Arch Appl Mech (2011) 81: 1031–1040 DOI 10.1007/s00419-010-0464-1
O R I G I NA L
Roushan Kumar · Rajesh Prasad · Santwana Mukhopadhyay
Some theorems on two-temperature generalized thermoelasticity
Received: 8 November 2009 / Accepted: 14 July 2010 / Published online: 10 August 2010 © Springer-Verlag 2010
Abstract 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. Keywords Two-temperature thermoelasticity · Generalized thermoelasticity · Reciprocal principle
1 Introduction In classical coupled dynamical theory of thermoelasticity (Biot [1]), the infinite speed of propagation of thermal signal contradicts the physical fact. During last few decades various generalized theories of thermoelasticity involving finite speed of heat transportation in elastic solids have therefore been developed to remove this ‘so called paradox’. In this regard, we recall the extended thermoelasticity theory proposed by Lord and Shulman [20] and the temperature-rate dependent thermoelasticity theory developed by Green and Lindsay [12]. In the theory of Lord and Shulman, the heat conduction equation is based on the generalized version of Fourier law (Cattaneo [3,4]) and this theory contains one thermal relaxation parameter, whereas the theory of Green and Lindsay contains two constants that act as thermal relaxation times and modifies all the equations of the coupled theory, not only the heat equation. The classical Fourier law of heat conduction is not violated in this theory if the medium under consideration has a center of symmetry. In the mechanics of continuous media, a material is said to have hereditary characteristics or memory if the behavior of the material at time t is specified in terms of the experience of the body up to the time t. Coleman [10] formulated a theory of materials with memory. He defined a simple material to be one for which the internal energy, the entropy, the heat flux and the stress at a point x are determined in terms of the history of the deformation gradient, the history of the temperature, and the present value of the temperature gradient at x. Thereafter, Gurtin and Williams [13] slightly generalized the definition of simple materials by including a second temperature. They pointed out that there is no a priori ground for assuming that the second law of thermodynamics for continuous bodies involves only a single temperature and that it is more logical to assume a second law in which the entropy contribution due to heat—conduction is governed by one temperature (θ ), R. Kumar · R. Prasad · S. Mukhopadhyay (B) Department of Applied Mathematics, Institute of Technology, Banaras Hindu University, Varanasi 221005, India E-mail:
[email protected];
[email protected] Tel.: 0542-2575512 Fax: 0542-2368174 R. Kumar Birla Institute of Technology, Mesra, Deoghar Campus, Jasidih 714142, India
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and that of the heat supply by another (ϕ). ϕ is the conductive temperature and θ is the thermodynamic temperature. Gurtin and Williams [13] defined a simple material to be one for which the stress, the energy, the entropy, the heat-flux and the thermodynamic temperature at a given time depend on the histories up to that time of the deformation gradient, the conductive temperature and the gradient of this temperature. Later on, Chen and Gurtin [7] and Chen et al. [8,9] formulated the theory of heat conduction on a deformable body which depends on two different temperatures—the conductive temperature, φ and the thermodynamic temperature, θ . Chen et al. [8] have suggested that the difference between these two temperatures is proportional to heat supply and in case of the absence of heat supply the two temperatures are equal for time-independent situation. However, for time dependent cases the two temperatures are in general different, regardless of heat supply. This two-temperature thermoelasticity theory has gained much attention of the researchers in recent years. The existence, structural stability, convergence and spatial behavior of two-temperature thermoelasticity have been reported by Quintanilla [26]. Puri and Jordan [25] investigated the propagation of harmonic plane waves in two temperature theory in a detailed way. Recently, Youssef [28] extended this theory in the frame of generalized theory of heat conduction by introducing thermal relaxation parameters in the constitutive relations and formulated a two-temperature theory of generalized thermoelasticity. The uniqueness theorem on two-temperature generalized thermoelasticity (2TGT) is also provided by Youssef [28]. Magana and Quintanilla [21] studied the uniqueness and growth of solutions of Youssef theory (2TGT). Subsequently, Youssef [29], Youssef and Al-Lehaibi [30], Mukhopadhyay and Kumar [22] and Kumar and Mukhopadhyay [18] carried out some investigations on two-temperature generalized thermoelasticity and indicated some significant features of 2TGT. Kumar et al. [19] reported the convolutional type variational principle on 2TGT. By reciprocity theorem, it is possible to deduce various methods of integrating the elasticity equations by means of Green’s function. Betti’s type reciprocal theorem of isothermal elasticity allows one to determine the deformation of a body due to one system of body forces and surface tractions if the deformation due to another system of body forces and surface tractions are known. Reciprocal principles are not only of mathematical interest but also of practical utility [11,16,23,24]. Reciprocity theorems in the case of classical coupled thermoelasticity have been established Biot [2] and later on by Ionescu-Cazimir [15]. Ie¸san [14] established the reciprocity theorem in the case of two-temperature thermoelasticity (2TT). Later on, the reciprocity theorem on generalized theories of thermoelasticity have been reported by Chandrasekharaiah and Srikantaiah [6], Wang et al. [27] and Chandrasekharaiah [5]. In elastostatics, Somigliana’s theorem and Green’s theorem provide the relations between the displacements in the interior of the body and the tractions on its surface (Nowacki [23]). Generalizations of Somogliana and Green formulae to dynamic coupled thermoelasticity and thermoelasticity with one relaxation parameter are presented by Nowacki [23] and Khomyakevich and Rudenko [17], respectively. These formulae connect the displacements and temperature in the interior of the body with displacements, tractions, temperature and its gradient on the surface of the body provided that the Green functions of displacement and temperature are known. The main objective of the present work is to establish a reciprocity theorem for the theory of two-temperature generalized thermoelasticity by providing the proof of the theorem. It should be mentioned here that Kumar, et al. [19] established a reciprocity theorem based on an alternative formulation of mixed boundary—initial value problem in which the initial conditions are incorporated into the field equations. This approach for two temperature thermoelasticity was first given by Ie¸san [14]. However, in the present work we have established the reciprocity relation without using any alternative formulation. This is known as reciprocity relation of Betti type, one of the most interesting theorems of the theory of elasticity [15,23]. We will establish our reciprocity relation by using Laplace transform technique. We also derive extensions of the formulae of Somigliana and Green theorem to the case of two-temperature generalized thermoelasticity. To the best of the authors’ knowledge, no work in this direction has yet been reported for the theory of two-temperature thermoelasticity in the frame of a generalized theory.
2 Basic equations In this section, we summarize the main governing equations, constitutive relations and field equations for twotemperature theory of generalized thermoelasticity [28]. We employ a rectangular co-ordinate system xk and usual indicial notation throughout our paper. We consider V as a regular region of space bounded by a closed and bounded surface ∂ V and occupied by homogeneous and isotropic thermoelastic material. We assume n i
Some theorems on two-temperature generalized thermoelasticity
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are the components of outward drawn unit normal to ∂ V . The basic equations are therefore as follows [28]: Equations of motion : σi j, j + Fi = ρ u¨ i Symmetry relation of stress tensor : σi j = σ ji Equation of entropy : Q − ρφ0 S˙ = qi,i
(1) (2) (3)
Constitutive relations: σi j = λekk δi j + 2μei j − γ θ δi j qi + τ0 q˙i = −kϕ,i ρφ0 S = ρc E θ + φ0 γ ekk ϕ − θ = aϕ,ii Geometrical equations: 1 u i, j + u j,i ei j = 2
(4) (5) (6) (7) (8)
In these equations we have used the following notations: u i —components of displacement, σi j —components of stress tensor, ei j —components of strain tensor, Fi —components of body force vector, ϕ—conductive temperature measured from a constant reference temperature φ0 , θ —thermodynamic temperature measured from the constant reference temperature φ0 . a > 0 is the two temperature parameter and called the temperature discrepancy [8]. qi —component of heat flux vector, Q—heat source per unit volume, S—entropy per unit mass, λ, μ—Lame’s elastic constants, ρ—mass density, k—thermal conductivity of the material, c E —specific heat at constant strain, τ0 —thermal relaxation parameter, γ = (3λ+2μ)αt , αt —coefficient of linear thermal expansion. The over-headed dots denote partial derivative with respect to time variable t and the subscripted comma notations are used to represent the partial derivatives with respect to the space variables. It is assumed that the displacements and temperatures are functions of the class C (2) whereas the stresses and strains are functions of the class C (1) for all x ∈ V + ∂ V. Taking divergence of Eq. (5) and using Eqs. (3) and (6) we have kϕ,ii =
∂ ∂2 + τ0 2 ∂t ∂t
∂Q (ρc E θ + γ φ0 ekk ) − Q + τ0 ∂t
(9)
From Eqs. (1), (4) and (8) we also get λu j, ji + μ u i, j j + u j, ji − γ θ,i +Fi = ρ u¨ i
(10)
To the above system of field equations, we adjoin the homogeneous initial conditions u i (x, 0) = 0, u˙ i (x, 0) = 0, x ∈ V ϕ(x, 0) = 0, ϕ(x, ˙ 0) = 0, x ∈ V
(11a) (11b)
and also the following prescribed boundary conditions: σi (x, t) = σ ji n j = pi (x, t) x ∈ ∂ V, t > 0, ϕ(x, t) = ϕ0 (x, t) x ∈ ∂ V, t > 0,
(12a) (12b)
where pi are the components of surface traction and ϕ0 (x, t) is the temperature prescribed on the surface ∂ V bounding the body.
3 Reciprocity theorem We now present the reciprocal principle of Betti [15,23] type in the form of the following theorem. Theorem 1 Consider two initial boundary value problems associated with Eq. (1)–(12) in which a homogeneous, isotropic bounded thermoelastic body is subjected to the action of two different systems of thermoelastic
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loadings (consisting of the body forces Fi , surface tractions pi , heat source Q and the heating of the surface, ϕ0 ) (α) (α) (α) α = 1, 2 (13) L (α) = Fi , pi ; Q (α) , ϕ0 and the two corresponding thermoelastic configurations I (α) = u (α) , ϕ (α) α = 1, 2
(14)
Now for α, β = 1, 2, let L αβ = k
t
(α) ϕ0(β) (x, τ ) ϕ,n d A(x) (x, t − τ ) dτ
∂V
0
+
d V (x)
t
Q (α) (x, τ ) θ (β) (x, t − τ ) dτ
0
V
t
+ τ0
d V (x)
Q (α) (x, τ )
0
V
t
+ φ0
d A(x)
∂V
0
+ φ0 τ0
t
Fi 0
V
(β)
t
d V (x)
(x, τ ) dτ ∂τ 2
(α)
(x, t − τ )
∂ 2ui
(x, τ ) dτ ∂τ
(α)
(x, t − τ )
(β)
Fi 0
V
(x, τ ) dτ ∂τ
(α)
∂u i
0
d V (x)
+ φ0 τ0
(β)
pi
t
+ φ0
pi(β) (x, t − τ )
d A(x)
∂V
∂ (β) θ (x, t − τ ) dτ ∂τ
∂u i
∂ 2 u i(α) (x, τ ) dτ (x, t − τ ) ∂τ 2
(15)
where ϕ,n = ϕ,i n i is the derivative of the temperature ϕ in the direction of the normal to the surface ∂ V . Then, L 12 = L 21
(16)
Proof By hypothesis, the stress functions associated with the two problems considered are given by the field Eq. (4) and (7) as
(α) (α) (α) (α) σi j = 2μei j + λekk − γ ϕ (α) − aϕ,ii δi j , α = 1, 2 (17) Now we perform Laplace transform integral over the boundary conditions (12b) and the Eq. (17) defined by f¯(x, s) =
∞
f (x, t)e−st dt, s > 0
0
Therefore we get (for α = 1, 2) p¯ i (x, t) = σ¯ ji n j x ∈ ∂ V, t > 0 ϕ(x, ¯ t) = ϕ¯ 0 (x, t) x ∈ ∂ V, t > 0,
(18a) (18b)
Some theorems on two-temperature generalized thermoelasticity
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(α) (α) (α) (α) δi j σ¯ i j = 2μe¯i j + λe¯kk − γ ϕ¯ (α) − a ϕ¯ ,ii (2)
(19)
(1)
We multiply (19) for α = 1 by e¯i j and for α = 2 by e¯i j . Then subtracting the results, integrating over volume V and using the identity (1) (1) (2) (2) (2) (1) 2μe¯i j + λe¯kk δi j e¯i j = 2μe¯i j + λe¯kk δi j e¯i j we arrive at the relation
(1) (2) (2) (1) (2) (1) (1) (2) σ¯ i j e¯i j − σ¯ i j e¯i j d V = γ ϕ¯ (2) − a ϕ¯ ,ii e¯kk − ϕ¯ (1) − a ϕ¯ ,ii e¯kk d V V
(20)
V
Now taking Laplace transform of Eq. (8) we get from Eq. (20)
(1) (2) (2) (1) (2) (1) (1) (2) σ¯ i j u¯ i, j − σ¯ i j u¯ i, j d V = γ ϕ¯ (2) − a ϕ¯ ,ii e¯kk − ϕ¯ (1) − a ϕ¯ ,ii e¯kk d V V
(21)
V
Performing Laplace transform over Eq. (1) and using the homogeneous initial conditions (11a) we obtain σ¯ i j, j + F¯i = ρs 2 u¯ i
x∈V
(22)
Then by using Gauss’s divergence theorem on left hand side of Eq. (21) and combining the Eq. (22) we arrive at (1) (2) (2) (1) (1) (2) (2) (1) p¯ i u¯ i − p¯ i u¯ i dA + F¯i u¯ i − F¯i u¯ i dV ∂V
V
(1) (2) (2) (1) ϕ¯ (1) − a ϕ¯ ,ii e¯kk − ϕ¯ (2) − a ϕ¯ ,ii e¯kk d V = 0 +γ
(23)
V
Equation (23) contains only the cause of a mechanical nature: body forces and surface tractions. Now, we examine the heat conduction equation associated with thermoelastic configurations I (α) = u (α) , ϕ (α) α = 1, 2 Firstly, performing Laplace transform over the Eqs. (7) and (9) we obtain after simplifications ρc E s + τ0 s 2 γ φ0 s + τ0 s 2 1 + τ0 s (α) (α) (α) ϕ¯ + e¯kk Q¯ (α) , α = 1, 2 ϕ¯,ii = − 2 2 k + aρc E s + τ0 s k + aρc E s + τ0 s k + aρc E s + τ0 s 2 (24) Multiplying (24) for α = 1 by ϕ¯ (2) and for α = 2 by ϕ¯ (1) , subtracting the results and integrating over the volume V we arrive at the identity γ φ0 (s + τ0 s 2 ) (1) (2) (2) (1) (2) (1) (1) (2) ϕ¯ ϕ¯ ,ii − ϕ¯ ϕ¯ ,ii d V = e ¯ dV ϕ ¯ − e ¯ ϕ ¯ kk kk k + aρc E (s + τ0 s 2 ) V V 1 + τ0 s (2) ¯ (1) (1) ¯ (2) − ϕ ¯ dV (25) Q Q − ϕ ¯ k + aρc E (s + τ0 s 2 ) V
Application of Gauss’s divergence theorem on left hand side of Eq. (25) yields γ φ0 (s + τ0 s 2 ) (2) (1) (1) (2) (1) (2) (2) (1) ϕ¯0 ϕ¯ ,n − ϕ¯ 0 ϕ¯ ,n d A = e ¯ dV ϕ ¯ − e ¯ ϕ ¯ kk kk k + aρc E (s + τ0 s 2 ) ∂V V 1 + τ0 s (1) ¯ (2) (2) ¯ (1) + ϕ ¯ dV Q Q − ϕ ¯ k + aρc E (s + τ0 s 2 ) V
(26)
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(1)
Now multiplying (24) for α = 1 by ϕ¯ ,ii and for α = 2 by ϕ¯ ,ii , subtracting, using (26) and integrating over the volume V we obtain ρc E (1) (2) (2) (1) (2) (1) (1) (2) e¯kk ϕ¯ ,ii − e¯kk ϕ¯ ,ii d V = ϕ¯ 0 ϕ¯ ,n dA − ϕ¯ 0 ϕ¯ ,n γ φ0 V ∂V 1 + τ0 s (2) ¯ (1) (1) ¯ (2) + ϕ ¯ dV (27) − ϕ ¯ Q Q ,ii ,ii (s + τ0 s 2 )γ φ0 V
Equations (26) and (27) contain the thermal cause: the and heating heat source of
the surface V. (1) (1) (2) (2) (1) (2) After eliminating the integral V ϕ¯ − a ϕ¯ ,ii e¯kk − ϕ¯ − a ϕ¯ ,ii e¯kk d V from Eqs. (23), (26) and (27) we finally obtain (2) (1) (1) (2) ϕ¯ 0 ϕ¯ ,n d A + (1 + τ0 s) Q¯ (1) θ¯ (2) − Q¯ (2) θ¯ (1) d V − ϕ¯ 0 ϕ¯ ,n k ∂V
= φ0 s + τ0 s
2
V
(1) (2) p¯ i u¯ i
−
(2) (1) p¯ i u¯ i
d A + φ0 s + τ0 s
∂V
2
(1) (2) (2) (1) F¯i u¯ i − F¯i u¯ i dV
(28)
V
To invert Laplace transforms involved in Eq. (28) we shall use the following convolution theorem of the Laplace transform: L
−1
f¯1 ( p) f¯2 ( p) =
t
t f 1 (τ ) f 2 (t − τ )dτ =
0
f 2 (τ ) f 1 (t − τ )dτ 0
After the inversion of Laplace transforms of (28) by employing the above convolution theorem, we get Eq. (16). This completes the proof of the reciprocity theorem for the two-temperature generalized thermoelasticity. 4 Generalizations of the Somigliana and Green theorems to the problems of two-temperature generalized thermoelasticity Now, we employ above reciprocity theorem to derive the extensions of the formulae of Somigliana and Green for the case of two-temperature generalized thermoelasticity. To determine the functions u i (x, t) , x ∈ V, t > 0, we consider at a point ξ , the concentrated force Fi(2) = δ(x − ξ )δi j δ(t) applied in the direction of the x j -axis in the infinite thermoelastic medium. The dis( j) (2) placement and temperature which arise due to the action of this force are denoted by u i = Ui (x, ξ, t) and ϕ (2) = ψ ( j) (x, ξ, t). These functions constitute the solution of equation of motion and heat conduction equation
( j) ( j) Dik [Uk (x, ξ, t)] + δ(x − ξ )δi j δ(t) = γ ψ ( j) (x, ξ, t) − aψ,ii (x, ξ, t) (29) ,i γ φ0 ∂ ∂2 ( j) (30) + τ0 2 Uk,k (x, ξ, t) = 0 x, ξ ∈ V, t > 0 D32 ψ ( j) (x, ξ, t) − k ∂t ∂t with the homogenous initial conditions ( j) ( j) Ui (x, ξ, 0) = 0, U˙ i (x, ξ, 0) = 0, ψ ( j) (x, ξ, 0) = 0, ψ˙ ( j) (x, ξ, 0) = 0, x, ξ ∈ V, t = 0 (31)
Where the operators entering (29) and (30) are as follows: ∂ ∂ (i, j = 1, 2, 3) ∂ xi ∂ x j 2 ∂ ∂ ∂2 ∂2 aρc E ∂ 2 − ρc E + τ0 2 + τ0 2 D3 = 1 + k ∂t ∂t ∂x2 ∂t ∂t 2 2 ∂ 1 ∂ μ 22 = − 2 2 , c22 = ∂ xi ∂ xi ρ c2 ∂t
Di j = μδi j 22 + (λ + μ)
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( j)
Now we assume that the functions Ui and ψ ( j) constituting the Green functions for infinite medium are (2) (2) known. Thus assuming Q (2) = 0 and introducing Laplace transform of Fi = δ(x − ξ )δi j δ(t) and u i = ( j) Ui (x, ξ, t), ϕ (2) = ψ ( j) (x, ξ, t) into the reciprocity relation (28) in the Laplace transform domain and finally inverting the Laplace transform we get u˙ j (ξ, t) + τ0 u¨ j (ξ, t) t t ( j) ( j) ∂Ui (x, ξ, τ ) ∂ 2 Ui (x, ξ, τ ) Fi (x, t − τ ) Fi (x, t − τ ) d V (x) d V (x) + τ0 dτ = dτ ∂τ ∂τ 2 0
−
0
V
1 φ0
dτ
τ0 − φ0
0
V
t
0
V
dτ ∂V
0
t +τ0
∂V
0
t
k − φ0 ( j)
∂ ( j) ( j) ψ (x, ξ, τ ) − aψ,ii (x, ξ, τ ) d V (x) ∂τ
( j) ( j) ∂Ui (x, ξ, τ ) ∂ pi (x, ξ, τ ) pi (x, t − τ ) − u i (x, ξ, t − τ ) d A(x) ∂τ ∂τ
dτ
V
( j) Q(x, t − τ ) ψ ( j) (x, ξ, τ ) − aψ,ii (x, ξ, τ ) d V (x)
Q(x, t − τ )
dτ
t +
t
( j) ( j) ∂ 2 Ui (x, ξ, τ ) ∂ 2 pi (x, ξ, τ ) pi (x, t − τ ) d A(x) − u i (x, ξ, t − τ ) ∂τ 2 ∂τ 2
( j) ϕ,n (x, τ )ψ ( j) (x, ξ, t − τ ) − ϕ0 (x, τ )ψ,n (x, ξ, t − τ ) d A(x) dτ
(32)
∂V
0 ( j)
( j)
(2)
Here, pi = σki n k where σik are the stresses on the surface due to the force Fi = δ(x − ξ )δi j δ(t). The conductive temperature ϕ(x, t), x ∈ V, t > 0 can be determined by adopting the heat source Q (2) = δ(x − ξ )δ(t), acting at a point ξ of the unbounded thermoelastic media. The displacement components and (2) temperature due to this heat source are denoted by u i = wi (x, ξ, t) and ϕ (2) = H (x, ξ, t), respectively and satisfy the equations Dik w j (x, ξ, t) = γ [H (x, ξ, t) − a H,ii (x, ξ, t)],i (i, j = 1, 2, 3) (33) 2 ∂ γ ϕ0 ∂ D23 H (x, ξ, t) − (34) + τ0 2 wk,k (x, ξ, t) = 0, x, ξ ∈ V, t > 0 k ∂t ∂t with the homogenous initial conditions wi (x, ξ, 0) = 0, w˙ i (x, ξ, 0) = 0,
H (x, ξ, 0) = 0,
H˙ (x, ξ, 0) = 0, x, ξ ∈ V, t = 0
(2)
(35) (2)
Now assuming Fi = 0 and introducing Laplace transform of functions Q (2) = δ(x − ξ )δ(t), u i = (2) wi (x, ξ, t) and ϕ = H (x, ξ, t) into reciprocity relation (28) and taking the inversion of Laplace transform thereafter, we obtain θ (ξ, t) + τ0 θ˙ (ξ, t) t t ∂ Q(x, τ )H1 (x, ξ, t − τ )d V (x) + τ0 dτ Q(x, τ ) H1 (x, ξ, t − τ )d V(x) = dτ ∂τ 0
0
V
t −φ0
dτ 0
V
V
∂wi (x, ξ, τ ) Fi (x, t − τ ) d V (x) − τ0 φ0 ∂τ
t
Fi (x, t − τ )
dτ 0
V
∂ 2 wi (x, ξ, τ ) d V (x) ∂τ 2
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t −φ0
dτ ∂V
0
t −τ0 φ0
∂V
0
∂ 2 pi(Q) (x, ξ, τ ) ∂ 2 wi (x, ξ, τ ) pi (x, t − τ ) d A(x) − u i (x, ξ, t − τ ) ∂τ 2 ∂τ 2
(ϕ,n (x, t − τ )H (x, ξ, τ ) − ϕ0 (x, t − τ )H,n (x, ξ, τ )) d A(x)
dτ 0
(x, ξ, τ ) d A(x) ∂τ
t +k
dτ
(Q)
∂p ∂wi (x, ξ, τ ) pi (x, t − τ ) − u i (x, ξ, t − τ ) i ∂τ
(36)
∂V (Q)
(Q)
(Q)
where ϕ is given by Eq. (6) and H1 = H − a H,ii . Here pi = σik n k , σik being the components of stress on the surface at the point x ∈ ∂ V due to the concentrated instantaneous heat source Q (2) = δ(x − ξ )δ(t). The relations (32) and (36) are the extensions of the Somigliana formulae for the case of dynamic problems of two-temperature generalized thermoelasticity. These relations enable one to determine the displacements and the temperature inside the body if the displacements, conductive temperature and conductive temperature gradient are known on the boundary surface provided that the Green functions ( j) Ui (x, ξ, t) , ψ ( j) (x, ξ, t) , wi (x, ξ, t) and H (x, ξ, t) are known. ( j) Now we consider the case when the functions Ui (x, ξ, t) , wi (x, ξ, t) and H (x, ξ, t) refer to a bounded (2) medium. In this case it is assumed that at the point ξ ∈ V , the forces Fi = δ(x − ξ )δi j δ(t) are applied ( j) (2) and these functions produce the displacements and conductive temperature u i = Ui (x, ξ, t) and ϕ (2) = ψ ( j) (x, ξ, t), respectively and satisfy the Eqs. (29), (30) with initial condition (31) as well as boundary conditions ( j) Ui (x, ξ, t) = 0, ψ ( j) (x, ξ, t) = 0, ψ˙ ( j) (x, ξ, t) = 0 on ∂ V ; x ∈ ∂ V ; ξ ∈ V, t > 0
(37)
(2)
Similarly, we assume that the displacements u i = wi (x, ξ, t) and the conductive temperature ϕ (2) = H (x, ξ, t) arise due to the existence of the concentrated instantaneous heat source Q (2) = δ(x − ξ )δ(t) and satisfy the Eqs. (33), (34) with initial condition (35) as well as the boundary conditions wi (x, ξ, t) = 0,
H˙ (x, ξ, t) = 0 on ∂ V ; x ∈ ∂ V ; ξ ∈ V, t > 0
H (x, ξ, t) = 0,
In this case, Eqs. (32) and (36) can be written as follows:
t u˙ j (ξ, t) + τ0 u¨ j (ξ, t) =
( j)
Fi (x, t − τ )
dτ 0
V
t +τ0
−
1 φ0 τ0 φ0
−
V
t dτ 0
( j) Q(x, t − τ ) ψ ( j) (x, ξ, τ ) − aψ,ii (x, ξ, τ ) d V (x)
Q(x, t − τ )
dτ 0
V
dτ
0
∂ 2 Ui (x, ξ, τ ) d V (x) ∂τ 2
V
t
t
( j)
Fi (x, t − τ )
dτ 0
−
∂Ui (x, ξ, τ ) d V (x) ∂τ
∂V
∂ ( j) ( j) ψ (x, ξ, τ ) − aψ,ii (x, ξ, τ ) d V (x) ∂τ ( j)
∂ p (x, ξ, τ ) u i (x, ξ, t − τ ) i d A(x) ∂τ
(38)
Some theorems on two-temperature generalized thermoelasticity
t −τ0
θ (ξ, t) + τ0 θ˙ (ξ, t) =
( j)
u i (x, ξ, t − τ )
dτ ∂V
0
+
t
k φ0
1039
∂ 2 pi (x, ξ, τ ) d A(x) ∂τ 2
( j)
ϕ0 (x, τ )ψ,n (x, ξ, t − τ )d A(x)
dτ
(39)
∂V
0
t
Q(x, τ )H1 (x, ξ, t − τ )d V (x)
dτ 0
V
t +τ0
Q(x, τ )
dτ 0
V
t −φ0
Fi (x, t − τ )
dτ 0
V
−φ0 τ0
Fi (x, t − τ )
dτ 0
V
t
∂ pi
∂V
t +τ0 φ0
∂V
0
(x, ξ, τ ) d A(x) ∂τ (Q)
u i (x, ξ, t − τ )
dτ
∂ 2 pi
(x, ξ, τ ) d A(x) ∂τ 2
t
ϕ0 (x, t − τ )H,n (x, ξ, τ )d A(x)
dτ 0
∂ 2 wi (x, ξ, τ ) d V (x) ∂τ 2 (Q)
u i (x, ξ, t − τ )
dτ 0
−k
∂wi (x, ξ, τ ) d V (x) ∂τ
t
+φ0
∂ H1 (x, ξ, t − τ )d V (x) ∂τ
(40)
∂V
The formula (39) and (40) enable one to determine the displacements and temperature inside a body provided they are known on the surface of the body. These formulae are the extension of Green’s theorem to dynamic problems of two temperature generalized thermoelasticity.
5 Conclusions We have established a reciprocal principle of Betti type in the context of linear theory of two-temperature generalized thermoelasticity [28,29] for homogeneous and isotropic body. On the basis of our reciprocal principle, generalizations of the theorems of Somigliana and Green to two-temperature generalized thermoelasticity are also established. It should be mentioned here that our derived reciprocal relation, i.e., Eq. (16) and the formulae of Somigliana and Green given by Eqs. (32, 36) and (39,40) are the more generalized forms and they contain the corresponding relations of the previous works given in [23] and [17] as special cases. If we assume a = 0, i.e. when θ = φ, then our formulae (16), (32), (36), (39) and (40) reduce to the corresponding formulae (1.11), (2.8), (2.4), (3.5) and (3.3) with some changes of notations under generalized thermoelasticity with one relaxation parameter as reported by Khomyakevich and Rudenko [17]. In the case when we assume τ0 = 0, the results correspond to the formulae under the theory of two temperature thermoelasticity. Furthermore, if we put τ0 = 0 as well as a = 0 then the relation (16) reduce to the formula (21) with some changes of notations as given by Nowacki [23] (see Sect. 1.12, p. 54) and the formulae (32), (36), (39) and (40) reduce to the corresponding formulae (11), (12), (16) and (17) as obtained by Nowacki [23] (see Sect. 1.13, pp. 63–65) under classical coupled dynamical thermoelasticity theory.
1040
R. Kumar et al.
Acknowledgments The authors are thankful to the reviewers for their valuable and constructive comments and suggestions which have improved the quality of the paper. One of the authors (RK) thankfully acknowledges the support and encouragement of BIT, Mesra (Deoghar), India.
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