Applied & Computational Mathematics
Bhave et al., J Appl Computat Math 2015, 4:1
http://dx.doi.org/10.4172/2168-9679.1000205
Research Article
Open Access
Three Dimensional Steady State Thermal Stress Analyses in a Clamped
Solid Sphere
Asha Bhave1*, Deshmukh KC1 and Kulkarni VS2
1
2
Research Scholar, Department of Mathematics, R.T.M. Nagpur University, Nagpur-440010, Maharashtra, India
Department of Mathematics, University of Mumbai, Mumbai-400098, Maharashtra India
Abstract
The present work deals with the three dimensional axisymmetric problem in a spherical coordinate system. A
clamped solid sphere is subjected to heat flux on its plane boundary surface. Heat is generated within the sphere.
The attempt made to analyze heat transfer and thermal stresses of solid sphere under steady temperature field.
The Legendre's transforms are used for the solution of governing heat conduction equation. The solution of Navier's
equation in terms of Goodier's thermoelastic displacement potential and the Boussinesq's harmonic function
for spherical co-ordinate system have been used to determine displacement and thermal stresses within solid.
The results for temperature change, displacement and stresses have been computed numerically and illustrated
graphically.
Keywords: Solid sphere; Heat transfer analysis; Steady-state; Thermal
stress analysis
Introduction
Thermoelasticity theories that admit finite speeds for thermal
signals have aroused much interest in the last three decades. Spherically
thermoelastic isotropy is a special case of orthogonal anisotropy in the
framework of a spherical polar coordinate system and such materials
have their physical properties transversely isotropic about the radius
vector. For such spherically isotropic materials, a few theoretical
treatments have been accomplished. The problem of thermoelastic field
has been discussed by Nowinski et al. [1], however the temperature
field is not considered. The problems of a solid sphere and a spherical
cavity in an infinite solid are treated [2] under the assumption that
the constant temperature is suddenly applied on the surface. Cheung
et al. [3] has analyzed the transient thermal stresses in a solid sphere
(Figure 1). Their assumptions include the isotropic homogeneous
material and a local surface heating which results in a non-uniform
temperature distribution [4] obtained the thermal stress distribution
and the residual stresses in a spherical body where the time dependent
temperature distribution is symmetrical with respect to the center of
the sphere. Tanigawa et al. [5] considered the transient thermal stress
problems of a spherical region under the condition that continuum
medium possesses spherically isotropic thermoelastic property (Figure
2). Chandrasekharaiah et al. used the linear theory of thermoelasticity
without energy dissipation to study waves emanating from the boundary
of a spherical cavity in a homogeneous and isotropic unbounded
thermoelastic body. Yapici has analyzed transient temperature and
thermally-induced stress distributions in a hollow steel sphere heated
by a moving uniform heat source applied on a certain zenithal segment
(the heated zenithal segment), of its outer surface (the processed
surface) under stagnant ambient conditions numerically (Figure 3).
In the present paper, we consider the three dimensional
axisymmetric problem in a spherical coordinate system. A solid
sphere is subjected to heat flux on its plane boundary surface. Heat is
generated within the sphere. The attempt made to analyze heat transfer
and thermal stresses of solid hemisphere under steady temperature
field. The Legendre's transforms are used for the solution of governing
heat conduction equation. The solution of Navier's equation in terms
Figure 2: The displacement f along radial direction.
*Corresponding author: Asha Bhave, Research Scholar, Department of
Mathematics, RTM Nagpur University, Nagpur-440010, Maharashtra, India, Tel:
0712 253 3452; E-mail: [email protected]
Received December 14, 2014; Accepted February 12, 2015; Published February
20, 2015
Citation: Bhave A, Deshmukh KC, Kulkarni VS (2015) Three Dimensional Steady
State Thermal Stress Analyses in a Clamped Solid Sphere. J Appl Computat Math
4: 205. doi:10.4172/2168-9679.1000205
Figure 1: The temperatureT along radial direction.
J Appl Computat Math
ISSN: 2168-9679 JACM, an open access journal
Copyright: © 2015 Bhave A, et al. This is an open-access article distributed under
the terms of the Creative Commons Attribution License, which permits unrestricted
use, distribution, and reproduction in any medium, provided the original author and
source are credited.
Volume 4 • Issue 1 • 1000205
Citation: Bhave A, Deshmukh KC, Kulkarni VS (2015) Three Dimensional Steady State Thermal Stress Analyses in a Clamped Solid Sphere. J Appl
Computat Math 4: 205. doi:10.4172/2168-9679.1000205
Page 2 of 6
∇ 2 Φ = kτ 2
(5)
∇ ϕ =0 (6)
∇ 2 Ψ =0 (7)
Where
∇ 2=
k
=
∂2 2 ∂ 1 ∂2
1
∂
+
+ 2 2 + 2 cot θ
2
∂r
r ∂r r ∂θ r
∂θ
(8)
1+ v
α 1− v
(9)
Where
Figure 3: The radial displacement r μ along radial direction.
K is restraint coefficient.
ν is poison’s ratio.
α is thermal conductivity.
The stress components in the spherical coordinate system are
Figure 4: The angular displacement q μ along radial direction.
of Goodier's thermoelastic displacement potential and the Boussinesq's
harmonic function for spherical co-ordinate system have been used to
determine displacement and thermal stresses within solid. The results
for temperature change, displacement and stresses have been computed
numerically and illustrated graphically (Figure 4).
Figure 5: The radial thermal stresses rrs along radial direction.
Formulation of the Problem
Following Noda et al. [6], Navier’s equation of thermoelasticity for
axisymmetric problems may be expressed as
(λ + 2µ)
∂e
2µ ∂ (ωφ sin θ)
∂τ
−
− β + Fr = 0 ∂r r sin θ
∂θ
∂r
(1)
(λ + 2µ)
1 ∂e 2µ ∂ (r ωφ )
1 ∂τ
+
−β
+ Fθ = 0 r ∂θ r
∂r
r ∂θ
(2)
Where
Figure 6: The angular thermal stresses qq s along radial direction.
1  ∂ (r µ θ ) ∂µ r 
=
ωφ
−
2r  ∂r
∂θ 
The solution of the Navier’s equation without the body force
for axisymmetric problems in the spherical coordinate system can
be expressed [7-9], for example, by the Goodier’s thermoelastic
displacement potential f and the Boussinesq harmonic functions ϕ and
Ψ (Figure 5).
µ r=
∂φ ∂ϕ
∂Ψ
+
+ r cos θ
− (3 − 4 v)Ψ cos θ ∂r ∂r
∂r
µ=
θ
1 ∂φ ∂ϕ
∂Ψ
+
+ cos θ
− (3 − 4 v)Ψ sin θ r ∂θ ∂θ
∂θ
(3)
Where the three functions must satisfy the equations (Figure 6)
J Appl Computat Math
ISSN: 2168-9679 JACM, an open access journal
(4)
Figure 7: The angular thermal stresses ff s along radial direction.
Volume 4 • Issue 1 • 1000205
Citation: Bhave A, Deshmukh KC, Kulkarni VS (2015) Three Dimensional Steady State Thermal Stress Analyses in a Clamped Solid Sphere. J Appl
Computat Math 4: 205. doi:10.4172/2168-9679.1000205
Page 3 of 6
represented in terms of three functions φ, ϕ and Ψ are (Figure 7)
 ∂ 2Φ ∂ 2ϕ

∂2Ψ
∂Ψ
1
∂Ψ
=
σrr 2G  2 + 2 + r cos θ 2 − 2(1 − v) cos θ
+ 2v sin θ
− Kτ 
r
∂r
∂r
∂r
∂θ
 ∂r

(10)
 1 ∂φ 1 ∂ 2 Φ 1 ∂ϕ 1 ∂ 2 ϕ

∂Ψ 1
∂2Ψ
∂Ψ
1
=
σθθ 2G 
+ 2 2 +
+
+ (1 − 2 v) cos θ
+ cos θ
+ 2(1 − v) sin θ
− Kτ 
∂r r
∂θ
∂θ
r ∂r r 2 ∂θ2
r
 r ∂r r ∂r

(11)
 1 ∂φ

1 ∂ 2 Φ 1 ∂ϕ
1 ∂ 2ϕ
∂Ψ
1 ∂Ψ
=
σθθ 2G 
+ cot θ 2 2 +
+ cot θ 2 2 + (1 − 2 v) cos θ
+ (cos θ cot θ + 2v sin θ)
− Kτ 
r ∂r
r ∂r
r ∂θ
∂r
r ∂θ
 r ∂r

(12)
∂2 φ
∂ ϕ
∂Ψ
∂2Ψ
( )+
( ) + (1 − 2 v) sin θ
+ cos θ
∂r∂θ r ∂r∂θ r
∂r
∂r∂θ
1
∂Ψ
− 2(1 − v) cos θ
]
r
∂θ
(13)
=
σrθ 2G[
Figure 10: The displacement f along radial direction.
The mechanical boundary conditions on the traction free surface
r=b are (Figure 8)
σrr = 0, σrθ = 0 on r = b (14)
The equations (1) to (14) constitute thermoelastic formulation of
the problem for spherical coordinate system.
Governing Equation of Heat Conduction
Consider a sphere of radius b defined by 0 ≤ r ≤ b, 0 ≤ μ ≤1,0 ≤ f ≤ 2p
which is subjected to arbitrary heat flux on its plane boundary surface
r=b. Heat is generated within the sphere. Assume the boundary surface
of a sphere is free from traction. Under these prescribed conditions, the
thermal stresses are required to be determined.
A steady state temperature of a sphere satisfies the heat conduction
equation (Figure 9),
∂T 
∂ 2T g 0
1 ∂  2 ∂T  1 ∂ 
1
+= 0in 0 ≤ r ≤ b, 0 ≤ µ ≤ 1, 0 ≤ φ ≤ 2π
(1 − µ 2 )  + 2
r
+
∂µ  r (1 − µ 2 ) ∂φ2 k
r 2 ∂r  ∂r  r 2 ∂µ 
(15)
Figure 11: The radial displacement r μ along radial direction.
∂T
= f (µ, φ) at r = b ∂r
(16)
where =
T T(r, µ, φ) at=
r b (17)
subject to
Solution
The solution for temperature change
The Poisson’s equation (15) can be reduced to Laplace equation by
defining a new dependent variable θ(r, µ, φ) as
g r2
T(r, µ, φ) =θ(r, µ, φ) − 0
(18)
k 6
Substituting equation (18) into equation (15) and (16), one obtains
the Laplace equation with one inhomogeneous boundary condition
(Figure 10)
Figure 8: The resultant rq s along radial direction.
∂θ 
∂ 2θ
1 ∂  2 ∂θ  1 ∂ 
1
=0
(1 − µ 2 )  + 2
r
+ 2

2
2
∂µ  r (1 − µ ) ∂φ2
r ∂r  ∂r  r ∂µ 
∂θ g r
With = 0 + f (µ, φ) at=
r b ∂r 3k
(19)
(20)
The partial derivative with respect to the φ variable can be removed
from the differential equation with Fourier transform. Since the range
of independent variable φ is from 0 to 2 π and the temperature function
is cyclic with period of 2 π, we define the Fourier transform and the
corresponding inversion formula with respect to the φ variable of the
temperature function (r, μ, φ ) as (Figure 11)
θ(r,
=
µ, m)
2π
∫ cos m(φ − φ′).θ(r, µ, φ′) d φ′ (21)
φ=0
Figure 9: The temperatureT along radial direction.
J Appl Computat Math
ISSN: 2168-9679 JACM, an open access journal
θ(r, µ,=
φ)
1 ∞
∑ θ(r, µ, m) π m =0
(22)
Volume 4 • Issue 1 • 1000205
Citation: Bhave A, Deshmukh KC, Kulkarni VS (2015) Three Dimensional Steady State Thermal Stress Analyses in a Clamped Solid Sphere. J Appl
Computat Math 4: 205. doi:10.4172/2168-9679.1000205
Page 4 of 6
By applying Integral transform defined in the equation (21) to
governing heat conduction equation (15) along with the boundary
condition as defined in equation (16), one obtain
∂  2 ∂θ  ∂ 
∂θ  m 2
(1 − µ 2 )  −
θ =0
r
+

∂r  ∂r  ∂µ 
∂µ  1 − µ 2
∂θ
∂r
=
With
∂θ
=
∂r
2π
in 0 ≤ r ≤ b, −1 ≤ µ ≤ 1
∂θ
φ′ at r
∫ cos m(φ − φ′). ∂r d=
b
(23)
(24)
φ=0
2π
 g0r

∫ cos m(φ − φ′).  3k + f (µ, φ′) dφ′ (25)
∂θ
=
f (µ, m) at r =
b ∂r
(26)
φ=0
Figure 13: The radial thermal stresses rrs along radial direction.
The partial derivative with respect to space variable μ can be
removed by means of Legendre transform. Since the range of the space
variable is -1≤ μ ≤1, define Legendre transform and the corresponding
inversion formula for the temperature function T (r, μ, m) with respect
to μ variable as (Figure 12).
θ(r,=
n, m)
1
∫
µ′ =−1
=
, m)
And θ(r, µ
K mn (µ′) θ(r, µ′, m) du ′ ∞
∑K
n
m
n
(µ) θ(µ, n, m) where the
kernel K mn (µ)
=
2n + 1 (n − m)! m
=
p n (µ) n 0,1, 2,...
2 (n + m)!
(27)
(28)
(29)
Figure 14: The angular thermal stresses qq s along radial direction.
By applying Legendre transform defined in the equation (27) to
n
governing heat conduction equation (23) along with the boundary
f (n, m) n b  r 
r
=
θ
=
  f (n, m) condition as defined in equation (26), one obtain (Figure 13).
nb n −1
nb
d  2 dθ 
1) θ 0in 0 ≤ r ≤ b r
 − n(n +=
dr  ∂r 
(30)
using boundary condition prescribed in equation (31), one obtain
C1
f (n, m)
nb n −1
hence equation (32) gives
∞
∑K
n
(31)
=
θ(r, µ, φ)
n
m
n
b r 
(µ)   f (n, m) nb
n
1
b r 
∑ ∑ K mn (µ) n  b  f (n, m) π m =0 n
∞
∞
(35)
(36)
n
on solving above equation (30), one obtain
θ =C1r n hence equation (28) gives
m)
θ(r, µ,=
1
dθ
m
at r b
With = ∫ K n (µ′) f(µ′, m) du ′ =
dr
′
µ
=−
1
dθ
= f=
(n, m) at r b dr
(34)
=
T(r, µ, φ)
(32)
g r2
1 ∞ ∞ m
b r 
K n (µ)   f (n, m) − 0
∑
∑
π m =0 n
nb
k 6
(37)
using equation (37), under steady state temperature condition
t=T
(33)
=
τ
∞
∞
∑∑A
m =0 n
mn
Where A mn =
=
f (n, m)
r n p nm (µ) −
g0 r 2
k 6
(38)
1 2n + 1 (n − m)! 1
f (n, m)
π
2 (n + m)! nb n −1
2n + 1 (n − m)!
 m + 1  m − (2 n + 1) 
cos mwφ.T.n!.π 


2 (n + m)!
 m + 2  m − 2n 
(39)
The goodiers thermoelastic Displacement Potential Φ
The Goodier’s thermoelastic displacement Φ potential satisfying
the equation (Figure 14).
Figure 12: The angular displacement q μ along radial direction.
J Appl Computat Math
ISSN: 2168-9679 JACM, an open access journal
∇ 2 Φ = kτ (40)
subject to ∂Φ = Φ = 0 at r = b ∂r
(41)
Volume 4 • Issue 1 • 1000205
Citation: Bhave A, Deshmukh KC, Kulkarni VS (2015) Three Dimensional Steady State Thermal Stress Analyses in a Clamped Solid Sphere. J Appl
Computat Math 4: 205. doi:10.4172/2168-9679.1000205
Page 5 of 6
using value of temperature difference from equation (38), one obtain
∞ ∞
g r2 
∂Φ 2 ∂Φ
=
+
k  ∑ ∑ A mn r n p nm (µ) − 0  k 6
∂r r ∂r
 m =0 n
(42)
on solving, one obtains the Goodier’s thermoelastic displacement
potential Φ as
∞ ∞

g 1
1
k  ∑ ∑ A mn
(r − 1) n + 2 Pnm (µ) − 0
(r − 1) 4 
(n
3)(n
2)
k
120
+
+
 m =0 n

Φ
(43)
The first component of displacement and thermal stresses
Figure 15: The angular thermal stresses ff s along radial direction.
The first component of displacement and thermal stress with
respect to Goodiers thermoelastic displacement potential function Ф
can be obtained as
∞ ∞ 1

g 1
∂Φ
A mn (r − 1) n +1 p nm (µ) − 0
(r − 1)3 
= k ∑ ∑
(n
3)
k
30
∂r
+
 m =0 n

µ=
r
=
µθ
∞ ∞
1 ∂Φ
1
1
= K ∑∑
A mn (r − 1) n +1
[(n + m) p nm−1 (µ) − n µ p mn (µ)]
r ∂θ
(n + 2)(1 − µ 2 )
m =1 n (n + 3)
(44)
(45)
 ∂ 2Φ

∞ ∞ 2

g 1
A mn (r − 1) n p mn (µ) + 0 (r − 1) 2 
=
σ rr 2G  2=
− kτ  2Gk  ∑ ∑
k 15
 m =0 n (n + 3)

 ∂r

 1 ∂Φ 1 ∂ 2 Φ

=
σθθ 2G 
+ 2
− Kτ 
2
 r ∂r r ∂θ

(46)
∂ Φ



2
1
1
= 2GK  =
− kτ  2Gk  ∑ ∑
A mn (r − 1) n  (− n − 2) p mn +
{(m + n)
2
(n + 2) (1 − µ 2 ) 2
 m =0 n (n + 3)

 ∂r

g0 4
m
m
2 2
2
m
(m + n − 1) Pn − 2 (µ) − (m + n)(2 n − 3)µ Pn −1 (µ) + (n µ − n − n µ ) Pn (µ)}] +
(r − 1) 2 ]
k 30
(47)
2
∞
Figure 16: The resultant rq along radial direction.
∞
∞ ∞ 1


g 4
1
1
A mn (r − 1) n  (− n − 2) +
cot θ[(n + m) p nm−1 (µ) − n µ p nm (µ)]) + 0
(r − 1) 2 
= 2GK  ∑ ∑
(n + 2) (1 − µ 2 )
k 30

 m =0 n (n + 3)

 ∂ 2  Φ 
σrθ =
2G 
 
 ∂r∂θ  r  
(48)
(49)
=
m 0=
n 0
=
Ψ
∞
1,n
− (n − 4 + 4 v) D1,n − 2 ](r − 1) n }Pnm (cos θ)
∞
∑ ∑ [(2 n + 1) D1, n − 1(r − 1)
=
m 0=
n 0
(50)
]Pnm (cos θ)
(51)
The second component of displacement and thermal stresses
The displacements and stresses due to displacement function φ and
Ѱ are (Figure 15)
∞
∞
∑ ∑ {[C
m 0=
n 0
=
1,n
− (n − 4 + 4 v) D1,n − 2 ]n(r − 1) n −1 + [(2 n + 1) D1,n −1 ](r − 1) n µ[n − (3 − 4v)]} p mn (µ)
1
∂Ψ 
−2(1 − v) (1 − µ 2 )

r
∂µ 
∞
∞
= 2G ∑ ∑ {[C1,n − (n − 4 + 4 v) D1,n − 2 ]n(n − 1)(r − 1) n − 2 [nPnm (µ) −
=
m 0=
n 0
 1 ∂ϕ

∂Ψ
µ θ = −(1 − µ 2 )1/2 
+µ
− (3 − 4 v)Ψ 
r
∂µ
∂µ


J Appl Computat Math
ISSN: 2168-9679 JACM, an open access journal
µ
[(m + n) Pnm−1 (µ) − n µ Pnm (µ)]
1 − µ2
+(1 − µ 2 )[(m + n)(m + n − 1) Pnm− 2 (µ) − (m + n)(2 n − 3)µ Pnm−1 (µ) + (n 2 µ 2 − n − n µ 2 ) Pnm (µ)]]
+
µ2
[(m + n) Pnm−1 (µ) − n µ Pnm (µ)]
1 − µ2
µ
[(m + n)(m + n − 1) Pnm− 2 (µ) − (m + n)(2 n − 3)µ Pnm−1 (µ) + (n 2µ 2 − n − n µ 2 ) Pnm (µ)]
(1 − µ 2 )
−2(1 − v)[(m + n) Pnm−1 (µ)]]}
(55)
1 ∂ϕ
1 ∂ϕ
∂Ψ
1 ∂Ψ
=
σφφ 2G{
−µ 2
+ (1 − 2 v)µ
− [2 v + (1 − 2 v)µ 2 ]
}
r ∂r
r ∂µ
∂r
r ∂µ
∞
∞
= 2G ∑ ∑ {[C1,n − (n − 4 + 4 v) D1,n − 2 ](r − 1) n − 2 [nPnm (µ) −
+[(2 n + 1) D1,n −1 (r − 1) n −1[n µ Pnm (µ) −
(52)
(54)
2
2
 ∂ϕ
∂Ψ µ  ∂ψ
2 ∂ ϕ
2 ∂ Ψ
 −µ ∂µ + (1 − µ ) ∂µ 2  + (1 − 2 v)µ ∂r + r  −µ ∂µ + (1 − µ ) ∂µ 2 




=
m 0=
n 0
∂Ψ
 ∂Ψ

=
µ
+ µ r
− (3 − 4 v)Ψ 
∂r
 ∂r

=
=
m 0=
n 0
(r − 1) n −1[n µ(n − 1) Pnm (µ) − 2 n µ(1 − v) Pnm (µ) − 2 v[(m + n) Pnm−1 (µ) − n µ Pnm (µ)]]}
+[(2 n + 1) D1,n −1 (r − 1) n −1[(1 − 2 v) n µ Pnm (µ) −
n
(53)
∞
1 ∂ϕ 1
=
σθθ 2G 
+ 2
 r ∂r r
The solution of Laplace equation satisfying conditions given in
equation (6) and (7) are
∞
(r − 1) n
[(m + n) Pnm−1 (µ) − n µ Pnm (µ) − (3 − 4 v)(1 − µ 2 ) Pnm (µ)]}
1 − µ2
 ∂ 2ϕ
∂2Ψ
∂Ψ
∂Ψ 
1
=
σrr 2G  2 + µr 2 − 2(1 − v)µ
− 2v (1 − µ 2 )
∂r
∂r
∂µ 
r
 ∂r
∞
The solution of laplace equation
∞
1
[(m + n) Pnm−1 (µ) − n µ Pnm (µ)]
1 − µ2
= 2G ∑ ∑ {[C1,n − (n − 4 + 4 v) D1,n − 2 ]n(n − 1)(r − 1) n − 2 Pnm (µ) + [(2 n + 1) D1,n −1 ]
∞ ∞
n +1
1 
2GK  ∑ ∑
A mn (r − 1) n
[(n + m) Pnm−1 (µ) − n µ Pnm (µ)]
1 − µ 2 
 m =0 n (n + 3)(n + 2)
∑ ∑{[C
∞
=
m 0=
n 1
+[(2 n + 1) D1,n −1 ]
1 ∂Φ
 1 ∂Φ

=
σφφ 2G 
+ cot θ 2
− Kτ 
r ∂r
 r ∂r

=
ϕ
∞
= −(1 − µ 2 )1/2 ∑ ∑ {[C1,n − (n − 4 + 4 v) D1,n − 2 ](r − 1) n −1
+
µ
[(m + n) Pnm−1 (µ) − n µ Pnm (µ)]]
1 − µ2
2v
(m + n) Pnm (µ)]
1 − µ2
2nv
µ2
µPnm (µ) −
(1 − 2 v)[(m + n) Pnm−1 (µ) − n µ Pnm (µ)]]}
2
1− µ
1− µ
(56)
Volume 4 • Issue 1 • 1000205
Citation: Bhave A, Deshmukh KC, Kulkarni VS (2015) Three Dimensional Steady State Thermal Stress Analyses in a Clamped Solid Sphere. J Appl
Computat Math 4: 205. doi:10.4172/2168-9679.1000205
Page 6 of 6
=
σrθ 2G(1 − µ 2 )1/2 [−
∞
∂ ϕ
∂Ψ
∂2Ψ
µ ∂Ψ
−µ
+ 2(1 − v)
]
  + (1 − 2 v)
∂r∂µ  r 
∂r
∂r∂µ
r ∂µ
∞
2G(1 − µ 2 )1/2 ∑ ∑ {[C1,n − (n − 4 + 4 v) D1,n − 2 ](r − 1) n − 2
m 0=
n 0
=
Concluding Remarks
µ
1 − µ2
m
n
[(m + n) P (µ) − n µ P (µ)]]} (57)
The displacement and thermal stresses
The displacements and thermal stresses defined in equations (3),
(4) and (10) to (14) can be obtain by adding (44) to (49) and (52) to (57)
respectively as (Figure 16)
µr = µr + µr
µθ = µθ + µθ
In this paper a clamped solid sphere is considered and determined
the expressions for temperature, displacement and thermal stresses
at different values of μ and f. As a special case mathematical model is
constructed for iron (pure) solid sphere subjected to heat flux at the
spherical boundary with the material properties specified as above.
In order to study the characteristic of temperature variation,
displacement and stresses within solid sphere we have performed
numerical calculation for special case and illustrated these variations
graphically along radial direction.
Due to internal heat generation and heat flux applied on spherical
boundary surface, heat flow can be observed towards the outer
spherical boundary of the sphere from center of sphere. At the same
Time radial and angular displacements seen at the center. It means
solid displacement and heat flow are opposite to each other.
σrr = σrr + σrr
σθθ = σθθ + σθθ
σrθ = σrθ + σrθ
The unknowns used in equations (49) and (50), can be determined
by solving equilibrium equation (14) as
C1,n = 0
gk
2KA mn b 2
b−n +4
k
+ 0
−
D1,n − 2 =
n(n − 1)(n + 3)(n − 4 + 4 v) 15k (n − 1)(n − 4 + 4 v) Pnm (µ) 2n + 1
 2
 m + n 
 n µ − 3nµ + 4nµv − 2v  n    A b  2
(n + 1)

   mn 

+

1
 n 2 − 2vn 2
 2mv + 2nv    (n + 3)  n
(n + 2)(1 − µ 2 ) 2

 
 n + m − n 2µ − µ + 2µv − 
n2



D1,n −1
1.35
(20.39)(10−6 ),
0.65
G=0.5
(n − 1)
[(m + n) Pnm−1 (µ)
1 − µ2
−nµPnm (µ)] + [(2 n + 1) D1,n −1 ](r − 1) n −1[(1 − 2 v) nPnm (µ) + [2(1 − v) − n]
m
n −1
K=


−K
1
(n + 1)
 A mn b  2
+

1
2n + 1  n 2 − 2vn 2
 2mv + 2nv    (n + 3)  n
+
− µ2 ) 2
(n
2)(1
−
µ
+
µ
−
2
v


 

2
n2


n + m − n µ
Numerical Calculations
The tensile thermal stresses can be observed around center of
sphere where as in the rotation of sphere tensile as well compressive
thermal stresses are seen. All stresses are negligible towards the outer
spherical boundary of the sphere.
It may be concluded that, clamped spherical boundary restrict
expansion at outer surface hence displacement and thermal stresses
developed around center which may results in vertical expansion of
solid sphere.
References

− n +3
 + g0 b
}
 15k nPnm (µ)


− n +3
 + g0 b
}
 15k nPnm (µ)

1. Nowinski J (1959) Nte on a thermoelastic problem for a transversely isotropic
hollow sphere embedded in an elastic medium. J Appl Mech 26: 649-650.
2. Boley B, Weiner J (1960) Theory of thermal stresses. Wiley, New York.
3. Cheung JB, Chen TS, Thirumalai K (2012) Transient thermal stresses in a
sphere by local heating. J Appl Mech 41: 930-934.
4. Hetnarski R, Reza E (2009) Thermal stresses-advanced theory and
applications. In: solid mechanics and its applications. 158.
5. Tanigawa Y, Takeuti Y, Ueshima Y (1984) Thermal stresses of solid and hollow
spheres with spherically isotropic thermoelastic properties. Ingenieur-archive
54: 259-267.
Special case
Setting f (=
µ, φ) T0 (1 − µ 2 )δ(φ − Ψ (φ′))
Heat transfer coefficient h=10
Strengthening of heat flux 0 T=100°C
Material properties
Thermal diffusivity a 20.34×10 (m s ),
-6
2 -1
Thermal Conductivity k=72.7(W/mK)
Lame constant μ=0.5,
6. Noda N, Hetnarski R, Tanigawa Y (2003) In: Taylor, Francis (eds.) Thermal
stresses (2ndedn).
7. Yapici H, Ozisik G, Genc S (2010) Non-uniform temperature gradients and
thermal stresses produced by a moving heat flux applied on a hollow sphere.
Indian academy of sciences. Sadhana 35: 195-213.
8. Chandrasekharaiah DS, Srinath KS (2000) Thermoelastic waves without
energy dissipation in an unbounded body with a spherical cavity. International
Journal of Mathematics and Mathematical Sciences 23: 555-562.
9. Ozisik MN (1968) Boundary value problems of heat conduction, international
textbook company, Scranton, Pennsylvania.
Poisson ration=0.35,
Heat transfer coefficient h=10
Strengthening of heat flux 0 T=100°C
Dimensions
Radius of Sphere b=1m,
J Appl Computat Math
ISSN: 2168-9679 JACM, an open access journal
Citation: Bhave A, Deshmukh KC, Kulkarni VS (2015) Three Dimensional
Steady State Thermal Stress Analyses in a Clamped Solid Sphere. J Appl
Computat Math 4: 205. doi:10.4172/2168-9679.1000205
Volume 4 • Issue 1 • 1000205