Nonlinear buckling response of embedded piezoelectric cylindrical shell reinforced with BNNT under electro–thermo-mechanical loadings using HDQM

Composites: Part B 44 (2013) 722–727

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Nonlinear buckling response of embedded piezoelectric cylindrical shell reinforced with BNNT under electro–thermo-mechanical loadings using HDQM A.A. Mosallaie Barzoki a, A. Ghorbanpour Arani a,b,⇑, R. Kolahchi a, M.R. Mozdianfard c, A. Loghman a a

Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Islamic Republic of Iran Institute of Nanoscience & Nanotechnology, University of Kashan, Kashan, Islamic Republic of Iran c Department of Chemical Engineering, University of Kashan, Kashan, Islamic Republic of Iran b

a r t i c l e

i n f o

Article history: Received 5 November 2011 Received in revised form 8 January 2012 Accepted 10 January 2012 Available online 20 January 2012 Keywords: A. Nanostructures A. Polymer–matrix composites (PMCs) B. Buckling C. Numerical analysis

a b s t r a c t Nonlinear buckling response of a composite cylindrical shell made of polyvinylidene fluoride (PVDF), is investigated. A two-dimensional smart model surrounded by an elastic foundation subjected to combined electro–thermo-mechanical loading is considered. The nonlinear strain terms based on Donnell’s theory are taken into account using the first shear deformation theory. The Hamilton’s principle is employed to obtain coupled differential equations, containing displacement and electric potential terms. Harmonic differential quadrature method (HDQM) is applied to obtain the critical buckling load for clamped supported mechanical and free electric potential boundary conditions at both ends of the smart cylinder. Results indicate that the critical buckling load increases when piezoelectric effect is considered. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction System stability is commonly studied using various analytical, semi-analytical and numerical methods including finite element method (FEM), finite difference method (FDM), boundary element method (BEM), and harmonic differential quadrature method (HDQM). Buckling analysis of components has been investigated by many researchers. Von Karman and Tsien [1] introduced an expression for linear buckling load and analyzed postbuckling equilibrium path of an axially compressed thin homogeneous cylindrical shell. Buckling strength of the cylindrical shell and tank subjected to axially compressive loads are investigated by Kim and Kim [2]. Both geometrically perfect and imperfect shells and tanks are studied. They showed that the buckling strength of the shell and tank decreases significantly as the amplitude of initial geometric imperfection increases. Axial stability of cylindrical shell with an elastic core was investigated by Ghorbanpour Arani et al. [3] using energy method. Critical load curves versus the thickness aspect ratio are illustrated. It has been concluded that the application of an elastic core increases elastic stability and significantly reduces the weight of cylindrical shells. Later they [4] considered elastic buckling analysis of ring and stringer-stiffened cylindrical shells under general pressure and axial compression. All the above studies have con⇑ Corresponding author at: Department of Mechanical Engineering, Faculty of Engineering, University of Kashan, Kashan, Islamic Republic of Iran. Tel.: +98 3615912447; fax: +98 3615559930. E-mail addresses: [email protected], [email protected] (A. Ghorbanpour Arani). 1359-8368/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.compositesb.2012.01.052

ducted linear buckling and have not considered nonlinear terms in their governing equations. Huang and Han [5] however worked on nonlinear elastic buckling and postbuckling of axially compressed functionally graded cylindrical shells. The material properties of the shells vary smoothly through the shell thickness according to a power law distribution based on the volume fraction of the constituent materials. In another attempt the nonlinear buckling of torsion-loaded functionally graded cylindrical shells in the existence of thermal environment were studied by the same authors [6]. They discussed various effects of the inhomogeneous parameter, the dimensional parameters and the thermal environment. Jiang et al. [7] also used differential quadrature element method (DQEM) in investigating the buckling analysis of stiffened circular cylindrical shell panels. They studied about circular cylindrical panels with a stringer stiffener and the results were compared with previously published data to verify the established methodology and procedures. None of the above mentioned works have so far considered nano and composite structures, as well as the piezoelectric effect on buckling behavior of cylindrical shells. Recently, in view of extensive applications of nanoscaled conveying fluid structures, Wang et al. [8] studied buckling instability of double-walled carbon nanotubes (DWCNTs) conveying fluid flow. It is found that the resonant frequencies depend on the fluid flow velocity, and that buckling instability of the DWCNTs occurs at a critical flow velocity. Considering the immense advantages offered by composites, parametric study is performed by Shadmehri et al. [9]. They studied buckling behavior of conical composite shells and illustrated the effect of cone angle and fiber orientation

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on the critical buckling load of the conical composite shells. Panda and Singh [10] investigated thermal buckling of a composite cylindrical shell. The shell panel is represented with the non-linear finite element model. The influence of different parameters such as lamination scheme, modular ratio, amplitude ratio, and thickness ratio were investigated. Postbuckling and collapse experiments of stiffened composite cylindrical shells subjected to axial and torsional loads were conducted by Bisagni and Cordisco [11] who showed the strength capacity of such structures in the postbuckling mode. In their experiments the axial compressive load versus lateral displacement and the torsional load versus rotation are recorded in real time during the tests. Smart nanocomposites, with PVDF as matrix and Boron nitride nanotubes (BNNTs) as the reinforcement have received plenty of interests amongst researchers [12–15] due to their ability in maintaining their mechanical properties at elevated temperature. Buckling of BNNTs in a PVDF elastic medium subjected to combined electro–thermo-mechanical loadings was investigated by SalehiKhojin and Jalili [16] who showed that applying direct and reverse voltages to BNNT changed buckling loads for any axial and circumferential wave-numbers. Recently, in another study by Mosallaie Barzoki et al. [17], torsional linear buckling of a PVDF cylindrical shell reinforced by BNNTs with an elastic core under the same loading condition as [16] were investigated, indicating that buckling strength increased substantially as harder foam cores were employed. However, to date, no report has been found in the literature on the nonlinear electro–thermo-mechanical buckling behavior of a piezoelectric polymeric cylindrical shell surrounded by elastic medium. Also, considering the nonlinear higher order terms of strains and coupling effect of electro mechanical relation based on the charge equation can enhance the accuracy of the results, with the background that there is no closed form solution for smart composite cylinders. Motivated by these considerations, in order to improve optimum design of a smart composite polymeric cylindrical shell we aim to study nonlinear buckling of an isotropic, clamped supported PVDF cylindrical shell, reinforced by BNNTs subjected to combined electro–thermo-mechanical loadings embedded in a Pasternak foundation is investigated using HDQM. 2. Constitutive equations for piezoelectric materials A schematic diagram, of a piezoelectric polymeric cylindrical shell with two fixed ends is shown in Fig. 1 in which geometrical parameters of length, L radius, R and thickness h are also indicated.

Applying an electric field to a piezoelectric material will yield a strain proportional to the displacement field, and vice versa. In piezoelectric materials, the constitutive equation includes stresses r and strains e tensors on the mechanical side, as well as flux density D and field strength E tensors on the electrostatic side, which may be arbitrarily combined as follows [18–20]:

8 9 rxx > > > > > > < =

2

C 11

C 12

0

6C rhh 6 12 C 22 0 ¼6 > 4 0 r 0 C 66 xh > > > > > :

Dxx

;

e11

e12

0

9 38 e11 > exx  axx DT > > > > > < = e12 7 7 ehh  ahh DT ; 7 > 2exh 0 5> > > > > : ; 211 Exx

ð1Þ

where Cij, eij, 2ii (i, j = 1, . . . , 6) are elastic constants, piezoelectric constants, dielectric constants, respectively (these are obtained by micromechanical model [17]), and akk (k = x, h), DT and Exx representing respectively, thermal expansion coefficient, thermal gradient and electric field, the latter of which is defined as a function of electric potential, /xx as below:

Exx ¼ 

@/xx : @x

ð2Þ

2.1. Strain displacement relationships In order to calculate the middle-surface strain and curvatures, using Kirchhoff–Law assumptions, the displacement components of an arbitrary point may be written as [21]:

@wðx; hÞ ; @x @wðx; hÞ ; vðx; h; zÞ ¼ v 0 ðx; hÞ  z @h wðx; h; zÞ ¼ wðx; hÞ:

uðx; h; zÞ ¼ u0 ðx; hÞ  z

ð3Þ

Using Donnell’s theory, strains may be obtained by a combination of linear, nonlinear and curvature change terms as:

1 2 v ;h w 1 2 z ehh ¼ þ þ 2 w;h  2 w;hh ; R 2R R R u;h w;x w;h 2z þ v ;x þ  w;xy ; 2exh ¼ R R R

exx ¼ u;x þ w2;x  zw;xx ;

ð4Þ

where x and h denote axial and circumferential direction of coordinate system, respectively, z is the distance from an arbitrary point to the middle surface and R is the radius of the shell. 3. Energy formulation The total potential energy, V, of the piezoelectric polymeric cylindrical shell conveying fluid is the sum of strain energy, U and the work done by the applied elastic medium load, W modeled using spring Winkler and shear Pasternak constants. Considering the governing Eq. (1) and strain displacement Eq. (4), and assuming longitudinally polarized PVDF (i.e. Eh = Ez = 0), U and W may be expressed as [17]:

U¼

Z

h 2

2h

Z A

þ rhh þ rxh

Fig. 1. Polymeric piezoelectric cylindrical shell reinforced with BNNTs embedded in an elastic medium.

and

!  2 @u 1 @w @2w þ z 2 @x 2 @x @x !  2 @ v w 1 @w @2w z 2 2 þ þ R@h R 2 R@h R @h ! ! @u @ v @w @w @2w @/ þ  2z þ Dx dAdz; þ R@h @x R@h @x R@h@x @x

rxx

ð5Þ

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A.A. Mosallaie Barzoki et al. / Composites: Part B 44 (2013) 722–727

W ¼

Z

ðF e ÞwdA ¼ 

Z

ðK W w  K g r2 wÞwdA;

ð6Þ

where dA is the surface element. Dimensionless parameter are then defined as:

h x h fu1 ; v 1 ; w1 g ; L L R h  C kij N C ij ¼ ; i; j ¼ 1; 2; 6 Nx0 ¼ x0 C 11 LC 11 sffiffiffiffiffiffiffiffi hkW kg /1 C 11 ; KW ¼ ; Kg ¼ ; U¼ ; U0 ¼ h C 11 hC 11 U0 211 R t



ð7Þ

! !  @ w   @2w  @w   @ u @ 2 v @w @w þ þ b c2 þ c bC þ 12 s @h @n@h @n@h @n @n2 @n @n2 ! 2 2 2  @ w   @2w  @ v  @w  @ u @w @w þb þ þ bC 66 b 2 þ þb @n @h2 @n @h @n@h @n@h @h þc

@2U @n2

¼ 0;

12

c2

 @4w @n4

 C 12 b2



@n2

þ e11

ðmÞ

ð2Þ  ðxk ; hj Þ Aik u

þ

k¼1

Nx X

ð1Þ  k ; hj Þ Aik wðx

k¼1 Nh Nx X X

þ cbC 12

b

Nh X

Nh X

! !  @2w  b @w  @2w   @w be12 @ 2 v @ w ¼ 0: þ þ c þ þ @n @n2 c @n@h @n c @h @n@h @n2

 @2u

ð11Þ As can be seen, these are nonlinear equation which could not be solved analytically. Hence, HDQM is employed which in essence approximates the partial derivative of a function, with respect to a spatial variable at a given discrete point, as a weighted linear sum of the function values at all discrete points chosen in the solution domain of the spatial variable [22]. Let F be a function repre; v ; w  and U with respect to variables n and h in the senting u following domain of (0 < n < L, 0 < h < 2p) having Nn  Nh grid points along these variables. The nth-order partial derivative of F(n, h) with respect to n, the mth-order partial derivative of F(n, h) with respect to h and the (n + m)th-order partial derivative of F(n, h) with respect to both n and h may be expressed discretely [19] at the point (ni, hi) as:

Nh Nx X X

p¼1

ð1Þ ð1Þ Aik Bjp v ðxk ; hj Þ

k¼1 p¼1

ð1Þ  k ; hj Þ Aik wðx

Nh X

ð2Þ ðxk ; hj Þ þ Bjp u

p¼1

k¼1 Nh X

!

ð1Þ ð1Þ  k ; hj Þ Aik Bjp wðx

ð2Þ  ðxk ; hj Þ þ Bjp u

ð1Þ  k ; hj Þ Bjp wðx

p¼1

@n2 @h2

Nh Nx X X k¼1 p¼1

þ bC 66 b

þb

ð1Þ  k ; hj Þ Aik wðx

k¼1

p¼1

þb

ð2Þ  k ; hj Þ Aik wðx

k¼1

ð1Þ ð1Þ Aik Bjp v ðxk ; hj Þ þ

ð1Þ  k ; hj Þ Bjp wðx

Nx X

!

Nx X

Nx X

k¼1 p¼1

!  2 !    1 @4w @4w cbC 12 @ u c @ w c2 b2 C 12 2 2  b4 C 22 4  þ þ 12 @n 2 @n 3 @n @h @h  2 !  @ v b2 @ w þ  bC 22 þ bw @h 2 @h " #   @2w @2w 2 2 2 2  ðb axx þ b C 12 ahh Þ 2  ðc axx þ c C 12 ahh Þ 2 DT @h @n ! 2 2 2    @ w @ w @ w @U   K g c2 2 þ b2 2 þ cNx 2 þ cbe12 þ KW w ¼ 0; ð10Þ @n @n @h @n @2U

Nx X

c

!

ð14Þ

where Aik and Bjl are the weighting coefficients associated with nth-order partial derivative of F(n, h) with respect to n at the discrete point ni and mth-order derivative with respect to h at hi, respectively, whose recursive formulae can be found in [23]. A more superior choice for the positions of the grid points is Chebyshev polynomials as expressed in [23]. Applying HDQM and using Eqs. (12)–(14), (8)–(10) and (11) results the following governing equations:

ð8Þ

 @4w

ð13Þ

Nn X Nh Fðni ; hj Þ X ðnÞ ðmÞ ¼ Aik Bjl Fðnk ; hl Þ; n m dn dh k¼1 l¼1

2

! !  @2w   @2w    @2u @w @ 2 v @ w @w bC 12 þ b þ þ þ b2 C 22 @h @h2 @n@h @n @n@h @h2 @h !  b@ w  @2w  @ 2 v b@ 2 w  @w  b@ 2 u @2U þ be12 þ þ cC 66 þ 2þ ¼ 0; ð9Þ 2 @h @n @n@h @n @h@n @n @n@h

c2

N

ðnÞ

ekj eij ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi; C 11 211

2

2 e11

h d Fðni ; hj Þ X ðmÞ ¼ Bjl Fðni ; hl Þ m ¼ 1; . . . ; Nh  1; m dh l¼1

d

Applying Hamilton principle ðdW  dUÞdt ¼ 0 and rear0 ranging the governing equation in mechanical displacement directions (U, V and W) as well as electric potential (U), yield the following four coupled electro–thermo-mechanical equations: 2

m

ð12Þ

nþm

 ¼ c ¼ ; n ¼ ; b ¼ fu ; v ; wg

i; j ¼ 1; 2:

N

n

n d Fðni ; hj Þ X ðnÞ ¼ Aik Fðnk ; hj Þ n ¼ 1; . . . ; Nn  1; dnn k¼1

Nx X

ð1Þ  k ; hj Þ Aik wðx

k¼1

Nh Nx X X

!

ð1Þ ð1Þ Aik Bjp v ðxk ; hj Þ

k¼1 p¼1 Nx X

þ c2 e11

ð2Þ

Aik Uðxk ; hj Þ ¼ 0;

ð15Þ

k¼1

Nh Nx X X

bC 12

ð1Þ ð1Þ  ðxk ;hj Þ þ Aik Bjp u

k¼1 p¼1

ð1Þ  k ;hj Þ Aik wðx

ð1Þ ð1Þ  ðxk ;hj Þ þ b Aik Bjp u

Nx X

k¼1 p¼1

þ cC 66 b

Nh Nx X X

Nh Nx X X

ð1Þ ð1Þ  ðxk ;hj Þ þ Aik Bjp u

þb

ð1Þ ð1Þ  k ;hj Þ Aik Bjp wðx

Nx X

þ be12

Nh Nx X X

! ð1Þ ð1Þ  k ;hj Þ Aik Bjp wðx

k¼1 p¼1 ð2Þ Aik v ðxk ;hj Þ

ð1Þ  k ;hj Þ Aik wðx

k¼1

ð1Þ  k ;hj Þ Bjp wðx

p¼1

! ð1Þ ð1Þ  k ;hj Þ Aik Bjp wðx

k¼1

k¼1 p¼1 Nh X

ð1Þ  k ;hj Þ Aik wðx

k¼1 Nx X

k¼1 p¼1

þb

Nh Nx X X k¼1 p¼1

k¼1

Nh Nx X X

þ bC 12

Nx X

Nx X

!

ð2Þ  k ;hj Þ Aik wðx

k¼1

Nh Nx X X

ð1Þ ð1Þ

Aik Bjp Uðxk ;hj Þ ¼ 0;

ð16Þ

k¼1 p¼1

c2

2

12

c

Nx X

ð4Þ  k ; hj Þ Aik wðx

1 c2 b2 C 12 þ 12 

c b C 12 3

 C 12 b

Nh Nx X X

! ð2Þ ð2Þ  k ; hj Þ Aik Bjp wðx

k¼1 p¼1

k¼1

2 2

2

Nh Nx X X k¼1 p¼1

ð2Þ ð2Þ  k ; hj Þ Aik Bjp wðx

4

 b C 22

Nh X

! ð4Þ  k ; hj Þ Bjp wðx

p¼1

Nx Nx X cX ð1Þ ð1Þ ð1Þ  ðxk ; hj Þ þ  k ; hj Þ  k ; hj Þ Aik u Aik wðx Aik wðx 2 k¼1 k¼1 k¼1

Nx X

!

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A.A. Mosallaie Barzoki et al. / Composites: Part B 44 (2013) 722–727 Nh X

 bC 22

ð1Þ  k ;hj Þ þ Bjp v ðxk ;hj Þ þ bwðx

p¼1

 ðb2 axx þ b2 C 12 ahh Þ

Nh X

Nh Nh X b2 X ð1Þ ð1Þ  k ;hj Þ  k ;hj Þ Bjp wðx Bjp wðx 2 p¼1 p¼1

ð2Þ  k ;hj Þ  ðc2 axx þ c2 C 12 ahh Þ Bjp wðx

p¼1 Nx X

ð2Þ  k ;hj Þ þ b2 Aik wðx



ð2Þ  k ;hj Þ þ cbe12 Aik wðx

ð2Þ Aik Uðxk ; hj Þ þ e11

Nx X



ð1Þ

Nx X

c

Nx X

0.9

ð1Þ  k ; hj Þ Aik wðx

0.85

k¼1

0.8

ð1Þ ð1Þ Aik Bjp v ðxk ; hj Þ þ

Nx X

! ð1Þ  k ; hj Þ þ Aik wðx

k¼1

ð1Þ  k ; hj Þ Bjp wðx

p¼1

1 0.95

ð2Þ ðxk ; hj Þ þ c Aik u

k¼1 p¼1 Nh Nx X X

b

ð18Þ

1.4

According to HDQM, mechanical clamped and free electrical boundary conditions at both ends of the shell may be written as:

1.3

8 > > > > < wi1 ¼ v i1 ¼ ui1 ¼ 0;

1.2

Nh P

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.5

¼ 0:

> > > > : wNx i ¼ v Nx i ¼ uNx i ¼ 0;

0.1

Fig. 2. Effect of aspect ratio L/R on the critical buckling load.

c

ð1Þ ð1Þ Aik Bjp v ðxk ; hj Þ

Δ Δ Δ Δ Δ

T=-50 K T=-20 K T=0.0 K T=20 K T=50 K

cr*

N NL

A2j wji ¼ 0

j¼1

0

x/L

k¼1 p¼1

Nh P

L/R=4.0

1.05

ð17Þ

ð2Þ  k ; hj Þ Aik wðx

Nh Nx X be12 X

Nh X

1.1

Aik Uðxk ;hj Þ ¼ 0

k¼1

þ

L/R=3.0 L/R=3.5

!

ð2Þ  k ;hj Þ Bjp wðx

k¼1

Nx X

L/R=2.5

1.15

k¼1

k¼1



Nh X p¼1

k¼1

Nx X

ð2Þ  k ;hj Þ DT Aik wðx

cr*

Nx X

L/R=2.0

1.2

!

k¼1

k¼1

þ cN x

1.25

N NL

c

 k ;hj Þ  K g þ K W wðx

2

Nx X

!

for i ¼ 1; . . . ; Nh :

1.1

AðNx 1Þj wji ¼ 0

j¼1

1

ð19Þ Applying these boundary conditions into the governing Eqs. (15)–(18) yields the following coupled assembled matrix equations:

"

K beq K beq

K dG

a

K dG

a

#(

b

d

d

d

)

0.9 0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L

¼ 0;

ð20Þ

where db and dd represent boundary and domain points expressed as: b

i1 ; v i1 ; w  i1 ; w  i2 ; Ui1 ; u  iNh ; v iNh ; w  iNh ; w  iðNh 1Þ ; UiNh g fd g ¼ fu i ¼ 1; . . . ; Nx ;

ð21Þ

d

ij ; v ij ; w  iðjþ1Þ ; Uij g j ¼ 2; . . . ; Nx  1: fd g ¼ fu

4. Numerical results and discussion   for In order to obtain the nonlinear critical buckling load N cr NL a PVDF cylindrical shell reinforced with BNNTs, embedded in the Pasternak foundation, HDQM was used in conjunction with a program being written in MATLAB, where the effect of dimensionless parameters such as aspect ratios of length to radius of the shell, (L/ R), temperature gradient, (DT), Winkler, KW and Pasternak, (Kg) modules as well as the material type of the shell, were investigated. The materials tested include polyethylene (PE) and PVDF as matrix and carbon nanotube (CNT) and BNNT as matrix reinforcer whose properties are taken from [14]. Fig. 2 illustrates the effects of aspect ratio (L/R) on the critical buckling load along the dimensionless shell length. It is evident that an increase in the aspect ratio does not affect the minimum  critical value of N cr NL . However the minimum takes place at farther distance along the shell (x). This is because the effect of boundary condition is more significant at lower shell length but gradually for

Fig. 3. Effect of thermal gradient DT on the critical buckling load.

longer shells this minimum approaches the middle of the shell. Fig. 3 demonstrates the influence of thermal gradient DT on the  critical buckling load N cr NL along the shell length. Changing temperature gradients (DT) slightly alters the position of minimum criti cal load. It could be mentioned however, that the N cr NL decreases slightly as DT is increased at higher x/L’s. The reason is that the equivalent stiffness in eigenvalue problem decreases with increasing temperature gradient. In realizing the influence of material  type, Fig. 4 shows how N cr NL changes along the length of the shell for four different composites. For both smart matrix (PVDF) and  smart reinforcement (BNNT), the N cr NL is higher than non smart composites, i.e. when PE and CNT are used as matrix and reinforcement, respectively. This is most likely due to the fact that in piezoelectric material, the direction of polarization for both reinforcements and matrix was the same. It should be noted that irrespective of the material type, being smart or not, the minimum  N cr NL , lies within the short range of 0.15 < x/L < 0.20, perhaps because none of the shell geometrical parameters change. Figs. 5 and 6 illustrate the influence of elastic medium, includ ing Winkler and Pasternak modules, on N cr NL , along the length of the shell. As can be seen, for lower shell length the elastic medium has not a significant effect on nonlinear critical buckling load due to high stability of the composite shell however, for longer shell length the effect of Pasternak and Winkler dimensionless spring constants become considerable. The higher the Winkler and

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A.A. Mosallaie Barzoki et al. / Composites: Part B 44 (2013) 722–727 1.6

5. Conclusion PE+CNTs PE+BNNTs PVDF+CNTs PVDF+BNNTs

1.5 1.4

cr*

N NL

1.3 1.2 1.1 1 0.9 0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L Fig. 4. The value of critical buckling load for four different composites.

Acknowledgements The authors are grateful to University of Kashan for supporting this work by Grant No. 65475/2. They would also like to thank the Iranian Nanotechnology Development Committee for their financial support.

1.5

Kg =4.5e-6 Kg =4.0e-6

1.4

Electro–thermo-mechanical nonlinear buckling of a cylindrical shell made with PVDF, reinforced by BNNTs and embedded on elastic medium has been investigated here. Using HDQM the derived governing equations were discretized, and solved to obtain the  critical buckling load N cr NL with clamped boundary conditions. The results have indicated that increasing the aspect ratio did not affect  the minimum value of N cr the posiNL . Changing DT altered slightly  cr  tion of minimum N NL along 0.15 < x/L < 0.20 and N cr NL decreased slightly as D T is increased at higher x/L’s. Application of piezoelectric materials such as PVDF and BNNTs (be it as matrix or reinforce)  in the composite increased N cr NL as compared with non smart materials such as PE and CNT. When analyzing the effect of elastic medium, the higher the spring constants were for Winkler and  Pasternak, the higher was the N cr NL .

Kg =3.5e-6

References

Kg =3.0e-6

1.3

Kg =0.0 cr*

N NL

1.2 1.1 1 0.9 0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L Fig. 5. Effect of Winkler constant KW on the critical buckling load.

1.6

Kw =4e4 1.5

Kw =3e4 Kw =2e4

1.4

Kw =1e4 Kw =0.0

N NL

cr*

1.3 1.2 1.1 1 0.9 0.8

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x/L Fig. 6. Effect of Pasternak constant Kg on the critical buckling load.



Pasternak constants, the higher is the N cr NL although the minimum  N cr NL happens at a distance farther away from the left-hand-side of the shell. This is perhaps because increasing Winkler and Pasternak coefficient increases the shell stiffness.

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