Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

56, 1, pp. 107-122, Warsaw 2018
DOI: 10.15632/jtam-pl.56.1.107

AN ANALYTICAL INVESTIGATION OF A 2D-PPMS HOLLOW INFINITE
CYLINDER UNDER THERMO-ELECTRO-MECHANICAL (TEM) LOADINGS

Mohsen Meshkini

School of Science and Engineering, Sharif University of Technology, International Campus, Kish Island, Iran

e-mail: meshkini@kish.sharif.edu

Keikhosrow Firoozbakhsh

Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran

e-mail: firoozbakhsh@sharif.edu

Mohsen Jabbari

Department of Mechanical Engineering, South Tehran Branch, Islamic Azad University, Iran

e-mail: m jabbari@azad.ac.ir

Ali SelkGhafari

School of Science and Engineering, Sharif University of Technology, International Campus, Kish Island, Iran

e-mail: a selkgafari@sharif.edu

The analytical solution of steady-state asymmetric thermo-electro-mechanical loads of a
hollow thick infinite cylinder made of porous piezoelectric materials (2D-PPMs) based on
two-dimensional equationsof thermoelasticity is considered.Thegeneral formof thermal and
mechanical boundary conditions is considered on the inside and outside surfaces. A direct
method is used to solve the heat conduction equation and the non-homogenous system
of partial differential Navier equations using the complex Fourier series and the power-
exponential law functions method. The material properties are assumed to depend on the
radial and circumferential variable and are expressed as power-exponential law functions
along the radial and circumferential direction.

Keywords: piezoelectric, porothermoelastisity, 2D-PPMs, hollow cylinder, TEM

1. Introduction

Porouspiezoelectricmaterials (PPMs)have lower acoustic impedanceandcanbe incorporated in
medical ultrasonic imaging devices. They are widely used for applications such as low frequency
hydrophones, accelerometers, vibratory sensors and contact microphones. The classical method
of analysis is to combine equilibrium equations with stress-strain and strain-displacement rela-
tions to arrive at governing equations in terms of displacement components, namely the Navier
equations (Hetnarski and Eslami, 2009). Li et al. (2003) presented fabrication and evaluation
of porous piezoelectric ceramics and poroussity-graded piezoelectric actuators. Zielinski (2010)
discussed the fundamentals of multi physics modeling of piezo-poro-elastic structures. The pro-
cessing and properties of porous piezoelectric materials with high hydrostatic figures of merit
was given by Bowen et al. (2004).The porous piezoelectric composites with extremely high re-
ception was discussed by Topolov and Turik (2001). Ciarletta and Scarpetta (1996) gave some
results on thermoelasticity for porous piezoelectric materials. Batifol et al. (2007) presented a
finite-element study of a piezoelectric/poroelastic sound package concept. Zeng et al. (2007)
have discussed the processing and piezoelectric properties of porous PZT ceramics. Ivanov et al.
(2002) used the porous piezoelectric ceramicsmaterials for ultrasonic flawdetection andmedical



108 M.Meshkini et al.

diagnostics. Ding et al. (2004) presented an analytical solution of a special non-homogeneous
pyroelectric hollow cylinder for piezothermoelastic axisymmetric plane strain dynamicproblems.
Akbari Alashti et al. (2013) presented thermo-elastic analysis of a functionally graded spheri-
cal shell with piezoelectric layers by differential quadrature method. Jabbari et al. (2012, 2016)
studied mechanical and thermal stresses in FGPPM hollow cylinders. Meshkini et al. (2017)
studied a asymmetric mechanical and thermal stresses in 2D-FGPPMs hollow cylinder. The
applied separation of variables and the complex Fourier series to solve the heat conduction and
Navier equations.
In this study, an analytical method is presented for mechanical and thermal stress analysis

for a hollow infinite cylindermade of fluid saturated porous piezoelectric materials (2D-PPMs).
In present study, the material properties are assumed to be expressed by power functions in
the radial and circumferential direction. The effects of compressibility, pore volume fraction
(porosity), and electric potential coefficient on displacements, electric potential and stresses
are studied. Temperature distribution is considered in the steady state asymmetric case and
mechanical and thermal boundaryconditions by satisfying the stress anddisplacementboundary
condition.

2. Governing equations

2.1. Stress analysis

The strain-displacement relations and electric intensity are (Ding et al., 2004)

εrr =
∂u

∂r
εθθ =

1

r

∂v

∂θ
+
u

r
εrθ =

1

2

(1
r

∂u

∂θ
+
∂v

∂r
−
v

r

)

Er =
∂ψ

∂r
Eθ =

1

r

∂ψ

∂θ

(2.1)

Stress-strain relations of a 2D-PPM cylinder for the asymmtric condition are (Meshkini et al.,
2017)

σrr = C11εrr+C12εθθ +e21Er−γp−C
T
1 T(r,θ)

σθθ = C12εrr +C22εθθ +e22Er−γp−C
T
2 T(r,θ)

σzz = C12(εrr+εθθ)+e23Er−γp−C
T
3 T(r,θ)

σrθ =2C44εrθ+e24Eθ Drr = e21εrr+e22εθθ−ε22Er +g21T(r,θ)

Dθθ =2e24εrθ −ε21Eθ+g22T(r,θ)

(2.2)

where p is related to Biot’s modulus, volumetric strain and the variation of the fluid content.
Considering the undrained conditions (ξ =0) as (Jabbari et al., 2012)

p = M(ξ −γ(εrr +εθθ)=−Mγ(εrr +εθθ) (2.3)

Using relations (2.2) and (2.3), the stress-strain relations of the 2D-PPM for the asymmtric
condition are (Meshkini et al., 2017)

σrr = Ĉ11εrr+ Ĉ12εθθ +e21Er−C
T
1 T(r,θ)

σθθ = Ĉ12εrr + Ĉ22εθθ +e22Er−C
T
2 T(r,θ)

σzz = Ĉ12(εrr+εθθ)+e23Er−C
T
3 T(r,θ)

σrθ =2Ĉ44εrθ+e24Eθ Drr = e21εrr+e22εθθ−ε22Er +g21T(r,θ)

Dθθ =2e24εrθ −ε21Eθ+g22T(r,θ)

(2.4)



An analytical investigation of a 2D-PPMs hollow infinite cylinder... 109

and

Ĉ11 = C11+CM Ĉ12 = C12+CM Ĉ22 = C22+CM Ĉ44 = C44 (2.5)

where CM = Mγ
2 and CTi are thermal moduli which can be expressed by elastic constants and

linear thermal expansion coefficients αi (Ding et al., 2004)

CT1 = C11αr+2C12αθ C
T
2 =2C12αr +C22αθ (2.6)

under consideration αr =αθ = α (Hetnarski and Eslami, 2009). Therefore,

CT1 =(C11+2C12)α C
T
2 =(2C12+C22)α C

T
3 = C

T
1 (2.7)

The equilibrium equations in the radial and circumferential direction, disregarding the body
force and the inertia terms, are (Ding et al., 2004)

∂σrr
∂r
+
1

r

∂σrθ
∂θ
+
1

r
(σrr −σθθ)= 0

∂σrθ
∂r
+
1

r

∂σθθ
∂θ
+
2

r
σrθ =0

∂Drr
∂r
+
1

r

∂Dθθ
∂θ
+
1

r
Drr =0

(2.8)

To obtain the equilibrium equations in terms of displacement components for the 2D-PPM
cylinder, the functional relationship of the material properties must be known. Because the
cylinder material is assumed to be graded along the radial and circumferential direction, the
coefficient of thermal expansion and electric constants are assumed to be described with the
power-exponential laws as

α = α0r̃
m1en1θ Cij = Cijr̃

m2en2θ K = k0r̃
m3en3θ

e2i = e2ir̃
m4en4θ ε2i = ε2ir̃

m5en5θ g2i = g2ir̃
m6en6θ

(2.9)

where r̃ = r/a and a is the inner radius.

Fig. 1. Geometric model of a 2D-PPMhollow cylinder under two dimensional inner and outer
Thermo-Electro-Mechanical (TEM) loads

Using relations (2.4) and (2.9) into (2.8), the Navier equations in terms of the displacement
components are



110 M.Meshkini et al.

u,rr+

(
m2+1+(m2−1)

Ĉ12

Ĉ11

)
1

r
u,r+

m2Ĉ12− Ĉ22

Ĉ11

1

r2
u+

n2C44

Ĉ11

1

r
v,r−

n2C44

Ĉ11

1

r2
v

+
C44

Ĉ11

1

r2
u,θθ+

n2C44

Ĉ11

1

r2
u,θ+

C12+C44

Ĉ11

1

r
v,rθ+

m2C12−C22−C44

Ĉ11

1

r2
v,θ

+

(
e21

Ĉ11
ψ,rr+

(m4+1)e21−e22

Ĉ11

1

r
ψ,r+

e24

Ĉ11

1

r2
ψ,θθ+

n4e24

Ĉ11

1

r2
ψ,θ

)
r̃m4−m2e(n4−n2)θ

=

(
(m1+m2+1)C11+2(m1+m2)C12−C22

Ĉ11

1

r
T +

C11+2C12

Ĉ11
T,r

)
α0r̃
m1en1θ

v,rr+(m2+1)
1

r
v,r− (m2+1)

1

r2
v+n2

Ĉ22

C44

1

r2
v,θ+

Ĉ22

C44

1

r2
v,θθ

+

(
m2+1+

Ĉ22

C44

)
1

r2
u,θ+n2

Ĉ12

C44

1

r
u,r+

(
1+

Ĉ12

C44

)
1

r
u,rθ+n2

Ĉ22

C44

1

r2
u

+

(
n4
e22

C44

1

r
ψ,r+

e22+e24

C44

1

r
ψ,rθ+(m4+2)

e24

C44

1

r2
ψ,θ

)
r̃m4−m2e(n4−n2)θ

=

(
(n1+n2)

2C12+C22

C44

1

r
T +
2C12+C22

C44

1

r
T,θ

)
α0r̃
m1en1θ

(2.10)

ψ,rr+(m5+1)
1

r
ψ,r+n5

ε21
ε22

1

r2
ψ,θ+

ε21
ε22

1

r2
ψ,θθ−

(
e21
ε22

u,rr+
(m4+1)e21+e22

ε22

1

r
u,r

+
m4e22
ε22

1

r2
u+

n4e24
ε22

1

r
v,r−

n4e24
ε22

1

r2
v+
(m4+1)e22−e24

ε22
)
1

r2
v,θ

+
e24
ε22

1

r
v,rθ

)
r̃m4−m5e(n4−n5)θ =

(
(m6+1)g21+n6g22

ε22

1

r
T +

g22
ε22

∂T

∂r

+
g22
ε22

1

r

∂T

∂θ

)
r̃m6−m5e(n6−n5)θ

Navier equations (2.10) are a non-homogeneous system of partial differential equations with
non-constant coefficients.

2.2. Heat conduction problem

The first law of thermodynamics for energy equation in the steady-state condition for the
2D-PPM two dimensional cylinder is

1

r
(krT,r),r +

1

r2
(kT,θ),θ =0 a ¬ r ¬ b −π ¬ θ ¬+π (2.11)

where T(r,θ) is temperature distribution, k(r,θ) is the thermal conduction coefficient and a
comma denotes partial differentiation with respect to the space variable.
The thermal boundary conditions are assumed as

S11T(a,θ)+S12T,r(a,θ)= f1(θ) S21T(b,θ)+S22T,r(b,θ)= f2(θ) (2.12)

we assume that the non-homogeneous thermal conduction coefficient k(r,θ) is a power function
of the radial and circumferential coordinates (r,θ) as k(r,θ)= k0r̃

m3en3θ.
Using the definition for the material properties, the temperature equation becomes

T,rr+(m3+1)
1

r
T,r+

1

r2
(n3T,θ+T,θθ)= 0 (2.13)



An analytical investigation of a 2D-PPMs hollow infinite cylinder... 111

The solution to Eq. (2.13) is written in the form of complex Fourier series, as

T(r,θ)=
∞∑

q=−∞

Tq(r)e
iqθ (2.14)

Substituting Eq. (2.14) into Eq. (2.13), the following equation is obtained

T ′′q (r)+(m3+1)
1

r
T ′q(r)+

1

r2
(iqn3−q

2)Tq(r)= 0 (2.15)

Equation (2.15) is the Euler equation and has solutions in the form of

Tq(r)= Aqr
β (2.16)

Substituting Eq. (2.16) into Eq. (2.15), the following characteristic equation is obtained

β2+m3β +(iqn3−q
2)= 0 (2.17)

the roots of Eq. (2.17) are

βq1,2 =
−m3
2
∓

√
m23
4
+q2− iqn3 (2.18)

Thus

Tq(r)= Aq1r
βq1 +Aq2r

βq2 (2.19)

Substituting Eq. (2.19) into Eq. (2.14) gives

T(r,θ)=
∞∑

q=−∞

(Aq1r
βq1 +Aq2r

βq2)eiqθ (2.20)

The constants Aq1 and Aq2 are presented in Appendix.

3. Solution of the Navier equation

u(r,θ) =
∞∑

q=−∞

uq(r)e
(iq+n1)θ v(r,θ) =

∞∑

q=−∞

vq(r)e
(iq+n1)θ

ψ(r,θ) =
∞∑

q=−∞

ψq(r)e
(iq+n1)θ

(3.1)

Substituting Eqs. (2.20) and (3.1) into Eqs. (2.10) yields

u′′q + ζ1
1

r
u′q+(τ2+iτ3)

1

r2
uq+(τ4+iτ5)

1

r
v′q+(τ6+iτ7)

1

r2
vq+ τ8ψ

′′

q + τ9
1

r
ψ′q

+(τ10+iτ11)
1

r2
ψq =

1

am1

[
(τ12+βq1τ13)Aq1r

m1+βq1−1+(τ12+βq2τ13)Aq2r
m1+βq2−1

]

v′′q + τ14
1

r
v′q − (τ15− iτ16)

1

r2
vq +(τ17+iτ18)

1

r
u′q+(τ19+iτ20)

1

r2
uq+(τ21+iτ22)

1

r
ψ′q

+(τ23+iτ24)
1

r2
ψq =

1

am1
(τ25+iτ26)

(
Aq1r

βq1+m1−1+Aq2r
βq2+m1−1

)

ψ′′q + τ27
1

r
ψ′q +(τ28+iτ29)

1

r2
ψq− τ30u

′′

q − τ31
1

r
u′q− τ32

1

r2
uq +(τ33+iτ34)

1

r
v′q

+(τ35+iτ36)
1

r2
vq =

1

am1

[
(τ37+iτ38+βq1τ39)Aq1r

βq1+m1−1

+(τ37+iτ38+βq2τ39)Aq2r
βq2+m1−1

]

(3.2)



112 M.Meshkini et al.

Equations (3.2) are a system of ordinary differential equations having general and particular
solutions.

The general solutions are assumed as

ugq(r)= Dr
η vgq(r)= Er

η ψgq(r)= Fr
η (3.3)

Substituting Eqs. (3.3) into Eqs. (3.2) yields

[η(η −1)+ τ1η+ τ2+iτ3]D+[τ4η+τ5+i(τ6η+ τ7)
]
E

+[η(η −1)τ8+ τ9η+τ10+iτ11]F =0

[τ19+ τ17η+i(τ18η+ τ20)]D+[η(η−1)+ τ14η− τ15+iτ16]E

+[τ21η+ τ23+i(τ22η+ τ24)]F =0

η(η −1)τ30− τ31η− τ32]D+[τ33η+ τ35+i(τ34η+ τ36)]E

+[η(η −1)+ τ27η+ τ28+iτ29]F =0

(3.4)

The constants τi are presented in Appendix.

A nontrivial solution is obtained by setting the determinant of the coefficients of Eqs. (3.4)
equal to zero, where a six-order polynomial characteristic equation is obtained. It gives six
eigenvalues ηq1 to ηq6. Thus, the general solutions are

ugq(r)=
6∑

j=1

Dqjr
ηqj ⇒ ugq(r)=

6∑

j=1

Dqjr
ηqj

vgq(r)=
6∑

j=1

Eqjr
ηqj ⇒ vgq(r)=

6∑

j=1

XqjDqjr
ηqj

ψgq(r)=
6∑

j=1

Fqjr
ηqj ⇒ ψgq(r)=

6∑

j=1

YqjDqjr
ηqj

(3.5)

whereXqj is the relationbetween constantsDqj andEqj andYqj is the relationbetween constants
Dqj and Fqj. It is obtained from Eqs. (3.4). The constants are presented in Appendix.

The particular solutions upq(r) and v
p
q(r) are assumed as

upq(r)= Iq1r
βq1+m1+1+ Iq2r

βq2+m1+1 vpq(r)= Iq3r
βq1+m1+1+ Iq4r

βq2+m1+1

ψpq(r)= Iq5r
βq1+m1+1+ Iq6r

βq2+m1+1
(3.6)

Substituting Eqs. (3.6) into the non-homogeneous form of Eqs. (3.2) gives Iq1 to Iq6, as they are
presented in Appendix. The complete solutions for uq(r), vq(r) and ψq(r) are the sum of the
general and particular solutions

uq(r)=
6∑

j=1

Dqjr
ηqj + Iq1r

βq1+m1+1+ Iq2r
βq2+m1+1

vq(r)=
6∑

j=1

XqjDqjr
ηqj + Iq3r

βq1+m1+1+ Iq4r
βq2+m1+1

ψq(r)=
6∑

j=1

YqjDqjr
ηqj + Iq5r

βq1+m1+1+ Iq6r
βq2+m1+1

(3.7)



An analytical investigation of a 2D-PPMs hollow infinite cylinder... 113

Substituting Eqs. (3.7) into Eqs. (3.1) gives

u(r,θ) =
∞∑

q=−∞
q 6=0

(
6∑

j=1

Dqjr
ηqj +Iq1r

βq1+m1+1+ Iq2r
βq2+m1+1

)
e(iq+n1)θ

v(r,θ) =
∞∑

q=−∞
q 6=0

(
6∑

j=1

XqjDqjr
ηqj +Iq3r

βq1+m1+1+ Iq4r
βq2+m1+1

)
e(iq+n1)θ

ψ(r,θ) =
∞∑

q=−∞
q 6=0

(
6∑

j=1

YqjDqjr
ηqj + Iq5r

βq1+m1+1+ Iq6r
βq2+m1+1

)
e(iq+n1)θ

(3.8)

Substituting Eqs. (3.8) into Eqs. (2.1), the strains and electric intensity are obtained as

εrr =
∞∑

q=−∞
q 6=0

(
6∑

j=1

ηqjDqjr
ηqj−1+(βq1 +m1+1)Iq1r

βq1+m1

+(βq2 +m1+1)Iq2r
βq2+m1

)
e(iq+n1)θ

εθθ =
∞∑

q=−∞
q 6=0

(
6∑

j=1

(iq +n1)(Xqj +1)Dqjr
ηqj−1+[(iq +n1)Iq3 + Iq1]r

βq1+m1

+[(iq +n1)Iq4 + Iq2]r
βq2+m1

)
e(iq+n1)θ

εrθ =
1

2

∞∑

q=−∞
q 6=0

(
6∑

j=1

[iq+n1+(ηqj −1)Xqj]Dqjr
ηqj−1+[(iq +n1)Iq1

+(βq1 +m1)Iq3]r
βq1+m1 +[(iq +n1)Iq2 +(βq2 +m1)Iq4]r

βq2+m1

)
e(iq+n1)θ

Er =
∞∑

q=−∞
q 6=0

(
6∑

j=1

ηqjYqjDqjr
ηqj−1+(βq1 +m1+1)Iq5r

βq1+m1

+(βq2 +m1+1)Iq6r
βq2+m1

)
e(iq+n1)θ

Eθ =
∞∑

q=−∞
q 6=0

(
6∑

j=1

(iq +n1)YqjDqjr
ηqj−1+(iq +n1)Iq5r

βq1+m1 +(iq +n1)Iq6r
βq2+m1

)
e(iq+n1)θ

(3.9)

Substituting Eqs. (3.9) into Eqs. (2.4), the stresses and electric displacement are obtained as

σrr =
1

am2

∞∑

q=−∞
q 6=0

{
6∑

j=1

(
Ĉ11[ηqjDqjr

ηqj+m2−1+(βq1 +m1+1)Iq1r
βq1+m1+m2

+(βq2 +m1+1)Iq2r
βq2+m1+m2]−

α0
am1

C11(Aq1r
βq1+m1+m2 +Aq2r

βq2+m1+m2)

+ Ĉ12[(iq +n1)(Xqj +1)Dqjr
ηqj+m2−1+

(
(iq +n1)Iq3 +Iq1

)
rβq1+m1+m2

+
(
(iq +n1)Iq4 +Iq2

)
rβq2+m1+m2]−

2α0
am1

C12(Aq1r
βq1+m1+m2 +Aq2r

βq2+m1+m2)

)
en2θ



114 M.Meshkini et al.

+e21[ηqjYqjDqjr
ηqj+m2−1+(βq1 +m1+1)Iq5r

βq1+m1+m2

+(βq2 +m1+1)Iq6r
βq2+m1+m2]en2θ

}
e(iq+n1)θ

σθθ =
1

am2

∞∑

q=−∞
q 6=0

{
6∑

j=1

(
Ĉ12[ηqjDqjr

ηqj+m2−1+(βq1 +m1+1)Iq1r
βq1+m1+m2

+(βq2 +m1+1)Iq2r
βq2+m1+m2]−

α0
am1

C12(Aq1r
βq1+m1+m2 +Aq2r

βq2+m1+m2)

+ Ĉ22[(iq +n1)(Xqj +1)Dqjr
ηqj+m2−1+

(
(iq +n1)Iq3 +Iq1

)
rβq1+m1+m2

+
(
(iq +n1)Iq4 +Iq2

)
rβq2+m1+m2]−

2α0
am1

C22(Aq1r
βq1+m1+m2 +Aq2r

βq2+m1+m2)

)
en2θ

+e22[ηqjYqjDqjr
ηqj+m2−1+(βq1 +m1+1)Iq5r

βq1+m1+m2

+(βq2 +m1+1)Iq6r
βq2+m1+m2]en2θ

}
e(iq+n1)θ

σrθ =
1

am2

∞∑

q=−∞
q 6=0

{
6∑

j=1

C44

(
[(iq +n1)+(ηqj −1)Xqj]Dqjr

ηqj+m2−1+[(iq +n1)Iq1

+(βq1 +m1)Iq3]r
βq1+m1+m2 +[(iq +n1)Iq2 +(βq2 +m1)Iq4]r

βq2+m1+m2

)
en2θ

−e24[(iq +n1)YqjDqjr
ηqj+m2−1+(iq +n1)Iq5r

βq1+m1+m2

+(iq +n1)Iq6r
βq2+m1+m2]en2θ

}
e(iq+n1)θ

(3.10)

σzz =
1

am2

∞∑

q=−∞
q 6=0

{
6∑

j=1

(
Ĉ12[ηqjDqjr

ηqj+m2−1+(βq1 +m1+1)Iq1r
βq1+m1+m2

+(βq2 +m1+1)Iq2r
βq2+m1+m2 +

(
(iq +n1)Iq3 + Iq1

)
rβq1+m1+m2

+
(
(iq +n1)Iq4 +Iq2

)
rβq2+m1+m2 −

3α0
am1

C12(Aq1r
βq1+m1+m2 +Aq2r

βq2+m1+m2)]

)
en2θ

+e23[ηqjYqjDqjr
ηqj+m2−1+(βq1 +m1+1)Iq5r

βq1+m1+m2

+(βq2 +m1+1)Iq6r
βq2+m1+m2]en2θ

}
e(iq+n1)θ

Drr =
1

am2

∞∑

q=−∞
q 6=0

{
6∑

j=1

(
e21[ηqjDqjr

ηqj+m2−1+(βq1 +m1+1)Iq1r
βq1+m1+m2

+(βq2 +m1+1)Iq2r
βq2+m1+m2]+e22[(iq +n1)(Xqj +1)Dqjr

ηqj+m2−1

+
(
(iq +n1)Iq3 +Iq1

)
rβq1+m1+m2 +

(
(iq +n1)Iq4 + Iq2

)
rβq2+m1+m2]

)
en2θ

−ε22[ηqjYqjDqjr
ηqj+m2−1+(βq1 +m1+1)Iq5r

βq1+m1+m2

+(βq2 +m1+1)Iq6r
βq2+m1+m2]en2θ

+
g21
am1
(Aq1r

βq1+m1+m2 +Aq2r
βq2+m1+m2)e(n1+n2)θ

}
e(iq+n1)θ



An analytical investigation of a 2D-PPMs hollow infinite cylinder... 115

Dθθ =
1

am2

∞∑

q=−∞
q 6=0

{
6∑

j=1

e24

(
[iq +n1+(ηqj −1)Xqj]Dqjr

ηqj+m4−1+[(iq +n1)Iq1

+(βq1 +m1)Iq3]r
βq1+m1+m4 +[(iq +n1)Iq2 +(βq2 +m1)Iq4]r

βq2+m1+m4

)
en4θ

−ε21

(
(iq +n1)YqjDqjr

ηqj+m5−1+(iq+n1)Iq5r
βq1+m1+m5

+(iq +n1)Iq6r
βq2+m1+m5

)
en5θ+

g22
am1
(Aq1r

βq1+m1+m6 +Aq2r
βq2+m1+m6)en6θ

}
e(iq+n1)θ

To determine the constants Dqj, any general from of boundary conditions for displacements,
stresses and potential electric is considered as

u(a,θ)= w1(θ) u(b,θ)= w2(θ) v(a,θ)= w3(θ)

v(b,θ)= w4(θ) σrr(a,θ)=w7(θ) σrr(b,θ)= w8(θ)

σrθ(a,θ)= w9(θ) σrθ(b,θ)= w10(θ) ψ(a,θ)= w5(θ)

ψ(b,θ)= w6(θ) Drr(a,θ)= w11(θ) Drr(b,θ)=w12(θ)

(3.11)

It is recalled that Eqs. (3.9) and (3.10) contain six unknowns, Dq1,Dq2, . . . ,Dq6. Assume that
the six boundary conditions are specified from list of Eqs. (3.11). The boundary conditionsmay
be either the given displacements and electric potential or stresses, or combinations. Expanding
the given boundary conditions in complex Fourier series gives

wj(θ)=
∞∑

n=−∞

Wj(q)e
(iq+n1)θ j =1, . . . ,6 (3.12)

where

Wj(q)=
1

2π

π∫

−π

wj(q)e
−(iq+n1)θ dθ j =1, . . . ,6 (3.13)

Using the selected six boundary conditions of Eqs. (3.11) with the help of Eqs. (3.12) and (3.13),
the six unknown coefficients Dq1 to Dq6 are calculated.

4. Results and discussion

Consider a thick hollow cylinder of inner radius a = 1m and outer radius b = 1.2m of
Ba2NaNb5O15 material with properties given in Table 1.
The thermal boundary conditions are substituted into Eq. (2.12) to obtain the temperature

distribution, where the constants of integration are obtained from the equations given in Ap-
pendix. The stress and displacement and electric potential boundary conditions are assumed
to be selected such that the mathematical strength of the proposed method can be examined.
These type of boundary conditions may not be handled with the potential function method.
The constant coefficients of the series expansions are obtained from Eq. (3.13). Here, B is the
compressibility coefficient, sometimes named the skemptonporepressure coefficient, andφ is the
pore volume fraction and is pore per total volume, respectively, which are given in Appendix.
Using Eqs. (3.11) and (3.12), the boundary conditions given in terms of the radial and shear
stresses as well as electric potential appear in Table 2. These boundary conditions are expanded
by the integral series and the unknown coefficients Dqj are determined.



116 M.Meshkini et al.

Table 1.Material constants of Ba2NaNb5O15 for 2D-PPM (Akbari Alashti et al., 2013; Jabbari
et al., 2012)

Parameter Value Parameter Value Parameter Value

α0 [1/
◦C] 1.2 ·10−6 C11 [GPa] 239 e22 [C/m

2] −0.3

γ 0.75 C12 [GPa] 104 e24 [C/m
2] 3.4

ν 0.25 C22 [GPa] 247 ε21 [C
2/Nm2] 1.96 ·10−9

νu 0.3 C44 [GPa] 76 ε22 [C
2/Nm2] 2.01 ·10−9

k0 [W/mK] 13.9 e21 [C/m
2] −0.4 g21 [C/m

2K] 5.4 ·10−5

m1,m2, . . . ,m6 m n1,n2, . . . ,n6 n g22 [C/m
2K] 5.4 ·10−5

Table 2.Boundary condition for 2D-PPM (Jabbari et al., 2012)

T(a,θ)
T(b,θ)

σrr(a,θ) σrθ(a,θ)
u(b,θ) v(b,θ)

ψ(a,θ)
[◦C] [MPa] [MPa] [W/A]

60sin(2|θ|) 0 400sin
(
θ2

4
−|θ|

)
50θ2cosθ 0 0 ψ0θ

2cos(2θ)

Fig. 2. Temperature distribution in the (a) radial at θ = π/3 and (b) circumferential direction at r = r

Fig. 3. (a) Circumferential distribution of radial thermo-electro-mechanicalal stresses σrr at r = r.
(b) Radial distribution of shear thermo-electro-mechanicalal stresses σrθ at θ = π/3



An analytical investigation of a 2D-PPMs hollow infinite cylinder... 117

Figure 2a and 2b shows the effect of the power-exponential law index on the temperature
distribution in thewall thickness along the radial andcircumferential directions.The effect of the
power-exponential law index on the distribution of the radial thermo-electro-mechanical stresses
is shown in Fig. 3a. It is shown that as m,n increases, the radial, hoop, shear and axial thermal
stresses are increased.Thisfigure is aplot of stresses versusθ at r = r =1.1,where r is theavrege
inner radius a and the outer radius b. Figure 3b shows the shear thermo-electro-mechanical
stresses in the cross section of the cylinder, where the pore compressibility coefficient B is
changed and the other parameters are fixed. Figure 4a shows the radial displacement in the
cross section of the cylinder, where the based on the pore volume fraction φ is changing. Also
the electric potential constant in Figs. 1 to 4a is ψ0 =60V. Figure 4b shows the circumferential
displacements in the cross section of the cylinder,where thebasedon theversus electric potential
coefficient ψ0 is changing.

Fig. 4. (a) Radial distribution of u with diffrent porosity cofficient at θ = π/3. (b) Circumferential
distribution of v with electric potential coefficient at r = r

5. Conclusions

In the present work, an attempt is made to study the problem of analitical solution for the
Thermo-Electro-Mechanical (TEM) in a thick 2D-PPM hollow infinite cylinder where the two-
dimensional asymmetric steady-state loads are implied. The method of solution is based on
the direct method and uses the power series, rather than the potential function method. The
advantage of thismethod is itsmathematical power to handle both simple and complicatedma-
thematical functions for the thermal andmechanical stresses boundary conditions.Thepotential
function method is capable of handling complicated mathematical functions as the boundary
conditions. The proposed method does not have mathematical limitations to deal with general
types of boundary conditions, which usually occur in the potential function method.

Appendix

d1 =(βq1 +m1+1)(βq1 +m1)+
(
(m2+1)+(m2−1)

Ĉ12

Ĉ11

)
(βq1 +m1+1)

+
m2Ĉ12− Ĉ22

Ĉ11
+[(n1+n2)n1+iq(2n1+n2)− q

2]
C44

Ĉ11



118 M.Meshkini et al.

d2 =(βq2 +m1+1)(βq2 +m1)+
(
(m2+1)+(m2−1)

Ĉ12

Ĉ11

)
(βq2 +m1+1)

+
m2Ĉ12− Ĉ22

Ĉ11
+[(n1+n2)n1+iq(2n1+n2)− q

2]
C44

Ĉ11

d3 =
(
(iq +n1)

Ĉ12

Ĉ11
+(iq +n1+n2)

C44

Ĉ11

)
(βq1 +m1+1)

+(iq +n1)
m2Ĉ12− Ĉ22

Ĉ11
− (iq +n1+n2)

C44

Ĉ11

d4 =
(
(iq +n1)

Ĉ12

Ĉ11
+(iq +n1+n2)

C44

Ĉ11

)
(βq2 +m1+1)

+(iq +n1)
m2Ĉ12− Ĉ22

Ĉ11
− (iq +n1+n2)

C44

Ĉ11

d5 =
e21

Ĉ11
(βq1 +m1+1)(βq1 +m1)+

(m4+1)e21−e22

Ĉ11
(βq1 +m1+1)

+[(n1+n4)n1+iq(2n1+n4)−q
2]
e24

Ĉ11

d6 =
e21

Ĉ11
(βq2 +m1+1)(βq2 +m1)+

(m4+1)e21−e22

Ĉ11
(βq2 +m1+1)

+[(n1+n4)n1+iq(2n1+n4)−q
2]
e24

Ĉ11

d7 =
((m1+m2+1)C11+2(m1+m2)C12−C22

Ĉ11
+
C11+2C12

Ĉ11
βq1

) α0
am1

Aq1

d8 =
((m1+m2+1)C11+2(m1+m2)C12−C22

Ĉ11
+
C11+2C12

Ĉ11
βq2

) α0
am1

Aq2

d9 =(βq1 +m1+1)(βq1 +m1)+(m2+1)(βq1 +m1+1)

− (m2+1)+[(n1+n2)+ iq(n2+2)−q
2]
Ĉ22

C44

)

d10 =(βq2 +m1+1)(βq2 +m1)+(m2+1)(βq2 +m1+1)

− (m2+1)+[(n1+n2)+ iq(n2+2)−q
2]
Ĉ22

C44

d11 =
(
(iq +n1)+(iq +n1+n2)

Ĉ12

C44

)
(βq1 +m1+1)

+(iq +n1)(m2+1)+(iq+n1+n2)
Ĉ22

C44

d12 =
(
(iq +n1)+(iq +n1+n2)

Ĉ12

C44

)
(βq2 +m1+1)

+(iq +n1)(m2+1)+(iq+n1+n2)
Ĉ22

C44

d13 =
(
(iq +n1)

e24

C44
+(iq +n1+n4)

e22

C44

)
(βq1 +m1+1)+(iq+n1)(m4+2)

e24

C44



An analytical investigation of a 2D-PPMs hollow infinite cylinder... 119

d14 =
(
(iq +n1)

e24

C44
+(iq +n1+n4)

e22

C44

)
(βq2 +m1+1)+(iq+n1)(m4+2)

e24

C44

d15 =(n1+n2+iq)
2C12+C22

C44

α0
am1

Aq1

d16 =(n1+n2+iq)
2C12+C22

C44

α0
am1

Aq2

d17 =(βq1 +m1+1)(βq1 +m1)+(βq1 +m1+1)(m5+1)

+[(n1+n5)n1+iq(n5+2n1)−q
2]
ε21
ε22

d18 =(βq2 +m1+1)(βq2 +m1)+(βq2 +m1+1)(m5+1)

+[(n1+n5)n1+iq(n5+2n1)−q
2]
ε21
ε22

d19 =−
e21
ε22
(βq1 +m1+1)(βq1 +m1)−

(m4+1)e21+e22
ε22

(βq1 +m1+1)−
m4e22
ε22

d20 =−
e21
ε22
(βq2 +m1+1)(βq2 +m1)− (

(m4+1)e21+e22
ε22

(βq2 +m1+1)−
m4e22
ε22

d21 =(iq+n1+n4)
e24
ε22
(βq1 +m1+1)+(iq +n1)(m4+1)

e22
ε22
− (iq+n1+n4)

e24
ε22

d22 =(iq+n1+n4)
e24
ε22
(βq2 +m1+1)+(iq +n1)(m4+1)

e22
ε22
− (iq+n1+n4)

e24
ε22

d23 =
(
(m6+1)

g21
ε22
+(iq +n6+βq1)

g22
ε22

) 1
am1

Aq1

d24 =
(
(m6+1)

g21
ε22
+(iq +n6+βq2)

g22
ε22

) 1
am1

Aq2

N̂lqj = ηqj(ηqj −1)+ τ1η+τ2+ τ3+iτ4 N̂2qj = τ5ηqj + τ7+i(τ6+ τ8ηqj)

N̂3qj = ηqj(ηqj −1)τ9+τ10ηqj + τ11+iτ12 N̂4qj = τ16+ τ18ηqj +i(τ19ηqj + τ21)

N̂5qj = ηqj(ηqj −1)+ τ15ηqj − τ16+iτ17 N̂6qj = τ26ηqj + τ28+i(τ27ηqj + τ29)

N̂7qj = ηqj(ηqj −1)τ35− τ36ηqj − τ37 N̂8qj = τ38ηqj + τ40+i(τ39ηqj + τ41)

N̂9qj = ηqj(ηqj −1)− τ32ηqj + τ33+iτ34


N̂1qj N̂2qj N̂3qj
N̂4qj N̂5qj N̂6qj
N̂7qj N̂8qj N̂9qj








Dqj
Eqj
Fqj



=




0
0
0



 ⇒

∣∣∣∣∣∣∣

N̂1qj N̂2qj N̂3qj
N̂4qj N̂5qj N̂6qj
N̂7qj N̂8qj N̂9qj

∣∣∣∣∣∣∣
=0

Xqj =
Eqj
Dqj
=
N̂1qjN̂6qj − N̂3qjN̂4qj

N̂3qjN̂5qj − N̂2qjN̂6qj
Yqj =

Fqj
Dqj
=
N̂4qjN̂8qj − N̂5qjN̂7qj

N̂5qjN̂9qj − N̂6qjN̂8qj
j =1, . . . ,6

Iq1 =
d7d9d17−d3d15d17−d5d9d23−d7d13d19+d3d13d23+d5d15d19
d1d9d17−d3d11d17−d1d13d19−d5d9d21+d5d11d19+d3d13d21

Iq2 =
d8d10d18−d4d16d18−d6d10d24−d8d14d20+d4d14d24+d6d16d20
d2d10d18−d4d12d18−d2d14d20−d6d10d22+d6d12d20+d4d14d22

Iq3 =
d1d15d17−d7d11d17−d1d13d23−d5d15d21+d5d11d23+d7d13d21
d1d9d17−d3d11d17−d1d13d19−d5d9d21+d5d11d19+d3d13d21

Iq4 =
d2d16d18−d8d12d18−d2d14d24−d6d16d22+d6d12d24+d8d14d22
d2d10d18−d4d12d18−d2d14d20−d6d10d22+d6d12d20+d4d14d22

Iq5 =
d1d9d23−d1d15d19−d3d11d23−d7d9d21+d7d11d19+d3d15d21
d1d9d17−d3d11d17−d1d13d19−d5d9d21+d5d11d19+d3d13d21

Iq6 =
d2d10d24−d2d16d20−d4d12d24−d8d10d22+d8d12d20+d4d16d22
d2d10d18−d4d12d18−d2d14d20−d6d10d22+d6d12d20+d4d14d22



120 M.Meshkini et al.

τ1 =(m2+1)+(m2−1)
Ĉ12

Ĉ11
τ2 =

m2Ĉ12− Ĉ22+[(n
2
1+n1n2)−q

2]C44

Ĉ11

τ3 =
(2n1+n2)C44

Ĉ11
q τ4 =

n1Ĉ12+(n1+n2)C44

Ĉ11
τ5 =

Ĉ12+C44

Ĉ11
q

τ6 =
(m2Ĉ12− Ĉ22)n1− (n1+n2)C44

Ĉ11
τ7 =

m2Ĉ12− Ĉ22−C44

Ĉ11
q

τ8 =
e21

Ĉ11
τ9 =

(m4+1)e21−e22

Ĉ11
τ10 = [(n

2
1+n1n4)−q

2]
e24

Ĉ11

τ11 =(2n1+n4)
e24

Ĉ11
q τ12 =

α0[(m1+m2+1)C11+2(m1+m2)C12−C22]

Ĉ11

τ13 = α0
C11+2C12

Ĉ11
τ14 = m2+1 τ15 =(m2+1)− [(n1+n2)− q

2]
Ĉ22

C44

τ16 =(n2+2)
Ĉ22

C44
q τ17 = n1+(n1+n2)

Ĉ12

C44
τ18 =

(
1+

Ĉ12

C44

)
q

τ19 =(m2+1)n1+(n1+n2)
Ĉ22

C44
τ20 =

(
(m2+1)+

Ĉ22

C44

)
q

τ21 =
(n1+n4)e22+n1e24

C44
τ22 =

e22+e24

C44
q τ23 =(m4+2)n1

e24

C44

τ24 =(m4+2)q
e24

C44
τ25 = α0(n1+n2)

2C12+C22

C44

τ26 = α0
2C12+C22

C44
q τ27 = m5+1 τ28 = [(n

2
1+n1n5)−q

2]
ε21
ε22

τ29 =(2n1+n5)q
ε21
ε22

ζ30 =−
e21
ε22

τ31 =
(m4+1)e21+e22

ε22

τ32 =
m4e22
ε22

τ33 =(n1+n4)
e24
ε22

τ34 =
e24
ε22

q

τ35 =
(m4+1)n1e22− (n1+n4)e24

ε22
τ36 =

(m4+1)e22−e24
ε22

q

τ37 =
(m6+1)g21+n6g22

ε22
τ38 =

g22
ε22

q τ39 =
g22
ε22

Also

B =
3(νu−ν)

(1−2ν)(1+νu)
0¬ B ¬ 1 φ =

γ(B −kf)

B[(1−α)+k]

wherekf andk are thebulkmodulusof thefluidphase and thebulkmodulusof theporouselastic
medium under the drained condition, respectively

M =
2G(νu−ν)

γ2(1−2ν)(1−2νu)

where M and γ are Biot’s modulus, Biot’s coefficient of the porouselastic medium, respectively.



An analytical investigation of a 2D-PPMs hollow infinite cylinder... 121

Using the boundary conditions (2.12) to determine the constants Aq1, Aq2

∞∑

q=−∞

[
(S11a

βq1 +S12βq1a
βq1 −1)Aq1 +(S11a

βq2 +S12βq2a
βq2 −1)Aq2

]
eiqθ = f1(θ)

∞∑

q=−∞

[
(S21b

βq1 +S22βq1b
βq1 −1)Aq1 +(S21b

βq2 +S22βq2b
βq2 −1)Aq2

]
eiqθ = f2(θ)

(S11a
βq1 +S12βq1a

βq1 −1)Aq1 +(S11a
βq2 +S12βq2a

βq2 −1)Aq2 =
1

2π

π∫

−π

f1(θ)e
−iqθ dθ

(S21b
βq1 +S22βq1b

βq1 −1)Aq1 +(S21b
βq2 +S22βq2b

βq2 −1)Aq2 =
1

2π

π∫

−π

f1(θ)e
−iqθ dθ

Aq1 =
1

2π

π∫

−π

1

Ŝ1− Ŝ2

[
(S21b

βq2 +S22βq1b
βq2−1)f1(θ)

− (S11a
βq2 +S12βq2a

βq2−1)f2(θ)
]
e−iqθ dθ

Aq2 =
1

2π

π∫

−π

1

Ŝ1− Ŝ2

[
(S11a

βq1 +S12βq1a
βq1−1)f2(θ)

− (S21b
βq1 +S22βq1a

βq1−1)f1(θ)
]
e−iqθ dθ

Ŝ1 =(S11a
βq1 +S12βq1a

βq1 −1)(S21b
βq2 +S22βq2b

βq2 −1)

Ŝ2 =(S11a
βq2 +S12βq2a

βq2 −1)(S21b
βq1 +S22βq1b

βq1 −1)

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Manuscript received January 4, 2017; accepted for print July 25, 2017