CHAPTER 2

THEORETICAL ANALYSIS OF THE

THERMOELASTICITY THEORY BASED ON
EXACT HEAT CONDUCTION LAW WITH
SINGLE DELAY

2.1 Some Theorems on Linear Theory of Thermoe-

lasticity based on Exact Heat Conduction Model

with Single Delay for Anisotropic Medium1

2.1.1 Introduction

In the literature, some pioneering work on thermoelasticity theory have been reported
by eminent researchers like, Nickel and Sackman (1968), Ieşan (1966; 1974), Ignaczak
(1963), Gurtin (1964), etc. and it has been shown that the state of dynamics of a ther-
moelastic system can be determined by using the variational method which describes
it as the extremum of a functional or function. Ignaczak (1963) and Gurtin (1964)
explained the variational principles for the initial-boundary value problems by incorpo-
rating the initial conditions into the field equations. With the help of this formulation,
1
The content of this subchapter is published in (Ghosh D., Giri D., Mohapatra R., Savas E.,
Sakurai K., Singh L. (eds)) Mathematics and Computing. ICMC 2018. Communications in Computer
and Information Science,834, Springer, Singapore,
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

Ieşan (1966; 1974) and then Nickell and Sackman (1968), established convolution type
variational principle for the linear coupled thermoelasticity. Subsequently, the varia-
tional theorem of Gurtin type for solids with micro-structure was presented by Ieşan
(1967). It is worth to be mentioned that the variational principle also plays an im-
portant role in the theoretical foundation of the numerical techniques for solving the
various thermo-mechanical problems.

Moreover, it has been observed that the reciprocity theorem is used to derive various
methods of integrating the elasticity equations in terms of Green’s function. It has sig-
nificant practical applications in finding the numerical solution of engineering problems
(Nowacki (1975b)) as reciprocity theorem states the relation between two sets of ther-
moelastic loadings and the corresponding thermoelastic configurations. Maizel (1951)
developed the Betti–Maxwell reciprocity theorem for the static problems in theory of
thermoelasticity. Later, the reciprocity theorem was extended to uncoupled thermoe-
lasticity, coupled thermoelasticity and coupled thermoelasticity for anisotropic homoge-
neous material by Predeleanu (1959), Ionescu-Cazimir (1964) and Nowacki (1975b), re-
spectively. Ieşan (1967) presented the first reciprocal relation without using the Laplace
transform. Convolution type reciprocity theorems were also derived by Ieşan (1966;
1974). Scalia (1990) used a method to deduce reciprocity relations without using the
Laplace transform and without incorporation of the initial data in the field equations.
An exhaustive treatment of the variational principles in thermoelasticity is available
in the books by Lebon (1980), Carlson (1973), Hetnarski and Ignaczak (2010), and
Hetnarski et al. (2009). Recently, the convolution type variational principles and re-
ciprocal relations on different theories of thermoelasticity were given by Chiriţă and
Ciarletta (2010), Mukhopadhyay et al. (2011b), Kothari and Mukhopadhyay (2013a),
and Kumari and Mukhopadhyay(2017c). Further, Shivay and Mukhopadhyay (2019)
presented Somigliano and Green’s theorem based on reciprocity theorem in the con-
text of the generalized thermoelasticity model with single delay. Moreover, Jangid

36
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

and Mukhopadhyay (2020) focused on variational and reciprocal principles for modified
temperature-rate dependent two-temperature thermoelasticity theory.

The present chapter of the thesis aims at analyzing the thermoelasticity theory
based on the exact heat conduction model with a single delay as proposed by Quin-
tanilla (2011) very recently. In the article, Quintanilla (2011) presented one modified
heat conduction model to overcome the stability complexities under the three-phase-lag
heat conduction theory (Roychoudhuri (2007a)) as mentioned by researchers including
Dreher et al. (2009). The author proved the well posedness of the problem under
this heat conduction theory and elaborated the recurrent scheme to obtain the explicit
form of the solution. Later, the author extended the results of well posedness to the
system of equations in thermoelasticity theory and proposed an alternative thermoe-
lasticity theory with a single delay (τ ) time parameter. Subsequently, Leseduarte and
Quintanilla (2013) represented Phragmén Lindelöf type alternative for the forward-in-
time (Eq. (1.3.25)) and backward-in-time (Eq. (1.3.26)) version of the model given by
Quintanilla (2011).

In this subchapter, some important theorems are established in the context of this
new thermoelasticity model (2011) for homogeneous and anisotropic medium. The sub-
chapter starts with describing the basic governing equations and constitutive relations
for anisotropic medium in the context of the present theory and considers a mixed
initial-boundary value problem with non-homogeneous initial conditions. Then, the
work is progressed in the direction to prove the uniqueness of solution of the mixed
problem by using the specific internal energy function. Next, the alternative formula-
tion of the mixed initial-boundary value problem using convolution is presented. The
benefit of this formulation is that it incorporates the initial conditions into the field
equations, due to which there is no need to consider the initial conditions separately.
Lastly, using this formulation, the variational principle of convolution type and a reci-
procity theorem is exhibited.

37
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

2.1.2 Basic Equations and Problem Formulation

Following Quintanilla (2011) and Leseduarte and Quintanilla (2013), the basic govern-
ing equations and the constitutive relations in context of of Quintanilla’s thermoelas-
ticity model for a homogeneous and anisotropic material can be written as follows:

The equation of motion:

σij,j + ρHi = ρüi . (2.1.1)

The equation of energy:

ρT0 Ṡ = −qi,i + ρR. (2.1.2)

The constitutive relations:

σij = Cijkl ekl −βij θ, (2.1.3)
θ
ρS = ρcE + βij eij , (2.1.4)
T0
τ 2 ∂2
 
∂ ∂
∗
q˙i = − Kij + Kij (1 − τ + ) ηj . (2.1.5)
∂t ∂t 2 ∂t2

The geometrical relations:

ηj = θ,j , (2.1.6)
1
eij = (ui,j + uj,i ) = u(i,j) . (2.1.7)
2

In this system of equations, a rectangular coordinate system xk in three dimen-

sional Euclidean space with usual indicial notations and ηi denotes the components of
temperature gradient.

Mixed Initial Boundary Value Problem

Now, considering V as the closure of an open, bounded, connected domain with bound-

38
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

ary, ∂V, enclosing an homogeneous and anisotropic thermoelastic material. Let V

denote the interior of V and ni be the components of an outward drawn unit normal
to ∂V . Let Bi , (i = 1, 2, 3, 4) be the subsets of ∂V such that B1 ∪ B2 = B3 ∪ B4 = ∂V
and B1 ∩ B2 = B3 ∩ B4 = φ. The motion relative to an undistorted stress free reference
state is considered for the present study.

For a mixed initial and boundary value problem, the field equations and constitutive
relations are given by Eqs. (2.1.1-2.1.7) defined on V ×[0, ∞) together with the following
initial conditions and boundary conditions:

Initial conditions: On V


ui (x, 0) = u0i (x), u̇i (x, 0) = vi (x),


(2.1.8)
θ(x, 0) = θ0 (x), θ̇(x, 0) = θ1 (x), qi (x, 0) = q0i (x). 


Boundary conditions:


ui = ũi (x, t) on B1 × [0, ∞), 




σi = σij nj = σ̃i (x, t) on B2 × [0, ∞), 


(2.1.9)
q = qi ni = q̃(x, t) on B3 × [0, ∞),  




θ = θ̃(x, t) on B4 × [0, ∞).



Here, u0i , vi , θ0 , θ1, q0i represent the specified initial displacement component, veloc-
ity component, temperature, rate of temperature, and heat-flux, respectively together
with ũi , σ̃i , θ̃, q̃, which denote the known surface displacement component, component
of traction vector, temperature and normal heat-flux, respectively. The smoothness
requirements and other regularity assumptions on the ascribable functions are also con-
sidered as hypotheses on data. Also, assumptions are made that u0i , vi , θ0 , θ1 , q0i are
continuous on V , Hi , and R are continuously differentiable on V × [0, ∞). q̃ and σ̃
are piecewise continuous on B3 × [0, ∞) and B2 × [0, ∞), respectively. ũi and θ̃ are
continuous on B1 × [0, ∞) and B4 × [0, ∞), respectively.

39
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

Further, Cijkl , βij , Kij , and Kij∗ are assumed to be smooth on V and satisfy

Cijkl = Cklij = Cjikl = Cijlk , βij = βji , Kij = Kji , Kij∗ = Kji∗ , (2.1.10)

Cijkl eij ekl > 0, for all eij on V × [0, ∞), (2.1.11)

Kij ϕi ϕj > 0 for any real ϕi on V × [0, ∞), (2.1.12)

Kij∗ ψi ψ j > 0 for any real ψi on V × [0, ∞). (2.1.13)

The material constants and delay time parameters satisfy the following inequalities:

ρ > 0, cE > 0, T0 > 0, τ > 0, Kij − τ Kij∗ > 0 on V. (2.1.14)

Now, defining an admissible state as R = {ui , θ, ηi , eij , σij , qi , S}, which is an ordered
array of functions ui , θ, ηi , eij , σij , qi , S defined on V × [0, ∞) with the properties that
ui ∈ C 2,2 , θ ∈ C 1,2 , ηi ∈ C 0,2 , σij ∈ C 1,0 , qi ∈ C 1,1 , S ∈ C 0,1 and eij = eji , σij = σji
on V × [0, ∞). Further, defining two operations, addition of two admissible states and
multiplication of an admissible state with a scalar as follows:
0 0 0
R + R = {ui + ui , θ + θ , ......, S + S },
0
λ∗ R = {λ∗ ui , λ∗ θ, ....., λ∗ S}, where λ∗ is any scalar. Then the set of all admissible
states is clearly a linear space.

Further, an admissible state is the solution of the present mixed problem if it satisfies
all the field Eqs. (2.1.1-2.1.7), the initial conditions (2.1.8) and the boundary conditions
(2.1.9).

2.1.3 Uniqueness of Solution

For the uniqueness of solution, the specific internal energy for the present initial-
boundary value problem is considered which is in the form
1 ρcE 2
E = Cijkl ėkl ėij + θ̇ . (2.1.15)
2 2T0

Clearly, from Eq. (2.1.11) and Eq. (2.1.14), it can be stated that the specific internal

40
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

energy (Eq. (2.1.15)) is positive definite and using Eq. (2.1.3), Eq. (2.1.4), and Eq.
(2.1.10), it is attained that

Ė = σ̇ij ëij + ρS̈ θ̇. (2.1.16)

Now, relations (2.1.2), (2.1.5), (2.1.6), and (2.1.7) together give

1 ρṘ
Ė = σ̇ij üi,j − q̇i,i θ̇ + θ̇
T0 T0
1 1 ρṘ
= (σ̇ij üi ),j − σ̇ij,j üi − (q̇i θ̇)i + (q̇i η̇i ) + θ̇
T0 T0 T0
1 ρṘ ...
= (σ̇ij üi ),j − (q̇i θ̇),i + ρḢi üi + θ̇ − ρ u i üi
T0 T0
τ2 ∗
 
η̇i
− ∗ ∗
Kij η˙j + Kij ηj − τ Kij η˙j + Kij η̈j . (2.1.17)
T0 2

Integrating both sides of Eq. (2.1.17) over V , using divergence theorem and applying
(2.1.1), acquire

ˆ  ˆ
Kij∗ τ 2 Kij∗

∂ ρ 1
Kij − τ Kij∗ η̇i η̇j dV


E + üi üi + ηi ηj + η̇i η̇j dV +
∂t 2 2T0 4T0 T0
V V
ˆ ! ˆ  
ρṘθ̇ ˙ 1 ˙
= ρḢi üi + dV + ei üi − θ̇qe dA. (2.1.18)
σ
T0 T0
V A

Now, the uniqueness of solution of the present mixed initial-boundary value problem is
established by the following uniqueness theorem.

Theorem-2.1.3.1 (Uniqueness theorem):

Statement: The mixed initial-boundary value problem given by Eqs. (2.1.1-2.1.7),

which satisfies the initial conditions (2.1.8) and boundary conditions (2.1.9) has at
most one solution.

(γ) (γ) (γ) (γ)

Proof: Assume that there are two sets of solutions ui , θ(γ) , eij , σij , qi , S (γ) for
γ = 1, 2. and construct the difference between these two sets of functions as

41
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

(1) (2)
ui = ui − ui , θ = θ(1) − θ(2) , ..............., S = S (1) − S (2) . (2.1.19)

Since, the set of all admissible states is a linear space, so the difference functions defined
by (2.1.19) also satisfy the Eqs. (2.1.1-2.1.7) with zero body forces and heat source,
the initial conditions (2.1.8) and the boundary conditions (2.1.9) in their homogeneous
form and hence, Eq. (2.1.18) too. Therefore, Eq. (2.1.18) yields
ˆ  ˆ
Kij∗ τ 2 Kij∗

∂ ρ 1
Kij − τ Kij∗ η̇ i η̇ j dV = 0.


E + üi üi + ηiηj + η̇ i η̇ j dV +
∂t 2 2T0 4T0 T0
V V
(2.1.20)

Integrating the above equation over time interval (0, t) after interchanging the variable
t with ξ and using the homogeneous initial conditions for difference functions give the
following equation:
ˆ  ˆt ˆ
Kij∗ τ 2 Kij∗

ρ 1
Kij − τ Kij∗ η̇ i η̇ j dV dξ = 0.


E + üi üi + ηη + η̇ η̇ dV +
2 2T0 i j 4T0 i j T0
V 0 V
(2.1.21)

From Eq. (2.1.11), Eq. (2.1.12), Eq. (2.1.13), and Eq. (2.1.14), it is observed that the
component in each term present on the left hand side of Eq. (2.1.21) is non-negative.
Thus it can be concluded that each term in Eq. (2.1.21) must be zero which implies
that

üi = 0, θ˙ = 0 on V × [0, ∞). (2.1.22)

i.e.,
∂ 2 ui ∂θ
2
= 0, = 0, on V × [0, ∞). (2.1.23)
∂t ∂t

Therefore, in view of the initial conditions ui (x, 0) = 0, u̇i (x, 0) = 0, and θ(x, 0) = 0,
Eq. (2.1.23) yields
ui = 0, θ = 0 on V × [0, ∞),

i.e.,
(1) (2)
ui = ui , θ(1) = θ(2) on V × [0, ∞).

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 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

This completes the proof of the uniqueness theorem.

2.1.4 Alternative Formulation of Mixed Problem

This subsection discusses the alternative formulation of the above mixed initial-boundary
value problem in which the initial conditions are combined into the field equations
(Gurtin (1964)). For this purpose, the following results are used:

Let φ and ψ be two functions defined on V × [0, ∞) such that both are continuous on
[0,∞) for each x ∈ V. Then the convolution φ ∗ ψ of φ and ψ is defined as
ˆt
[φ ∗ ψ](x, t) = φ(x, t − τ )ψ(x, τ )dτ, (x, t) ∈ V × [0, ∞).
0

The commutativity, associativity, and distributivity properties of convolution and the

property that

φ ∗ ψ = 0 ⇒ φ = 0 or ψ = 0 (2.1.24)

are used.

Now, the functions g and l are defined on [0,∞) as

g(t) = t, l(t) = 1. (2.1.25)

Also, let functions fi and W be defined on V × [0, ∞) as

fi = g ∗ ρHi + ρ(tvi + u0i ), (2.1.26)

ρR θ0
W =l∗ + ρcE + βij u0i,j , (2.1.27)
T0 T0

and let

τ2 ∗ τ2
Ni = l ∗ (tq0i + tθ0,j Kij − tτ θ0,j Kij∗ + tθ1,j Kij + θ0,j Kij∗ ). (2.1.28)
2 2

Consider p(x, t) and ṗ(x, t), two functions defined on V × [0, ∞) such that both are
continuous and differentiable on [0,∞). Then the following results hold clearly:

g ∗ p̈(x, t) = p(x, t) − [tṗ(x, 0) + p(x, 0)], (2.1.29)

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 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

l ∗ ṗ(B, t) = p(x, t) − p(x, 0), (2.1.30)

g ∗ ṗ(x, t) = l ∗ (l ∗ ṗ(x, t)) = l ∗ [p(x, t) − p(x, 0)] = l ∗ p(x, t) − tp(x, 0). (2.1.31)

Using this formulation, the following theorem is obtained that characterizes the con-
sidered mixed problem in an alternative way.

Theorem -2.1.4.1:

Statement: The function ui , θ, ηi , eij , σij , qi , S satisfy Eq. (2.1.1), Eq. (2.1.2) and
Eq. (2.1.5) and the initial conditions (2.1.8) if and only if

g ∗ σij,j + fi = ρui , (2.1.32)

qi,i
ρS = −l ∗ + W, (2.1.33)
T0

L1 ∗ qi = −L1 ∗ Kij ηj − L2 ∗ Kij∗ ηj + Ni , (2.1.34)

τ2
where, L1 = l ∗ l and L2 = l ∗ (g + τ l + 2
), fi , W and Ni are given by Eq. (2.1.26),
Eq. (2.1.27), and Eq. (2.1.28), respectively.

Proof: Firstly, assuming that the governing Eq. (2.1.1), Eq. (2.1.2) and Eq. (2.1.5),
and initial conditions (2.1.8) hold good. Then, taking the convolution of Eq. (2.1.1)
with g and using the results from Eq. (2.1.29) and Eq. (2.1.8), the Eq. (2.1.32) is
obtained. Similarly, taking the convolution of the Eq. (2.1.2) with l and using Eq.
(2.1.30), Eq. (2.1.4), and Eq. (2.1.8), the Eq. (2.1.33) is acquired. Again, taking the
convolution of Eq. (2.1.5) with l ∗ g, and using the relation from (2.1.29), (2.1.31) and
(2.1.8), the Eq. (2.1.34) is yielded.

Similarly, the converse of the above theorem can be proved using reverse arguments.
Hence, presenting the following theorem.

Theorem-2.1.4.2:

Statement: Let R = {ui , θ, ηi , eij , σij , qi , S} be an admissible state. Then R is a

44
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

solution of the mixed problem if and only if it satisfies the Eqs. (2.1.32-2.1.34), Eq.
(2.1.3), Eq. (2.1.4), Eq. (2.1.6), Eq. (2.1.7) and the boundary conditions (2.1.9).

2.1.5 Variational Theorem

Using the alternative formulation and the theorem established in the previous subsec-
tion, a variational principle on linear theory of thermoelasticity for anisotropic and
homogeneous medium under the present heat conduction model given by Quintanilla
(2011) is formulated in the following way:

Theorem -2.1.5.1:

Statement: Let Λ be a linear space of all admissible states with addition and scalar
multiplication as describe in Subsection-2.1.2. If for each t ∈ [0, ∞) and for every
R = {ui , θ, ηj , eij , σij , qi , S} ∈ Λ, a functional Ft {R} on Λ is defined by

Ft {R}
ˆ 
1 1 1
= L1 ∗ g ∗ Cijkl ekl ∗ eij − L1 ∗ ρui ∗ ui − L1 ∗ g ∗ σij ∗ eij − L1 ∗ g ∗ l ∗ qi ∗ ηi
2 2 T0
V
 
qi,i
+ L1 ∗ ui ∗ (ρui − g ∗ σij,j − fi ) − L1 ∗ g ∗ θ ∗ ρS + l ∗ −W
T0
 
1 1 1 ∗
+ g∗l∗ −L1 ∗ Kij ηj − L2 ∗ Kij ηj + Ni ∗ ηi
T0 2 2
 ˆ
T0
+ L1 ∗ g ∗ (ρS − βrs ers ) ∗ (ρS − βij eij ) dV + L1 ∗ g ∗ ũi ∗ σi dA
2ρcE
B1
ˆ ˆ
1
+ L1 ∗ g ∗ (σi − σ̃i ) ∗ ui dA + L1 ∗ g ∗ l ∗ q ∗ θ̃dA
T0
B2 B3
ˆ
1
+ M1 ∗ g ∗ l ∗ (q − q̃) ∗ θdA, (2.1.35)
T0
B4

then the variation of this functional,

δFt {R} = 0, t ∈ [0, ∞), (2.1.36)

45
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

if and only if, R is a solution of the mixed initial-boundary value problem given by Eqs.
(2.1.1-2.1.7) with the initial conditions (2.1.8) and the boundary conditions (2.1.9).
0 0 0 0 0 0 0 0 0
Proof: Let R = {ui , θ , ηi , eij , σij , qi , S } ∈ Λ, which implies that R + λR ∈ Λ, for
every real λ. Then, Eq. (2.1.35) together with properties of convolution, definition of
variation, and the divergence theorem, implies

δR0 Ωt {R}
ˆ   
T0 βij 0
= L1 ∗ g ∗ Cijkl ekl − (ρS − βrs ers ) − σij ∗ eij
ρcE
V
 
T0 0
+ L1 ∗ g ∗ ((ρS − βrs ers )) − θ ∗ ρS
ρcE

1 ∗

0
+g ∗ l ∗ −L1 ∗ Kij ηj − L2 ∗ Kij ηj + Ni − L1 ∗ qi ∗ ηi dV
T0
ˆ    
0 qi,i 0
− L1 ∗ (g ∗ σij,j + fi − ρui ) ∗ ui + L1 ∗ g ∗ ρS + l ∗ − W ∗ θ dV
T0
V
ˆ  

0 1 0
− L1 ∗ g ∗ eij − u(i,j) ∗ σij − L1 ∗ g ∗ l ∗ (θ,i − ηi ) ∗ qi dV
T0
ˆ
V
ˆ
0 0
+ L1 ∗ g ∗ (ũi − ui ) ∗ σi dA + L1 ∗ g ∗ (σi − σ̃i ) ∗ ui dA
B1 B2
ˆ ˆ
1  1  0 0
+ L1 ∗ g ∗ l ∗ θ̃ − θ ∗ q dA + L1 ∗ g ∗ l ∗ (q − q̃) ∗ θ dA, (2.1.37)
T0 T0
B3 B4

for all t ∈ [0, ∞).

Firstly, assuming that R is a solution of the mixed initial-boundary value problem, then
from Theorem-2.1.4.2, the relations (2.1.32) to (2.1.34) and the boundary conditions
(2.1.9), the following is obtained:

δR0 Ωt {R} = 0, t ∈ [0, ∞) (2.1.38)

0 0 0 0 0 0 0 0
for every R = {ui , θ , γi , eij , σij , qi , S } ∈ Λ, and therefore yield Eq. (2.1.36). This
completes the proof of the necessary part of the Theorem-2.1.5.1.
0
Conversely, let Eq. (2.1.36) holds true and hence, Eq. (2.1.38) holds for every R =

46
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

0 0 0 0 0 0 0
{ui , θ , ηi , eij , σij , qi , S } ∈ Λ. Then, it is to be shown that R is a solution of mixed
initial-boundary value problem.
0 0 0 0
Since, Eq. (2.1.38) holds for every R ∈ Λ, let R = {ui , 0, 0, 0, 0, 0, 0} and let ui
along with all the space derivatives vanish on ∂V × [0, ∞). Therefore, Eq. (2.1.37) and
Eq. (2.1.38) yield
ˆ
0
L1 ∗ (g ∗ σij,j + fi − ρui ) ∗ ui dV = 0, for t ∈ [0, ∞). (2.1.39)
V

Further, by using Lemma-1 (see Gurtin (1964)) and convolution properties, it is found
that Eq. (2.1.32) holds.
0 0 0
Again, by choosing R = {0, θ , 0, 0, 0, 0, 0} and letting θ along with all the space
derivatives vanish on ∂V × [0, ∞), Eq. (2.1.37) and Eq. (2.1.38) imply the following:
ˆ  
qi,i 0
L1 ∗ g ∗ ρS + l ∗ − W ∗ θ dV = 0, for t ∈ [0, ∞).
T0
V

Therefore, by using Lemma-1 (see Gurtin (1964)) and convolution properties, Eq.
(2.1.33) is obtained.
0
Similarly, by substituting appropriate choices of R into Eq. (2.1.37) and with the
help of three lemmas (1-3) (Gurtin (1964)) it can be proved that R also satisfies the
Eq. (2.1.33), Eq. (2.1.34), Eq. (2.1.3), Eq. (2.1.4), Eq. (2.1.6), Eq. (2.1.7) and the
boundary conditions (2.1.9). Therefore, from Theorem-2.1.4.2, R is the solution of the
present mixed problem. Hence, the proof of the above theorem is complete.

2.1.6 Reciprocity Theorem

Now, considering two different systems of thermoelastic loadings

 (a)

(a) (a) (a) (a) (a) (a) (a) (a)
La = Hi , R(a) , ũi (a) , θ̃ , q̃ i , σ̃i , u0i , vi , θ0 , θ1 , q0i , a = 1, 2. (2.1.40)

The corresponding thermoelastic configurations are denoted as

(a)
I a = (ui , θ(a) ), (2.1.41)

that satisfy Eqs. (2.1.32-2.1.34), Eq. (2.1.3), Eq.(2.1.4), Eq. (2.1.6), Eq. (2.1.7) and

47
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

the boundary conditions (2.1.9).

The aim is to establish a reciprocity theorem that states the relation between these two
sets of thermoelastic loading and thermoelastic configurations. For this, the following
notations are used:
 
(a) (a) (a) (a)
fi = ρ g ∗ Hi + tvi + u0i , (2.1.42)

(a)
ρR(a) θ (a)
W (a) = l ∗ + ρcE 0 + βij u0i,j , (2.1.43)
T0 T0

(a) (a) (a) (a) τ 2 ∗ (a) τ 2 ∗ (a)

Ni = l ∗ (tq0i + tKij θ0,j − tτ Kij∗ θ0,j + t K θ + Kij θ0,j ), (2.1.44)
2 ij 1,j 2

for a = 1, 2. Then, the reciprocity theorem is given as below.

Theorem -2.1.6.1 (Reciprocity theorem):

Statement: If a thermoelastic solid is associated with two different systems of ther-

moelastic loadings, La , (a = 1, 2) and the corresponding thermoelastic configurations,
I a , (a = 1, 2) , then the following reciprocity relation holds:

ˆ ˆ  
h
(1) (2) (1) (2)
i
(1) (2) 1 (1) (2)
L1 ∗ fi ∗ ui − g ∗ W ∗ θ dV + L1 ∗ g ∗ σi ∗ ui + l ∗ q ∗ θ dA
T0
V A
ˆ   ˆ
1 (1) (2)
h
(2) (1)
i
− g∗l∗ Ni ∗ ηi dV = L1 ∗ fi ∗ ui − g ∗ W (2) ∗ θ(1) dV
T0
V V
ˆ   ˆ  
(2) (1) 1 1 (2) (1)
+ L1 ∗ g ∗ σi ∗ ui + l ∗ q ∗ θ (2) (1)
dA − g ∗ l ∗ N ∗ ηi dV, (2.1.45)
T0 T0 i
A V

(a) (a)
where, fi , W (a) , Ni (a = 1, 2) associated with two systems are given by Eq.
(2.1.42), Eq. (2.1.43), and Eq. (2.1.44), respectively.

Proof: Using Eq. (2.1.3), it can be written as

(a) (a)
σij = Cijkl ekl − βij θ(a) . (2.1.46)

(2) (1)
Next on taking convolution of Eq. (2.1.46) for a = 1 with eij and for a = 2 with eij
and then subtracting the results yield

48
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...
     
(1) (1) (2) (2) (2) (1) (1) (2) (2) (1)
σij + βij θ ∗ eij = σij + βij θ ∗ eij + Cijkl ekl ∗ eij − ekl ∗ eij .

Hence, the symmetric properties of Cijkl give

 
(1) (2) (2) (1) (1) (2) (1) (2)
Cijkl ekl ∗ eij − ekl ∗ eij = Cijkl ekl ∗ eij − Cklij ekl ∗ eij = 0. (2.1.47)

Therefore,
   
(1) (2) (2) (1)
σij + βij θ(1) ∗ eij = σij + βij θ(2) ∗ eij . (2.1.48)

Again from Eq. (2.1.4), it can be written as

(a) θ(a)
ρS (a) − βij eij = ρcE , a = 1, 2. (2.1.49)
T0

Taking convolution of Eq. (2.1.49) for a = 1 with θ(2) and for a = 2 with θ(1) and
subtracting, yield the equation as
   
(1) (2)
ρS (1) − βij eij ∗ θ(2) = ρS (2) − βij eij ∗ θ(1) . (2.1.50)

Eq. (2.1.48) and Eq. (2.1.50) yield

   
(1) (2) (2) (1)
(1)
σij ∗ eij − ρS ∗ θ (2) (2)
= σij ∗ eij − ρS ∗ θ (1)
. (2.1.51)

Next, introducing the notation

ˆ h i
(a) (b)
Lab = L1 ∗ g ∗ σij ∗ eij − ρS (a) ∗ θ(b) dV, a, b = 1, 2. (2.1.52)
V

Now, Eq. (2.1.7) and Eqs. (2.1.32-2.1.34) give

 
(a) (b)
L1 ∗ g ∗ σij ∗ eij − ρS (a) ∗ θ(b)
(a)
!
(a) (b) q i,i
= L1 ∗ g ∗ σij ∗ ui,j − L1 ∗ g ∗ −l ∗ + W (a) ∗ θ(b)
T0
   
(a) (b) (a) (b)
= L1 ∗ g ∗ σij ∗ ui ,j − L1 ∗ g ∗ σij,j ∗ ui
1 
(a)
 1 (a) (b)
+ L1 ∗ g ∗ l ∗ qi ∗ θ(b) ,i − L1 ∗ g ∗ l ∗ qi ∗ ηi
T0 T0
− L1 ∗ g ∗ W (a) ∗ θ(b) ,

49
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

 
(a) (b)
L1 ∗ g ∗ σij ∗ eij − ρS (a) ∗ θ(b)
 
(a) (b) (a) (b) (a) (b)
= L1 ∗ g ∗ σij ∗ ui ,j − L1 ∗ ρui ∗ ui + L1 ∗ fi ∗ ui
1 
(a) (b)
 1 
(a)
+ L 1 ∗ g ∗ l ∗ qi ∗ θ ,i + g ∗ l ∗ L1 ∗ Kij ηj
T0 T0
(a)

(b) 1 (a) (b)
+L2 ∗ Kij∗ ηj ∗ ηi − g ∗ l ∗ Ni ∗ ηi
T0
− L1 ∗ g ∗ W (a) ∗ θ(b) . (2.1.53)

Therefore, from Eq. (2.1.52) and Eq. (2.1.53), the following is obtained:

Lab
ˆ ˆ  
h
(a) (b) (a) (b)
i
(a) (b) 1 (a) (b)
= L1 ∗ fi ∗ ui − g ∗ W ∗ θ dV + L1 ∗ g ∗ σi ∗ ui + l ∗ qi ∗ θ dA
T0
V ∂V
ˆ  
(a) (b) 1 (a) (b) 1 ∗ (a) (b)
− L1 ∗ ρui ∗ ui − g ∗ l ∗ L1 ∗ Kij ηj ∗ ηi − g ∗ l ∗ L2 ∗ Kij ηj ∗ ηi dV
T0 T0
V
ˆ  
1 (a) (b)
− g ∗ l ∗ Ni ∗ ηi dV. (2.1.54)
T0
V

Clearly, Eq. (2.1.51) and Eq. (2.1.52) imply

L12 = L21 . (2.1.55)

Hence, Eq. (2.1.54) and Eq. (2.1.55) prove the reciprocity relation (2.1.45), which
completes the proof of the Theorem-2.1.6.1.

2.1.7 Conclusion

In the present subchapter, some important theorems under generalized thermoelasticity

model by Quintanilla (2011) are established. Uniqueness of the solution for mixed
initial-boundary problem for homogeneous and anisotropic thermoelastic medium is
obtained. Variational theorem of convolution type using an alternative formulation of
the problem followed by reciprocity theorem is presented .

50
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

2.2 Galerkin-Type Solution for the Theory of Ther-

moelasticity under an Exact Heat Conduction Law

with Single Delay2

2.2.1 Introduction

In this subchapter, the theoretical analysis of Quintanilla’s thermoelastic model is fur-

ther pursued by deriving the representation of solution into elementary functions for
isotropic and homogeneous thermoelastic material under linear theory. This represen-
tation simplifies the original complicated system of differential equations and further
helps to find the solution of original problem in terms of elementary functions such
as harmonic, biharmonic, metaharmonic, etc. This provides aid in solving various
boundary value problems in the field of elasticity and thermoelasticity. Proceeding
with addressing all the governing equations and constitutive relations, a Galerkin-type
solution of equations of motion under the thermoelasticity model is presented followed
by a Galerkin-type solution for the system of equations of steady oscillations. Lastly,
the general solution for the homogeneous system of equations for steady oscillations is
acquired.

The related works available in the literature are stated as following. The Galerkin-
type solution (Galerkin (1930)) of the equations of classical elastokinetics was given
by Iacovache (1949). Nowacki (1964; 1969a) and Sandru (1966) discussed the repre-
sentation of solutions namely Galerkin’s and Papkovitch’s in the classical theory of
thermodynamics and micropolar elasticity. Representations of a solution such as the
Boussinesq-Somigliana-Galerkin (BSG), Boussinesq-Papkovitch-Neuber (BPN), Green-
Lame (GLa), and Cauchy-Kovlevski-Somigliana (CKS) (Gurtin (1972), Nowacki (1975b),
2
The content of this subchapter is presented in International Conference on Engineering, Computers
and Natural Sciences, 2018

51
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

Kupradze et al. (1976)) are well established (Scalia and Svanadze (2006)) in the con-
text of classical elasticity. Chandrasekharaiah (1987a; 1989) thoroughly presented the
BPN, GLa, and CKS forms of the solution in the theory of voids. Ciarletta (1991;
1995; 1999) provided the Galerkin-type representation of solutions in case of the the-
ory of thermoelastic materials with voids, micropolar thermoelasticity without energy
dissipation and the dynamical theory of binary mixture consisting of gas and an elas-
tic solid, respectively. In the theory of binary mixtures of elastic solids and theory
of porous media, Svanadze (1993) and Svanadze and De Boer (2005) presented the
Galerkin-type representation of general solutions. Scalia and Svanadze (2006) gave
the representation of general as well as steady oscillation solutions in the theory of
thermoelasticity with micro-temperatures. Mukhopadhyay et al. (2010) presented the
representation of solutions for the linear theory of three-phase-lag thermoelasticity the-
ory (Roychoudhuri (2007a)). Later, Kothari and Mukhopadhyay (2012) established the
representation theorem for the generalized theory of thermoelastic diffusion (Sherief et
al. (2004)). Recently, Svanadze (2014; 2017) derived the Galerkin-type solution in the
case of linear thermoviscoelasticity theory for Kelvin-Voigt materials with voids and
linear theory of micropolar viscoelasticity, respectively. For understanding applications
of this representation of solution, it is worth referring the recent article by Giorgashvili
et al. (2015).

2.2.2 Governing Equations

Let x = (x1 , x2 , x3 ) represents an arbitrary point in three-dimensional Euclidean space

and t be the time variable. An isotropic elastic homogeneous medium is considered to
analyze a thermoelasticity theory. The medium occupies a bounded region Ξ of Eu-
clidean three-dimensional space at t = 0. Following Quintanilla (2011) and Leseduarte
and Quintanilla (2013), the basic equations in the context of considered thermoelasticity
theory are as follows:

52
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

Heat conduction law:

τ 2 ∂2
  
∂ ∂
q̇ = − K ∗
+K 1−τ + grad θ. (2.2.1)
∂t ∂t 2 ∂ t2

Energy equation:

− divq + ρR = ρ T0 Ṡ. (2.2.2)

Entropy equation:

T0 ρ S = ρ cE θ + β T0 trE. (2.2.3)

Equation of motion:

divΓ + ρ H = ρ ü. (2.2.4)

Stress-strain-temperature relation:
Γ = λtrE I + 2µE − βθI. (2.2.5)

Strain-displacement relation:
1
gradu + (gradu)T . (2.2.6)


E=
2

Further, eliminating q, E, Γ, and S from Eqs. (2.2.1-2.2.6) gives the following field
equations in the context of thermoelasticity theory under the heat conduction model
given by Quintanilla (2011):

µ ∇2 u + (λ + µ) grad divu − βgrad θ + ρ H = ρ ü, (2.2.7)

τ 2 ∂2
  
∂ ∂
K ∗
+K 1−τ + ∇2 θ = β T0 div ü + ρ cE θ̈ − r. (2.2.8)
∂t ∂t 2 ∂ t2

where, r = ρṘ is the external rate of heat source.

Now, introducing the following notations and operators:

53
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

τ 2 ∂2
  
2 2 2 ∂ ∗
2 ∂
∇2 − ρ cE T 2 ,



`1 ∇ , T = m2 ∇ − T , `2 ∇ , T = K +K 1−τ + 2
∂t ∂t 2 ∂t
2
 
∂ ∂ λ+µ µ β
T = , T 2 = 2 , m1 = , m2 = , m3 = .
∂t ∂t ρ ρ ρ

Therefore, Eq. (2.2.7) and Eq. (2.2.8) take the forms as follows:

m1 grad div u + `1 u − m3 grad θ = −H, (2.2.9)

`2 θ − β T0 T 2 div u = −r. (2.2.10)

2.2.3 Galerkin-Type Solution of Equations of Motion

In virtue of Eq. (2.2.9) and Eq. (2.2.10), presenting the matrix differential operator as
following:

 
(1) (2)
 Ω Ω
Ω (D x , T ) = 


Ω(3) Ω(4)
h i h i
(1)
 (1)  (2) (2) (3) (3)
Ω (D x , T ) = Ωpq 3×3 , Ω = Ωp1 , Ω = Ω1q , Ω(4) = [Ω44 ]1×1 ,
3x1 1×3
∂2
Ω(1)
pq (D x , T ) = ` δ
1 pq + m 1 ,
∂xp ∂xq
(2) ∂
Ωp1 (D x , T ) = −m3 ,
∂xp
(3)
∂
Ω1q (D x , T ) = −β T0 T 2 , Ω44 (D x , T ) = `2 . (2.2.11)
∂xq

 
where, the notations are defined as; D x = ∂
, ∂ , ∂
∂x1 ∂x2 ∂x3
and δpq as the Kronecker
delta for p, q = 1, 2, 3.

Therefore, Eq. (2.2.9) and Eq. (2.2.10), can be written as

Ω (D x , T ) U (x, t) = F (x, t) , (2.2.12)

where, U = (u, θ), F = (−H, −r) and (x, t) ∈ Ξ × (0, +∞).

Now, the following system of equations is introduced:

54
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

m1 grad div u + `1 u − β T0 T 2 grad θ = F 0 , (2.2.13)

`2 θ − m3 div u = F0 , (2.2.14)

where, F 0 = (F10 , F20 , F30 ) is the vector function with F0 and Fi0 (i = 1, 2, 3) as scalar
functions on Ξ × (0, +∞) .

Hence, in term of matrix operator, system (2.2.13-2.2.14) can be expressed in the form

ΩT (D x , T ) U (x, T ) = H (x, t) . (2.2.15)

where, ΩT is the transpose of matrix Ω and H = (F 0 , F0 ).

Next, taking the divergence of Eq. (2.2.13) yields

B1 div u − β T0 T 2 ∇2 θ = div F 0 , (2.2.16)

 
where, B1 (∇2 , T ) = λ+2µ
ρ
∇2 − T 2 .

Therefore, the matrix representation of Eq. (2.2.14) and Eq. (2.2.16) is derived as
follows:
B ∇2 , T V = F̃ . (2.2.17)

where, V = (div u, θ), F̃ = (div F 0 , F0 ), and

 
2 2
 B1 −β T0 T ∇ 
B(∇2 , T ) = Bpq ∇2 , T 2×2 = 


.
−m3 `2

System (2.2.17) implies

χ1 ∇2 , T V = Φ, (2.2.18)

with,

2
X
B∗pq fp , χ1 (∇2 , T ) = detB ∇2 , T , (2.2.19)


Φ = (Φ1 , Φ2 ) , Φq =
p=1

55
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

where, q = 1, 2 and B∗pq is the co-factor of the element Bpq of the matrix B.

Now, operating χ1 (∇2 , T ) to Eq. (2.2.13), and using Eq. (2.2.18), gives the following
relation:

χ1 ∇2 , T `1 u = Φ0 , (2.2.20)

where,

Φ0 = χ1 F 0 − grad m1 Φ1 − β T0 T 2 Φ2 . (2.2.21)
 

Further, in view of Eq. (2.2.18) and Eq. (2.2.20), it is acquired that

χ ∇2 , T U (x, t) = Φ̃, (2.2.22)

where, Φ̃ = (Φ0 , Φ2 ) and

χ(∇2 , T ) = χpq (∇2 , T ) 4×4

 

χjj = χ1 ∇2 , T `1 , j = 1, 2, 3

χ44 = χ1 (∇2 , T ), χpq = 0, p, q = 1, 2, 3, 4 p 6= q. (2.2.23)

Further, introducing the operators

np1 ∇2 , T = − m1 B∗p1 − β T0 T 2 B∗p2 ,



np2 (∇2 , T ) = B∗p2 , p = 1, 2, (2.2.24)

it can be obtained from Eq. (2.2.19), Eq. (2.2.21) that

Φ0 = (χ1 I + n11 grad div) F 0 + n21 grad F0 , (2.2.25)

Φ2 = n12 div F 0 + n22 F0 . (2.2.26)

56
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

Thus, in view of Eq. (2.2.25) and Eq. (2.2.26), it is found that

Φ̃(x,t) = LT (Dx ,T ) H (x, t) , (2.2.27)

where,
 
(1)
 L L(2) 
L=  ,
L(3) L(4)
4×4
h i h i
(2) (3)
L(1) = L(1) (2)
, L(3) = L1q , L(4) = [L44 ]1×1 ,
 
pq 3×3 , L = L p1
3x1 1×3
∂2 (2)
∂
L(1) 2 2
, Lp1 (D x , T ) = n12 ∇2 , T



pq (D x , T ) = χ1 ∇ , T δpq + n11 ∇ , T ,
∂xp ∂xq ∂xp
(3) ∂
L1q (D x , T ) = n21 (∇2 , T ) , L44 = n22 (∇2 , T ), p, q = 1, 2, 3. (2.2.28)
∂xq

Next, using Eq. (2.2.15), Eq. (2.2.22), and Eq. (2.2.27), following is obtain

χU = LT ΩT U ,

which implies LT ΩT = χ and hence,

Ω(D x , T )L(D x , T ) = χ(∇2 , T ). (2.2.29)

Thus, the following lemma is proved.

Lemma-2.2.3.1:

Statement: If the matrix differential operators Ω, L, and χ are defined by Eq.

(2.2.11), Eq. (2.2.28), and Eq. (2.2.23), respectively, then Ω, L, and χ satisfy Eq.
(2.2.29).
Now, let Hj0 (x, t), (j = 1, 2, 3) and h(x, t) be functions on Ξ × (0, +∞) with H 0 =
(H10 , H20 , H30 ), and H
f = (H 0 , h). Then, the subsequent theorem provides a Galerkin-type

solution to the system by Eq. (2.2.3) and Eq. (2.2.4).

57
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

Theorem-2.2.3.1:

Statement: Let

u = L(1) H 0 + L(2) h, (2.2.30)

θ = L(3) H 0 + L(4) h, (2.2.31)

where, the fields Hj0 of class C 6 and h of class C 4 satisfy

χ1 (∇2 , T ) `1 H 0 = −H, (2.2.32)

χ1 (∇2 , T ) h = −r, (2.2.33)

on Ξ×(0, +∞). Then U = (u, θ) is the solution of Eq. (2.2.9) and Eq. (2.2.10).

Proof: Eq. (2.2.30) and Eq. (2.2.31) yield

U (x, t) = L(Dx ,T )H(x,

f t). (2.2.34)

On the other hand, from Eq. (2.2.32) and Eq. (2.2.33), it is obtained that

χ(∇2 , T )H(∇
f 2 , T ) = F (∇2 , T ). (2.2.35)

In view of Eq. (2.2.29), Eq. (2.2.34), and Eq. (2.2.35), it is acquired that

ΩU = ΩLH f = F , which finalizes the proof of the theorem.

f = χH

2.2.4 Galerkin-Type Solution of System of Equations for Steady

Oscillations

In this subsection, the steady state oscillations are considered. Hence, the solution and
external loads can be assumed in the following forms:
u(x, t) = Re[ũ(x) e−iωt ], H(x, t) = Re[H̃(x) e−iωt ],

θ(x, t) = Re[θ̃(x) e−iωt ], r(x,t) = Re[r̃(x) e−iωt ].

58
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

Therefore, from Eq. (2.2.7) and Eq. (2.2.8), the system of equations of the steady
oscillations for the assumed thermoelasticity theory are derived as follows:

µ ∇2 ũ + (λ + µ)grad div ũ − βgrad θ̃ + ρ H̃ = −ω 2 ρ ũ,

(2.2.36)
ω2τ K ∗
  
K− − i ω (K − K τ ) ∇ + ρ cE ω θ̃ + β T0 ω 2 div ũ = −r̃.
∗ 2 2
(2.2.37)
2

√
where, (x, t) ∈ Ξ ×(0, +∞), i = −1, and ω(> 0) denotes the frequency of oscillation.

The above system can further be expressed as

ρ ω 2 + µ ∇2 ũ + (λ + µ)grad div ũ − β grad θ̃ = −ρ H̃, (2.2.38)

 

ω2τ K ∗
  
K− − i ω (K − K τ ) ∇ + ρ cE ω θ̃ + β T0 ω 2 div ũ = −r̃.
∗ 2 2
(2.2.39)
2

In the following, the underneath notations are used

C(∇2 ) = |Cpq (∇2 )|2x2

 
2 2 2 2
 ρ ω + (λ + 2µ)∇ β T0 ω ∇
= .

h i 
ω2 τ K∗
−β K− 2
− i ω (K − K ∗ τ ) ∇2 + ρ cE ω 2
2×2

Now, let

χ̃1 (∇2 ) = det C(∇2 ),

mp1 (∇2 ) = − (λ + µ)C∗p1 + β T0 ω 2 C∗p2 ,

 

mp2 (∇2 ) = C∗p2 , p = 1, 2.

It can be easily verified that if λ21 and λ22 are the roots of the equation χ̃1 (−λ∗ ) = 0,
then χ̃1 (∇2 ) = (∇2 + λ21 )(∇2 + λ22 ).

Next, the matrix differential operators M and χ̃ are defined by

59
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

 
(1)
 M M(2) 
M=  ,
(3) (4)
M M
4×4
h i h i h i
(1) (2) (3)
M(1) = Mlj , M(2) = Ml1 , M(3) = M1l , M(4) = [M44 ]1×1 ,
3×3 3×1 3×1
∂2 (2) ∂
M(1) 2 2
pq (D x ) = χ̃1 (∇ )δpq + m11 (∇ ) , Mp1 (D x ) = m12 (∇2 ) ,
∂xp ∂xq ∂xp
(3) ∂
M1p (D x ) = m21 (∇2 ) , M44 = m22 (∇2 ), p, q = 1, 2, 3. (2.2.40)
∂xp

χ̃(∇2 , T ) = χpq (∇2 ) 4×4 ,

 

χ̃jj = χ̃1 (∇2 )[ρ ω 2 + µ ∇2 ], j = 1, 2, 3,

χ̃44 = χ̃1 (∇2 ), χ̃pq = 0, p, q = 1, 2, 3, 4 p 6= q. (2.2.41)

If Q̃j , (j = 1, 2, 3) and q be functions on Ξ with Q̃ = (Q̃1 , Q̃2 , Q̃3 ), and Q =

(Q̃, q) then, in accordance with the Theorem-2.2.3.1, the following theorem provides a
Galerkin-type solution to system by Eq. (2.2.36) and Eq. (2.2.37).

Theorem-2.2.4.1:

Statement: Let

ũ = M(1) Q̃ + M(2) q, (2.2.42)

θ̃ = M(3) Q̃ + M(4) q, (2.2.43)

where, the fields Q̃j of class C 6 and q of class C 4 on Ω satisfy

60
 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

χ̃1 (∇2 ) ρ ω 2 + µ ∇2 Q̃ = −H̃, (2.2.44)

 

χ̃1 (∇2 )q = −r̃, (2.2.45)

 
on Ξ. Then ũ, θ̃ is the solution of Eq. (2.2.38) and Eq. (2.2.39).

2.2.5 General Solution of System of Equations for Steady Os-

cillations

In the absence of any body force and external heat source, the Eq. (2.2.38) and Eq.
(2.2.39) can be written as

ρ ω + µ ∇2 ũ + (λ + µ)grad div ũ − β grad θ̃ = 0, (2.2.46)

 2 

ω2τ K ∗
  
K− − i ω (K − K τ ) ∇ + ρ cE ω θ̃ + β T0 ω 2 div ũ = 0.
∗ 2 2
(2.2.47)
2

Firstly, the following lemma in the context of above system of equations is required to
be proved:

Lemma-2.2.5.1:

Statement: If (ũ, θ̃) is a solution of Eq. (2.2.46) and Eq. (2.2.47), then

χ̃1 (∇2 )div ũ = 0, (2.2.48)

χ̃1 (∇2 )θ̃ = 0, (2.2.49)

ρ ω + µ ∇2 curl ũ = 0. (2.2.50)
 2 

Proof: Firstly, using the operator div to Eq. (2.2.46) acquires

ρ ω + (λ + 2µ) ∇2 div ũ − β ∇2 θ̃ = 0. (2.2.51)

 2 

Then, elimination of θ̃ from Eq. (2.2.51) and Eq. (2.2.47) gthe ives

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 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

χ̃1 div ũ = 0.

Again, from Eq. (2.2.51) and Eq. (2.2.47), eliminating div ũyields

χ̃1 θ̃ = 0.

Furthermore, by applying the operator curl to Eq. (2.2.46), the following is obtained

ρ ω + µ ∇2 curl ũ = 0.
 2 

Therefore, the Eqs. (2.2.48-2.2.50) are acquired, which completes the proof of Lemma
2.2.5.1.

Theorem-2.2.5.1:
 
Statement: If ũ, θ̃ is a solution of Eq. (2.2.46) and Eq. (2.2.47), then

2
X
ũ(x) = β grad ϕp (x) + Ψ (x), (2.2.52)
p=1
2
X
θ̃(x) = ap ϕp (x), (2.2.53)
p=1

where, ϕp (p = 1, 2) and Ψ = (Ψ1 , Ψ2 , Ψ3 ) satisfy the following equations:

∇2 + λ2p ϕp (x) = 0, (2.2.54)

ρ ω2
 
2
∇ + Ψ (x) = 0, x ∈ Ξ (2.2.55)
µ
div Ψ (x) = 0, (2.2.56)

and
ap = − (λ + 2µ) λ2p + ρ ω 2 where, p = 1, 2. (2.2.57)
 
Proof: Let Eq. (2.2.46) and Eq. (2.2.47) have ũ, θ̃ as solution. Then, taking into

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 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

account ∇2 ũ = grad div ũ − curl curl ũ, and using Eq. (2.2.46) give
1 n h i o
ũ = grad − (λ + 2µ) div ũ + β θ̃ +µ curl curl ũ . (2.2.58)
ρ ω2

Introducing the notation

µ
Ψ (x) = curl curl ũ, (2.2.59)
ρ ω2

and using Eq. (2.2.50), Eq. (2.2.55) and Eq. (2.2.56) can be directly obtained
asdiv curl ũ = 0 for x ∈ Ξ.

Now, let
 
2
Y 2
∇ + λ2p  θ̃, (2.2.60)


ϕj = bj 
p=1
p6=j

where,  −1
2
Y
λ2p − λ2j 


bj = aj , j = 1, 2,

p=1
p6=j

Therefore, in view of Eq. (2.2.49), the Eq. (2.2.60) yields Eq. (2.2.54) and Eq. (2.2.53).

Next, using Eq. (2.2.47), Eq. (2.2.53), Eq. (2.2.54), and Eq. (2.2.57), give
X2
div ũ = −β λ2p ϕp . (2.2.61)
p=1

Hence, Eq. (2.2.58) yields

( " 2
# )
1 X
ũ = grad (λ + 2µ) β λ2p ϕp + β θ̃ +µ curl curl ũ . (2.2.62)
ρ ω2 p=1

Further, simplifying the above equation using Eq. (2.2.57) and Eq. (2.2.59), the fol-
lowing result is obtained

2
X
ũ(x) = βgrad ϕp (x) + Ψ (x),
p=1

which completes the proof of Theorem 2.2.5.1.

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 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

Theorem-2.2.5.2:

 
Statement: If ũ, θ̃ is expressed as in Eq. (2.2.52) and Eq. (2.2.53), where ϕj and
 
Ψ satisfies Eqs. (2.2.54-2.2.56), then ũ, θ̃ is the solution of Eq. (2.2.46) and Eq.
(2.2.47) on Ξ.

Proof: From Eq. (2.2.52) and using Eq. (2.2.54) and Eq. (2.2.55), the following is
attained

2
2
X ρ ω2
∇ ũ = −β λ2p ϕp − Ψ,
p=1
µ
2
X
grad div ũ = −βgrad λ2p ϕp . (2.2.63)
p=1

Replacing ũ and θ̃ as given in Eq. (2.2.52) and Eq. (2.2.53) on the left-hand side of
Eq. (2.2.46) and using Eq. (2.2.54), Eq. (2.2.57), and Eq. (2.2.63), give

ρ ω 2 + µ ∇2 ũ + (λ + µ)grad divũ − β grad θ̃

 
" 2
# 2
X X
= ρ ω β grad
2
(λ + 2µ) λ2p + ap ϕp − ρ ω 2 Ψ .
 
ϕp + Ψ − βgrad
p=1 p=1

After simplification, the above equation yields

ρ ω + µ ∇2 ũ + (λ + µ)grad divũ − β grad θ̃ = 0,

 2 

which is the field Eq. (2.2.46).

Similarly, replacing ũ and θ̃ again on the left-hand side of Eq. (2.2.47) by the expression
given in (2.2.52) and (2.2.53) and using Eq. (2.2.54), Eq. (2.2.57) and Eq. (2.2.61),
the following is acquired

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 CHAPTER 2. Theoretical analysis of the thermoelasticity theory...

ω2τ K ∗
  
K− − i ω (K − K τ ) ∇ + ρ cE ω θ̃ + β T0 ω 2 div ũ
∗ 2 2
2
2
! 2
!
ω2τ K ∗
   X X
= K− − i ω (K − K ∗ τ ) ∇2 + ρ cE ω 2 ap ϕp + β 2 T0 ω 2 − λ2p ϕp
2 p=1 p=1
2
ω2τ K ∗
X     
= ap K − − i ω (K − K ∗ τ ) (−λ2p ) + ρ cE ω 2 − β 2 T0 ω 2 λ2p ϕp
p=1
2

= 0. by using χ̃1 −λ2p = 0, p = 1, 2

Thus acquiring Eq. (2.2.47).

Hence, it can be confirmed that the general solution of the system of homogeneous Eq.
(2.2.46) and Eq. (2.2.47) is attained in terms of the metaharmonic functions ϕp and Ψ .

2.2.6 Conclusion

The present subchapter investigates a non-classical thermoelasticity model under exact

heat conduction law with single delay. This includes Galerkin-type representation of
solution for the system of equations of motion in terms of elementary functions. A
theorem that represents Galerkin type solution of equations for steady state oscillations
in the context of considered linear thermoelasticity theory is established. Finally, the
representation of general solution of the system of equations in case of steady state
oscillations is also acquired in terms of metaharmonic functions.

65