THERMOELASTICITY WITHOUT ENERGY DISSIPATION FOR INITIALLY STRESSED BODIES

Thermoelastic equations without energy dissipation are formulated 
for a body which has previously received a large deformation and 
is at nonuniform temperature. A linear theory of thermoelasticity 
without energy dissipation for prestressed bodies is derived and 
the uniqueness theorem for a class of mixed initial-boundary 
value problems is established.


Introduction.
The so-called second sound effect has been given increasing attention in recent decades.This effect arises from the possible transport of heat by a wave propagation process rather than diffusion.Many articles have been devoted to the development of the generalized theory of thermoelasticity that predicts a finite speed for heat propagation.Lord and Shulman [11], employing a modified Fourier's law, developed what now is known as extended thermoelasticity.Green and Lindsay [6], based on an entropy production inequality proposed by Green and Laws [5], formulated temperature-rate dependent thermoelasticity that includes the temperature-rate among constitutive variables.Lebon [10] formulated heat-flux dependent thermoelasticity on the basis of a nonclassical approach to thermodynamics which includes the heat flux among the constitutive variables and assumes an equation of evolution for the heat flux.All these theories yield governing systems of hyperbolic equations and predict finite speed for heat propagation.
Recently, Green and Naghdi [7] reexamined the basic postulates of thermomechanics.They postulated three types of constitutive repose functions for the thermal phenomena and, accordingly, formulated three models of thermoelasticity.The nature of these three types of constitutive functions [8] is that when the respective theories are linearized, model I theory is the same as the classic heat conduction theory (based on Fourier's law); model II theory predicts a finite speed for heat propagation and involves no energy dissipation, now referred to as thermoelasticity without energy dissipation; model III theory permits propagation of thermal signals at both finite and infinite speeds and there is a structural difference between these field equations and those developed in [5,6,10,11].Ciarletta [3] later formulated a theory of micropolar thermoelasticity without energy dissipation.Detailed and comprehensive references to the developments of generalized thermoelasticity are found in two nice review papers by Chandrasekharaiah [1,2].
In this paper, we adapt the postulates made by Green and Naghdi [7] and formulate a thermoelasticity theory without energy dissipation for solids which have previously received a large deformation and are at nonuniform temperatures.The classical theory for prestressed thermoelastic bodies has been obtained by Green [4] for the isothermal case and by Iesan [9] for the nonisothermal case.
2. Nonlinear formulae.Consider a thermoelastic solid whose configuration changes continuously, under external mechanical actions and heating, from an original reference state Ω 0 , with uniform temperature T 0 and uniform thermal-displacement α 0 , to a deformed state Ω.Let ∂Ω 0 and ∂Ω be the closed surface of Ω 0 and Ω, respectively.Let X K denote the position of a particle X in the reference state, and the position in the deformed state, where t is the time.We assume that x i is sufficiently smooth and det(x i,A ) > 0 in order for this deformation to be possible.
In the material description, the basic nonlinear equations in thermoelasticity without energy dissipation take the form [2,3,8] T Ji,J + ρ 0 f i = ρ 0 ẍi , (2.1) in Ω 0 × [0,t 0 ), where t 0 is some time instant that may be infinite.In the above equations, T is the absolute temperature, S is the external rate of heat supply per unit mass and time, ρ 0 is the initial mass density, T Ji is the first Piola-Kirchhoff stress tensor (T Ji ≠ T iJ ), f i is the external body force per unit mass, η is the entropy per unit mass, Φ K is the internal entropy flux vector.If Q K denotes the flux of heat across surfaces in the deformed body that were originally coordinate planes perpendicular to the X K -axes through the point X, measured per unit time, then Q K = T Φ K .Through this paper, a superposed dot stands for the material time derivative while a comma followed by a subscript denotes partial derivatives with respect to the spatial coordinates, in the deformed configuration if the subscript is a minuscule, in the reference configuration if the subscript is a majuscule; that is, g ,J = ∂g/∂X J and g ,i = ∂g/∂x i .Einstein summation on repeated indices is also used throughout this paper.The constitutive variables are the strain tensor E AB , T , and α ,J with the geometric equations where α is the thermal-displacement (w α = T ) and δ AB is the Kronecker delta.
The constitutive equations are ) ) where Ψ is the Helmholtz free-energy per unit volume.Unlike other theories of hyperbolic thermoelasticity, in obtaining the above constitutive equations, the second law of thermodynamics in the form of an entropy production inequality was not used.Rather, this inequality is automatically satisfied [2,8].

Small deformation superposed on a large deformation.
In this section, we consider three states of the solid, the initial state Ω 0 , the first deformed state Ω, and the final deformed state Ω * corresponding respectively to the uniform temperature T 0 , uniform thermal-displacement α 0 and zero displacement; temperature T , thermaldisplacement α, and displacement v i ; and temperature T * , thermal displacement α * and displacement v i + u i .Following Iesan [9], we call Ω the primary state and Ω * the secondary state.Thermodynamic quantities and forces associated with Ω * are denoted by an asterisk.The position coordinates of the particle X at time t in Ω * will be denoted by We define the incremental displacement u i and incremental temperature θ as Here we consider the case in which u i and θ are small; that is, there exists a real parameter ε, small enough for its square and higher powers to be neglected, such that where u i and θ are independent of ε.
In the secondary state Ω * , the basic equations are of the same form as those in the primary state Ω.We have the equation of motion the energy equation the geometric equations 2E * AB = y i,A y i,B − δ AB , (3.5) and the constitutive equations Next, we derive the equations satisfied by u i and θ.Keeping (3.2) in mind, we get the following second-order approximation: where b IJ , b IJK , and d IJ have certain symmetries in their indices, and that Substituting (3.9) into (3.7) and (3.8), we obtain (3.12) Since it is required that Φ K vanishes at equilibrium, that is, for α ,K = 0 and T ,K = 0, we also must have that From geometric equations and the relation we get that where we have used the fact that u i s are small, and hence, the terms like u i,A u i,B have been neglected.Introducing the notation we find that we have Substituting (3.17) and (3.20) into (3.12),we find that then we have where 4. Linear theory of thermoelasticity for initially stressed bodies.In this section, we consider the special case when the primary state Ω of the solid is identical with that of the initial solid Ω 0 so that x 1 = X 1 , x 2 = X 2 , x 3 = X 3 , and we suppose that Ω 0 is subjected to an initial stress and an initial heat flux caused by the nonuniform initial temperature T 0 .Due to the action of external loadings and heating, the solid Ω 0 undergoes a deformation.There arise displacements u i = u i , thermal displacement α, and temperature increment θ = θ .Here, we systematically neglect all powers of above the first, except in the free energy function Ψ , where we retain quadratic terms of .
The work of Section 3 can be applied to this special case and yields a linear theory of thermoelasticity without energy dissipation in the presence of initial stress and heat flux.In this case, we have All coefficients defined in the previous section are now evaluated at E AB = 0, α = α 0 , and T = T 0 .For convenience, from now on, we change all the majuscule subscripts into minuscule subscripts.From (3.17), we find that The governing equations (3.27) become 3) The constitutive relations (3.26) reduce to where d ijmn = C ijmn + δ jm T in with δ jm being the usual Kronecker delta.The functions T in characterize the initial stress and they can be arbitrary functions apart from satisfying the energy equation for the static case and the condition that Ω 0 is in equilibrium.The presence of the initial heat flux vector is determined by the nonuniformity of the initial temperature T 0 .This fact implies the appearance of the coefficients b ijk and d k in the expression for ϕ k .If T 0 is constant, then b ijk = 0 and d k = 0.
Substituting (4.4) into (4.3),we find the governing equations in the linear theory of thermoelasticity for initially stressed bodies ) If T in = 0 and T 0 is a constant, for nonhomogeneous and anisotropic materials, the above equations reduce to For homogeneous and isotropic materials, the above equations reduce to the same set of governing equations as developed in [8].

Uniqueness theorem.
In this section, we establish a uniqueness theorem for a class of initial-boundary value problems associated with the governing equations (4.5) and (4.6) for prestressed solids.
In the context of the theory considered, an initial-boundary value problem consists in determining the functions u i (x, t) and θ(x, t) for x ∈ Ω and t > 0, by solving (4.5) and (4.6) with F i (x, t), s(x, t), T in , and T 0 being prescribed functions for x ∈ Ω and t > 0; under the initial conditions and the boundary conditions for t > 0 where ūi , pi , θ, and q are prescribed functions in the domains of their definition, ∂Ω i + ∂Ω c i = ∂Ω for i = 1, 2 and n i is the outward normal vector to ∂Ω.To establish the uniqueness theorem, we impose the following restrictions on the material constants: (i) the initial mass density ρ 0 and the specific heat d are strictly positive; that is, there exist positive constants r and d 0 with the same dimensions as of ρ 0 and d, respectively, such that (ii) d ijmn is positive definite in the sense that there exists a positive constant β with the same dimension of d ijmn such that (5.4) for all second order tensors ξ ij ; (iii) the heat conductivity tensor d km is symmetric and positive definite in the sense that there exists a positive constant b such that for all vectors ξ k ; (iv) there exists a constant h with the same dimension of b such that 4b 2  ijk < βh, h < b. (5.6) Theorem 5.1.If assumptions (i), (ii), (iii), and (iv) hold and T 0 > 0, then there exists at most one solution to the initial-boundary value problem defined by (4.5), (4.6), (5.1), and (5.2).
Proof.It suffices to show that for F i = 0, ṡ = 0, and homogeneous boundary conditions, the solution is trivial.

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: The coefficients C ijmn , b ij , b ijk , d km , d k , and d are functions of the position coordinates.The coefficients C ijmn have the symmetry as in (3.10), b ij and d km have the usual symmetry about their indices and that b ijk = b jik .