Journal of the Mechanics and Physics of Solids 60 (2012) 487–508
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Journal of the Mechanics and Physics of Solids
journal homepage: www.elsevier.com/locate/jmps
Postbuckling analysis and its application to stretchable electronics
Yewang Su a,1, Jian Wu a,1, Zhichao Fan a, Keh-Chih Hwang a,n, Jizhou Song b,
Yonggang Huang c,n, John A. Rogers d
a
AML, Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China
Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, FL 33124, USA
Departments of Civil and Environmental Engineering and Mechanical Engineering, Northwestern University, Evanston, IL 60208, USA
d
Department of Materials Science and Engineering, University of Illinois, Urbana, IL 61801, USA
b
c
a r t i c l e i n f o
abstract
Article history:
Received 13 September 2011
Received in revised form
12 November 2011
Accepted 19 November 2011
Available online 1 December 2011
A versatile strategy for fabricating stretchable electronics involves controlled buckling
of bridge structures in circuits that are configured into open, mesh layouts (i.e. islands
connected by bridges) and bonded to elastomeric substrates. Quantitative analytical
mechanics treatments of the responses of these bridges can be challenging, due to the
range and diversity of possible motions. Koiter (1945) pointed out that the postbuckling
analysis needs to account for all terms up to the 4th power of displacements in the
potential energy. Existing postbuckling analyses, however, are accurate only to the
2nd power of displacements in the potential energy since they assume a linear
displacement–curvature relation. Here, a systematic method is established for accurate
postbuckling analysis of beams. This framework enables straightforward study of the
complex buckling modes under arbitrary loading, such as lateral buckling of the islandbridge, mesh structure subject to shear (or twist) or diagonal stretching observed in
experiments. Simple, analytical expressions are obtained for the critical load at the
onset of buckling, and for the maximum bending, torsion (shear) and principal strains in
the structure during postbuckling.
Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved.
Keywords:
Postbuckling
Stretchable electronics
Higher-order terms in curvature
Lateral buckling
Diagonal stretching
1. Introduction
Recent work suggests that it is possible to configure high performance electronic circuits, conventionally found in rigid,
planar formats, into layouts that match the soft, curvilinear mechanics of biological tissues (Rogers et al., 2010). The
resulting capabilities open up many application opportunities that cannot be addressed using established technologies.
Examples include ultralight weight, rugged circuits (Kim et al., 2008a), flexible inorganic solar cells (Yoon et al., 2008) and
LEDs (Park et al., 2009), soft, bio-integrated devices (Kim et al., 2010a, 2010b, 2011a, 2011b; Viventi et al., 2010), flexible
displays (Crawford, 2005), eye-like digital cameras (Jin et al., 2004; Ko et al., 2008; Jung et al., 2011), structural health
monitoring devices (Nathan et al., 2000), and electronic sensors for robotics (Someya et al., 2004; Mannsfeld et al., 2010;
Takei et al., 2010) and even ‘epidermal’ electronics capable of mechanically invisible integration onto human skin (Kim
et al., 2011b). Work in this area emphasizes mechanics and geometry, in systems that integrate hard materials for the
active components of the devices with elastomers for the substrates and packaging components. (Rogers et al., 2010).
n
Corresponding authors.
E-mail addresses: huangkz@tsinghua.edu.cn (K.-C. Hwang), y-huang@northwestern.edu (Y. Huang).
1
Equal contribution to this paper.
0022-5096/$ - see front matter Crown Copyright & 2011 Published by Elsevier Ltd. All rights reserved.
doi:10.1016/j.jmps.2011.11.006
488
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
shear
diagonal stretch
Fig. 1. (a) A schematic diagram of island-bridge, mesh structure in stretchable electronics; (b) SEM image (colorized for ease of viewing) of the islandbridge, mesh structure under shear; and (c) SEM image of the island-bridge, mesh structure under diagonal stretch.
A particularly successful strategy to stretchable electronics uses postbuckling of stiff, inorganic films on compliant,
polymeric substrates (Bowden et al., 1998). Kim et al. (2008b) further improved by structuring the film into a mesh and
bonding it to the substrate only at the nodes, as shown in Fig. 1a. Once buckled, the arc-shaped interconnects between the
nodes can move freely out of the mesh plane to accommodate large applied strain (Song et al., 2009). The interconnects
undergo complex buckling modes, such as lateral buckling, when subject to shear (Fig. 1b) or diagonal stretching (451 from
the interconnects) (Fig. 1c). Different from Euler buckling, these complex buckling modes involve large torsion and out-ofplane bending. It is important to ensure the maximum strain in interconnects are below their fracture limit.
Koiter (1945, 1963, 2009) established a robust method for postbuckling analysis via energy minimization. The potential
energy was expanded up to 4th power of displacement (from the non-buckled state), and the 3rd and 4th power terms
actually governed the postbuckling behavior (Koiter, 1945). The existing analyses for postbuckling of plates (or shells),
however, only account for the 1st power of the displacement in the curvature (von Karman and Tsien, 1941; Budiansky,
1973), which translates to the 2nd power of the displacement in the potential energy. This can be illustrated via a beam,
00
whose curvature k is approximately the 2nd order derivative of the deflection w, i.e., k ¼w . The bending energy is
R 002
ðEI=2Þ w dx, where EI is the bending stiffness, and the integration is along the central axis x of the beam. The accurate
R
expression of the curvature is k ¼ w00 =ð1 þ w02 Þ3=2 w00 ½1ð3=2Þw02 , which gives bending energy ðEI=2Þ k2 dx R 002
R 002 02
02
ðEI=2Þ w ð13w Þdx. It is different from the approximate expression above by ð3EI=2Þ w w dx, which is the 4th
power of displacement, and may be important in the postbuckling analysis (Koiter, 1945).
The objective of this work is to establish a systematic method for postbuckling analysis of beams that may involve
rather complex buckling modes such as lateral buckling of the island-bridge, mesh structure in stretchable electronics.
It avoids the complex geometric analysis for lateral buckling (Timoshenko and Gere, 1961) and establishes a straightforward method to study any buckling mode. It accounts for all terms up to the 4th power of displacements in the potential
energy, as suggested by Koiter (1945). Examples are given to show the importance of the 4th power of displacement,
including the complex buckling patterns of the island-bridge, mesh structure for stretchable electronics (Fig. 1). Analytical
expressions are obtained for the amplitude of, and the maximum strain in, buckled interconnects, which are important to
the design of stretchable electronics.
2. Postbuckling analysis
2.1. Membrane strain and curvatures
Let Z denote the central axis of the beam before deformation. The unit vectors in the Cartesian coordinates (X,Y,Z) before
deformation are Ei (i ¼1,2,3). A point X¼(0,0,Z) on the central axis moves to X þU¼(U1,U2,U3 þZ) after deformation, where
Ui(Z) (i¼1,2,3) are the displacements. The stretch along the axis is
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 02
1 0 02
02
0 2
0
02
ðU þ U 02
l ¼ U 02
ð2:1Þ
1 þ U 2 þð1 þ U 3 Þ 1 þU 3 þ
2 Þ U 3 ðU 1 þU 2 Þ,
2 1
2
where ()0 ¼d()/dZ, and terms higher than the 3rd power of displacement are neglected because their contribution to the
potential energy are beyond the 4th power. The length dZ becomes ldZ after deformation.
The unit vector along the deformed central axis is e3 ¼d(XþU)/(ldZ). The other two unit vectors, e1 and e2, involve twist
angle f of the cross section around the central axis. Their derivatives are related to the curvature vector j of the central
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
489
axis by Love (1927)
e0i
l
¼ j ei
ði ¼ 1,2,3Þ,
ð2:2Þ
where k1 and k2 are the curvatures in the (e2,e3) and (e1,e3) surfaces, respectively, and the twist curvature k3 is related to
the twist angle f of the cross section by
k3 ¼
f0
:
l
ð2:3Þ
The unit vectors before and after deformation are related by the direction cosine aij
ei ¼ aij Ej
ði ¼ 1,2,3, summation over jÞ:
ð2:4Þ
Eq. (2.2) gives 6 independent equations, while the orthonormal conditions ei ej ¼ dij give another 6. These 12 equations are
solved to determine 9 direction cosines aij and 3 curvatures ki in terms of the displacements Ui and twist angle f as
9
9 8 1 2
9 8
8
ð2Þ
2 ðf þ U 02
12 U 01 U 02 þ c
U 01 U 03 fU 02 >
>
0
f U 01 >
1Þ
1 0 0>
>
>
>
>
= n
=
<
=
<
<
o
0
ð2Þ
2
12 ðf þ U 02
U 02 U 03 þ fU 01
þ að3Þ
aij ¼ 0 1 0 þ f 0 U 2 þ 12 U 01 U 02 c
,
ð2:5Þ
2Þ
ij
>
>
>
>
; >
>
; >
: U0 U0
:
;
:
0 0
0 0
02
02 >
1
0
0 0 1
1
2
U U
U U
ðU þU Þ
1
3
2
3
2
1
2
R
1 Z
ð2Þ
are the 3rd power of displacements and twist
where c ¼ 2 0 ðU 01 U 002 U 001 U 02 ÞdZ is the 2nd power of displacements, and að3Þ
ij
angle given in Appendix A. As to be shown in the next section, the work conjugate of bending moment and torque is
j^ ¼ lj, which is given in terms of the power of Ui and f by
9
8
00 9 8
00
0 0 0
>
=
= >
< fU 1 þ ðU 2 U 3 Þ >
< U 2 >
U 001
^ i ð3Þ g,
^ ig ¼
þ fU 002 ðU 01 U 03 Þ0 þ fk
fk
ð2:6Þ
>
>
>
>
;
: f0 ; :
0
^ i ð3Þ are the 3rd power of displacements given in Appendix A.
where k
2.2. Force, bending moment and torque
Let t ¼tiei and m ¼miei denote the forces and bending moment (and torque) in the cross section Z of the beam after
deformation. Equilibrium of forces requires
8 0
^ þt k
^ þ lp1 ¼ 0
t t k
>
< 1 2 3 3 2
t0
0
^
^ 1 þ lp2 ¼ 0 ,
t 2 þ t 1 k3 t 3 k
ð2:7Þ
þ p ¼ 0, or
>
l
: t 0 t k
^
^ 1 þ lp3 ¼ 0
1 2 þ t2 k
3
where p is the distributed force on the beam (per unit length after deformation). Equilibrium of moments requires
8 0
^ 3 þm3 k
^ 2 lt 2 þ lq1 ¼ 0
m m2 k
>
< 1
m0
0
^
þ
m
k
m
k
m
ð2:8Þ
þ e3 t þq ¼ 0, or
1 3
3 ^ 1 þ lt 1 þ lq2 ¼ 0 ,
2
>
l
: m0 m k
^
^ 1 þ lq3 ¼ 0
þm
1 2
2k
3
where q is the distributed moment on the beam (per unit length after deformation). Elimination of shear forces t1 and t2
from Eqs. (2.7) and (2.8) yields 4 equations for t3 and m. Principle of Virtual Work gives their work conjugates to be l 1
R
^ , respectively. For example, the virtual work of m is mUjds, where ds¼ ldZ represents the integration along the
and j
R
^ dZ in the initial
central axis in the current (deformed) configuration. This integral can be equivalently written as mUj
(undeformed) configuration.
2.3. Onset of buckling and postbuckling
o
o
Let t and m denote the critical forces, bending moments and torque at the onset of buckling, at which the deformation
o
o
is still small. The distributed force and moment at the onset of buckling are denoted by p and q. Equilibrium Eqs. (2.7)
and (2.8) at the onset of buckling are
o
o
t 0i þ pi ¼ 0
o
o
o
ði ¼ 1,2,3Þ,
m01 t 2 þq1 ¼ 0,
o
ð2:9Þ
o
o
m02 þ t 1 þ q2 ¼ 0,
o
o
m03 þ q3 ¼ 0:
ð2:10Þ
490
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
o
o
The forces, bending moments and torque during postbuckling can be written as t ¼ t þ Dt and m ¼ m þ Dm, where Dt and
Dm are the changes beyond the onset of buckling. Equilibrium Eqs. (2.7) and (2.8) become
8
o
o
o
>
>
Dt0 ðt2 þ Dt2 Þk^ 3 þðt3 þ Dt3 Þk^ 2 þ ðlp1 p1 Þ ¼ 0
>
>
< 1
o
o
o
ð2:11Þ
Dt02 þ ðt 1 þ Dt 1 Þk^ 3 ðt3 þ Dt3 Þk^ 1 þ ðlp2 p2 Þ ¼ 0 ,
>
>
>
o
o
o
>
: Dt 0 ðt þ Dt Þk
^ 1 þ ðlp3 p3 Þ ¼ 0
1
1 ^ 2 þðt 2 þ Dt 2 Þk
3
8
o
o
o
o
>
>
Dm01 ðm2 þ Dm2 Þk^ 3 þðm3 þ Dm3 Þk^ 2 ½ðl1Þt2 þ lDt2 þ ðlq1 q1 Þ ¼ 0
>
>
<
o
o
o
o
Dm02 þ ðm1 þ Dm1 Þk^ 3 ðm3 þ Dm3 Þk^ 1 þ ½ðl1Þt 1 þ lDt1 þðlq2 q2 Þ ¼ 0 :
>
>
>
o
o
o
>
:
Dm03 ðm1 þ Dm1 Þk^ 2 þðm2 þ Dm2 Þk^ 1 þ ðlq3 q3 Þ ¼ 0
ð2:12Þ
Elimination of Dt1 and Dt2 from Eqs. (2.11) and (2.12) yields 4 equations for Dt3 and Dm.
The linear elastic relations give Dt3 and Dm in terms of the displacements Ui and twist angle f by
Dt3 ¼ EAðl1Þ, Dm1 ¼ EI1 k^ 1 , Dm2 ¼ EI2 k^ 2 , Dm3 ¼ C k^ 3 ,
ð2:13Þ
where EA is the tensile stiffness, EI1 and EI2 are the bending stiffness, and C is the torsion stiffness. Substitution of Eq. (2.13)
into Eqs. (2.11) and (2.12) leads to four ordinary differential equations (ODEs) for Ui and f.
Examples in the following sections show that the buckling analysis (both onset of buckling and postbuckling) is
straightforward even for complex buckling modes such as the island-bridge, mesh structure in stretchable electronics in
Section 5. It is much simpler than the existing analysis for the onset of buckling, which works only for relatively simple
buckling modes (Timoshenko and Gere, 1961). It is also more accurate than the existing postbuckling analysis, which
neglects the 2nd and 3rd power of displacements in the curvature (von Karman and Tsien, 1941; Budiansky, 1973).
3. Euler-type buckling
Consider a beam of length L with one end clamped and the other simply supported. The beam is subject to uniaxial
o
compression force P. There are no distributed force and moment in the beam, p ¼q ¼0. Let P denote the critical
o
compression at the onset of buckling (to be determined), and DP the change of P beyond P . The forces, bending moments
o
o
o
o
o
and torque at the onset of buckling are t 1 ¼ t 2 ¼ 0, t 3 ¼ P and m ¼ 0. Without losing generality it is assumed the bending
^1 ¼k
^3 ¼0
stiffness EI1 4EI2 such that the beam buckles within the (X,Z) plane (Fig. 2). This gives U2 ¼ f ¼0, and therefore k
from Eq. (2.6) and Dm1 ¼ Dm3 ¼0 from Eq. (2.13). Eqs. (2.11) and (2.12) give Dt2 ¼0, and
o
Dt01 þ ðP þ Dt3 Þk^ 2 ¼ 0, Dt03 Dt 1 k^ 2 ¼ 0, Dm02 þ lDt1 ¼ 0:
ð3:1Þ
Elimination of Dt1, together with the linear elastic relation Eq. (2.13), give
0 0 o
k^
^ 2 ¼ 0, ðEAl2 þEI2 k
^ 22 Þ0 ¼ 0:
EI2 2 þ P EAðl1Þ k
ð3:2Þ
l
^ 2 on the order of U1. Its substitution into Eq. (3.1) gives Dt1 and Dm2 on the order of U1, and Dt3 on
Eq. (2.6) gives k
the order of (U1)2. The linear elastic relation Dt3 ¼ EA(l 1) in Eq. (2.13) and the stretch l in Eq. (2.1) suggest U3 on the
order of (U1)2
U 3 ðU 1 Þ2 :
ð3:3Þ
The stretch in Eq. (2.1) becomes
h
i
1
4
l ¼ 1 þU 03 þ U 02
1 þ O ðU 1 Þ :
2
ð3:4Þ
h
×
Z
P
b
h
X
L
2
L
2
Fig. 2. Illustration of beam buckling under compression, with one end clamped and the other simply supported.
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
^ i ð3Þ in Appendix A is
The only non-zero curvature obtained from Eq. (2.6) and k
0
h
i
1
k^ 2 ¼ U 001 U 01 U 03 þ U 03
þO ðU 1 Þ5 ,
1
3
491
ð3:5Þ
where the underlined are the higher-order terms in the curvature neglected by von Karman and Tsien (1941) and
Budiansky (1973). (Their curvatures neglected the 2nd and 3rd power of displacements in the postbuckling analysis of
plates.) Substitution of Eqs. (3.4) and (3.5) into (3.2) gives two equations for U1 and U3
o
o 0
0
h
i
000
1
P 00 EA 00
1
1 03 000
P
1
0 0
U1
U 1 1U 03 U 02
þ
U1 U 1 U 03 þ U 02
U 01 U 03 þ U 03
þ O ðU 1 Þ5 ¼ 0,
1
1 U1 U3 þ
1
2
EI2
2
3
3
EI2
EI2
0
h
i
EA
002
ð2U 03 þ U 02
þ O ðU 1 Þ4 ¼ 0:
1 Þ þ U1
EI2
ð3:6Þ
ð3:7Þ
The boundary conditions at the clamped end (Z ¼ L/2, Fig. 2) are
U 1 ¼ U 01 ¼ U 3 ¼ 0
at
Z ¼ L=2
ð3:8Þ
The vanishing displacement and bending moment Dm2 ¼0 at the simply supported end (Z¼L/2) gives
0
h
i
1
þ O ðU 1 Þ5 ¼ 0 at Z ¼ L=2,
U 1 ¼ U 001 U 01 U 03 þ U 03
1
3
ð3:9Þ
o
where Eqs. (2.13) and (3.5) have been used. The traction at this end is Dt 1 e1 þ ðP þ Dt 3 Þe3 ; its component along the axial
o
direction E3 in the initial (undeformed) configuration must be ðP þ DPÞ, which gives
o
o
Dt1 e1 UE3 þðP þ Dt3 Þe3 UE3 ¼ ðP þ DPÞ, at Z ¼ L=2,
ð3:10Þ
or using the direction cosines aij in Eq. (2.5) (and aðij3Þ in Appendix A)
h
i
o
o
1
at Z ¼ L=2
Dt 1 U 01 P 1 U 02
þ Dt 3 ¼ ðP þ DPÞ þO ðU 1 Þ4
2 1
ð3:10aÞ
Elimination of Dt1 and Dt3 in the above equation via Eqs. (3.1), (2.13), (3.4) and (3.5) gives
000
U 01 U 1 þ
o
h
i
P
EA
1
DP
U 02
U 03 þ U 02
þ O ðU 1 Þ4 ¼ 0
þ
1 þ
1
EI2
2
EI2
2EI2
at
Z ¼ L=2:
ð3:11Þ
The perturbation method is used to solve ODEs (3.6) and (3.7) with boundary conditions (3.8), (3.9) and (3.11). Let a be
the ratio of the maximum deflection U1max to the beam length L; a is small such that displacements U1 and U3 are
expressed as the power series of a. It can be shown that U1 and U3 correspond to the odd and even powers of a,
respectively, and can be written as
U 1 ¼ aU 1ð0Þ þa3 U 1ð1Þ þ. . .
U 3 ¼ a2 U 3ð0Þ þ . . .,
ð3:12Þ
3
where terms higher than the 3rd power of displacement (i.e., higher than a ) are neglected in the postbuckling analysis.
Similarly, DP is on the order of a2, and can be written as
DP ¼ a2 DP ð0Þ þ. . .:
ð3:13Þ
Substitution of Eqs. (3.12) and (3.13) into Eqs. (3.6)–(3.9) and (3.11) gives
8
00
o
>
>
00
>
< U 1ð0Þ þ EIP2 U 1ð0Þ ¼ 0
,
>
0
>
>
¼ U 1ð0Þ Z ¼ L=2 ¼ U 001ð0Þ ¼0
: U 1ð0Þ Z ¼ L=2 ¼ U 1ð0Þ Z ¼ L=2
ð3:14Þ
Z ¼ L=2
8
0
0
02
EI2 002
>
>
< 2U 3ð0Þ þ U 1ð0Þ þ EA U 1ð0Þ ¼ 0
o
000
DP ð0Þ 0
02
EI2 0
>
1 02
P
>
: U 3ð0Þ Z ¼ L=2 ¼ U 3ð0Þ þ 2 U 1ð0Þ þ EA U 1ð0Þ U 1ð0Þ þ 2EA U 1ð0Þ þ EA Z ¼ L=2
¼0
ð3:15Þ
492
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
for the leading powers of a, and
00 h
8
i0
o
000
>
00
0
02
EA 00
>
þ EI
U 1ð0Þ U 03ð0Þ þ 12 U 02
> U 1ð1Þ þ EIP2 U 1ð1Þ ¼ U 1ð0Þ U 3ð0Þ þ 12 U 1ð0Þ
1ð0Þ
>
2
>
>
>
<
000
0
o þ EIP2 U 01ð0Þ U 03ð0Þ þ 13 U 03
þ U 01ð0Þ U 03ð0Þ þ 13 U 03
1ð0Þ
1ð0Þ
>
>
>
>
h
0 i
>
>
>
: U 1ð1Þ 9
¼ U 01ð1Þ 9Z ¼ L=2 ¼ U 1ð1Þ Z ¼ L=2 ¼ U 001ð1Þ U 01ð0Þ U 03ð0Þ þ 13 U 03
1ð0Þ
Z ¼ L=2
Z ¼ L=2
ð3:16Þ
¼0
for the next power of a, where the underline represents the higher-order terms neglected in the existing postbuckling
analysis.
Eq. (3.14) constitutes an eigenvalue problem for U1(0). The critical buckling load is the eigenvalue, and is given by
2
o
P¼
k EI2
L2
¼
20:19EI2
L2
,
ð3:17Þ
where k¼4.493 is the smallest positive root of the equation
tan k ¼ k:
ð3:18Þ
The corresponding buckling mode is
kðL2ZÞ L sin k L2Z
2L
,
U 1ð0Þ ¼
4p
2p cos k
ð3:19Þ
and its maximum is L. Eq. (3.15) gives U 03ð0Þ as
U 03ð0Þ ¼ "
#2
#
o "
2
2
cos k L2Z
DPð0Þ
k
k P cos2 k L2Z
2L
2L
1
:
1
þ
cos k
EA
cos2 k
8p2
8p2 EA
ð3:20Þ
Its integration, together with U3(0)9Z ¼ L/2 ¼0 in Eq. (3.15), gives U3(0).
The higher-order displacement U1(1) could be obtained from Eq. (3.16). However, DP can be determined directly
o
without solving U1(1). This is because fU 001ð1Þ þ ½P =ðEI2 ÞU 1ð1Þ g00 in Eq. (3.16) is orthogonal to U1(0)
Z
L=2
o
U 001ð1Þ þ ½P =ðEI2 ÞU 1ð1Þ
00
U 1ð0Þ dZ ¼ 0
L=2
o
(also shown in Appendix A). Substitution of fU 001ð1Þ þ ½P =ðEI2 ÞU 1ð1Þ g00 in Eq. (3.16) into the above integral gives the equation
for DP
8 h 000 i0
9
0
1 02
EA 00
>
>
>
þ EI
U 1ð0Þ U 03ð0Þ þ 12 U 02
Z L=2 >
=
< U 1ð0Þ U 3ð0Þ þ 2 U 1ð0Þ
1ð0Þ
2
000
0 U 1ð0Þ dZ ¼ 0:
o ð3:21Þ
0
0
03
0
0
03
1
P
1
>
L=2 >
>
þ EI2 U 1ð0Þ U 3ð0Þ þ 3 U 1ð0Þ >
;
: þ U 1ð0Þ U 3ð0Þ þ 3 U 1ð0Þ
This gives
o
DP ð0Þ ¼ P
2
k
288p2
o
o
2
2
P
P
9k 25ð27k 3Þ EA
20 þ 12 EA
o
P
o
1P=ðEAÞ
2
2
k ð9k 2520Þ
,
288p2
ð3:22Þ
where the underline represents the contribution neglected in the existing postbuckling analysis, without which the
o
o
existing postbuckling analysis would give DPð0Þ ¼ 1:113P, which is 15% larger than DPð0Þ ¼ 0:971P obtained from Eq. (3.22).
The compression force P is then related to the maximum deflection U 1max during postbuckling by
"
#
2
2
o
o k2 ð9k2 2520Þ
o
k ð9k 2520Þ U 1max 2
¼
P
1þ
,
ð3:23Þ
P P þ a2 P
L
288p2
288p2
o
where a ¼ U 1max =L has been used. The critical compressive strain at the onset of buckling, P=ðEAÞ ¼ 20:19EI2 =ðEAL2 Þ, is
negligible. The applied (compressive) strain during postbuckling is
eapplied ¼
U 3 jZ ¼ L=2 U 3 jZ ¼ L=2
a2
¼
L
L
Z
L=2
L=2
U 03ð0Þ dZ 4 k
U 1max 2
U 1max 2
¼ 2:58
:
2
L
L
16p
ð3:24Þ
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
493
The compression force P is then related to the applied (compressive) strain eapplied by
!
2
o
o
9k 2520 P P 1þ
eapplied ¼ Pð1 þ0:3762eapplied Þ,
2
18k
ð3:25Þ
o
while the existing postbuckling analysis would give P P ð1 þ0:43129eapplied 9Þ.
o
The maximum membrane strain in the beam is P=ðEAÞUf1þ O½ðU 1max =LÞ2 g, which is on the order of h2/L2, where h is the
beam height. It is negligible as compared to the bending strain given below for the maximum deflection U 1max much larger
than h. The maximum bending strain in the beam is given by
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h
h emax ¼ maxU 001 ¼ 4:603
eapplied ,
ð3:26Þ
2
L
which is much smaller than the applied strain for U 1max bh.
4. Lateral buckling
In general, the following orders of displacement and rotation increments hold during postbuckling:
(i) The increment of axial displacement U3 is always 2nd order;
(ii) The deflection and rotations increments during postbuckling are 1st order if these increments are zero prior to the
onset of buckling (e.g., U1 and f in this section);
(iii) The deflection and rotations increments during postbuckling are 2nd order if these increments are not zero prior to
the onset of buckling (e.g., U2 in this section).
Fig. 3 shows a beam of length L with the left end clamped and the right end subject to a shear force P in the Y direction
without any rotation. Let Z denote the central axis of the beam before deformation, and with Z ¼0 at the beam center. The
o
beam buckles out of the Y–Z plane (i.e., lateral buckling) when the shear force P reaches the critical buckling load P (to be
h
b
Z
×
×P
h
X
L
L
2
2
P
h
Y
b
×
b
Z
L
L
2
2
symmetric
anti-symmetric
Fig. 3. Illustration of lateral buckling of a beam under shear along the thick direction ðb b hÞ of the cross section; (a) view along the direction of shear;
(b) view normal to the direction of shear; and (c) the symmetric and anti-symmetric buckling modes.
494
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
determined). As to be shown at the end of Section 4, this represents a special case of the island-bridge, mesh structure in
stretchable electronics (Kim et al., 2008b).
4.1. Equations for postbuckling analysis
There are no distributed force and moment, p ¼q ¼0. The force, bending moment and torque in the beam prior to the
onset of buckling are
o
o
o
o
t 1 ¼ t 3 ¼ m2 ¼ m3 ¼ 0,
o
o
t2 ¼ P,
o
o
o
m1 ¼ P Z
for
L
L
rZ r :
2
2
ð4:1Þ
Let DP ¼ PP denote the change of shear force during postbuckling. Only the displacements along Z and Y directions are
not zero prior to the onset of buckling. Therefore the displacement U1 and rotation f during postbuckling are 1st order,
and displacements U2 and U3 are 2nd order, i.e.,
f U 1 , U 2 U 3 ðU 1 Þ2 :
ð4:2Þ
The stretch in Eq. (2.1) becomes
h
i
1
4
l ¼ 1 þU 03 þ U 02
1 þ O ðU 1 Þ :
2
ð4:3Þ
ð3Þ
^i
Eq. (2.6) and k
in Appendix A give the curvatures
h
i
k
þO ðU 1 Þ4 ,
h
i
1
1
0
5
k^ 2 ¼ U 001 þ fU 002 ðU 01 U 03 Þ0 f2 U 001 ðU 03
1 Þ þ O ðU 1 Þ ,
2
3
^ 1 ¼ U 002 þ fU 001
k^ 3 ¼ f0 ,
ð4:4Þ
where the underlined are the higher-order terms in the curvatures neglected in the existing postbuckling analysis (von
Karman and Tsien, 1941; Budiansky, 1973). Substitution Eqs. (4.1)–(4.4) into Eqs. (2.11)–(2.13) gives the ODEs for U and f
h
i
o
1 2
1
00
0
5
¼ 0,
ð4:5Þ
C f þ ðEI1 EI2 ÞU 001 ðU 002 fU 001 ÞP Z U 001 þ fU 002 ðU 01 U 03 Þ0 f U 001 ðU 03
1 Þ þO ðU 1 Þ
2
3
000
1 02 0
1 03 0 00
02 00
0
00
0 0 0 1 2 00
U
ðU
EI2 U ð4Þ
U
U
þ
f
U
þ
f
U
ðU
U
Þ
f
U
Þ
1
3
1
2
1 3
1
1
2 1
2
3 1
h
i
h
i
00
0
00
0
02
00
00
00
00 0
00
00
EI1 f ðU 2 fU 1 Þ þ2f ðU 2 fU 1 Þ þC f ðU 2 fU 1 Þ þ f ðU 002 fU 001 Þ0 þ f U 001
o
h
i
o
o
1
1 02 0
0
0 0
0
0
U
f
þðP
Z
f
Þ
P
Z
f
U
þ
þ O ðU 1 Þ5 ¼ 0,
þ
P
EAU 001 U 03 þ U 02
3
2 1
2 1
ð4:6Þ
0 o
h
i
o
1
00
0 000
0
02
þ PZ f þO ðU 1 Þ4 ¼ 0,
EI1 ðU 002 fU 001 Þ00 þ EI2 ðf U 001 þ 2f U 1 ÞCðf U 001 Þ0 þ P U 03 þ U 02
1
2
ð4:7Þ
0 o
h
i
o
1
1
0
EA U 03 þ U 02
PðU 002 fU 001 Þ þ PZ f U 001 þO ðU 1 Þ4 ¼ 0:
þ EI2 U 002
1
1
2
2
ð4:8Þ
The increments of axial force Dt3 and bending moment and torque Dm can be obtained from Eqs. (2.13), (4.3) and (4.4).
The shear forces, needed in the boundary conditions, are given by
0
h
i
o
1
1
0
0
0
lDt 1 ¼ EI2 U 001 þ fU 002 ðU 01 U 03 Þ0 f2 U 001 ðU 03
þ ðEI1 CÞf ðU 002 fU 001 ÞP Z f þO ðU 1 Þ5
1Þ
2
3
h
i
o
1
0 00
00
00 0
ð4:9Þ
lDt 2 ¼ EI1 ðU 2 fU 1 Þ ðEI2 CÞf U 1 P U 03 þ U 02
þ O ðU 1 Þ4 :
2 1
The displacements U2 and U3 at the left end are zero
U2 ¼ U3 ¼ 0
at
Z ¼ L=2:
ð4:10Þ
The boundary conditions at the two ends are zero displacement
U1 ¼ 0
at
Z ¼ 7 L=2
ð4:11Þ
and zero rotation, ei ¼Ei (i¼1,2,3), and therefore direction cosines aij ¼ dij, which are expressed in terms of displacements
and rotation as
U 01 ¼ 0,
U 02 ¼ 0,
f þ cð2Þ þ cð3Þ ¼ 0 at Z ¼ 7L=2,
where
cð2Þ ¼
1
2
ZZ
0
ðU 01 U 002 U 001 U 02 ÞdZ
ð4:12Þ
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
495
given after Eq. (2.5) is in the order of O[(U1)3], and c(3) given in Appendix A is also in the order of O[(U1)3]. The traction
condition at the right end Z¼L/2 is
Dt2 ¼ DP, Dt3 ¼ 0 at Z ¼ L=2:
ð4:13Þ
Substitution of Dt2 in Eq. (4.9) and Dt3 from Eqs. (2.13) and (4.3) into the above equation gives
h
i
U 03 þ O ðU 1 Þ4 ¼ 0
o
ðU 002 fU 001 Þ0 þ
h
i
EI2 C 0 00
P 0 DP
f U1 þ
U3 þ
þ O ðU 1 Þ4 ¼ 0
EI1
EI1
EI1
at
Z ¼ L=2:
ð4:14Þ
4.2. Perturbation method
The perturbation method is used to solve the ODEs (4.5)–(4.8) with boundary conditions (4.10)–(4.12) and (4.14). Let a
denote a small non-dimensional parameter, such as the ratio of the maximum deflection U1max to the beam length L or the
maximum twist f. The displacements and rotation are expanded to the power series of a, with f and U1 to the odd powers
of a, and U2 and U3 to the even powers of a
f ¼ afð0Þ þ a3 fð1Þ þ . . .
U 1 ¼ aU 1ð0Þ þa3 U 1ð1Þ þ. . .
U 2 ¼ a2 U 2ð0Þ þ . . .
U 3 ¼ a2 U 3ð0Þ þ . . .,
ð4:15Þ
3
where terms higher than the 3rd power of displacement [i.e., o(a )] are neglected in the postbuckling analysis. Similarly,
DP is in the order of a2, and can be written as
DP ¼ a2 DP ð0Þ þ. . .:
ð4:16Þ
Substitution of the above two equations into the ODEs and boundary conditions gives
8
o
00
>
>
C fð0Þ P ZU 001ð0Þ ¼ 0
>
>
>
>
o
o
<
0
0
0
EI2 U ð4Þ
,
1ð0Þ þ P fð0Þ þðP Z fð0Þ Þ ¼ 0
>
>
>
>
0
>
¼ U 1ð0Þ ¼0
>
: fð0Þ Z ¼ 7 L=2 ¼ U 1ð0Þ Z ¼ 7 L=2
ð4:17Þ
Z ¼ 7 L=2
8
0 o
o
000
00
0
0
02
>
>
EI1 ðU 002ð0Þ fð0Þ U 001ð0Þ Þ00 þ EI2 ðfð0Þ U 001ð0Þ þ 2fð0Þ U 1ð0Þ ÞCðfð0Þ U 001ð0Þ Þ0 þ P U 03ð0Þ þ 12 U 02
>
1ð0Þ þ P Z fð0Þ ¼ 0
>
>
>
>
h i0 o
o
>
>
0
0
002
00
00
00
1 02
1
>
>
< EA U 3ð0Þ þ 2 U 1ð0Þ þ 2 EI2 U 1ð0Þ PðU 2ð0Þ fð0Þ U 1ð0Þ Þ þ PZ fð0Þ U 1ð0Þ ¼ 0
>
U ¼ U 3ð0Þ 9Z ¼ L=2 ¼ U 02ð0Þ 9Z ¼ 7 L=2 ¼ U 03ð0Þ ¼0
>
> 2ð0Þ Z ¼ L=2
Z ¼ L=2
>
>
>
>
>
o
>
DP
00
00
>
0
EI2 C 0
>
fð0Þ U 001ð0Þ þ EI11 PU 03ð0Þ
¼ EIð0Þ
: ðU 2ð0Þ fð0Þ U 1ð0Þ Þ þ EI
1
1
ð4:18Þ
Z ¼ L=2
for the leading powers of a, and
8
i
o
o h
00
2
>
0
>
C fð1Þ P ZU 001ð1Þ ¼ ðEI1 EI2 ÞU 001ð0Þ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ þ PZ fð0Þ U 002ð0Þ ðU 01ð0Þ U 03ð0Þ Þ0 12 fð0Þ U 001ð0Þ 13 ðU 03
>
1ð0Þ Þ
>
>
>
0
8 000 9
>
>
02
0
02
00
>
>
>
>
< U 1ð0Þ U 3ð0Þ þ 12 U 1ð0Þ þ fð0Þ U 1ð0Þ
=
>
o
o
>
>
0
0
>
h
i00
þ Pfð1Þ þðP Z fð1Þ Þ0 ¼ EI2
EI2 U ð4Þ
>
1ð1Þ
>
0
>
>
>
: fð0Þ U 002ð0Þ ðU 01ð0Þ U 03ð0Þ Þ0 12 f2ð0Þ U 001ð0Þ 13 ðU 03
;
>
>
1ð0Þ Þ
>
>
>
<
h
i
00
0
þEI1 fð0Þ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ þ 2fð0Þ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ0
>
>
h
i
>
>
00
0
02
00
00
00
00
00
0
>
>
> C fð0Þ ðU 2ð0Þ fð0Þ U 1ð0Þ Þ þ fð0Þ ðU 2ð0Þ fð0Þ U 1ð0Þ Þ þ fð0Þ U 1ð0Þ
>
>
>
>
o
0
>
>
>
0
0
1 02
>
þ EAU 001ð0Þ U 03ð0Þ þ 12 U 02
>
1ð0Þ þ P Z fð0Þ U 3ð0Þ þ 2 U 1ð0Þ
>
>
>
>
>
ð3Þ
0
>
: ½f1ð1Þ þ cð2Þ
¼0
ð0Þ þ cð0Þ 9Z ¼ 7 L=2 ¼ U 1ð1Þ 9Z ¼ 7 L=2 ¼ U 1ð1Þ Z ¼ 7 L=2
for the next power of a, where
cð2Þ
ð0Þ ¼
1
2
ZZ
0
ðU 01ð0Þ U 002ð0Þ U 001ð0Þ U 02ð0Þ ÞdZ
and
1
4
cð3Þ
ð0Þ ¼ fð0Þ
2 2
fð0Þ þ U 02
1ð0Þ :
3
ð4:19Þ
496
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
Eq. (4.17) constitutes an eigenvalue problem for f(0) and U1(0). Elimination of U1(0) yields the equation for f(0)
2
3
o
d 4 000
2 00
ðPZÞ2 0 5
¼ 0:
ð4:20Þ
fð0Þ fð0Þ þ
f
dZ
Z
EI2 C ð0Þ
It has two sets of solutions, corresponding to even and odd functions of Z, respectively, which are called symmetric and
anti-symmetric buckling modes in Sections 4.3 and 4.4. It can be shown that U1(0) is odd for an even f(0), and U1(0) is even
when f(0) is odd.
For the beam with a narrow cross section such that EI1 b EI2 ,C as in experiments, Eq. (4.18) can be significantly
simplified by neglecting EI2/(EI1) and C/(EI1)
8 00
00
00
>
< ðU 2ð0Þ fð0Þ U 1ð0Þ Þ ¼ 0
0
>
¼ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ0 ¼0
: U 2ð0Þ Z ¼ L=2 ¼ U 2ð0Þ Z ¼ 7 L=2
8
0
0
>
1 02
>
< U 3ð0Þ þ 2 U 1ð0Þ ¼ 0
0
>
>
: U 3ð0Þ Z ¼ L=2 ¼ U 3ð0Þ Z ¼ L=2
Z ¼ L=2
ð4:18aÞ
¼ 0,
o
pffiffiffiffiffiffiffiffiffiffi
where the deformation at the onset of buckling is negligible since P and DP(0) are proportional to EI2 C (to be shown in
Sections 4.3 and 4.4) and the bridge length L is much larger than the cross section dimension (e.g., thickness). The above
equation, together with f(0) and U1(0) being opposite even or odd functions, give U 002ð0Þ ¼ fð0Þ U 001ð0Þ and U 03ð0Þ ¼ U 02
1ð0Þ =2. Its
solution is
Z L=2
Z Z
Z L=2
U 2ð0Þ ¼ Z
fð0Þ U 001ð0Þ dZ 1 Z 1 fð0Þ U 001ð0Þ dZ 1 Z 1 fð0Þ U 001ð0Þ dZ 1
Z
1
U 3ð0Þ ¼ 2
Z
0
Z
0
1
U 02
1ð0Þ dZ 1 2
Z
0
L=2
0
U 02
1ð0Þ dZ 1 :
ð4:21Þ
Similar to Eq. (3.21), the normality condition (or the existence condition for f(1) and U1(1)) gives DP(0). The underlined
terms in Eq. (4.21) clearly show the importance of the 4th power of displacement in the potential energy: the
displacement U2(0) would be zero if the 4th power of displacement were not accounted for.
4.3. Symmetric buckling mode
The symmetric buckling mode corresponds to an even function for f(0), and an odd function for U1(0). Without losing
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o pffiffiffiffiffiffiffiffiffiffi
generality only half of the beam, 0 rZrL/2, is considered. By introducing a new variable z ¼ P = EI2 C Z, integration of
Eq. (4.20) and f(0) being an even function give
3
d fð0Þ
3
dz
2
2 d fð0Þ
z dz2
2
þz
dfð0Þ
¼ 0:
dz
It is a 2nd order ODE for df(0)/dz, and has the solution
!
dfð0Þ
z2
3=2
¼ B1 z J3=4
dz
2
ð4:22Þ
ð4:23Þ
for an even function f(0), where B1 is a constant to be determined, J is the Bessel function of the first kind (Gradshteyn and
Ryzhik, 2007). Integration of the above equation together with the boundary condition f(0)9Z ¼ L/2 ¼0 in Eq. (4.17) gives
!
"
!
!#
Z zmax
pffiffiffi
pffiffiffiffiffiffiffiffiffiffi
z2
z2
z2
fð0Þ ¼ B1
z3=2 J3=4
zmax J 1=4 max z J 1=4
dz ¼ B1
,
ð4:24Þ
2
2
2
z
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o pffiffiffiffiffiffiffiffiffiffi
where zmax ¼ P= EI2 C ðL=2Þ corresponds to the end Z ¼L/2. The parameter B1 is determined from maxðfð0Þ Þ ¼ 1 as
hpffiffiffiffiffiffiffiffiffiffi 2 pffiffi i1
2
Gð3=4Þ
zmax J 1 zmax
in terms of the G function (Gradshteyn and Ryzhik, 2007), to ensure the parameter a in
B1 ¼
2
4
Eq. (4.15) to be the maximum angle of twist, fmax.
Integration of the 1st equation in (4.17) gives U1(0) as
sffiffiffiffiffiffiffi"
! Z
! #
zmax
B1
C pffiffiffiffiffiffiffiffiffiffi
z2max
1
z2
pffiffiffi J 3=4
zmax ðzmax zÞJ 3=4
U 1ð0Þ ¼ L
z
dz :
2zmax EI2
2
2
z
z
ð4:25Þ
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
497
Its being an odd function requires U1(0) ¼ 0 at Z ¼0, which gives
!
z2max
J3=4
2
¼0
ð4:26Þ
and has the solution zmax ¼ 2:642. The critical buckling load is then given by
pffiffiffiffiffiffiffiffiffiffi
o
EI2 C
P ¼ 27:93
:
L2
The constant B1 ¼ 0.5423. Eqs. (4.24) and (4.25) can then be written explicitly as
!
pffiffiffi
z2
fð0Þ ¼ 0:3742þ 0:5423 zJ 14
2
2
2 3
sffiffiffiffiffiffiffi Z
sffiffiffiffiffiffiffi2
3
!
0:07986z0:07879z J 1=4 z2 H3=4 z2
zmax
U 1ð0Þ
C
1
z2
C 6
7
2
2
pffiffiffi J 3=4
¼ 0:1026
z
dz ¼
4
5,
z
z
EI2 z
EI2 þ 0:03051z3 J
L
2
z
3=4 2 sð7=4Þ,ð1=4Þ 2
ð4:27Þ
ð4:28Þ
where H is the Struve Function and s is the Lommel Function (Gradshteyn and Ryzhik, 2007). Eq. (4.21) then gives U2(0)
and U3(0).
The applied nominal shear strain during postbuckling is obtained from Eqs. (4.21) and (4.28) as
sffiffiffiffiffiffiffi
Z
2
U 2 jZ ¼ L=2 U 2 jZ ¼ L=2
2fmax L=2
C 2
¼
gapplied ¼
Z fð0Þ U 001ð0Þ dZ ¼ 0:2444
f :
ð4:29Þ
EI2 max
L
L
0
It would give gapplied ¼0 without the underlined terms, which clearly suggests the importance of higher-order terms in the
curvature in the postbuckling analysis.
For a rectangular section with height h 5 width b, EI1 ¼Ehb3/12, EI2 ¼Eh3b/12 and C ¼Gh3b/3, where G is the shear
modulus. The maximum bending strain, which is reached at the two ends of the beam, is the sum of bending strains at the
onset of buckling and during postbuckling, and is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
0
rffiffiffiffi
rffiffiffiffi1
rffiffiffiffi
o
00 h
b P ðL=2Þ h
G
h GA
h
G
@
7:474 gapplied
þ 13:97
7:474
,
ð4:30Þ
ebending ¼
þ max U 1 ¼
gapplied
2 EI1
2
L
E
b E
L
E
where the approximation on the 2nd line holds for fmax bh=b. The maximum shear strain due to torsion is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffi
0
h
E
:
ð4:31Þ
gtorsion ¼ hmax f ¼ 6:606
gapplied
L
G
4.4. Anti-symmetric buckling mode
The anti-symmetric buckling mode corresponds to an odd function for f(0), and an even function for U1(0). Without
losing generality only half of the beam, 0 rZ rL/2, is considered. Integration of Eq. (4.20) gives the same equation as (4.22)
except that the right-hand side is replaced by a constant of integration B2 (to be determined)
3
d fð0Þ
3
dz
2
2 d fð0Þ
z dz2
2
þz
dfð0Þ
¼ B2 :
dz
ð4:32Þ
It has the solution
dfð0Þ
z2
3=2
¼ B2 FðzÞ þB3 z J3=4
dz
2
!
for an odd function f(0), where the constant B3 is to be determined, and
sffiffiffiffiffiffi "
!
!#
1 2 1 2p
3
z2
z2
2
FðzÞ ¼ z þ
G
z H5=4
H1=4
:
3
4
4
z
2
2
Its integration together with the boundary condition f(0)9Z ¼ L/2 ¼0 in Eq. (4.17) gives
"
!
!#
Z zmax
pffiffiffi
pffiffiffiffiffiffiffiffiffiffi
z2max
z2
fð0Þ ¼ B2
FðzÞdzB3
zmax J 1=4
zJ1=4
:
2
2
z
ð4:33Þ
ð4:34Þ
ð4:35Þ
498
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
Its being an odd function requires f(0) ¼ 0 at Z ¼0, i.e.
!
Z zmax
pffiffiffiffiffiffiffiffiffiffi
z2max
FðzÞdz þB3 zmax J 1=4
B2
¼ 0:
2
0
ð4:36Þ
¼ 0 give
Integration of the 1st equation in (4.17) and the boundary condition U 01ð0Þ ¼ L=2
sffiffiffiffiffiffiffi( Z
"
!# Z)
Z
zmax
zmax
3
d
L
C
1 dFðzÞ
1 d
z2
z2 J 3=4
U 1ð0Þ ¼ dz þB3
B2
dz :
dz
2zmax EI2
d
d
z
z
z
z
2
z
z
It is an odd function (since U1(0) is even), which requires U 01ð0Þ ¼ 0 at Z ¼0. This gives
"
!#
Z zmax
Z zmax
1 dFðzÞ
1 d
z2
dz þ B3
B2
z3=2 J3=4
dz ¼ 0:
z dz
z dz
2
0
0
ð4:37Þ
ð4:38Þ
Eqs. (4.36) and (4.38) give two linear, homogeneous equations for B2 and B3. Their determinant must be zero in order to
have a non-trivial solution, which gives the equation for the critical buckling load
R
2 pffiffiffiffiffiffiffiffiffiffi
zmax
zmax J 1=4 zmax
0 FðzÞdz
2
h
i
ð4:39Þ
Rz
¼ 0:
R
2
z
3=2
z
max 1 d
max 1 dFðzÞ dz
d
z
J
z
0 z dz
3=4 2
0
z dz
It has the solution zmax ¼ 3:038. The critical buckling load is then given by
pffiffiffiffiffiffiffiffiffiffi
o
EI2 C
P ¼ 36:92
,
L2
which is larger than Eq. (4.27) for the symmetric mode.
Integration of Eq. (4.37) and the boundary condition U1(0)9Z ¼ L/2 ¼0 give
9
8 h
R zmax FðzÞ i
z z
sffiffiffiffiffiffiffi>
>
dz
>
> B2 max
zmax Fðzmax Þz z
=
z2
L
C <
2 R
2
:
U 1ð0Þ ¼
pffiffiffiffiffiffiffiffiffiffi
zmax 1
zmax
z
2zmax EI2 >
>
>
zmax ðzmax zÞ J 3=4 2 z z pffiffi J 3=4 2 dz >
;
: þ B3
ð4:40Þ
ð4:41Þ
z
The parameter a in Eq. (4.15) is the ratio of maximum displacement to L, i.e., a ¼U1max/L, which, together with
Eq. (4.41), give
"
!
rffiffiffiffiffiffiffi
#
z2max
2
3
EI2
3=2
¼ 2zmax
B2 Fðzmax Þ þ B3 zmax J 3=4
G
:
ð4:42Þ
2
p 4
C
pffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffi
Eqs. (4.36) and (4.42) determine B2 ¼ 3:850 EI2 =C and B3 ¼ 1:166 EI2 =C . Eqs. (4.35) and (4.41) can then be written
explicitly as
!#
rffiffiffiffiffiffiffi"
Z 3:038
pffiffiffi
EI2
z2
fð0Þ ¼ FðzÞdz þ 1:166 zJ 1=4
0:7250þ 3:850
C
2
z
!
Z 3:038
Z 3:038
U 1ð0Þ
FðzÞ
1
z2
ffiffi
ffi
p
¼ 0:85030:2799z0:6337z
d
z
þ
0:1919
z
ð4:43Þ
J
dz:
3=4
L
2
z
z
z
z2
Eq. (4.21) then gives U2(0) and U3(0).
The applied nominal shear strain during postbuckling is obtained from Eqs. (4.21) and (4.43) as
rffiffiffiffiffiffiffi
Z
U 2 jZ ¼ L=2 U 2 jZ ¼ L=2
2U 21max L=2
EI2 U 1max 2
00
¼
gapplied ¼
Z
f
U
dZ
¼
10:38
:
ð0Þ
1ð0Þ
L
C
L
L3
0
ð4:44Þ
It would also give gapplied ¼0 without the underlined terms, which confirms the importance of higher-order terms in the
curvature in the postbuckling analysis.
h
ipffiffiffiffiffiffiffiffiffi
2
The maximum bending strain at the onset of buckling is ebending ¼ 18:46 h =ðLbÞ
G=E. The maximum bending strain
and torsion (shear) strain due to postbuckling are
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffi
h
G
,
ð4:45Þ
ebending ¼ 6:833
gapplied
L
E
h
gtorsion ¼ hmaxf0 ¼ 8:632
L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffi
E
:
gapplied
G
ð4:46Þ
Their comparison with Eqs. (4.30) and (4.31) suggests that the anti-symmetric buckling mode gives a smaller bending
strain but a larger shear strain than the symmetric buckling mode.
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
499
P
Z
Z
Y
Lisland
A
Y
Z
X
B
L
h
b
D
C
L
Lisland
b
L
PZ
h
Substrate
Y
L
2
Z
b
h
X
anti-symmetric
compressing
stretching
symmetric
L
2
Fig. 4. Illustration of the island-bridge, mesh structure in the stretchable electronics subject to stretching/compressing along the 451 (diagonal)
direction; (a) top view; (b) side view; and (c) the symmetric and anti-symmetric buckling modes for both stretching and compressing.
Fig. 3c shows the symmetric and anti-symmetric buckling modes in Sections 4.3 and 4.4, respectively. The applied
nominal shear strain is gapplied ¼ 20%, and C/(EI2)¼1.57. The buckling mode for the anti-symmetric appears to be rather
similar to the experimentally observed buckling profile in Fig. 1b for the shear case.
The analysis in this section also holds for the island-bridge, mesh structure in stretchable electronics (Fig. 4) subject to
pure shear. For example, the bending and torsion strains in Eqs. (4.30), (4.31), (4.45) and (4.46) still hold if the applied
nominal shear strain gapplied to the beam is replaced by g(LþLisland)/L, where g is the shear strain applied to the islandbridge, mesh structure, and Lisland is the island size (Fig. 4).
500
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
o
PZ
Z +
L
2
o
Y
t2
o
t3
o
t3
A
B
Z
o
t2
o
t2
Z
Y
o
o
t2
o
t3
t3
o
t3
o
t3
o
t2
o
t2
b
D
C
o
PZ
Fig. 5. Free-body diagram of the island-bridge, mesh structure prior to the onset of buckling.
5. Buckling of island-bridge, mesh structure in stretchable electronics
Fig. 4a shows the top view of an island-bridge, mesh structure in stretchable electronics (Kim et al., 2008b). The square
islands of size Lisland are linked by bridges of length L. The islands are well attached to the substrate, but bridges are not, as
seen from the side view in Fig. 4b. The bridges buckle to shield the islands from large deformation. Let Z denote the central
axis of the bridge before deformation, with Z ¼0 at the bridge center (Fig. 4a), and Y axis in the plane of the island-bridge,
mesh structure and be normal to Z.
Euler buckling occurs in bridges for the island-bridge, mesh structure subject to compression along Z or Y directions.
Tension or compression along the diagonal direction (Z in Fig. 4a) of the square islands, as in experiments, is studied in the
following.
5.1. State prior to the onset of buckling
Let P Z denote the tension force (per unit length) along Z direction (Fig. 4a). There are no distributed force and moment,
o
o
o
p ¼q ¼0. The force, bending moment and torque in bridges t 1 ¼ m2 ¼ m3 ¼ 0, and the non-zero forces and bending moment
o
o
t2 , t3
o
and m1 are symmetric about Z and Y directions.
Fig. 5 shows the free body diagram of the left and lower half of the island-bridge, mesh structure that is cut along Y direction.
o
o
The forces t 2 and t 3 in the bridges along Z and Y directions satisfy the symmetry about Z and Y directions. Equilibrium of forces in
o
o
o
o
o
Z direction gives t 2 þt 3 P Z ðL þ Lisland Þ ¼ 0. Similarly, force equilibrium in Y direction gives t 2 t 3 ¼ 0. These give
o
t2 ¼
1o
P ðLþ Lisland Þ,
2 Z
o
t3 ¼
1o
P ðL þLisland Þ:
2 Z
ð5:1Þ
o
The islands do not rotate due to symmetry about Z and Y directions. The bending moment m1 can then be found as
o
m1 ¼
1o
P ðLþ Lisland ÞZ
2 Z
for
L
L
rZ r :
2
2
ð5:2Þ
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
501
5.2. Equations for postbuckling analysis
o
Let P Z denote the critical tension at the onset of buckling, and DP Z the changes during postbuckling. Only the
displacements along Z and Y directions are not zero prior to the onset of buckling. Therefore the orders of displacements
and rotation f during postbuckling are the same as Eq. (4.2). The stretch and curvatures are still given by Eqs. (4.3) and
(4.4). Their substitution into Eqs. (2.11)–(2.13) gives the ODEs for U and f
h
i
o
1 2
1
00
0
5
¼ 0,
ð5:3Þ
C f þðEI1 EI2 ÞU 001 ðU 002 fU 001 Þm1 U 001 þ fU 002 ðU 01 U 03 Þ0 f U 001 ðU 03
1 Þ þO ðU 1 Þ
2
3
00 000
1 02 0
1 2
1
02
0
0
U 1 f U 001 þ fU 002 ðU 01 U 03 Þ0 f U 001 ðU 03
EI2 U ð4Þ
1Þ
1 U 1 U 3 þ
2
2
3
h
i
h
i
o 0
1
00
0
00
0
02
þt 2 f
EI1 f ðU 002 fU 001 Þ þ 2f ðU 002 fU 001 Þ0 þ C f ðU 002 fU 001 Þ þ f ðU 002 fU 001 Þ0 þ f U 001 EAU 001 U 03 þ U 02
1
2
h
i
o
o
o
1
1 03 0
1 02 0
0 0
0
00
0 0 0 1 2 00
0
ðU
U
t 3 U 001 1 þU 03 þ U 02
f
U
ðU
U
Þ
f
U
Þ
f
Þ
m
f
U
þ
þ O ðU 1 Þ5 ¼ 0,
þ
þðm
1
1
1
2
1 3
1
1
3
1
2
2
3
2
ð5:4Þ
0
h
i
o
o
o
1
00
0 000
0
02
t 3 ðU 002 fU 001 Þ þ m1 f þ O ðU 1 Þ4 ¼ 0,
EI1 ðU 002 fU 001 Þ00 þ EI2 ðf U 001 þ2f U 1 ÞCðf U 001 Þ0 þ t 2 U 03 þ U 02
1
2
ð5:5Þ
0
h
i
o
o
1
1
0
t 2 ðU 002 fU 001 Þ þm1 f U 001 þ O ðU 1 Þ4 ¼ 0,
þ EI2 U 002
EA U 03 þ U 02
1
1
2
2
ð5:6Þ
where the underlined are the higher-order terms in the curvatures neglected in the existing postbuckling analysis
(von Karman and Tsien, 1941; Budiansky, 1973). The increments of axial force Dt3 and bending moment and torque Dm
can be obtained from Eqs. (2.13), (4.3) and (4.4). The shear forces are given by
0
h
i
o
1
1
0
0
0
lDt1 ¼ EI2 U 001 þ fU 002 ðU 01 U 03 Þ0 f2 U 001 ðU 03
þ ðEI1 CÞf ðU 002 fU 001 Þm1 f þ O ðU 1 Þ5
1Þ
2
3
h
i
o
1
ð5:7Þ
lDt2 ¼ EI1 ðU 002 fU 001 Þ0 ðEI2 CÞf0 U 001 t2 U 03 þ U 02
þO ðU 1 Þ4
1
2
o
o
Eqs. (5.3)–(5.7) hold for arbitrary equilibrium state t and m prior to the onset of buckling. For example, for the state in Eq.
(4.1), the above equations degenerate to Eqs. (4.5)–(4.9).
The displacements U2 and U3 at the midpoint are zero to remove the rigid body translation
U2 ¼ U3 ¼ 0
at
Z¼0
ð5:8Þ
The other boundary conditions (4.11) and (4.12) still hold. The traction conditions at the end Z ¼L/2 are Dt 2 ¼ Dt 3 ¼
DP Z ðLþ Lisland Þ=2, which can be expressed in terms of the displacements as
h
i
DP Z ðLþ Lisland Þ
þO ðU 1 Þ4 ¼ 0
U 03 2EA
ðU 002 fU 001 Þ0 þ
o
h
i
DP ðL þLisland Þ
EI2 C 0 00
t
f U 1 þ 2 U 03 þ Z
þ O ðU 1 Þ4 ¼ 0
EI1
EI1
2EI1
at
Z ¼ L=2:
ð5:9Þ
5.3. Perturbation method
The perturbation method is used to solve the ODEs (5.3)–(5.6) with boundary conditions (4.11), (4.12), (5.8), and (5.9).
Let a denote the ratio of the maximum deflection U1max to the beam length L or the maximum twist fmax. The
displacements and rotation are expanded to the power series of a as in Eq. (4.15). Similarly, DP Z is written as
DPZ ¼ a2 DP Zð0Þ þ. . .:
ð5:10Þ
Substitution of Eqs. (4.15) and (5.10) into Eqs. (5.3)–(5.6), (5.8) and (5.9) gives
8
o
00
00
>
>
> C fð0Þ m1 U 1ð0Þ ¼ 0
>
>
o 0
o
<
o
0
00
0
EI2 U ð4Þ
1ð0Þ þ t 2 fð0Þ t 3 U 1ð0Þ þðm1 fð0Þ Þ ¼ 0
>
>
>
>
>
¼ U 1ð0Þ ¼ U 01ð0Þ jZ ¼ 7 L=2 ¼ 0,
: fð0Þ Z ¼ 7 L=2
Z ¼ 7 L=2
ð5:11Þ
502
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
8
00
0
0
00
0
EI1 ðU 002ð0Þ fð0Þ U 001ð0Þ Þ00 þ EI2 ðfð0Þ U 001ð0Þ þ2fð0Þ U 000
>
1ð0Þ ÞCðfð0Þ U 1ð0Þ Þ
>
>
>
>
0 o
o
>
o
>
02
00
00
>
þt 2 U 03ð0Þ þ 12 U 02
>
1ð0Þ t 3 ðU 2ð0Þ fð0Þ U 1ð0Þ Þ þ m1 fð0Þ ¼ 0
>
>
>
>
>
i0 o
>
o
<h 0
0
002
00
00
00
1
EA U 3ð0Þ þ 12 U 02
1ð0Þ þ 2 EI2 U 1ð0Þ t 2 ðU 2ð0Þ fð0Þ U 1ð0Þ Þ þ m1 fð0Þ U 1ð0Þ ¼ 0
>
>
DP ðL þ Lisland Þ
>
>
>
U 2ð0Þ jZ ¼ 0 ¼ U 3ð0Þ jZ ¼ 0 ¼ U 02ð0Þ jZ ¼ 7 L=2 ¼ U 03ð0Þ jZ ¼ L=2 Zð0Þ 2EA
¼0
>
>
>
>
>
>
>
o
DP ðL þ Lisland Þ
>
00
00
00
0
EI C 0
0
>
1
>
¼ Zð0Þ 2EI1
: ðU 2ð0Þ fð0Þ U 1ð0Þ Þ þ EI2 1 fð0Þ U 1ð0Þ þ EI1 t 2 U 3ð0Þ
ð5:12Þ
Z ¼ L=2
for the leading powers of a, and
8
h
i
o
o
00
00
00
00
00
00
0
0
00
03
0 1 2
0
1
>
>
> C fð1Þ m1 U 1ð1Þ ¼ ðEI1 EI2 ÞU 1ð0Þ ðU 2ð0Þ fð0Þ U 1ð0Þ Þ þm1 fð0Þ U 2ð0Þ ðU 1ð0Þ U 3ð0Þ Þ 2 fð0Þ U 1ð0Þ 3 ðU 1ð0Þ Þ
>
>
>
>
o 0
o
o
>
0
00
0
>
>
EI2 U ð4Þ
>
1ð1Þ þ t 2 fð1Þ t 3 U 1ð1Þ þðm1 fð1Þ Þ
>
>
n
h
i00 o
>
0
>
02
0
00
00
0
0
00
03
0 1 2
0
>
1 02
1
>
¼ EI2 U 000
>
1ð0Þ U 3ð0Þ þ 2 U 1ð0Þ þ fð0Þ U 1ð0Þ fð0Þ U 2ð0Þ ðU 1ð0Þ U 3ð0Þ Þ 2 fð0Þ U 1ð0Þ 3 ðU 1ð0Þ Þ
>
>
>
<
h
i h
i
00
0
00
0
02
þEI1 fð0Þ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ þ 2fð0Þ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ0 C fð0Þ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ þ fð0Þ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ0 þ fð0Þ U 001ð0Þ
>
>
>
oh
i
>
>
00
0
00
0
0
00
03
0 1 2
0
1 02
1
>
þEAU 001ð0Þ U 03ð0Þ þ 12 U 02
>
1ð0Þ þt 3 U 1ð0Þ U 3ð0Þ þ 2 U 1ð0Þ þ fð0Þ U 2ð0Þ ðU 1ð0Þ U 3ð0Þ Þ 2 fð0Þ U 1ð0Þ 3 ðU 1ð0Þ Þ
>
>
>
>
0
>
o
>
0
>
þm1 fð0Þ U 03ð0Þ þ 12 U 02
>
1ð0Þ
>
>
>
>
>
ð3Þ 0
>
: ½fð1Þ þ cð2Þ
¼0
ð0Þ þ cð0Þ Z ¼ 7 L=2 ¼ U 1ð1Þ Z ¼ 7 L=2 ¼ U 1ð1Þ Z ¼ 7 L=2
ð5:13Þ
for the next power of a, where
cð2Þ
ð0Þ ¼
1
2
Z
Z
ðU 01ð0Þ U 002ð0Þ U 001ð0Þ U 02ð0Þ ÞdZ
0
and
1
4
cð3Þ
ð0Þ ¼ fð0Þ
2 2
fð0Þ þU 02
1ð0Þ :
3
Elimination of U1(0) in Eq. (5.11) yields the equation for f(0)
9
8
o
o
½P Z ðLþ Lisland ÞZ2 0
P Z ðL þLisland Þ 0 =
d < 000 2 00
f f þ
fð0Þ fð0Þ ¼ 0
;
dZ : ð0Þ Z ð0Þ
4EI2 C
2EI2
ð5:14Þ
for the state prior to the onset of buckling in Eqs. (5.1) and (5.2). The symmetric and anti-symmetric buckling modes
correspond to even and odd f(0), respectively, and U1(0) is odd for an even f(0), and is even for an odd f(0).
For bridges with a narrow cross section such that EI1 b EI2 ,C as in experiments, Eq. (5.12) can be significantly simplified
by neglecting EI2/(EI1) and C/(EI1)
8
8
0
00
00
00
0
>
>
1 02
>
>
< ðU 2ð0Þ fð0Þ U 1ð0Þ Þ ¼ 0
< U 3ð0Þ þ 2 U 1ð0Þ ¼ 0
ð5:12aÞ
,
,
0
>
>
¼ U 02ð0Þ ¼ ðU 002ð0Þ fð0Þ U 001ð0Þ Þ0 ¼0
U >
>
: 2ð0Þ Z ¼ 0
: U 3ð0Þ Z ¼ 0 ¼ U 3ð0Þ Z ¼ L=2 ¼ 0
Z ¼ 7 L=2
Z ¼ L=2
o
where the deformation at the onset of buckling is negligible since P Z and DP Zð0Þ are proportional to
pffiffiffiffiffiffiffiffiffiffi
EI2 C (to be shown in
Sections 5.4 and 5.5) and the bridge length L is much larger than the cross section dimension (e.g., thickness). The above
equation, together with f(0) and U1(0) being opposite even or odd functions, give U 002ð0Þ ¼ fð0Þ U 001ð0Þ and U 03ð0Þ ¼ U 02
1ð0Þ =2.
Its solution is
U 2ð0Þ ¼ Z
Z
L=2
fð0Þ U 001ð0Þ dZ 1 Z
U 3ð0Þ ¼ 1
2
Z
Z
Z
0
Z 1 fð0Þ U 001ð0Þ dZ 1
Z
0
U 02
1ð0Þ dZ 1 :
ð5:15Þ
For both symmetric or anti-symmetric buckling modes, U2(0) and U3(0) are always odd functions. Similar to Eq. (3.21), the
normality condition for Eq. (5.13) gives DP Zð0Þ . The displacement U2(0) would be zero in Eq. (5.15) without accounting for
the 4th power of displacement in the potential energy.
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
503
12
symmetric mode
anti-sym. mode
stretching
10
8
compressing
6
4
0.0
0.1
0.2
ν
0.3
0.4
0.5
Fig. 6. The normalized critical buckling load versus the Poisson’s ratio of the bridge for the symmetric and anti-symmetric buckling modes, subject to
stretching or compressing along 451 (diagonal) direction.
5.4. Symmetric buckling mode
For an even function f(0) and odd function U1(0), only half of the beam, 0 r Z rL=2, is considered. Integration of
Eq. (5.14) and f(0) being an even function give
3
d fð0Þ
3
dz
where z ¼
2
2 d fð0Þ
z dz2
2
þz
dfð0Þ
dfð0Þ
a
¼ 0,
dz
dz
ð5:16Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ffi
o
o
o
pffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
o P ðL þLisland Þ=ð2 EI2 C ÞZ, a ¼ C=ðEI2 ÞsignðP Þ, signðP Þ ¼ 1 for tension and signðP Þ ¼ 1 for compression.
Z
Z
Z
Z
The solution of the above equation is
i
pffiffiffi h
dfð0Þ
2
¼ B4 zRe M iða=4Þ,ð3=4Þ ðiz Þ
dz
ð5:17Þ
for an even function f(0), where B4 is a constant to be determined, Re stands for the real part, M is the Whittaker M function
pffiffiffiffiffiffiffi
with indices ia/4 and 3/4 (Gradshteyn and Ryzhik, 2007), and i ¼ 1. Integration of Eq. (5.17) together with the boundary
condition f(0)9Z ¼ L/2 ¼0 in Eq. (5.11) gives
Z zmax pffiffiffi
fð0Þ ¼ B4
zRe½Miða=4Þ,ð3=4Þ ðiz2 Þdz,
ð5:18Þ
z
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
o where zmax ¼ P Z ðL þLisland ÞL2 =ð8 EI2 C Þ corresponds to the end Z ¼L/2. The parameter B4 is determined from
nR
o1
pffiffiffi
z
2
, which ensures the parameter a in Eq. (4.15) to be the maximum
maxðfð0Þ Þ ¼ 1 as B4 ¼ 0 max zRe½M iða=4Þ,ð3=4Þ ðiz Þdz
angle of twist, fmax.
Integration of the 1st equation in (5.11) gives U1(0) as
( Z
)
h
i
i
zmax
aB4
1
zmax z h
2
2
p
ffiffiffiffiffiffiffiffiffi
ffi
z
Re
M
ði
z
Þ
d
z
ði
z
Þ
U 1ð0Þ ¼ L
Re
M
:
iða=4Þ,ð3=4Þ
iða=4Þ,ð3=4Þ
max
2zmax
zmax
z
z3=2
ð5:19Þ
Its being an odd function requires U1(0) ¼0 at Z ¼0, which gives the following simple equation for the critical buckling
load:
2
Re½M iða=4Þ,ð3=4Þ ðizmax Þ ¼ 0,
ð5:20Þ
which has the solution zmax ¼ zmax ðaÞ. The critical buckling load is
pffiffiffiffiffiffiffiffiffiffi
o P ¼ 8 EI2 C ½zmax ðaÞ2 :
ð5:21Þ
Z
ðLþ Lisland ÞL2
pffiffiffiffiffiffiffiffiffiffi
o The normalized critical buckling load, P Z ðLþ Lisland ÞL2 =ð8 EI2 C Þ, is shown versus the Poisson’s ratio n of the bridge in
o
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
Fig. 6, where a ¼ 2=ð1 þ nÞsignðP Z Þ for a rectangular cross section of the bridge. The critical buckling load for stretching is
clearly larger than that for compressing, and both are essentially independent of n.
504
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
Table 1
The coefficients b versus the Poisson’s ratio n of the bridge for the symmetric and anti-symmetric buckling modes under 451 stretching.
Poisson’s ratio
Stretching
Symmetric
n ¼ 0.1
n ¼ 0.2
n ¼ 0.3
n ¼ 0.4
n ¼ 0.5
Anti-symmetric
bsym
bending
3
bsym
bending
bsym
torsion
banti
bending
3
banti
bending
banti
torsion
11.55
10.96
10.46
10.01
9.614
6.796
6.622
6.467
6.328
6.201
10.07
10.20
10.33
10.45
10.56
16.23
15.43
14.71
14.11
13.56
8.111
7.775
7.671
7.262
6.897
12.75
12.94
13.11
13.29
13.46
Table 2
The coefficients b versus the Poisson’s ratio n of the bridge for the symmetric and anti-symmetric buckling modes under 451 compressing.
Poisson’s ratio
Compressing
Symmetric
n ¼ 0.1
n ¼ 0.2
n ¼ 0.3
n ¼ 0.4
n ¼ 0.5
Anti-symmetric
3
bsym
bending
bsym
bending
bsym
torsion
banti
bending
3
banti
bending
banti
torsion
7.570
7.316
7.087
6.880
6.691
5.503
5.410
5.324
5.246
5.173
6.216
6.426
6.623
6.807
6.981
7.290
7.149
7.016
6.891
6.773
4.639
4.557
4.481
4.409
4.342
7.328
7.680
8.008
8.313
8.600
Eq. (5.19) can then be simplified to
Z zmax
h
i
U 1ð0Þ
az
1
2
¼ B4
Re M iða=4Þ,ð3=4Þ ðiz Þ dz:
3=2
L
2zmax z
z
Eq. (5.15) then gives U2(0) and U3(0).
The applied strain along Z (451) direction during postbuckling is
!
(
2 )
2
Z L=2
Z L=2
Z zmax Z zmax 2
dfð0Þ
1
Lfmax
a
d U 1ð0Þ
e453 ¼
U 03 dZ þ
U 02 dZ ¼
dz2zmax
dz ,
L þ Lisland
dz
L þLisland zmax 0
dz
L
0
L=2
L=2
ð5:22Þ
ð5:23Þ
where the coefficient inside (von Karman and Tsien, 1941) depends only on a since zmax ¼ zmax ðaÞ.
For a rectangular section with height h 5 width b, EI1 ¼Ehb3/12, EI2 ¼Eh3b/12 and C ¼Gh3b/3. The maximum bending
strain is the sum of bending strains at the onset of buckling and during postbuckling, and is given by
!
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
3
h
L þLisland
h
sym
sym
ebending ¼
bbending
je451 j þ bbending
:
ð5:24Þ
L
b
L
The maximum shear strain due to torsion is given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h Lþ Lisland
gtorsion ¼ bsym
je451 j:
torsion L
L
ð5:25Þ
The coefficients b’s in the above two equations are shown versus the Poisson’s ratio n of the bridge in Tables 1 and 2 for
stretching and compressing along 451, respectively. They are essentially independent of n.
Without accounting for the 4th power of displacement in the potential energy one could not study postbuckling for
stretching along 451 because Eq. (5.23) would give a compressive strain e451 o0. For compressing along 451, the maximum
bending and shear strains would be overestimated by more than a factor of 2.
5.5. Anti-symmetric buckling mode
For an odd function f(0) and even function U1(0), only half of the beam, 0 rZrL/2, is considered. Integration of
Eq. (5.14) and f(0) being an odd function give
3
d fð0Þ
3
dz
2
2 d fð0Þ
z dz2
2
þz
dfð0Þ
dfð0Þ
a
¼ B5 ,
dz
dz
ð5:26Þ
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
where B5 is a constant of integration to be determined. The solution of the above equation is
i
pffiffiffi h
dfð0Þ
2
¼ B5 DðzÞ þ B6 zRe M iða=4Þ,ð3=4Þ ðiz Þ
dz
505
ð5:27Þ
for an odd function f(0), where B5 and B6 are constants to be determined, the function D is given in Appendix A, M is the
Whittaker M function with indices ia/4 and 3/4 (Gradshteyn and Ryzhik, 2007). Integration of Eq. (5.27) together with
the boundary condition f(0)9Z ¼ L/2 ¼0 in Eq. (5.11) gives
Z zmax
Z zmax pffiffiffi
fð0Þ ¼ B5
DðzÞdzB6
zRe½Miða=4Þ,ð3=4Þ ðiz2 Þdz:
ð5:28Þ
z
z
Its being an odd function requires f(0) ¼0 at Z ¼0, i.e.
Z zmax
Z zmax pffiffiffi
DðzÞdz þB6
zRe½Miða=4Þ,ð3=4Þ ðiz2 Þdz ¼ 0:
B5
0
ð5:29Þ
0
¼ 0 give
Integration of the 1st equation in (5.11) and the boundary condition U 01ð0Þ Z ¼ L=2
* Z
+
Z zmax
n
h
i
o
zmax
p
ffiffi
ffi
d
a
1 dDðzÞ
1 d
U 1ð0Þ ¼ L
B5
dz þ B6
zRe Miða=4Þ,ð3=4Þ ðiz2 Þ dz :
dz
2zmax
z dz
z dz
z
z
It is an odd function (since U1(0) is even), which requires U 01ð0Þ ¼ 0 at Z ¼0
Z zmax
Z zmax
io
1 dDðzÞ
1 d npffiffiffi h
B5
dz þ B6
zRe Miða=4Þ,ð3=4Þ ðiz2 Þ dz ¼ 0:
z dz
z dz
0
0
ð5:30Þ
ð5:31Þ
Eqs. (5.29) and (5.31) give two linear, homogeneous equations for B5 and B6. Their determinant must be zero in order to
have a non-trivial solution, which gives the equation for the critical buckling load
i
R
R zmax pffiffiffi h
zmax
zRe Miða=4Þ,ð3=4Þ ðiz2 Þ dz 0 DðzÞdz
0
npffiffiffi h
io ¼ 0:
ð5:32Þ
Rz
2
max 1 dDðzÞ dz R zmax 1 d
z
ði
z
Þ
dz Re
M
iða=4Þ,ð3=4Þ
0 z dz
0
z dz
Its solution zmax ¼ zmax ðaÞ is different from zmax after Eq. (5.20) for the symmetric buckling mode. The critical buckling load
is given by Eq. (5.21), but with the new zmax above for the anti-symmetric buckling mode. As shown in Fig. 6, the
pffiffiffiffiffiffiffiffiffiffi
o (normalized) critical buckling load, P Z ðL þLisland ÞL2 =ð8 EI2 C Þ, is essentially independent of n, and is larger for stretching
than for compressing. For the island-bridge, mesh structure under stretching along 451 direction, symmetric buckling
mode occurs since it gives a small buckling load than the anti-symmetric mode. For compressing along 451 direction,
however, both symmetric and anti-symmetric modes may occur since their corresponding buckling loads are very close.
Integration of Eq. (5.30) and the boundary condition U1(0)9Z ¼ L/2 ¼0 give
h
i
8 Rz
9
2
max
* " Z
+
#
1
>
< z z Re M iða=4Þ,ð3=4Þ ðiz Þ z3=2 dz >
=
zmax
U 1ð0Þ
a
DðzÞ
zmax z
h
i
:
ð5:33Þ
¼
B5 z
dz
Dðzmax Þ þB6
2
zmax
z
>
>
L
2zmax
zmax
ffiffiffiffiffiffiffi
Re M iða=4Þ,ð3=4Þ ðizmax Þ
z
z2
:p
;
z
max
The parameter a in Eq. (4.15) is the ratio of maximum displacement to L, i.e., a ¼U1max/L, which, together with
Eq. (5.33), give
npffiffiffiffiffiffiffiffiffiffi h
i
po 2zmax
B5 Dðzmax Þ þ B6
¼
zmax Re Miða=4Þ,ð3=4Þ ðiz2max Þ cos
:
ð5:34Þ
8
a
Eqs. (5.29) and (5.34) give two equations to determine B5 and B6. Eq. (5.15) then gives U2(0) and U3(0).
The applied strain along Z (451) direction during postbuckling is
(
2 )
2
Z zmax Z zmax dfð0Þ
U 21max
a
d U 1ð0Þ
e453 ¼
dz2zmax
dz :
dz
ðLþ Lisland ÞL zmax 0
dz
L
0
3
ð5:35Þ
2
The maximum bending strain at the onset of buckling is ebending ¼ banti
bending h =ðLbÞ. The maximum bending strain and torsion
(shear) strain due to postbuckling are
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h Lþ Lisland
ebending ¼ banti
je451 j,
ð5:36Þ
bending
L
L
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h L þ Lisland
gtorsion ¼ banti
je451 j:
ð5:37Þ
torsion
L
L
Here the coefficients b’s are given in Tables 1 and 2, and are essentially independent of the Poisson’s ratio n. Without
accounting for the 4th power of displacement, postbuckling for stretching along 451 could not be studied because e451 o 0
in Eq. (5.35), and the maximum bending and shear strains would be significantly overestimated for compressing along 451.
506
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
Fig. 4c shows the symmetric and anti-symmetric buckling modes in Sections 5.4 and 5.5, respectively. The applied
strain is e451 ¼ 7 10%UL=ðLþ Lisland Þ (for both stretching and compressing), and C/EI2 ¼ 1.57. The buckling mode for the
anti-symmetric appears to be rather similar to the experimentally observed buckling profile in Fig. 1c for the
stretching case.
5.6. Maximum strains
5.6.1. 451-Direction stretching
Comparison of Eqs. (5.36) and (5.37) with Eqs. (5.24) and (5.25), together with Tables 1 and 2, suggest that the
symmetric buckling mode always gives smaller bending and torsion strains than anti-symmetric buckling. The maximum
bending and torsion strains are then given by
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h L þ Lisland
h Lþ Lisland
ebending 6:5
je451 j, gtorsion 10
je451 j,
ð5:38Þ
L
L
L
L
where the bending strain at the onset of buckling is neglected, and the coefficient b’s for n ¼ 0.3 are used since their
variations are small ( 5%) in Tables 1 and 2. These maximum strains are not reached at the same point in the bridge. The
maximum principal strain is the same as the maximum bending strain, i.e.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h Lþ Lisland
je451 j:
emax 6:5
L
L
5.6.2. 451-Direction compressing
Eqs. (5.24), (5.25), (5.36) and (5.37), together with Tables 1 and 2, suggest that the symmetric buckling mode gives
larger bending strain and but smaller torsion strain than anti-symmetric buckling. The maximum bending and torsion
strains can then be bounded as
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h L þLisland
h L þLisland
je451 j, gtorsion r 8:6
je451 j:
ð5:39Þ
ebending r5:5
L
L
L
L
The maximum principal strain is the same as the maximum bending strain, i.e.
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
h Lþ Lisland
je451 j:
emax r 5:5
L
L
For Euler buckling, Song et al. (2009) used both analytical and finite element methods to show that the maximum strain
in the island is much smaller than that in the bridge, even at the stress and strain concentration at the island-bridge
interface. This is also expected to hold also for the complex buckling modes in the present study.
6. Concluding remarks
(i) All terms up to the 4th power of displacements in the potential energy are necessary in the postbuckling analysis
(Koiter, 1945). The existing postbuckling analyses, however, are accurate only to the 2nd power of displacements in
the potential energy since they assume a linear displacement–curvature relation.
(ii) A systematic method is established for postbuckling analysis of beams that may involve complex buckling modes
(e.g., lateral buckling under arbitrary loading). It avoids complex geometric analysis of deformation for postbuckling
(Timoshenko and Gere, 1961), and accounts for all terms up to the 4th power of displacements in the potential
energy.
(iii) The lateral buckling mode of an island-bridge, mesh structure subject to shear (or twist), which is used in stretchable
electronics, involves the Bessel Function of the first kind. Simple, analytical expressions are obtained for the critical
load at the onset of buckling in Eqs. (4.27) and (4.40), and for the maximum bending and torsion strains during
postbuckling in Eqs. (4.30), (4.31), (4.45) and (4.46).
(iv) Buckling of the island-bridge, mesh structure subject to diagonal stretching (and compressing) involves the Whittaker
M function. Simple, analytical expressions are obtained for the maximum bending, torsion (shear) and principal
strains during postbuckling in Eq. (5.38).
Acknowledgments
The authors acknowledge support from NSF Grant nos. ECCS-0824129 and OISE-1043143, and DOE, Division of
Materials Sciences Grant no.DE-FG02-07ER46453. The support from NSFC (Grant no. 11172146) and the Ministry of
Education, China is also acknowledged.
Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508
507
Appendix A
The 3rd order terms in the direction cosines aij and curvatures ki are
8
ð2Þ
ð3Þ
0 0
0
02
0 0 0
1
>
þ U 02
að3Þ
¼ 14 fðU 02
að3Þ
>
1 U3 ,
1 U 2 Þ þ U 1 U 2 U 3 þ c ,
11 ¼ 2 fU 1 U 2 fc
>
>
h12
i
>
>
ð2Þ 0
ð3Þ
0 0
0
02 1
02
02
1 2 0
>
>
> a13 ¼ 2 f U 1 þ fU 2 U 3 c U 2 U 1 U 3 2 ðU 1 þ U 2 Þ ,
>
>
< ð3Þ
ð3Þ
ð2Þ
02
0 0 0
0 0
0
1
að3Þ
þ U 02
a21 ¼ 14 fðU 02
1 U 2 Þ þ U 1 U 2 U 3 c ,
2 U3,
22 ¼ 2 fU 1 U 2 fc
h
i
>
>
ð2Þ 0
0 0
0
02 1
02
02
>
1 2 0
> að3Þ
>
23 ¼ 2 f U 2 fU 1 U 3 þ c U 1 U 2 U 3 2 ðU 1 þ U 2 Þ ,
>
>
>
h
i
h
i
>
>
ð3Þ
ð3Þ
ð3Þ
0
02
02
02
0
02
02
02
0
02
02
>
: a31 ¼ U 1 U 3 12 ðU 1 þ U 2 Þ , a32 ¼ U 2 U 3 12 ðU 1 þU 2 Þ , a33 ¼ U 3 ðU 1 þU 2 Þ,
8
ð2Þ 00
00 0
0 00
00 1 02
02
02
00 0 0
1 2 00
1 00 0 0
>
^ ð3Þ
>
1 ¼ c U 1 þ 2 f U 2 fðU 1 U 3 þ U 1 U 3 Þ þU 2 2 U 1 þU 2 U 3 þ 2 U 1 U 1 U 2 2U 3 U 2 U 3 ,
>k
>
<
ð2Þ 00
00 0
0 00
00 1 02
02
02
00 0 0
1 2 00
1 00 0 0
k^ ð3Þ
2 ¼ c U 2 2 f U 1 fðU 2 U 3 þU 2 U 3 ÞU 1 2 U 2 þ U 1 U 3 2 U 2 U 1 U 2 þ 2U 3 U 1 U 3 ,
>
>
>
>
:k
^ ð3Þ
3 ¼ 0,
ðA:1Þ
ðA:2Þ
where
cð3Þ ¼
Z
Z
0
i0
1 0 2 1h
02
00 0
0 00
0
f f fðU 02
þ
U
Þ
þ
ðU
U
U
U
ÞU
1
2
1 2
1 2
3 dZ:
2
4
The orthogonal condition before Eq. (3.21) can be proven as
0
100
20
100
0
100
3
o
o
o
Z L=2
Z L=2
P
P
P
@U 00 þ
4@U 00 þ
U 1ð1Þ A U 1ð0Þ dZ ¼
U 1ð1Þ A U 1ð0Þ @U 001ð0Þ þ
U 1ð0Þ A U 1ð1Þ 5dZ
1ð1Þ
1ð1Þ
EI2
EI2
EI2
L=2
L=2
L=2
000
000
¼ ðU 1ð1Þ U 1ð0Þ U 1ð1Þ U 1ð0Þ Þ
Z ¼ L=2
L=2
ðU 001ð1Þ U 01ð0Þ U 01ð1Þ U 001ð0Þ Þ
Z ¼ L=2
þ
o
L=2
P
ðU 01ð1Þ U 1ð0Þ U 1ð1Þ U 01ð0Þ Þ
¼ 0:
EI2
Z ¼ L=2
The function D in Eq. (5.27) is given by
Z z
Z z
T 2 ðzÞ
T 1 ðzÞ
dzT 2 ðzÞ
dz,
DðzÞ ¼ T 1 ðzÞ
dT 1 ðzÞ
dT 2 ðzÞ
dT 1 ðzÞ
dT 2 ðzÞ
0
T
ð
z
z1 dz T 2 ðzÞT 1 ðzÞ dz
2 ÞT 1 ðzÞ dz
dz
where
T 1 ðzÞ ¼
pffiffiffi
zRe½Miða=4Þ,ð3=4Þ ðiz2 Þ,
T 2 ðzÞ ¼
ðA:3Þ
ðA:4Þ
pffiffiffi
zRe½Miða=4Þ,ð3=4Þ ðiz2 Þ,
ðA:5Þ
which have the limits
3
T 1 ðzÞ ¼ z cos
i
5p h
a 2
1þ
z þ Oðz4 Þ
8
10
and
T 2 ðzÞ ¼ cos
ph
a
2
The lower limit of the 1st integral z1 in Eq. (A.4) is determined from
" Z
#
z
d
T 2 ðzÞ
z
d
z
lim
¼ 0:
dT 1 ðzÞ
z-0 dz
z1
T 2 ðzÞT 1 ðzÞ dT 2 ðzÞ
dz
4
1 z þ Oðz Þ
8
2
i
as
z-0:
ðA:6Þ
dz
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