nonlocal nonlinear rst-order shear deformable beam model for post-buckling...

Scientia Iranica F (2016) 23(6), 3099{3114

Sharif University of TechnologyScientia Iranica

Transactions F: Nanotechnologywww.scientiairanica.com

Nonlocal nonlinear �rst-order shear deformable beammodel for post-buckling analysis ofmagneto-electro-thermo-elastic nanobeams

R. Ansaria and R. Gholamib;�

a. Department of Mechanical Engineering, University of Guilan, Rasht, P.O. Box 41635-3756, Iran.b. Department of Mechanical Engineering, Lahijan Branch, Islamic Azad University, Lahijan, P.O. Box 1616, Iran.

Received 22 August 2015; received in revised form 6 April 2016; accepted 26 September 2016

KEYWORDSFirst-order sheardeformable nanobeam;Magneto-electro-thermo-elasticmaterials;Post-buckling;Nonlocal elasticitytheory;Small-scale e�ect.

Abstract. In this study, the size-dependent post-buckling behavior of Magneto-Electro-Thermo-Elastic (METE) nanobeams with di�erent edge supports is investigated. Based onthe nonlocal �rst-order shear deformation beam theory and considering the von K�arm�anhypothesis, a size-dependent nonlinear METE nanobeam model is developed, in whichthe e�ects of small-scale parameter and thermo-electro-magnetic-mechanical loadings areincorporated. A numerical solution procedure based on the Generalized Di�erentialQuadrature (GDQ) and pseudo arc-length continuation methods is utilized to describethe size-dependent post-buckling behavior of METE nanobeams under various boundaryconditions. The e�ects of di�erent parameters such as nonlocal parameter, externalelectric voltage, external magnetic potential, and temperature rise on the post-bucklingpath of METE nanobeams are explored. The results indicate that increasing the non-dimensional nonlocal parameter, imposed positive voltage, negative magnetic potential,and temperature rise decreases the critical buckling load and post-buckling load-carryingcapacity of METE nanobeams, while an increase in the negative voltage and positivemagnetic potential leads to a considerable increase of critical buckling load as well aspost-buckling strength of the METE nanobeams.© 2016 Sharif University of Technology. All rights reserved.

1. Introduction

Due to their inherent coupling of �eld quantities,the Magnet-Electro-Thermo-Elastic (METE) materialshave attracted much attention recently [1-5]. Thesuperior properties of METE structures enable them tobe used in a wide range of engineering applications suchas transducers, actuators, vibration control devices,resonators, robotics, and high-density information stor-age devices [6-11]. Also, due to the rapid advancementin modern nanoscience, nanotechnology, and Nano-and Micro-Electro-Mechanical Systems (NEMS and

*. Corresponding author. Tel./Fax: +98 1342222906E-mail address: gholami @liau.ac.ir (R. Gholami)

MEMS), the development of the new class of smart orintelligent nanostructures made of METE materials hasreceived considerable attention. These nanostructureshave the capability to be used in many novel nanoscaledevises such as sensors and NEMS devices [12,13].Due to the dependence of mechanical characteristicsof nanostructures on the size e�ects, which has beenproven in the experimental tests and molecular dynam-ics simulations [14-17], considering the size e�ects inthe continuum models is necessary.

In recent years, the size-dependent mechani-cal behaviors of nanostructures, such as nanobeams,nanoplates, and nanoshells, have attracted the at-tention of researchers [18-27]. For instance, theEringen's nonlocal elasticity theory [28-30] has been

3100 R. Ansari and R. Gholami/Scientia Iranica, Transactions F: Nanotechnology 23 (2016) 3099{3114

extended to the METE material to take the size e�ectsinto account [31-34]. Using a nonlocal Timoshenkobeam model, the thermoelectric-mechanical vibrationof piezoelectric nanobeams was examined by Ke andWang [35]. Further, considering the in uence ofmagnetic potential, Ke and Wang [36] developed a non-local METE beam model to investigate the e�ects ofimportant parameters such as the nonlocal parameter,temperature rise, and external electric and magneticpotentials on the free vibration of METE Timoshenkonanobeams with various boundary conditions. Ac-cording to the nonlocal theory and Kirchho� platetheory, Ke et al. [37] analyzed the free vibration of size-dependent METE nanoplates with simply-supportededge conditions. Also, based on the nonlocal Mindlinplate theory, the free vibration of piezoelectric rect-angular nanoplates with various boundary conditionsunder the thermo-electro-mechanical loading was stud-ied by Ke et al. [38]. A nonlocal magneto-electro-elastic Mindlin nanoplate model was proposed by Li etal. [39] for the analysis of size-dependent buckling andfree vibration of magneto-electro-elastic nanoplates.Recently, Ke et al. [40] developed a nonlocal METEcylindrical nanoshell model based on the Love's shelland Eringen's nonlocal elasticity theories. This modelincludes the in uences of the small-scale parameterand magneto-electro-thermal loadings. Moreover, a fewstudies have been performed to investigate the surfacee�ect on the mechanical characteristics of piezoelectricnanostructures [41-44].

Although several nonlocal beam, plate, andshell models have been developed to examine thefree vibration characteristics of METE nanostruc-tures, the buckling behavior of nanobeams with dif-ferent boundary conditions under the magneto-electro-thermo-mechanical loading has not been considered.Moreover, the investigations have been done by em-ploying the linear METE models and the geometricalnonlinearities in the METE nanosystems have beenabsolutely excluded. When the METE nanostructuresare subjected to a large amplitude vibration or abuckling instability, these linear size-dependent METEmodels are not capable of predicting the mechanicalcharacteristics of METE nanostructures. Therefore,the geometric nonlinearities must be considered.

To the authors' best knowledge, the size-dependent post-buckling analysis of the METEnanobeams has not been reported so far. Hence, inthis study, it is attempted to develop a size-dependentnonlinear �rst-order shear deformable nanobeam modelto examine the post-buckling characteristics of METEnanobeams with di�erent edge supports. By meansof the Eringen's nonlocal elasticity theory, �rst-ordershear deformation beam theory, and von K�arm�anhypothesis, the nonlinear governing equations andcorresponding boundary conditions are obtained. The

developed beam model contains the e�ects of small-scale parameter, thermo-electro-magnetic-mechanicalloadings, and transverse shear deformation. Then, theGeneralized Di�erential Quadrature (GDQ) method isutilized to discretize the nonlinear governing equations,which are then solved using the pseudo arc-lengthcontinuation method to obtain the post-buckling pathof the METE nanobeams under various edge supports.Finally, numerical results in the graphical forms areprovided to investigate the e�ects of various parame-ters such as the nonlocal parameter, external electricvoltage, external magnetic potential, and temperaturerise on the post-buckling path of METE nanobeams.

This paper is organized as follows; in Section 2,the nonlocal elasticity theory for the METE materi-als is introduced and the mathematical modeling ofthe METE nanobeam is presented. By consideringthe appropriate non-dimensional parameters, the non-dimensional form of size-dependent governing equa-tions is given in Section 3. In Section 4, the post-buckling problem of the METE nanobeams is numer-ically solved using the GDQ method and pseudo arc-length continuation algorithm. Results and discussionare presented in Section 5 and concluding remarks aregiven in Section 6.

2. Problem formulation

2.1. A brief overview of the nonlocal elasticitytheory for the METE materials

According to the Eringen's nonlocal elasticity the-ory [28-30] and considering this hypothesis that thestress at reference point in a continuum body is afunction of the strain �eld at all points in the body,the basic relations for a homogeneous and METE solidare written mathematically as:

�ij =ZV

�(jx� x0j; �)[cijkl"kl(x0)� emijEm(x0)

� qnijHn(x0)� �ij�T ]dV (x0); 8 x 2 V;(1a)

Di =ZV�(jx� x0j; �)[eikl"kl(x0) + simEm(x0)

+ dinHn(x0) + pi�T ]dV (x0); 8 x 2 V; (1b)

Bi =ZV

�(jx� x0j; �)[qikl"kl(x0) + dimEm(x0)

+ �inHn(x0) + �i�T ]dV (x0); 8 x 2 V; (1c)

where �ij , Di, and Bi represent the components ofstress, electric displacement, and magnetic induction,

R. Ansari and R. Gholami/Scientia Iranica, Transactions F: Nanotechnology 23 (2016) 3099{3114 3101

respectively; and "ij , Ei, and Hi denote the compo-nents of the strain, electric �eld, and magnetic �eld,respectively. Moreover, cijkl, emij , sim, qnij , dij , sij ,pi, and �i are, respectively, the elastic, piezoelectric,dielectric, piezomagnetic, magnetoelectric, magnetic,pyroelectric, and pyromagnetic constants; and �ijand �T denote the thermal moduli and temperaturechange, respectively. Also, �; jx � x0j, and � =e0a=l denote the nonlocal modulus, Euclidean distance,and scale coe�cient, respectively; and e0, a, and lare, respectively, a non-dimensional material property,internal characteristic length, and external character-istic length of the nanostructures. The parametere0a denotes the nonlocal parameter incorporating thesmall-scale e�ect into the response of the nano-scalestructures, which is obtained from molecular dynam-ics, experimental data, or comparison of calculationsbased on lattice dynamics, to calibrate the nonlocalcontinuum models.

According to Eringen's theory [29], the spatial in-tegral constitutive relation (Eqs. (1)) can be expressedas the equivalent di�erential constitutive equations as:

�ij � (e0a)2r2�ij

= cijkl"kl � emijEm � qnijHn � �ij�T; (2a)

Di�(e0a)2r2Di

=eikl"kl+simEm+dinHn+pi�T; (2b)

Bi�(e0a)2r2Bi

=qikl"kl+dimEm+�inHn+�i�T; (2c)

in which r2 represents the Laplace operator.Considering the quasi-static approximation,

which neglects the electric and magnetic chargedensities, the electric �eld vector and magneticintensity vector can be respectively obtained by meansof the Maxwell theory as follows [39]:

E = �r�; (3a)

H = �r; (3b)

in which � and denote the scalar electric andmagnetic potentials, respectively.

2.2. Governing equations and correspondingboundary conditions

The considered system is an METE nanobeam oflength, L, thickness, h, cross sectional area A, andmoment of inertia of the cross section, I, as depicted inFigure 1. The METE nanobeam is subjected to an elec-tric potential, �(x; z), a magnetic potential, (x; z), auniform temperature rise, �T , and a compressive axial

Figure 1. Schematic view of an METE nanobeam.

load, N0x . Based upon the �rst-order shear deformation

beam theory, the displacements of an arbitrary point inthe METE nanobeam along the axial direction, x, andlateral direction, z, respectively, de�ned by ux(x; z)and uz(x; z), are given by:

ux(x; z) = U(x) + z�x(x); uz(x; z) = W (x); (4)

where U is the axial displacement of the center ofsections; W stands for the lateral de ection of theMETE nanobeam; and �x denotes the rotation ofbeam cross section.

Assuming that the METE nanobeam is subjectedto small slopes after de ection, the von K�arm�an hy-pothesis can be utilized to express the nonlinear strain-displacement relations as follows:

"xx=duxdx

+12

�dWdx

�2

=dUdx

+zd�x

dx+

12

�dWdx

�2

;

xz =�

�x +dWdx

�: (5)

Moreover, the electric potential and magnetic potentialare, respectively, considered as [36]:

� = � cos(�z)�E(x) +2zhVE ; (6a)

= � cos(�z) H(x) +2zh

H ; (6b)

in which � = �=h; VE and H denote the initialexternal electric and magnetic potentials, respectively;and �E(x) and H(x) represent the variations of theelectric and magnetic potentials in the x-direction,respectively.

Substituting Eqs. (6a) and (6b) into Eqs. (3a)and (3b), the non-zero components of electric �elds(Ex, Ez) and magnetic �elds (Hx, Hz) can be easilyderived as follows:

Ex = �d�dx

= cos(�z)d�Edx

;


Ez = �d�dz

= �� sin(�z)�E � 2VEh; (7a)

Hx = �ddx

= cos(�z)d Hdx

;

Hz = �ddz

= �� sin(�z) H � 2Hh

: (7b)

The strain energy �s of METE nanobeams can beobtained as:

�s =12

LZ0

ZA

(�xx"xx + �xz xz �DxEx �DzEz

�BxHx �BzHz)dAdx: (8)

Inserting Eqs. (5) and (7) into Eq. (8) yields:

�s =12

LZ0

(Nx

"@U@x

+12

dWdx

!2#

+Mxd�x

dx+Qx

dWdx

+ �x

!)dx

+12

LZ0

ZA

�Dx cos(�z)

d�Edx

+Dz

� sin(�z)�E +

2VEh

!!dAdx

+12

LZ0

ZA

�Bx cos(�z)

d Hdx

+Bz

� sin(�z) H +

2Hh

!!dAdx; (9)

where the normal resultant force, Nx, transverse shearforce, Qx, and bending moment, Mx, present in Eq. (9),can be obtained by:

Nx =ZA

�xxdA; Mx =ZA

�xxzdA;

Qx = �sZA

�xzdA; (10)

in which �s = 5=6 is the shear correction factor.Moreover, the work done by applied external axial load,N0x , can be calculated from:

�ex =12

LZ0

N0x

�dWdx

�2

dx: (11)

By means of the principle of minimum total potentialenergy and fundamental lemma of the calculus of vari-ation (��s � ��ex = 0; � is the variational operator),the nonlinear governing equations of an METE beamcorresponding to the classical Timoshenko beam theoryare derived as:

dNxdx

= 0; (12a)

ddx

�Nx

dWdx

�+dQxdx

+N0xd2Wdx2 = 0; (12b)

dMx

dx�Qx = 0; (12c)Z

A

�cos(�z)

dDx

dx+ � sin(�z)Dz

�dA = 0; (12d)

ZA

�cos(�z)

dBxdx

+ � sin(�z)Bz�dA = 0: (12e)

Also, the mathematical expressions of associatedboundary conditions at beam ends (x = 0; L) arederived:

�U = 0 or Nx = 0; (13a)

�W = 0 or (Nx +N0x)dWdx

+Qx = 0; (13b)

��x = 0 or Mx = 0; (13c)

��E = 0 orZA

cos(�z)DxdA = 0; (13d)

� H = 0 orZA

cos(�z)BxdA = 0: (13e)

Now, to take the small-scale e�ect into account, theEringen's nonlocal elasticity theory is used. Accordingto Eqs. (2), (5), and (7), the nonlocal constitutiverelations for the METE beams under the plane stresscondition can be obtained as:

�xx �(e0a)2 d2�xxdx2

=~c11

"dUdx

+zd�x

dx+

12

�dWdx

�2#

+ ~e31

�� sin(�z)�E +

2VEh

�+ ~q31

�� sin(�z) H +

2Hh

�� ~�1�T; (14a)


�xz � (e0a)2 d2�xzdx2

= ~c44

��x +

dWdx

�� ~e15 cos(�z)

d�Edx

� ~q15 cos(�z)d Hdx

; (14b)

Dx � (e0a)2 d2Dx

dx2

= ~e15

��x +

dWdx

�+ ~s11 cos(�z)

d�Edx

+ ~d11 cos(�z)d Hdx

; (14c)

Dz � (e0a)2 d2Dz

dx2

= ~e31

"dUdx

+zd�x

dx+

12

�dWdx

�2#

� ~s33

�� sin(�z)�E +

2VEh

�� ~d33

�� sin(�z) H +

2Hh

�+ ~p3�T; (14d)

Bx � (e0a)2 d2Bxdx2

= ~q15

��x +

dWdx

�+ ~d11 cos(�z)

d�Edx

+ ~�11 cos(�z)d Hdx

; (14e)

Bz � (e0a)2 d2Bzdx2

= ~q31

"dUdx

+zd�x

dx+

12

�dWdx

�2#

� ~d33

�� sin(�z)�E +

2VEh

�� ~�33

�� sin(�z) H +

2Hh

�+ ~�3�T; (14f)

where ~cij , ~eij , ~sij , ~qij , ~dij , ~�ij , ~�i, ~pi, and ~�i denotethe reduced elastic, piezoelectric, dielectric constants,piezomagnetic, magnetoelectric, magnetic, thermalmoduli, pyroelectric, and pyromagnetic constants forthe METE nanobeam under the plane stress condition,respectively. These constants can be obtained as:

~c11 = c11 � c213c33

; ~c44 = c44;

~e31 = e31 � c13e33

c33; ~e15 = e15;

~q31 = q31 � c13q33

c33; ~s11 = s11;

~s33 = s33 +e2

33c33

; ~d11 = d11;

~d33 = d33 +q33e33

c33; ~�11 = �11;

~�33 = �33 +q233c33

; ~q15 = q15;

~�1 = �1 � c13�3

c33; ~p3 = p3 +

�3e33

c33;

~�3 = �3 +�3q33

c33: (15)

By use of Eq. (14), the following relations associatedwith the nonlocal form of Nx, Qx, Mx, Dx, Dz, Bx,and Bz are obtained as:

Nx � (e0a)2 d2Nxdx2

= A11

"dUdx

+12

�dWdx

�2#

+NE +NH +NT ;(16a)

Mx�(e0a)2 d2Mx

dx2 =D11d�x

dx+E31�E+Q31 H ;(16b)

Qx � (e0a)2 d2Qxdx2

=ksA44

��x+

dWdx

��sE15

d�Edx

��sQ15d Hdx

; (16c)ZA

�Dx � (e0a)2 d2Dx

dx2

�cos(�z)dA

= E15

��x +

dWdx

�+X11

d�Edx

+ Y11d Hdx

; (16d)ZA

�Dz � (e0a)2 d2Dz

dx2

�� sin(�z)dA

= E31d�x

dx�X33�E � Y33 H ; (16e)


ZA

�Bx � (e0a)2 d2Bx

dx2

�cos(�z)dA

= Q15

��x +

dWdx

�+ Y11

d�Edx

+ T11d Hdx

; (16f)ZA

�Bz � (e0a)2 d2Bz

dx2

�� sin(�z)dA

= Q31d�x

dx� Y33�E � T33 H : (16g)

The constant coe�cients present in Eq. (6) are de�nedas:

NE =2~e31VEA

h; NH =

2~q31HAh

;

NT = �~�1A�T; (17a)

A11 = ~c11A; A44 = ~c44A; D11 = ~c11I; (17b)

E31 =ZA

~e31� sin(�z)zdA;

Q31 =ZA

~q31� sin(�z)zdA; (17c)

E15 =ZA

~e15 cos(�z)dA;

Q15 =ZA

~q15 cos(�z)dA; (17d)

X11 =ZA

~s11 cos2(�z)dA;

Y11 =ZA

~d11 cos2(�z)dA;

T11 =ZA

~�11 cos2(�z)dA; (17e)

X33 =ZA

~s33[� sin(�z)]2dA;

Y33 =ZA

~d33[� sin(�z)]2dA;

T33 =ZA

~�33[� sin(�z)]2dA: (17f)

By substituting Eqs. (16a)-(16c) into Eqs. (12a)-(12c),the nonlocal normal resultant force, Nx, bending mo-ment, Mx, and transverse shear force, Qx, can beexplicitly expressed in the following form:

Nx = A11

"dUdx

+12

�dWdx

�2#

+NE +NH +NT ;(18a)

Mx =D11d�x

dx+ E31�E +Q31 H

� (e0a)2 ddx

�[Nx +N0

x ]dWdx

�; (18b)

Qx =ksA44

��x +

dWdx

�� sE15

d�Edx

� �sQ15d Hdx�(e0a)2 d2

dx2

�[Nx+N0

x ]dWdx

�:

(18c)

It should be noted that the nonlocal forms of Dx,Dz, Bx, and Bz cannot be explicitly obtained bysubstituting Eqs. (16d) and (16e) into Eqs. (12d)-(12g).However, the nonlocal form of governing equilibriumequations for the METE nanobeams can be expressedin terms of U , W , �x, �E , and H as follows:

A11

�d2Udx2 +

dW@x

d2Wdx2

�= 0; (19a)

ksA44

�d�x

dx+d2Wdx2

�+�N0x+NE+NH+NT

��d2Wdx2 �(e0a)2 d4W

dx4

��s

�E15

d2�Edx2 +Q15

d2 Hdx2

�+Z1�(e0a)2Z2

=0; (19b)

D11d2�x

dx2 � ksA44

��x +

dWdx

�+ (E31+�sE15)

d�Edx

+(Q31+�sQ15)d�Edx

=0; (19c)

E31d�x

dx+ E15

�d2Wdx2 +

d�x

dx

�+X11

d2�Edx2

+ Y11d2 Hdx2 �X33�E � Y33 H

= 0; (19d)


Q31d�x

dx+Q15

�d2Wdx2 +

d�x

dx

�+ Y11

d2�Edx2

+ T11d2 Hdx2 � Y33�E � T33 H

= 0; (19e)

where:

Z1 =A11

d2Udx2

dWdx

+dUdx

d2Wdx2 +

32

�dWdx

�2 d2Wdx2

!;

(20a)

Z2 = A11

d4Udx4

dWdx

+ 3d3Udx3

d2Wdx2

+ 3d2Udx2

d3Wdx3 +

dUdx

d4Wdx4

!+A11

"3�d2Wdx2

�2

+ 9dWdx

d2Wdx2

d3Wdx3

+32

�dWdx

�2 d4Wdx4

#: (20b)

Eqs. (19a)-(19e) represent the size-dependent nonlinearequilibrium equations of a �rst-order shear deformableMETE nanobeam modeled on the basis of the Eringen'snonlocal elasticity theory.

3. Non-dimensional form of size-dependentnonlinear governing equations

Introducing the following non-dimensional parameters:

� =xL; (u;w) =

(U;W )h

; �x = �x;

� =�E�0; =

H 0

; �0 =pA11=X33;

0 =pA11=T33; � =

e0aL; � =

Lh;

� �N0x ; �NT ; �NE ; �NH

�=�N0x

A11;NTA11

;NEA11

;NHA11

�;

� �A11; �A44; �D11�

=�A11

A11;A44

A11;D11

A11h2

�;

� �E15; �E31�

=�E15�0

A11h;E31�0

A11h

�;

� �Q15; �Q31�

=�Q15 0

A11h;Q31 0

A11h

�;

� �X11; �X33�

=�X11�2

0A11h2 ;

X33�20

A11

�;

� �T11; �T33�

=�T11 2

0A11h2 ;

T33 20

A11

�;

� �Y11; �Y33�

=�Y11�0 0

A11h2 ;Y33�0 0

A11

�; (21)

the non-dimensional equilibrium equations (Eqs. (19a)-(19e)) can be expressed as:

�A11

�d2ud�2 +

1�dwd�

d2wd�2

�= 0; (22a)

ks �A44

�d2wd�2 + �

d�xd�

�+� �N0

x + �NT + �NE + �NH��d2w

d�2 � �2 d4wd�4

�� ks �E15

d2 ��Ed�2 � ks �Q15

d2 � Hd�2 + �Z1 � �2 �Z2

= 0;(22b)

�D11d2�xd�2 �ks �A44�

�dwd�

+��x�

+�� E31+�s �E15

�d�d�

+ �� Q31 + �s �Q15

� d d�

= 0; (22c)

�E31�d�xd�

+ �E15

�d2wd�2 + �

d�xd�

�+ �X11

d2�d�2

+ �Y11d2 d�2 � �X33�2�� Y33�2

= 0; (22d)

�Q31�d�xd�

+Q15

�d2wd�2 + �

d�xd�

�+ �Y11

d2�d�2

+ �T11d2 d�2 � �Y33�2�� T33�2

= 0; (22e)

in which the nonlinear terms, �Z1 and �Z2, are de�nedas:

�Z1 =�A11

�

d2ud�2

dwd�

+32�

�dwd�

�2 d2wd�2 +

dud�d2wd�2

!;


�Z2 =�A11

�

�d4ud�4

dwd�

+3d3ud�3

d2wd�2 +3

d2ud�2

d3wd�3 +

dud�d4wd�4

�+

�A11

�2

"3�d2wd�2

�3

+9dwd�

d2wd�2

d3wd�3 +

32

�dwd�

�2 d4wd�4

#:(23)

Assuming the electric potential and magnetic potentialto be zero at the ends of the METE nanobeam, themathematical expressions of edge supports can bewritten in the non-dimensional forms as:

u = w = �x = � = = 0

at � = 0;

�Nx = w = �x = � = = 0

at � = 1; (24)

for a Clamped-Clamped (C-C) end condition:

u = w = �Mx = � = = 0

at � = 0;

�Nx = w = �Mx = � = = 0

at � = 1; (25)

for a Simply Supported-Simply Supported (SS-SS) endcondition, and:

u = w = �x = � = = 0

at � = 0;

�Nx = w = �Mx = � = = 0

at � = 1; (26)

for a Clamped-Simply Supported (C-SS) end condition,in which:

�Nx= �A11

"dud�

+12�

�dwd�

�2#+ �N0

x+ �NE+ �NH+ �NT ;

�Mx = �D11d�xd�

+ � �E31�+ � �Q31

� �2� � �N0x + �NE + �NH + �NT

� d2wd�2

� �2 �A11dd�

"dud�

+12�

�dwd�

�2#dwd�

!: (27)

It should be remarked that in the applications ofMETE devices such as sensors and actuators, two kinds

of boundary conditions for the electric and magneticpotential can be considered. One of them is opencircuit with the assumption of:ZA

cos(�z)DxdA =ZA

cos(�z)BxdA = 0:

This type of boundary conditions is for the sensorsituation [45]. Another type of boundary condition isshort circuit with the assumption of � = = 0, whichis for the actuator case [45]. In the present study, theshort circuit has been considered and it is assumed that� = = 0.

4. Numerical solution

To solve the post-buckling problem of METEnanobeams subjected to the thermo-electro-magnetic-mechanical loadings along with various end supports,the GDQ method [46] is �rst used to discretize thenonlinear coupled equilibrium equations and associatedboundary conditions. Then, the discretized equationsare solved on the basis of the pseudo arc-length contin-uation method.

If N is the number of grid points in the x-direction, the grid points, �i, on the basis of theshifted Chebyshev-Gauss-Lobatto grid points can beconsidered as:

�i =12

�1� cos

i� 1N � 1

��; i = 1; 2; � � � ; N: (28)

Therefore, the discretized forms of the �eld variablesu, w, �x, �, and can be respectively expressed ascolumn vectors U, W, �, �, and de�ned as follows:

U =�u1 u2 � � � uN

�T ;W =

�w1 w2 � � � wN

�T ;� =

��x1 �x2 � � � �xN

�T ;� =

��1 �2 � � � �N

�T ; =

� 1 2 � � � N

�T ; (29)

in which ui = u(�i), wi = w(�i), �xi = �x(�i), �i =�(�i), and 1 = (�i).

Moreover, on the basis of the GDQ method,the rth derivative of a continues function, f(�), withrespect to � can be approximately expressed as a linearsum of the function values as:

drf(�)d�r

=NXj=1

D(r)ij f(�j) = D(r)

� F; (30)


where [D(r)� ]ij = D(r)

ij denotes the weighting coe�cientsassociated with the rth-order derivative expressed inthe form of a recursion formula as:h

D(r)�

iij

=W(r)ij

=

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

Ix; r = 0

P(�i)(�i��j)P(�j) ;i 6= j and i; j = 1; � � � ; Nand r = 1

r�W(1)ij W(r�1)

ii � W(r�1)ij�i��j

�;

i 6= j and i; j = 1; � � � ; Nand r = 2; 3; � � �N � 1

� NPj=1;j 6=i

W(r)ij ;

i = j and i; j = 1; � � � ; Nand r = 1; 2; 3; � � �N � 1

(31)

where Ix denotes an N�N identity matrix and P(xi) =NQ

j=1;j 6=i(�i � �j).

In the case of post-buckling problem, the appliedexternal axial load is assumed to be �N0

x = �p.

Transforming the equilibrium equations (Eqs. (22a)-(22e)) into the discrete forms according to the GDQmethods gives:

KX� pKgX + N(X) = 0; (32)

where X, K, Kg, and N(X) are the �eld variablesvector, sti�ness matrix, geometric sti�ness matrix,and nonlinear part vector, respectively, which arecalculated by Eqs. (33a) to (33c) as shown in Box I.where N = ( �NT + �NE+ �NH)(D(2)

� ��2D(4)� ). Moreover,

the components of N(X) are de�ned in the followingforms:

Nu(X) =�A11

�

�D(1)� W

� � �D(2)� W

�; (34a)

Nw(X) =�A11

�

�D(1)� U

� � �D(2)� W

�+�D(2)� U

�� D(3)

� W�

+32�

�D(2)� W

� � �D(1)� W

�� D(1)

� W�!� �2( �A11

�

��D(4)� U

� � �D(1)� W

�+ 3

�D(3)� U

� � �D(2)� W

�+ 3

�D(2)� U

�� D(3)

� W��

+�A11

�

��D(1)� U

� � �D(4)� W

��

X =�UT WT �T �T T �T ; (33a)

K=

2666666666666664

�A11D(2)� 0 0 0 0

0 ks �A44D(2)� +N ks �A44�D

(1)� �ks �E15D

(2)� �ks �Q15D

(2)�

0 �ks �A44�D(1)� �ks �A44�2D(0)

� + �D11D(2)� �( �E31+�s �E15)D(1)

� �( �Q31+�s �Q15)D(1)�

0 �E15D(2)� ( �E31+ �E15)�D(1)

� � �X33�2D(0)� + �X11D

(2)� � �Y33�2D(0)

� + �Y11D(2)�

0 �Q15D(2)� ( �Q31+ �Q15)�D(1)

� � �Y33�2D(0)� + �Y11D

(2)� � �T33�2D(0)

� + �T11D(2)�

3777777777777775;(33b)

Kg =

2666640 0 0 0 00 D(2)

� � �2D(4)� 0 0 0

0 0 0 0 00 0 0 0 00 0 0 0 0

377775 ; (33c)

N(X) =�NTu (X) NT

w(X) NT� (X) NT

� (X) NT (X)

�T : (33e)

Box I


+�A11

�2

�3�D(2)� W

� � �D(2)� W

� � �D(2)� W

��+

�A11

�2

9�D(1)� W

� � �D(2)� W

� � �D(3)� W

�+

32

�D(1)� W

� � �D(1)� W

� � �D(4)� W

�!); (34b)

N� (X) = N�(X) = N (X) = 0; (34c)

where � is the Hadamard product de�ned in theAppendix. The boundary conditions can be discretizedin the same way. For instance, the discretized edgeconditions for an METE nanobeam with C-C endsupport (see Eq. (24)) can be written as:

U = W = � = � = = 0 at � = 0; 1: (35)

The governing equations of the domain equilibriumtogether with associated edge conditions represent aset of nonlinear problems of the form:

F : R5N+1 ! R5N ; F(p;X) = 0: (36)

The pseudo arc-length continuation algorithm [47]is utilized to estimate the post-buckling path ofMETE nanobeams under the magneto-electro-thermo-mechanical loading.

5. Results and discussion

In this section, numerical results are provided to ex-amine the size-dependent post-buckling behavior of the

METE nanobeams with SS-SS, C-SS, and C-C end sup-ports subjected to magneto-electro-thermo-mechanicalloading. It is assumed that the METE nanobeams aremade of the two-phase BiTiO3-CoFe2O4 compositeswith the material properties given in Table 1 [36,48-50]. The in uences of the non-dimensional nonlocal pa-rameter, external magnetic potential, external electricpotential, and temperature rise on the post-bucklingpath of the METE nanobeams are discussed in detail.

First of all, in order to verify validity of the pro-posed model and solution methodology, the �rst threenormalized critical buckling loads ( ~P = N0

xL2=12D11)of METE nanobeam with SS-SS edge supports corre-sponding to various nonlocal parameters are listed inTable 2 and compared with the analytical results givenby Li et al. [51]. It can be found that the resultsprovided in this investigation are in good agreementwith those of Li et al. [51].

The in uence of the non-dimensional nonlocalparameter, �, on the post-buckling path of METEnanobeams with SS-SS, C-SS, and C-C edge sup-ports corresponding to the �rst and second bucklingmodes, respectively, is represented in Figures 2 and 3,where the non-dimensional maximum de ection of theMETE nanobeam (wmax) versus the non-dimensionalapplied compressive axial load is plotted. The non-dimensional applied compressive axial load is de�nedas P = N0

xL2=D11. Moreover, in addition to theassociated critical buckling loads, the results of theclassical METE beam (� = 0) are given for a directcomparison. For all types of edge supports, an increasein the non-dimensional nonlocal parameter decreases

Table 1. Material properties of BiTiO3-CoFe2O4 composite materials.

Properties BiTiO3-CoFe2O4

Elastic (GPa) c11 = 226; c12 = 125; c13 = 124; c33 = 216; c44 = 44:2Piezoelectric (C/m2) e31 = 2:2; e33 = 9:3; e15 = 5:8Dielectric (109 C/V.m) s11 = 5:64; s33 = 6:35Piezomagnetic (N/A.m) q15 = 275; q31 = 290:1; q33 = 349:9Magnetoelectric (1012 Ns/VC) d11 = 5:367; d33 = 2737:5Magnetic (106 Ns2/C2) �11 = �297; �33 = 83:5Thermal moduli (105 N/K.m2) �1 = 4:74; �3 = 4:53Pyroelectric (106 C/N) p3 = 25Pyromagnetic (106 N/A.m.K) �3 = 5:19

Table 2. Normalized buckling loads ( ~P = N0xL2=12D11) of METE nanobeam with SS-SS edge conditions (L = 10 nm,

h = 0:4 nm, and VE = H = �T = 0).

Mode e0a = 0 nm e0a = 1 nm e0a = 2 nm e0a = 3 nmRef. [51] Present Ref. [51] Present Ref. [51] Present Ref. [51] Present

1 0.8189 0.8192 0.7198 0.7204 0.5564 0.5595 0.4089 0.41142 3.1595 3.1620 2.0462 2.0529 1.1026 1.1096 0.6365 0.63843 6.7046 6.7563 3.0463 3.0540 1.3102 1.3137 0.6877 0.6936


Figure 2. Post-buckling path of METE nanobeamsassociated with di�erent nonlocal parameters andboundary conditions corresponding to the �rst mode(L=h = 10, h = 10 nm, VE = �0:02 V, H = 0:02 A, and�T = 20�C).

the critical buckling load and for a given applied loadin the post-buckling domain, the maximum de ectionof METE nanobeam increases. The reason is that thenonlocal METE nanobeam model predicts the lowersti�ness values compared to the classical beam model.

Figure 3. Post-buckling path of METE nanobeamsassociated with di�erent nonlocal parameters andboundary conditions corresponding to the second mode(L=h = 10, h = 10 nm, VE = �0:02 V, H = 0:02 A, and�T = 20�C).

Moreover, compared to the METE nanobeams with C-SS and C-C end conditions, the nonlocal parameter haslesser in uence on the critical buckling load and post-buckling path of SS-SS METE nanobeams. Another�nding is that the e�ect of nonlocal parameter on thecritical buckling load and post-buckling path of METE


Figure 4. Post-buckling path of METE nanobeamsassociated with di�erent temperature changes andboundary conditions (L=h = 10, h = 10 nm, VE = 0 V,H = 0 A, and � = 0:08).

nanobeam associated with the second mode is moresigni�cant than in the �rst buckling mode.

Figure 4 shows the e�ect of temperature riseon the critical buckling load and post-buckling pathof METE nanobeams with SS-SS, C-SS, and C-Cboundary conditions. According to this �gure, withincreasing the temperature change, the critical buck-

ling load decreases and maximum de ection of METEnanobeam in the post-buckling domain increases. Infact, increasing the temperature rise generates higheraxial compressive loads, leading to more reductionin the sti�ness of METE nanobeams. Moreover, itcan be seen that the in uence of temperature rise forMETE nanobeams with SS-SS edge conditions is moreconsiderable than that for other ones.

The e�ect of external electric voltage on the post-buckling responses of METE nanobeams with SS-SS,C-SS, and C-SS edge conditions is examined in Fig-ure 5. It can be seen that, due to the imposed positiveelectric voltages, the METE nanobeam experienceslarger post-buckling deformations than in the caseswithout and with negative electric voltages. Moreover,the positive values of external electric voltage decreasethe critical buckling load of METE nanobeams, whilethe negative electric voltages have an increasing e�ect.In fact, the positive and negative external electricvoltages respectively generate compressive and tensilein-plane forces in the METE nanobeams leading todecrease and increase in the sti�ness of nanobeams,correspondingly.

Figure 6 depicts the post-buckling responsesof METE nanobeams subjected to various externalmagnetic potentials. The METE nanobeams sub-jected to the negative external magnetic potentialsundergo larger post-buckling deformations and lowercritical buckling loads, indicating that higher nega-tive magnetic potentials cause the stability of METEnanobeams to decrease. The reason is that negativemagnetic potentials generate a compressive axial forceand, consequently, lead to decrease in the sti�ness ofMETE nanobeams. Obviously, the positive externalmagnetic potentials can lead to high critical bucklingloads and a considerable increase in the post-bucklingload-carrying capacity in comparison with the negativemagnetic potentials.

6. Concluding remarks

In the present study, the size-dependent post-bucklingcharacteristics of METE nanobeams subjected to themagneto-electro-thermo-mechanical loading were ex-amined. To this end, a size-dependent nonlinear �rst-order shear deformable nanobeam model was devel-oped, which contained the e�ects of small-scale pa-rameter, thermo-electro-magnetic-mechanical loading,and transverse shear deformation. Using the Eringen'snonlocal elasticity theory, �rst-order shear deformationbeam theory, and von K�arm�an hypothesis, the nonlin-ear governing equations and corresponding boundaryconditions were obtained. The GDQ method wasused to discretize the nonlinear governing equations,which were then solved using the pseudo arc-lengthcontinuation method to obtain the post-buckling path


Figure 5. Post-buckling path of METE nanobeamsassociated with di�erent external electric voltages andboundary conditions (L=h = 10, h = 10 nm, H = 0 A,�T = 0�C, and � = 0:08).

of the METE nanobeams under various edge supports.Finally, the e�ects of various parameters such asnonlocal parameter, external electric voltage, externalmagnetic potential, and temperature rise on the post-buckling path of METE nanobeams were examined indetail. The results revealed that an increase in the non-dimensional nonlocal parameter as well as temperature

Figure 6. Post-buckling path of METE nanobeamsassociated with di�erent external magnetic potentials andboundary conditions (L=h = 10, h = 10 nm, VE = 0 V,�T = 0�C, and � = 0:08).

rise decreased the critical buckling load and post-buckling load-carrying capacity of METE nanobeams.Moreover, due to the imposed negative electric voltagesand positive magnetic potentials, the critical bucklingload as well as post-buckling strength of the METEnanobeams increased.


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Appendix

Hadamard Products: Let A = [Aij ]N�M and B =[Bij ]N�M ; then, the Hadamard product of these ma-trices can be expressed as A �B = [AijBij ]N�M .

Biographies

Reza Ansari received his PhD degree from Universityof Guilan, Iran, in 2008, where he is currently a facultymember in the Department of Mechanical Engineering.During his PhD program. He was also a visitingfellow at Wollongong University, Australia, from 2006to 2007. He has authored more than 200 refereedjournal papers and 12 book chapters. His researchbackground and interests include computational me-chanics; numerical analysis; continuum mechanics; andnano-, micro-, and macro-mechanics.

Raheb Gholami received his BS, MS, and PhDdegrees in Mechanical Engineering from University of


Guilan, Iran, in 2008, 2010, and 2015, respectively.He is currently a faculty member in the Departmentof Mechanical Engineering, Lahijan Branch, IslamicAzad University. His research background and interests

include computational micro- and nano-mechanics; nu-merical techniques; nonlinear analyses; and predictionof mechanical behavior of beam, plate, and shell-typestructures.

nonlocal nonlinear rst-order shear deformable beam model for post-buckling...

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