THERMO-ELASTIC PROBLEM OF A HALF-SPACE
Sukhwinder Kaur Bhullar1
Department of Mathematics
Panjab University, Chandigarh – 160014,
India
Abstract
We consider a homogeneous, isotropic elastic semi-infinite space, which is at
temperature T0 initially and whose boundary surface is subjected to heat source and load
moving with finite velocity. Temperature and stress distribution occurring due to heating
or cooling and have been determined using appropriate initial and boundary conditions.
Numerical data for stainless steel is considered for the numerical calculations and the
results obtained are shown through contour maps.
Keywords: Thermal conductivity, stress, displacement, temperature.
1. INTRODUCTION
Thermoelastic problems play an important part in different branches of technical
Sciences. In engineering practice, most structures contain internal interface when heat
flow in a structure is disturbed by some defects, such as holes and
cracks, the local
temperature gradient around the defects is increased and the temperature is often
discontinuous across the defects. Thermal disturbances of this type may produce material
failure. Therefore, thermal analysis for such structures is very important. As well as
loading is concerned generally two types of loading mechanical and thermal is subjected
on the surface of structure. The mechanical loading consists of pressure and shear
tractions, and thermal loading is caused by frictional heating and results in thermoelastic
stress. The knowledge of thermoelastic stress field in a structure is essential for failure
prevention and life prediction because the total stress consists of thermoelastic stress and
elastic stress and due to this phenomena frictional heating significantly influence the
failure of components in contacts under relative motions e.g. thermo-cracking of breaks
1
sbhullar_2000@hotmail.com
1
and face sears, and scuffing in gears. The problem of thermoelasticity for a half-space is
being used in many applications. Some of the important and significant work on
thermoelastic problems in half planes, plates and half spaces have been done by different
authors1-10. Several authors have studied the disturbances produced in a half space due to
application of time dependent loading or heating applied to the boundary. A number of
studies dealing with flaw induced thermal stresses in infinite regions have been done by
Olesiak and Sneddonh11,Sih12 and Florence and Goodier13.Clements14studied thermal
stresses in anisotropic elastic half space. Fox15 has studied the uncoupled problem of a
moving line temperature pulse using complex variables technique neglecting inertia term.
Eason and Sneddon16 has investigated coupled problem concerning pulses using multiple
Fourier transforms. Boley and Tolins17 studied the disturbances in the context of linear
coupled thermoelasticity. Lord and Shulman18, Popov19 and Narwood and Warren20
employed with both linear and second order equations of the coupled theory based on
modified law of heat conduction. Johnsons21 has made a comprehensive comparative
study for the results obtained by Lord and Shulman, and Popov
18-19
. Lamb22, cole and
Huth23 and Sneddon24 have studied steady state response to moving loads in an elastic
solid medium in the classical theory of thermoelasticity. By taking specific input pulse
Roberts25 solved a steady state problem of line loads moving over a coupled thermoelastic
half space and obtained the stresses and displacements as single Fourier inversion
integrals. Chanderashekhar26 studied one-dimensional dynamical problems in a
thermoelastic half space with plane boundary due to application of a step in strain or
temperature on the boundary in the context of Green Lindsay thermoelastic theory. Mann
and Blackburn27 obtained solution for a nonlinear steady state temperature problem of a
semi-infinite strip. Aggarwala and Nasim28 have constructed explicit solutions for
discontinuous boundary value problems for steady-state temperatures in a quarter plane.
Rubin29 obtained Solution of boundary value problems in dynamic thermoelasticity for a
half-space with a uniformly moving boundary in the case of boundary functions of a
general form. Germanovich30 et al have solved the problem of heat conduction for a halfspace and obtained approximate solution based on the asymptotic expansion of
temperature and stresses for small times. Kerchman31 formulated the problems of coupled
thermoelasticity and consolidation theory accounting for the porefluid compressibility,
2
and constructed solutions of the basic plane and axisymmetric problems for the halfspace
with the aid of auxiliary functions which generalize the McNamee and Gibson functions
and discussed the importance of the coupling effects in formulating and solving some
classes of boundary value problems. Bukatar and Lenjuk32 worked on a dynamical
problem of thermoelasticity of a half-space in the case of slowly changing temperature of
the bottom. Hetnarski and Ignaczak33 obtained closed-form solutions to the initialboundary
value
problems
of
one-dimensional
generalized
dynamic
coupled
thermoelasticity. Sherief34 obtained the distributions of temperature, displacement, and
stress by taking the equations of generalized thermoelasticity with one relaxation time for
one-dimensional problems.
With above background in the present paper, we have studied a two
dimensional problem of thermoelastic half space by using equations of generalized theory
of thermoelasticity with inertia term.
2. FORMULATION OF THE PROBLEM
Consider a half space y  0 , initially at the temperature T0 and in the stress free state. A
variation in temperature, displacement and stress fields will occur due to action of
external loadings. Assuming that the displacement will be along x -axis, y -axis function
of space co-ordinate x , y and time t .
Equation of heat conduction in the elastic half space is given, by Lord and Shulman18 as
following
 T

 2T 
2 
K  2T   0 ce 
  0 2   3  2   0 T0    0 2 
t 
t 
 t
 t
Equation of motion in the medium considered is given by

2 u

2 
     . u     u   0 2  3  2   0  T
t
(1)
(2)
Constitutive relation is given by

ij
 2  u i , j  u
j ,i
   
ij
 0 3   2  T  T0  i j
3
(3)


where,    . u , where    . u , T is temperature , ce is specific heat,  0 is density,
 0 is coefficient of thermal expansion,  0 is relaxation time, K is thermal conductivity

u  (u , v, w) is displacement vector ,where  i j are the components of stress
and
tensor and  i j is kronecker delta and ( i , j  1,2,3 ).
Let us consider a moving heat source on the surface of the half space with the following
Initial conditions as
T  x, y , t   f  x  vt ,
(4)
 xy  x, y, t   0   yy  x, y , t ,
Along with the moving source, we consider a moving load with the following initial
conditions
 yy ( x , y , t )  g ( x  vt) ,
 xy ( x, y, t )  0 ,
(5)
T
 hT  0,
t
where, h is the surface heat transfer coefficient and f and g are arbitrary function and
v be the velocity of motion of both source and load.
Introducing following non-dimensional variables as
t 
c12
t,
k1
 i j 
xi 
c1
xi ,
k1
1
i j ,
c 3  2   T0
2
1
0
 c3

u  1
u,
k1 3  2   T0
T 
T  T0
  2
K
2
and k1 
where, c1 
0
 0 ce
T0
Using these quantities into equations (1) - (3), we get
 T 
2 T    
2 
  
 ,
 T    0 ce 


 t  2    t 
 t  2 
  t
2

2u 
  0,
t  2 
   .u     2u    2    T   

 ij  u  i , j  u 
where,  
j ,i
  c1

2

 2    ij  T  ij ,

3  2   0 2T0
 0 ce   2  
and    0
(6)
(7)
(8)
2
c1
.
k1
4
In the subsequent discussions we omit the primes for convenience.
3. SOLUTION OF THE PROBLEM
Since we are considering two-dimensional problem therefore we shall consider

u  (u , v , 0) , where u and v are function of x and y only and y  0,
- < x <  . Introducing potentials  and  as follows
 

,
x y
(9)
 

,
y x
(10)
u
v
Using (9) and (10) in equations (6) and (7), we get
 2 
2 
  
 2  T  
t
t 

 2 
  
t


2 
  2   2 ,
t 
 t
(11)

   T ,

(12)
 2 1 2 
   2 2   0 ,
c t 

where, c 2 

  2
(13)
.
We change to coordinate system moving with input by shifting origin to the position of
input
x 
c1
( x  m t ) ,
k1
(14)
y   y  ,
1 
2
(15)
2
2

,
 x  2  y  2
where, m 
(16)
v
is dimensionless loading speed and co-ordinates x  and y  move in
c1
positive direction of x-axis with speed ‘m’.
Now,using (14)- (16) in equation (9)- (11) we get
5
2
 2

 2

2 

2 
 1  m


 
 m 2
T



m


m
2 
2  1

 x 
 x 
 x  
 x  


(17)
 2
2 
 1  m 2
  T
 x  2 

(18)
 2 m2  2 
 1  2
  0
2 


c

x


(19)
Omitting the double primes and writing  to 1, we get
 2

2
   m
 m 2
x
 x2




2
 T     m
 m 2
x
 x2


 2
 

(20)
 2
2
   m 2
 x2


  T

(21)
 2 m2  2
   2
c  x2


  0

(22)
We see that equations (20) and (21) are coupled in T and  , while equation (22) is
independent in  .
In order to solve the coupled equations, we eliminate T or  and obtain the following
equation

2 
2  
2  
  2  m   m 2     2  m 2      m      2   , T   0.
x

x 2  
x 2   x
x 2  

(23)
To solve the equations (22) and (23) we take
  Ae
ikx y
  Be
ikx    y
,
(i)
,
where, A and B in general are complex constant and k , 
and   are unknown
quantities to be determined .
Substituting for  and  in (22) and (23) we get

m2 
  2  1  2  k 2 ,
 c 
(24)
6
 2  (1  m2 )k 2  imk  2  (1  m2 )k 2   imk  m2k 2  2  s 2  0
(25)
 m2 
For   to be real , 1  2   0 and c 2  m 2 and,
 c 
 m2 
 3  k 1  2  is root of equation (24).
 c 
Equation (25) is bi-quadratic in  hence has four roots. We shall consider the roots with
positive real parts .Let these be  1 and  2 are given by

   k 2  i k  i m k   2i m k   k 2 ,
1,2
(26)
where


1
2  m 2 1   1    ,
2
1
i m 1    , 
2


 
1
1      1   2 ,   1 m 2 1  2 1     
2
2
 

2

1
1   2 ,
2
1   2  .
Thus from relation (i) we have


A
ke
i k x 
3
y
,
(27)
k  

i k x  y
i k x  y 
 
1
2 

,
B
e

C
e

k
 k

k   

T 
(28)
   2
ikx   y
ikx   y 
2
2
1  C  2  k 2 1  m 2  e
2
  Bk  1  k 1  m  e

k 2








k   
(29)
The constants Ak, Bk and Ck can be determined by using of
boundary conditions.
Stress components can be written in terms of potentials as follows
2 
 2  2
 x y 2


,
 x y  x 2  y 2
(30)
  2
 1
 
 2  
,
 x x  2

 2T  
 2  m
2  x y 
2
x

x
c




7
(31)
  2
2  
1
 
  2T  
.
 y y  2 2 
 2 m
2

 y x 
c
 y
 y
(32)
Case 1. Moving heat source
Boundary conditions for moving heat source are given by (4). Using (14)-(16) in the (4)
after omitting double primes they are written at y = 0, as
T  x, 0, t   f  x  ,
 xy  x,0, t   0   yy  x,0, t  ,
(33)
In first equation of (33) taking
T x, 0, t   f x  

ikx ,
 ak e
k 

2
where, a 
f x e ikx dx , and f x   e  x .

k 

1
Hence from (33),
 B 
k 
k  
k
2
1

 k 2 1 m 2



 Ck 
2
2



a e
 k 2 1 m 2 e i k x 
k  
ik x
k
,



2
2 2  
2
  Ak  2 i  3  Bk   1m  2c ( 1   1m  k m )   i kx
0 ,
 
e
k  C   m  2c 2 ( 2   m  k 2 m 2 ) 



2
2
 k  2


   2
2
 ikx  0
  Ak  k   3   2 Bk i k x  1  2C k i k x  2  e




k 
Hence,

Bk 
2
1


 k 2 1 m 2


 Ck 
2
2

 k 2 1 m 2
  a
k
.
,
(34)

2
 A k  3  k 2  2B k i k  1  2 Ck i k 2  0 ,
 2 Ak i 3 kc2  Bk
 
1

m  2c 2 
2
1
c
(35)
  1 m  k 2m2
  C  
k
2
2
From equations (34), (35) and (36) by using Cramer’s rule we get
8


m  2c 2  2  k 2   2 m
 0.
c2
(36)
2
Ak 
B k
Ck 


 4 i ak k  1  2   1 
2
2 c 2 k 2  23 k 2  


 k 2   2 k 2m2



2
2

2
2
a k  4  1 3 k 2  k 2  

 ak  2 m  
3

c2
2  k
2
3

m  2c2 
c 
1
2
,
2
1

2
3
m  k 2m2
  1m  k 2 m 2
 ,




2

,

2
3
2
2
2 
 2
   1  2 k   2 k  1  m   1k   2  k  1  m   
4k 3 
   2  k 2     m  2 c 2   2  k 2   m   2  k 2  1  m 2


 2
2
2  1

3
2


c

   2  k 2  1  m 2   m  2 c 2   2  k 2 m 2   m  

1  
  2
 1
 1






  .

 




Case 2. Moving Load
Boundary conditions for moving Load source are given by (5). Using (14)-(16) in the (5)
and after omitting double primes they are written at y = 0, as

( x , 0 , t )  g ( x  vt) , 
0 ,
yy
xy
T
 hT  0.
t
(37)
where, h is the surface heat transfer coefficient.
In first equation of (37) taking

 yy  x 0, t   g  x    b e ikx , and
k  k

2
1
b 
g x e ikx dx , and g x   e  x .

k 

Hence from (37) we get
 B 

k  
k
2
1

 k 2 1  m2


 C k 
2
2

 k 2 1  m2
9
h  ik e
ik x
0
,
(38)


 
2
2
2 2  
2

  Ak  2 i c k 3  Bk   1m  2c ( 1   1m  k m )   i kx 2 k 
c  b
 
e
k
k  C    m  2c 2 ( 2   m  k 2 m 2 ) 

k 


2
2
 k  2


(39)
   2
2
 ikx  0 .
  Ak  k   3   2 Bk i k x  1  2C k i k x  2  e




k 
(40)
From equations (38)-(40) by using Theory of Matrices, we get

 y  
  y 
1        2  k 2   1  m 2   k e
2 
 2 ib      2  k 2   1  m 2   k e



k 
2 1
1
2






 



A 
k




  2 y 


 b    2  k 2     2  k 2  1  m 2  e

k 
3
2
 

 ,

B 
k




  1 y 
b    2  k 2     2  k 2  1  m 2  k e

k 
3
  1

 ,
C 
k



 y
 y
     2  k 2  1  m 2   e 1     2  k 2  1  m 2  e 2 

2 1
2 2






  y


1


   4k 2 3   e 1    2  k 2     m  2 c 2   2  k 2   m   2  k 2   1  m 2    
3
2
2  1



   

c2 



  y 2

 e 2    k 2  1  m 2   1m  2 c 2   2  k 2 m 2   1m  

 2

 1









4. NUMERICAL CALCULATIONS AND CONCLUSION
In order to study temperature and stress distribution in a homogeneous, isotropic elastic
half space, we have computed them for a specific model. For this purpose the values of
relevant parameters for Stainless steel are in Table 1. The contour maps are shown in to
10
express the variation of temperature and stresses for different time interval. The range of
motion of heat source and load is taken  2  x  2 .
1. Due to moving heat source, the temperature distribution is symmetrically
2
distributed with respect to x since f x   e  x symmetric function. As the
source is moving in the x -direction the contours also tend to move in the same
direction.
2. Due to moving load the temperature distribution is symmetrically distributed
2
with respect to x since g x   e  x is symmetric function. As the load is
moving in the
x -direction the contours also tend to move in the same direction.
3. As well as the variation of stress component xy , due to moving heat source
concerned we see that variation of stress component xy is more in left side and
less in right side.
4. Due to moving heat source the stress component yy is almost symmetrically
distributed but less in case of high temperature zone.
5. Similar stress distribution is observed for stress component xy and yy
respectively due to Moving load.
The contour maps are shown in figures 1(a) to 1(c) express the variation of temperature
and stresses for different time interval. Figures 4(a)-4(c) representing
variation of
temperature due to Moving Load.In both cases to show the variation of temperature ,time
intervals and time relaxation constant are taken same. Figures 2(a)- 2(c) demonstrate
variation of stress component xy and Figures 3(a)- 3(c) variation of stress component yy
due to Moving heat source. Figures 5(a)- 5(c) represent variation of stress component xy
and Figures 6(a)- 6(c) variation of stress component yy respectively due to Moving heat
Load.
11
Figure Captions
Figure 1(a): Temperature variation due to moving heat source at time, t= 0.5
Figure 1(b) :Temperature variation due to moving heat source at time, t= 1.0
Figure 1(c) :Temperature variation due to moving heat source at time, t= 1.5
Figure 2(a) : Stress variation of xy due to moving heat source at time,t = 0.5
Figure 2(b) : Stress variation of xy due to moving heat source at time, t =1.0
Figure 2(c) : Stress variation of xy due to moving heat source at time, t = 1.5
Figure 3(a) : Stress variation of yy due to moving heat source at time, t = 0.5
Figure 3(b) : Stress variation of yy due to moving heat source at time, t = 1.0
Figure 3(c) : Stress variation of yy due to moving heat source at time, t = 1.5
Figure 4(a):Temperature variation due to moving Load at time t =0 .5
Figure 4(b) : Temperature variation due to moving Load at time t =1. 0
Figure 4(c): Temperature variation due to moving Load at time t = 1.5
Figure 5(a) : Stress variation of xy due to moving Load at time t = 0.5
Figure 5(b) : Stress variation of xy due to moving Load at time
t = 1.0
Figure 5(c) : Stress variation of xy due to moving Load at time t = 1.5
Figure 6(a) : Stress variation of yy due to moving Load at time t = 0.5
Figure 6(b) : Stress variation of yy due to moving Load at tme t = 1.0
Figure 6(c) : Stress variation of yy due to moving Load at tme t = 1.5
Table1
Material Constant
Steel

9.2091010

6.4531010
Linear thermal expansion 
17.710-6
Mass density 
7.97103
Specific Heat at constant vol. C
0.560103
Thermal Conductivity K
19.5
12
AcknowledgementI express my sincere gratitude to Professor Harinder Singh, Panjab University
Chandigarh, for guidance and encouragement during the preparation of this paper.
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15
2.4.1 MOVING HEAT SOURCE
T1
T
0.8
T
1
0.8
0.8
0.2
0.6
< 0.2
0.6
0.4
-1
0
1
2
x
0.6
0.8
> 0.8
0.2
0.2
0
-2
0.2
0.4
0.4
0.8
0.2
1
0
-2
-1
0
1
2
Figure 1(a)
Figure 1(b)
Temperature Variation
Temperature Variation
due to Moving Heat
due to Moving Heat
Source at time=0.5
Source at time, t=1.0
x
0
-2
-1
0
1
Figure 1(c)
16
Temperature Variation
due to Moving Heat
Source at time, t=1.5
2
x
1
1
1
0.8
0.8
0.8
< 0.2
0.6
xy
< 0.2
xy
0.4
0.6
xy
0.4
> 0.8
0.2
-2
-1
0
1
2
x
0.4
> 0.8
0.2
0
< 0.2
0.6
> 0.8
0.2
0
-2
-1
0
1
x
2
0
-2
-1
0
1
Figure 2(a)
Figure 2(b)
Figure 2(c)
Variation of xy due to
Variation of xy due to
Variation of xy due to
moving heat source at
moving heat source at time,
moving heat source at
time, t= 0.5
t= 1.0
time, t=1.5
17
2
x
1
1
1
0.8
0.8
0.8
< 0.2
0.6
< 0.2
yy
yy
0.4
0.6
yy
0.4
0.4
> 0.8
0.2
> 0.8
0.2
0
-2
-1
0
1
2
x
< 0.2
0.6
> 0.8
0.2
x
0
-2
-1
0
1
2
0
-2
-1
0
1
Figure 3(a)
Figure 3(b)
Figure 3(c)
Variation of yy due to
Stress variation of yy
Stress variation of yy
moving heat source at
due to moving heat
due to moving heat
time, t= 0.5
source at time, t= 1.0
source at time, t= 1.5
18
x
2
2.4.2 MOVING LOAD
T
T
T
1
1
1
0.8
0.8
0.8
< 0.2
< 0.2
< 0.2
0.6
0.6
0.6
0.4
0.4
0.4
> 0.8
0.2
> 0.8
0.2
x
0
-2
-1
0
1
2
> 0.8
0.2
0
-2
-1
0
1
2
x
0
-2
-1
0
1
Figure 4(a)
Figure 4(b)
Figure 4(c)
Temperature variation
Temperature variation
Temperature variation
of due
of due
of due
to moving Load at
to moving Load at
to moving Load at
time,t=0.5
time,t=1.0
time,t=1.5
19
2
x
1
1
1
0.8
0.8
0.8
< 0.2
0.6
xy
xy
0.4
< 0.2
0.6
xy
0.4
0.4
> 0.8
0.2
> 0.8
0.2
0
-2
-1
0
1
2
x
< 0.2
0.6
> 0.8
0.2
x
0
-2
-1
0
1
2
x
0
-2
-1
0
1
Figure 5(a)
Figure 5(b)
Figure 5(c)
Variation of xy due to
Variation of xy due to
Variation of xy due to
moving Load at time,
moving Load at time,
moving Load at time,
t= 0.5
t= 1.0
t= 1.5
20
2
1
1
1
0.8
0.8
0.8
< 0.2
0.6
< 0.2
< 0.2
0.6
yy
0.6
yy
0.4
yy
0.4
0.4
> 0.8
0.2
> 0.8
0.2
0
-2
-1
0
1
2 x
> 0.8
0.2
x
0
-2
-1
0
1
2
0
-2
-1
0
1
Figure 6(a)
Figure 6(b)
Figure 6(c)
Variation of yy due to
Variation of yy due to
Variation of yy due to
moving Load at time,
moving Load at time,
moving Load at time,
t= 0.5
t= 1.0
t=1.5
21
2
x