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Applied Mathematical Modelling 36 (2012) 2983–2995
Contents lists available at SciVerse ScienceDirect
Applied Mathematical Modelling
journal homepage: www.elsevier .com/locate /apm
Electro-thermo-mechanical torsional buckling of a
piezoelectricpolymeric cylindrical shell reinforced by DWBNNTs with
an elastic core
A.A. Mosallaie Barzoki a, A. Ghorbanpour Arani a,b,⇑, R.
Kolahchi a, M.R. Mozdianfard ca Department of Mechanical
Engineering, Faculty of Engineering, University of Kashan, Kashan,
Islamic Republic of Iranb Institute of Nanoscience and
Nanotechnology, University of Kashan, Kashan, Islamic Republic of
Iranc Department of Chemical Engineering, University of Kashan,
Kashan, Islamic Republic of Iran
a r t i c l e i n f o
Article history:Received 23 May 2011Received in revised form 26
September2011Accepted 29 September 2011Available online 10 October
2011
Keywords:DWBNNTPiezoelectric polymerCylindrical shellElastic
coreElectro-thermo-torsional buckling
0307-904X/$ - see front matter � 2011 Elsevier
Incdoi:10.1016/j.apm.2011.09.093
⇑ Corresponding author at: Department of MechTel.: +98
3615912425; fax: +98 3615559930.
E-mail addresses: [email protected], a_gho
a b s t r a c t
The effect of partially filled poly ethylene (PE) foam core on
the behavior of torsional buck-ling of an isotropic, simply
supported piezoelectric polymeric cylindrical shell made
frompolyvinylidene fluoride (PVDF), and subjected to combined
electro-thermo-mechanicalloads has been analyzed using energy
method. The shell is reinforced by armchair doublewalled boron
nitride nanotubes (DWBNNTs). The core is modeled as an elastic
environ-ment containing Winkler and Pasternak modules. Using
representative volume element(RVE) based on micromechanical
modeling, mechanical, electrical and thermal character-istics of
the equivalent composite were determined. Critical buckling load is
calculatedusing strains based on Donnell theory, the coupled
electro-thermo-mechanical governingequations and principle of
minimum potential energy. The results indicate that
bucklingstrength increases substantially as harder foam cores are
employed i.e. as Ec/Es is increased.The most economic in-fill foam
core is at g = 0.6, as cost increases without much
significantimprovement in torsional buckling at higher g’s.
� 2011 Elsevier Inc. All rights reserved.
1. Introduction
Composites offer advantageous characteristics of different
materials with qualities that none of the constituents
possess.Nanocomposites developed in recent years, have received
much attention amongst researchers due to provision of
newproperties and exploiting unique synergism between materials.
PVDF is an ideal piezoelectric matrix due to
characteristicsincluding flexibility in thermoplastic conversion
techniques, excellent dimensional stability, abrasion and corrosion
resis-tance, high strength, and capability of maintaining its
mechanical properties at elevated temperature. It has therefore
foundmultiple applications in nanocomposites in a wide range of
industries including oil and gas, petrochemical, wire and
cable,electronics, automotive, and construction. Boron nitride
nanotubes (BNNTs) used as the matrix reinforcers, apart from
hav-ing high mechanical, electrical and chemical properties,
present more resistant to oxidation than other conventional
nano-reinforcers such as carbon nanotubes (CNTs). Hence, they are
used for high temperature applications [1–6].
Regarding research development into the application of foam
core, Karam and Gibson [7] analyzed elastic buckling of athin
cylindrical shell supported by an elastic core and reported
significant weight saving compared with a hollow cylinderusing this
structural configuration. Agrawal and Sobel [8] investigated the
weight compressions of cylindrical shells withvarious stiffness
under axial compression and showed that honeycomb sandwiches offer
a substantial weight advantage over
. All rights reserved.
anical Engineering, Faculty of Engineering, University of
Kashan, Kashan, Islamic Republic of Iran.
[email protected] (A. Ghorbanpour Arani).
http://dx.doi.org/10.1016/j.apm.2011.09.093mailto:[email protected]:[email protected]://dx.doi.org/10.1016/j.apm.2011.09.093http://www.sciencedirect.com/science/journal/0307904Xhttp://www.elsevier.com/locate/apm
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2984 A.A. Mosallaie Barzoki et al. / Applied Mathematical
Modelling 36 (2012) 2983–2995
a significant load range. Hutchinson and He [9] studied buckling
of cylindrical shells with metal foam cores under similarload and
obtained optimal outer shell thickness, core thickness and core
density by minimizing the weight of geometricallyperfect shell with
a specified load carrying capacity. Elastic stability of
cylindrical shell with an elastic core under axial com-pression was
investigated by Ghorbanpour Arani et al. [10] using energy method.
They reported increased elastic stabilityand significant weight
reduction of the cylindrical shells.
The above studies have assumed solid foam core or one with a
fixed thickness, and have not considered necessarily theoptimum
design arrangement. Partial or complete filling of the foam core,
can significantly increase buckling resistance ofthe shell, the
extent of which needs to be optimized for design purposes in terms
of weight and cost, at different circum-stances. Ye et al. [11],
however, investigated buckling of a thin-walled cylindrical shell
with foam core of various thicknessunder axial compression and
suggested that despite enhancing the resistance to buckling
failure, increase in foam core thick-ness beyond 10% of the outer
radius is inefficient due to extra cost and weight involved.
With respect to developmental works on buckling of the
cylindrical shells, it should be noted that none of the
researchmentioned above, have considered smart composites and their
specific characteristics. Active control of laminated cylindri-cal
shells using piezoelectric fiber reinforced composites was studied
by Ray and Reddy [12] using Mori–Tanaka model.However, the
reinforced materials used were CNTs which are not smart. Also,
Mori–Tanaka models for the thermal conduc-tivity of composites with
interfacial resistance and particle size distributions were studied
by Bohm and Nogales [13]. Micro-mechanical modeling which has the
potential to take into account the electrical load was used by Tan
and Tong [14] forstudying an imperfect textile composite. However,
neither the matrix nor the reinforced material used in the
composite em-ployed in this work was smart. Buckling of boron
nitride nanotube reinforced piezoelectric polymeric composites
subjectedto combined electro-thermo-mechanical loadings was
investigated by Salehi-Khojin and Jalili [15] and showed that
applyingdirect and reverse voltages to BNNT changed buckling loads
for any axial and circumferential wave-numbers. These studieswho
have taken into account smart composites in buckling of the
cylindrical shells, have not considered application of
foamcore.
In order to investigate the effect of an elastic core on the
torsional buckling of a cylindrical shell, in this research, the
effectof partially filled poly ethylene (PE) foam core on the
behavior of electro-thermo-mechanical torsional buckling of an
isotro-pic, simply supported PVDF shell, reinforced by DWBNNTs has
been analyzed using energy method and the principle of min-imum
potential energy. Simultaneous applications of DWBNNTs and PVDF
here are important in providing smartcomposites.
2. Formulation
2.1. Constitutive equations for piezoelectric materials
In a piezoelectric material, application of an electric field to
it will cause a strain proportional to the mechanical
fieldstrength, and vice versa. The constitutive equation for
stresses r and strains e matrix on the mechanical side, as well as
fluxdensity D and field strength E matrix on the electrostatic
side, may be arbitrarily combined as follows [16–18]:
rxrhrzshzsxzsxh
2666666664
3777777775¼
C11 C12 C13 0 0 0C12 C22 C23 0 0 0C13 C23 C33 0 0 00 0 0 C44 0
00 0 0 0 C55 00 0 0 0 0 C66
2666666664
3777777775
exehez
2ehz2exz2exh
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;�
axahaz000
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
DT
0BBBBBBBB@
1CCCCCCCCA�
e11 0 0e12 0 0e13 0 00 e24 00 0 e350 0 0
2666666664
3777777775
ExEhEz
8><>:
9>=>;; ð1Þ
DxDhDz
264
375 ¼
e11 e12 e13 0 0 00 0 0 e24 0 00 0 0 0 e35 0
264
375
exehez
2ehz2exz2exh
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;�
axahaz000
8>>>>>>>><>>>>>>>>:
9>>>>>>>>=>>>>>>>>;
DT
0BBBBBBBB@
1CCCCCCCCA�
�11 0 00 �22 00 0 �33
264
375
ExEhEz
8><>:
9>=>;; ð2Þ
where Cij, eij, �ii (i, j = 1, . . . ,6), ak (k = x,h,z) and DT
are elastic constants, piezoelectric constants, dielectric
constants, thermalexpansion coefficient and temperature difference,
respectively.
BNNTs in general have two highly symmetrical structures; zigzag
and armchair. For uniaxial strain, zigzag tubes exhibit
alongitudinal piezoelectric response [19], while the armchair tubes
have an electric dipole moment linearly coupled to tor-sion. Hence,
for investigating torsional buckling behaviour of the smart
composite in this study, the armchair structure ofBNNTs was
selected. Double-wall BNNTs were chosen over single-wall BNNT
primarily because of their superior stabilityand durability in
applications requiring mechanical strength, hardness and high
thermal conductivity.
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A.A. Mosallaie Barzoki et al. / Applied Mathematical Modelling
36 (2012) 2983–2995 2985
2.2. Strain displacement relationships
In order to calculate the middle-surface strain and curvatures,
using Kirchhoff-Love assumptions, the displacement com-ponents of
an arbitrary point anywhere are written as [20]:
uðx; h; zÞ ¼ u0ðx; hÞ � zowðx; hÞ
ox;
vðx; h; zÞ ¼ v0ðx; hÞ � zowðx; hÞ
oh;
wðx; h; zÞ ¼ wðx; hÞ:
ð3Þ
where, u, v, w are the displacements of a arbitrary point of the
shell in the axial, circumferential and radial directions,
respec-tively, u0, v0, w0 are the displacements of points on the
middle surface of the shell and z is the distance of the arbitrary
pointof the shell from the middle surface.
Assuming the total strain tensor to be the sum of mechanical and
thermal strains, i.e.
e ¼ eMech þ eTherm; ð4Þ
where mechanical (eMech) and thermal (eTherm) strains are
defined as:
emech ¼exxehhexh
8><>:
9>=>;; eTherm ¼
�axT�ahT0
0B@
1CA: ð5Þ
The mechanical strain components exx, ehh, exh at an arbitrary
point of the shell are related to the middle surface strains
ex,0,eh,0, exh, 0 and changes in the curvature and torsion of the
middle surface kx, kh, kxh as follows:
exx ¼ ex;0 þ zkx;ehh ¼ eh;0 þ zkh;exh ¼ exh;0 þ zkxh;
ð6Þ
where z is, the distance from the arbitrary point to the middle
surface and assume Donnell’s hypothesis and z�Rs, where Rs isthe
radius of the shell, the expressions for the middle surface strains
and the changes in the curvature and torsion of themiddle surface
with using Eq. (3) becomes:
eMech ¼
ou0ox � z o
2wox2
1Rs
ov0oh þw� �
� zR2s
o2woh2
12Rs
ou0oh þ Rs
ov0ox � 2z o
2woxoh
� �0BBB@
1CCCA: ð7Þ
2.3. Micro-electromechanical models
In this work, PVDF and DWBNNTs were used respectively, as matrix
and reinforced materials in shell for the polymericpiezoelectric
fiber reinforced composites (PPFRC), with their constituents
assumed to be orthotropic and homogeneous withrespect to their
principal axes. To evaluate the effective properties of a PPFRC
unit cell, using approach adopted by Tan and
Fig. 1. A schematic of RVE and DBNNTs reinforced composite.
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2986 A.A. Mosallaie Barzoki et al. / Applied Mathematical
Modelling 36 (2012) 2983–2995
Tong [14] in which they use RVE base on micromechanical models,
first the properties of the required strips made from
pie-zoelectric fiber reinforced composite (PFRC) are obtained using
the appropriate ‘X model’ in association with the ‘Y model’
(orvice-versa). Then, properties of a PPFRC unit cell are
calculated using ‘XY (or YX) rectangle model’ (see Fig. 1). The
closed-formformula used in ‘X model’ (or ‘Y model’) expressing the
mechanical, thermal and electrical properties of the composite
asexplained in Eqs. (1) and (2) above are [14]:
C11 ¼Cr11C
m11
qCm11 þ ð1� qÞCr11
; ð8Þ
C12 ¼ C11qCr12Cr11þþð1� qÞC
m12
Cm11
� �; ð9Þ
C13 ¼ C11qCr13Cr11þþð1� qÞC
m13
Cm11
� �; ð10Þ
C22 ¼ qCr22 þ ð1� qÞCm22 þ
C212C11� qðC
r12Þ
2
Cr11� ð1� qÞðC
m12Þ
2
Cm11; ð11Þ
C23 ¼ qCr23 þ ð1� qÞCm23 þ
C12C13C11
� qCr12C
r13
Cr11� ð1� qÞC
m12C
m13
Cm11; ð12Þ
C44 ¼ qCr44 þ ð1� qÞCm44; ð13Þ
C55 ¼A
B2 þ AC; ð14Þ
C66 ¼Cr66C
m66
qCm66 þ ð1� qÞCr66
; ð15Þ
e31 ¼ C11qer31Cr11þþð1� qÞe
m31
Cm11
� �; ð16Þ
e32 ¼ qer32 þ ð1� qÞem32 þC12e31
C11� qC
r12e
r31
Cr11� ð1� qÞC
m12e
m31
Cm11; ð17Þ
e33 ¼ qer33 þ ð1� qÞem33 þC13e31
C11� qC
r13e
r31
Cr11� ð1� qÞC
m13e
m31
Cm11; ð18Þ
e24 ¼ qer24 þ ð1� qÞem24; ð19Þ
e15 ¼B
B2 þ AC; ð20Þ
�11 ¼C
B2 þ AC; ð21Þ
�22 ¼ q�r22 þ ð1� qÞ�m22; ð22Þ
�33 ¼ q�r33 þ ð1� qÞ�m33 �e231C11þ qðe
r31Þ
2
Cr11þ ð1� qÞðe
m31Þ
2
Cm11; ð23Þ
where
A ¼ qCr55
ðer15Þ2 þ Cr55�r11
þ ð1� qÞCm55
ðem15Þ2 þ Cm55�m11
; ð24Þ
B ¼ qer15
ðer15Þ2 þ Cr55�r11
þ ð1� qÞem15
ðem15Þ2 þ Cm55�m11
; ð25Þ
C ¼ q�r11
ðer15Þ2 þ Cr55�r11
þ ð1� qÞ�m11
ðem15Þ2 þ Cm55�m11
; ð26Þ
Superscripts r and m refer to the reinforced and matrix
components of the composite, respectively. q is also the vol% of
thereinforced DWBNNTs in matrix.
2.4. Energy method
The total potential energy, V, of the PPFRC cylindrical shell
with a foam core under torsional moment is the sum of strainenergy,
U, the work W done by the applied load, and the strain energy X
stored in the foam core is expressed as:
V ¼ U þW þX; ð27Þ
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A.A. Mosallaie Barzoki et al. / Applied Mathematical Modelling
36 (2012) 2983–2995 2987
where, the strain energy is:
U ¼ 12
ZeT � ETn o r
D
� dv ; ð28Þ
where dv is volume element and superscript T corresponds to the
transposed matrix. Considering Eqs. (1) and (2), as well asthe
armchair structure for DWBNNTs employed here, and the longitudinal
arrangement of strips in matrix, makes Eh = Ez = 0.Hence, Eq. (28)
becomes:
U ¼ 12
Zex eh 2exh � Exf g
C11 C12 0 e11C12 C22 0 e120 0 C66 0e11 e12 0 ��11
26664
37775
exeh2exh�Ex
8>>><>>>:
9>>>=>>>;
dv ; ð29Þ
where transformed elastic constants are defined as:
½C� ¼ ½R�½C�½R�T ; ð30Þ
where [R] is the transfer matrix defined as [21]:
½R� ¼
cos2ðhÞ sin2ðhÞ 0 0 0 � sinð2hÞsin2ðhÞ cos2ðhÞ 0 0 0 sinð2hÞ
0 0 1 0 0 00 0 0 cosðhÞ sinðhÞ 00 0 0 � sinðhÞ cosðhÞ 0
sinðhÞ cosðhÞ � sinðhÞ cosðhÞ 0 0 0 cos2ðhÞ � sin2ðhÞ
26666666664
37777777775; ð31Þ
here, h is the angle between the global and local cylindrical
co-ordinates, which corresponds to the orientation angle be-tween
DWBNNTs and the main axis of the matrix.
Strain energy by combining Eqs. (4)–(7) and Eq. (29), may be
written as:
U ¼ 12
ZC11 �z
o2wox2� axDTð Þ
!2þ 2C12 �z
o2wox2� axDTð Þ
!wR� z
R2o2w
oh2� ahDT
! !þ C22
wR� z o
2wox2� axDTð Þ
!224
þC66 �zRs
o2woxoh
!2� 2E1e11 �z
o2wox2� axDTð Þ
!þ 2E1e12
wR� z
R2o2w
oh2� ahDT
!� �11E21
35dv: ð32Þ
The second type of total energy to be verified is the work done
by applied force, expressed as:
W ¼Z
Nxex þ Nheh þ 2Nxhexhð Þds; ð33Þ
where ds is surface element. It is noted that the torsional load
(Nxh), makes axial and circumferential loads equal to zeros(Nx = Nh
= 0). Using strain relation from Eq. (7), W rewritten as:
W ¼ 2Z
Nxh �zRs
o2woxoh
!ds: ð34Þ
The third type of total energy needing to be verified is the
energy stored in the core, X, expressed as:
X ¼Z
Fowds; ð35Þ
where Fo, the interfacial force per unit length is:
Fo ¼ Poð2pRsÞ ¼ kww� kGr2w� �
; ð36Þ
where Kw and Kg are Winkler and Pasternak modules, respectively.
Po is also the pressure generated on the foam core outerinterface
due to shell buckling. Based on the assumption of a linear,
homogeneous and isotropic foam core the pressure Pomay be expressed
as [22]:
Po ¼Ec
1� tcwRs
1� g21þ g2 ; ð37Þ
where Ec, tc, R s, are elastic module of the core, Poisson’s
ratio of the core and radius of the shell, respectively. The
in-fill ratiog (corresponding to the thickness of the foam core) is
also defined as g = Rc/Rs. The displacement term in z direction can
onlybe defined once the boundary condition of the cylindrical shell
is determined. The boundary condition considered in this
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2988 A.A. Mosallaie Barzoki et al. / Applied Mathematical
Modelling 36 (2012) 2983–2995
study includes a simply supported PPFRC cylindrical shell with
an elastic core subjected to a torsional moment. Hence,
thedisplacement caused by the pre-buckling force, which determines
our boundary condition is [19]:
w ¼ C sinðqx� nhÞ; ð38Þ
where q ¼ PpL as well as C, L, P and n are arbitrary constant,
length of cylinder, half axial and circumferential wave
number,respectively.
Replacing Eq. (38) into Eqs. (36) and (37) yields Fo defined
as:
Fo ¼ Kw þkGn2
R2sþ KgP
2
L2
! !¼ 2p1� g
2
1þ g2Ec
1� m2c: ð39Þ
At this stage, various components of total potential energy can
be presented by replacing Eqs. (32), (34) and (35) into Eq.(27),
and integrating with respect to the distance z within the limits of
Rs and Rs + h, which yields the expression for the totalpotential
energy V as:
V ¼ 12
Z 2p0
Z L0
C11Rsþhð Þ3�R3s
3o2wox2
!2þh axDTð Þ2þ Rsþhð Þ2�R2s
� �o2wox2
axDT
0@
1A
24
þ2C12 � Rsþhð Þ2�R2s� �o2w
ox2w
2Rsþ Rsþhð Þ
3�R3s3R2s
o2wox2
o2w
oh2þ Rsþhð Þ
2�R2s2
ahDTo2wox2�axhDT
wRs
þ Rsþhð Þ2�R2s� �axDT
2R2s
o2w
oh2þaxahhDT2
!
þC22wRs
�2hþ Rsþhð Þ
3�R3s3R4s
o2w
oh2
!2þh ahDTð Þ2� Rsþhð Þ2�R2s
� �wR3s
o2w
oh2�2 w
RshaxDTþ
Rsþhð Þ2�R2sR2s
o2w
oh2ahDT
0@
1A
þC664 Rsþhð Þ3�R3s� �
3R2s
o2woxoh
!20@1Aþ2E1e11 Rsþhð Þ2�R2s2 o
2wox2þaxhDT
!
þ2E1e12hwRs� Rsþhð Þ
2�R2s2R2s
o2w
oh2�ahhDT
!� �11E21h�2Nxh
Rsþhð Þ2�R2s� �
Rs
o2woxoh
!0@1Aþ2p1�g2
1þg2Ecw
1�m2c
35Rsdxdh:
ð40Þ
2.5. Minimum potential energy principle
In order to determine the critical torsional buckling load,
minimum potential energy principle [10] is used, in which thetotal
potential energy is minimum with respect to arbitrary constants in
the boundary condition. By replacing the boundarycondition Eq. (38)
into Eq. (40) and differentiating twice the latter with respect to
the arbitrary constant C, the critical tor-sional load is obtained
as below:
Ncritxh ¼ C11Rs þ hð Þ3 � R3s
� �Rsp3P3
12nL3
0@
1Aþ C12 Rs þ hð Þ
2 � R2s� �
pP
4Lnþ
Rs þ hð Þ3 � R3s� �
RsnpP
6RsL
0@
1A
þ C22Lh
4pPnRsþ q4
Rs þ hð Þ3 � R3s� �
Ln3
12pPR3sþ
Rs þ hð Þ2 � R2s� �
Ln
4pPR2s
0@
1Aþ C66 Rs þ hð Þ
3 � R3s� �
pPn
3RsL
0@
1A
�C11ax þ C12ah� �
hDT � e11E1h� �
RspP
Ln�
C12ax þ C22ah� �
hDT � e12E1h� �
nLP
Rspn
þ 2LRsPpn
2p1� g2
1þ g2Ecw
1� m2c
�: ð41Þ
In this study, the critical torsional buckling load ðNcritxh Þ
is normalized by multiplying it to (1/(Esh)). Hence, the
dimensionlesscritical torsional buckling load is N�xh ¼ N
critxh =Ech.
3. Numerical results and discussion
Having obtained Eq. (41) above, the influence of the extent of
in-fill core, that is the in-fill ratio g, on the
dimensionlesscritical torsional buckling load could be investigated
considering parameters including: the vol% of DWBNNTs in the
matrix
-
Fig. 2. Hollow circular cylindrical composite shell with
core.
Table 1Mechanical, electrical, and thermal properties of PVDF,
DBNNT and PE.
PVDF DWBNNT PE
C11 = 238.24 (GPa) E = 1.8 (TPa) E = 125 (GPa)C22 = 23.6 (GPa) t
= 0.34 t = 0.30C12 = 3.98 (GPa) e11 = 0.95 (C/m2) q = 1.45
(kg/m3)C66 = 6.43 (GPa) ax = 1.2 � 10�6 (1/K)e11 = �0.135 (C/m2) ah
= 0.6 � 10�6 (1/K)e12 = �0.145 (C/m2)� = 1.1068 � 10�8 (F/m)ax =
7.1 � 10�5 (1/K)ah = 7.1 � 10�5 (1/K)
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 11
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
η
N* xθ
With electric field, Present workWithout electric field, Ye et
al. [11]
Fig. 3. Dimensionless critical stress versus g for Ec/Es = 10�1
in the presence and absence of electric field.
A.A. Mosallaie Barzoki et al. / Applied Mathematical Modelling
36 (2012) 2983–2995 2989
(q), the orientation angle between DWBNNTs and the main axis of
the matrix (h), dimensionless aspect ratios of length toradius of
the shell (Ls/Rs), dimensionless aspect ratios of elasticity
modules of core to shell (Ec/Es), the core Poisson’s ratio
-
0 2 4 6 8 10 12 14 16 18 200
0.05
0.1
0.15
0.2
0.25
P
Nxθ*
ρ=0% & η=0.8ρ=25% & η=0.8ρ=50% & η=0.8ρ=0% &
η=0.2ρ=25% & η=0.2ρ=50% & η=0.2
Fig. 4. Effect of g and q on the buckling load with respect to
half axial wave number P.
1 1.5 2 2.5 3 3.5 4 4.5 50
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
L/R
Nxθ*
η=0, "Solid Core"
η=0.25
η=0.5
η=0.75
η≅1, "Without Core"
Fig. 5. Effect of aspect ratio L/Rs on the buckling load for
different g.
2990 A.A. Mosallaie Barzoki et al. / Applied Mathematical
Modelling 36 (2012) 2983–2995
(mc), electrical field (E), half axial (P) and circumferential
(n) wave numbers. Fig. 2 illustrates PPFRC cylindrical shell with
theelastic core in which geometrical parameters of length, Ls,
radius, Rs, and thickness h are also indicated. Mechanical,
electricaland thermal characteristics of PVDF matrix, DWBNNTs
reinforce, and PE foam core are presented in Table 1 [15].
In the present work, the torsional buckling of PPFRC with an
elastic core has been studied. Since, no reference to such awork is
found to-date in the literature, its validation is not possible.
However, in an attempt to validate this work as far aspossible,
axial buckling of PPFRC with an elastic core was studied which in
the absence of electric field and considering q = 0,Es = 200 GPa,
ts = 0.3, h = 0.1524 mm, Rs = 76.2 mm, Ls = 100 mm and tf = 0.1 is
similar to that presented by Ye et al. [11]. Forthis purpose, the
displacement satisfying our boundary condition is [11]:
w ¼ q � sin p a� xL
� �h isin p x
L� 1
� �h i; ð42Þ
-
0 0.5 1 1.5 2 2.5 3 3.50.008
0.01
0.012
0.014
0.016
0.018
0.02
0.022
0.024
0.026
0.028
θ
Nxθ*
η=0, "Solid Core" η=0.25η=0.5η=0.75η≅1, "Without Core"
Fig. 6. Influence of the orientation angle of DBNNT’s h, on the
torsional buckling load.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.019
0.0192
0.0194
0.0196
0.0198
0.02
0.0202
0.0204
η
Nx θ*
Ec/Es=10-1
Ec/Es=5*10-1
Ec/Es=100
Ec/Es=5*100
Ec/Es=101
Fig. 7. Influence of elastic modulus in the form of an aspect
ratio Ec/Es on the torsional buckling load.
A.A. Mosallaie Barzoki et al. / Applied Mathematical Modelling
36 (2012) 2983–2995 2991
where q and a are amplitude and circumferential wave number,
respectively. At this stage, shell critical stress ðrcritx Þ is
deter-mined by dividing Ncritx to the thickness of shell (h).
ðrcritx Þ is then normalized, by dividing it to r0 defined as:
r0 ¼1ffiffiffi3p �
Esffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1� m2p h
Rs
�: ð44Þ
Fig. 3 illustrates the results of validation exercise by
plotting rcritx =r0 versus g for Ec/Es = 10�1 in the presence and
absence of
electric field. As can be seen, in case of no electric field,
the results obtained are the same as those expressed in [11],
indi-cating validation of our work. In the presence of electric
field however, the normalized critical axial buckling stress
increases,indicating the important influence of the electric field
discussed in more details later in Fig. 8.
-
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.0191
0.0191
0.0191
0.0191
0.0191
0.0191
0.0192
0.0192
0.0192
0.0192
η
Nxθ*
νc=0
νc=0.1
νc=0.2
νc=0.3
νc=0.4
Fig. 8. Influence of the core Poisson’s ratio mc on the
torsional buckling load.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.114
0.115
0.116
0.117
0.118
0.119
0.12
0.121
0.122
η
Nxθ*
E=+50 vE=+25 vE=0 vE=-25 vE=-50 v
Fig. 9. Effect of direct and reverse electric field on the
torsional buckling load.
2992 A.A. Mosallaie Barzoki et al. / Applied Mathematical
Modelling 36 (2012) 2983–2995
3.1. Effect of q and g
Fig. 4 illustrates the influences of g and q on the
dimensionless critical torsional buckling load N�xh with respect to
halfaxial wave number P. As can seen, N�xh is directly related to
q. Also, the influence of q is more significant than g, which
isperhaps due to the fact that at a specific q, the buckling load
does not vary much with changes in g. However, at a specificg; N�xh
does vary considerably with changes in q. At small P values, N
�xh is very high; as P increases, critical buckling loads
decrease sharply first to a minimum between P values ranging
from 2 to 8 where the minimum N�xh takes place, before theyincrease
slightly again. It is also worth mentioning that at a specific q,
the influence of in-fill ratio g at lower P values aremore apparent
than higher P’s.
-
05
1015
20
02
46
8100
0.2
0.4
0.6
0.8
1
1.2
1.4
Pn
Nxθ*
Fig. 10. Buckling load versus to half axial wave number P and
circumferential wave number n for g = 0.2.
05
1015
20
02
46
8100
0.2
0.4
0.6
0.8
1
1.2
1.4
Pn
Nxθ*
Fig. 11. Buckling load versus to half axial wave number P and
circumferential wave number n for g = 0.8.
A.A. Mosallaie Barzoki et al. / Applied Mathematical Modelling
36 (2012) 2983–2995 2993
3.2. Effect of Ls/Rs and g
Fig. 5 demonstrates the graph of dimensionless critical
torsional buckling load versus the aspect ratio Ls/Rs for
differentg’s. In lower values of Ls/Rs, critical buckling load is
high and reduces sharply down to Ls/Rs = 1.5, where a minimum
isobserved, before N�xh increases again slightly. Interestingly, as
g increases, minimum N
�xh occurs at higher Ls/Rs values, and
for coreless cylinder (g = 1), N�xh does not vary much after
minimum point, irrespective of higher Ls/Rs.
3.3. Effects of h and g
The influence of the orientation angle of DWBNNTs (h), on the
dimensionless critical torsional buckling load is shown inFig. 6
for different values of g. As can be seen, the critical buckling
load curves are periodic functions with a period of h = 3.14(or p).
In the main period, there are both a maximum and a minimum buckling
loads, which take place at higher h as theorientation angle, g is
increased. In other words, as g increases (i.e. the core thickness
decreases), minimum critical buckling
-
2994 A.A. Mosallaie Barzoki et al. / Applied Mathematical
Modelling 36 (2012) 2983–2995
takes place at higher h. The same observation could also be made
from both Figs. 4 and 5. The maximum N�xh is of interest
inindustrial applications which is different for various g’s and
occurs in the range of h = 0 to h = 1.
3.4. Effect of Ec and g
The influence of elastic modulus in the form of an aspect ratio
Ec/Es on the graph of dimensionless critical torsional buck-ling
load versus g is shown in Fig. 7. The results indicate that
buckling strength increases substantially as harder foam coresare
employed i.e. as Ec/Es is increased. If the core is soft (i.e.
Ec/Es = 10�1), the thickness of the core has little effect on the
N
�xh.
3.5. Effect of tc and g
Fig. 8 shows the influence of the core Poisson’s ratio (mc) on
the dimensionless critical torsional buckling load, where mc
isdirectly related to the N�xh. The critical buckling load is
maximum for solid core, (g = 0) and does not vary significantly
withchanges in g. This is because mc is low in value and will not
affect the outcomes of the N�xh calculations.
For practical design purposes, cost optimization, reduced weight
and increased efficiency are important, all of which, areaffected
by g. Figs. 7 and 8 do not show a clear optimum point, but indicate
that, there is little improvement for g < 0.6.Hence, this may be
considered as the maximum allowable economic g.
3.6. Effect of E and g
Fig. 9, shows the effect of direct and reverse electric field on
N�xh along different g’s. As can be seen, reverse electrical
fieldincreases critical buckling load, possibly due to its
longitudinal direction of polarization. This is the same as
observationsmade by [10]. Also, there is little change in N�xh at 0
< g < 0.2, while for 0.2 < g < 1, critical buckling
load decreases sharply.
3.7. Effect of P, n and g
Figs. 10 and 11, show three dimensional illustrations of N�xh in
terms of axial half wave number P and circumferential wavenumber n
for g = 0.2 and g = 0.8, respectively. As expected, for less
in-fill core (i.e. g = 0.8 in Fig. 11), the N�xh is less as
shellstability is reduced.
4. Conclusion
In this study, the critical torsional buckling load of a smart
composite cylinder (a PVDF piezoelectric polymer reinforcedwith
DWBNNTs) is evaluated using the principle of minimum potential
energy. This work furthers previous studies in threeaspects; the
influence of in-fill foam core on the critical torsional buckling
load, evaluating the composite characteristicsusing RVE based on
micromechanical model, and using DWBNNTs as reinforcer. The results
indicated that the higher thein-fill core (i.e. lower g), the
higher is dimensionless critical torsional buckling load N�xh, and
the harder the foam core, thehigher the N�xh. However, g has little
significant effect on N
�xh in softer cores. Indeed, the most economic in-fill foam
core
is at g > 0.6, as cost increases without much significant
improvement in critical torsional buckling at higher g’s.
Furthermore,minimum critical torsional buckling load occurs at
axial half wave numbers ranging from 2 to 8 and optimum
orientationangle of DWBNNTs takes place for 0 < h < 1.5
radian. It is worth noting that compared to direct one, if reverse
electric fieldis being applied to the cylindrical composite, N�xh
will increase. The results of this study are validated as far as
possible by theaxial buckling of cylindrical shell with an elastic
core in the absence of electric field, as presented by Ye et al.
[11].
Acknowledgments
The authors would like to thank the referees for their valuable
comments. They would also like to thank the Iranian Nano-technology
Development Committee for their financial support.
References
[1] L. MerhariHybrid, Nanocomposites for Nanotechnology,
Springer Science, New York, 2009.[2] M. Schwartz, SMART MATERIALS
by John Wiley and Sons, A Wiley-Interscience Publication Inc., New
York, 2002.[3] R. Kotsilkova, Thermoset Nanocomposites for
Engineering Applications, Smithers Rapra Technology, USA, 2007.[4]
V. Yu, T. Christopher, R. Bowen, Electromechanical Properties in
Composites Based on Ferroelectrics, Springer-Verlag, London,
2009.[5] J. Vang, The Mechanics of Piezoelectric Structures, World
Scientific Publishing Co., USA, 2006.[6] T.H. Brockmann, Theory of
adaptive fiber composites from piezoelectric material behavior to
dynamics of rotating structures, Solid Mech. Appl. (2009).[7] G.N.
Karam, L.J. Gibson, Elastic buckling of cylindrical shells with
elastic cores. I. Analysis, Int. J. Solids Struct. 32 (1995)
1259–1283.[8] B.L. Agarwal, L.H. Sobel, Weight comparisons of
optimized stiffened, unstiffened, and sandwich cylindrical shells,
AIAA J. 14 (1977) 1000–1008.[9] J.W. Hutchinson, M.Y. He, Buckling
of cylindrical sandwich shells with metal foam cores, Int. J.
Solids Struct. 37 (2000) 6777–6794.
[10] A. Ghorbanpour Arani, S. Golabi, A. Loghman, H. Daneshi,
Investigating elastic stability of cylindrical shell with an
elastic core under axial compressionby energy method, J. Mech. Sci.
Technol. 21 (2007) 693–698.
[11] L. Ye, G. Lun, L.S. Ong, Buckling of a thin-walled
cylindrical shell with foam core under axial compression, Thin
Wall. Struct. 49 (2011) 106–111.
-
A.A. Mosallaie Barzoki et al. / Applied Mathematical Modelling
36 (2012) 2983–2995 2995
[12] M.C. Ray, J.N. Reddy, Active control of laminated
cylindrical shells using piezoelectric fiber reinforced composites,
Compos. Sci. Technol. 65 (2005)1226–1236.
[13] H.J. Bohm, S. Nogales, Mori–Tanaka models for the thermal
conductivity of composites with interfacial resistance and particle
size distributions,Compos. Sci. Technol. 68 (2008) 1181–1187.
[14] P. Tan, L. Tong, Micro-electromechanics models for
piezoelectric-fiber-reinforced composite materials, Compos. Sci.
Technol. 61 (2001) 759–769.[15] A. Salehi-Khojin, N. Jalili,
Buckling of boron nitride nanotube reinforced piezoelectric
polymeric composites subject to combined electro-thermo-
mechanical loadings, Compos. Sci. Technol. 68 (2008)
1489–1501.[16] A. Ghorbanpour Arani, R. Kolahchi, A.A. Mosallaie
Barzoki, Effect of material inhomogeneity on
electro-thermo-mechanical behaviors of functionally
graded piezoelectric rotating cylinder, J. Appl. Math. Model. 35
(2011) 2771–2789.[17] N. Jalili, Piezoelectric-Based Vibration
Control from Macro to Micro/Nano Scale Systems, Springer, Boston,
2010.[18] A. Ghorbanpour Arani, R. Kolahchi, A.A. Mosallaie
Barzoki, A. Loghman, Electro-thermo-mechanical behaviors of FGPM
spheres using analytical
method and ANSYS software, J. Appl. Math. Model. 36 (2012)
139–157.[19] N. Sai, E.J. Mele, Microscopic theory for nanotube
piezoelectricity, Phys. Rev. B. 68 (2003) 1–3.[20] D.O. Brush, B.O.
Almroth, Buckling of Bars, Plates and Shells, McGraw-Hill, New
York, 1975.[21] W. Michel lai, D. Rubin, E. Kremple, Continuum
Mechanics, Elsevier Science, USA, 1993.[22] S.P. Timoshenko, Theory
of Elasticity, McGraw-Hill, New York, 1951.
Electro-thermo-mechanical torsional buckling of a piezoelectric
polymeric cylindrical shell reinforced by DWBNNTs with an elastic
core1 Introduction2 Formulation2.1 Constitutive equations for
piezoelectric materials2.2 Strain displacement relationships2.3
Micro-electromechanical models2.4 Energy method2.5 Minimum
potential energy principle
3 Numerical results and discussion3.1 Effect of ρ and η3.2
Effect of Ls/Rs and η3.3 Effects of θ and η3.4 Effect of Ec and
η3.5 Effect of υc and η3.6 Effect of E and η3.7 Effect of P, n and
η
4 ConclusionAcknowledgmentsReferences