2321-7871
Weekly Science Research Journal
Vol-1, Issue-44, 22th May 2014
Original Article
Quasi-Static transient Thermal Stresses in a Robin's thin
Rectangular plate with internal moving heat source
D. T. Solanke and M. H. Durge
D. T. Solanke and M. H. Durge
ABSTRACT
This paper concern with the transient non-homogeneous thermoelastic problem
with Robin’s boundary condition in thin rectangular plate of isotropic material,
occupying region R: 0 £
x£
a, 0 £
y£
b, 0 £
z£
c where b <
a, c <
b, c is very small as
compared to b . Initial temperature of the plate is f ( x, y , z ) and plate is placed in an
ambient temperature zero. The plate is subjected to the activity of moving point
heat source at point ( x¢
u1i +
u2 j +
u3 k
, y¢
, z¢
) moving with constant velocity vector u =
where u1 , u 2 , u3 are the component of velocity vector along x, y, z axes respectively.
The heat conduction equation containing heat generation term is solved by
applying integral transform technique and Green’s theorem is adopted in deriving
the solution of heat conduction equation. The solution is obtained in a series form
of trigonometric function and the thermal stresses are derived.
Keywords:
Robin's thin rectangular plate, moving heat source, thermal stresses, Green's
theorem.
2.Introduction:
During the second half of 20th century, non-isothermal problems of the theory
of elasticity became increasingly important. This is due to their wide application in
diverse fields. The high velocities of modern aircraft give rise to aerodynamic heating,
which produces intense thermal stresses that reduce the strength of aircraft structure.
Recently D. T. solanke, S. M. Durge have studied the thermal stresses, in thin solid
cylinder and hollow cylinder with Dirichlet's, Neumann's and Robin's boundary
condition and rectangular plates from [1] to [6]. Nobody previously have studied such
type of three dimensional temperature distribution and thermal stresses with moving
heat source in thin rectangular plate with Robin's boundary condition. This is new
contribution to the field of thermoelasticity. In this present paper we determine
temperature distribution and thermal stresses in thin rectangular plate with moving
point heat source with Robin's type boundary condition. The heat conduction equation
containing heat generation term is solved by applying integral transform technique and
Green's theorem is adopted in deriving the solution of heat conduction equation. The
solution is obtained in a series form of trigonometric function and the thermal stresses
are derived.
3.Formulation of the problem:
Consider a thin rectangular plate of isotropic material of length a , breadth b and
height c ; occupying the region R: 0 £
x£
a ,0 £
y£
b,0 £
z£
c where b <
a, c <
b , c is
very small as compared to b . Initial temperature of the plate is f ( x, y, z ) placed in
an ambient temperature zero. The plate is subjected to the activity of instantaneous
moving point heat source at the point ( x¢
, y¢
, z¢
) which changes its place along x, y , z
axes moving with constant velocity vector u =
u1i +
u2 j +
u3k where u1 , u 2 , u3 are
Page No-1
From
Sudhakar Naik and Umashankar Khetan
college Akola, Maharashtra State, India.
Anand Niketan Collge, Warora,
Maharashtra State, India.
Article Is Published On May 2014
Issue & Available At
www.weeklyscience.org
DOI : 10.9780/ 2321-7871/1202013/53
2321-7871
th
Vol-1, Issue-44, 22 May 2014
Quasi-Static transient Thermal Stresses in a Robin's thin Rectangular plate with internal moving heat source
component of velocity vector along x , y , z axes respectively. The activity of moving heat
source or initial temperature of the plate may cause the generation of heat due to nuclear
interaction that may be a function of position and time in the form g ( x, y, z, t ) w/s3. The
temperature distribution of the rectangular plate is described by the differential equation of
heat conduction with heat generation term as in [7] page no. 9 is given by
1
1 ¶
T
2
Ñ
T+
g =
k
a
¶
t
T=
T ( x, y , z ) is temperature distribution, k is thermal conductivity of the material
k
of the plate, a
=is thermal diffusivity, r
is density, c p is specific heat of the material
r
cp
where
2
and g is volumetric energy( heat) generation term. Ñ
is Laplacian operator in rectangular
coordinates in three dimension. Now consider an instantaneous moving point heat source
at a point ( x¢
, y¢
, z¢
) and releasing its heat spontaneously at time t
. Such volumetric
moving
heat
source
in
rectangular
coordinates
is
given
by
g (x , y , z,t) =
g p id
(x x¢
)d
(y y¢
)d
(z z¢
)d
(t t
)
Hence above equation reduces to
1
1 ¶
T
2
Ñ
T +
g p id
(x x¢
)d
(y y¢
)d
(z z¢
)d
(t t
)=
k
a
¶
t
(3.1)
Where x¢
=
u1t , y ¢
=
u 2t ,
z¢
=
u3 t
(3.2)
With initial and boundary condition
¶
T
-=
hT
0
¶
x
at
x=
0
(3.3)
¶
T
+
hT =
0
¶
x
at
x=
a
(3.4)
¶
T
hT =
0
¶
y
at
y=
0
(3.5)
at
y=
a
(3.6)
¶
T
+
hT =
0
¶
y
¶
T
hT =
0
¶
z
at
z=
0
(3.7)
¶
T
at
+
hT =
0
¶
z
z=
a
(3.8)
t=
0, t
=
¥
(3.9)
T=
f ( x , y, z )
at
4. Formulation of the thermoelastic problem:
Let us introduce a thermal stress function c
related to component of stress in the
rectangular coordinates system as in [8] where c
=
c
c
c +
p , c
c is complementary solution
and c
p is particular solution. c
c and c
p are governed by a linear homogeneous differential
equation and linear non-homogeneous differential equation
4
Ñ
c
=
0
c
(4.1)
2
Ñ
c
a
EG
p =
(4.2)
Page No-2
2321-7871
th
Vol-1, Issue-44, 22 May 2014
Quasi-Static transient Thermal Stresses in a Robin's thin Rectangular plate with internal moving heat source
Where
G
is temperature change G
=
TTi
where Ti is initial temperature
2
2
¶
¶
2
Ñ
=+
2
¶
x
¶
y2
since plate is thin z is negligible
2
¶
c
s
xx =
¶
y2
(4.3)
2
¶
c
s
yy =
¶
x2
(4.4)
2
¶
c
s
=
xy
¶
x¶
y
(4.5)
The boundary condition is s
s
0 at
xy =
yy =
y=
b (4.6)
5.Solution:
We define integral transform of T ( x, y, z, t ) by
(5.1)
T (a
T ( x , y , z , t ) X (a
m,b
m,g
m ,t) =
m x )Y (b
m y ) Z (g
m z ) dv
ò
R
And its inverse integral transform by
¥
T (a
m , b
m ,g
m , t ) X (a
m x )Y ( b
m y ) Z (g
m z)
T ( x , y, z , t) =
å
h
m=
0
m
(5.2)
Where
X ( aaa
H s in (a
m x) =
m cos(
m x) +
m x)
(5.3)
Y (b
b
H sin ( b
m y) =
m co s( b
m y) +
m y)
(5.4)
Z (g
g
H s in (g
m z) =
m c o s (g
m z) +
m z)
(5.5)
h
H =
k
(5.6)
h
m is product of the normalization integral and
( ) ( ) ( )
2
2
2
é
é
é
ha
N ( m)N(b
a
H 2 a2 +
Hù
b
H 2 b2 +
Hù
g
H 2 c2 +
Hù
m =
m ) N (g
m )) =
m +
m +
m +
ë
û
ë
û
ë
û
(5.7)
2a
H
m
a
are roots of the transcendental equation tan(a
m
ma) =
2
2
(5.8)
2b
mH
b
are roots of the transcendental equation tan ( b
m
mb ) =
2
2
(5.9)
a
H
m -
b
H
m -
2g
mH
g
are roots of the transcendental equation tan (g
m
mc ) =
2
2
(5.10)
g
H
m Taking integral transform of equation (3.1) and using boundary condition and following
Green's theorem we obtain
N
é¶
ù
¶
y
T
2
2
k
Ñ=
T
y
dv
T
Ñ
y
dv
+
y
T
dsi
k
k
å
ê
k
ú
ò
ò
ò
¶
ni
¶
ni û
i=
1 si ë
R
R
Page No-3
(5.11)
2321-7871
th
Vol-1, Issue-44, 22 May 2014
Quasi-Static transient Thermal Stresses in a Robin's thin Rectangular plate with internal moving heat source
dT
a
2
2
2
¢
¢ m z¢
+
a
(a
b
g
g p i X (ab
)d
(t t
)
m +
m +
m )T =
m x )Y ( m y )Z (g
dt
k
a
é
ù
(a
+
b
+
g
)t
a
(a
+
b
+
g
)t
T =
f (a
g p i X (ab
)Y ( m y ¢
) Z (g
) ea
em,b
m ,g
m)+
mx¢
m z¢
ê
ú
k
ë
û
Taking inverse integral transform we obtain
2
m
2
2
m
2
m
2
m
2
m
m
¥
X (ab
a
(a
+
b
+
g
)t
m x )Y ( m y ) Z ( g
mz)
T =
y
eå
h
m=
0
m
2
2
m
m
(5.12)
2
m
a
é
ù
(a
+
b
+
g
)t
¢
¢ m z¢
y
=
f (a
g p i X (ab
) ea
m,b
m,g
m)+
m x )Y ( m y ) Z (g
ê
ú
k
ë
û
2
Where
2
m
m
(5.13)
2
m
¥
X (a
x )Y ( b
y ) Z (g
z)
m
m
m
G
=
y
f (t )
å
h
m=
0
m
2
2
(5.14)
2
a
(a
b
g
m
m +
m )t
é
f (t ) =
e-+
1ù
ë
û
Where
(5.15)
6.Solution of Thermoelastic problem:
Let the suitable form of
c
c satisfying (4.1) be
¥
(6.1)
a
a
my
my
my
my
ù
ù
c
yé
Aea
+
Be cos(a
yé
Ce a
+
De sin(a
å
c =
m x) +
m x)
ë
û
ë
û
m=
0
Let the suitable form of
c
p satisfying (4.2) be
¥
X (ab
m x ) Y ( m y ) Z (g
mz)
c
a
Eå
y
f (t )
p =
2
2
(a
b
m=
m +
m )h
m
(6.2)
¥
X (a
a
y
a
y
a
y
a
y
m x )Y ( bg
m y )Z ( m z )
c
=
yé
Ae m +
Be m ù
cos(a
yé
Ce m +
De m ù
sin(a
a
E
y
f (t )
å
m x) +
m x) +
2
2
ë
û
ë
û
(a
+
b
)h
m=
0
m
m
(6.3)
m
From (4.3) and (6.3) we obtain
¥
2 a
y
a
y
a
y
a
y ù
y
a
y
a
y
a
y ù
é
é
s
y(
Aaa
+
B m2e+
2(
Aa
Ba
cosaaa
x+
y(
C m2ea
+
D m2e+
2(
Ca
Da
sina
x)
)
)
)
xx =
å
me
me
me
me
me
ë
ûm ë
ûm
m=
0
m
m
m
m
m
m
m
m
2
a
Eb
m X (ab
m x )Y ( m y ) Z ( g
mz)
y
f (t )
2
2
(a
+
b
)
h
m
m
m
(6.4)
From (4.4)and (6.3) we obtain
2
¥
a
Ea
X (a
x)Y (b
y)Z (g
z)
a
2
a
2
m
m
m
m
my
my
my
my
s
y Aea
+
Bea
+
y Cea
+
Dea
y
f (t) (6.5)
(
(
å
yy =
m cos(a
m x) )
m sin(a
m x) )
2
2
(a
b
m=
0
m +
m )h
m
(
)
(
)
From (4.5) and (6.3) we obtain
¥
(
)
(
)
(
)
(
)
y
y
a
y
y
y
a
y
2 a
2 2 a
2 é
ù
é
ù
s
=
y Aa
e yBa
ea
+
Aaa
ea
+
B m esin a
xy Ca
e yDa
ea
+
Ca
ea
+
Da
ecosa
xå
xy
m
m
m
m
m
ëm
ûm ë m
ûm
m=
0
m
m
m
m
m
m
m
m
2
¢m y ) Z (g
a
E[a
s in (a
H aa
m
m x) +
m cos(
m )]Y ( b
m z)
y
f (t )
2
2
(a
+
b
m
m )h
m
Applying condition (4.6) to (6.5) and (6.6) we obtain
Page No-4
(6.6)
2321-7871
th
Vol-1, Issue-44, 22 May 2014
Quasi-Static transient Thermal Stresses in a Robin's thin Rectangular plate with internal moving heat source
a
b
a
EY (b
b ) Z (g
z )e m
A = 2 m 2
y
f (t ) [
1b (a
H )]
m 2
2 b (a
+
b
)
h
m
m
m
(6.7)
m
(6.8)
a
b
a
EY (b
b ) Z (g
m z )e
B =
-2 m 2
y
f (t ) [
1+
b (a
H )]
m +
2
2b (a
+
b
)h
m
m
m
m
a
b
a
E H Y (b
m b )Z (g
m z )e
C = 2
y
f (t ) [
1b (a
H )]
m 2
2
2b a
+
b
m (a
m
m )h
m
(6.9)
b
a
E H Y (b
b )Z (g
z )e a
m
m
D =
-2
y
f (t ) [
1+
b (a
H )]
m +
2
2
2b a
+
b
)h
m (a
m
m
m
(6.10)
m
m
Substituting these value in above equations we obtain
¥
ab
E Y ( m b ) Z (g
m z )
s
y
f (t )
xx =
å
2
2
b 2 ( ab
+
)h
m =
0
m
m
m
2
ì
ü
é
a
sinha
b)ba
b)+
bHsinha
b))
+
2a
cosha
b) ba
b)+
bH cosha
b))
ù
cosa
m y(
m( y m cosha
m( y m( y m(
m( y m sinha
m( y m( y mx +
ë
û
ï
ï
í
ý
2
2
é
ù
aa
y
H
sinh
(
y
b
)
b
a
H
cosh
a
(
y
b
)
+
bH
sinh
a
(
y
b
)
+
2
H
cosh
a
(
y
b
)
b
a
H
sinh
a
(
y
b
)
+
bH
cosh
a
(
y
b
)
sin
a
x
ï
(
)
(
)
m
m
m
m
m
m
m
m
m
ë
ûm ï
î
þ
(6.11)
2
a
Eb
m X (a
m x )Y ( b
m y ) Z (g
m z )
y
f (t )
2
2
(a
+
b
m
m )h
m
(
)
2
ì
ü
baa
b) sinh a
b) bH sinh a
b )]
yaa
[
(6.12)
m co sh
m( y m(y m(y m c os( m x +
Y (b
m b)
ï
ï
a
E Z (g
ï
ï
2
mz )
2
s
y
f (t ) í
b
é
ù
b
a
H
cosh
a
(
y
b
)
H
sinh
a
(
y
b
)
bH
sinh
a
(
y
b
)
y
a
sin(
a
x
)
ý
yy =
å
(
)
2
2
m
m
m
m
m
m
ë
û
b
m=
0 (a
m +
m )h
m
ï
ï
2
ï
a
ï
m X (a
m x )Y ( b
m y)
î
þ
¥
a
EY (b
m b ) Z (g
m z)
s
y
f (t )
xy =
2
2
b 2 (a
+
b
m
m )h
m
2
ì
ü
é
ù
a
cosha
b) ba
b) +
bH cosha
b) )
+
a
sinha
b) ba
b) +
bH sinha
b) )
sina
m y(
m (y m sinh a
m(y m( y m(
m( y m cosha
m(y m( y mx +
ë
û
ï
ï
í
ý
2
2
é
ù
a
y
b
aa
H
sinh
(
y
b
)
H
cosh
a
(
y
b
)
bH
cosh
a
(
y
b
)
+
b
aa
H
cosh
(
y
b
)
+
bH
sinh
a
(
y
b
)
H
sinh
a
(
y
b
)
cos
a
x
ï
)
(
)
ï
m (
m
m
m
m
m
m
m
m
ë
ûm þ
î
aa
EX ¢
( m x )Y ¢
(b
m y ) Z (g
m z)
y
f (t )
2
2
(a
+
b
m
m )h
m
(6.13)
8. References:
1.D. T. Solanke, M. H. Durge Quasi-Static transient Thermal Stresses in a Dirichlet's thin
Solid
cylinder with internal moving heat source (IOSR-JM) Volume 10,Issue 2 ver.1 (Mar-Apr.2014 PP 5155.
2.D. T. Solanke, M. H. Durge Quasi-Static transient Thermal Stresses in a Neumann's thin
Solid cylinder with internal moving heat source (AJER) Volume-03, Issue-03, pp-75-79
3.D. T. Solanke, M. H. Durge Quasi-Static transient Thermal Stresses in a Robin's thin Solid
cylinder with internal moving heat source AJCEM 3 :2 March-April (2014) pp 8-11
4.D. T. Solanke, M. H. Durge Quasi-Static Transient Thermal Stresses in a Dirichlet's thin Hollow
Cylinder with Internal Moving Heat Source Internatioal journal of Physics and Mathematics
ISSN:2277-2111 2014 vol.4 (1) January- March pp 188-192
5.D. T. Solanke, M. H. Durge Quasi-Static Transient Thermal Stresses in a Neumann's thin Hollow
Cylinder with Internal Moving Heat Source
6.D. T. Solanke, M. H. Durge Quasi-Static Transient Thermal Stresses in a Robin's thin Hollow
Cylinder with Internal Moving Heat Source. International journal of Mathematics Trends and
Technology vol. 8 number 1- April 2104
7.M. Necati Ozisik `Heat conduction, Second Edition, A Wiley-Interscience Publication John Wiley
and Sons, inc. New-York.
8. N. Noda, R. B. Hetnarski, Y. Tanigawa, Thermal Stresses, second edition, 2002
Page No-5

GROUPE 1

German-Norwegian workshop on numerical modelling in

October

a. − b. c. 2 d.