Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 4, pp. 1369-1380, Warsaw 2017
DOI: 10.15632/jtam-pl.55.4.1369

THERMOELASTIC STRAIN AND STRESS FIELDS DUE TO A SPHERICAL

INCLUSION IN AN ELASTIC HALF-SPACE

Kuldip Singh, Renu Muwal

Guru Jambheshwar University of Science and Technology, Department of Mathematics, Hisar, Haryana, India

e-mail: renumuwal66@gmail.com

In this paper, closed form analytical expressions for thermoelastic strain and stress compo-
nents due to a spherical inclusion in an elastic half-space are obtained. These expressions
are derived in the context of steady-state uncoupled thermoelasticity using thermoelastic
displacement potential functions. The thermal strain and stress fields are generated due
to differences in the coefficients of linear thermal expansion between a subregion and the
surrounding material. The strain and stress components for exterior points of the spheri-
cal inclusion are same as those of the center of dilatation. Variations of strain and stress
components for exterior and interior points of the spherical inclusion are shown graphically.

Keywords: uncoupled thermoelasticity, strain and stress fields, potential functions, spherical
inclusion

1. Introduction

Thermoelasticity is an extension of theory of elasticity to include thermal effects. Theory of
thermoelasticity is concerned with the interaction between the thermal field and elastic bodies.
Comprehensive treatises on the topics of thermoelasticity have been covered in the classical texts
(Nowinski, 1978; Truesdell, 1984; Nowacki, 1986; Boley and Weiner, 1997; Boresi et al., 2011;
etc.). The study of thermoelasticity has begunwith Duhamel (1837) andNeumann (1885), who
postulated equations of linear thermoelasticity for isotropic bodies. Goodier (1937) studied the
static problem of uncoupled thermoelasticity by employing the method of superposition using
displacement potential functions.With the help ofGoodier’smethod,Mindlin andCheng (1950)
obtained the thermal stress in a semi-infinite elastic solid with the traction free surface for the
centre of dilatation using Galerkin’s vector stress function. The result was then applied to the
case of a spherical inclusion in a semi-infinite medium. Nowacki (1957) considered the problem
of an instantaneous source of heat in an infinite isotropic elastic space and determined the state
of stress using the potential of thermoelastic strain. Also, stress components in a semi-infinite
elastic space due to an instantaneous source of heatwas obtained using themethod ofGalerkin’s
displacement function and the Fourier integral under stress free surface boundary conditions.
Sen (1957) presented a direct method for solving problems of circular holes in an isotropic
elastic plate having prescribed displacements on the edge. The problem of deformation in an
infinite isotropic plate due to the centre of dilatation in the form of a nucleus of thermoelastic
strain at a finite distance from the holewas also discussed. Sharma (1957) obtained deformation
and stress fields due to a nucleus of thermoelastic strain in an infinite solid having a spherical
cavity using interior and exterior spherical harmonics. Another problem of a solid sphere at zero
temperature having a heated nucleus inside was also discussed. Biharmonic solutions to steady
state thermoelastic problems in three dimensions were derived by Nowinski (1961). Nowinski
(1963) proved amean value theorem for an arbitrary steady-state thermoelastic problem for an



1370 K. Singh, R. Muwal

isotropic elastic solid sphere. The stress components at the center of the sphere were expressed
in terms of temperature on the surface of the sphere.

Guell andDundurs (1967) obtained elastic fields due to a subregion between two joined half-
-spaces by differentiation from the corresponding fields for the whole space using the solutions
for the center of dilatation. The Papkovich-Neuber displacement potentials were used when the
interface between two joined half-spaces was a smooth interface and perfectly bonded. Those
resultswere then applied to a spherical inclusion.The case of the force on the center of dilatation
was also discussed. Hu (1989) considered the problem of a parallelepipedic thermal inclusion in
a three-dimensional half-space and derived analytical solution using Goodier’s method of the
nucleus of thermal strain or the center of dilatation. Wang and Huang (1991) studied some
thermoelastic problems in the half-space by using Goodier’s thermoelastic potential functions
and general solutions of elasticity based on theBoussinesq solutions (consisting of different har-
monic functions). Yu et al. (1992) and Yu and Sanday (1992), respectively, derived analytical
solutions for thermoelastic displacements and stresses due to an inclusion in adissimilarmedium
consisting of two joined semi-infinite elastic solids and for the centre of dilatation in a plate.
The stress-deformation state of an elastic half-space due to an inhomogeneous spheroidal ther-
mal inclusion in the form of a prolate or oblate ellipsoid of revolution under stress-free surface
boundary conditions was discussed by Kolesov et al. (1992) using thermoelastic displacement
potential functions and theTeredzavemethod.Yu et al. (2002) calculated thermal stresses indu-
ced by an ellipsoidal inclusion with uniform dilatational eigenstrains in one of the two perfectly
bonded semi-infinite solids using the method of dilatation centres and potential functions. Da-
vies (2003) derived the elastic field due to a non-uniform temperature or a coherentlymisfitting
inclusion in a semi-infinite region from the corresponding field in an infinite region.

Using Green’s function method and series expansion techniques, closed form solutions for
displacement and stress fields due to a hemispherical inclusion with a uniform eigenstrain in
a semi-infinite isotropic elastic medium were obtained by Linzhi (2003). Liu and Wang (2005)
obtained the elastic field caused by eigenstrains in a half-space using two types of numerical
techniques – discrete correlation and fast Fourier transform as well as discrete convolution and
fast Fourier transform. To obtain that field, analytical solutions for influence coefficients were
expressed in terms of derivatives of four key integrals. Zhou et al. (2009) introduced a fast me-
thod for solving the problem of three-dimensional arbitrarily shaped inclusions in an isotropic
half-space using a combination of 2D and 3D fast Fourier transform. A fast method for solving
contact problems for a half-space containing multiple inhomogeneities such as voids, cavities,
inclusions and fibers was presented by Leroux et al. (2010). The displacement and stress fields
due to eigenstrains of all spherical inclusion were obtained using Eshelby’s equivalent inclusion
method along with 2D and 3D fast Fourier transforms. Itou (2014) derived basic equations for
thermoelastic plane stress conditions and thermoelastic plane strain conditions. Two problems
were solvedusing thermoelastic displacementpotential functions: (i) axisymmetric thermal stres-
ses for a hollow thin disk, (ii) thermal stresses for an infinite thin platewith a circular hole under
uniform heat flow. Kumagai et al. (2014) presented different representations of a seismic sphe-
rical source. Those different representations consisted of a spherical source (S1), spherical crack
source (S2), isotropic source of threemutually perpendicular vector dipoles (S3) or threemutu-
ally perpendicular tensile cracks (S4), Eshelby’s spherical source with stress-free strain (S5) and
Eshelby’s spherical source with strain-free stress (S6). Static displacement fields due to those
sources in an infinite medium and in a half-space were also derived.

In the present paper,we obtain thermoelastic strain and stress fields due to a spherical inclu-
sion in an elastic half-space in the context of steady-state uncoupled thermoelasticity. Following
the method opted by Davies (2003), we derive thermoelastic strain and stress components for
an infinite region fromwhich corresponding fields can be derived in the semi-infinite region. The
expressions for these fields are obtained for the axisymmetric problem in cylindrical coordinates



Thermoelastic strain and stress fields due to a spherical inclusion... 1371

using thermoelastic displacement potential functions. These results are in good agreement with
the results found byMindlin and Cheng (1950).

2. Theory

In the linear theory of thermoelasticity, the total strain can bewritten as the sumofmechanical
and thermal strains (Goodier, 1937; and Nowinski, 1978)

eij = e
(M)
ij +e

(T)
ij (2.1)

inwhich for an isotropicmaterial, the thermal strain takes the form e
(T)
ij =αTδij, whereα is the

coefficient of linear thermal expansion, T is temperature difference, δij is Kronecker delta. Then
in the context of uncoupled linear thermoelasticity, the displacement-temperature equation of
equilibrium in the absence of body forces can be written as

∇2u+
1

1−2ν
∇(∇·u)=

2(1+ν)

1−2ν
α∇T (2.2)

where eij are components of the strain tensor; u is the displacement vector and ν is Poisson’s
ratio.
The uncoupled heat conduction equation for a steady state temperature field T with Q as

the heat supply and λ0 as the thermal conductivity can be written as

∇2T =
−Q

λ0
(2.3)

The solution to inhomogeneous equation (2.2) can be expressed as

u=uc+up (2.4)

whereuc is the complementary function satisfying the homogeneous equation of (2.2) andup re-
presents the particular solution to the displacement field generated by the temperature field T .
According to Goodier’s method (Timoshenko andGoodier, 1951), the displacement u(∞)(r)

for an infinite solid is given by u(∞) = ∇ϕ, where the potential function ϕ satisfies Poisson’s
equation

∇2ϕ=
1+ν

1−ν
αT(r) (2.5)

Then the function ϕ is obtained as

ϕ(r) =
−1

4π

1+ν

1−ν
α

∫

T(r′)

|r−r′|
d3(r′) (2.6)

where |r− r′| = |(x,y,z)− (ξ,η,ς)| =
√

(x− ξ)2+(y−η)2+(z− ς)2 is the distance between
the points (x,y,z) and (ξ,η,ς).
Then the displacement, strain and stress components in the cylindrical polar coordinates

(r,θ,z) are expressed in terms of the potential function ϕ as (Barber, 2002; Sadd, 2005)

ur =
∂ϕ

∂r
uθ =

1

r

∂ϕ

∂θ
uz =

∂ϕ

∂z
(2.7)

and

err =
∂ur
∂r

eθθ =
1

r

(

ur+
∂uθ
∂θ

)

erθ =
1

2

(1

r

∂ur
∂θ
+
∂uθ
∂r
−
uθ
r

)

ezz =
∂uz
∂z

eθz =
1

2

(∂uθ
∂z
+
1

r

∂uz
∂θ

)

ezr =
1

2

(∂ur
∂z
+
∂uz
∂r

)

(2.8)



1372 K. Singh, R. Muwal

and

1

2µ
σrr =

∂2ϕ

∂r2
−∇2ϕ

1

2µ
σθθ =

1

r

∂ϕ

∂r
+
1

r2
∂2ϕ

∂θ2
−∇2ϕ

1

2µ
σzz =

∂2ϕ

∂z2
−∇2ϕ

1

2µ
σrθ =

∂

∂r

(1

r

∂ϕ

∂θ

) 1

2µ
σθz =

1

r

∂2ϕ

∂θ∂z

1

2µ
σzr =

∂2ϕ

∂z∂r

(2.9)

where the Laplacian operator in the cylindrical polar coordinates (r,θ,z) is given by

∇2ϕ=
( ∂2

∂r2
+
1

r

∂

∂r
+
1

r2
∂2

∂θ2
+
∂2

∂z2

)

ϕ (2.10)

3. Formulation and solution of the problem

We consider an axisymmetric problem of a spherical inclusion in the upper half-space (z ­ 0)
having a different coefficient of thermal expansion to that of the half-space but the same elastic
constants as in Wang and Huang (1991). Due to this difference in the coefficients of thermal
expansion between the sub region and its surrounding material, say η0, a thermoelastic stress
field is generated. Let radius of the spherical inclusion be a and center of it be located at the
point r=0 and z= h, where h>a, as shown in Fig. 1. The surface z =0 is taken as traction
free surface. We take the axis of symmetry as the z axis, then for the axisymmetric problem
in the cylindrical coordinates (r,θ,z), all quantities are independent of θ. Then using Eq. (2.7),
the displacement vector u has the form u=(ur,0,uz). Therefore, the components of the strain
and stress tensor can be written in the following form, see equations (2.8) and (2.9)

err =
∂ur
∂r

eθθ =
u

r
ezz =

∂uz
∂z

ezr =
1

2

(∂ur
∂z
+
∂uz
∂r

)

erθ =0 eθz =0

(3.1)

and

1

2µ
σrr =

∂2ϕ

∂r2
−∇2ϕ

1

2µ
σθθ =

1

r

∂ϕ

∂r
−∇2ϕ

1

2µ
σzz =

∂2ϕ

∂z2
−∇2ϕ

1

2µ
σzr =

∂2ϕ

∂z∂r
σrθ =0 σθz =0

(3.2)

where

∇2ϕ=
( ∂2

∂r2
+
1

r

∂

∂r
+
∂2

∂z2

)

ϕ (3.3)

Fig. 1. A spherical inclusion in a thermoelastic half-space



Thermoelastic strain and stress fields due to a spherical inclusion... 1373

Then according toWang and Huang (1991), the thermoelastic potential function ϕ satisfies
the followingPoisson’s equations,when temperature of the semi-infinite region increases up toT0

∇2ϕ=







1+ν

1−ν
αT =

1+ν

1−ν
η0T0 for R1 ¬ a

0 for R1 >a
(3.4)

whereR21 = r
2+(z−h)2 is the distance of the point (r,z) from (0,h). Then, the functionϕ for

this problem is taken as (Wang and Huang 1991)

ϕ=















1

6
KR21 for R1 ¬ a

1

6
K
(

3a2−2a
3

R1

)

for R1 >a

(3.5)

where

K =
1+ν

1−ν
η0T0 (3.6)

Considering the boundary conditions, we assume that the boundary z=0 of the half-space
is traction free surface i.e.

σzz =σrz =0 at z=0 (3.7)

Now according to Davies (2003), the strain and stress components within the semi-infinite
region z ­ 0 with the traction free surface in terms of strain and stress components for an
infinite region for the axisymmetric problem in the rz-plane can be reduced to the form

err = e
(∞)
rr +(3−4ν)e

(∞)
rr +2z

∂

∂z
e(∞)rr eθθ = e

(∞)
θθ
+(3−4ν)e

(∞)
θθ
+2z

∂

∂z
e
(∞)
θθ

ezz = e
(∞)
zz − (1−4ν)e

(∞)
zz +2z

∂

∂z
e(∞)zz erz = e

(∞)
rz −e

(∞)
rz −2z

∂

∂z
e(∞)rz

erθ =0= eθz

(3.8)

and

σrr =σ
(∞)
rr +(3−4ν)σ

(∞)
rr −4νσ

(∞)
zz +2z

∂

∂z
σ(∞)rr

σθθ =σ
(∞)
θθ
+(3−4ν)σ

(∞)
θθ
−4νσ(∞)zz +2z

∂

∂z
σ
(∞)
θθ

σrθ =0=σθz

σzz =σ
(∞)
zz −σ

(∞)
zz +2z

∂

∂z
σ(∞)zz σrz =σ

(∞)
rz −σ

(∞)
rz −2z

∂

∂z
σ(∞)rz

(3.9)

Then the displacement, strain and stress fields in the infinite region and those at the image
point for exterior points (R1 >a) of the spherical inclusion (where∇

2ϕ=0) are obtained using
u(∞) =∇ϕ and equations (3.1)-(3.3)

u(∞)r =
∂ϕ

∂r
=
1

3
Ka3
( r

R31

)

u
(∞)
θ
=
1

r

∂ϕ

∂θ
=0 u(∞)z =

∂ϕ

∂z
=
1

3
Ka3
(z−h

R31

)

(3.10)

e(∞)rr =
∂2ϕ

∂r2
=
1

3
Ka3
( 1

R31
−
3r2

R51

)

e
(∞)
θθ
=
ur
r
=
1

3
Ka3
( 1

R31

)

e
(∞)
rθ
= e
(∞)
θz
=0

e(∞)zz =
∂2ϕ

∂z2
=
1

3
Ka3
( 1

R31
−
3(z−h)2

R51

)

e(∞)rz = e
(∞)
zr =

∂2ϕ

∂r∂z
=−Ka3

r(z−h)

R51

(3.11)



1374 K. Singh, R. Muwal

1

2µ
σ(∞)rr =

1

3
Ka3
( 1

R31
−
3r2

R51

) 1

2µ
σ
(∞)
θθ
=
1

3
Ka3
( 1

R31

)

σ
(∞)
rθ
=0=σ

(∞)
θz

1

2µ
σ(∞)zz =

1

3
Ka3
( 1

R31
−
3(z−h)2

R51

) 1

2µ
σ(∞)rz =−Ka

3r(z−h)

R51

(3.12)

u(∞)r =
1

3
Ka3
( r

R32

)

u
(∞)
θ
=0 u(∞)z =

−1

3
Ka3
(z+h

R32

)

(3.13)

e(∞)rr =
1

3
Ka3
( 1

R32
−
3r2

R52

)

e
(∞)
θθ
=
1

3
Ka3
( 1

R32

)

e
(∞)
rθ
= e
(∞)
θz
=0

e(∞)zz =
1

3
Ka3
( 1

R32
−
3(z+h)2

R52

)

e(∞)rz =Ka
3r(z+h)

R52

(3.14)

1

2µ
σ(∞)rr =

1

3
Ka3
( 1

R32
−
3r2

R52

) 1

2µ
σ
(∞)
θθ
=
1

3
Ka3
( 1

R32

)

σ
(∞)
rθ
=0=σ

(∞)
θz

1

2µ
σ(∞)zz =

1

3
Ka3
( 1

R32
−
3(z+h)2

R52

) 1

2µ
σ(∞)rz =Ka

3r(z+h)

R52

(3.15)

where (0,−h) is the image of point (0,h) and R22 = r
2 +(z+h)2 is the distance of the point

(r,z) from (0,−h).
Substituting equations (3.11), (3.14) and (3.12), (3.15) into (3.8) and (3.9), respectively, the

strain and stress components in the thermoelastic half-space for exterior points (R1 >a) of the
spherical inclusion can be expressed as

err =
1

3
Ka3
[ 1

R31
+
3−4ν

R32
−
6z(z+h)

R52
−3r2

( 1

R51
+
3−4ν

R52
−
10z(z+h)

R72

)]

eθθ =
1

3
Ka3
( 1

R31
+
3−4ν

R32
−
6z(z+h)

R52

)

ezz =
1

3
Ka3
( 1

R31
−
1−4ν

R32
−
18z(z+h)

R52
−
3(z−h)2

R51
+
3(1−4ν)(z+h)2

R52
+
30z(z+h)3

R72

)

erz =−Ka
3r
(z−h

R51
+
3z+h

R52
−
10z(z+h)2

R72

)

(3.16)

and

σrr =2µ
Ka3

3

[ 1

R31
+
3−8ν

R32
−
6z(z+h)

R52
+
12ν(z+h)2

R52
−3r2

( 1

R51
+
3−4ν

R52
−
10z(z+h)

R72

)]

σθθ =2µ
Ka3

3

( 1

R31
+
3−8ν

R32
−
6z(z+h)

R52
+
12ν(z+h)2

R52

)

σzz =2µ
Ka3

3

( 1

R31
−
1

R32
−
18z(z+h)

R52
−
3(z−h)2

R51
+
3(z+h)2

R52
+
30z(z+h)3

R72

)

σrz =−2µKa
3r
(z−h

R51
+
3z+h

R52
−
10z(z+h)2

R72

)

(3.17)

Further, from equation (3.17), we can see that σzz = σrz =0 at the boundary z =0 of the
half-space, which is in accordance with the boundary conditions as in equation (3.7).
Also, for the interior points (R1 ¬ a) of the spherical inclusion, as in Mindlin and Cheng

(1950)

uint =uext+
a3KR1
3

( 1

a3
−
1

R31

)

(3.18)



Thermoelastic strain and stress fields due to a spherical inclusion... 1375

Using this, the following relations between the strain components of the exterior points
(R1 >a) and the interior points (R1 ¬ a) of the spherical inclusion are given by

eintrr = e
ext
rr +

Ka3

3

( 1

a3
−
1

R31

)

+Ka3
r2

R51
eintθθ = e

ext
θθ +

Ka3

3

( 1

a3
−
1

R31

)

eintzz = e
ext
zz +

Ka3

3

( 1

a3
−
1

R31

)

+Ka3
(z−h)2

R51
eintrz = e

ext
rz +Ka

3r(z−h)

R51

(3.19)

Also, the relations between the stress components of the exterior points (R1 > a) and the
interior points (R1 ¬ a) of the spherical inclusion are given below, seeGuell andDundurs (1967)

σintrr =σ
ext
rr −2µK

( 2

a3
+
1

R31
−
3r2

R51

)

σintθθ =σ
ext
θθ −2µK

( 2

a3
+
1

R31

)

σintzz =σ
ext
zz −2µK

( 2

a3
+
1

R31
−
3(z−h)2

R51

)

σintrz =σ
ext
rz +6µK

r(z−h)

R51

(3.20)

Substituting equations (3.16) and (3.17) into (3.19) and (3.20), respectively, the strain and
stress components in the thermoelastic half-space for the interior points (R1 ¬ a) of the spherical
inclusion can be expressed as

err =
1

3
Ka3
[ 1

a3
+
3−4ν

R32
−
6z(z+h)

R52
−3r2

(3−4ν

R52
−
10z(z+h)

R72

)]

eθθ =
1

3
Ka3
( 1

a3
+
3−4ν

R32
−
6z(z+h)

R52

)

ezz =
1

3
Ka3
( 1

a3
−
1−4ν

R32
−
18z(z+h)

R52
+
3(1−4ν)(z+h)2

R52
+
30z(z+h)3

R72

)

erz =−Ka
3r
(3z+h

R52
−
10z(z+h)2

R72

)

(3.21)

and

σrr =2µ
Ka3

3

[−2

a3
+
3−8ν

R32
−
6z(z+h)

R52
+
12ν(z+h)2

R52
−3r2

(3−4ν

R52
−
10z(z+h)

R72

)]

σθθ =2µ
Ka3

3

(−2

a3
+
3−8ν

R32
−
6z(z+h)

R52
+
12ν(z+h)2

R52

)

σzz =2µ
Ka3

3

(−2

a3
−
1

R32
−
18z(z+h)

R52
+
3(z+h)2

R52
+
30z(z+h)3

R72

)

σrz =−2µKa
3r
(3z+h

R52
−
10z(z+h)2

R72

)

(3.22)

The results obtained above are in good agreement to those of Mindlin and Cheng (1950)
for the spherical inclusion in the interior of a thermoelastic semi-infinite solid using the usual
thermoelastic relation.

4. Numerical results and discussion

In this Section, graphical representations of the strain and stress components at the points
(0,0) and (0,h− a) just outside the spherical inclusion in the thermoelastic half-space are
obtained using MATLAB software programming. The numerical computations are carried out
for Poisson’s ratios ν = 0.1 to 0.5. Figures 2a and 2b respectively show the variation of strain



1376 K. Singh, R. Muwal

components err(= eθθ) and ezz at the point (0,0) for exterior points of the spherical inclusion
along the distance and varyingPoisson’s ratio from 0.1 to 0.5. FromFig. 2a, we observe that the
strain err(= eθθ) decreases smoothly with increasing values of the distance h/a, and it vanishes
at infinitely large values of h/a. Also, as Poisson’s ratio ν increases from 0.1 to 0.5, this strain
increases quantitatively. From Fig. 2b, we observe that the strain ezz assumes negative values
for all the values of Poisson’s ratio ν ranging from 0.1 to 0.5. Further, it continuously increases
with the increasing distance h/a assuming the value of zero at infinitely large values of h/a. In
Figs. 3a and 3b, respectively, the variation of strain components err(= eθθ) and ezz at the point
(0,h−a) just outside the spherical inclusion are presented for exterior points. From Fig. 3b, it
can be seen that the strain ezz first increases and thendecreases continuously as the distanceh/a
increases. Further, this strain approaches a finite negative value (depending on Poisson’s ratio)
at infinitely large values of the distance h/a.

Fig. 2. (a) Strain component err(= eθθ) at the point (0,0) for exterior points of the spherical inclusion,
(b) strain component ezz atthe point (0,0) for exterior points of the spherical inclusion

Fig. 3. (a) Strain component err(= eθθ) at the point (0,h−a) just outside the spherical inclusion for
exterior points, (b) strain component ezz at the point (0,h−a) just outside the spherical inclusion for

exterior points

Figure 4 shows thevariation of the stress componentσrr(=σθθ) at thepoint (0,0) for exterior
points of the spherical inclusion in the thermoelastic half-space. FromFig. 4, we observe that the
stress component σrr(= σθθ) decreases gradually as the distance h/a increases and it vanishes
at infinitely large of h/a. Also, as Poisson’s ratio ν increases from 0.1 to 0.5, this stress increases



Thermoelastic strain and stress fields due to a spherical inclusion... 1377

quantitatively as shown in the figure. Figures 5a and 5b, respectively, show the variation of the
stress component σrr(= σθθ) and σzz at the point (0,h−a) just outside the spherical inclusion
in the thermoelastic half-space. From Fig. 5a, we observe that the stress component σrr(=σθθ)
also decreases gradually with the increasing distance h/a but it assumes finite positive values
(depending on Poisson’s ratio) at infinitely large values of the distance h/a. From Fig. 5b, we
observe that as Poisson’s ratio increases from 0.1 to 0.5, the stress decreases more rapidly with
increasing values of the distance h/a and tends to finite negative values (depending on Poisson’s
ratio) at infinitely large values of the distance h/a.

Fig. 4. Stress component σrr(=σθθ) at the point (0,0) for exterior points of the spherical inclusion

Fig. 5. (a) Stress component σrr(=σθθ) at the point (0,h−a) just outside the spherical inclusion for
exterior points, (b) stress component σzz at the point (0,h−a) just outside a spherical inclusion for

exterior points

Figures 6a and 6b, respectively, show the variation of strain components err(= eθθ) and ezz
at the point (0,h) for interior points of the spherical inclusion. As observed from Fig. 6a, the
strain err(= eθθ) increases continuously with the increasing distance h/a and with increasing
values of Poisson’s ratio ν from0.1 to 0.5. FromFig. 6b, we observe that the strain ezz decreases
gradually for all values ofPoisson’s ratio ν =0.1 to 0.5 as thedistanceh/a increases. It is noticed
from Figs. 6a and 6b that these strains assume finite positive values (depending on Poisson’s
ratio) at infinitely large values of the distance h/a. Figures 7a and 7b, respectively, show the
variation of stress components σrr(= σθθ) and σzz at the point (0,h) for interior points of the
spherical inclusion. It is noticed from Figs. 7a and 7b that the stresses decrease continuously



1378 K. Singh, R. Muwal

as the distance h/a increases and these assume finite negative values (depending on Poisson’s
ratio) at infinitely large values of distance h/a.

Fig. 6. (a) Strain component err(= eθθ) at the point (0,h) for interior points of the spherical inclusion,
(b) strain component ezz at the point (0,h) for interior points of the spherical inclusion

Fig. 7. (a) Stress component σrr(=σθθ) at the point (0,h) for interior points of the spherical inclusion,
(b) stress component σzz at the point (0,h) for interior points of the spherical inclusion

5. Conclusion

In this paper, closed form analytical expressions for thermoelastic strain and stress components
due to a spherical inclusion in a thermoelastic half-space are obtained. These expressions are
derived in axisymmetric cylindrical coordinates in the context of steady-state uncoupled ther-
moelasticity using thermoelastic displacement potential functions. The thermoelastic strain and
stress components for the exterior points of the spherical inclusion are the same as those of the
center of dilatation. The variations of the thermoelastic strain and stress components for the
exterior and interior points of the spherical inclusion are also shown graphically for different
values of Poisson’s ratios. It is observed that the strain and stress components at the points
(0,h−a) and (0,h) approach finite values depending on Poisson’s ratio at infinitely large values
of the distance h/a, whereas at the point (0,0) these quantities tend to zero as h/a tends to
infinity. Thus, Poisson’s ratio has a significant effect on the strain and stress components due to
the spherical inclusion in the thermoelastic half-space.



Thermoelastic strain and stress fields due to a spherical inclusion... 1379

Acknowledgement

We are thankful to the University Grants Commission, New Delhi, for financial support. We also

thank the reviewer for useful suggestions in the improvement of our paper.

References

1. Barber J.R., 2002,Elasticity, Second Edition, Kluwer Academic Publishers, Dordrecht

2. Boley B.A., Weiner J.H., 1997,Theory of Thermal Stresses, Dover Publications, NewYork

3. Boresi A.P., ChongK.P., Lee J.D., 2011,Elasticity in EngineeringMechanics, ThirdEdition,
JohnWiley & Sons, Inc., Hoboken, New Jersey

4. Davies J.H., 2003, Elastic field in a semi-infinite solid due to thermal expansion or a coherently
misfitting inclusion, Journal of Applied Mechanics, 70, 5, 655-660

5. Duhamel J.M.C., 1837, Second mémoire sur les phénomènes thermo-mécaniques, Journal de
l’Ecole Polytechnique, 15, 25, 1-57

6. Goodier J.N., 1937, On the integration of the thermoelastic equations,Philosophical Magazine,
7, 23, 1017-1032

7. Guell D.L., Dundurs J., 1967, Further results on center of dilatation and residual stresses in
joined elastic half-spaces, [In:] Developments in Theoretical and Applied Mechanics, W.A. Shaw
(Edit.), 3, 105-115, PergamonPress, Oxford

8. HuS.M., 1989, Stress from a parallelepipedic thermal inclusion in a semispace, Journal of Applied
Physics, 66, 6, 2741-2743

9. Itou S., 2014, Plane strain and plane stress problems in thermoelasticity, [In:] Encyclopedia of
Thermal Stresses, Richard B. Hetnarski (Eds.), Springer Netherlands, 3921-3930

10. Kolesov V.S., Vlasov N.M., Tisovskii L.O., Shatskii I.P., 1992, The stress-deformation
state of an elastic half-spacewith a spheroidal thermal inclusion, International Applied Mechanics,
28, 7, 426-434

11. Kumagai H., Maeda Y., Ichihara M., Kame N., Kusakabe T., 2014, Seismic moment and
volume change of a spherical source,Earth Planets and Space, 66, 7, 1-10

12. Leroux J., Fulleringer B., Nelias D., 2010, Contact analysis in presence of spherical inho-
mogeneities within a half-space, International Journal of Solids and Structures, 47, 3034-3049

13. Linzhi W., 2003, The elastic field induced by a hemispherical inclusion in the half-space, Acta
Mechanica Sinica, 19, 3, 253-262

14. Liu S., Wang Q., 2005, Elastic fields due to eigenstrains in a half-space, Journal of Applied
Mechanics, 72, 871-878

15. Mindlin R.D., Cheng D.H., 1950, Thermoelastic stress in the semi-infinite solid, Journal of
Applied Physics, 21, 931-933

16. Neumann F., 1885, Vorlesungen über die Theorie der Elasticität der festen Körper und des
Lichtäthers, B.G. Teubner, Leipzig

17. Nowacki W., 1957, State of stress in an infinite and semi-infinite elastic space due to an instan-
taneous source of heat,Bulletin de L’Academic Polonaise des Sciences, IV, 5, 2, 77-86

18. Nowacki W., 1986, Thermoelasticity, Second Edition, Pergamon Press, PWN-Polish Scientific
Publishers,Warszawa

19. Nowinski J., 1961, Biharmonic solutions to the steady-state thermoelastic problems in three
dimensions, Zeitschrift für angewandte Mathematik und Physik (ZAMP),XII, 2, 132-149

20. Nowinski J.L., 1963, Amean value theorem for an arbitrary steady-state thermoelastic problem
for a solid sphere, Journal of the Society for Industrial and Applied Mathematics, 11, 3, 623-631



1380 K. Singh, R. Muwal

21. Nowinski J.L., 1978,Theory of Thermoelasticity with Applications, AlphenAanDenRijn Sijthoff
and Noordhoff International Publishers

22. Sadd M.H., 2005,Elasticity – Theory, Applications and Numerics, Elsevier Academic Press Inc.,
UK

23. SenB., 1957,Note onadirectmethod of solvingproblemsof elastic plateswith circularboundaries
having prescribed displacement, Kurze Mitteilungen-Brief Reports-Communications breves, VIII,
307-309

24. SharmaB.D., 1957, Stresses due to a nucleus of thermo-elastic strain (i) in an infinite elastic solid
with spherical cavity and (ii) in a solid elastic sphere, Zeitschrift für angewandte Mathematik und
Physik (ZAMP),VIII, 142-150

25. Timoshenko S., Goodier J.N., 1951, Theory of Elasticity, Second Edition, McGraw-Hill, New
York

26. Truesdell C. (Edit.), 1984,Mechanics of Solids, Volume II, Linear Theories of Elasticity and
Thermoelasticity, Springer-Verlag Berlin Heidelberg.

27. Wang M., Huang K., 1991, Thermoelastic problems in the half space – An application of the
general solution in elasticity,Applied Mathematics and Mechanics, 12, 9, 849-862

28. Yu H.Y., Sanday S.C., 1992, Centre of dilatation and thermal stresses in an elastic plate, Pro-
ceedings of the Royal Society of London A, 438, 103-112

29. Yu H.Y., Sanday S.C., Rath B.B., 1992, Thermoelastic stresses in bimaterials, Philosophical
Magazine A, 65, 5, 1049-1064

30. Yu J.H., Kuang Z.B., Gu M.Y., 2002, Thermal stresses in joined semi-infinite solids, Journal
of Thermal Stresses, 25, 9, 889-905

31. Zhou K., Chen W.W., Keer L.M., Wang Q. J., 2009, A fast method for solving three-
dimensional arbitrarily shaped inclusions in a half space,Computer Methods in Applied Mechanics
and Engineering, 198, 9-12, 885-892

Manuscript received December 13, 2016; accepted for print June 24, 2017