses are fundamentally important in engineering processes and have paved the way for technologies which operate in high temperatures such as nuclear reactors, aerodynamic structures, etc. The classical coupled **thermoelasticity** **theory** finds its first mention in Biot [1]. In non-classical theories of **thermoelasticity**, the Fourier heat conduction equation is generalized with the introduction of one relaxation time obtained by Lord and Shulman [2]. Various authors [3-8] contributed to the problems on generalized **thermoelasticity**. Recently, a lot of interest has developed in fractional order **theory** of **thermoelasticity** [9-15].

Green and Naghdi (G-N) have formulated three models (I, II, III) of **thermoelasticity** for homogeneous and an isotropic materials [1, 2]. Model I of G-N **theory** after linearization reduced to the classical **thermoelasticity** **theory**. Model II of G-N **theory** [3] does not suction dissipation of the thermoelastic energy. In this model, the constitutive equations are derived by starting with the reduced energy equation and by including the thermal displacement gradient among the constitutive variables. Chandrasekharaiah [4] used Laplace method to study the one-dimensional thermal wave propagation in a half space based on G-N **theory** of type II due to a sudden application of the temperature to the boundary. The disturbances produced in a half space by the application of a mechanical point load and thermal source acting on a boundary of the half space is investigated in ref. [5]. Model III of G-N **theory** confesses a dissipation of energy, where the constitutive equations are derived starting with a reduced energy equation. It includes the thermal displacement gradient, in addition to the temperature gradient among its independent constitutive variables.

1. Introduction. **Thermoelasticity** theories that admit ﬁnite speeds for thermal sig- nals have aroused much interest in the last three decades. In contrast to the conven- tional coupled **thermoelasticity** **theory** based on a parabolic heat equation [1], which predicts an inﬁnite speed for the propagation of heat, these theories involve hyper- bolic heat equations and are referred to as generalized **thermoelasticity** theories. For details about the physical relevance of these theories and a review of the relevant literature, see [2].

The absence of any elasticity term in the heat conduction equation for uncoupled **thermoelasticity** appears to be unrealistic, since due to the mechanical loading of an elastic body, the strain so produced causes variation in the temperature field. Moreover, the parabolic type of the heat conduction equation results in an infinite velocity of thermal wave propagation, which also contradicts the actual physical phenomena. Introducing the strain-rate term in the uncoupled heat conduction equation, Biot extended the analysis to incorporate coupled **thermoelasticity** [1]. In this way, although the first shortcoming was over, there remained the parabolic type partial differential equation of heat conduction, which leads to the paradox of infinite velocity of the thermal wave. To overcome this paradox, generalized **thermoelasticity** **theory** was developed subsequently. Due to the advancement of pulsed lasers, fast burst nuclear reactors and particle accelerators, etc. which can supply heat pulses with a very fast time-rise [2,3], generalized **thermoelasticity** **theory** is receiving serious attention. The development of the second sound effect has been nicely reviewed by Chandrasekharaiah [4]. At present, mainly two different models of generalized **thermoelasticity** are being extensively used- one proposed by Lord and Shulman [5] and the other proposed by Green and Lindsay [6]. LS (Lord and Shulman) **theory** introduces one relaxation time and according to this **theory**, only Fourier’s heat conduction equation is modified. While GL (Green and Lindsay) **theory** introduces two relaxation times and both the energy equation and the equation of motion are modified.

The **theory** of Thermo elasticity deals with the effect of mechanical and thermal disturbances on an elastic body. In the nineteenth century, Duhamel (1837) was the first to consider elastic problems with heat changes. In 1855 Neumann (1841) re derived the equations obtained by Duhamel using a different approach. Their **theory**, the **theory** of uncoupled thermo elasticity consists of the heat equation which is independent of mechanical effects and the equation of motion which contains the temperature as a known function. There are two defects of the **theory**. First, the fact that the mechanical state of the elastic body has no effect on the temperature. This is not in accord with true physical experiments. Second, the heat equation being parabolic, predicts an infinite speed of propagation for the temperature which again contradicts physical observations.

Motivated by the above results, in the present work, we study the well-posedness and asymp- totic behaviour of solutions to the laminated beam (1.1) in **thermoelasticity** of type III. By using semigroup method and Lumer-Philips theorem, we prove the existence and uniqueness of the solution. By using the perturbed energy method and construct some Lyapunov functionals, we then obtain the exponential decay result for the case of equal wave speeds, i.e., ρ G

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The Classical Fourier law of heat conduction and conse- quent mathematical models for temperature dynamics constructed on the basis of parabolic partial differential equations assumes that the thermal disturbances propa- gate at infinite speeds. However, the assumption may lead to an inaccurate response of the super large-scale space structures, since a time lag of the propagation of the thermal disturbances in such structures could not be dis- regarded. The literature dedicated to coupled and general- ized theories of **thermoelasticity** theories is quite large and its detailed review can be found in Nowacki (1975, 1986), Chadwick (1960, 1979), and Chadwick and Seet (1970).

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Speciﬁc heat at constant strain. t = Time variable. K = Coeﬃcient of thermal conductivity. γ = (3 λ + 2 μ ) β ∗ . β ∗ = Coeﬃcient of volume expansion. β = Coeﬃcient of stress temperature. k ∗ = Material constants characteris- tic of the **theory**. H ( t ) = Heaviside unit step function.

Caring is the essence of nursing which is highly applicable to any type of clients. However, the concept of caring explored in the study focused on the elderly care and how this is tion of cultural differences and backgrounds. Objective: This study developed a substantive **theory** on elderly care known as the Elderly Care **Theory** that defined what and how elderly caring is based on Filipino context of caring. Methods: This was using grounded **theory** with ten care providers interviewed and an actual observation of the The theoretical assumptions developed were: (1) ocess and care agent which ultimately leads to the development of the elderly care satisfaction and quality of life; (2) Caring elements for the elderly are dependent on the quality/extent/status of delivery of caring culture, caring process and the psychological-spiritual and political factors; (3) The care culture has its own elements which interact interdependently with each other fect enhances the flourishing culture in an elderly facility; and (4) The care process and agents have interactive elements such as confident, enduring and strategic care and its combined effect creates the caring self. Conclusion: The care for is a holistic and specialized care. The elderly care elements necessary for the provision of cultural, and spiritual factors of care and the personal

Nonlinear nonstationary heat conduction problem for infinite circular cylinder under a complex heat transfer taking into account the temperature dependence of thermophysical characteristics of materials is solved numerically by the method of lines. Directing it to the Cauchy’s problem for systems of ordinary differential equations studied feature which takes place on the cylinder axis. Taken into account the dependence on the temperature coefficient of heat transfer that the different interpretation of its physical content makes it possible to consider both convective and convective-ray or heat ray. Using the perturbation method, the corresponding **thermoelasticity** problem taking into account the temperature dependence of mechanical properties of the material is construed. The influence of the temperature dependence of the material on the distribution of temperature field and thermoelastic state of infinite circular cylinder made of titanium alloy Ti-6Al-4V by radiant heat transfer through the outer surface has been analyzed.

is a material constant characteristic of the **theory**. It may be noted that the third model represented by (2.5) of Green and Naghdi [5] for heat transport in solids accommodates infinite thermal wave speed due to the presence of third-order mixed derivative term present on the right-hand side of (2.5) and it involves thermal damping. As such, the corresponding thermoelastic model admits coupled damped thermoelastic waves.

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perature is slightly changed and the differences between C-D, L-S, and G-L are very small (tiny). The coupled **theory** (C-D) may give results with small relative error compared with those given by Lord and Shulman’s (L-S) and Green and Lindsay’s (G-L) theories. However, the results of L-S and G-L are much closed to each other (see Figure 3). 2) The plots of results given by Lord and Shulman’s

Among the authors who contribute to developing this **theory**, Quintanilla studied existence, structural stability, convergence and spatial behavior for this **theory** [6], Youssef constructed the generalized Fourier’s law to the two-temperature **theory** of **thermoelasticity** and proved its uniqueness of solution for homogeneous isotropic material [7]. Puri and Jordan studied the propagation of plane harmonicwaves, recently [8], Magaña and Quintanilla [9] have studied the uniqueness and growth solutions for the model proposed by Youssef [7]. A new **theory** of generalized **thermoelasticity** has been constructed based on two-temperature generalized thermo- elasticity **theory** for anisotropic and homogeneous body without energy dissipation by Youssef [10]. This new theorem has been constructed in the context of Green and Naghdi model of type II of linear **thermoelasticity**. Also, a theorem of general uniqueness is proved for two-temperature generalized **thermoelasticity** without energy dissipation [10].

A large amount of work has been devoted for solving **thermoelasticity** problems with the consideration of the coupling effect between temperature and strain rate. Stress waves in a half-space induced by variations of surface strain, temperature, or stress were studied by [Boley and Tolins, 1962] and [Chandrasekhariaiah and Srinath, 1979]. [Mozina and Dovc, 1994] attempted to use the Laplace transform to solve the thermoelastic stress wave induced by volumetric heating. Due to the difficulty in finding analytical Green`s functions, only solution for locations on the surface was obtained .

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2. **Thermoelasticity** of type III: uniqueness. The aim of this section is to obtain a uniqueness result for the solutions of system (1.6) and (1.7). From now on, we assume that the functions ρ and c are greater than or equal to a positive constant and that the following inequality

mathematical model of fractional heat conduction law in which the generalized Fouriers law of heat conduction is modified by using the new Taylors series expansion of time fractional order developed by Jumarie [6]. Recently, Youssef [7] derived a new **theory** of **thermoelasticity** with fractional order strain which is considered as a new modification to Duhamel- Neumanns stress-strain relation. In this paper, the author postulated a new unified system of equations that govern seven different models of **thermoelasticity** in the context of one-temperature and two-temperature and one dimensional problem for an isotropic and homogeneous elastic half-space.