Influence of Gravity Field and Rotation on a

JOURNAL OF THERMOELASTICITY ISSN 2328-2401 (print) ISSN 2328-241X (Online)

VOL.1 NO. 4 December 2013 http://www.researchpub.org/journal/jot/jot.html

Influence of Gravity Field and Rotation on a Generalized Thermoelastic Medium using a Dual-Phase-Lag Model Mohamed I. A. Othman, W. M. Hasona and Ebtesam E. M. Eraki 

compatible with physical observations was introduced "by Biot

Abstract— In the present paper, The normal mode analysis

[1]". First, the equation of heat conduction of this theory does

is used to study the deformation of a rotating generalized

not contain any elastic terms. Second, the heat equation is of a

thermoelastic medium under the influence of gravity subjected

to

thermal

loading

in

the

context

parabolic type, predicting infinite speeds of propagation for

of

heat waves. The governing equations for Biot theory are

dual-phase-lag thermoelastic model. The exact expressions

coupled, eliminating the first paradox of the classical theory.

for the temperature, displacement components, and stress

However, both theories share the second shortcoming since the

components are given in the physical domain and

heat equation for the coupled theory is also parabolic.

illustrated graphically. These expressions are calculated

Thermoelasticity theories that predict a finite speed for the

numerically for the problem. Comparisons are made with

propagation of thermal signals have aroused much interest in

the results predicted by Lord-Shulman theory and

the last three decades. These theories are known as generalized

dual-phase-lag model in the presence and absence of

thermoelasticity theories. The first generalization of the

rotation as well as gravity.

thermo-elasticity theory is due to "Lord and Shulman [2]", who introduced the theory of generalized thermoelasticity with one

Keywords — (Thermoelasticity, generalized thermoelasticity, rotation, gravity, dual- phase-lag model, normal mode analysis

relaxation time by postulating a new law of heat conduction to replace the classical Fourier’ law. This law contains the heat flux vector as well as its time derivative. It contains also a new constant that acts as a relaxation time. The heat equation of this

I. INTRODUCTION

theory is of the wave-type, ensuring finite speeds of

The theory of coupled thermoelasticity to overcome the first shortcoming

in

the

classical

uncoupled

theory

propagation for heat and elastic waves. The remaining

of

governing equations for this theory, namely, the equations of

thermoelasticity where it predicts two phenomena not

motion and the constitutive relations remain the same as those for the coupled and the uncoupled theories. This theory was M. I. Othman, Department of Mathematics, Faculty of Science, P.O. Box 44519, Zagazig University, Zagazig, Egypt (Email: [email protected]) W. M. Hasona, Department of Mathematics, Faculty of Science, P.O. Box 44519, Zagazig University, Zagazig, Egypt (Email: [email protected]) Ebtesam E. Erakil, Department of Mathematics, Faculty of Science, P.O. Box 44519, Zagazig University, Zagazig, Egypt (Email: [email protected])

extended "by Dhaliwal and Sherief [3]" to general anisotropic media in the presence of heat sources. A generalization of this inequality was proposed "by Green and Laws [4]". Another version of the constitutive equations was obtained "by Green and Lindsay [5]". The theory

12

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VOL.1 NO. 4 December 2013 http://www.researchpub.org/journal/jot/jot.html

of thermoelasticity without energy dissipation is another

conducting elastic half-space was presented "by Chand [11]".

generalized theory and was formulated "by Green and Naghdi

The effect of rotation on elastic waves has been studied "by

[6]". It includes the thermal displacement gradient among its

many authors [12-15]". The effect of rotation on different types

independent constitutive variables, and differs from the

of wave propagating in a thermoelastic medium was discussed

previous theories in that it does not accommodate dissipation of

"by Sharma and his co-workers [16-17]". The effect of rotation

thermal energy.

in a magneto-thermoelastic medium was discussed by "Othman

The dual-phase-lag (DPL) model, which describes the

and Song [18]". The effect of rotation on propagation of plane

interactions between phonons and electrons on the microscopic

waves in generalized thermoelasticity has been studied "by

level as retarding sources causing a delayed response on the

Singh and Tomer [19]". The effect of rotation on

macroscopic scale, was proposed "by Tzou [7,8]". For

two-dimensional problem of fiber-reinforced thermoelastic

macroscopic formulation, it would be convenient to use the

with one relaxation time was investigated "by Othman and said

DPL model for investigation of the micro-structural effect on

[20]".

the behavior of heat transfer. The physical meanings and the

In the classical theory of elasticity, the gravity effect is

applicability of the DPL model have been supported "by the

generally neglected. The effect of gravity in the problem of

experimental results [9]". The dual-phase-lag is such a

propagation of waves in solids, in particular on an elastic globe,

modification of the classical thermoelastic model in which the

was first studied "by Bromwich in [21]". Subsequently, an

Fourier's law is replaced by an approximation to a modified

investigation of the effect of gravity was considered "by Love

Fourier's law with two different time translations: a phase-lag

in [22]", who showed that the velocity of Rayleigh waves is

 q and a phase-lag of temperature gradient   .

increased to a significant extent by gravitational field when

of the heat flux

wavelengths are large. The effect of gravity on surface waves,

A Taylor series approximation of the modified Fourier's law,

on the propagation of waves in an elastic layer has been studied

together with the remaining field equations leads to a complete

"by De and Sengupta in [23-24]". Surface waves under the

system of equations describing a dual-phase-lag thermoelastic

influence of gravity in a non-homogeneous elastic solid

model. The model transmits thermoelastic disturbance in a

medium was investigated "by Das et al. [25]". Wave

wave-like manner if the approximation is linear with respect to

 q and   , and 0 ≤   <  q

; or quadratic in

q

propagation in a non-homogeneous orthotropic elastic medium and linear in

under the influence of gravity was discussed "by Abd-Alla and

  , with  q > 0 and   > 0. This theory is developed in a Ahmed [26]". The effects of rotation and gravity in generalized rational way to produce a fully consistent theory which is able

thermoelastic medium has been depicted "by Ailawalia and

to incorporate thermal pulse transmission in a very logical

Narah [27]". The influences of rotation, magnetic field, initial

manner.

stress and gravity on Rayleigh in a homogeneous orthotropic

Some researches in past have investigated different problems

elastic half space was studied "by Abd-Alla [28]". Recently,

of rotating media. The propagation of plane harmonic waves in

The effect of magnetic field and rotation of the 2-D problem of

a rotating elastic medium without a thermal field has been

a fiber-reinforced thermoelastic under three theories with

studied "by Schoenberg and Censor [10]". It was shown there

influence of gravity was discussed "by Othman and Lotfy

that the rotation causes the elastic medium to be depressive and

[29]".

of

The present paper is concerned with the investigations related

deformation, stresses and magnetic field in a uniformly

to the effect of gravity field and rotation on a generalized

rotating, homogeneous, isotropic, thermally and electrically

thermoelastic medium based on Dual-Phase-Lag model by

anisotropic.

An

investigation

on

the

distribution

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VOL.1 NO. 4 December 2013 http://www.researchpub.org/journal/jot/jot.html

qi (x ,t   q )  KT ,i (x ,t    ).

applying the normal mode analysis. Also, the effects of rotation and gravity field on the physical quantities are discussed

Where

numerically and illustrated graphically.

(5)

q i is the heat flux vector. The model transmits

thermoelastic disturbances in a wave-like-manner if “equation (5)" is approximated by

II. FORMULATION OF THE PROBLEM AND BASIC EQUATIONS We consider a homogeneous generalized thermoelastic half-space rotating uniformly with an angular velocity

(1   q

  )q i   K (1    )T ,i . t t

Where

0     q .

(6)

Ω   n, where n is a unit vector representing the direction Hence, we get the heat conduction equation in the context of of the axis of rotation. All quantities considered will be functions of the time variable

t and of the coordinates x

dual-phase-lag model in the form

and

K (1   

z . The displacement equation of motion in the rotating frame

Moreover, if we put  

has two additional terms (Schoenberg and Censor [10]): centripetal acceleration, Ω  (Ω  u) due to time varying motion only and Corioli's acceleration dynamic displacement vector

  T e )T ,ii  (1   q )( C E  T 0 ). (7) t t t t

 0 and  q   (the first relaxation

time), then the fundamental

2Ω  u where the equations will be possible for the Lord and Shulman's theory.

is u  (u , 0,w ) ,

and

Where

, 

Ω  (0,  , 0) is the angular velocity. These terms do not distribution, We consider normal source acting at the plane surface of expansion,

generalized thermo-elastic half-space under the influence of

k

temperature,

gravity. The system of governing equations of a linear thermoelasticity

vector,

is the coefficient of linear thermal

is the thermal conductivity, T 0 is the reference

 ij

is the components of the stress tensor,

, C E

 ij

is

are the density and specific heat

Using the summation convection, we note that the second equation of motion in "equation (4)" is identically satisfied and (2)

the equations of motion under the influence of gravity become:

e T   x x (8) w 2 g   (u   u  2 w ), x e T  2w  (   )   z z (9) u 2  g   (w   w  2u ), x

The constitutive laws

 2u  (   ) (3)

Substituting "equation (3) into equation (1)", We obtain

[u i  {Ω  Ω  u}i  (2Ω  u)i ].

displacement

phase-lag of temperature gradient.

(1)

The strain-displacement relation

 u i , jj  (   )u j ,ij  T ,i 

the

respectively,  q is the phase-lag of the heat flux and   is the

Equations of motion

 ij  2e ij  [e  T ] ij .

is

the Kronecker delta,

with rotation and without body forces consists of:

e ij  12 (u i , j  u j ,i ).

u

  (3  2 )t , t

appear in non-rotating media.

 ij , j  [u i  {Ω  (Ω  u)}i  (2Ω  u)i ].

are the lame's constants, T is the temperature

(4)

The Chandrasekaraiah and Tzou theory (DPL) is such a modified of classical thermoelasticity model in which the Fourier's law is replaced by an approximation of the equation

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 2 ) T  t   C E (1   q )T  T 0 (1   q )e . t t

VOL.1 NO. 4 December 2013 http://www.researchpub.org/journal/jot/jot.html Substituting "equations (19) and (20) in equations (16) - (18)",

K (1   

we obtain

(10)

2     2 )  ( g  2 )    0, (21) 2 t x t 2   1  (g  2 )  ( 2  2  2   2 )  0, (22) x t  t      (1    )2  (1   q )   (1   q ) 2 . t t t t t (2 

The constitutive relations can be written as

 xx  yy  zz  xz For

 (  2 )u ,x  w ,z  T ,

(11)

 e  T ,

(12)

 u ,x  (  2 )w ,z  T ,

(13)

  (u,z  w ,x ),  xy   yz  0.

(14)

simplifications,

we

shall

use

the

(23)

following

The constitutive relations are

non-dimensional variables:

 xx  u ,x  (1 

c  x i  x i , u i  0 u i , c0 T 0 

*

*

 yy  (1 

 T {t , q ,  }   *{t , q ,  },   ,  ij  ij , T 0 T0

 zz  (1 

    * , where *  C E c 02 / K ,   (3  2 )t ,  2 (15) c 0  (  2 ) /  and i, j  1,3.

 xz 

In terms of non-dimensional quantities defined in "equation

Where

1

2

2



2

2

2

2

2

)w ,z   ,

(24)

) 2   ,

(25)

)u ,x  w ,z   ,

(26)

(u ,z  w ,x ),  xy   yz  0.

2 

  2 

and



(27)

 2T 0 .  K *

(15)", the above governing "equations (8) -(10)" reduce to (dropping the dashed for convenience)

III. THE SOLUATION OF THE PROBLEM

 2 (   ) e  u  c 02 c 02 x x g

The solution of the considered physical variables can be decomposed in terms of normal mode analysis in the

w  (u   2u  2w ), x

 2 (   ) e w  2 c 0 c 02 z 

[u ,w , e , , , ,  ij ](x , z , t ) 

(16)

u  g  (w   2w  2u ), z x  2T 0    (1    )2  (1   q )  (1   q )e . * t t K  t We introduce the displacement potentials

following form

[u * ,w * , e * , * , * , * ,  ij* ](z ) exp(t  iax ).

(17)

Where



is a complex constant,

number in the (18)

and

 (x , z , t ) and

 ij*

(28)

i  1 , a is the wave

x  direction and u * ,w * , e * , * , * , *

are the amplitudes of the field quantities.

Using "equations (28), equations (21) - (27)" take the form

 (x , z , t ) which are related to displacement components,

(D2  b1 ) *  b2 *   *  0,

(29)

we obtain

b3 *  (D2  b4 ) *  0,

(30)

 b5 (D2  a 2 ) *  (D2  b6 ) *  0.

(31)

u  ,x ,z , w  ,z ,x

e   2 ,

u ,z w ,x   2 .

(19)

Where

(20)

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d , b1  a 2   2   2 , b2  iag  2 , dz b3  b2  2 , b4  a 2   2 ( 2   2 ), 1  1     , 2 2 , and b 6  a 2  . 2  1   q , b5  1 1

 (z )    2 jR je *

D=

 xx*  iau *  (1  2

 yy*  (1 



2

 zz*  ia (1  

* xz

1





2

)Dw*   * , 2

(32)

)(D2  a 2 ) *   * ,

(33)



)u *  Dw*   * , 2

(Du  iaw ),

2

Where

1 j 

R j are some parameters and b5 (k j2  a 2 ) (k  b6 ) 2 j



* xy

u *   H1j R je

k j z



* yz

 0.

w *   H 2 j R j e

(35)

,

(42)

k j z

,

(43)

j 1

3

k j z

,

(44)

,

(45)

,

(46)

j 1

We get the following sixth order differential equation for

3

 yy*   H 4 j R j e

 (z ) *

k j z

j 1

[D6  A D4  B D2  C ] * (z )  0.

3

   H 5 j R je

(36)

* zz

In a similar manner, we arrive to

[D6  A D4  B D2  C ]{ * (z ),  * (z )}  0.

k j z

j 1

3

 xz*   H 6 j R j e

(37)

k j z

.

(47)

j 1

Where

Where

A  b1  b 4  b5  b6 ,

H 1 j  ia  k j  2 j , H 2 j  k j  ia 2 j ,

B  b1 (b4  b6 )  b4 (b5  b6 )  b2b3  a 2b5 ,

H 3 j  iaH 1 j  1 j  (1 

C  b1b4b6  b2b3b6  a 2b4b5.

H 4 j  (1 

"Equation (36)" can be factorized as

(D2  k 12 )(D2  k 22 )(D2  k 32 ) * (z )  0. Where

b3 , j  1, 2,3. b4  k j2

j 1

(34)

*  * and  * between "equations (29) - (31),"  xx   H 3 j R j e

Eliminating

, 2 j 

equations (32) - (35)" respectively, we obtain

3

*

(41)

Substituting from "equations (39) - (41) into equation (19) and

2

*

.

j 1

3



k j z

H6j 

equation of "equation (38)". The solution of "equation (38)", which is bounded as

at

.

(39)

)( k j2  a 2 )  1 j ,

2

2

) H 1 j  k j H 2 j  1 j ,

(k j H 1 j  iaH 2 j ),

j  1, 2,3.

in

order

to

determine

the

parameters

(1). Thermal boundary condition that the surface of the

In a similar manner, we get k j z

z 0

R j ( j  1, 2,3).

j 1

3

2

)k j H 2 j ,

In this section, we need to consider the boundary conditions

k j z

 * (z )  1jR j e

1

2

2

IV. THE BOUNDARY CONDITIONS OF THE PROBLEM

z  , is given by 3



H 5 j  ia (1 

(38)

k j2 , ( j  1, 2,3) are the roots of the characteristic

 * (z )   R j e

2

2

half-space subjected to thermal shock

.

(40)

 (x , 0, t )  f (x , 0, t )  f * exp(t  iax ).

j 1

16

(48)

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VOL.1 NO. 4 December 2013 http://www.researchpub.org/journal/jot/jot.html

(2) A mechanical boundary condition that the surface of the

  8954kg .m 3 , C E  383.1J .kg 1.k 1 ,

half-space is traction free

 xx (x , 0, t )   xz (x , 0, t )  0.

T 0  293K , a  1.5,

(49)

Where f ( x , t ) is an arbitrary function of x , t , and

f

*

f

*

 1,    0.3,  q  1.01,

  0  i  , 0  5,   0.9.

is a

The computations were carried out for a value of time t  0.1.

constant.

The numerical technique, outlined above, was used for the

Using the expressions of the variables considered into the above boundary conditions (48) and (49), we can obtain the

distribution of the real part of the temperature

following equations satisfied by the parameters:

displacement components

3

3

1 j R j  f * ,

H

j 1 3

H j 1

6j

j 1

3j

 xx ,  zz ,

R j  0,

and

 xz

,

the

u,w and the stress components

for the problem. Here all variables are

taken in non-dimensional form. The results are shown in Figs.1-12. The graph shows the curves predicted by L-S theory

R j  0.

(50)

and DPL model. In these figures, the solid lines represent the solution in the dual-phase-lag model and dashed lines represent

Solving the above system of "equations in (50)", we get the parameters

R j ( j  1,2,3) defined as follows:

 R1  1 , 

 R2  2 , 

 R3  3 . 

the solution derived using the generalized Lord and Shulman theory. Due to the boundary conditions, the stress components

 xx

(51)

and

 xz based on both L-S theory and DPL model start

from zero and terminate at a zero value.

Where

  11[H 32 H 63  H 33H 62 ]

Figures 1-6 show the comparisons between the displacement

 12 [H 31H 63  H 33H 61 ]

components

 13[H 31H 62  H 32 H 61 ],

components

 xx ,  zz ,

  0.5

for

1  f *[H 32 H 63  H 33H 62 ], 2  f *[H 31H 63  H 33H 61 ],

u,w the temperature  , and the stress and

three

 xz in the presence of rotation i.e. different

values

of

gravity

(g =0, 5, 9.8) i.e. in the absence and presence of gravity in

3  f *[H 31H 62  H 32 H 61 ].

the context of L-S theory and DPL model. Figure 1 depicts that the distribution of the horizontal

V. NUMERICAL RESULTS AND DISCUSSION

u, in the context of L-S theory and DPL model,

To illustrate the theoretical results obtained in the preceding

displacement

section, to compare these in the context of the Lord and

always begins from positive values for g =0, 5, 9.8. It

Shulman's theory L-S and the dual-phase-lag model DPL, and

shows that in the presence of gravity (i.e. g =5, 9.8 ), the

to study the effect of rotation and gravity on wave propagation,

values of

we now present some numerical results. For this purpose,

u based on both L-S theory and DPL model decrease

in the range

Copper is taken as the thermoelastic material for which we take

0  z  0.7 and increase in the range

the following values of the different physical constants

0.7  z  1.5. However, in the absence of gravity (i.e.

  7.76 1010 N .m 2 ,

g =0 ), the values of u based on L-S theory and DPL model

  3.86 1010 kg .m 1.s 2 ,

decrease in the ranges

K  386w .m 1.k 1 , t  1.78 105 k 1 ,

0  z  0.2 and 0.8  z  1.5,

while they increase in the range

17

0.2  z  0.8. The values of

JOURNAL OF THERMOELASTICITY ISSN 2328-2401 (print) ISSN 2328-241X (Online)

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u based on L-S theory and DPL model converge to zero with distribution of  xx is decreasing with the presence of gravity. increasing of the distance z at z  1.5 for g =0, 5, 9.8. The values of

We notice that the gravity acts to increase the magnitude of the real part of

 xx

based on L-S theory and DPL model

converge to zero with increasing of the distance z

u and the values of the horizontal displacement u

at

z  1.5 for g =0, 5, 9.8.

based on the DPL model are large compared to the values based

Figure 5 exhibits that the distribution of the stress component

on the L-S theory for g =0, 5, 9.8. Figure 2 shows that based

 zz

on L-S theory and DPL model, the values of the vertical

always begins from positive value (which is the same

w increase in the range 0  z  0.22 and point for the same value of g ) in the context of L-S theory and decrease in the range 0.22  z  1.5 for g =0, 5, 9.8. The DPL model for g =0, 5, 9.8. However, it decreases to a displacement

minimum value in the range

values of w converge to zero with increasing of the distance

0  z  0.3, then increases in

z at z  1.5 for g =0, 5, 9.8 in both L-S theory and DPL the range 0.3  z  1.5. In the context of L-S theory and model. Figure 3 demonstrates that the behavior of the temperature



based on both L-S theory and DPL model for the different values of

increase in the range



distance z at

z  1.5 for g =0, 5, 9.8.

 xz

0  z  0.35 and

converge to zero with increasing of the

begins from zero in the context of L-S theory and DPL

model for g =0, 5, 9.8. In the context of both L-S theory and

0.35  z  1.5. However, based on the

DPL model, the values of

 zz

Figure 6 depicts that the distribution of the stress component

g being similar. It shows that the values of  based

on the L-S theory decrease in the range

DPL model,

DPL model and in the absence of gravity (i.e. g =0 ), the

decrease in the range values of

0  z  1.5. The values of the temperature  converge to

 xz

start with increasing to a maximum value in the

z  1.5 for range 0  z  0.12, then decrease in the range g =0, 5, 9.8. We notice that the gravity has no great effect 0.12  z  1.5. However, in the context of both L-S theory

zero with increasing of the distance z at

on the distribution of the temperature

and DPL model and in the presence of gravity (i.e.

.

g =5, 9.8 ), the values of  xz start with decreasing in the

Figure 4 depicts that in the presence of gravity (i.e.

g =5, 9.8 .), the values of the stress component  xx based on range 0  z  0.12, and then increase in the range both L-S theory and DPL model decrease in the range

0.12  z  1.5.

0  z  0.2 and increase in the range 0.2  z  1.5.

Figures 7-12 represent the comparisons between the

However, in the absence of gravity (i.e. g =0 ) the values of

displacement components

 xx

stress components

based on both L-S theory and DPL model decrease in the

0  z  0.15

u,w, the temperature  and the

 xx ,  zz

and

 xz

in the presence of

range

gravity i.e. g =9.8 for two different values of rotation

0.15  z  1.5. We notice that the values of the stress

(  0,1) i.e. in the absence and presence of rotation in the

range

component

 xx

and

increase

in

the

context of L-S theory and DPL model.

based on the L-S theory are large compared to

Figure 7 depicts that the distribution of the horizontal

the values based on the DPL model for g =0, 5, 9.8 and the

displacement

18

u,

in the context of L-S theory and DPL model,

JOURNAL OF THERMOELASTICITY ISSN 2328-2401 (print) ISSN 2328-241X (Online) always begins from positive values for that the values of

u

VOL.1 NO. 4 December 2013 http://www.researchpub.org/journal/jot/jot.html

  0, 1. It shows DPL model,  zz converge to zero with increasing of the

based on both L-S theory and DPL model

distance z at

z  1.5 for   0, 1.

0  z  1.5. We notice that the rotation Figure 12 depicts that the distribution of the stress component acts to decrease the magnitude of the real part of u and the  xz begins from zero and satisfies the boundary condition at values of u based on DPL model are large compared to the decrease in the range

z  0 in the context of L-S theory and DPL model for

values based on the L-S theory.

  0, 1. In the context of both L-S theory and DPL model,

Figure 8 shows that in the context of both L-S theory and DPL model, the values of

w start

with increasing to a maximum

the values of

0.2  z  1.5 for   0, 1. We also notice that the vertical

0.12  z  1.5.

w is increasing with the presence of rotation.

Figure 9 demonstrates that the distribution of the temperature



start with decreasing to a minimum value in

0  z  0.2, then decrease in the range the range 0  z  0.12, then increase in the range

value in the range

displacement

 xz

VI. CONCLUSIONS According to the above results, we can conclude that:

begins from a positive value (which is the same point) for

  0, 1, in the context of both L-S theory and DPL model. 1. We found that, the parameters  q and   have significant We notice that the behavior of the temperature different values of





effects on all the fields.

for two

2.

being similar.

The phenomenon of finite speeds of propagation is

Figure 10 depicts that the distribution of the stress component

manifested in all these figures.

 xx

3.

begins from zero point and satisfies the boundary

thermoelastic medium in solids have been developed and

condition at z  0, in the context of L-S theory and DPL model for

utilized.

  0,1. The values of  xx , in the context of L-S 4. The value of all the physical quantities converges to zero with an increase in distance

theory and DPL model, start with decreasing to a minimum value in the range

the real part of

z

and

all

functions

are

0  z  0.2, then increase in the range continuous. 5.

0.2  z  1.5. We notice that the rotation acts to decrease

 xx

and the values of

 xx

The presence of rotation and gravity plays a significant

role in all the physical quantities except temperature. The

based on the L-S

amplitude of the physical quantities changes while

the

rotation and gravity increase. Therefore, the presence of

theory are large compared to the values based on the DPL model for

Analytical solutions based upon normal mode analysis for

  0, 1.

rotation and influence of gravity in the current model is of

Figure 11 exhibits that the distribution of the stress component

significance.

 zz

6.

All the physical quantities satisfy the boundary conditions.

7.

The comparison of different theories of thermoelasticity,

always begins from positive value (which is the same

point for the same value of and DPL model for

 ) in the context of L-S theory

  0, 1. It shows that  zz decreases to Tzou (DPL) model is carried out.

a minimum value in the range the range

i.e. Lord and Shulman theory and Chandrasekharaiah and

0  z  0.3, then increases in 8. Deformation of a body depends on the nature of the applied force as well as the type of boundary conditions.

0.3  z  1.5. In the context of L-S theory and

19

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VOL.1 NO. 4 December 2013 http://www.researchpub.org/journal/jot/jot.html

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presence of gravity

 xx

in the absence and

0.6

0.2 DPL L-S

g=9.8

DPL L-S

0.5

0.15

0.4 0.3

u

0.1

zz

g=5

g=0

0.05

0.2 g=0

g=5

g=9.8

0.1 0

0

-0.1

-0.05

0

0.5

1

1.5

-0.2

z

0

0.5

1

1.5

z

Fig. 1 Horizontal displacement distribution u in the absence

Fig. 5 Distribution of stress component

and presence of gravity presence of gravity

21

 zz in the absence and

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VOL.1 NO. 4 December 2013 http://www.researchpub.org/journal/jot/jot.html

0.06

0.1 DPL L-S

g=0

0.04

DPL L-S

0.05

 =0 0.02

0 g=5

xx

-0.05

xz

0

-0.02

-0.1

-0.04

-0.15 g=9.8

-0.06

-0.08

 =1

-0.2

0

0.5

1

-0.25

1.5

0

0.5

z

1

1.5

z

Fig. 6 Distribution of stress component

 xz in the absence and

Fig. 10 Distribution of stress component

presence of gravity

 xx

in the absence

and presence of rotation

0.3

0.7 DPL L-S

0.25

DPL L-S

0.6 0.5

0.2

0.4

 =0

0.3

u

zz

0.15

0.1

0.2

 =0

0.1

 =1

 =1

0

0.05

-0.1 0 -0.2 -0.05

0

0.5

1

-0.3

1.5

0

0.5

z

1

1.5

z

Fig. 7 Horizontal displacement distribution u in the absence

Fig. 11 Distribution of stress component

and presence of rotation

 zz in the absence

and presence of rotation

0.07

 =1

DPL L-S

0.06

0.01 DPL L-S

0

 =0

0.05

-0.01 -0.02 -0.03

0.03

xz

w

0.04

-0.04

 =0

0.02 -0.05 0.01

-0.06 -0.07

0 -0.01

-0.08 0

0.5

1

 =1

1.5 -0.09

z

0

0.5

1

1.5

z

Fig. 8 Vertical displacement distribution w in the absence and Fig. 12 Distribution of stress component

presence of rotation

and presence of rotation

22 0.3 DPL L-S

0.25

0.2



0.15

0.1

0.05

0

-0.05

0

0.5

1

1.5

z

Fig. 9 Temperature distribution

 in the absence and presence

of rotation

22

 xz in the absence