STATE-SPACE APPROACH TO 3D GENERALIZED THERMOVISCOELASTICITY UNDER GREEN-NAGDHI
THEORY
N. Sarkar, M. Bachher, and A. Lahiri
(Received 2 October, 2015)
Abstract. The present paper is concerned at investigating the effects of viscos- ity on generalized thermoelastic interactions in a three-dimensional homoge- neous isotropic semi-infinite medium whose surface is subjected to a thermal shock and is assumed to be traction free. The formulation is applied to the generalized thermoelasticity based on the G-N II and III theories. The normal mode analysis together with state-space approach is used to obtain the exact analytical expressions for the studied field variables. Numerical computations are done for a specific material and obtained numerical results are plotted in the form of two- and three-dimensional figures. Comparisons are performed with and without the effects of viscosity. A comparison is also made between G-N II and G-N III models to show the effects of viscosity.
1. Introduction
In order to eliminate the paradox of infinite speeds of propagation of thermal wave [2], Lord & Shulman [1] (L-S model) and Green & Lindsay [3] (G-L model) formulated two model of generalized thermoelasticity in the year 1967 and 1972 respectively. Later Green & Nagdhi [4-6] proposed three theories of generalized thermoelasticityy. The first model (G-N I) is exactly the same as Biot’s theory [2]. The second and third model are labeled as G-N II and G-N III model. In G-N II and G-N III models, the thermal wave propagates with finite speeds which agrees with physical situations. An important feature of G-N II theory is that this theory does not accommodates dissipation of thermal energy whereas G-N III theory accommodates dissipation of thermal energy. Several works based on G-N II and G-N III models can be found in the references [7-15].
Due to the rapid development of polymer science and plastic industry as well as the large use of materials under high temperature in modern technology, the investigation and application in thermo-visco-elastic solid materials have become a significant task for researchers in the field of mechanics of solid body . Keeping this fact in mind, many researchers [16-27] investigated various types of problems on in linear thermo-visco-elastic and electro-magneto-thermo-visco-elastic solid body.
In this paper, we introduce a state-space approach to investigated the effects of viscosity on wave propagation in a homogeneous isotropic three-dimensional ther- moelastic semi-infinite medium. The free surface of the medium is subjected to a thermal shock and is assumed to be traction free. The formulation is applied to the
2010 Mathematics Subject Classification 75F05.
Key words and phrases: Thermo-visco-elasticity, G-N model, Energy dissipation, Normal mode analysis, State-space formulation.
generalized thermoelasticity based on the G-N model II and III. The normal mode analysis [9, 28] together with state-space approach [28-32] is used to obtain the exact analytical solutions for the studied field variables. Numerical computations are done for a specific material and obtained numerical results are plotted in the form of two- and three-dimensional figures. Comparisons are performed with and without the effects viscosity. A comparison is also made between G-N II and G-N III model to show the effects of viscosity.
2. Governing Equations
Following [5, 6], the strain-displacement relations, the stress-strain-temperature relations, the equations of motion in absence of body forces and the heat conduction equation in absence of heat source for a homogeneous isotropic thermally conducting viscoelastic solid may be written as
σij = λuk,kδij+ 2µeij− γT δij, (1)
σij,j= ρ¨ui, (2)
k∗+ k∂
∂t
∇2T = ρCET + γT¨ 0¨e, (3) where
eij= 1
2(ui,j+ uj,i) , and
λ = λe
1 + λ0
∂
∂t
, µ = µe
1 + µ0
∂
∂t
, γ = γe
1 + γ0
∂
∂t
, γe= (3λe+ 2µe)αT, γ0= (3λ0λe+ 2µ0µe)αT
γe
, (i, j, k = x, y, z).
In the above equations, λ, µ are Lame’s constants, λ0, µ0, γ0 are viscoelastic parameters, ρ is the constant mass density of the medium, CE is the specific heat at constant strain, σij are the stress components, eij are the strain components, ui are displacement components, t is the time variable, x, y, z are the space variables, T is the absolute temperature, T0is the initial temperature of the medium assumed to be such that
θ T0
<< 1, γ = (3λ + 2µ)αT, a material constant, αT is the coefficient of linear thermal expansion, k∗ is a material constant characteristic of the theory, k is the thermal conductivity.
The subscript comma denotes spatial derivatives and a superimposed dot repre- sents differentiation with respect to time t.
3. Formulation of the Problem
We consider a homogenous isotropic thermo-visco-elastic semi-infinite medium in the three-dimensional region: Ω = {(x, y, z) : 0 ≤ x < ∞, −∞ < y < ∞, −∞ <
z < ∞}. We assume the homogeneous initial conditions. We are using the orthogo- nal Cartesian coordinates system (x, y, z). The displacement components thus have
the form: ui = (u, v, w). Hence the equations of motion and the expressions for normal stress components can be written from Eqs. (2) and (1) as:
ρ¨u =
(λe+ 2µe) + (λ0λe+ 2µ0µe)∂
∂t
u,xx+ µe
1 + µ0
∂
∂t
(u,yy+ u,zz) +
(λe+ µe) + (λ0λe+ µ0µe)∂
∂t
(v,xy+ w,xz) − γe
1 + γ0
∂
∂t
T,x, (4)
ρ¨v =
(λe+ 2µe) + (λ0λe+ 2µ0µe)∂
∂t
v,yy+ µe
1 + µ0
∂
∂t
(v,zz+ v,xx) +
(λe+ µe) + (λ0λe+ µ0µe)∂
∂t
(w,yz+ u,yx) − γe
1 + γ0 ∂
∂t
T,y, (5)
ρ ¨w =
(λe+ 2µe) + (λ0λe+ 2µ0µe)∂
∂t
w,zz+ µe
1 + µ0∂
∂t
(w,xx+ w,yy) +
(λe+ µe) + (λ0λe+ µ0µe)∂
∂t
(u,zx+ v,zy) − γe
1 + γ0
∂
∂t
T,z, (6)
k∗+ k∂
∂t
(T,xx+ T,yy+ T,zz) = ρCET + γ¨ eT0
1 + γ0
∂
∂t
¨
e, (7)
σxx= 2µe
1 + µ0∂
∂t
u,x+ λe
1 + λ0∂
∂t
e − γe
1 + γ0∂
∂t
T, (8)
σyy = 2µe
1 + µ0∂
∂t
v,y+ λe
1 + λ0∂
∂t
e − γe
1 + γ0∂
∂t
T, (9)
σzz= 2µe
1 + µ0∂
∂t
w,z+ λe
1 + λ0∂
∂t
e − γe
1 + γ0∂
∂t
T, (10)
where
e = ekk= (u,x+ v,y+ w,z) . (11) To make the above equations dimensionless, we introduce the following non- dimensional variables:
(x0, y0, z0) = 1
l(x, y, z), t0= c1t
l , (u0, v0, w0) =λ0+ 2µ0
γ0T0l (u, v, w), θ = T
T0, σ0ij= σij
γT0, (λ00, µ00, γ00) =c1
l (λ0, µ0, γ0), where l is assumed to be standard length.
Using the above parameters, Eqs. (4)-(11) become (suppressing the primes for convenience):
¨ u = β
1 + µ0∂
∂t
∇2u +
(1 − β) + {λ0(1 − 2β) + βµ0}∂
∂t
e,x
−
1 + γ0∂
∂t
θ,x, (12)
¨ v = β
1 + µ0
∂
∂t
∇2v +
(1 − β) + {λ0(1 − 2β) + βµ0}∂
∂t
e,y
−
1 + γ0
∂
∂t
θ,y, (13)
¨ w = β
1 + µ0∂
∂t
∇2w +
(1 − β) + {λ0(1 − 2β) + βµ0}∂
∂t
e,z
−
1 + γ0
∂
∂t
θ,z, (14)
CT2∇2θ + k0∇2θ = ¨˙ θ + ε
1 + γ0
∂
∂t
¨
e, (15)
σxx= 2β
1 + µ0∂
∂t
u,x+ (1 − 2β)
1 + λ0∂
∂t
e −
1 + γ0 ∂
∂t
θ, (16)
σyy = 2β
1 + µ0∂
∂t
v,y+ (1 − 2β)
1 + λ0∂
∂t
e −
1 + γ0 ∂
∂t
θ, (17)
σzz= 2β
1 + µ0∂
∂t
w,z+ (1 − 2β)
1 + λ0∂
∂t
e −
1 + γ0∂
∂t
θ, (18)
e = (u,x+ v,y+ w,z) , (19)
where β = λ µe
e+2µe, ε = ρC γe2T0
E(λe+2µe) is the dimensionless thermoelastic coupling parameter, c1=q
λe+2µe
ρ is the longitudinal waves speed, c3=q
k∗
ρCE is the finite thermal waves speed of G-N theory, CT = cc3
1 is the non-dimensional finite thermal wave speed of G-N theory and ∇2≡
∂2
∂x2 +∂y∂22 +∂z∂22 .
Differentiating Eqs. (12)-(14) with respect to x, y, z respectively and then adding we obtain
1 + {λ0− 2β(λ0− µ0)}∂
∂t
∇2e −
1 + γ0
∂
∂t
∇2θ = ¨e. (20) We define
σ = 1
3(σxx+ σyy+ σzz) (21)
to be the mean stress [9].
Adding Eqs. (16)-(18) and then using Eqs. (19) and (21), we get after some ma- nipulations
σ =
1 − 4β
3 +
(1 − 2β)λ0+2β 3 µ0
∂
∂t
e −
1 + γ0
∂
∂t
θ. (22)
4. Normal Mode Analysis
The solution of the field variables can be decomposed in terms of normal modes [9, 28] as:
[u, v, w, e, θ, σ, σij] (x, y, z, t) =u∗, v∗, w∗, e∗, θ∗, σ∗, σij∗ (x)exp [ωt + i(ay + bz)] , (23)
where u∗(x) etc. are the amplitude of the function u(x, y, t) etc., i = √
−1, ω (complex) is the angular frequency and a, b are the wave number in the y and z direction respectively.
Using the normal modes (23) in Eqs. (15), (20), (22) and then eliminating e∗(x) from the resulting equations we get the following equations, namely:
D2θ∗= C1θ∗+ C2σ∗, (24) D2e∗= D1θ∗+ D2σ∗, (25) where
D ≡ d
dx, C1=
a2+ b2+ ω2
CT2 +εω2(1 + γ0ω)2 α (CT2+ k0ω)
, C2= εω2(1 + γ0ω) α (CT2+ k0ω), D1= ω2(1 + γ0ω)
d −ω2(d − α)(1 + γ0ω){α + ε(1 + γ0ω)2} dα (CT2 + k0ω)
,
D2=
a2+ b2+ω2
d −εω2(d − α)(1 + γ0ω)2 dα (CT2+ k0ω)
, d = [1 + ω{λ0− 2β(λ0− µ0)}], α =
1 −4β
3 + ω
λ0(1 − 2β) +2β 3 µ0
. We choose θ∗(x) and σ∗(x) as the state variables in the x-direction and write Eqs. (24) and (25) in a vector-matrix differential equation as:
D2V (x) = A~~ V (x), (26)
where
V (x) =~ θ∗ σ∗
, A = C1 C2
D1 D2
.
The initial conditions of the problem are assumed to be homogeneous. Now, we consider the following boundary conditions in non-dimensional form:
˙
qn+ νθ(0, y, z, t) = r(y, z, t), (27) σ(0, y, z, t) = σii(0, y, z, t) = 0, (28) where qn denotes the normal component of the heat flux, ν is the Biot’s number and r(y, z, t) represents the intensity of the applied heat sources at x = 0.
The regularity condition is that all the field variables are bounded for x → +∞.
In order to use the boundary condition (27), we need to consider the following non-dimensional Fourier’s law of heat conduction of G-N model [5], namely
˙
qn= −∂θ
∂n. (29)
Substituting from Eq. (29) in Eq. (27) and then using the normal modes (23), we get:
νθ∗(0) − Dθ∗(0) = r∗, (30) σ∗(0) = σxx∗ (0) = σyy∗ (0) = σzz∗ (0) = 0. (31)
5. State Space Approach
The formal solution of the vector-matrix differential equation (26) can be written as (see [27-30] for details)
V (x) = exp~ h
−√
A xi ~V (0) (32)
where
V (0) =~ θ∗(0) σ∗(0)
=θ0∗ 0
, and the constant θ∗0 is to be determined using (30).
We have omitted the positive exponential part in (32) to obtain a bounded solu- tion for large x. Now, our first task is to obtain the matrix form of exph
−√ A xi
. The characteristic equation of A is given by
λ2− (C1+ D2)λ + (C1D2− C2D1) = 0. (33) Let λ1 and λ2be the roots (distinct) of the Eq. (33), where
λj =(C1+ D2) + (−1)jp(C1− D2)2+ 4C2D1
2 , j = 1, 2.
Following Simmons [33], the spectral decomposition of the matrix A can be written as
A = λ1E + λ2F, (34)
where E and F are called the projectors of A satisfying the following relations (see [33] for details):
E + F = I, EF = F E = O, E2= E, F2= F. (35) Since√
A and A has the same projectors ([33]) and if p1, p2are the eigenvalues of√
A then pj =pλj, j = 1, 2. Hence the spectral decomposition of the matrix
√A is given by
√A = p1E + p2F, (36)
where
E = 1
λ1− λ2
C1− λ2 C2
D1 D2− λ2
, F = 1 λ2− λ1
C1− λ1 C2
D1 D − λ1
, (37)
and E, F satisfy Eq. (35).
Thus, we find A∗=√
A = 1
p1+ p2
C1+ p1p2 C2
D1 D2+ p1p2
. (38)
With help of Eq. (38), the Taylor series expansion of the matrix exponential in Eq. (32) may take the following form
exph
−√ A xi
= exp [−A∗x] =
∞
X
n=0
[−A∗x]n
n! . (39)
Using the Cayley-Hamilton theorem, we can write the second and higher orders of the matrix A∗ in terms of I and A∗. Thus, Eq. (39) reduces to
exp [−A∗x] = a0(x)I + a1(x)A∗, (40) where the coefficients a0 and a1 depends on x only.
By Cayley-Hamilton theorem, the characteristic roots p1 and p2 of the matrix A∗ must satisfy Eq. (40) and thus we get
e−p1x= a0(x) + a1(x)p1, (41) e−p2x= a0(x) + a1(x)p2. (42) By solving the above system of Eqs. (41) and (42), we obtain
a0(x) = p1e−p2x− p2e−p1x
(p1− p2) , a1(x) = e−p1x− e−p2x
(p1− p2) . (43) From Eqs. (40) and (43), we get
exph
−√ Axi
=A11 A12
A21 A22
, (44)
where
A11=(λ1− C1)e−
√λ2x− (λ2− C1)e−
√λ1x
(λ1− λ2) , A12= C2(e−
√λ1x− e−√λ2x) (λ1− λ2) , (45) A21=D1(e−
√λ1x− e−√λ2x)
(λ1− λ2) , A22= (λ1− D2)e−
√λ2x− (λ2− D2)e−
√λ1x
(λ1− λ2) . (46) Thus the solution of Eq. (26) can be written as
V (x) = A~ jkV (0), j, k = 1, 2.~ (47) From Eqs. (45)-(47), the solutions for θ∗(x) and σ∗(x) can be obtained as
θ∗(x) = θ1e−
√λ1x− θ2e−
√λ2x, (48)
σ∗(x) = D1θ∗0 (λ1− λ2)
h e−
√λ1x− e−
√λ2xi
. (49)
where
θ1= θ0∗ C1− λ2
λ1− λ2
, θ2= θ0∗ C1− λ1
λ1− λ2
.
The boundary condition (30) gives the value of the constant θ∗0 as θ0∗= r∗(λ1− λ2)
ν(λ1− λ2) +√
λ1(C1− λ2) −√
λ2(C1− λ1).
Using Eqs. (23), (48) and (49) in Eq. (22), the cubical dilatational e∗(x) is ob- tained as
e∗(x) = e1e−
√k1x− e2e−
√k2x, (50)
where
e1= [θ∗0D1+ (C1− λ2)(1 + γ0ω)]
α(λ1− λ2) , e2= [θ∗0D1+ (C1− λ1)(1 + γ0ω)]
α(λ1− λ2) .
Substituting from Eqs. (23), (48) and (50) into the Eq. (12), we get D2− λ2u u∗(x) =
2
X
j=1
(−1)j−1uj(λ2j− λ2u)e−
√
λjx, (51)
where
λ2u=
a2+ b2+ ω2 β(1 + µ0ω)
,
uj =pλj[ej{(1 − β) + ω(λ0(1 − 2β) + µ0β)} − (1 + γ0ω)θj]
β(1 + µ0ω)(λ2j− λ2u) , j = 1, 2.
The general solution of Eq. (51) can be written as u∗(x) = u0e−λux+
2
X
j=1
(−1)j−1uje−
√λjx, (52)
where λ21 6= λ22 6= λ2u and u0 is a constant to be determined from the boundary conditions (31).
By using Eqs. (23), (48), (50) and (52) in Eq. (16), we find the solution for the stress component σ∗xx(x) as
σxx∗ (x) =
2
X
j=1
(−1)j−1σje−
√λjx+ σ3e−λux, (53)
where
σj= −2βujpλj(1 + µ0ω) + ej(1 − 2β)(1 + λ0ω) − θj(1 + γ0ω), j = 1, 2, σ3= −2βλuu0(1 + µ0ω).
With the help of the boundary condition (31) and Eq. (53), we find the constant u0 as
u0= (σ1− σ2) 2βλu(1 + µ0ω). 6. Numerical Example and Discussions
Since we have ω = ω0+ iζ, where i is the imaginary unit, eωt = eω0t(cos ζt + i sin ζt) and for small values of time, we can take ω = ω0(real). For the discussions of the nature of dependence of all the physical variables on viscosity, we shall compute them numerically for a particular model. For this purpose, we choose the following numerical values of the relevant parameters for copper like material:
λe= 7.76 × 1010N/m2, µe= 3.86 × 1010N/m2, λ0= 0.06 s, µ0= 0.09 s, αt= 1.78 × 10−5 K−1, CE= 383.1 m2/K, ρ = 8954 kg/m3, T0= 293 K, ε = 0.0168, β = 0.25, ω = 3, a = 1.2, b = 1.3, ν = 50, r∗= 100, CT = 2.
Using the above numerical values, the variations of the temperature distribution θ, the mean stress σ, the displacement component u and the stress component σxx
along x axis at two different plane y = z = 0.0 and y = z = 0.4 for a particular time instant t = 0.25 have been shown for (i) generalized thermoviscoelastic solid (GTVE) by solid line at y = z = 0.0, t = 0.25, (ii) generalized thermoviscoelastic
solid (GTVE) by solid-dot line at y = z = 0.4, t = 0.25, (iii) generalized thermoelas- tic solid (GTE) by solid-dot (bold) line at y = z = 0.0, t = 0.25 and (iv) generalized thermoelastic solid (GTE) by dashed line at y = z = 0.4, t = 0.25. These variations are shown in Figs. 1–4.
From Figs. 1–4 it is clear that y and z have decreasing effect on θ, σ, u and σxx
for both GTVE and GTE model for fixed t. Also it is depicted that the numerical values of σ, u and σxx are greater in GTVE model than GTE model for fixed x, y, z and t but Fig. 1 shows that the viscosity has no significant effect on θ. The maximum value of all the physical quantities attain in the case of GTVE at the plane y = z = 0.0.
Figs. 5–8 show the distributions of θ, σ, u and σxx for the GTVE model at y = z = 0.0 and y = z = 0.4 for two different time instants t = 0.2 and t = 0.3.
It is clear from all these figures that t has increasing effects on all the physical quantities.
Figs. 9–12 display the temperature θ, the mean stress σ, the displacement u and the stress σxx distributions at y = z = 0.4 with wide range of 0 ≤ x ≤ 4.0 and 0.1 ≤ t ≤ 0.5. From these figures it can be noted that the speed of the wave propagation of all the physical quantities are finite and coincide with the physical behavior of elastic materials. Also we can see from all the figures that the boundary conditions (30) and (31) are satisfied.
It is clear from Figs. 1–12 that all the distributions considered have a non-zero value only in a bounded region of the space Ω. Outside of this region all the values vanish identically and this means that the region has not felt thermal disturbance yet. Behavior of all the physical variables at y = z = 0.0 and y = z = 0.4 are likely to be similar but only differences are lie in the magnitudes.
Figs. 13-16 are plotted to show the effects of the viscosity parameter on θ, σ, u and σxx with different positions of the displacement x at (y, z, t) = (0.4, 0.4, 0.25).
In all these figures, solid line (-) represent G-N III model with viscosity (G-N III WV), solid star line (-*) represent G-N II model with viscosity (G-N II WV), solid line with ’o’(-o) represent G-N III model without viscosity (G-N III WOV) and dashed line (- -) represent G-N II model without viscosity (G-N II WOV). These figures exhibit that the presence of viscosity effects increase the values of all the studied field variables in both the theories (G-N II and G-N III). We can also notice that the magnitudes of all the physical quantities are greater for G-N III model than G-N II model. Figs. 14 and 15 show that the mean stress σ∗(x) and the normal stress σxx∗ (x) starts with zero value at the boundary x = 0 which agree with the boundary condition (31) of our problem. Figs. 13-16 depict that the behavior of all the four curves predicted by G-N II and G-N III model in the presence and absence of the viscosity parameter are same in nature. Finally all the curves in Figs. 13-16 converges to zero for some large x which agree with the characteristic of the generalized thermoelasticity.
0 0.5 1 1.5 2 2.5 3 0
0.5 1 1.5 2 2.5 3
x
θ
GTVE,y=z=0.0 GTVE,y=z=0.4 GTE,y=z=0.0 GTE,y=z=0.4
Fig. 1 Temperature distribution θ vs. x at t = 0.25.
0 0.5 1 1.5 2 2.5 3
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05 0
x
σ
GTVE,y=z=0.0 GTVE,y=z=0.4 GTE,y=z=0.0 GTE,y=z=0.4
Fig. 2 Mean stress σ distribution vs. x at t = 0.25.
0 0.5 1 1.5 2 2.5 3
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2
x
u
GTVE,y=z=0.0 GTVE,y=z=0.4 GTE,y=z=0.0 GTE,y=z=0.4
Fig. 3 Displacement distribution u vs. x at t = 0.25.
0 0.5 1 1.5 2 2.5 3 0
0.5 1 1.5 2 2.5 3 3.5 4
x σ xx
GTVE,y=z=0.0 GTVE,y=z=0.4 GTE,y=z=0.0 GTE,y=z=0.4
Fig. 4 Stress distribution σxxvs. x at t = 0.25.
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3 3.5 4
x
θ
t=0.2,y=z=0.0 t=0.2,y=z=0.5 t=0.3,y=z=0.0 t=0.3,y=z=0.5
Fig. 5 Temperature distribution θ vs. x for two time instants.
0 0.5 1 1.5 2 2.5 3
−0.45
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05 0
x
σ
t=0.2,y=z=0.0 t=0.2,y=z=0.4 t=0.3,y=z=0.0 t=0.3,y=z=0.4
Fig. 6 Mean stress σ distribution vs. x for two time instants.
0 0.5 1 1.5 2 2.5 3
−1.6
−1.4
−1.2
−1
−0.8
−0.6
−0.4
−0.2 0 0.2
x
u
t=0.2,y=z=0.0 t=0.2,y=z=0.4 t=0.3,y=z=0.0 t=0.3,y=z=0.4
Fig. 7 The displacement distribution u vs. distance x for two time instants.
0 0.5 1 1.5 2 2.5 3
0 0.5 1 1.5 2 2.5 3 3.5 4
x σxx
t=0.2,y=z=0.0 t=0.2,y=z=0.4 t=0.3,y=z=0.0 t=0.3,y=z=0.4
Fig. 8 The stress distribution σxxvs. x for two time instants.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0
1 2 3 4 0 1 2 3 4 5 6
x t
θ
Fig. 9 Temperature distribution vs. x and t at y = z = 0.4.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0
1 2 3 4
−0.8
−0.6
−0.4
−0.2 0
t x
σ
Fig. 10 Mean stress σ distribution vs. x and t at y = z = 0.4.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0
1 2 3 4
−2.5
−2
−1.5
−1
−0.5 0 0.5
t x
u
Fig. 11Displacement u distribution vs. x and t at y = z = 0.4.
0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0
1 2 3 4
−1 0 1 2 3 4 5
t x
σxx
Fig. 12 Stress distribution σxxvs. x and t at y = z = 0.4.
0 0.5 1 1.5 2 2.5 3 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
x
θ
G−N III WV G−N II WV G−N III WOV G−N II WOV
Fig. 13 Temperature distribution θ vs. x for G-N II and III theory.
0 0.5 1 1.5 2 2.5 3
−0.45
−0.4
−0.35
−0.3
−0.25
−0.2
−0.15
−0.1
−0.05 0
x
σ
G−N III WV G−N II WV G−N III WOV G−N II WOV
Fig. 14 Mean stress σ distribution vs. x for G-N II and III theory.
0 0.5 1 1.5 2 2.5 3
−2
−1.5
−1
−0.5 0 0.5
x
u
G−N III WV G−N II WV G−N III WOV G−N II WOV
Fig. 15 The displacement distribution u vs. distance x for G-N II and III theory.
0 0.5 1 1.5 2 2.5 3 0
0.5 1 1.5 2 2.5 3 3.5 4 4.5
x σ xx
G−N III WV G−N II WV G−N III WOV G−N II WOV
Fig. 16 The stress distribution σxxvs. x for G-N II and III theory.
7. Concluding Remark
Many researchers in the field of generalized thermoelasticity have applied state- space approach only for one-dimensional thermoelastic problem and very few of them can successfully applied for two-diemsional case. In the present paper, we apply state-space approach for solving a three-dimensional generalized thermoelas- tic problem for the first time. We tried to implement such a very useful technique which may be applied to solve a three-dimensional generalized thermoviscoelastic problem.
Acknowledgements
The authors are heartily thankful to the referees for their valuable comments and suggestions which improve the quality of this paper.
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N. Sarkar
Department of Applied Mathematics,
University of Calcutta, 92, A.P.C. Road, Kolkata-700 009, India.
M. Bachher
Department of Mathematics, Jadavpur University, Kolkata-700 032, India.
A. Lahiri
Department of Mathematics, Jadavpur University, Kolkata-700 032, India.
