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Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells
Composites Part B 132 (2018) 258e274
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Composites Part B journal homepage: www.elsevier.com/locate/compositesb
Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells S. Sahmani, M.M. Aghdam* Department of Mechanical Engineering, Amirkabir University of Technology, P.O. Box 15875-4413 Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 April 2017 Received in revised form 2 September 2017 Accepted 2 September 2017 Available online 8 September 2017
The present study deals with the size-dependent nonlinear buckling and postbuckling characteristics of magneto-electro-elastic cylindrical composite nanoshells incorporating simultaneously the both of hardening-stiffness and softening-stiffness size effects. To accomplish this purpose, the nonlocal strain gradient elasticity theory is applied to the classical shell theory. Via the virtual work's principle, the sizedependent governing differential equations are constructed including the coupling terms between the axial mechanical compressive load, external magnetic potential and external electrical potential. The nonlinear prebuckling deformations and the large postbuckling deflections are taken into consideration based upon the boundary layer theory of shell buckling. Finally, an improved perturbation technique is employed to achieve explicit analytical expressions for nonlocal strain gradient stability curves of magneto-electro-elastic nanoshells under various surface electric and magnetic voltages. It is seen that a positive electric potential and a negative magnetic potential cause to increase both of the nonlocality and strain gradient size dependencies in the nonlinear instability behavior of axially loaded magneto-electroelastic composite nanoshells, while a negative electric potential and a positive magnetic potential play an opposite role. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Nanomechanics Magneto-electro-elastic materials Nonlocality Strain gradient size dependency Nonlinear instability
1. Introduction In those early years, a new group of composite materials containing simultaneously the piezoelectric and piezomagnetic phases have been attracted the attention of research community due to their smart characteristics. This attraction has been even somehow more than that dedicated previously to functionally graded composite materials [1e3]. The novel magneto-electro-elastic coupling feature of these smart composites makes them more sensitive and adaptive as an eminent candidate in numerous areas of technology such as semiconducting capacitors, robotics, structural health monitoring, sensors and actuators and so on [4e7]. In recent years, the mechanical responses of magneto-electro-elastic macrostructures have been investigated extensively based on the classical continuum mechanics [8e15].
* Corresponding author. E-mail address: [email protected] (M.M. Aghdam). http://dx.doi.org/10.1016/j.compositesb.2017.09.004 1359-8368/© 2017 Elsevier Ltd. All rights reserved.
The synthesis of nanotechnology-based products is one of the important fields of engineering sciences because of their wide range of application. It is well recognized that nanostructures possess unique characteristics which are greatly depend on their shape as well as their crystalline arrangement. Accordingly, several non-classical continuum theories of elasticity have been proposed and employed to anticipate different size dependencies in mechanical behaviors of nanostructures [16e46]. Recently, some studies have been carried out to predict the mechanical behavior of nanoscaled structures made of magneto-electro-elastic (MEE) composite materials. Ke and Wang [47] analyzed the sizedependent free vibrations of MME composite Timoshenko nanobeams based on the nonlocal elasticity theory. Ke et al. [48,49] reported the size-dependent natural frequencies of MEE composite nanoplates and cylindrical nanoshells via nonlocal theory of
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
elasticity implemented in, respectively, the Kirchhoff plate theory and Love's shell theory. Li et al. [50] analyzed bending, buckling and free vibration of MEE composite Timoshenko nanobeams on the basis of nonlocal continuum theory. Farajpour et al. [51] established a nonlocal plate model to study the size effect on the nonlinear vibration behavior of MEE composite nanoplate under external electromagnetic loading condition. Jamalpoor et al. [52] used a nonlocal plate model to investigate the free vibration and biaxial buckling double-MEE nanoplate-system subjected to electric and magnetic potentials. Generally, in the previous investigations, it has been demonstrated that the small scale effect in type of the stress nonlocality results in softening-stiffness influence, while the strain gradient size dependency leads to a hardening-stiffness effect. As a consequence, Lim et al. [53] proposed a new size-dependent elasticity theory namely as nonlocal strain gradient theory which includes the both softening and stiffening influences to describe the size dependency in a more accurate way. Subsequently, a few studies have been performed on the basis of nonlocal strain gradient elasticity theory. Li and Hu [54] reported the size-dependent critical buckling loads of nonlinear EulerBernoulli nanobeams based upon nonlocal strain gradient theory of elasticity. They also presented the size-dependent frequency of wave motion on fluid-conveying carbon nanotubes via nonlocal strain gradient theory [55]. Yang et al. [56] established a nonlocal strain gradient beam model to evaluate the critical voltages corresponding to pull-in instability FG carbon nanotube reinforced actuators at nanoscale. Simsek [57] used nonlocal strain gradient theory to capture the size effects on the nonlinear natural frequencies of FGM Euler-Bernoulli nanobeams. Farajpour et al. [58] proposed a new size-dependent plate model for buckling of orthotropic nanoplates based on nonlocal strain gradient elasticity theory. Tang et al. [59] studied the wave propagation in a viscoelastic carbon nanotube via nonlocal strain gradient elasticity theory. Ebrahimi and Dabbagh [60] employed nonlocal strain gradient elasticity theory to study the flexural wave propagation in functionally graded MEE higher-order shear deformable nanoplates. Li et al. [61] utilized the nonlocal strain gradient elasticity theory within the framework of the Euler-Bernoulli beam theory to explore bending, buckling and free vibration of axially functionally graded nanobeams. Sahmani and Aghdam [62] employed the nonlocal strain gradient elasticity theory within the framework of the hyperbolic shear deformation shell theory to analyze the size-dependent nonlinear instability of a microtubule embedded in an elastic foundation related to the cytoplasm of a living cell. The objective of this work is to capture simultaneously the nonlocality and strain gradient size dependencies on the nonlinear buckling and postbuckling behavior of axially loaded MEE composite nanoshells under external electric and magnetic potentials. Thereby, the nonlocal strain gradient elasticity theory is incorporated to the classical shell theory to develop a more comprehensive size-dependent shell model. On the basis of the variational approach, the non-classical differential equations are derived. Afterwards, the boundary layer theory of shell buckling and an improved perturbation technique are put to use to achieve explicit analytical expressions for the nonlocal strain gradient stability curves of MEE composite nanoshells under combination of axial compressive load, electric and magnetic potentials.
259
2. Nonlocal strain gradient MEE shell model As illustrated in Fig. 1, an MEE composite cylindrical nanoshell with length L, radius R and thickness h is supposed under axial compressive load combined with external electric and magnetic potentials. Based upon the nonlocal strain gradient elasticity theory, the total nonlocal strain gradient stress tensor L can be expressed as below [53].
Lij ¼ sij Vs*ij ;
i; j ¼ x; y
(1)
where V is the gradient symbol, s and s* represent, respectively, the stress and higher-order stress tensors which can be introduced for an MEE composite material as follows
sij ¼
Z n h 0 0 0 91 X X cijkl εkl X emij E m X U
0 io dU qnij M n X
s*ij ¼ l2
(2a)
Z n h 0 0 0 92 X X cijkl Vεkl X emij E m X U
0 io dU qnij M n X
(2b)
in which cijkl ; emij ; qnij in order are the elastic, piezoelectric and piezomagnetic constants. Also, 91 and 92 are, respectively, the principle attenuation kernel function including the nonlocality and additional kernel function associated with the nonlocal effect of the 0 first-order strain gradient field, X and X in order denote a point and any point else in the body, and l stands for the internal strain gradient length scale parameter. E and M are the electric field and magnetic intensity, respectively. In a similar way, the other basic equations for an MEE composite material can be expressed based upon the nonlocal strain gradient elasticity theory as below
Di ¼
Z n h 0 0 0 91 X X eikl εkl X sim E m X U
0 io dU din M n X
D*i ¼ l2
(3a)
Z n h 0 0 0 92 X X eikl Vεkl X sim E m X U
0 io dU din M n X
(3b)
260
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
Fig. 1. MEE composite nanoshell under combination of axial compressive load, electric and magnetic potentials.
Bi ¼
Z n h 0 0 0 91 X X qikl εkl X dim E m X U
0 io dU bin M n X
B*i ¼ l2
(3c)
Z n h 0 0 0 92 X X qikl Vεkl X dim E m X U
0 io dU bin M n X
(3d)
where Di and D*i are, respectively, the electric displacement and higher-order electric displacement vectors, Bi and B*i in order represent the magnetic induction and higher-order magnetic induction vectors. Also, sij ; dij ; bij denote dielectric, magnetoeletric and magnetic constants, respectively. Following the method of Eringen, the constitutive relationship corresponding to the total nonlocal strain gradient stress tensor, electric displacement vector and magnetic induction vector for a two dimensional MEE composite material can be expressed as
1 m2 V2
8 9 < sxx =
9 2 38 c 11 b 0 < εxx = c 12 b syy ¼ 1 l2 V2 4 bc 12 bc 22 0 5 εyy : :g ; sxy ; 0 0 b c 66 xy 9 38 2 0 0 b e 31 < E x = 40 0 b e 32 5 E y : ; Ez 0 0 0 9 38 2 0 0 b q 31 < M x = 5 4 M 0 0 b q 32 : y; Mz 0 0 0
9 9 8 2 38 0 0 < εxx = < D x = 0 2 2 2 2 4 1m V 0 0 0 5 εyy D ¼ 1l V :g ; : y; b e 31 b Dz e 32 0 xy 9 38 2 b 0 0
(4b)
33
8 9 9 2 38 0 0 < εxx = < B x = 0 0 0 5 εyy 1 m2 V2 B ¼ 1 l 2 V2 4 0 : y; :g ; b q 31 b Bz q 32 0 xy 2 38 9 b d 11 0 0
(4c)
in which m stands for the nonlocal parameter and
(4a)
b c 22 ¼ c11 c 11 ¼ b
c213 ; c33
b q 32 ¼ q31 q 31 ¼ b b d 11 ¼ b d 22 ¼ d11 ;
b c 12 ¼ c12
c13 q33 ; c33
c213 ; c33
b c 66 ¼ c66 ;
bs 11 ¼ bs 22 ¼ s11 ;
d33 ¼ d33 þ
q33 e33 ; c33
b e 31 ¼ b e 32 ¼ e31
b s 33 ¼ s33 þ
b b 11 ¼ b b 22 ¼ b11 ;
c13 e33 c33
e233 c33
(5) 2
q b b 33 ¼ b33 þ 33 c33
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In accordance with the classical shell theory, the displacement field for an arbitrary point in the shell can be given as [63].
vwðx; yÞ ux ðx; y; zÞ ¼ uðx; yÞ z vx
(6a)
vwðx; yÞ uy ðx; y; zÞ ¼ vðx; yÞ z vy
(6b)
uz ðx; y; zÞ ¼ wðx; yÞ
(6c)
where uðx; yÞ; vðx; yÞ; wðx; yÞ represents the middle surface displacements along x; y and z axis, respectively. Now, based upon the Maxwell's equation, the electric field and magnetic intensity vectors can be written as the gradients of scalar electric and magnetic potentials as follows [64].
E ¼ VE~ ðx; y; zÞ
(7a)
~ ðx; y; zÞ M ¼ VM
(7b)
~ are the electric potential and magnetic potential, in which E~ and M
Ps ¼
1 2
Z S
h Z2
261
respectively, that can be defined as [64].
pz 2z40 4ðx; yÞ þ E~ ðx; y; zÞ ¼ cos h h
(8a)
~ ðx; y; zÞ ¼ cos pz jðx; yÞ þ 2zj0 M h h
(8b)
By taking von Karman geometric nonlinearity into account, the nonlinear strain-displacement relationships for a perfect cylindrical nanoshell can be expressed as
vu 1 vw 2 v2 w þ z 2 vx 2 vx vx vv w 1 vw 2 v2 w þ ¼ z 2 vy R 2 vy vy
εxx ¼ ε0xx þ zkxx ¼ εyy ¼ ε0yy þ zkyy
gxy ¼ g0xy þ zkxy ¼
(9)
vu vv vw vw v2 w þ þ 2z vy vx vx vy vxvy
As a result, the total strain energy related to the initial configuration of a nonlocal strain gradient MEE composite nanoshell can be calculated as
sij εij þ s*ij Vεij Di E i Bi M i dzdS
h 2
¼
1 2
Z n o Nxx ε0xx þ Nyy ε0yy þ Nxy g0xy þ Mxx kxx þ Myy kyy þ Mxy kxy dS S
1 þ 2
Z S
h ) Z2 ( vE~ vE~ vE~ Dx þ Dy þ Dz dzdS vx vy vz h 2
h Z2 ~ ~ ~ 1 vM vM vM Bx þ þ By þ Bz dzdS 2 vx vy vz S h 2 Z o 1 n ¼ Nxx ε0xx þ Nyy ε0yy þ Nxy g0xy þ Mxx kxx þ Myy kyy þ Mxy kxy dS 2 Z
S
1 þ 2
Z
h Z2 Dx
S
1 þ 2
Z
h 2
h Z2 Bx
S
h 2
pz pz v4 v4 p4 pz 240 cos Dy cos þ Dz sin þ dzdS vx h vy h h h h
pz pz vj vj pj pz 2j0 By cos þ Bz þ cos sin dzdS h h h vx vy h h
(10)
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S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
The first two differential equations can be already satisfied by introducing the Airy stress function f ðx; yÞ as follows
where the stress resultants are in the following forms
9 8 9 8 M9 8 9 2 38 0 E > εxx l2 V2 ε0xx > b Nxx m2 V2 Nxx > Nxx c 12 h 0 c h b > > > > > Nxx > > > > > > > > > = > = < = 6 11 < = < < 7 0 2 2 0 2 2 E M 7 6 Nyy m V Nyy ¼ 4 b þ Nyy þ Nyy c 12 h b c 22 h 0 5 εyy l V εyy > > > > > > > > > > > > > > ; > ; : ; > > > : : b 0 0 c 66 h : g0xy l2 V2 g0xy ; Nxy m2 V2 Nxy 0 0 3 2 b c 11 h3 b c 12 h3 0 7 6 12 12 9 8 9 9 6 9 8 8 78 7> kxx l2 V2 kxx > > M E > > MM > 6 Mxx m2 V2 Mxx > > > > > 7> 6 < = > = = 6 = > < xx > < xx > < 3 3 7 b b c 12 h c 22 h 7 kyy l2 V2 kyy þ ME M Myy m2 V2 Mxx ¼ 6 þ Myy 0 7 6 yy > > > > 6 12 > > 12 > > > > > > > > > 7> ; 6 : 7 : k l 2 V2 k ; : 0 ; : 0 ; Mxy m2 V2 Mxx xy xy 7 6 4 b c 66 h3 5 0 0 6
(11)
in which
h8 h 8 8 9 9 9 9 8 Lxx > Lxx > Nxx > Mxx > > > > > 2 2 > > > > > > > > Z Z < < < = = = = < Nyy ¼ Lyy dz ; Myy ¼ Lyy zdz > > > > > > > > > > > > > > > > : : : ; ; ; ; : Nxy Lxy Mxy Lxy h h 2 2 9 ( 8 ) 8 M9 ( ) E = = < < Nxx Nxx 2b e 31 40 2b q 31 j0 ¼ ¼ ; : NM ; : NE ; 2b e 4 2b q j yy
32 0
9 8 2hb e 31 > > > > 4ðx; yÞ > > = < p ¼ ; E :M ; > > > > > > 2hb e 32 yy ; : 4ðx; yÞ 9 8 E = < Mxx
p
Nxx ¼
9 8 M= < Mxx : MM yy
;
¼
Nyy ¼
v2 f ðx; yÞ ; vx2
Nxy ¼
v2 f ðx; yÞ vxvy
8 9 2h b q 31 > > > > j ðx; yÞ > > < =
p
> > > > > > q 32 : 2h b jðx; yÞ ;
p
(12) With the aid of virtual work's principle, the governing differential equations for an axially loaded MEE composite nanoshell are derived as
vNxx vNxy þ ¼0 vx vy
(13a)
vNxy vNyy þ ¼0 vx vy
(13b)
2 0 2 0 v2 ε0xx v εyy v gxy ¼ þ vxvy vy2 vx2
!2 v2 w v2 w v2 w 1 v2 w 2 vxvy vx vy2 R vx2
¼0 (13c) h
(13d)
2h
h
Z2 pz vB pz pB pz vBx y z þ þ dz ¼ 0 cos cos sin h h h vx vy h
2h
(13e)
(15)
Subsequently, by inserting equation (14) in the inverse of equation (11) in conjunction with equations (13aee) and (15), the nonlocal strain gradient governing differential equations for an axially loaded MEE composite nanoshell can be rewritten in terms of the mid-plane displacement components as below
!
v4 f 1 2b c 12 h þ 2 2 vx4 2 2 b c 66 h ðb ðb c 11 hÞ ðb c 11 hÞ ðb c 12 hÞ c 12 hÞ ! b v4 f 1 v2 w c 11 h þ þ 2 2 vy4 R vx2 c 12 hÞ ðb c 11 hÞ ðb !2 v2 w v2 w v2 w ¼ 2 vxvy vx vy2 b c 11 h
!
v4 f vx2 vy2
(16a)
v2 Mxy v2 Myy Nyy v2 Mxx v2 w v2 w v2 w þ Nyy 2 þ þ Nxx 2 þ 2Nxy þ2 þ 2 2 vxvy vxvy R vx vy vx vy
Z2 pz vD pz pD pz vDx y z þ þ dz ¼ 0 cos cos sin h h h vx vy h
(14)
Additionally, for a perfect shell-type structure, the compatibility equation corresponding to the mid-plane strain components can be expressed as
32 0
yy
v2 f ðx; yÞ ; vy2
b ðb c 12 þ 2b c 66 Þh3 v4 w c 11 h3 v4 w 1 l 2 V2 þ 12 vx4 6 vx2 vy2 ! ! b 1 v2 f 2h v2 4 v2 4 c h3 v4 w b b þ 11 þ e e 31 32 R vx2 12 vy4 p vx2 vy2 ! 2h v2 j v2 j b q 32 2 q 31 2 þ b p vx vy ! v2 w v2 f v2 w v2 f v2 w v2 f 2 2 þ ¼ 1m V 2 vxvy vxvy vy2 vx2 vx2 vy2
(16b)
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
Z2pR( ! ! b b b bs 22 h v2 4 d 11 h v2 j d 22 h v2 j s 11 h v2 4 þ þ þ 2 2 2 2 vx2 vy2 vx2 vy2 ! ! 2b e h v2 w 2b e 32 h v2 w p2 s33 p2 d33 4 31 j p p 2h 2h vx2 vy2 ¼0
263
!
! b v2 f v2 f w c 12 h þ 2 2 vx2 2 2 vy2 R c 12 hÞ c 12 hÞ ðb c 11 hÞ ðb ðb c 11 hÞ ðb 0 ! b 1 vw 2 c 12 h E M þ þN N x x 2 2 2 vy c 12 hÞ ðb c 11 hÞ ðb ) ! b c 11 h E M dy þN N y y 2 2 c 12 hÞ ðb c 11 hÞ ðb b c 11 h
¼0 (16c)
(18) In addition, the unit shortening associated to the movable ends of the MEE composite nanoshell can be evaluated by
! b d 11 h v2 4 þ 2 vx2
! ! ! b b b d 22 h v2 4 b 11 h v2 j b 22 h v2 j þ þ 2 2 2 vy2 vx2 vy2 ! ! ! ! 2b q h v2 w 2b q 32 h v2 w p2 bd 33 p2 bb 33 4 31 j p p 2h 2h vx2 vy2
Dx
1 ¼ 2pRL L
Z2pR ZL 0
1 ¼ 2pRL
Z2pR ZL (
þ
(16d) Moreover, the boundary conditions at the left and right ends of the MEE composite nanoshell are supposed to be clamped. As a consequence, one will have: w ¼ 0; vw vx ¼ 0 at x ¼ 0; L On the other hand, the equilibrium of applied loads in the x-axis direction can be expressed as
þ
b c 12 h
!
v2 f vx2
2 2 ðb c 11 hÞ ðb c 12 hÞ ! b v2 f 1 vw 2 c 11 h 2 2 vy2 2 vx c 12 hÞ ðb c 11 hÞ ðb ! b c 11 h NxE þ NxM 2 2 c 12 hÞ ðb c 11 hÞ ðb ! ) b c 12 h E M Ny þ Ny dxdy 2 2 ðb c 11 hÞ ðb c 12 hÞ 0
¼0
0
vu dxdy vx
0
(19)
3. Solving methodology for asymptotic solutions
Z2pR
Nxx dy þ 2pRhsxx ¼ 0
(17)
0
For a closed shell-type structure, the periodicity condition yields the following relation
X¼
px
;
y Y¼ ; R
b¼
L
;
h¼
L2
;
ε¼
The following dimensionless parameters are taken into consideration to obtain the solution of the problem in a more general framework
p2 Rh
L2 p h n o b c 11 h b c h b c h b c h b c h b c h * * * A*11 ; A*12 ; A*66 ; H11 ¼ ; H12 ; H66 ; 12 ; 66 ; 11 ; 12 ; 66 A00 A00 A00 12A00 12A00 6A00 n o b b q j b q j e 31 400 e 32 400 b * * * * E31 ¼ ; E32 ; Q31 ; Q32 ; ; 31 00 ; 32 00 A00 A00 A00 A00 ( ) o n b b b d 11 400 j00 d 22 400 j00 d 33 400 j00 ; ; D*11 ; D*22 ; D*33 ¼ A00 h A00 h A00 h ( ) n o b s 11 4200 b s 42 b s 42 b b j2 b b j2 b b j2 S*11 ; S*22 ; S*33 ; B*11 ; B*22 ; B*33 ¼ ; 22 00 ; 33 00 ; 11 00 ; 22 00 ; 33 00 A00 h A00 h A00 h A00 h A00 h A00 h W¼
L
εw ; h
F¼
ε2 f
; A00 h2
pR
F¼
ε2 R4 ; 400 h
2 2
3.1. Boundary layer theory of nonlocal strain gradient shell buckling
J¼
ε2 Rj ; j00 h
Px ¼
sxx R ; 2A00
dx ¼
Dx R 2Lh
(20)
264
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
where
A00 ¼ b c 11 h;
sffiffiffiffiffiffiffiffiffiffi A00 h ; ¼ b s 33
400
j00
1 2p
sffiffiffiffiffiffiffiffiffiffi A00 h ¼ b b
0
0
1
4 A*11 B Cv F B 1 @ 2 2 A 4 þ @ * vX A66 A*11 A*12 1
0
1
4 4 2A*12 A*11 B C 2 v F C 4v F þ @ 2 2 A b 2 2 Ab 2 2 vX vY vY 4 A*11 A*12 A*11 A*12
þ
v2 W vX 2
¼b
v2 W vXvY
2
!2
W v2 W vX 2 vY 2
2v
b
2
2
1p G þ
2 2 2V
"
4 * 4v W H11 b 4
ε
#!
vY
* b2 2Q32
p
v2 J vY 2
2
¼ 1p G
* H11
2
2 2 1V
v4 W v4 W * * þ 2 H12 þ H66 b2 2 2 4 vX vX vY
* * b * v2 F 2E31 v2 F 2E32 v2 F 2Q31 v2 J 2 2 2 p p p vX vX vY vX 2
b
2
v2 W v2 F v2 W v2 F v2 W v2 F þ 2 vXvY vXvY vX 2 vY 2 vY 2 vX 2
!
(21b)
v2 F S*11 2 vX
þ
2 2v F S*22 b vY 2 2
þ
* hb 4E32
p
v2 W vY 2
þ
v D*11
!#
2
J
vX 2
þ
(22)
80 0 1 1 Z2p> < * * 2 2 A11 A12 Cv F B C 2v F B A 2 @ Ab @ 2 2 2 2 > vX vY 2 : A*11 A*12 A*11 A*12 0 20 1 2 2 * A12 b vW 6B C * þW þ 4@ 2 2 A 2E31 F0 2 vY * * A11 A12 0 1 * A11 C * * * þ 2Q31 J0 B @ 2 2 A 2E32 F0 þ 2Q32 J0 A*11 A*12 9 3 > = 7 5ε1 dY > ;
(23)
2
v2 F dY þ 2εP x ¼ 0 vY 2
¼0 (21a)
b2
The dimensionless form of the periodicity condition yields
33
As a consequence, the dimensionless form of the nonlocal strain gradient governing differential equations for an axially loaded MEE composite nanoshell can be rewritten as
0
Z2p
2v D*22 b
2
J
vY 2
" ε
2
* h 4E31 v2 W p vX 2
!
Additionally, the dimensionless unit shortening of the electromagnetic actuated MEE composite nanoshell under axial compressive load takes the following form
80 1 Z2p Zp > < * 2 A11 1 C 2v F B dx ¼ 2 @ 2 2 A b > 4p ε vY 2 : A*11 A*12 0 0 0 2 1 2 * 2 A12 B 6 C v F 1 vW @ 4 2 2 A vX 2 2 vX * * A11 A12 0 1 * A11 B C * * @ 2 2 A 2E31 F0 þ 2Q31 J0 A*11 A*12 9 0 1 3 > = * A12 B C 7 1 * * 2E ε F þ 2Q J þ @ dXdY A 5 2 2 32 0 32 0 > ; A*11 A*12 (24)
p2 S*33 hF p2 D*33 hJ
¼0 (21c) " ! * 2 2 2 2 v2 F 2v F 2v J * * v J * 2 4Q31 h v W þ D b þ B þ B b ε 22 11 22 p vX 2 vX 2 vY 2 vX 2 vY 2 !# * hb2 4Q32 v2 W þ p2 D*33 hF p2 B*33 hJ p vY 2
D*11
¼0 (21d) Furthermore, the clamped edge supports at the left (X ¼ 0) and right (X ¼ p) ends of the MEE composite nanoshell can be given in dimensionless form as W ¼ 0; vW vX ¼ 0. Also, the boundary layer-type equilibrium requirement for an axially loaded MEE composite nanoshell can be given as
3.2. Solving process based upon a two-stepped perturbation technique As it was explained, by taking the small perturbation parameter ε into consideration, the boundary layer-type nonlocal strain gradient governing equation (21aed) were established for axially loaded MEE composite nanoshells. Thereafter, a two-stepped perturbation technique [65e71] is employed, based upon which the independent variables are assumed as the summations of the regular and boundary layer solutions as below
~ c ðX; Y; ε; 2Þ W ¼ WðX; Y; εÞ þ WðX; Y; ε; xÞ þ W
(25a)
~ Y; ε; xÞ þ b F ðX; Y; ε; 2Þ F ¼ FðX; Y; εÞ þ FðX;
(25b)
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
~ ðX; Y; εÞ þ F b ðX; Y; εÞ F ¼ FðX; Y; εÞ þ F
(25c)
~ ðX; Y; εÞ þ J b ðX; Y; εÞ J ¼ JðX; Y; εÞ þ J
(25d)
where the accent characters e, ~ and ^ denote, respectively, the regular solution, and the boundary layer solutions relevant to the left (X ¼ 0) and right (X ¼ p) ends of the MEE composite nanoshell. After that, each part of the solutions can be introduced in the form of the perturbation expansions as follows
265
the dimensionless nonlocal strain gradient governing equation (21aed) to achieve the perturbation sets of equations associated with both of the regular and boundary layer solutions. Thereafter, the expressions with similar order of ε are collected to obtain the required relationships for capturing the asymptotic solutions. The assumption for the initial buckling mode shape of MEE composite nanoshell is considered as below
W 2 ðX; YÞ ¼ A
ð2Þ 00
þA
ð2Þ 11
sinðmXÞsinðnYÞ þ A
P
εi W i ðX; YÞ;
i¼0
c ðX; Y; 2Þ; c ðX; Y; ε; 2Þ ¼ P εiþ1 W W iþ1 i¼0
b ðX; Y; ε; 2Þ ¼ F
P
i¼0
b ðX; Y; 2Þ; εiþ2 F iþ1
i¼0 X
b F ðX; Y; ε; 2Þ ¼ b ðX; Y; ε; 2Þ ¼ J
X ; ε1=2
2¼
pX ε1=2
(26)
εiþ2 b F iþ2 ðX; Y; 2Þ
i¼0 X
b εiþ2 J iþ2 ðX; Y; 2Þ
i¼0
in which x and 2 represent the boundary layer variables in the following forms
x¼
cosð2nYÞ (28)
X FðX; Y; εÞ ¼ εi F i ðX; YÞ i¼0 i¼0 X P FðX; Y; εÞ ¼ εi Fi ðX; YÞ; JðX; Y; εÞ ¼ εi Ji ðX; YÞ i¼1 i¼1 X P iþ1 ~ ~ ~ ε W iþ1 ðX; Y; xÞ; FðX; Y; ε; xÞ ¼ εiþ2 F~iþ2 ðX; Y; xÞ WðX; Y; ε; xÞ ¼ i¼0 i¼0 X ~ ðX; Y; ε; xÞ ¼ P εiþ2 F ~ ðX; Y; xÞ; J ~ ~ ðX; Y; ε; xÞ ¼ F εiþ2 J iþ2 iþ2 ðX; Y; xÞ WðX; Y; εÞ ¼
ð2Þ 02
(27)
Subsequently, equations (25aed) and (26) are substituted into
By performing some mathematical calculations, the asymptotic solutions corresponding to each independent variable of the problem are extracted as presented in Appendix A. Then, the solutions are inserted in the size-dependent equations (22) and (24), the obtained relations are rearranged with respect to the second ð2Þ perturbation parameter (A 11 ε) which results in explicit analytical expressions for the nonlocal strain gradient load-deflection and load-shortening stability paths of the MEE composite nanoshell
Fig. 2. A brief description of the two-stepped perturbation solving process.
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S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
under axial mechanical compressive load in conjunction with external electric and magnetic potentials, respectively, as follows
Table 1 Mechanical, electric and magnetic properties of BiTiO3CoFe2O4 composite material [72,73]. Elastic properties (GPa) c11¼c22 c12 c13 c33 c66
e31¼e32 e33
dxðiÞ A
A
i ð2Þ 11 ε
A
2 ð2Þ 11 ε
i ð2Þ 11 ε
i¼0;2;4;… ð0Þ
5:64 109
E
ð2Þ
M
q31¼q32 q33
Magnetoelectric properties
290.1 349.9
N:s V:C
d11¼d22
5:367 1012
d33
2737:5 1012
A
N:s2 C2
b11¼b22
297 106
b33
83:5 106
ð4Þ
þPx
A
4 ð2Þ 11 ε
þ…
(29)
dEx dM x
A
2 ð2Þ 11 ε
ð4Þ
þ dx
A
4 ð2Þ 11 ε
þ…
(30)
The different weight coefficients in the above equations are defined in Appendix B. On the basis of this assumption that the maximum deflection of the MEE composite nanoshell occurs at the point with dimensionless coordinates of ðX; YÞ ¼ ðp=2m; p=2nÞ, it yields
6:35 109
N A:m
Piezomagnetic properties
¼ dx dx dx þ dx
s11¼s22 s33
ð2Þ
X
dx ¼
C V:m
Dielectric properties
Magnetic properties
ð0Þ
¼Px þPx
2.2 9.3
ðiÞ
Px
i¼0;2;4;…
C m2
Piezoelectric properties
X
Px ¼
226 125 124 216 44.2
ð2Þ 11 ε
¼
wm þS h
2
þS
1
w 2 m þS 2 h
(31)
in which wm stands for the maximum deflection of the MEE composite nanoshell and the symbols S 1 and S 2 are defined in Appendix B. Also, the present solving process is shown in Fig. 2 briefly as a flowchart.
Table 2 Comparison of the nonlinear critical buckling loads for perfect FGM cylindrical shells made of the mixture of ZrO2 and Ti6Al4V (L=R ¼ 2; R=h ¼ 200). Material property gradient index
Critical buckling load (MPa) Present work
Ref. [74]
0.2 1 5
439.860 380.707 332.981
441.053 382.955 335.512
4. Results and discussion After proposing explicit analytical expressions for nonlocal strain gradient stability curves of MEE nanoshells under axial compression combined with external electromagnetic potential, selected numerical results are presented in this section. It is assumed that the MEE nanoshells are made of two-phase BiTiO3CoFe2O4 composite material whose material properties are given in Table 1. Also, it is supposed that R ¼ 50h, L ¼ 2R, and the two ends
Fig. 3. Dimensionless nonlocal strain gradient load-deflection stability paths of an axially loaded MEE composite nanoshell (F0 ¼ J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
of nanoshells are clamped. Firstly, the efficiency as well as accuracy of the present solution methodology is checked. Because in accordance with the best authors' knowledge, there is no study in the open literature in which the size-dependent nonlinear instability of MEE nanoshells is investigated, the nonlocal strain gradient terms associated with the constitutive relations removed and the nonlinear critical buckling
267
loads of FGM cylindrical shells at usual scale (macroscale) made from a mixture of zirconia (ZrO2) and titanium alloy (Ti6Al4V) with simply supported end conditions and subjected to axial compression are tabulated in Table 2 and compared with those obtained by Huang and Han [74] using Galerkin method. An excellent agreement is found between two types of solving process which confirms the validity of the given preceding numerical results.
Fig. 4. Dimensionless nonlocal strain gradient load-shortening stability paths of an axially loaded MEE composite nanoshell (F0 ¼ J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
Fig. 5. Influence of external electric potential on the nonlocal strain gradient load-deflection stability path of an axially loaded MEE composite nanoshell (J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
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Figs. 3 and 4 depict, respectively, the dimensionless nonlocal strain gradient load-deflection and load-shortening stability curves of axially loaded MEE composite nanoshells with different small scale parameters. It is seen that the softening-stiffness effect of nonlocality causes to reduce both of the critical buckling load and the minimum postbuckling load, while the hardening-stiffness of strain gradient size dependency leads to enhance them. In addition, it can be observed that both types of size effect has no influence on the slope of prebuckling part of the load-shortening response, but
the nonlocal size effect causes to decrease the end-shortening of axially loaded MEE composite nanoshell at the critical buckling point, while the strain gradient size effect leads to increase it. Moreover, it is obvious that the both types of size dependency have approximately no influence on the width of postbuckling domain, so the maximum deflection associated with the minimum postbuckling load of MEE composite nanoshells remains approximately constant for all values of the small scale parameters. Figs. 5 and 6 in order display the dimensionless nonlocal strain
Fig. 6. Influence of external electric potential on the nonlocal strain gradient load-shortening stability path of an axially loaded MEE composite nanoshell (J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
269
Fig. 7. Influence of external magnetic potential on the nonlocal strain gradient load-deflection stability path of an axially loaded MEE composite nanoshell (F0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
gradient load-deflection and load-shortening stability curves of MEE composite nanoshells subjected to axial compression combined with external electric potential corresponding to various values of small scale parameters. It is found that a positive electric potential causes to enhance the critical axial compressive load, but a negative electric potential leads to reduce it. However, both of the positive and negative electric potentials have no influence on the minimum postbuckling load of axially loaded MEE composite nanoshell. Moreover, it is shown that the positive and negative electric potentials induce, respectively, initial shortening and initial extension in the MEE composite nanoshell. As a consequence, the positive electric potential increases the critical end-shortening of an axially loaded MEE composite nanoshell, but the negative electric potential decreases it. Also, it is displayed that by applying a positive electric potential, the postbuckling domain of the axially loaded MEE composite nanoshell increases, while the negative electric potential causes to reduce it. In Figs. 7 and 8, the dimensionless nonlocal strain gradient loaddeflection and load-shortening stability curves of MEE composite nanoshell under axial compressive load with combination of magnetic potential are illustrated, respectively, corresponding to various small scale parameters. It is revealed that a positive external magnetic potential reduces the critical axial compressive load, but a negative external magnetic potential enhances it. However, the external magnetic potential with both of the positive and negative signs has no influence on the minimum postbuckling load of the axially loaded MEE composite nanoshell. As a result, a negative magnetic potential increases the width of the postbuckling domain, while a positive one causes to decrease it. Additionally, it can be seen that the positive and negative magnetic potentials induce, respectively, initial extension and initial shortening in the MEE composite nanoshell which in order lead to decrease and increase the critical end-shortening of an axially loaded MEE composite nanoshell. In Figs. 9 and 10, the error of the classical shell model in
prediction of the critical buckling load of an axially loaded MEE composite nanoshell is demonstrated corresponding to different external electric potentials and magnetic potentials, respectively. The value of error is calculated as below
classical buckling load nonclassical buckling load Error ð%Þ ¼ classical buckling load 100 It is obvious that by increasing the value of nonlocal and strain gradient parameters, the size dependency becomes more significant, so the error of the classical model increases. Furthermore, it is indicated that applying a negative external electric potential causes to reduce the error of the classical model, while a positive one leads to enhance it. However, this pattern is vice versa for external magnetic potential as the negative external magnetic potential increases the error of the classical model, but a positive value decreases it. In other words, nonlocal strain gradient size dependency plays the most important role in nonlinear instability of MEE composite nanoshells under external positive electric potential and negative magnetic potential. 5. Concluding remarks The objective of this work was to analyze the size-dependent buckling and postbuckling behavior of MEE composite nanoshells under combination of axial compressive loaded and electromagnetic external potential including the both of hardening-stiffness and softening-stiffness size effects. To this end, the nonlocal strain gradient theory of elasticity was incorporated to the classical shell theory. After that, an improved perturbation technique in conjunction with the boundary layer theory of shell buckling was employed to propose explicit analytical expressions for nonlocal strain gradient stability paths of MEE composite nanoshells. It was revealed that the softening-stiffness effect of nonlocality
270
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Fig. 8. Influence of external magnetic potential on the nonlocal strain gradient load-shortening stability path of an axially loaded MEE composite nanoshell (F0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
causes to reduce both of the critical buckling load and the minimum postbuckling load, while the hardening-stiffness of strain gradient size dependency leads to enhance them. Moreover, it was observed that the positive electric potential and negative magnetic potential cause enhance the critical axial compressive load, but their opposite sign leads to decrease the buckling resistance of the axially loaded MEE composite nanoshell. Additionally, it was seen that both of the electric and magnetic potentials with any sign have no influence on the value of minimum postbuckling load.
Furthermore, the positive electric potential and negative magnetic potential induce an initial shortening in the MEE composite nanoshell that leads to increase the critical end-shortening, while the negative electric potential and positive magnetic potential induce an initial extension which causes to reduce the critical endshortening of the axially loaded MEE composite nanoshell. In addition, it was demonstrated that a positive electric potential and a negative magnetic potential cause to increase both types of the size dependency in the nonlinear instability behavior of axially
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
271
Fig. 9. Error of the classical model in prediction of critical buckling load of axially loaded MEE composite nanoshell under different external electric fields (J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
Fig. 10. Error of the classical model in prediction of critical buckling load of axially loaded MEE composite nanoshell under different external magnetic fields (F0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
loaded MEE composite nanoshells, while a negative electric potential and a positive magnetic potential play an opposite role.
Appendix A The solutions relevant to each independent variable of the problem are achieved in asymptotic forms as below.
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S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
h
ð4Þ ð4Þ F ¼ ε4 G11 sinðmXÞsinðnYÞ þ G02
cosð2nYÞ þ O ε5 i
h
i
ð4Þ ð4Þ J ¼ ε4 H11 sinðmXÞsinðnYÞ þ H02 cosð2nYÞ þ O ε5
A
ð1Þ 00
b2 n2 8
A
ð1Þ 2 11
þ b2 n2 A
ð2Þ 2 02
(A7)
(A4) Appendix B
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi urffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 u A*11 A*22 ðA*12 Þ u t A*11 H *11
(A5)
2 0
A
¼
(A3)
where
G¼
ð4Þ 00
1
20
¼
ð2Þ
1 ¼ 2
1
A*12 A*11 6B C C 2 2 AP x þ 4@ 2 2 A * * * * A11 A12 A11 A12 0 1 * A12 C * * * 2E32 F0 þ 2Q32 J0 B @ 2 2 A 2E31 F0 * * A11 A12 3 * þ 2Q31 J0 7 5ε3
B ¼ 2@
(A6)
o 1n 2 K0 ε1 þ ðK5 þ K6 K12 þ K7 K13 Þb ε 2
ð0Þ
Px
Px
(
! 3K02 K16 ε1 K2 þ K3 K14 þ K4 K15 2
þ
(B1)
K10 K16 2
2K0 K12 b ð1 K6 K16 Þ þ 2K0 K13 b ð1 K7 K16 Þ K2 þ K3 K14 þ K4 K15
2 J 2 2 K0 K8 K16 20 þ 4K0 K9 K16 þ 4K0 K9 K16 J20 þ K0 K16 J20 4ðK0 J20 K8 Þ ! ) 2 þ 8K K K K02 K16 0 9 16 þ ε 8ðK0 K8 Þ
þ
(B2)
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
P
ð4Þ x
1 ¼ 2
(
0
3K03 K11 K16 J13
S
ðK0 J13 K11 ÞðK2 þ K3 K14 þ K4 K15 Þ2 K03 K16 ðK0 þ 2K11 Þ
þ
ðK0 K11 ÞðK2 þ K3 K14 þ K4 K15 Þ2 ! ) 2 2K 3 K 2 6K02 K16 1 0 16 þ ε ðK2 þ K3 K14 þ K4 K15 Þ2 0
dxð0Þ
(B3)
0
1
2 * B A G P 2x 12 B B C ¼ @ 2 AP x þ B 2 2 2 @ A*11 A*12 p A*11 A*12 1 2 * C 2 A12 P x C 1=2 ε 2 2 C A * * * pGA11 A11 A12
dxð2Þ ¼ dxð4Þ ¼
8p2 ðK2 þ K3 K14 þ K4 K15 Þ
2
m2 4
K0 b n2 þ 4K9 4ðK0 K8 Þ
!2 ε3 (B6)
00
1
A*12 B C * dEx ¼ B @@ 2 2 AE32 * * A11 A12 0 1 1 B @
A*11 C * C 2 2 2 AE31 Aε F0 * * A11 A12
00
(B7)
1
A*12 BB C * ¼ @@ 2 2 AQ32 A*11 A*12 0 1 1 B @
A*11 C * C 2 2 2 AQ31 Aε J0 A*11 A*12
(B8)
where
m2 þ b2 n2 ; J20 ¼ 1 þ 4p2 G 21 m2 ¼ 1 þ p2 G 21 m2 þ 9b2 n2 ; I11 ¼ 1 þ p2 G 22 m2 þ b2 n2
J11 ¼ 1 þ p2 G J13
2 1
(B9) where Ki ði ¼ 0; …; 16Þ are constant parameters extracted via the perturbation sets of assssssss in terms of mechanical, magnetic and electric properties.
S
1
¼
0
K0 B ε1 þ 2@ K2 þ K3 K14 þ K4 K15
1 A*11
A*12 2
A*12
1
References
(B5)
þ 2
20
(B11)
m2 ε 16
GK02 ε3=2
1
A*12 A*11 B C ð0Þ 6B C ¼ 2@ 2 2 AP x 4@ 2 2 A A*11 A*12 A*11 A*12 0 1 * A C * * * 12 2E32 F0 þ 2Q32 J0 B @ 2 2 A 2E31 F0 * * A11 A12 3 * þ 2Q31 J0 7 5ε3
A*11 2
(B4)
dM x
2
273
C ð2Þ 2 AP x (B10)
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Contents lists available at ScienceDirect
Composites Part B journal homepage: www.elsevier.com/locate/compositesb
Nonlocal strain gradient shell model for axial buckling and postbuckling analysis of magneto-electro-elastic composite nanoshells S. Sahmani, M.M. Aghdam* Department of Mechanical Engineering, Amirkabir University of Technology, P.O. Box 15875-4413 Tehran, Iran
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 April 2017 Received in revised form 2 September 2017 Accepted 2 September 2017 Available online 8 September 2017
The present study deals with the size-dependent nonlinear buckling and postbuckling characteristics of magneto-electro-elastic cylindrical composite nanoshells incorporating simultaneously the both of hardening-stiffness and softening-stiffness size effects. To accomplish this purpose, the nonlocal strain gradient elasticity theory is applied to the classical shell theory. Via the virtual work's principle, the sizedependent governing differential equations are constructed including the coupling terms between the axial mechanical compressive load, external magnetic potential and external electrical potential. The nonlinear prebuckling deformations and the large postbuckling deflections are taken into consideration based upon the boundary layer theory of shell buckling. Finally, an improved perturbation technique is employed to achieve explicit analytical expressions for nonlocal strain gradient stability curves of magneto-electro-elastic nanoshells under various surface electric and magnetic voltages. It is seen that a positive electric potential and a negative magnetic potential cause to increase both of the nonlocality and strain gradient size dependencies in the nonlinear instability behavior of axially loaded magneto-electroelastic composite nanoshells, while a negative electric potential and a positive magnetic potential play an opposite role. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Nanomechanics Magneto-electro-elastic materials Nonlocality Strain gradient size dependency Nonlinear instability
1. Introduction In those early years, a new group of composite materials containing simultaneously the piezoelectric and piezomagnetic phases have been attracted the attention of research community due to their smart characteristics. This attraction has been even somehow more than that dedicated previously to functionally graded composite materials [1e3]. The novel magneto-electro-elastic coupling feature of these smart composites makes them more sensitive and adaptive as an eminent candidate in numerous areas of technology such as semiconducting capacitors, robotics, structural health monitoring, sensors and actuators and so on [4e7]. In recent years, the mechanical responses of magneto-electro-elastic macrostructures have been investigated extensively based on the classical continuum mechanics [8e15].
* Corresponding author. E-mail address: [email protected] (M.M. Aghdam). http://dx.doi.org/10.1016/j.compositesb.2017.09.004 1359-8368/© 2017 Elsevier Ltd. All rights reserved.
The synthesis of nanotechnology-based products is one of the important fields of engineering sciences because of their wide range of application. It is well recognized that nanostructures possess unique characteristics which are greatly depend on their shape as well as their crystalline arrangement. Accordingly, several non-classical continuum theories of elasticity have been proposed and employed to anticipate different size dependencies in mechanical behaviors of nanostructures [16e46]. Recently, some studies have been carried out to predict the mechanical behavior of nanoscaled structures made of magneto-electro-elastic (MEE) composite materials. Ke and Wang [47] analyzed the sizedependent free vibrations of MME composite Timoshenko nanobeams based on the nonlocal elasticity theory. Ke et al. [48,49] reported the size-dependent natural frequencies of MEE composite nanoplates and cylindrical nanoshells via nonlocal theory of
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elasticity implemented in, respectively, the Kirchhoff plate theory and Love's shell theory. Li et al. [50] analyzed bending, buckling and free vibration of MEE composite Timoshenko nanobeams on the basis of nonlocal continuum theory. Farajpour et al. [51] established a nonlocal plate model to study the size effect on the nonlinear vibration behavior of MEE composite nanoplate under external electromagnetic loading condition. Jamalpoor et al. [52] used a nonlocal plate model to investigate the free vibration and biaxial buckling double-MEE nanoplate-system subjected to electric and magnetic potentials. Generally, in the previous investigations, it has been demonstrated that the small scale effect in type of the stress nonlocality results in softening-stiffness influence, while the strain gradient size dependency leads to a hardening-stiffness effect. As a consequence, Lim et al. [53] proposed a new size-dependent elasticity theory namely as nonlocal strain gradient theory which includes the both softening and stiffening influences to describe the size dependency in a more accurate way. Subsequently, a few studies have been performed on the basis of nonlocal strain gradient elasticity theory. Li and Hu [54] reported the size-dependent critical buckling loads of nonlinear EulerBernoulli nanobeams based upon nonlocal strain gradient theory of elasticity. They also presented the size-dependent frequency of wave motion on fluid-conveying carbon nanotubes via nonlocal strain gradient theory [55]. Yang et al. [56] established a nonlocal strain gradient beam model to evaluate the critical voltages corresponding to pull-in instability FG carbon nanotube reinforced actuators at nanoscale. Simsek [57] used nonlocal strain gradient theory to capture the size effects on the nonlinear natural frequencies of FGM Euler-Bernoulli nanobeams. Farajpour et al. [58] proposed a new size-dependent plate model for buckling of orthotropic nanoplates based on nonlocal strain gradient elasticity theory. Tang et al. [59] studied the wave propagation in a viscoelastic carbon nanotube via nonlocal strain gradient elasticity theory. Ebrahimi and Dabbagh [60] employed nonlocal strain gradient elasticity theory to study the flexural wave propagation in functionally graded MEE higher-order shear deformable nanoplates. Li et al. [61] utilized the nonlocal strain gradient elasticity theory within the framework of the Euler-Bernoulli beam theory to explore bending, buckling and free vibration of axially functionally graded nanobeams. Sahmani and Aghdam [62] employed the nonlocal strain gradient elasticity theory within the framework of the hyperbolic shear deformation shell theory to analyze the size-dependent nonlinear instability of a microtubule embedded in an elastic foundation related to the cytoplasm of a living cell. The objective of this work is to capture simultaneously the nonlocality and strain gradient size dependencies on the nonlinear buckling and postbuckling behavior of axially loaded MEE composite nanoshells under external electric and magnetic potentials. Thereby, the nonlocal strain gradient elasticity theory is incorporated to the classical shell theory to develop a more comprehensive size-dependent shell model. On the basis of the variational approach, the non-classical differential equations are derived. Afterwards, the boundary layer theory of shell buckling and an improved perturbation technique are put to use to achieve explicit analytical expressions for the nonlocal strain gradient stability curves of MEE composite nanoshells under combination of axial compressive load, electric and magnetic potentials.
259
2. Nonlocal strain gradient MEE shell model As illustrated in Fig. 1, an MEE composite cylindrical nanoshell with length L, radius R and thickness h is supposed under axial compressive load combined with external electric and magnetic potentials. Based upon the nonlocal strain gradient elasticity theory, the total nonlocal strain gradient stress tensor L can be expressed as below [53].
Lij ¼ sij Vs*ij ;
i; j ¼ x; y
(1)
where V is the gradient symbol, s and s* represent, respectively, the stress and higher-order stress tensors which can be introduced for an MEE composite material as follows
sij ¼
Z n h 0 0 0 91 X X cijkl εkl X emij E m X U
0 io dU qnij M n X
s*ij ¼ l2
(2a)
Z n h 0 0 0 92 X X cijkl Vεkl X emij E m X U
0 io dU qnij M n X
(2b)
in which cijkl ; emij ; qnij in order are the elastic, piezoelectric and piezomagnetic constants. Also, 91 and 92 are, respectively, the principle attenuation kernel function including the nonlocality and additional kernel function associated with the nonlocal effect of the 0 first-order strain gradient field, X and X in order denote a point and any point else in the body, and l stands for the internal strain gradient length scale parameter. E and M are the electric field and magnetic intensity, respectively. In a similar way, the other basic equations for an MEE composite material can be expressed based upon the nonlocal strain gradient elasticity theory as below
Di ¼
Z n h 0 0 0 91 X X eikl εkl X sim E m X U
0 io dU din M n X
D*i ¼ l2
(3a)
Z n h 0 0 0 92 X X eikl Vεkl X sim E m X U
0 io dU din M n X
(3b)
260
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Fig. 1. MEE composite nanoshell under combination of axial compressive load, electric and magnetic potentials.
Bi ¼
Z n h 0 0 0 91 X X qikl εkl X dim E m X U
0 io dU bin M n X
B*i ¼ l2
(3c)
Z n h 0 0 0 92 X X qikl Vεkl X dim E m X U
0 io dU bin M n X
(3d)
where Di and D*i are, respectively, the electric displacement and higher-order electric displacement vectors, Bi and B*i in order represent the magnetic induction and higher-order magnetic induction vectors. Also, sij ; dij ; bij denote dielectric, magnetoeletric and magnetic constants, respectively. Following the method of Eringen, the constitutive relationship corresponding to the total nonlocal strain gradient stress tensor, electric displacement vector and magnetic induction vector for a two dimensional MEE composite material can be expressed as
1 m2 V2
8 9 < sxx =
9 2 38 c 11 b 0 < εxx = c 12 b syy ¼ 1 l2 V2 4 bc 12 bc 22 0 5 εyy : :g ; sxy ; 0 0 b c 66 xy 9 38 2 0 0 b e 31 < E x = 40 0 b e 32 5 E y : ; Ez 0 0 0 9 38 2 0 0 b q 31 < M x = 5 4 M 0 0 b q 32 : y; Mz 0 0 0
9 9 8 2 38 0 0 < εxx = < D x = 0 2 2 2 2 4 1m V 0 0 0 5 εyy D ¼ 1l V :g ; : y; b e 31 b Dz e 32 0 xy 9 38 2 b 0 0
(4b)
33
8 9 9 2 38 0 0 < εxx = < B x = 0 0 0 5 εyy 1 m2 V2 B ¼ 1 l 2 V2 4 0 : y; :g ; b q 31 b Bz q 32 0 xy 2 38 9 b d 11 0 0
(4c)
in which m stands for the nonlocal parameter and
(4a)
b c 22 ¼ c11 c 11 ¼ b
c213 ; c33
b q 32 ¼ q31 q 31 ¼ b b d 11 ¼ b d 22 ¼ d11 ;
b c 12 ¼ c12
c13 q33 ; c33
c213 ; c33
b c 66 ¼ c66 ;
bs 11 ¼ bs 22 ¼ s11 ;
d33 ¼ d33 þ
q33 e33 ; c33
b e 31 ¼ b e 32 ¼ e31
b s 33 ¼ s33 þ
b b 11 ¼ b b 22 ¼ b11 ;
c13 e33 c33
e233 c33
(5) 2
q b b 33 ¼ b33 þ 33 c33
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In accordance with the classical shell theory, the displacement field for an arbitrary point in the shell can be given as [63].
vwðx; yÞ ux ðx; y; zÞ ¼ uðx; yÞ z vx
(6a)
vwðx; yÞ uy ðx; y; zÞ ¼ vðx; yÞ z vy
(6b)
uz ðx; y; zÞ ¼ wðx; yÞ
(6c)
where uðx; yÞ; vðx; yÞ; wðx; yÞ represents the middle surface displacements along x; y and z axis, respectively. Now, based upon the Maxwell's equation, the electric field and magnetic intensity vectors can be written as the gradients of scalar electric and magnetic potentials as follows [64].
E ¼ VE~ ðx; y; zÞ
(7a)
~ ðx; y; zÞ M ¼ VM
(7b)
~ are the electric potential and magnetic potential, in which E~ and M
Ps ¼
1 2
Z S
h Z2
261
respectively, that can be defined as [64].
pz 2z40 4ðx; yÞ þ E~ ðx; y; zÞ ¼ cos h h
(8a)
~ ðx; y; zÞ ¼ cos pz jðx; yÞ þ 2zj0 M h h
(8b)
By taking von Karman geometric nonlinearity into account, the nonlinear strain-displacement relationships for a perfect cylindrical nanoshell can be expressed as
vu 1 vw 2 v2 w þ z 2 vx 2 vx vx vv w 1 vw 2 v2 w þ ¼ z 2 vy R 2 vy vy
εxx ¼ ε0xx þ zkxx ¼ εyy ¼ ε0yy þ zkyy
gxy ¼ g0xy þ zkxy ¼
(9)
vu vv vw vw v2 w þ þ 2z vy vx vx vy vxvy
As a result, the total strain energy related to the initial configuration of a nonlocal strain gradient MEE composite nanoshell can be calculated as
sij εij þ s*ij Vεij Di E i Bi M i dzdS
h 2
¼
1 2
Z n o Nxx ε0xx þ Nyy ε0yy þ Nxy g0xy þ Mxx kxx þ Myy kyy þ Mxy kxy dS S
1 þ 2
Z S
h ) Z2 ( vE~ vE~ vE~ Dx þ Dy þ Dz dzdS vx vy vz h 2
h Z2 ~ ~ ~ 1 vM vM vM Bx þ þ By þ Bz dzdS 2 vx vy vz S h 2 Z o 1 n ¼ Nxx ε0xx þ Nyy ε0yy þ Nxy g0xy þ Mxx kxx þ Myy kyy þ Mxy kxy dS 2 Z
S
1 þ 2
Z
h Z2 Dx
S
1 þ 2
Z
h 2
h Z2 Bx
S
h 2
pz pz v4 v4 p4 pz 240 cos Dy cos þ Dz sin þ dzdS vx h vy h h h h
pz pz vj vj pj pz 2j0 By cos þ Bz þ cos sin dzdS h h h vx vy h h
(10)
262
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The first two differential equations can be already satisfied by introducing the Airy stress function f ðx; yÞ as follows
where the stress resultants are in the following forms
9 8 9 8 M9 8 9 2 38 0 E > εxx l2 V2 ε0xx > b Nxx m2 V2 Nxx > Nxx c 12 h 0 c h b > > > > > Nxx > > > > > > > > > = > = < = 6 11 < = < < 7 0 2 2 0 2 2 E M 7 6 Nyy m V Nyy ¼ 4 b þ Nyy þ Nyy c 12 h b c 22 h 0 5 εyy l V εyy > > > > > > > > > > > > > > ; > ; : ; > > > : : b 0 0 c 66 h : g0xy l2 V2 g0xy ; Nxy m2 V2 Nxy 0 0 3 2 b c 11 h3 b c 12 h3 0 7 6 12 12 9 8 9 9 6 9 8 8 78 7> kxx l2 V2 kxx > > M E > > MM > 6 Mxx m2 V2 Mxx > > > > > 7> 6 < = > = = 6 = > < xx > < xx > < 3 3 7 b b c 12 h c 22 h 7 kyy l2 V2 kyy þ ME M Myy m2 V2 Mxx ¼ 6 þ Myy 0 7 6 yy > > > > 6 12 > > 12 > > > > > > > > > 7> ; 6 : 7 : k l 2 V2 k ; : 0 ; : 0 ; Mxy m2 V2 Mxx xy xy 7 6 4 b c 66 h3 5 0 0 6
(11)
in which
h8 h 8 8 9 9 9 9 8 Lxx > Lxx > Nxx > Mxx > > > > > 2 2 > > > > > > > > Z Z < < < = = = = < Nyy ¼ Lyy dz ; Myy ¼ Lyy zdz > > > > > > > > > > > > > > > > : : : ; ; ; ; : Nxy Lxy Mxy Lxy h h 2 2 9 ( 8 ) 8 M9 ( ) E = = < < Nxx Nxx 2b e 31 40 2b q 31 j0 ¼ ¼ ; : NM ; : NE ; 2b e 4 2b q j yy
32 0
9 8 2hb e 31 > > > > 4ðx; yÞ > > = < p ¼ ; E :M ; > > > > > > 2hb e 32 yy ; : 4ðx; yÞ 9 8 E = < Mxx
p
Nxx ¼
9 8 M= < Mxx : MM yy
;
¼
Nyy ¼
v2 f ðx; yÞ ; vx2
Nxy ¼
v2 f ðx; yÞ vxvy
8 9 2h b q 31 > > > > j ðx; yÞ > > < =
p
> > > > > > q 32 : 2h b jðx; yÞ ;
p
(12) With the aid of virtual work's principle, the governing differential equations for an axially loaded MEE composite nanoshell are derived as
vNxx vNxy þ ¼0 vx vy
(13a)
vNxy vNyy þ ¼0 vx vy
(13b)
2 0 2 0 v2 ε0xx v εyy v gxy ¼ þ vxvy vy2 vx2
!2 v2 w v2 w v2 w 1 v2 w 2 vxvy vx vy2 R vx2
¼0 (13c) h
(13d)
2h
h
Z2 pz vB pz pB pz vBx y z þ þ dz ¼ 0 cos cos sin h h h vx vy h
2h
(13e)
(15)
Subsequently, by inserting equation (14) in the inverse of equation (11) in conjunction with equations (13aee) and (15), the nonlocal strain gradient governing differential equations for an axially loaded MEE composite nanoshell can be rewritten in terms of the mid-plane displacement components as below
!
v4 f 1 2b c 12 h þ 2 2 vx4 2 2 b c 66 h ðb ðb c 11 hÞ ðb c 11 hÞ ðb c 12 hÞ c 12 hÞ ! b v4 f 1 v2 w c 11 h þ þ 2 2 vy4 R vx2 c 12 hÞ ðb c 11 hÞ ðb !2 v2 w v2 w v2 w ¼ 2 vxvy vx vy2 b c 11 h
!
v4 f vx2 vy2
(16a)
v2 Mxy v2 Myy Nyy v2 Mxx v2 w v2 w v2 w þ Nyy 2 þ þ Nxx 2 þ 2Nxy þ2 þ 2 2 vxvy vxvy R vx vy vx vy
Z2 pz vD pz pD pz vDx y z þ þ dz ¼ 0 cos cos sin h h h vx vy h
(14)
Additionally, for a perfect shell-type structure, the compatibility equation corresponding to the mid-plane strain components can be expressed as
32 0
yy
v2 f ðx; yÞ ; vy2
b ðb c 12 þ 2b c 66 Þh3 v4 w c 11 h3 v4 w 1 l 2 V2 þ 12 vx4 6 vx2 vy2 ! ! b 1 v2 f 2h v2 4 v2 4 c h3 v4 w b b þ 11 þ e e 31 32 R vx2 12 vy4 p vx2 vy2 ! 2h v2 j v2 j b q 32 2 q 31 2 þ b p vx vy ! v2 w v2 f v2 w v2 f v2 w v2 f 2 2 þ ¼ 1m V 2 vxvy vxvy vy2 vx2 vx2 vy2
(16b)
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
Z2pR( ! ! b b b bs 22 h v2 4 d 11 h v2 j d 22 h v2 j s 11 h v2 4 þ þ þ 2 2 2 2 vx2 vy2 vx2 vy2 ! ! 2b e h v2 w 2b e 32 h v2 w p2 s33 p2 d33 4 31 j p p 2h 2h vx2 vy2 ¼0
263
!
! b v2 f v2 f w c 12 h þ 2 2 vx2 2 2 vy2 R c 12 hÞ c 12 hÞ ðb c 11 hÞ ðb ðb c 11 hÞ ðb 0 ! b 1 vw 2 c 12 h E M þ þN N x x 2 2 2 vy c 12 hÞ ðb c 11 hÞ ðb ) ! b c 11 h E M dy þN N y y 2 2 c 12 hÞ ðb c 11 hÞ ðb b c 11 h
¼0 (16c)
(18) In addition, the unit shortening associated to the movable ends of the MEE composite nanoshell can be evaluated by
! b d 11 h v2 4 þ 2 vx2
! ! ! b b b d 22 h v2 4 b 11 h v2 j b 22 h v2 j þ þ 2 2 2 vy2 vx2 vy2 ! ! ! ! 2b q h v2 w 2b q 32 h v2 w p2 bd 33 p2 bb 33 4 31 j p p 2h 2h vx2 vy2
Dx
1 ¼ 2pRL L
Z2pR ZL 0
1 ¼ 2pRL
Z2pR ZL (
þ
(16d) Moreover, the boundary conditions at the left and right ends of the MEE composite nanoshell are supposed to be clamped. As a consequence, one will have: w ¼ 0; vw vx ¼ 0 at x ¼ 0; L On the other hand, the equilibrium of applied loads in the x-axis direction can be expressed as
þ
b c 12 h
!
v2 f vx2
2 2 ðb c 11 hÞ ðb c 12 hÞ ! b v2 f 1 vw 2 c 11 h 2 2 vy2 2 vx c 12 hÞ ðb c 11 hÞ ðb ! b c 11 h NxE þ NxM 2 2 c 12 hÞ ðb c 11 hÞ ðb ! ) b c 12 h E M Ny þ Ny dxdy 2 2 ðb c 11 hÞ ðb c 12 hÞ 0
¼0
0
vu dxdy vx
0
(19)
3. Solving methodology for asymptotic solutions
Z2pR
Nxx dy þ 2pRhsxx ¼ 0
(17)
0
For a closed shell-type structure, the periodicity condition yields the following relation
X¼
px
;
y Y¼ ; R
b¼
L
;
h¼
L2
;
ε¼
The following dimensionless parameters are taken into consideration to obtain the solution of the problem in a more general framework
p2 Rh
L2 p h n o b c 11 h b c h b c h b c h b c h b c h * * * A*11 ; A*12 ; A*66 ; H11 ¼ ; H12 ; H66 ; 12 ; 66 ; 11 ; 12 ; 66 A00 A00 A00 12A00 12A00 6A00 n o b b q j b q j e 31 400 e 32 400 b * * * * E31 ¼ ; E32 ; Q31 ; Q32 ; ; 31 00 ; 32 00 A00 A00 A00 A00 ( ) o n b b b d 11 400 j00 d 22 400 j00 d 33 400 j00 ; ; D*11 ; D*22 ; D*33 ¼ A00 h A00 h A00 h ( ) n o b s 11 4200 b s 42 b s 42 b b j2 b b j2 b b j2 S*11 ; S*22 ; S*33 ; B*11 ; B*22 ; B*33 ¼ ; 22 00 ; 33 00 ; 11 00 ; 22 00 ; 33 00 A00 h A00 h A00 h A00 h A00 h A00 h W¼
L
εw ; h
F¼
ε2 f
; A00 h2
pR
F¼
ε2 R4 ; 400 h
2 2
3.1. Boundary layer theory of nonlocal strain gradient shell buckling
J¼
ε2 Rj ; j00 h
Px ¼
sxx R ; 2A00
dx ¼
Dx R 2Lh
(20)
264
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
where
A00 ¼ b c 11 h;
sffiffiffiffiffiffiffiffiffiffi A00 h ; ¼ b s 33
400
j00
1 2p
sffiffiffiffiffiffiffiffiffiffi A00 h ¼ b b
0
0
1
4 A*11 B Cv F B 1 @ 2 2 A 4 þ @ * vX A66 A*11 A*12 1
0
1
4 4 2A*12 A*11 B C 2 v F C 4v F þ @ 2 2 A b 2 2 Ab 2 2 vX vY vY 4 A*11 A*12 A*11 A*12
þ
v2 W vX 2
¼b
v2 W vXvY
2
!2
W v2 W vX 2 vY 2
2v
b
2
2
1p G þ
2 2 2V
"
4 * 4v W H11 b 4
ε
#!
vY
* b2 2Q32
p
v2 J vY 2
2
¼ 1p G
* H11
2
2 2 1V
v4 W v4 W * * þ 2 H12 þ H66 b2 2 2 4 vX vX vY
* * b * v2 F 2E31 v2 F 2E32 v2 F 2Q31 v2 J 2 2 2 p p p vX vX vY vX 2
b
2
v2 W v2 F v2 W v2 F v2 W v2 F þ 2 vXvY vXvY vX 2 vY 2 vY 2 vX 2
!
(21b)
v2 F S*11 2 vX
þ
2 2v F S*22 b vY 2 2
þ
* hb 4E32
p
v2 W vY 2
þ
v D*11
!#
2
J
vX 2
þ
(22)
80 0 1 1 Z2p> < * * 2 2 A11 A12 Cv F B C 2v F B A 2 @ Ab @ 2 2 2 2 > vX vY 2 : A*11 A*12 A*11 A*12 0 20 1 2 2 * A12 b vW 6B C * þW þ 4@ 2 2 A 2E31 F0 2 vY * * A11 A12 0 1 * A11 C * * * þ 2Q31 J0 B @ 2 2 A 2E32 F0 þ 2Q32 J0 A*11 A*12 9 3 > = 7 5ε1 dY > ;
(23)
2
v2 F dY þ 2εP x ¼ 0 vY 2
¼0 (21a)
b2
The dimensionless form of the periodicity condition yields
33
As a consequence, the dimensionless form of the nonlocal strain gradient governing differential equations for an axially loaded MEE composite nanoshell can be rewritten as
0
Z2p
2v D*22 b
2
J
vY 2
" ε
2
* h 4E31 v2 W p vX 2
!
Additionally, the dimensionless unit shortening of the electromagnetic actuated MEE composite nanoshell under axial compressive load takes the following form
80 1 Z2p Zp > < * 2 A11 1 C 2v F B dx ¼ 2 @ 2 2 A b > 4p ε vY 2 : A*11 A*12 0 0 0 2 1 2 * 2 A12 B 6 C v F 1 vW @ 4 2 2 A vX 2 2 vX * * A11 A12 0 1 * A11 B C * * @ 2 2 A 2E31 F0 þ 2Q31 J0 A*11 A*12 9 0 1 3 > = * A12 B C 7 1 * * 2E ε F þ 2Q J þ @ dXdY A 5 2 2 32 0 32 0 > ; A*11 A*12 (24)
p2 S*33 hF p2 D*33 hJ
¼0 (21c) " ! * 2 2 2 2 v2 F 2v F 2v J * * v J * 2 4Q31 h v W þ D b þ B þ B b ε 22 11 22 p vX 2 vX 2 vY 2 vX 2 vY 2 !# * hb2 4Q32 v2 W þ p2 D*33 hF p2 B*33 hJ p vY 2
D*11
¼0 (21d) Furthermore, the clamped edge supports at the left (X ¼ 0) and right (X ¼ p) ends of the MEE composite nanoshell can be given in dimensionless form as W ¼ 0; vW vX ¼ 0. Also, the boundary layer-type equilibrium requirement for an axially loaded MEE composite nanoshell can be given as
3.2. Solving process based upon a two-stepped perturbation technique As it was explained, by taking the small perturbation parameter ε into consideration, the boundary layer-type nonlocal strain gradient governing equation (21aed) were established for axially loaded MEE composite nanoshells. Thereafter, a two-stepped perturbation technique [65e71] is employed, based upon which the independent variables are assumed as the summations of the regular and boundary layer solutions as below
~ c ðX; Y; ε; 2Þ W ¼ WðX; Y; εÞ þ WðX; Y; ε; xÞ þ W
(25a)
~ Y; ε; xÞ þ b F ðX; Y; ε; 2Þ F ¼ FðX; Y; εÞ þ FðX;
(25b)
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
~ ðX; Y; εÞ þ F b ðX; Y; εÞ F ¼ FðX; Y; εÞ þ F
(25c)
~ ðX; Y; εÞ þ J b ðX; Y; εÞ J ¼ JðX; Y; εÞ þ J
(25d)
where the accent characters e, ~ and ^ denote, respectively, the regular solution, and the boundary layer solutions relevant to the left (X ¼ 0) and right (X ¼ p) ends of the MEE composite nanoshell. After that, each part of the solutions can be introduced in the form of the perturbation expansions as follows
265
the dimensionless nonlocal strain gradient governing equation (21aed) to achieve the perturbation sets of equations associated with both of the regular and boundary layer solutions. Thereafter, the expressions with similar order of ε are collected to obtain the required relationships for capturing the asymptotic solutions. The assumption for the initial buckling mode shape of MEE composite nanoshell is considered as below
W 2 ðX; YÞ ¼ A
ð2Þ 00
þA
ð2Þ 11
sinðmXÞsinðnYÞ þ A
P
εi W i ðX; YÞ;
i¼0
c ðX; Y; 2Þ; c ðX; Y; ε; 2Þ ¼ P εiþ1 W W iþ1 i¼0
b ðX; Y; ε; 2Þ ¼ F
P
i¼0
b ðX; Y; 2Þ; εiþ2 F iþ1
i¼0 X
b F ðX; Y; ε; 2Þ ¼ b ðX; Y; ε; 2Þ ¼ J
X ; ε1=2
2¼
pX ε1=2
(26)
εiþ2 b F iþ2 ðX; Y; 2Þ
i¼0 X
b εiþ2 J iþ2 ðX; Y; 2Þ
i¼0
in which x and 2 represent the boundary layer variables in the following forms
x¼
cosð2nYÞ (28)
X FðX; Y; εÞ ¼ εi F i ðX; YÞ i¼0 i¼0 X P FðX; Y; εÞ ¼ εi Fi ðX; YÞ; JðX; Y; εÞ ¼ εi Ji ðX; YÞ i¼1 i¼1 X P iþ1 ~ ~ ~ ε W iþ1 ðX; Y; xÞ; FðX; Y; ε; xÞ ¼ εiþ2 F~iþ2 ðX; Y; xÞ WðX; Y; ε; xÞ ¼ i¼0 i¼0 X ~ ðX; Y; ε; xÞ ¼ P εiþ2 F ~ ðX; Y; xÞ; J ~ ~ ðX; Y; ε; xÞ ¼ F εiþ2 J iþ2 iþ2 ðX; Y; xÞ WðX; Y; εÞ ¼
ð2Þ 02
(27)
Subsequently, equations (25aed) and (26) are substituted into
By performing some mathematical calculations, the asymptotic solutions corresponding to each independent variable of the problem are extracted as presented in Appendix A. Then, the solutions are inserted in the size-dependent equations (22) and (24), the obtained relations are rearranged with respect to the second ð2Þ perturbation parameter (A 11 ε) which results in explicit analytical expressions for the nonlocal strain gradient load-deflection and load-shortening stability paths of the MEE composite nanoshell
Fig. 2. A brief description of the two-stepped perturbation solving process.
266
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
under axial mechanical compressive load in conjunction with external electric and magnetic potentials, respectively, as follows
Table 1 Mechanical, electric and magnetic properties of BiTiO3CoFe2O4 composite material [72,73]. Elastic properties (GPa) c11¼c22 c12 c13 c33 c66
e31¼e32 e33
dxðiÞ A
A
i ð2Þ 11 ε
A
2 ð2Þ 11 ε
i ð2Þ 11 ε
i¼0;2;4;… ð0Þ
5:64 109
E
ð2Þ
M
q31¼q32 q33
Magnetoelectric properties
290.1 349.9
N:s V:C
d11¼d22
5:367 1012
d33
2737:5 1012
A
N:s2 C2
b11¼b22
297 106
b33
83:5 106
ð4Þ
þPx
A
4 ð2Þ 11 ε
þ…
(29)
dEx dM x
A
2 ð2Þ 11 ε
ð4Þ
þ dx
A
4 ð2Þ 11 ε
þ…
(30)
The different weight coefficients in the above equations are defined in Appendix B. On the basis of this assumption that the maximum deflection of the MEE composite nanoshell occurs at the point with dimensionless coordinates of ðX; YÞ ¼ ðp=2m; p=2nÞ, it yields
6:35 109
N A:m
Piezomagnetic properties
¼ dx dx dx þ dx
s11¼s22 s33
ð2Þ
X
dx ¼
C V:m
Dielectric properties
Magnetic properties
ð0Þ
¼Px þPx
2.2 9.3
ðiÞ
Px
i¼0;2;4;…
C m2
Piezoelectric properties
X
Px ¼
226 125 124 216 44.2
ð2Þ 11 ε
¼
wm þS h
2
þS
1
w 2 m þS 2 h
(31)
in which wm stands for the maximum deflection of the MEE composite nanoshell and the symbols S 1 and S 2 are defined in Appendix B. Also, the present solving process is shown in Fig. 2 briefly as a flowchart.
Table 2 Comparison of the nonlinear critical buckling loads for perfect FGM cylindrical shells made of the mixture of ZrO2 and Ti6Al4V (L=R ¼ 2; R=h ¼ 200). Material property gradient index
Critical buckling load (MPa) Present work
Ref. [74]
0.2 1 5
439.860 380.707 332.981
441.053 382.955 335.512
4. Results and discussion After proposing explicit analytical expressions for nonlocal strain gradient stability curves of MEE nanoshells under axial compression combined with external electromagnetic potential, selected numerical results are presented in this section. It is assumed that the MEE nanoshells are made of two-phase BiTiO3CoFe2O4 composite material whose material properties are given in Table 1. Also, it is supposed that R ¼ 50h, L ¼ 2R, and the two ends
Fig. 3. Dimensionless nonlocal strain gradient load-deflection stability paths of an axially loaded MEE composite nanoshell (F0 ¼ J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
of nanoshells are clamped. Firstly, the efficiency as well as accuracy of the present solution methodology is checked. Because in accordance with the best authors' knowledge, there is no study in the open literature in which the size-dependent nonlinear instability of MEE nanoshells is investigated, the nonlocal strain gradient terms associated with the constitutive relations removed and the nonlinear critical buckling
267
loads of FGM cylindrical shells at usual scale (macroscale) made from a mixture of zirconia (ZrO2) and titanium alloy (Ti6Al4V) with simply supported end conditions and subjected to axial compression are tabulated in Table 2 and compared with those obtained by Huang and Han [74] using Galerkin method. An excellent agreement is found between two types of solving process which confirms the validity of the given preceding numerical results.
Fig. 4. Dimensionless nonlocal strain gradient load-shortening stability paths of an axially loaded MEE composite nanoshell (F0 ¼ J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
Fig. 5. Influence of external electric potential on the nonlocal strain gradient load-deflection stability path of an axially loaded MEE composite nanoshell (J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
268
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
Figs. 3 and 4 depict, respectively, the dimensionless nonlocal strain gradient load-deflection and load-shortening stability curves of axially loaded MEE composite nanoshells with different small scale parameters. It is seen that the softening-stiffness effect of nonlocality causes to reduce both of the critical buckling load and the minimum postbuckling load, while the hardening-stiffness of strain gradient size dependency leads to enhance them. In addition, it can be observed that both types of size effect has no influence on the slope of prebuckling part of the load-shortening response, but
the nonlocal size effect causes to decrease the end-shortening of axially loaded MEE composite nanoshell at the critical buckling point, while the strain gradient size effect leads to increase it. Moreover, it is obvious that the both types of size dependency have approximately no influence on the width of postbuckling domain, so the maximum deflection associated with the minimum postbuckling load of MEE composite nanoshells remains approximately constant for all values of the small scale parameters. Figs. 5 and 6 in order display the dimensionless nonlocal strain
Fig. 6. Influence of external electric potential on the nonlocal strain gradient load-shortening stability path of an axially loaded MEE composite nanoshell (J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
269
Fig. 7. Influence of external magnetic potential on the nonlocal strain gradient load-deflection stability path of an axially loaded MEE composite nanoshell (F0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
gradient load-deflection and load-shortening stability curves of MEE composite nanoshells subjected to axial compression combined with external electric potential corresponding to various values of small scale parameters. It is found that a positive electric potential causes to enhance the critical axial compressive load, but a negative electric potential leads to reduce it. However, both of the positive and negative electric potentials have no influence on the minimum postbuckling load of axially loaded MEE composite nanoshell. Moreover, it is shown that the positive and negative electric potentials induce, respectively, initial shortening and initial extension in the MEE composite nanoshell. As a consequence, the positive electric potential increases the critical end-shortening of an axially loaded MEE composite nanoshell, but the negative electric potential decreases it. Also, it is displayed that by applying a positive electric potential, the postbuckling domain of the axially loaded MEE composite nanoshell increases, while the negative electric potential causes to reduce it. In Figs. 7 and 8, the dimensionless nonlocal strain gradient loaddeflection and load-shortening stability curves of MEE composite nanoshell under axial compressive load with combination of magnetic potential are illustrated, respectively, corresponding to various small scale parameters. It is revealed that a positive external magnetic potential reduces the critical axial compressive load, but a negative external magnetic potential enhances it. However, the external magnetic potential with both of the positive and negative signs has no influence on the minimum postbuckling load of the axially loaded MEE composite nanoshell. As a result, a negative magnetic potential increases the width of the postbuckling domain, while a positive one causes to decrease it. Additionally, it can be seen that the positive and negative magnetic potentials induce, respectively, initial extension and initial shortening in the MEE composite nanoshell which in order lead to decrease and increase the critical end-shortening of an axially loaded MEE composite nanoshell. In Figs. 9 and 10, the error of the classical shell model in
prediction of the critical buckling load of an axially loaded MEE composite nanoshell is demonstrated corresponding to different external electric potentials and magnetic potentials, respectively. The value of error is calculated as below
classical buckling load nonclassical buckling load Error ð%Þ ¼ classical buckling load 100 It is obvious that by increasing the value of nonlocal and strain gradient parameters, the size dependency becomes more significant, so the error of the classical model increases. Furthermore, it is indicated that applying a negative external electric potential causes to reduce the error of the classical model, while a positive one leads to enhance it. However, this pattern is vice versa for external magnetic potential as the negative external magnetic potential increases the error of the classical model, but a positive value decreases it. In other words, nonlocal strain gradient size dependency plays the most important role in nonlinear instability of MEE composite nanoshells under external positive electric potential and negative magnetic potential. 5. Concluding remarks The objective of this work was to analyze the size-dependent buckling and postbuckling behavior of MEE composite nanoshells under combination of axial compressive loaded and electromagnetic external potential including the both of hardening-stiffness and softening-stiffness size effects. To this end, the nonlocal strain gradient theory of elasticity was incorporated to the classical shell theory. After that, an improved perturbation technique in conjunction with the boundary layer theory of shell buckling was employed to propose explicit analytical expressions for nonlocal strain gradient stability paths of MEE composite nanoshells. It was revealed that the softening-stiffness effect of nonlocality
270
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
Fig. 8. Influence of external magnetic potential on the nonlocal strain gradient load-shortening stability path of an axially loaded MEE composite nanoshell (F0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
causes to reduce both of the critical buckling load and the minimum postbuckling load, while the hardening-stiffness of strain gradient size dependency leads to enhance them. Moreover, it was observed that the positive electric potential and negative magnetic potential cause enhance the critical axial compressive load, but their opposite sign leads to decrease the buckling resistance of the axially loaded MEE composite nanoshell. Additionally, it was seen that both of the electric and magnetic potentials with any sign have no influence on the value of minimum postbuckling load.
Furthermore, the positive electric potential and negative magnetic potential induce an initial shortening in the MEE composite nanoshell that leads to increase the critical end-shortening, while the negative electric potential and positive magnetic potential induce an initial extension which causes to reduce the critical endshortening of the axially loaded MEE composite nanoshell. In addition, it was demonstrated that a positive electric potential and a negative magnetic potential cause to increase both types of the size dependency in the nonlinear instability behavior of axially
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
271
Fig. 9. Error of the classical model in prediction of critical buckling load of axially loaded MEE composite nanoshell under different external electric fields (J0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
Fig. 10. Error of the classical model in prediction of critical buckling load of axially loaded MEE composite nanoshell under different external magnetic fields (F0 ¼ 0). (a) l ¼ 0 nm, (b) m ¼ 0 nm.
loaded MEE composite nanoshells, while a negative electric potential and a positive magnetic potential play an opposite role.
Appendix A The solutions relevant to each independent variable of the problem are achieved in asymptotic forms as below.
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S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
h
ð4Þ ð4Þ F ¼ ε4 G11 sinðmXÞsinðnYÞ þ G02
cosð2nYÞ þ O ε5 i
h
i
ð4Þ ð4Þ J ¼ ε4 H11 sinðmXÞsinðnYÞ þ H02 cosð2nYÞ þ O ε5
A
ð1Þ 00
b2 n2 8
A
ð1Þ 2 11
þ b2 n2 A
ð2Þ 2 02
(A7)
(A4) Appendix B
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi urffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 u A*11 A*22 ðA*12 Þ u t A*11 H *11
(A5)
2 0
A
¼
(A3)
where
G¼
ð4Þ 00
1
20
¼
ð2Þ
1 ¼ 2
1
A*12 A*11 6B C C 2 2 AP x þ 4@ 2 2 A * * * * A11 A12 A11 A12 0 1 * A12 C * * * 2E32 F0 þ 2Q32 J0 B @ 2 2 A 2E31 F0 * * A11 A12 3 * þ 2Q31 J0 7 5ε3
B ¼ 2@
(A6)
o 1n 2 K0 ε1 þ ðK5 þ K6 K12 þ K7 K13 Þb ε 2
ð0Þ
Px
Px
(
! 3K02 K16 ε1 K2 þ K3 K14 þ K4 K15 2
þ
(B1)
K10 K16 2
2K0 K12 b ð1 K6 K16 Þ þ 2K0 K13 b ð1 K7 K16 Þ K2 þ K3 K14 þ K4 K15
2 J 2 2 K0 K8 K16 20 þ 4K0 K9 K16 þ 4K0 K9 K16 J20 þ K0 K16 J20 4ðK0 J20 K8 Þ ! ) 2 þ 8K K K K02 K16 0 9 16 þ ε 8ðK0 K8 Þ
þ
(B2)
S. Sahmani, M.M. Aghdam / Composites Part B 132 (2018) 258e274
P
ð4Þ x
1 ¼ 2
(
0
3K03 K11 K16 J13
S
ðK0 J13 K11 ÞðK2 þ K3 K14 þ K4 K15 Þ2 K03 K16 ðK0 þ 2K11 Þ
þ
ðK0 K11 ÞðK2 þ K3 K14 þ K4 K15 Þ2 ! ) 2 2K 3 K 2 6K02 K16 1 0 16 þ ε ðK2 þ K3 K14 þ K4 K15 Þ2 0
dxð0Þ
(B3)
0
1
2 * B A G P 2x 12 B B C ¼ @ 2 AP x þ B 2 2 2 @ A*11 A*12 p A*11 A*12 1 2 * C 2 A12 P x C 1=2 ε 2 2 C A * * * pGA11 A11 A12
dxð2Þ ¼ dxð4Þ ¼
8p2 ðK2 þ K3 K14 þ K4 K15 Þ
2
m2 4
K0 b n2 þ 4K9 4ðK0 K8 Þ
!2 ε3 (B6)
00
1
A*12 B C * dEx ¼ B @@ 2 2 AE32 * * A11 A12 0 1 1 B @
A*11 C * C 2 2 2 AE31 Aε F0 * * A11 A12
00
(B7)
1
A*12 BB C * ¼ @@ 2 2 AQ32 A*11 A*12 0 1 1 B @
A*11 C * C 2 2 2 AQ31 Aε J0 A*11 A*12
(B8)
where
m2 þ b2 n2 ; J20 ¼ 1 þ 4p2 G 21 m2 ¼ 1 þ p2 G 21 m2 þ 9b2 n2 ; I11 ¼ 1 þ p2 G 22 m2 þ b2 n2
J11 ¼ 1 þ p2 G J13
2 1
(B9) where Ki ði ¼ 0; …; 16Þ are constant parameters extracted via the perturbation sets of assssssss in terms of mechanical, magnetic and electric properties.
S
1
¼
0
K0 B ε1 þ 2@ K2 þ K3 K14 þ K4 K15
1 A*11
A*12 2
A*12
1
References
(B5)
þ 2
20
(B11)
m2 ε 16
GK02 ε3=2
1
A*12 A*11 B C ð0Þ 6B C ¼ 2@ 2 2 AP x 4@ 2 2 A A*11 A*12 A*11 A*12 0 1 * A C * * * 12 2E32 F0 þ 2Q32 J0 B @ 2 2 A 2E31 F0 * * A11 A12 3 * þ 2Q31 J0 7 5ε3
A*11 2
(B4)
dM x
2
273
C ð2Þ 2 AP x (B10)
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