snap-through and bifurcation of nano-arches on elastic foundation by the strain gradient and...

SNAP-THROUGH AND BIFURCATION OF NANO-ARCHES

ON ELASTIC FOUNDATION BY THE STRAIN GRADIENT

AND NONLOCAL THEORIES

FAREED KAVIANI* and HAMID REZA MIRDAMADI†

Department of Mechanical Engineering

Isfahan University of Technology, Isfahan 8415683111, Iran*[email protected]†[email protected]

Received 26 June 2012

Accepted 3 October 2012

Published 10 April 2013

This paper presents the snap-through and bifurcation elastic stability analysis of nano-arch type

structures with the Winkler foundation under transverse loadings by the strain gradient and

stress gradient (nonlocal) theories. The equations of equilibrium are derived by using thevariational method and virtual displacement theorem of minimum total potential energy. In

the elastic stability analysis, von Karman's nonlinear strain component is included, with the

deformation represented by a series solution. It is concluded that in general, the strain gradient

theory pushes the system away from instability as compared to the classical theory. However,the nonlocal theory does the reverse and causes the system to experience instability earlier than

that of the classical theory. Moreover, theories with di®erent small-size considerations change

the mechanism of instability in di®erent ways. For example, in similar conditions, the straingradient theory causes the system to reach a snap-through point, while the nonlocal theory

causes the system to stop at a bifurcation critical point.

Keywords: Snap-through; bifurcation; stability analysis; nano-arch type structure; strain

gradient; nonlocal.

1. Introduction

Carbon nano-tubes are among the most important nano-materials that have been

developed by nanotechnologists. They are ¯nding a wide range of applications. The

superior properties of these materials, such as mechanical, thermal, and electronic

properties used in nano-electronics, nano-devices, and nano-composites are among

the major reasons for their wide applications in various industries.1,2 Investigation

and analysis of nano-tube stability are among the engineering analyses of most

interest in the nano-community.3 There are di®erent structural models in use for the

†Corresponding author.

International Journal of Structural Stability and DynamicsVol. 13, No. 5 (2013) 1350022 (21 pages)

#.c World Scienti¯c Publishing Company

DOI: 10.1142/S0219455413500223

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http://dx.doi.org/10.1142/S0219455413500223

investigation of stability of nano-size structures.4�6 The models that are frequently

used for investigating the nano-size structural problems include the strain gradient7

and Eringen's nonlocal theory of elasticity.8 In the literature, researchers have pre-

sented diverse results on the problem of stability of nano-tubes by the nonlocal

theory of elasticity. Aydogdu9 used a generalized nonlocal beam theory to study the

buckling of nano-beams. He showed that by increasing the nonlocal parameter the

critical buckling load will decrease. Murmu and Pradhan10 investigated the buckling

of single-walled carbon nano-tube embedded in an elastic medium based on the

nonlocal elasticity with Timoshenko beam. They showed that the nonlocal par-

ameter can reduce the buckling load and stability of the beam. Beskou et al.11

analyzed the bending and stability of an Euler�Bernoulli beam based on a simple

linear theory of gradient elasticity with the surface energy taken into account. They

showed that this theory can increase the stability of elastic beams.

As a higher-order continuum theory, the classical coupled stress elasticity theory

originated by Mindlin, Toupin and Koiter contains four material constants for iso-

tropic materials.12,13 Fleck and Hutchinson14 reformulated and extended the classical

coupled stress theory, and renamed it as the strain gradient theory. Recently, a

modi¯ed coupled stress theory has been proposed by Yang et al.15 In this theory,

only one additional internal material parameter related to length scale is involved in

the constitutive equation, apart from those with two material constants as previously

de¯ned. Lam et al.7 proposed a modi¯ed strain gradient elasticity theory in which a

new additional equilibrium equation governs the behavior of higher-order stresses

and momentum, aside from the classical equilibrium equation. Curved beams have

found interesting applications in areas such as switches and actuators.16 Accordingly,

many researchers have studied the behavior of these structures.17 As for arch beams,

snap-through instability is a phenomenon deserving special attention, which has

been studied by numerous researchers in the ¯eld of MEMS.18 In macro size tech-

nology, this subject has been frequently studied in the static and/or dynamic

environments.19�21

The following areas are the potential areas for application of curved and snap-

through (bi-stable) micro/nano-structure devices: small-scale micro/nano- bi-stable

transformers and transducers, bi-stable actuators and sensors, micro/nano- bi-stable

switches, bi-stable relays, bi-stable multi-vibrators, bi-stable latches, etc. This is also

true for initially-curved arches and shell structures. Due to the existence of a snap-

through (bi-stable) equilibrium, they may be used as a tool for achieving pull-in

current. Other applications of this type of micro/nano-structures may be found in

Refs. 22�24.

In this research, we use two nonclassical theories, i.e. strain gradient and nonlocal

theories, to investigate the static nonlinear snap-through and bifurcation behavior of

nano shallow arches. We use a new strain gradient theory proposed by Lam et al.7 In

the nonlinear formulation, von Karman's terms of nonlinear strain are adopted for

modeling moderate rotations. A series solution procedure is proposed for solving the

F. Kaviani & H. R. Mirdamadi

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nonlinear equilibrium equation. The results showed that small size e®ect for di®erent

theories can have di®erent directions of sti®ness change. Including the strain gra-

dient theory in the continuum formulation can improve the stability of arch beams;

however, the same is not true for the nonlocal theory. Furthermore, in some cases,

these nonclassical theories can have di®erent mechanisms of instability, as compared

with the classical theory. Thus, \small size" has an important e®ect on the analysis of

nano-arch-type structures. In other words, nonclassical analysis of these structures is

mandatory.

The following sections of the paper are arranged as follows: In Sec. 2, the non-

classical theories are formulated, the equation of motion and related boundary

conditions for both theories are derived, and the analytical solutions for equations of

motion are presented. In Sec. 3, numerical results are presented, and ¯nally in Sec. 4,

some conclusions are drawn.

2. Nonclassical Theories

2.1. Strain gradient theory

2.1.1. Basic relations

According to the strain gradient theory proposed by Lam et al.,7 the strain energy

density is a function of the symmetric strain tensor, dilatation gradient vector,

deviatoric stretch gradient tensor, and symmetric rotation gradient tensor. The

strain energy density function (Us) for linear, isotropic, gradient-dependent elastic

material occupying region S may be written as25

Us ¼1

2

ZS

ð�ij"ij þ pi�j þ �1ijk�

1ijk þmij�ijÞdv: ð1Þ

In this relation "ij, �i, �1ijk, and �ij denote respectively the strain tensor, dilatation

gradient vector, deviatoric stretch, and symmetric rotation gradient tensor, which

are de¯ned as follows:

"ij ¼ 1

2ðui;j þ uj;iÞ;

�j ¼ "kk;i;

�1ijk ¼ 1

3ð"jk;i þ "ki;j þ "ij;kÞ �

1

15ð�ijð"ll;k þ 2"lk;lÞ

þ �jkð"ll;i þ 2"li;lÞ þ �kið"ll;j þ 2"lj;lÞÞ;�ij ¼ 1

2ðeipq"qj;p þ ejpq"qi;pÞ;

ð2Þ

where ui is the displacement vector, �ij, and eipq are, respectively, the Kronecker

delta and alternating tensor. The classical stress (�ijÞ and higher-order stresses

Snap-through and Bifurcation of Nano-Arches on Elastic Foundation

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(pi, �1ijk, and mijÞ are de¯ned as

�ij ¼ �"ij�ij þ 2G~"ij;

pi ¼ 2l20G�i;

� 1ijk ¼ 2l21G�1ijk;

mij ¼ 2l22G�ij:

ð3Þ

In the above equation, � and G are the bulk and shear moduli, respectively, l0, l1,

and l2 are the material length scale parameters related to dilatation, deviatoric

stretch, and rotation gradients, respectively. The deviatoric strain (~"ij) is given by

~"ij ¼ "ij �1

3"pp�ij: ð4Þ

Consider a slender arch of small initial curvature in Fig. 1. The assumption

of a slender beam is satis¯ed except that it has an initial shape with the un-

deformed mid-line characterized by w0ðxÞ. Using uðxÞ and wðxÞ to denote the

displacements in longitudinal and transversal directions, the only strain component

is de¯ned as

"11 ¼@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� z@ 2w

@x2� @ 2w0

@x2

� �: ð5Þ

Fig. 1. (Color online) Nano-arch shallow beam resting on an elastic foundation under transverse loadings.


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By substituting Eq. (5) into Eqs. (2)�(4), one obtains the nonzero components of

Eq. (2) as

�1 ¼@ 2u

@x2þ @w

@x

@ 2w

@x2� @w0

@x

@ 2w0

@x2� z

@ 3w

@x3� @ 3w0

@x3

� �;

�3 ¼ � @ 2w

@x2� @ 2w0

@x2

� �;

�12 ¼ �21 ¼ � 1

2

@ 2w

@x2� @ 2w0

@x2

� �;

�1111 ¼2

5

@ 2u

@x2þ @w

@x

@ 2w

@x2� @w0

@x

@ 2w0

@x2� z

@ 3w

@x3� @ 3w0

@x3

� �� ;

�1113 ¼ �1311 ¼ �1131 ¼ � 4

15

@ 2w

@x2� @ 2w0

@x2

� �;

�1122 ¼ �1133 ¼ �1212 ¼ �1221 ¼ �1313 ¼ �1331

¼ � 1

5

@ 2u

@x2þ @w

@x

@ 2w

@x2

�� @w0

@x

@ 2w0

@x2� z

@ 3w

@x3� @ 3w0

@x3

� ��;

�1223 ¼ �1232 ¼ �1322 ¼1

15

@ 2w

@x2� @ 2w0

@x2

� �;

�1333 ¼1

5

@ 2w

@x2� @ 2w0

@x2

� �:

ð6Þ

Consequently, by substituting Eq. (6) into Eq. (3), one obtains the nonzero com-

ponents of classical and higher-order stresses as

�11 ¼ E"11 ¼ E@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� z@ 2w

@x2� @ 2w0

@x2

� �� ;

p1 ¼ 2l20G@ 2u

@x2þ @w

@x

@ 2w

@x2� @w0

@x

@ 2w0

@x2� z

@ 3w

@x3� @ 3w0

@x3

� �� ;

p3 ¼ �2l20G@ 2w

@x2� @ 2w0

@x2

� �;

m12 ¼ m21 ¼ �l22G@ 2w

@x2� @ 2w0

@x2

� �;

� 1111 ¼

4

5l21G

@ 2u

@x2þ @w

@x

@ 2w

@x2� @w0

@x

@ 2w0

@x2� z

@ 3w

@x3� @ 3w0

@x3

� �� ;

� 1113 ¼ � 1

131 ¼ � 1311 ¼ � 8

15l21G

@ 2w

@x2� @ 2w0

@x2

� �;


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� 1122 ¼ � 1

133 ¼ � 1212 ¼ � 1

221 ¼ � 1313 ¼ � 1

331

¼ � 2

5l21G

@ 2u

@x2

�� @w

@x

@ 2w

@x2� @w0

@x

@ 2w0

@x2� z

@ 3w

@x3� @ 3w0

@x3

� ��;

� 1223 ¼ � 1

232 ¼ � 1322 ¼

2

15l21G

@ 2w

@x2� @ 2w0

@x2

� �;

� 1333 ¼

2

5l21G

@ 2w

@x2� @ 2w0

@x2

� �:

ð7ÞBy substituting the stresses and strains into Eq. (1), the strain energy density

function is calculated as

Us ¼1

2

ZS

�ij"ij þ pi�j þ �1ijk�

1ijk þmij�ij

� �dv

¼ 1

2

Z L

0

ZA

Ez2 þG 2l20 þ

120

225l21 þ l22

� �" #@ 2w

@x2� @ 2w0

@x2

!2

(

þ z2G 2l20 þ4

5l21

� �@ 3w

@x3 � @ 3w0

@x3

!2

þ E@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2" #

2

þG 2l20 þ4

5l21

� �@u

@xþ 1

2

@w

@x

� �2

"

� 1

2

@w0

@x

� �2

� z@ 2w

@x2� @ 2w0

@x2

!#2)dAdx: ð8Þ

By neglecting the initial curvature w0 of the beam, Eq. (8) would reduce to the strain

energy density function derived by Kahrobaiyan et al.26

2.1.2. Equilibrium equations for strain gradient

Here, the governing equations are derived using the principle of virtual displacements

as given below:

�UT ¼ �Us þ �Up ¼ 0; ð9Þ

where UT , Us, and Up are the total potential energy density function, strain energy

density function, and the work done by the conservative external loads acting on the

structure, respectively. The work done by the external loads acting on the nano-arch

beam resting on a Winkler elastic foundation and the transverse load P ðxÞ acting on


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the beam is de¯ned as

Up ¼Z L

0

P ðxÞw� 1

2Kwðw� w0Þ2

� �dx; ð10Þ

where P ðxÞ is the transversal distributed force per unit length, andKw is the Winkler

foundation modulus. By substituting Eqs. (10) and (8) into Eq. (9) and separating

the resulting equation with respect to the virtual displacements �u and �w, the

equilibrium equations for the strain gradient theory can be obtained as follows:

(a) Equilibrium in the u direction:

@

@xEA

@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� ��

� GA 2l20 þ4

5l21

� �@ 2

@x2

@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� ��¼ 0; ð11aÞ

(b) Equilibrium in the w direction:

� @

@xEA

@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� ��

� GA 2l20 þ4

5l21

� �@ 2

@x2

@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� ��@w

@x

�

þ ðEI þ �Þ @ 4w

@x4� @ 4w0

@x4

� ��

@ 6w

@x6� @ 6w0

@x6

� �

� P ðxÞ þKwðw� w0Þ ¼ 0; ð11bÞwhere A is the arch cross-sectional area and I the area moment of inertia of the nano-

arch cross-section, and the parameters � and are de¯ned as

� ¼ GA 2l20 þ8

15l21 þ l22

� �;

¼ GI 2l20 þ4

5l21

� �:

ð12Þ

Let us denote N as

N ¼EA@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� �

�GA 2l20 þ4

5l21

� �@ 2

@x2

@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� �: ð13Þ

From Eq. (11a) and Eq. (13), one has

@N

@x¼ 0: ð14Þ


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Accordingly, N should be constant and can be obtained from Eq. (15) as

N ¼ 1

L

Z L

0

EA@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� ��

�GA 2l20 þ4

5l21

� �@ 2

@x2

@u

@xþ 1

2

@w

@x

� �2

� 1

2

@w0

@x

� �2

� ��dx: ð15Þ

It is noted that, if the supports are immovable, then uð0Þ ¼ uðLÞ ¼ 0. Thus, the

expression for N; after averaging by integration over the length, becomes

N ¼ 1

2L

Z L

0

EA@w

@x

� �2

� @w0

@x

� �2

� ��

�GA 2l20 þ4

5l21

� �@ 2

@x2

@w

@x

� �2

� @w0

@x

� �2

� ��dx ð16Þ

By substituting Eq. (16) into Eq. (11b), the equation of equilibrium can be ob-

tained as

� @

@x

@w

@x

1

2L

Z L

0

EA@w

@x

� �2

� @w0

@x

� �2

� ��

� GA 2l20 þ4

5l21

� �@ 2

@x2

@w

@x

� �2

� @w0

@x

� �2

� ��dx

�þ ðEI þ �Þ

� @ 4w

@x4� @ 4w0

@x4

� ��

@ 6w

@x6� @ 6w0

@x6

� �� P ðxÞ þKwðw� w0Þ ¼ 0: ð17Þ

We note that Eq. (17) is a sixth-order di®erential equation of a shallow nano-arch

based on strain gradient theory, as opposed to the fourth-order di®erential equation

of a shallow macro-arch developed based on classical continuum theory. In Eq. (17),

by neglecting the material length scale parameters, one has � ¼ 0 and ¼ 0, which is

the case for the classical form of the equilibrium equation of the macro-arch beam

under transverse loads, as was presented in Ref. 27. In this study, only simply-

supported nano-arch beams are considered. For a simply-supported nano-arch beam,

one has the following geometrical boundary conditions:

wð0Þ � w0ð0Þ ¼ wðLÞ � w0ðLÞ ¼ 0: ð18Þ

The partial nonclassical natural boundary conditions for a partial strain gradient

theory and for a simply-supported nano-arch are based on zero bending moments at

the two ends, i.e. Mx¼0;L ¼ 0. The bending moment is a superposition of two e®ects:

a classical but higher-order bending moment, M1 ¼RAðz�1 þ p3 þ �1113 þm12ÞdA;

and a nonclassical bending moment, M2 ¼RAzðp1 þ �1111ÞdA. The corresponding

terms of bending moment in the strain energy density function are UM ¼ � 12

RA

½M1@ 2w@x2 þM2

@ 3w@x 3 �dx. After integration by parts, the ¯nal result for the classical but


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higher-order part of bending moment is as follows:

EI þGA 2l20 þ8

15l21 þ l22

� �� @ 2w

@x2� @ 2w0

@x2

� ��

�GI 2l20 þ4

5l21

� �@ 4w

@x4� @ 4w0

@x4

� ��x¼0;L

¼ 0: ð19aÞ

By using the de¯nitions in Eq. (12), one has

ðEI þ �Þ @ 2w

@x2� @ 2w0

@x2

� ��

@ 4w

@x4� @ 4w0

@x4

� �� x¼0;L

¼ 0: ð19bÞ

Before one proceeds with the analytical solution, it is convenient to express all of the

parameters in a nondimensional form. The nondimensional parameters are de¯ned as

�x ¼ x

L; Wð�xÞ ¼ wðxÞ

�; �Kw ¼ KwL4

4EI; �Pð�xÞ ¼ P ðxÞL4

�4EI;

� ¼ �

EI; � ¼ 2

L2EI; � ¼ �2

L2; �N ¼ NL2

2EI; � ¼

ffiffiffiffiffiI

A

r;

ð20Þ

where � is the radius of gyration for the nano-arch area beam.The nondimensional

strain gradient equilibrium equation is obtained by substituting the nondimensional

parameters in Eq. (20) into Eq. (17), namely,

��@ 6W

@�x 6� @ 6W0

@�x 6

� �þ ð1þ �Þ @ 4W

@�x 4� @ 4W0

@�x 4

� �

� �N@ 2W

@�x 2þ �Kw W �W0ð Þ � �PðxÞ ¼ 0: ð21Þ

Therefore, the expression for �N becomes

�N ¼ 1

2

Z

0

@W

@�x

� �2

� @W0

@�x

� �2

� ��

@ 2

@�x 2

@W

@�x

� �2

� @W0

@�x

� �2

� �� d�x: ð22Þ

The boundary conditions are as follows:

Wð0Þ �W0ð0Þ ¼ WðLÞ �W0ðLÞ ¼ 0

ð1þ �Þ @ 2W

@�x 2� @ 2W0

@�x 2

� ��

@ 4W

@�x 4� @ 4W0

@�x 4

� �� x¼0;L

¼ 0:ð23Þ

2.1.3. Analytical solution for strain gradient

An initial curvature of half-sine is assumed for a shallow nano-arch that is simply-

supported and loaded by a half-sine spatially-distributed force, i.e.

W0 ¼ c0 sinð�xÞ; ð24Þ


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where c0 is the initial rise parameter. The expression for the loading is

P ð�xÞ ¼ P0 sinð�xÞ: ð25ÞThe de°ection may be obtained by an in¯nite sine series expansion for a simply-

supported nano-arch beam, namely,

W ¼ c0 sinð�xÞ þX1n¼1

cn sinðn�xÞ: ð26Þ

Consequently, �N can be obtained as follows

�N ¼ 1

4

X1n

ðn2c2nÞ þ �n4c2n �þ 2c0c1

( ): ð27Þ

Substituting Eqs. (26) and (27) into Eq. (21) yields the following equation of

equilibrium:

�X1n

n6cn sinðn�xÞ �þ ð1þ �Þ

X1n

n4cn sinðn�xÞ �þ �Kw

X1n

cn sinðn�xÞ½ �

� P0 sinð�xÞ þ1

4

X1m

ðm2c2mÞ þ �m4c2m �þ 2c0c1

( ) !

� c0 sinð�xÞ þX1n

n2cn sinðn�xÞ � !

¼ 0: ð28Þ

By separating the sine coe±cients, the above equation breaks into two coupled

equations as in Eq. (29). In the ¯rst equation, the integer n takes the value one and in

the second equation, n takes values higher than one, that is, we could have two

distinguished modes for system deformations, as follows:

(1) Equilibrium for the ¯rst mode of nano-arch deformations,

ð1þ �þ � þ �KwÞc1 � P0

þ 1

4

X1m


( )c0 þ c1ð Þ ¼ 0; n ¼ 1: ð29aÞ

(2) Equilibrium for the higher modes of nano-arch deformations,

�n6 þ ð1þ �Þn4 þ �Kw þ 1

4

X1m


( )n2

( )cn ¼ 0;

n ¼ 2; 3; . . . :

ð29bÞEquations (29a) and (29b) could represent the basic formulation of the nano-

system including strain gradient theory for a discussion on system stability, shortly.


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2.1.4. Stability analysis for strain gradient

As far as the stability of the nano-arch beam is concerned, the equilibrium equations

derived in Eqs. (29a) and (29b) represent the exact equilibrium equations developed

based on the strain gradient theory. Equation (29) shows the coupled behavior of the

¯rst mode with the higher modes of deformation of the nano-arch beam. Accordingly,

one can analyze Eqs. (29a) and (29b) for two possible modes of equilibrium. In the

¯rst mode of equilibrium, only the ¯rst term of the transverse displacement (c1Þ isallowed to exist, while the higher terms (cn;n ¼ 2; 3; . . .) are set to zero. In the other

mode of equilibrium, in addition to the ¯rst term of transverse displacement, there

could exist one higher term cnðn1Þ that should not be set to zero.

In the sequel, for the ¯rst mode of equilibrium, ðn ¼ 1Þ, Eq. (29b) is satis¯ed

automatically. By using Eq. (29a), the load�displacement relation can be derived as

1þ �þ � þ �Kw þ 1

2c20

� �c1 þ

3

4þ �

� �c0c

21 þ

1

4þ �

� �c31 ¼ P0: ð30Þ

For a stability analysis, the snap-through would occur at the point where sti®ness of

the nano-arch beam would be diminished to zero, or would be at the minimal point of

the load�displacement curve.

For the second mode of equilibrium, to satisfy Eq. (29b), for n ¼ j > 1, one has

�j6 þ ð1þ �Þj4 þ �Kw þ 1

4ðj2c2jÞ þ �j4c2j �þ 2c0c1�

j2 ¼ 0 j > 1: ð31Þ

By using Eq. (31), one can obtain from Eq. (29a) the following result:

ð1þ �þ � þ �KwÞc1 � P0 ��j6 þ ð1þ �Þj4 þ �Kw

j2

� �c0 þ c1ð Þ ¼ 0 j > 1: ð32Þ

Equation (32) would be established when the higher terms of displacements would

not be set to zero, which could express the behavior of the nano-arch beam based on

the strain gradient theory, together with Eq. (30). In summary, Eqs. (30) and (32)

would govern the equilibrium and stability behavior of a nano-arch resting on a

Winkler foundation and subjected to transverse loading, accounting for strain

gradient theory. Equations. (30) and (32) could represent the stability behavior

of the system for the ¯rst- and higher-order modes of the system deformations,

respectively.

2.2. Eringen's nonlocal theory

2.2.1. Basic relations

According to Eringen,8 the stress ¯eld at a point x in an elastic continuum is

dependent not only on the strain ¯eld at the point under consideration, but also on

the strains at all other points of the body. The most general form of constitutive

equation for the nonlocal elasticity involves a convolution integral over the entire

region of interest. This integral contains a kernel function that describes the relative


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in°uences of strains at various locations on the stress for a given location. Thus, the

nonlocal stress tensor � at a point x is expressed as

� ¼ZV

K x 0 � xj j; �ð Þtðx 0Þdðx 0Þ; ð33Þ

where tðxÞ is the classical stress tensor at point x and the kernel function

(Kðjx 0 � xj; �ÞÞ represents a nonlocal modulus. jx 0 � xj is the distance in Euclidean

norm and � is a material constant that depends on the internal and external

characteristic lengths of the structure. According to the Hookean solid modeling, the

nonlocal constitutive equation can be represented as follows:

ð1� � 2l2r2Þ� ¼ C : "; � ¼ e0a

l2; ð34Þ

where C is the elasticity tensor and e0, l, and a are the material constant, external

characteristic length, and internal characteristic length, respectively. For the nano-

arch beam based on the Euler�Bernoulli beam theory, the only nonzero nonlocal

stress is de¯ned as

�xx � �@ 2�xx

@x2¼ E"xx; � ¼ ðe0aÞ2; ð35Þ

where � is the nonlocal parameter. When the nonlocal parameter is zero, one obtains

the constitutive relations of the local theory, as they should be.

2.2.2. Equilibrium equations for Eringen's nonlocal theory

According to the nonlocal constitutive relation, the force resultant of axial stress is

de¯ned as

N � �@ 2N

@x2¼ EA

@u

@xþ 1

2

@w

@x

� �2

� @w0

@x

� �2

� �: ð36Þ

The bending moment M, resulting from the nonlocal stress is given by

M � �@ 2M

@x2¼ �EI

@w

@x

� �2

� @w0

@x

� �2

� �: ð37Þ

By substituting Eqs. (36) and (37) into the equilibrium equation, one obtains

� 1� �@ 2

@x2

� �@

@x

1

2L

Z L

0

EA@w

@x

� �2

� @w0

@x

� �2

� �� dx

@w

@x

� �

þ EI@ 4w

@x4� @ 4w0

@x4

� �� 1� �

@ 2

@x2

� �P ðxÞ þ 1� �

@ 2

@x2

� �Kwðw� w0Þ ¼ 0:

ð38Þ


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Similar to the strain gradient theory, there are two geometrical boundary conditions

and two natural boundary conditions for the partial nonlocal theory, as given below:

wð0Þ � w0ð0Þ ¼ wðLÞ � w0ðLÞ ¼ 0;

@ 2w

@x2� @ 2w0

@x2

� �x¼0;L

¼ 0:ð39Þ

By substituting the nondimensional parameters in Eq. (20) into Eq. (38), the non-

dimensional nonlocal equilibrium equation is expressed as follows:

@ 4W

@�x 4� @ 4W0

@�x 4

� �� ð1� �

@ 2

@�x 2Þ �N @ 2W

@�x 2

þ 1� �@ 2

@�x 2

� ��Kw W �W0ð Þ � 1� �

@ 2

@�x 2

� ��PðxÞ ¼ 0: ð40Þ

Therefore, the expression for �N becomes

�N ¼ 1

2

Z

0

@W

@�x

� �2

� @W0

@�x

� �2

� �� d�x: ð41Þ

The boundary conditions are converted to

W ð0Þ �W0ð0Þ ¼ W ðLÞ �W0ðLÞ ¼ 0;

@ 2W

@�x 2� @ 2W0

@�x 2

� �x¼0;L

¼ 0:ð42Þ

2.2.3. Analytical solution for Eringen's nonlocal theory

The axial force �N can be obtained as follows:

�N ¼ 1

4

X1n

ðn2c2nÞ �þ 2c0c1

( ): ð43Þ

By substituting Eqs. (26) and (43) into Eq. (40), one obtains the equilibrium

equation as

X1n

½n4cn sinðn�xÞ� þ �Kw

X1n

½ð1þ �n2Þcn sinðn�xÞ� � ð1þ �ÞP0 sinð�xÞ

þ 1

4

X1m

½ðm2c2mÞ� þ 2c0c1

( ) !ð1þ �Þc0 sinð�xÞð

þX1n

ð1þ �n2Þn2cn sinðn�xÞ �! ¼ 0: ð44Þ

By separating the sine coe±cients, Eq. (44) breaks into two coupled equations as

given by Eqs. (45a) and (45b). In Eq. (45a), the integer n is allowed to take the value

one while in Eq. (45b) the integer n can take values other than one; namely, we could


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have two distinguished modes for system deformations, as follows:

(1) Equilibrium for the ¯rst mode of nano-arch deformations,

ð1þ ð1þ �Þ �KwÞc1 � ð1þ �ÞP0

þ ð1þ �Þ4

X1m

ðm2c2mÞ �þ 2c0c1

( )c0 þ c1ð Þ ¼ 0; n ¼ 1: ð45aÞ

(2) Equilibrium for the higher modes of nano-arch deformations,

ðn4 þ ð1þ �n2Þ �Kw þ ð1þ �n2Þ4

X1m

ðm2c2mÞ �þ 2c0c1

( )n2

( )cn ¼ 0;

n ¼ 2; 3; . . . : ð45bÞ

Equations (45a) and (45b) would establish the basic formulation of the nano-system

including Eringen's nonlocal theory for the coming discussion of system stability.

2.2.4. Stability analysis for Eringen's nonlocal theory

The equilibrium equations derived in Eqs. (45a) and (45b) represent the exact

equilibrium equations developed based on the Eringen's nonlocal theory. Equations

(45a) and (45b) would depict the coupled behavior of the ¯rst mode with the higher

modes of deformations of the nano-system. Therefore, one could analyze Eqs. (45a)

and (45b) for two possible equilibrium modes. In the ¯rst mode of equilibrium, only

the ¯rst term of the transverse displacement (c1) is allowed to exist, while the higher-

order terms (cn;n ¼ 2; 3; . . .) are set to be zero. In the other mode of equilibrium, in

addition to the ¯rst term of transverse displacement, there exists one higher-order

term cnðn > 1Þ that should not be set to zero for observing more mechanisms of

stable and unstable equilibria.

Next, for the ¯rst mode of equilibrium, ðn ¼ 1Þ, Eq. (45b) is satis¯ed automati-

cally. By using Eq. (45a), the load�displacement relation can be obtained as

1

1þ �þ �Kw þ 1

2c20

� �c1 þ

3

4c0c

21 þ

1

4c31 ¼ P0: ð46Þ

For the second mode of equilibrium, to satisfy Eq. (45b), for n ¼ j, one has

j4 þ ð1þ �j2Þ �Kw þ ð1þ �j2Þ4

j2c2j þ 2c0c1�

j2 ¼ 0 j > 1: ð47Þ

By using preceding equation, one obtains from Eq. (45a) the following relation:

ð1þ ð1þ �Þ �KwÞc1 � ð1þ �ÞP0 �ð1þ �Þð1þ �j2Þ j4 þ ð1þ �j2Þ �Kw

� c0 þ c1ð Þ ¼ 0 j > 1:

ð48Þ


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Equation (48) would be established when the higher-order terms of displacements

would not set to zero. This equation expresses the behavior of the nano-arch beam

based on Eringen's nonlocal theory, besides Eq. (46). In summary, Eqs. (46) and (48)

would govern the stability behavior of a nano-arch resting on a Winkler foundation

and subjected to transverse loading, including Eringen's nonlocal theory.

3. Numerical Results

3.1. Validation

To validate the analytical solutions derived based on strain gradient and Eringen's

nonlocal theories, the e®ect of the nonclassical theories are discarded herein, and the

resulting equation is compared with that of Ref. 27. The derived formulations for the

strain gradient theory, i.e.Eqs. (30) and (32), andEringen's nonlocal theory, i.e. Eqs. (46)

and (48), without the e®ects of small-size could reduce to the equation appearing in the

Ref. 27, developed for the classical (local) structural systems, as follows:

j2c2j þ c0 þ c1ð Þ2� ¼ c0ð Þ2 � 4n4 þ �Kw

n2: ð49Þ

In this study, the equilibrium equations were derived for both the strain gradient and

Eringen's nonlocal theories, which could be used to investigate the existence of higher-

order deformationmodes. In what follows, we use the equilibrium equations (29b) for the

strain gradient theory and (45b) for Eringen's nonlocal theory.

3.2. Strain gradient theory

For the strain gradient theory, one has

j2 þ �j4½ �1þ �

c2j þc0

1þ �þ c1

� �2

� �¼ c0

1þ �

� �2

� 4�n6 þ ð1þ �Þn4 þ �Kw

ð1þ �Þn2: ð50Þ

Equation (50) represents an elliptical shape of the equilibrium equation in c1 � cjspace, which limits the existence of cj mode. According to this equation, the existence

of cj mode is possible only when the right side of the equation is positive, namely,

c01þ �

� �2

> 4�n6 þ ð1þ �Þn4 þ �Kw

ð1þ �Þn2: ð51Þ

In one special case, all of the characteristic lengths, l0, l1, and l2, could be selected the

same. Accordingly, the nondimensional small size parameters in the strain gradient

theory (� and �) are related to each other by �=� ¼ 149=126.

3.3. Eringen's nonlocal theory

For the nonlocal theory, Eq. (51) can be expressed as follows:

ðc0Þ2 > 4n2

ð1þ �n2Þ þ�Kw

n2

� �: ð52Þ


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3.4. Classical (local) theory

For the classical analysis, Eq. (50) can be shown as follows:

ðc0Þ2 > 4 n2 þ�Kw

n2

� �: ð53Þ

3.5. Comparison of di®erent theories

The possible positions of displacements, for both Eringen's nonlocal and strain

gradient theories, are shown in Fig. 2. For this example, n ¼ 2, �Kw ¼1, c0 ¼ 5.5,

� ¼ 0:05, and � ¼ 0:05 are considered. It is observed that under the external loading,

the absolute value of a1 will increase. When both of Eqs. (52) and (53) are satis¯ed,

after c1 reaches a critical value, cn-modes will appear. This point is a bifurcation

point where the nano-arch will be unstable after this point. It is observed that for the

strain gradient theory, the stability of nano-arch beam will last longer than that of

the classical theory, and the critical load corresponding to the bifurcation will occur

after passing the value corresponding to the classical theory. However, for the clas-

sical theory, the nano-arch beam will last longer, as a stable system, compared with

the nonlocal theory. The results shown in Fig. 2 indicate a complete agreement of the

present theories with those of Ref. 27 for the case that the small-size e®ect was not

considered.

Let us consider ¯rst the case where higher modes of deformation do not exist in

the behavior of the nano-arch beam. According to the classical theory, there are two

variants for this case, one for c0 � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffikw þ 1

pand the other for c0 > 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffikw þ 1

p. For

the ¯rst condition with c0 � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffikw þ 1

p, there is a one-to-one correspondence between

the loading and the deformation. Accordingly, neither snap-through nor bifurcation

Fig. 2. (Color online) Higher-order term (n ¼ 2) of the deformation sine series versus the ¯rst-order term

(n ¼ 1) of the deformation sine series for the classical, strain gradient, and Eringen's nonlocal theories.


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will occur. For the second condition with c0 > 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffikw þ 1

p, there is no such one-to-one

correspondence and the nano-arch beam will encounter instability for that special

loading. Figure 3 shows the condition with c0 � 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffikw þ 1

p, for which the e®ects of

the nonclassical theories are investigated. It can be observed that for the nonlocal

analysis, the nano-arch beam can become unstable and the snap-through can occur,

while the local analysis shows that the nano-arch is stable.

Figure 4 shows the second condition with c0 > 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffikw þ 1

p, for which the e®ect of

non-classical theories is investigated. It can be observed that for the strain gradient

Fig. 3. (Color online) Nondimensional load versus the ¯rst-order term (n ¼ 1) of the deformation sine

series for the classical and nonclassical theories, for c0 ¼ 3:4, �Kw ¼ 2:0, � ¼ 0:5, and � ¼ 0:5.




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analysis the nano-arch beam is stable, while for the classical analysis, it can become

unstable.

One of the most important cases in instability analysis of the nano-arch beams is

related to the condition where the higher-order terms do exist. This case will occur in

the classical analysis for the condition of ðc0Þ2 > 4ðn2 þ �Kw

n2 Þ. To investigate this case,one can have recourse to Eqs. (48) and (46). In Fig. 5, this case is investigated for the

classical and nonclassical theories. For each theory, there are two curves, one related


series for the classical and nonclassical theories, for c0 ¼ 5:0, �Kw ¼ 1:0, � ¼ 0:01, and � ¼ 0:01:




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to only c1 and another to c1 with due account to cn. The intersection of the two

curves is a bifurcation point, to which the corresponding load is the critical load.

After passing this critical load, the nano-arch beam becomes unstable. The strain

gradient theory predicts a higher critical load and a more stable nano-arch beam,

while the nonlocal theory predicts a lower critical load.

In the case study of Fig. 6, there exist some values of the strain gradient par-

ameter, which indicates the possibility of bifurcation by the classical theory. How-

ever, it changes to the type of snap-through instability using the strain gradient

theory.

Figure 7 shows that the critical load corresponding to the snap-through

instability can occur before that corresponding to the bifurcation instability. For the

classical theory, this can occur when c0 < ½2ð3n2 � 1Þ � 2 �Kwð1� 3n2Þ� 12 . For some

values of the nonlocal parameter, when the instability is of the snap-through type

based on the classical theory, the nano-arch beam will lose its stability by the

bifurcation type based on the nonlocal theory.

4. Conclusions

In this paper, a nonlinear size-dependent formulation for shallow nano-arches resting

on a Winkler elastic foundation was derived based on both the strain gradient and

Eringen's nonlocal theories. To investigate the elastic stability of a nano-arch, von

Karman's geometric nonlinear terms were considered. The transverse loading applied

on the nano-arch was assumed to have a sinusoidal distribution. To solve the non-

linear equilibrium equations, a sine series solution was adopted for the displacement

Fig. 7. (Color online) Nondimensional load versus the ¯rst-order term (n ¼ 1) of the deformation sineseries for the classical and nonclassical theories, for c0 ¼ 4:6, �Kw ¼ 1:0, � ¼ 0:03, and � ¼ 0:02.


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of the simply-supported boundary arch. To investigate di®erent small-size e®ects,

both the stress gradient theory (Eringen's nonlocal) and the strain gradient theories

were adopted. The results showed that the instability in the nano-arch can occur in

two di®erent modes either by snap-through or by bifurcation instabilities. The modes

of deformation of the nano-arch are related to the system parameters, such as the

initial curvature of the nano-arch, the foundation modulus, and the small-size par-

ameters used in the nonclassical strain gradient and Eringen's nonlocal theories.

Therefore, for di®erent system parameters, the system showed di®erent modes of

deformation. Overall, two cases are possible for the deformation of the nano-arch. In

the ¯rst case, the nano-arch is deformed by the ¯rst-order term of the sine series

solution, and in the second case, only, one higher-order term together with the ¯rst-

order term of the deformation can exist. The system parameters can prescribe the

deformation of the problem considered. For the ¯rst case, the only possible type of

instability is snap-through, but for the second case, the instability mechanism can be

either snap-through or bifurcation. The system parameter might specify the mech-

anism of instability in both cases. The small-size parameter in the strain gradient

theory can make the systemmore stable. By increasing the strain gradient parameter,

the critical load will increase. However, Eringen's nonlocal theory can reduce the

stability of the nano-arch. By increasing the nonlocal parameter inmost occasions, the

critical load will decrease. The foundation modulus can increase the stability of

the system and in some cases can change the type of mechanism for instability.

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snap-through and bifurcation of nano-arches on elastic foundation by the strain gradient and...

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