applied mathematical modelling · 2017. 3. 5. · ⇑ corresponding author at: department of...

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

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.

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.


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


þ ð1� qÞem15


; ð25Þ

C ¼ q�r11


þ ð1� qÞ�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


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


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


Ln3

12pPR3sþ


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.


(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.


(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.


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.


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.


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


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.


[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

applied mathematical modelling · 2017. 3. 5. · ⇑ corresponding author at: department of...

Documents