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Journal of the Mechanics and Physics of Solids

Journal of the Mechanics and Physics of Solids 60 (2012) 487–508

0022-50

doi:10.1

n Corr

E-m1 Eq

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, Chinab Department of Mechanical and Aerospace Engineering, University of Miami, Coral Gables, FL 33124, USAc Departments of Civil and Environmental Engineering and Mechanical Engineering, Northwestern University, Evanston, IL 60208, USAd Department of Materials Science and Engineering, University of Illinois, Urbana, IL 61801, USA

a r t i c l e i n f o

Article history:

Received 13 September 2011

Received in revised form

12 November 2011

Accepted 19 November 2011Available online 1 December 2011

Keywords:

Postbuckling

Stretchable electronics

Higher-order terms in curvature

Lateral buckling

Diagonal stretching

96/$ - see front matter Crown Copyright & 2

016/j.jmps.2011.11.006

esponding authors.

ail addresses: huangkz@tsinghua.edu.cn (K.-C

ual contribution to this paper.

a b s t r a c t

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 island-

bridge, 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.

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). Theresulting 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) andLEDs (Park et al., 2009), soft, bio-integrated devices (Kim et al., 2010a, 2010b, 2011a, 2011b; Viventi et al., 2010), flexibledisplays (Crawford, 2005), eye-like digital cameras (Jin et al., 2004; Ko et al., 2008; Jung et al., 2011), structural healthmonitoring 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 (Kimet al., 2011b). Work in this area emphasizes mechanics and geometry, in systems that integrate hard materials for theactive components of the devices with elastomers for the substrates and packaging components. (Rogers et al., 2010).

011 Published by Elsevier Ltd. All rights reserved.

. Hwang), y-huang@northwestern.edu (Y. Huang).

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 island-

bridge, mesh structure under shear; and (c) SEM image of the island-bridge, mesh structure under diagonal stretch.

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508488

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 andbonding it to the substrate only at the nodes, as shown in Fig. 1a. Once buckled, the arc-shaped interconnects between thenodes can move freely out of the mesh plane to accommodate large applied strain (Song et al., 2009). The interconnectsundergo complex buckling modes, such as lateral buckling, when subject to shear (Fig. 1b) or diagonal stretching (451 fromthe interconnects) (Fig. 1c). Different from Euler buckling, these complex buckling modes involve large torsion and out-of-plane 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 potentialenergy was expanded up to 4th power of displacement (from the non-buckled state), and the 3rd and 4th power termsactually 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,whose curvature k is approximately the 2nd order derivative of the deflection w, i.e., k¼w

00

. The bending energy isðEI=2Þ

Rw002dx, where EI is the bending stiffness, and the integration is along the central axis x of the beam. The accurate

expression of the curvature is k¼w00=ð1þw02Þ3=2�w00½1�ð3=2Þw02�, which gives bending energy ðEI=2Þ

Rk2dx�

ðEI=2ÞR

w002ð1�3w02Þdx. It is different from the approximate expression above by ð3EI=2ÞR

w002w02dx, which is the 4thpower 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 involverather 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 straightfor-ward method to study any buckling mode. It accounts for all terms up to the 4th power of displacements in the potentialenergy, 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). Analyticalexpressions are obtained for the amplitude of, and the maximum strain in, buckled interconnects, which are important tothe 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) beforedeformation are Ei (i¼1,2,3). A point X¼(0,0,Z) on the central axis moves to XþU¼(U1,U2,U3þZ) after deformation, whereUi(Z) (i¼1,2,3) are the displacements. The stretch along the axis is

l¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiU021 þU022 þð1þU03Þ

2q

� 1þU03þ1

2ðU021 þU022 Þ�

1

2U03ðU

021 þU022 Þ, ð2:1Þ

where ()0 ¼d()/dZ, and terms higher than the 3rd power of displacement are neglected because their contribution to thepotential energy are beyond the 4th power. The length dZ becomes ldZ after deformation.

The unit vector along the deformed central axis is e3¼d(XþU)/(ldZ). The other two unit vectors, e1 and e2, involve twistangle 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)

e0il¼ j� ei ði¼ 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 tothe twist angle f of the cross section by

k3 ¼f0

l: ð2:3Þ

The unit vectors before and after deformation are related by the direction cosine aij

ei ¼ aijEj ði¼ 1,2,3, summation over jÞ: ð2:4Þ

Eq. (2.2) gives 6 independent equations, while the orthonormal conditions ei � ej¼dij give another 6. These 12 equations aresolved to determine 9 direction cosines aij and 3 curvatures ki in terms of the displacements Ui and twist angle f as

aij

� �¼

1 0 0

0 1 0

0 0 1

8><>:

9>=>;þ

0 f �U01�f 0 �U02U01 U02 0

8><>:

9>=>;þ

� 12 ðf

2þU021 Þ �

12 U01U02þc

ð2Þ U01U03�fU02

� 12 U01U02�c

ð2Þ� 1

2 ðf2þU022 Þ U02U03þfU01

�U01U03 �U02U03 � 12 ðU

021 þU022 Þ

8>><>>:

9>>=>>;þ að3Þij

n o, ð2:5Þ

where cð2Þ ¼ 12

R Z0 ðU

01U002�U001U02ÞdZ is the 2nd power of displacements, and að3Þij are the 3rd power of displacements and twist

angle given in Appendix A. As to be shown in the next section, the work conjugate of bending moment and torque isj¼ lj, which is given in terms of the power of Ui and f by

fkig ¼

�U002U001f0

8><>:

9>=>;þ

fU001þðU02U03Þ

0

fU002�ðU01U03Þ

0

0

8><>:

9>=>;þfki

ð3Þg, ð2:6Þ

where kið3Þ are the 3rd power of displacements given in Appendix A.

2.2. Force, bending moment and torque

Let t¼tiei and m¼miei denote the forces and bending moment (and torque) in the cross section Z of the beam afterdeformation. Equilibrium of forces requires

t0

lþp¼ 0, or

t01�t2k3þt3k2þlp1 ¼ 0

t02þt1k3�t3k1þlp2 ¼ 0

t03�t1k2þt2k1þlp3 ¼ 0

,

8><>: ð2:7Þ

where p is the distributed force on the beam (per unit length after deformation). Equilibrium of moments requires

m0

lþe3 � tþq¼ 0, or

m01�m2k3þm3k2�lt2þlq1 ¼ 0

m02þm1k3�m3k1þlt1þlq2 ¼ 0

m03�m1k2þm2k1þlq3 ¼ 0

,

8><>: ð2:8Þ

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�1and j, respectively. For example, the virtual work of m is

RmUjds, where ds¼ldZ represents the integration along the

central axis in the current (deformed) configuration. This integral can be equivalently written asR

mUjdZ in the initial(undeformed) configuration.

2.3. Onset of buckling and postbuckling

Let to

and mo

denote the critical forces, bending moments and torque at the onset of buckling, at which the deformationis still small. The distributed force and moment at the onset of buckling are denoted by p

oand q

o. Equilibrium Eqs. (2.7)

and (2.8) at the onset of buckling are

t0io

þpo

i ¼ 0 ði¼ 1,2,3Þ, ð2:9Þ

m01o

�t2

oþq

o1 ¼ 0, m02

o

þt1

oþq

o2 ¼ 0, m03

o

þqo

3 ¼ 0: ð2:10Þ

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508490

The forces, bending moments and torque during postbuckling can be written as t ¼ toþDt and m¼m

oþDm, where Dt and

Dm are the changes beyond the onset of buckling. Equilibrium Eqs. (2.7) and (2.8) become

Dt01�ðt2

oþDt2Þk3þðt3

oþDt3Þk2þðlp1�p1

oÞ ¼ 0

Dt02þðt1

oþDt1Þk3�ðt3

oþDt3Þk1þðlp2�p2

oÞ ¼ 0

Dt03�ðt1

oþDt1Þk2þðt2

oþDt2Þk1þðlp3�p3

oÞ ¼ 0

,

8>>>><>>>>:

ð2:11Þ

Dm01�ðm2oþDm2Þk3þðm3

oþDm3Þk2�½ðl�1Þt2

oþlDt2�þðlq1�q1

oÞ ¼ 0

Dm02þðm1oþDm1Þk3�ðm3

oþDm3Þk1þ½ðl�1Þt1

oþlDt1�þðlq2�q2

oÞ ¼ 0

Dm03�ðm1oþDm1Þk2þðm2

oþDm2Þk1þðlq3�q3

oÞ ¼ 0

:

8>>>><>>>>:

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 EAðl�1Þ, Dm1 ¼ EI1k1, Dm2 ¼ EI2k2, Dm3 ¼ Ck3, 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) isstraightforward even for complex buckling modes such as the island-bridge, mesh structure in stretchable electronics inSection 5. It is much simpler than the existing analysis for the onset of buckling, which works only for relatively simplebuckling modes (Timoshenko and Gere, 1961). It is also more accurate than the existing postbuckling analysis, whichneglects 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

compression force P. There are no distributed force and moment in the beam, p¼q¼0. Let Po

denote the critical

compression at the onset of buckling (to be determined), and DP the change of P beyond Po. The forces, bending moments

and torque at the onset of buckling are t1

o¼ t2

o¼ 0, t3

o¼�P

o

and mo¼ 0. Without losing generality it is assumed the bending

stiffness EI14EI2 such that the beam buckles within the (X,Z) plane (Fig. 2). This gives U2¼f¼0, and therefore k1 ¼ k3 ¼ 0

from Eq. (2.6) and Dm1¼Dm3¼0 from Eq. (2.13). Eqs. (2.11) and (2.12) give Dt2¼0, and

Dt01þð�PoþDt3Þk2 ¼ 0, Dt03�Dt1k2 ¼ 0, Dm02þlDt1 ¼ 0: ð3:1Þ

Elimination of Dt1, together with the linear elastic relation Eq. (2.13), give

EI2k02l

� �0þ P

o�EAðl�1Þ

� �k2 ¼ 0, ðEAl2

þEI2k22Þ0¼ 0: ð3:2Þ

Eq. (2.6) gives k2 on the order of U1. Its substitution into Eq. (3.1) gives Dt1 and Dm2 on the order of U1, and Dt3 onthe order of (U1)2. The linear elastic relation Dt3¼EA(l�1) in Eq. (2.13) and the stretch l in Eq. (2.1) suggest U3 on theorder of (U1)2

U3 � ðU1Þ2: ð3:3Þ

The stretch in Eq. (2.1) becomes

l¼ 1þU03þ1

2U021 þO ðU1Þ

4h i

: ð3:4Þ

2L

×

2L

hh

X

Z Pb

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) 487–508 491

The only non-zero curvature obtained from Eq. (2.6) and kið3Þ in Appendix A is

k2 ¼U001� U01U03þ1

3U031

� �0þO ðU1Þ

5h i

, ð3:5Þ

where the underlined are the higher-order terms in the curvature neglected by von Karman and Tsien (1941) andBudiansky (1973). (Their curvatures neglected the 2nd and 3rd power of displacements in the postbuckling analysis ofplates.) Substitution of Eqs. (3.4) and (3.5) into (3.2) gives two equations for U1 and U3

U000

1 1�U03�1

2U021

� �� �0þ

Po

EI2U001�

EA

EI2U001 U03þ

1

2U021

� �� U01U03þ

1

3U031

� �000�

Po

EI2U01U03þ

1

3U031

� �0þO ðU1Þ

5h i

¼ 0, ð3:6Þ

EA

EI2ð2U03þU021 ÞþU0021

� �0þO ðU1Þ

4h i

¼ 0: ð3:7Þ

The boundary conditions at the clamped end (Z¼�L/2, Fig. 2) are

U1 ¼U01 ¼U3 ¼ 0 at Z ¼�L=2 ð3:8Þ

The vanishing displacement and bending moment Dm2¼0 at the simply supported end (Z¼L/2) gives

U1 ¼U001� U01U03þ1

3U031

� �0þO ðU1Þ

5h i

¼ 0 at Z ¼ L=2, ð3:9Þ

where Eqs. (2.13) and (3.5) have been used. The traction at this end is Dt1e1þð�Po

þDt3Þe3; its component along the axial

direction E3 in the initial (undeformed) configuration must be �ðPoþDPÞ, which gives

Dt1e1UE3þð�PoþDt3Þe3UE3 ¼�ðP

oþDPÞ, at Z ¼ L=2, ð3:10Þ

or using the direction cosines aij in Eq. (2.5) (and a 3ð Þij in Appendix A)

�Dt1U01�Po

1�1

2U021

� �þDt3 ¼�ðP

oþDPÞþO ðU1Þ

4h i

at Z ¼ L=2 ð3:10aÞ

Elimination of Dt1 and Dt3 in the above equation via Eqs. (3.1), (2.13), (3.4) and (3.5) gives

U01U000

1þPo

2EI2U021 þ

EA

EI2U03þ

1

2U021

� �þDP

EI2þO ðU1Þ

4h i

¼ 0 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 bethe ratio of the maximum deflection U1max to the beam length L; a is small such that displacements U1 and U3 areexpressed 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

U1 ¼ aU1ð0Þ þa3U1ð1Þ þ . . .

U3 ¼ a2U3ð0Þ þ . . ., ð3:12Þ

where terms higher than the 3rd power of displacement (i.e., higher than a3) are neglected in the postbuckling analysis.Similarly, DP is on the order of a2, and can be written as

DP¼ a2DPð0Þ þ . . .: ð3:13Þ

Substitution of Eqs. (3.12) and (3.13) into Eqs. (3.6)–(3.9) and (3.11) gives

U001ð0Þ þPo

EI2U1ð0Þ

� �00¼ 0

U1ð0Þ

Z ¼ �L=2

¼U01ð0Þ

Z ¼ �L=2

¼U1ð0Þ

Z ¼ L=2

¼U001ð0Þ

Z ¼ L=2

¼ 0,

8>>><>>>:

ð3:14Þ

2U03ð0Þ þU021ð0Þ þEI2EA U0021ð0Þ

�0¼ 0

U3ð0Þ

Z ¼ �L=2

¼ U03ð0Þ þ12 U021ð0Þ þ

EI2EA U01ð0ÞU

000

1ð0Þ þPo

2EA U021ð0Þ þDPð0Þ

EA

� �Z ¼ L=2

¼ 0

8>><>>: ð3:15Þ

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508492

for the leading powers of a, and

U001ð1Þ þPo

EI2U1ð1Þ

� �00¼ U

000

1ð0Þ U03ð0Þ þ12 U021ð0Þ

�h i0þ EA

EI2U001ð0Þ U03ð0Þ þ

12 U021ð0Þ

�

þ U01ð0ÞU03ð0Þ þ

13 U031ð0Þ

�000þ P

o

EI2U01ð0ÞU

03ð0Þ þ

13 U031ð0Þ

�0U1ð1Þ9Z ¼ �L=2 ¼U01ð1Þ9Z ¼ �L=2 ¼U1ð1Þ

Z ¼ L=2

¼ U001ð1Þ� U01ð0ÞU03ð0Þ þ

13 U031ð0Þ

�0h iZ ¼ L=2

¼ 0

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

ð3:16Þ

for the next power of a, where the underline represents the higher-order terms neglected in the existing postbucklinganalysis.

Eq. (3.14) constitutes an eigenvalue problem for U1(0). The critical buckling load is the eigenvalue, and is given by

Po¼

k2EI2

L2¼

20:19EI2

L2, ð3:17Þ

where k¼4.493 is the smallest positive root of the equation

tank¼ k: ð3:18Þ

The corresponding buckling mode is

U1ð0Þ ¼kðL�2ZÞ

4p �Lsin k L�2Z

2L

� 2pcosk

, ð3:19Þ

and its maximum is L. Eq. (3.15) gives U03ð0Þ as

U03ð0Þ ¼ �k2

8p2

cos k L�2Z2L

� cosk

�1

" #2

þk2P

o

8p2EA

cos2 k L�2Z2L

� cos2 k

�1

" #�DPð0Þ

EA: ð3:20Þ

Its integration, together with U3(0)9Z¼�L/2¼0 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

without solving U1(1). This is because fU001ð1Þ þ½Po=ðEI2Þ�U1ð1Þg

00 in Eq. (3.16) is orthogonal to U1(0)

Z L=2

�L=2U001ð1Þ þ½P

o=ðEI2Þ�U1ð1Þ

� �00U1ð0ÞdZ ¼ 0

(also shown in Appendix A). Substitution of fU001ð1Þ þ½Po=ðEI2Þ�U1ð1Þg

00 in Eq. (3.16) into the above integral gives the equationfor DP

Z L=2

�L=2

U000

1ð0Þ U03ð0Þ þ12 U021ð0Þ

�h i0þ EA

EI2U001ð0Þ U03ð0Þ þ

12 U021ð0Þ

�þ U01ð0ÞU

03ð0Þ þ

13 U031ð0Þ

�000þ P

o

EI2U01ð0ÞU

03ð0Þ þ

13 U031ð0Þ

�08>><>>:

9>>=>>;U1ð0ÞdZ ¼ 0: ð3:21Þ

This gives

DPð0Þ ¼ Po k2

288p2

9k2�25�ð27k2

�3Þ Po

EA� 20þ12 Po

EA

� �

1�Po=ðEAÞ

� Po k2ð9k2�25�20Þ

288p2, ð3:22Þ

where the underline represents the contribution neglected in the existing postbuckling analysis, without which the

existing postbuckling analysis would give DPð0Þ ¼ 1:113Po, which is 15% larger than DPð0Þ ¼ 0:971P

oobtained from Eq. (3.22).

The compression force P is then related to the maximum deflection U1max during postbuckling by

P� Poþa2P

o k2ð9k2�25�20Þ

288p2¼ P

o1þ

k2ð9k2�25�20Þ

288p2

U1max

L

� �2" #

, ð3:23Þ

where a¼U1max=L has been used. The critical compressive strain at the onset of buckling, �Po

=ðEAÞ ¼ �20:19EI2=ðEAL2Þ, is

negligible. The applied (compressive) strain during postbuckling is

eapplied ¼U3jZ ¼ L=2�U3jZ ¼ �L=2

L¼

a2

L

Z L=2

�L=2U03ð0ÞdZ ��

k4

16p2

U1max

L

� �2

¼�2:58U1max

L

� �2

: 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

P� Po

1þ9k2�25�20

18k2eapplied

!¼ P

oð1þ0:3762 eapplied

Þ, ð3:25Þ

while the existing postbuckling analysis would give P� Poð1þ0:43129eapplied9Þ.

The maximum membrane strain in the beam is �Po=ðEAÞUf1þO½ðU1max=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 U1max much largerthan h. The maximum bending strain in the beam is given by

emax ¼h

2max U001

¼ 4:603h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieapplied

q, ð3:26Þ

which is much smaller than the applied strain for U1maxbh.

4. Lateral buckling

In general, the following orders of displacement and rotation increments hold during postbuckling:

(i)

Fig. 3(b) vi

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 directionwithout any rotation. Let Z denote the central axis of the beam before deformation, and with Z¼0 at the beam center. The

beam buckles out of the Y–Z plane (i.e., lateral buckling) when the shear force P reaches the critical buckling load Po

(to be

2

L

×

2

L

h× P

Z

X

h

b×

2

L

2

L

b

Z

Y

P

h

b

symmetric anti-symmetric

. Illustration of lateral buckling of a beam under shear along the thick direction ðbbhÞ of the cross section; (a) view along the direction of shear;

ew normal to the direction of shear; and (c) the symmetric and anti-symmetric buckling modes.

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508494

determined). As to be shown at the end of Section 4, this represents a special case of the island-bridge, mesh structure instretchable electronics (Kim et al., 2008b).

4.1. Equations for postbuckling analysis

There are no distributed force and moment, p¼q¼0. The force, bending moment and torque in the beam prior to theonset of buckling are

t1

o¼ t3

o¼m2

o¼m3

o¼ 0, t2

o¼ P

o, m1

o¼ P

oZ for �

L

2rZr

L

2: ð4:1Þ

Let DP¼ P�Po

denote the change of shear force during postbuckling. Only the displacements along Z and Y directions arenot 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�U1, U2 �U3 � ðU1Þ2: ð4:2Þ

The stretch in Eq. (2.1) becomes

l¼ 1þU03þ1

2U021 þO ðU1Þ

4h i

: ð4:3Þ

Eq. (2.6) and kið3Þ in Appendix A give the curvatures

k1 ¼�U002þfU001þO ðU1Þ4

h i,

k2 ¼U001þfU002�ðU01U03Þ

0�

1

2f2U001�

1

3ðU031 Þ

0þO ðU1Þ

5h i

, k3 ¼f0, ð4:4Þ

where the underlined are the higher-order terms in the curvatures neglected in the existing postbuckling analysis (vonKarman and Tsien, 1941; Budiansky, 1973). Substitution Eqs. (4.1)–(4.4) into Eqs. (2.11)–(2.13) gives the ODEs for U and f

Cf00 þðEI1�EI2ÞU001ðU002�fU001 Þ�P

o

Z U001þfU002�ðU01U03Þ

0�

1

2f2U001�

1

3ðU031 Þ

0

� �þO ðU1Þ

5h i

¼ 0, ð4:5Þ

EI2 Uð4Þ1 �U000

1 U03þ1

2U021

� �0�f02U001þ fU002�ðU

01U03Þ

0�

1

2f2U001�

1

3ðU031 Þ

0

� �00� �

�EI1 f00ðU002�fU001 Þþ2f0ðU002�fU001 Þ0

h iþC f00ðU002�fU001 Þþf

0ðU002�fU001 Þ

0þf02U001

h i�EAU001 U03þ

1

2U021

� �þP

of0 þðP

oZf0Þ0�P

oZf0 U03þ

1

2U021

� �0þO ðU1Þ

5h i

¼ 0, ð4:6Þ

EI1ðU002�fU001 Þ

00þEI2ðf

00U001þ2f0U000

1 Þ�Cðf0U001Þ0þP

oU03þ

1

2U021

� �0þP

oZf02þO ðU1Þ

4h i

¼ 0, ð4:7Þ

EA U03þ1

2U021

� �þ

1

2EI2U0021

� �0�P

oðU002�fU001 ÞþP

oZf0U001þO ðU1Þ

4h i

¼ 0: 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

lDt1 ¼�EI2 U001þfU002�ðU01U03Þ

0�

1

2f2U001�

1

3ðU031 Þ

0

� �0þðEI1�CÞf0ðU002�fU001 Þ�P

oZf0 þO ðU1Þ

5h i

lDt2 ¼�EI1ðU002�fU001 Þ

0�ðEI2�CÞf0U001�P

oU03þ

1

2U021

� �þO ðU1Þ

4h i

: ð4:9Þ

The displacements U2 and U3 at the left end are zero

U2 ¼U3 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 7L=2 ð4:11Þ

and zero rotation, ei¼Ei (i¼1,2,3), and therefore direction cosines aij¼dij, which are expressed in terms of displacementsand rotation as

U01 ¼ 0, U02 ¼ 0, fþcð2Þ þcð3Þ ¼ 0 at Z ¼ 7L=2, ð4:12Þ

where

cð2Þ ¼1

2

ZZ

0

ðU01U002�U001U02ÞdZ

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 tractioncondition at the right end Z¼L/2 is

Dt2 ¼DP, 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

U03þO ðU1Þ4

h i¼ 0

ðU002�fU001 Þ0þ

EI2�C

EI1f0U001þ

Po

EI1U03þ

DP

EI1þO ðU1Þ

4h i

¼ 0 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 themaximum twist f. The displacements and rotation are expanded to the power series of a, with f and U1 to the odd powersof a, and U2 and U3 to the even powers of a

f¼ afð0Þ þa3fð1Þ þ . . .

U1 ¼ aU1ð0Þ þa3U1ð1Þ þ . . .

U2 ¼ a2U2ð0Þ þ . . .

U3 ¼ a2U3ð0Þ þ . . ., ð4:15Þ

where terms higher than the 3rd power of displacement [i.e., o(a3)] are neglected in the postbuckling analysis. Similarly,DP is in the order of a2, and can be written as

DP¼ a2DPð0Þ þ . . .: ð4:16Þ

Substitution of the above two equations into the ODEs and boundary conditions gives

Cf00ð0Þ�Po

ZU001ð0Þ ¼ 0

EI2Uð4Þ1ð0Þ þPof0ð0Þ þðP

oZf0ð0ÞÞ

0¼ 0

fð0ÞZ ¼ 7L=2

¼U1ð0Þ

Z ¼ 7 L=2

¼U01ð0Þ

Z ¼ 7 L=2

¼ 0

,

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

ð4:17Þ

EI1ðU002ð0Þ�fð0ÞU

001ð0Þ Þ

00þEI2ðf

00

ð0ÞU001ð0Þ þ2f0ð0ÞU

000

1ð0ÞÞ�Cðf0ð0ÞU001ð0ÞÞ

0þP

o

U03ð0Þ þ12 U021ð0Þ

�0þP

o

Zf02ð0Þ ¼ 0

EA U03ð0Þ þ12 U021ð0Þ

�þ 1

2 EI2U0021ð0Þ

h i0�P

o

ðU002ð0Þ�fð0ÞU001ð0Þ ÞþP

o

Zf0ð0ÞU001ð0Þ ¼ 0

U2ð0Þ

Z ¼ �L=2

¼U3ð0Þ9Z ¼ �L=2 ¼U02ð0Þ9Z ¼ 7L=2 ¼U03ð0Þ

Z ¼ L=2

¼ 0

ðU002ð0Þ�fð0ÞU001ð0Þ Þ

0þ

EI2�CEI1

f0ð0ÞU001ð0Þ þ

1EI1

Po

U03ð0Þ

� �Z ¼ L=2

¼�DPð0ÞEI1

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

ð4:18Þ

for the leading powers of a, and

Cf00ð1Þ�PoZU001ð1Þ ¼ �ðEI1�EI2ÞU

001ð0ÞðU

002ð0Þ�fð0ÞU

001ð0Þ ÞþP

oZ fð0ÞU

002ð0Þ�ðU

01ð0ÞU

03ð0ÞÞ

0� 1

2f2ð0ÞU

001ð0Þ�

13 ðU

031ð0ÞÞ

0h i

EI2Uð4Þ1ð1Þ þPo

f0ð1Þ þðPo

Zf0ð1ÞÞ0¼ EI2

U000

1ð0Þ U03ð0Þ þ12 U021ð0Þ

�0þf02ð0ÞU

001ð0Þ

� fð0ÞU002ð0Þ�ðU

01ð0ÞU

03ð0ÞÞ

0� 1

2f2ð0ÞU

001ð0Þ�

13 ðU

031ð0ÞÞ

0h i00

8><>:

9>=>;

þEI1 f00ð0ÞðU002ð0Þ�fð0ÞU

001ð0Þ Þþ2f0ð0ÞðU

002ð0Þ�fð0ÞU

001ð0Þ Þ

0h i

�C f00ð0ÞðU002ð0Þ�fð0ÞU

001ð0Þ Þþf

0

ð0ÞðU002ð0Þ�fð0ÞU

001ð0Þ Þ

0þf02ð0ÞU

001ð0Þ

h i

þEAU001ð0Þ U03ð0Þ þ12 U021ð0Þ

�þP

oZf0ð0Þ U03ð0Þ þ

12 U021ð0Þ

�0½f1ð1Þ þc

ð2Þð0Þ þc

ð3Þð0Þ�9Z ¼ 7L=2 ¼U1ð1Þ9Z ¼ 7L=2 ¼U01ð1Þ

Z ¼ 7L=2

¼ 0

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

ð4:19Þ

for the next power of a, where

cð2Þð0Þ ¼

1

2

ZZ

0

ðU01ð0ÞU002ð0Þ�U001ð0ÞU

02ð0ÞÞdZ and cð3Þ

ð0Þ ¼ �1

4fð0Þ

2

3f2ð0Þ þU021ð0Þ

� �:

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508496

Eq. (4.17) constitutes an eigenvalue problem for f(0) and U1(0). Elimination of U1(0) yields the equation for f(0)

d

dZf000

ð0Þ�2

Zf00ð0Þ þ

ðPo

ZÞ2

EI2Cf0ð0Þ

24

35¼ 0: ð4:20Þ

It has two sets of solutions, corresponding to even and odd functions of Z, respectively, which are called symmetric andanti-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 evenwhen f(0) is odd.

For the beam with a narrow cross section such that EI1bEI2,C as in experiments, Eq. (4.18) can be significantlysimplified by neglecting EI2/(EI1) and C/(EI1)

ðU002ð0Þ�fð0ÞU001ð0Þ Þ

00¼ 0

U2ð0Þ

Z ¼ �L=2

¼U02ð0Þ

Z ¼ 7 L=2

¼ ðU002ð0Þ�fð0ÞU001ð0Þ Þ

0Z ¼ L=2

¼ 0

8><>:

U03ð0Þ þ12 U021ð0Þ

�0¼ 0

U3ð0Þ

Z ¼ �L=2

¼U03ð0Þ

Z ¼ L=2

¼ 0,

8>><>>: ð4:18aÞ

where the deformation at the onset of buckling is negligible since Po

and DP(0) are proportional toffiffiffiffiffiffiffiffiffiffiEI2C

p(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 aboveequation, together with f(0) and U1(0) being opposite even or odd functions, give U002ð0Þ ¼fð0ÞU

001ð0Þ and U03ð0Þ ¼ �U021ð0Þ=2. Its

solution is

U2ð0Þ ¼ �Z

Z L=2

Zfð0ÞU

001ð0ÞdZ1�

Z Z

0Z1fð0ÞU

001ð0ÞdZ1�

Z L=2

0Z1fð0ÞU

001ð0ÞdZ1

U3ð0Þ ¼ �1

2

Z Z

0U021ð0ÞdZ1�

1

2

Z L=2

0U021ð0ÞdZ1: ð4:21Þ

Similar to Eq. (3.21), the normality condition (or the existence condition for f(1) and U1(1)) gives DP(0). The underlinedterms in Eq. (4.21) clearly show the importance of the 4th power of displacement in the potential energy: thedisplacement 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

generality only half of the beam, 0rZrL/2, is considered. By introducing a new variable z¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPo=ffiffiffiffiffiffiffiffiffiffiEI2C

pqZ, integration of

Eq. (4.20) and f(0) being an even function give

d3fð0Þdz3

�2

zd2fð0Þ

dz2þz2 dfð0Þ

dz¼ 0: ð4:22Þ

It is a 2nd order ODE for df(0)/dz, and has the solution

dfð0Þdz¼ B1z

3=2J3=4z2

2

!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 andRyzhik, 2007). Integration of the above equation together with the boundary condition f(0)9Z¼L/2¼0 in Eq. (4.17) gives

fð0Þ ¼ �B1

Z zmax

zz3=2J3=4

z2

2

!dz¼ B1

ffiffiffiffiffiffiffiffiffiffizmax

pJ�1=4

z2max

2

!�

ffiffiffiz

pJ�1=4

z2

2

!" #, ð4:24Þ

where zmax ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPo=ffiffiffiffiffiffiffiffiffiffiEI2C

pqðL=2Þ corresponds to the end Z¼L/2. The parameter B1 is determined from maxðfð0ÞÞ ¼ 1 as

B1 ¼ffiffiffiffiffiffiffiffiffiffizmax

pJ�1

4

z2max2

��

ffiffi2p

Gð3=4Þ

h i�1

in terms of the G function (Gradshteyn and Ryzhik, 2007), to ensure the parameter a in

Eq. (4.15) to be the maximum angle of twist, fmax.Integration of the 1st equation in (4.17) gives U1(0) as

U1ð0Þ ¼ LB1

2zmax

ffiffiffiffiffiffiffiC

EI2

s ffiffiffiffiffiffiffiffiffiffizmax

pðzmax�zÞJ3=4

z2max

2

!�zZ zmax

z

1ffiffiffiz

p J3=4z2

2

!dz

" #: ð4:25Þ

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508 497

Its being an odd function requires U1(0)¼0 at Z¼0, which gives

J3=4z2

max

2

!¼ 0 ð4:26Þ

and has the solution zmax ¼ 2:642. The critical buckling load is then given by

Po¼ 27:93

ffiffiffiffiffiffiffiffiffiffiEI2C

pL2

: ð4:27Þ

The constant B1¼�0.5423. Eqs. (4.24) and (4.25) can then be written explicitly as

fð0Þ ¼ 0:3742þ0:5423ffiffiffiz

pJ�1

4

z2

2

!

U1ð0Þ

L¼ 0:1026

ffiffiffiffiffiffiffiC

EI2

szZ zmax

z

1ffiffiffiz

p J3=4z2

2

!dz¼

ffiffiffiffiffiffiffiC

EI2

s0:07986z�0:07879z3 J�1=4

z2

2

�H�3=4

z2

2

�þ0:03051z3J3=4

z2

2

�sð�7=4Þ,ð1=4Þ

z2

2

�264

375, 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

gapplied ¼U2jZ ¼ L=2�U2jZ ¼ �L=2

L¼�

2f2max

L

Z L=2

0Zfð0ÞU

001ð0ÞdZ ¼ 0:2444

ffiffiffiffiffiffiffiC

EI2

sf2

max : ð4:29Þ

It would give gapplied¼0 without the underlined terms, which clearly suggests the importance of higher-order terms in thecurvature in the postbuckling analysis.

For a rectangular section with height h 5 width b, EI1¼Ehb3/12, EI2¼Eh3b/12 and C¼Gh3b/3, where G is the shearmodulus. The maximum bending strain, which is reached at the two ends of the beam, is the sum of bending strains at theonset of buckling and during postbuckling, and is given by

ebending ¼b

2

PoðL=2Þ

EI1þ

h

2max U001

¼ h

L7:474

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigapplied

ffiffiffiffiG

E

rsþ13:97

h

b

ffiffiffiffiG

E

r0@

1A� 7:474

h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigapplied

ffiffiffiffiG

E

rs, ð4:30Þ

where the approximation on the 2nd line holds for fmaxbh=b. The maximum shear strain due to torsion is given by

gtorsion ¼ hmax f0 ¼ 6:606

h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigapplied

ffiffiffiffiE

G

rs: ð4:31Þ

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). Withoutlosing generality only half of the beam, 0rZrL/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)

d3fð0Þdz3

�2

zd2fð0Þ

dz2þz2 dfð0Þ

dz¼ B2: ð4:32Þ

It has the solution

dfð0Þdz¼ B2FðzÞþB3z

3=2J�3=4z2

2

!ð4:33Þ

for an odd function f(0), where the constant B3 is to be determined, and

FðzÞ ¼�1

3z2þ

1

4

ffiffiffiffiffiffi2pz

sG

3

4

� �z2H5=4

z2

2

!�H1=4

z2

2

!" #: ð4:34Þ

Its integration together with the boundary condition f(0)9Z¼L/2¼0 in Eq. (4.17) gives

fð0Þ ¼ �B2

Z zmax

zFðzÞdz�B3

ffiffiffiffiffiffiffiffiffiffizmax

pJ1=4

z2max

2

!�

ffiffiffiz

pJ1=4

z2

2

!" #: ð4:35Þ

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508498

Its being an odd function requires f(0)¼0 at Z¼0, i.e.

B2

Z zmax

0FðzÞdzþB3

ffiffiffiffiffiffiffiffiffiffizmax

pJ1=4

z2max

2

!¼ 0: ð4:36Þ

Integration of the 1st equation in (4.17) and the boundary condition U01ð0Þ

Z ¼ L=2

¼ 0 give

d

dzU1ð0Þ ¼ �

L

2zmax

ffiffiffiffiffiffiffiC

EI2

sB2

Z zmax

z

1

zdFðzÞ

dzdzþB3

Z zmax

z

1

zd

dzz

32J�3=4

z2

2

!" #dz

( ): ð4:37Þ

It is an odd function (since U1(0) is even), which requires U01ð0Þ ¼ 0 at Z¼0. This gives

B2

Z zmax

0

1

zdFðzÞ

dzdzþB3

Z zmax

0

1

zd

dzz3=2J�3=4

z2

2

!" #dz¼ 0: ð4:38Þ

Eqs. (4.36) and (4.38) give two linear, homogeneous equations for B2 and B3. Their determinant must be zero in order tohave a non-trivial solution, which gives the equation for the critical buckling loadR zmax

0 FðzÞdzffiffiffiffiffiffiffiffiffiffizmax

pJ1=4

z2max2

�R zmax

01z

dFðzÞdz dz

R zmax

01z

ddz z3=2 J�3=4

z2

2

�h idz

¼ 0: ð4:39Þ

It has the solution zmax ¼ 3:038. The critical buckling load is then given by

Po

¼ 36:92

ffiffiffiffiffiffiffiffiffiffiEI2C

pL2

, ð4:40Þ

which is larger than Eq. (4.27) for the symmetric mode.Integration of Eq. (4.37) and the boundary condition U1(0)9Z¼L/2¼0 give

U1ð0Þ ¼L

2zmax

ffiffiffiffiffiffiffiC

EI2

s B2zmax�zzmax

FðzmaxÞ�zR zmax

zFðzÞz2 dz

h iþB3

ffiffiffiffiffiffiffiffiffiffizmax

pðzmax�zÞ J�3=4

z2max2

��zR zmax

z1ffiffizp J�3=4

z2

2

�dz

� �8>><>>:

9>>=>>;: ð4:41Þ

The parameter a in Eq. (4.15) is the ratio of maximum displacement to L, i.e., a¼U1max/L, which, together withEq. (4.41), give

B2FðzmaxÞþB3 z3=2max J�3=4

z2max

2

!�

2

pG

3

4

� �" #¼ 2zmax

ffiffiffiffiffiffiffiEI2

C

r: ð4:42Þ

Eqs. (4.36) and (4.42) determine B2 ¼ 3:850ffiffiffiffiffiffiffiffiffiffiffiffiEI2=C

pand B3 ¼�1:166

ffiffiffiffiffiffiffiffiffiffiffiffiEI2=C

p. Eqs. (4.35) and (4.41) can then be written

explicitly as

fð0Þ ¼ �

ffiffiffiffiffiffiffiEI2

C

r0:7250þ3:850

Z 3:038

zFðzÞdzþ1:166

ffiffiffiz

pJ1=4

z2

2

!" #

U1ð0Þ

L¼ 0:8503�0:2799z�0:6337z

Z 3:038

z

FðzÞz2

dzþ0:1919zZ 3:038

z

1ffiffiffiz

p J�3=4z2

2

!dz: ð4:43Þ

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

gapplied ¼U2jZ ¼ L=2�U2jZ ¼ �L=2

L¼�

2U21max

L3

Z L=2

0Zfð0ÞU

001ð0ÞdZ ¼ 10:38

ffiffiffiffiffiffiffiEI2

C

rU1max

L

� �2

: ð4:44Þ

It would also give gapplied¼0 without the underlined terms, which confirms the importance of higher-order terms in thecurvature in the postbuckling analysis.

The maximum bending strain at the onset of buckling is ebending ¼ 18:46 h2=ðLbÞh i ffiffiffiffiffiffiffiffiffi

G=Ep

. The maximum bending strainand torsion (shear) strain due to postbuckling are

ebending ¼ 6:833h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigapplied

ffiffiffiffiG

E

rs, ð4:45Þ

gtorsion ¼ hmax f0 ¼ 8:632

h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffigapplied

ffiffiffiffiE

G

rs: ð4:46Þ

Their comparison with Eqs. (4.30) and (4.31) suggests that the anti-symmetric buckling mode gives a smaller bendingstrain but a larger shear strain than the symmetric buckling mode.

A B

C Db

LL

h

b

Y

ZP

ZP

ZY

ZX

h Y Z

XSubstrate

bh

islandL

2L

L

islandL

2L

symmetric anti-symmetric

stre

tchi

ngco

mpr

essi

ng

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.

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508 499

Fig. 3c shows the symmetric and anti-symmetric buckling modes in Sections 4.3 and 4.4, respectively. The appliednominal shear strain is gapplied¼20%, and C/(EI2)¼1.57. The buckling mode for the anti-symmetric appears to be rathersimilar 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 topure shear. For example, the bending and torsion strains in Eqs. (4.30), (4.31), (4.45) and (4.46) still hold if the appliednominal shear strain gapplied to the beam is replaced by g(LþLisland)/L, where g is the shear strain applied to the island-bridge, mesh structure, and Lisland is the island size (Fig. 4).

A

C D

b

o

ZP

BZ

Y

ot3

ot3

ot2

ot2Y

ot3ot3ot2

ot2

ot3

ot3

ot2

ot2

o

ZP

2

LZ +

Z

Fig. 5. Free-body diagram of the island-bridge, mesh structure prior to the onset of buckling.

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508500

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 squareislands of size Lisland are linked by bridges of length L. The islands are well attached to the substrate, but bridges are not, asseen from the side view in Fig. 4b. The bridges buckle to shield the islands from large deformation. Let Z denote the centralaxis of the bridge before deformation, with Z¼0 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 thefollowing.

5.1. State prior to the onset of buckling

Let PZ

denote the tension force (per unit length) along Z direction (Fig. 4a). There are no distributed force and moment,

p¼q¼0. The force, bending moment and torque in bridges t1

o¼m2

o¼m3

o¼ 0, and the non-zero forces and bending moment

t2

o, t3

oand m1

oare 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.

The forces t2

oand t3

oin the bridges along Z and Y directions satisfy the symmetry about Z and Y directions. Equilibrium of forces in

Z direction gives t2

oþt3

o�P

o

ZðLþLislandÞ ¼ 0. Similarly, force equilibrium in Y direction gives t2

o�t3

o¼ 0. These give

t2

o¼

1

2Po

ZðLþLislandÞ, t3

o¼

1

2Po

ZðLþLislandÞ: ð5:1Þ

The islands do not rotate due to symmetry about Z and Y directions. The bending moment m1o

can then be found as

m1o¼

1

2Po

ZðLþLislandÞZ for �L

2rZr

L

2: ð5:2Þ

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508 501

5.2. Equations for postbuckling analysis

Let Po

Z denote the critical tension at the onset of buckling, and DPZ the changes during postbuckling. Only thedisplacements along Z and Y directions are not zero prior to the onset of buckling. Therefore the orders of displacementsand 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

Cf00 þðEI1�EI2ÞU001ðU002�fU001 Þ�m1

oU001þfU002�ðU

01U03Þ

0�

1

2f2U001�

1

3ðU031 Þ

0

� �þO ðU1Þ

5h i

¼ 0, ð5:3Þ

EI2 Uð4Þ1 �U000

1 U03þ1

2U021

� �0�f02U001þ fU002�ðU

01U03Þ

0�

1

2f2U001�

1

3ðU031 Þ

0

� �00� �

�EI1 f00ðU002�fU001 Þþ2f0ðU002�fU001 Þ0

h iþC f00ðU002�fU001 Þþf

0ðU002�fU001 Þ

0þf02U001

h i�EAU001 U03þ

1

2U021

� �þt2

of0

�t3

oU001 1þU03þ

1

2U021

� �þfU002�ðU

01U03Þ

0�

1

2f2U001�

1

3ðU031 Þ

0

� �þðm1

of0Þ0�m1

of0 U03þ

1

2U021

� �0þO ðU1Þ

5h i

¼ 0, ð5:4Þ

EI1ðU002�fU001 Þ

00þEI2ðf

00U001þ2f0U000

1 Þ�Cðf0U001Þ0þt2

oU03þ

1

2U021

� �0�t3

oðU002�fU001 Þþm1

of02þO ðU1Þ

4h i

¼ 0, ð5:5Þ

EA U03þ1

2U021

� �þ

1

2EI2U0021

� �0�t2

oðU002�fU001 Þþm1

of0U001þO ðU1Þ

4h i

¼ 0, 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 Dmcan be obtained from Eqs. (2.13), (4.3) and (4.4). The shear forces are given by

lDt1 ¼�EI2 U001þfU002�ðU01U03Þ

0�

1

2f2U001�

1

3ðU031 Þ

0

� �0þðEI1�CÞf0ðU002�fU001 Þ�m1

of0 þO ðU1Þ

5h i

lDt2 ¼�EI1ðU002�fU001 Þ

0�ðEI2�CÞf0U001�t2

oU03þ

1

2U021

� �þO ðU1Þ

4h i

ð5:7Þ

Eqs. (5.3)–(5.7) hold for arbitrary equilibrium state to

and mo

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 ¼U3 ¼ 0 at Z ¼ 0 ð5:8Þ

The other boundary conditions (4.11) and (4.12) still hold. The traction conditions at the end Z¼L/2 are Dt2 ¼Dt3 ¼

DPZðLþLislandÞ=2, which can be expressed in terms of the displacements as

U03�DP

ZðLþLislandÞ

2EAþO ðU1Þ

4h i

¼ 0

ðU002�fU001 Þ0þ

EI2�C

EI1f0U001þ

t2

o

EI1U03þ

DPZðLþLislandÞ

2EI1þO ðU1Þ

4h i

¼ 0 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. Thedisplacements and rotation are expanded to the power series of a as in Eq. (4.15). Similarly, DPZ is written as

DPZ ¼ a2DPZð0Þ þ . . .: ð5:10Þ

Substitution of Eqs. (4.15) and (5.10) into Eqs. (5.3)–(5.6), (5.8) and (5.9) gives

Cf00ð0Þ�m1o

U001ð0Þ ¼ 0

EI2Uð4Þ1ð0Þ þt2

of0ð0Þ�t3

oU001ð0Þ þðm1

of0ð0ÞÞ

0¼ 0

fð0ÞZ ¼ 7L=2

¼U1ð0Þ

Z ¼ 7 L=2

¼U01ð0ÞjZ ¼ 7 L=2 ¼ 0,

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

ð5:11Þ

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508502

EI1ðU002ð0Þ�fð0ÞU

001ð0Þ Þ

00þEI2ðf

00

ð0ÞU001ð0Þ þ2f0ð0ÞU

0001ð0ÞÞ�Cðf0ð0ÞU

001ð0ÞÞ

0

þt2

oU03ð0Þ þ

12 U021ð0Þ

�0�t3

oðU002ð0Þ�fð0ÞU

001ð0Þ Þþm1

of02ð0Þ ¼ 0

EA U03ð0Þ þ12 U021ð0Þ

�þ 1

2 EI2U0021ð0Þ

h i0�t2

oðU002ð0Þ�fð0ÞU

001ð0Þ Þþm1

of0ð0ÞU

001ð0Þ ¼ 0

U2ð0ÞjZ ¼ 0 ¼U3ð0ÞjZ ¼ 0 ¼U02ð0ÞjZ ¼ 7 L=2 ¼U03ð0ÞjZ ¼ L=2�DP

Zð0ÞðLþ LislandÞ

2EA ¼ 0

ðU002ð0Þ�fð0ÞU001ð0Þ Þ

0þ

EI2�CEI1

f0ð0ÞU001ð0Þ þ

1EI1

t2

oU03ð0Þ

� �Z ¼ L=2

¼�DP

Zð0ÞðLþ LislandÞ

2EI1

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

ð5:12Þ

for the leading powers of a, and

Cf00ð1Þ�m1o

U001ð1Þ ¼ �ðEI1�EI2ÞU001ð0ÞðU

002ð0Þ�fð0ÞU

001ð0Þ Þþm1

ofð0ÞU

002ð0Þ�ðU

01ð0ÞU

03ð0ÞÞ

0� 1

2f2ð0ÞU

001ð0Þ�

13 ðU

031ð0ÞÞ

0h i

EI2Uð4Þ1ð1Þ þt2

of0ð1Þ�t3

oU001ð1Þ þðm1

of0ð1ÞÞ

0

¼ EI2 U0001ð0Þ U03ð0Þ þ12 U021ð0Þ

�0þf02ð0ÞU

001ð0Þ� fð0ÞU

002ð0Þ�ðU

01ð0ÞU

03ð0ÞÞ

0� 1

2f2ð0ÞU

001ð0Þ�

13 ðU

031ð0ÞÞ

0h i00n o

þEI1 f00ð0ÞðU002ð0Þ�fð0ÞU

001ð0Þ Þþ2f0ð0ÞðU

002ð0Þ�fð0ÞU

001ð0Þ Þ

0h i

�C f00ð0ÞðU002ð0Þ�fð0ÞU

001ð0Þ Þþf

0

ð0ÞðU002ð0Þ�fð0ÞU

001ð0Þ Þ

0þf02ð0ÞU

001ð0Þ

h iþEAU001ð0Þ U03ð0Þ þ

12 U021ð0Þ

�þt3

oU001ð0Þ U03ð0Þ þ

12 U021ð0Þ

�þfð0ÞU

002ð0Þ�ðU

01ð0ÞU

03ð0ÞÞ

0� 1

2f2ð0ÞU

001ð0Þ�

13 ðU

031ð0ÞÞ

0h i

þm1of0ð0Þ U03ð0Þ þ

12 U021ð0Þ

�0½fð1Þ þc

ð2Þð0Þ þc

ð3Þð0Þ� Z ¼ 7L=2 ¼U1ð1Þ

Z ¼ 7L=2

¼U01ð1Þ

Z ¼ 7 L=2

¼ 0

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

ð5:13Þ

for the next power of a, where

cð2Þð0Þ ¼

1

2

Z Z

0ðU01ð0ÞU

002ð0Þ�U001ð0ÞU

02ð0ÞÞdZ and cð3Þ

ð0Þ ¼ �1

4fð0Þ

2

3f2ð0Þ þU021ð0Þ

� �:

Elimination of U1(0) in Eq. (5.11) yields the equation for f(0)

d

dZf000ð0Þ�

2

Zf00ð0Þ þ

½Po

ZðLþLislandÞZ�2

4EI2Cf0ð0Þ�

Po

ZðLþLislandÞ

2EI2f0ð0Þ

8<:

9=;¼ 0 ð5:14Þ

for the state prior to the onset of buckling in Eqs. (5.1) and (5.2). The symmetric and anti-symmetric buckling modescorrespond 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 EI1bEI2,C as in experiments, Eq. (5.12) can be significantly simplifiedby neglecting EI2/(EI1) and C/(EI1)

ðU002ð0Þ�fð0ÞU001ð0Þ Þ

00¼ 0

U2ð0Þ

Z ¼ 0¼U02ð0Þ

Z ¼ 7 L=2

¼ ðU002ð0Þ�fð0ÞU001ð0Þ Þ

0Z ¼ L=2

¼ 0,

U03ð0Þ þ12 U021ð0Þ

�0¼ 0

U3ð0Þ

Z ¼ 0¼U03ð0Þ

Z ¼ L=2

¼ 0,

8>><>>:

8>><>>: ð5:12aÞ

where the deformation at the onset of buckling is negligible since Po

Z and DPZð0Þ are proportional toffiffiffiffiffiffiffiffiffiffiEI2C

p(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 U002ð0Þ ¼fð0ÞU001ð0Þ and U03ð0Þ ¼ �U021ð0Þ=2.

Its solution is

U2ð0Þ ¼ �Z

Z L=2

Zfð0ÞU

001ð0ÞdZ1�

Z Z

0Z1fð0ÞU

001ð0ÞdZ1

U3ð0Þ ¼ �1

2

Z Z

0U021ð0ÞdZ1: ð5:15Þ

For both symmetric or anti-symmetric buckling modes, U2(0) and U3(0) are always odd functions. Similar to Eq. (3.21), thenormality condition for Eq. (5.13) gives DP

Z 0ð Þ. The displacement U2(0) would be zero in Eq. (5.15) without accounting forthe 4th power of displacement in the potential energy.

4

6

8

10

12

compressing

ν

symmetric modeanti-sym. modestretching

0.0 0.50.40.30.20.1

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.

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508 503

5.4. Symmetric buckling mode

For an even function f(0) and odd function U1(0), only half of the beam, 0rZrL=2, is considered. Integration ofEq. (5.14) and f(0) being an even function give

d3fð0Þdz3

�2

zd2fð0Þ

dz2þz2 dfð0Þ

dz�a

dfð0Þdz¼ 0, ð5:16Þ

where z¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPo

Z

ðLþLislandÞ=ð2

ffiffiffiffiffiffiffiffiffiffiEI2C

pÞ

sZ, a¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC=ðEI2Þ

psignðP

o

ZÞ, signðPo

ZÞ ¼ 1 for tension and signðPo

ZÞ ¼�1 for compression.

The solution of the above equation is

dfð0Þdz¼ B4

ffiffiffiz

pRe Miða=4Þ,ð3=4Þðiz

2Þ

h ið5:17Þ

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

with indices ia/4 and 3/4 (Gradshteyn and Ryzhik, 2007), and i¼ffiffiffiffiffiffiffi�1p

. Integration of Eq. (5.17) together with the boundary

condition f(0)9Z¼L/2¼0 in Eq. (5.11) gives

fð0Þ ¼ �B4

Z zmax

z

ffiffiffiz

pRe½Miða=4Þ,ð3=4Þðiz

2Þ�dz, ð5:18Þ

where zmax ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiPo

Z

ðLþLislandÞL

2=ð8ffiffiffiffiffiffiffiffiffiffiEI2C

pÞ

scorresponds to the end Z¼L/2. The parameter B4 is determined from

maxðfð0ÞÞ ¼ 1 as B4 ¼�R zmax

0

ffiffiffiz

pRe½Miða=4Þ,ð3=4Þðiz

2Þ�dz

n o�1, which ensures the parameter a in Eq. (4.15) to be the maximum

angle of twist, fmax.Integration of the 1st equation in (5.11) gives U1(0) as

U1ð0Þ ¼ �LaB4

2zmaxzZ zmax

z

1

z3=2Re Miða=4Þ,ð3=4Þðiz

2Þ

h idz�

zmax�zffiffiffiffiffiffiffiffiffiffizmax

p Re Miða=4Þ,ð3=4Þðiz2maxÞ

h i( ): ð5:19Þ

Its being an odd function requires U1(0)¼0 at Z¼0, which gives the following simple equation for the critical bucklingload:

Re½Miða=4Þ,ð3=4Þðiz2maxÞ� ¼ 0, ð5:20Þ

which has the solution zmax ¼ zmaxðaÞ. The critical buckling load is

Po

Z

¼ 8

ffiffiffiffiffiffiffiffiffiffiEI2C

pðLþLislandÞL

2½zmaxðaÞ�2: ð5:21Þ

The normalized critical buckling load, Po

Z

ðLþLislandÞL

2=ð8ffiffiffiffiffiffiffiffiffiffiEI2C

pÞ, is shown versus the Poisson’s ratio n of the bridge in

Fig. 6, where a¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2=ð1þnÞ

psignðP

o

ZÞ 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.

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 Anti-symmetric

b3

symbending

bsymbending bsym

torsion b3

antibending

bantibending banti

torsion

n¼0.1 11.55 6.796 10.07 16.23 8.111 12.75

n¼0.2 10.96 6.622 10.20 15.43 7.775 12.94

n¼0.3 10.46 6.467 10.33 14.71 7.671 13.11

n¼0.4 10.01 6.328 10.45 14.11 7.262 13.29

n¼0.5 9.614 6.201 10.56 13.56 6.897 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 Anti-symmetric

b3

symbending

bsymbending bsym

torsion b3

antibending

bantibending banti

torsion

n¼0.1 7.570 5.503 6.216 7.290 4.639 7.328

n¼0.2 7.316 5.410 6.426 7.149 4.557 7.680

n¼0.3 7.087 5.324 6.623 7.016 4.481 8.008

n¼0.4 6.880 5.246 6.807 6.891 4.409 8.313

n¼0.5 6.691 5.173 6.981 6.773 4.342 8.600

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508504

Eq. (5.19) can then be simplified to

U1ð0Þ

L¼�B4

az2zmax

Z zmax

z

1

z3=2Re Miða=4Þ,ð3=4Þðiz

2Þ

h idz: ð5:22Þ

Eq. (5.15) then gives U2(0) and U3(0).The applied strain along Z (451) direction during postbuckling is

e453 ¼1

LþLisland

Z L=2

�L=2U03dZþ

Z L=2

�L=2U02dZ

!¼

Lf2max

LþLisland

azmax

Z zmax

0

dfð0Þdz

� �2

dz�2zmax

Z zmax

0

d

dzU1ð0Þ

L

� �� �2

dz

( ), ð5:23Þ

where the coefficient inside (von Karman and Tsien, 1941) depends only on a since zmax ¼ zmaxðaÞ.For a rectangular section with height h 5 width b, EI1¼Ehb3/12, EI2¼Eh3b/12 and C¼Gh3b/3. The maximum bending

strain is the sum of bending strains at the onset of buckling and during postbuckling, and is given by

ebending ¼h

Lbsym

bending

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

rþb

3symbending

h

b

!: ð5:24Þ

The maximum shear strain due to torsion is given by

gtorsion ¼ bsymtorsion

h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r: 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 forstretching 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 forstretching along 451 because Eq. (5.23) would give a compressive strain e451o0. For compressing along 451, the maximumbending 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, 0rZrL/2, is considered. Integration ofEq. (5.14) and f(0) being an odd function give

d3fð0Þdz3

�2

zd2fð0Þ

dz2þz2 dfð0Þ

dz�a

dfð0Þdz¼ B5, ð5:26Þ

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508 505

where B5 is a constant of integration to be determined. The solution of the above equation is

dfð0Þdz¼ B5DðzÞþB6

ffiffiffiz

pRe Miða=4Þ,ð�3=4Þðiz

2Þ

h ið5:27Þ

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 theWhittaker M function with indices ia/4 and �3/4 (Gradshteyn and Ryzhik, 2007). Integration of Eq. (5.27) together withthe boundary condition f(0)9Z¼L/2¼0 in Eq. (5.11) gives

fð0Þ ¼ �B5

Z zmax

zDðzÞdz�B6

Z zmax

z

ffiffiffiz

pRe½Miða=4Þ,ð�3=4Þðiz

2Þ�dz: ð5:28Þ

Its being an odd function requires f(0)¼0 at Z¼0, i.e.

B5

Z zmax

0DðzÞdzþB6

Z zmax

0

ffiffiffiz

pRe½Miða=4Þ,ð�3=4Þðiz

2Þ�dz¼ 0: ð5:29Þ

Integration of the 1st equation in (5.11) and the boundary condition U01ð0Þ

Z ¼ L=2

¼ 0 give

d

dzU1ð0Þ ¼ �L

a2zmax

B5

Z zmax

z

1

zdDðzÞ

dzdzþB6

Z zmax

z

1

zd

dz

ffiffiffiz

pRe Miða=4Þ,ð�3=4Þðiz

2Þ

h in odz

* +: ð5:30Þ

It is an odd function (since U1(0) is even), which requires U01ð0Þ ¼ 0 at Z¼0

B5

Z zmax

0

1

zdDðzÞ

dzdzþB6

Z zmax

0

1

zd

dz

ffiffiffiz

pRe Miða=4Þ,ð�3=4Þðiz

2Þ

h in odz¼ 0: ð5:31Þ

Eqs. (5.29) and (5.31) give two linear, homogeneous equations for B5 and B6. Their determinant must be zero in order tohave a non-trivial solution, which gives the equation for the critical buckling loadR zmax

0 DðzÞdzR zmax

0

ffiffiffiz

pRe Miða=4Þ,ð�3=4Þðiz

2Þ

h idzR zmax

01z

dDðzÞdz dz

R zmax

01z

ddz

ffiffiffiz

pRe Miða=4Þ,ð�3=4Þðiz

2Þ

h in odz

¼ 0: ð5:32Þ

Its solution zmax ¼ zmaxðaÞ 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

(normalized) critical buckling load, Po

Z

ðLþLislandÞL

2=ð8ffiffiffiffiffiffiffiffiffiffiEI2C

pÞ, 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 bucklingmode 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)9Z¼L/2¼0 give

U1ð0Þ

L¼�

a2zmax

B5 zZ zmax

z

DðzÞz2

dz�zmax�zzmax

DðzmaxÞ

" #þB6

zR zmax

z Re Miða=4Þ,ð�3=4Þðiz2Þ

h i1

z3=2 dz

�zmax�zffiffiffiffiffiffiffi

zmax

p Re Miða=4Þ,ð�3=4Þðiz2maxÞ

h i8><>:

9>=>;

* +: ð5:33Þ

The parameter a in Eq. (4.15) is the ratio of maximum displacement to L, i.e., a¼U1max/L, which, together withEq. (5.33), give

B5DðzmaxÞþB6

ffiffiffiffiffiffiffiffiffiffizmax

pRe Miða=4Þ,ð�3=4Þðiz

2maxÞ

h i�cos

p8

n o¼

2zmax

a : ð5:34Þ

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

e453 ¼U2

1max

ðLþLislandÞL

azmax

Z zmax

0

dfð0Þdz

� �2

dz�2zmax

Z zmax

0

d

dzU1ð0Þ

L

� �� �2

dz

( ): ð5:35Þ

The maximum bending strain at the onset of buckling is ebending ¼ b3

antibendingh2=ðLbÞ. The maximum bending strain and torsion

(shear) strain due to postbuckling are

ebending ¼ bantibending

h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r, ð5:36Þ

gtorsion ¼ bantitorsion

h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r: ð5:37Þ

Here the coefficients b’s are given in Tables 1 and 2, and are essentially independent of the Poisson’s ratio n. Withoutaccounting for the 4th power of displacement, postbuckling for stretching along 451 could not be studied because e451o0in Eq. (5.35), and the maximum bending and shear strains would be significantly overestimated for compressing along 451.

Y. Su et al. / J. Mech. Phys. Solids 60 (2012) 487–508506

Fig. 4c shows the symmetric and anti-symmetric buckling modes in Sections 5.4 and 5.5, respectively. The appliedstrain is e451 ¼ 710%UL=ðLþLislandÞ (for both stretching and compressing), and C/EI2¼1.57. The buckling mode for theanti-symmetric appears to be rather similar to the experimentally observed buckling profile in Fig. 1c for thestretching 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 thesymmetric buckling mode always gives smaller bending and torsion strains than anti-symmetric buckling. The maximumbending and torsion strains are then given by

ebending � 6:5h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r, gtorsion � 10

h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r, ð5:38Þ

where the bending strain at the onset of buckling is neglected, and the coefficient b’s for n¼0.3 are used since theirvariations are small (�5%) in Tables 1 and 2. These maximum strains are not reached at the same point in the bridge. Themaximum principal strain is the same as the maximum bending strain, i.e.

emax � 6:5h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r:

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 giveslarger bending strain and but smaller torsion strain than anti-symmetric buckling. The maximum bending and torsionstrains can then be bounded as

ebending r5:5h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r, gtorsionr8:6

h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r: ð5:39Þ

The maximum principal strain is the same as the maximum bending strain, i.e.

emaxr5:5h

L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiLþLisland

Le451j j

r:

For Euler buckling, Song et al. (2009) used both analytical and finite element methods to show that the maximum strainin the island is much smaller than that in the bridge, even at the stress and strain concentration at the island-bridgeinterface. 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 inthe 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 potentialenergy.

(iii)

The lateral buckling mode of an island-bridge, mesh structure subject to shear (or twist), which is used in stretchableelectronics, involves the Bessel Function of the first kind. Simple, analytical expressions are obtained for the criticalload at the onset of buckling in Eqs. (4.27) and (4.40), and for the maximum bending and torsion strains duringpostbuckling 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 WhittakerM function. Simple, analytical expressions are obtained for the maximum bending, torsion (shear) and principalstrains during postbuckling in Eq. (5.38).

Acknowledgments

The authors acknowledge support from NSF Grant nos. ECCS-0824129 and OISE-1043143, and DOE, Division ofMaterials Sciences Grant no.DE-FG02-07ER46453. The support from NSFC (Grant no. 11172146) and the Ministry ofEducation, 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

að3Þ11 ¼�12fU01U02�fc

ð2ÞþU021 U03, að3Þ12 ¼

14fðU

021 �U022 ÞþU01U02U03þc

ð3Þ,

að3Þ13 ¼12f

2U01þfU02U03�cð2ÞU02�U01 U023 �

12 ðU

021 þU022 Þ

h i,

að3Þ21 ¼14fðU

021 �U022 ÞþU01U02U03�c

ð3Þ, að3Þ22 ¼12fU01U02�fc

ð2ÞþU022 U03,

að3Þ23 ¼12f

2U02�fU01U03þcð2ÞU01�U02 U023 �

12 ðU

021 þU022 Þ

h i,

að3Þ31 ¼U01 U023 �12 ðU

021 þU022 Þ

h i, að3Þ32 ¼U02 U023 �

12 ðU

021 þU022 Þ

h i, að3Þ33 ¼U03ðU

021 þU022 Þ,

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

ðA:1Þ

kð3Þ1 ¼cð2ÞU001þ 12f

2U002�fðU001U03þU01U003ÞþU002

12 U021 þU022 �U023

�þ 1

2 U001U01U02�2U003U02U03,

kð3Þ2 ¼cð2ÞU002� 12f

2U001�fðU002U03þU02U003Þ�U001

12 U022 þU021 �U023

�� 1

2 U002U01U02þ2U003U01U03,

kð3Þ3 ¼ 0,

8>>>><>>>>:

ðA:2Þ

where

cð3Þ ¼Z Z

0�

1

2f0f2�

1

4fðU021 þU022 Þh i0

þðU001U02�U01U002ÞU03

� �dZ:

The orthogonal condition before Eq. (3.21) can be proven as

Z L=2

�L=2U001ð1Þ þ

Po

EI2U1ð1Þ

0@

1A00

U1ð0ÞdZ ¼

Z L=2

�L=2U001ð1Þ þ

Po

EI2U1ð1Þ

0@

1A00

U1ð0Þ� U001ð0Þ þPo

EI2U1ð0Þ

0@

1A00

U1ð1Þ

24

35dZ

¼ ðU000

1ð1ÞU1ð0Þ�U1ð1ÞU000

1ð0ÞÞ

L=2

Z ¼ �L=2�ðU001ð1ÞU

01ð0Þ�U01ð1ÞU

001ð0ÞÞ

L=2

Z ¼ �L=2þ

Po

EI2ðU01ð1ÞU1ð0Þ�U1ð1ÞU

01ð0ÞÞ

L=2

Z ¼ �L=2¼ 0: ðA:3Þ

The function D in Eq. (5.27) is given by

DðzÞ ¼ T1ðzÞZ z

z1

T2ðzÞdT1ðzÞ

dz T2ðzÞ�T1ðzÞ dT2ðzÞdz

dz�T2ðzÞZ z

0

T1ðzÞdT1ðzÞ

dz T2ðzÞ�T1ðzÞ dT2ðzÞdz

dz, ðA:4Þ

where

T1ðzÞ ¼ffiffiffiz

pRe½Miða=4Þ,ð3=4Þðiz

2Þ�, T2ðzÞ ¼

ffiffiffiz

pRe½Miða=4Þ,ð�3=4Þðiz

2Þ�, ðA:5Þ

which have the limits

T1ðzÞ ¼ z3 cos5p8

1þa

10z2þOðz4

Þ

h iand T2ðzÞ ¼ cos

p8

1�a2z2þOðz4

Þ

h ias z-0:

The lower limit of the 1st integral z1 in Eq. (A.4) is determined from

limz-0

d

dzzZ z

z1

T2ðzÞdT1ðzÞ

dz T2ðzÞ�T1ðzÞ dT2ðzÞdz

dz

" #¼ 0: ðA:6Þ

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