Chi Chen
Department of Mechanical and
Aerospace Engineering,
University of Miami,
Coral Gables, FL 33146
Weiming Tao
Institute of Applied Mechanics,
Zhejiang University,
Hangzhou 310027, China
Yewang Su
Department of Mechanical Engineering and
Department of Civil and
Environmental Engineering,
Northwestern University,
Evanston, IL 60208;
Center for Mechanics and Materials,
Tsinghua University,
Beijing 100084, China
Jian Wu
AML,
Department of Engineering Mechanics,
Tsinghua University,
Beijing 100084, China;
Center for Mechanics and Materials,
Tsinghua University,
Beijing 100084, China
Lateral Buckling of Interconnects
in a Noncoplanar Mesh Design
for Stretchable Electronics
Analytical models have been established to study the lateral buckling of interconnects
under shear in a noncoplanar mesh design for stretchable electronics. Analytical expressions are obtained for the critical load and buckling shape at the onset of buckling
by solving the equilibrium equations. The postbuckling behavior is studied by energy
minimization of the potential energy, including up to fourth power of the displacement. A
simple expression of the amplitude characterizing the deformation after buckling is
obtained. These results agree well with the finite element simulations without any parameter fitting. The models in this paper may provide a route to study complex buckling
modes of interconnects, such as diagonal compression/stretching involving both compression and shear. [DOI: 10.1115/1.4023036]
Keywords: lateral buckling, finite element analysis, energy method
Jizhou Song1
Department of Mechanical and
Aerospace Engineering,
University of Miami,
Coral Gables, FL 33146
e-mail: [email protected]
1
Introduction
Stretchable electronics is an emerging technology that enables
an electronic system with large stretchability. It has various applications, including flexible displays [1], electronic eye cameras
[2–4], biointegrated devices [5–7], electronic sensors for robotics
[8,9], and structural health-monitoring devices [10]. Organic
semiconductor materials can sustain relatively large deformation
and have been used to develop stretchable electronics, but their
electrical performance is relatively poor comparing to conventional inorganic semiconductor materials. Compatibility with
well-developed, high-performance inorganic semiconductor materials represents a key advantage in this area. The challenge is to
make these inorganic semiconductor materials stretchable without
sacrificing the electrical performance, since they are brittle and
cannot sustain large deformation. One successful approach is to
use noncoplanar mesh design [11] on a compliant substrate, which
is based on the interconnect-island concept.
Figure 1(a) schematically shows the noncoplanar mesh design,
where the islands are chemically bonded to the compliant
substrate, while the interconnects are loosely bonded. The interconnects between islands can buckle to accommodate the external
strains. When the compression (or the release of the prestretch in
the substrate) is applied along the interconnect direction, the interconnects undergo Euler buckling, which has been studied thor1
Corresponding author.
Manuscript received October 26, 2012; final manuscript received November 19,
2012; accepted manuscript posted November 22, 2012; published online May 23,
2013. Editor: Yonggang Huang.
Journal of Applied Mechanics
oughly [12]. When shear (Fig. 1(b)) or diagonal stretch (45 deg
from the interconnects) is applied, the interconnects may undergo
lateral buckling. Different from Euler buckling, where the
displacement takes the simple sinusoidal form, lateral buckling
involves large torsion and out-of-plane bending with a very
complex form of displacements. The solution of lateral buckling is
important to guide the design of stretchable electronics. Most of
existing results on lateral buckling [13,14], however, are based on
numerical methods, and closed-form solutions only exist for initial
buckling, but not for postbuckling. Recently, Su et al. [15] established a systematic method based on the nonlinear equilibrium
equations for postbuckling of beams that may involve rather complex buckling modes, such as later buckling. The perturbation
method is used to obtain the amplitude of buckled interconnects.
The objective of this paper is to derive analytical solutions for
Fig. 1 (a) A schematic diagram of noncoplanar mesh design
with islands chemically bonded to the compliant substrate and
the interconnects loosely bonded; (b) SEM image of noncoplanar mesh design under shear
C 2013 by ASME
Copyright V
JULY 2013, Vol. 80 / 041031-1
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Fig. 2 A schematic diagram of lateral buckling of a beam
lateral bucking (initial buckling and postbuckling) of interconnects
under shear. The initial buckling of interconnects is investigated
through the equilibrium equations to find the critical buckling load
and buckling shape. The postbuckling behavior is studied by
energy minimization of the potential energy, including up to fourth
power of the displacement. The paper is outlined as follows: The
initial buckling analysis is described in Sec. 2, while the postbuckling analysis is presented in Sec. 3. Section 4 describes the finite
element simulations, and the results are discussed in Sec. 5.
The deflected beam’s constitutive equations are
8
d2 v
>
>
EIn 2 ¼ Mn
>
>
>
dz
>
<
d2 u
EIg 2 ¼ Mg
>
dz
>
>
>
>
>
: C d/ ¼ M1
dz
2
where C ¼ Ea3 b=½6ð1 þ Þ is the torsional rigidity, E is the
Young’s modulus, and is the Poisson’s ratio, and In ¼ ab3 =12
and Ig ¼ a3 b=12 are the moment of inertia about axes n and g,
respectively. The boundary conditions are
Initial Buckling Analysis
2.1 Equations for Initial Buckling. The interconnect can be
modeled as an L-long beam with a thin rectangular cross-section
a b, where b a. The left end is fixed, and the right end is subjected to a shear or vertical displacement v0 with no rotations (Fig.
2) (due to the rigid island). To investigate the initial lateral buckling of the beam, we start from the equilibrium equations [16].
Two sets of coordinate axes (x, y, z) and (n, g, f) are defined as
shown in Fig. 2. The first set of coordinate axes (x, y, z) is fixed in
space with the axis z coinciding with the centroidal axis of the
undeformed beam and the x and y coinciding with the principal
axes of the cross-section. The second is a set of moving axes (n, g,
f) fixed to a point on the centroidal axis of the beam and moves
with it. The axes n and g are the principal axes of the cross-section
on the deformed beam, and f coincides with the tangent to the
deformed centroidal axis of the beam. The deformation of the
beam is defined by the components u, v, and w of the displacement
of the center in the x, y, and z directions and by the rotation / of
the cross-section (Fig. 2). Components u and v are taken positive
in the positive direction of the corresponding axes, and the angle /
is taken positive about the z axis according to the right-hand rule.
The bending moments can be easily obtained as
8
Mx ¼ py z
>
>
>
<
L
(1)
My ¼ my þ px
z
>
2
>
>
:
Mz ¼ t py u þ px v
where py denotes the load in the y direction to cause v0 on the right
end, px the load in the x direction on the right end, my the bending
moment in the y direction on the right end, and t the torque in the
z direction on the right end. The following orders of forces and
displacement hold during buckling: the applied force and displacement (i.e., py and v) are zero order, while the corresponding
force, moment, and displacement (i.e., px, my, t, u, and /) resulting from lateral buckling are first order [15]. Projecting the above
moments to the axes (n, g, f) and neglecting any terms higher than
the second order yields
8
Mn ¼ py z
>
>
>
>
>
L
<
Mg ¼ py z/ þ my þ px
z
(2)
2
>
>
>
du
dv
L
>
>
: Mf ¼ py z þ px
z þ t py u þ px v
dz dz
2
041031-2 / Vol. 80, JULY 2013
8
0
0
>
< uðL=2Þ ¼ u ðL=2Þ ¼ uðL=2Þ ¼ u ðL=2Þ ¼ 0
vðL=2Þ ¼ v0 ðL=2Þ ¼ v0 ðL=2Þ ¼ 0 vðL=2Þ ¼ v0
>
:
/ðL=2Þ ¼ /ðL=2Þ ¼ 0
(3)
(4)
By introducing a nondimensional coordinate s ¼ 2z=L, where
1 s 1, Eqs. (3) and (4) become
8
3
>
d2 v
L
>
> EIn
¼
py s
>
2
>
ds
2
>
>
>
<
d2 u
L 3
2
(5)
EI
m
¼
p
s/
þ
þ
p
ð
1
s
Þ
g
y
y
x
>
ds2
2
L
>
>
>
>
> d/ L
du dv
>
>
:C
¼
py s þ px ð1 sÞ þ t py u þ px v
ds
2
ds ds
and
8
0
0
>
< uð1Þ ¼ u ð1Þ ¼ uð1Þ ¼ u ð1Þ ¼ 0
vð1Þ ¼ v0 ð1Þ ¼ v0 ð1Þ ¼ 0vð1Þ ¼ v0
>
:
/ð1Þ ¼ /ð1Þ ¼ 0
(6)
After eliminating u from the second and third equations in Eq. (5)
and applying In Ig , we obtain
4
4
p2y
py
d2 /
L
L
2
2
m
þ
s
/
¼
s
þ
p
ð
1
s
Þ
y
x
ds2 CEIg 2
2 CEIg L
(7)
The solution / of Eq. (7) is either symmetric or antisymmetric.
2.2 Symmetric Buckling Mode. The symmetric buckling
mode corresponds to an even function for / and an odd function
for u. Integration of the second part of Eq. (5) from 1 to 1 gives
2my =L þ px ¼ 0. Then, Eq. (7) becomes
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4
4
p2y
d2 /
L
L px py 2
2
þ
s
/
¼
s
ds2 CEIg 2
2 CEIg
The above equation can be solved as
2 2 ffiffiffiffi
p
bL 2
bL 2
px
4
/ðsÞ ¼ s2 AJ1=4
s þ BJ1=4
s
8
8
py
(8)
(9)
where A and B are constants to be determined by boundary
conditions,
J is the Bessel function of the first kind, and b
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
¼ p2y = EIg C . After applying the boundary conditions /0 ðsÞ ¼ 0
and /ð1Þ ¼ 0, /ðsÞ in Eq. (9) becomes
2 pffiffiffiffi
2 bL 2
bL
4
/ðsÞ ¼ A s2 J1=4
s J1=4
(10)
8
8
The displacement uðsÞ can then be obtained from the second part
of Eq. (5) as
uðsÞ ¼ 2 ð ð f qffiffiffiffiffi
Apy L 3 s
bL 2
4
n n2 J1=4
n dn df
EIg 2
8
1
1
The boundary uð1Þ ¼ 0 yields
2
bL
J3=4
¼0
8
(11)
(12)
which can be solved for L2 bcr =8. The critical load for lateral
buckling is then given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
pffiffiffiffiffiffiffiffiffiffi
Ea3 b
1
cr
b
(13)
py ¼ EIg Cbcr ¼
6
2ð1 þ Þ cr
The displacement vðsÞ can be obtained from the first part of
Eq. (5) and second part of Eq. (6) as
vðsÞ ¼ p y L3 3
s 3s 2
48EIn
(14)
The corresponding critical displacement vcr
0 is then given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L3 pcr
L3 a2 bcr
1
y
cr
(15)
v0 ¼
¼
2ð1 þ Þ
12EIn
6b2
By introducing a nondimensional parameter, a ¼ bcr L2 , the buckling shape is characterized by /ðsÞ in Eq. (10) and uðsÞ in
Eq. (11), which become
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
L
1
uðsÞ
(16)
/ðsÞ ¼ A/ðsÞ; and uðsÞ ¼ A
4 2ð1 þ Þ
where /ðsÞ and uðsÞ are independent of the materials and geometries, and they only depend on the nondimensional coordinate s as
a
a p
ffiffiffiffi
4
; and
/ðsÞ ¼ s2 J1=4 s2 J1=4
8
8
ð f
ðs
qffiffiffiffiffi
a 4
n n2 J1=4 n2 dn df
uðsÞ ¼ a
(17)
8
f¼1
n¼1
We refer A as the amplitude in this paper, which can be obtained
by postbuckling analysis in Sec. 3.
2.3 Antisymmetric Buckling Mode. The antisymmetric
buckling mode corresponds to an odd function for / and an even
function for u. From Eq. (7), px ¼ 0. Then, Eq. (7) becomes
4
3
p2y
d2 /
L
L py my
2
þ
s
/
¼
s
ds2 CEIg 2
2 CEIg
The solution of Eq. (19) is obtained as
2 2 pffiffi
bL 2
bL 2
/ðsÞ ¼ s AJ1=4
s þ BJ1=4
s
8
8
2 3
L
b
bL 2
pffiffiffiffiffiffiffiffiffiffi my s3 /p
s
þ
2
8
CEIg
3
2 2 2 ð 1
pffiffi
bL
bL 2
2 7
4
s sJ1=4
s ds
bL 2 7
8
8
s¼0
3
þ
s 7 (21)
2 s /p
2 2 ð 1
5
8
bL
bL 2
4
14
s /p
s ds
8
8
s¼0
Journal of Applied Mechanics
(19)
where /p ð xÞ is
8
9
9=4
>
>
823=4 x1 F2 ð3=4; 5=4; 7=4; x2 =4ÞJ1=4 ðxÞCð3=4Þ
>
>
>
>
=
pffiffiffi 7=4
1 < pffiffiffi 2
/ p ð xÞ ¼ J
ðxÞJ
ðxÞ
þ
3
2
px
J
ðxÞJ
ðxÞS
ðxÞ
6
2
px
1=4
1=4
1=4
3=4
1=4;7=4
>
48x2 >
>
>
pffiffiffi
>
>
: pffiffiffi 3=4
;
9 2px J1=4 ðxÞJ3=4 ðxÞS5=4;3=4 ðxÞ þ 6 2px7=4 J1=4 ðxÞJ1=4 ðxÞS5=4;3=4 ðxÞ
F is the hypergeometric function, C is the c function, and S is the
Lommel function. It should be noted that Eq. (19) is only valid for
0 s 1. The expression for 1 s 0 can be easily obtained
as /ðsÞ, since / is an odd function. After applying the boundary conditions /ð0Þ ¼ 0 and u0 ð1Þ ¼ 0, /ðsÞ becomes
2
2 6pffiffi
bL 2
/ðsÞ ¼ A6
s
J
s
1=4
4
8
(18)
(20)
From /ð1Þ ¼ 0, we have
J1=4
bL
8
2
2 2 2 ð 1
pffiffi
bL
bL 2
4
s sJ1=4
s ds 2 bL
8
8
s¼0
þ
¼0
2 /p
2 2 ð 1
8
bL
bL
4
2
14
s /p
s ds
8
8
s¼0
(22)
which can be solved for L2 bcr =8. The critical load in Eq. (13)
and critical displacement in Eq. (15) for symmetric mode
still hold for antisymmetric mode. The buckling shape can also
be given by Eq. (16), except /ðsÞ and uðsÞ are different and
given by
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a
2 ð 1 pffiffi
a 4
s sJ1=4 s2 ds
a a p
ffiffi
8 s¼0
8
/ðsÞ ¼ sJ1=4 s2 þ
s3 /p s2
ð1
2
8
8
a
a
s4 /p s2 ds
14
8 s¼0
8
(23)
C1 ¼
C3 ¼
3
Postbuckling Analysis
u1 ðx; y; zÞ ¼ uðzÞ /ðzÞy
v1 ðx; y; zÞ ¼ vðzÞ þ /ðzÞx
(25)
0
w1 ðx; y; zÞ ¼ wðzÞ u ðzÞx v ðzÞy
The longitudinal and shear strains due to buckling can be calculated by
8
@w1 1 @u1 2 1 @v1 2 1 @w1 2
>
>
>
¼
þ
þ
þ
e
< zz
2 @z
2 @z
2 @z
@z
>
1
@v
@w
@u
@u
@v
@w1 @w1
>
1
1
1
1
1 @v1
>
: eyz ¼
þ
þ
þ
þ
2 @z
@y
@y @z
@y @z
@y @z
(26)
The strain energy of the beam can be obtained by [18]
Ustrain ¼
ð L 2
ð 1
1
d/
r0z ezz þ Ee2zz þ r0yz 2eyz dV þ C
dz
2
2 z¼0 dz
V
(27)
where r0z ¼ py yz=In is the initial normal stress in the cross-section
due to bending, r0yz ¼ py =ðabÞ is the initial shear stress, py
pffiffiffiffiffiffiffiffiffiffi
¼ EIg Cbcr , C is the torsional rigidity, and V is the volume of
the beam. The displacement v(z) is negligible
Ð due to buckling.
The force equilibrium in z direction A Eezz dA ¼ 0 gives
1 02 2
w0 ¼ 12 u02 24
/ ða þ b2 Þ. Substituting Eq. (16) to the energy
expression in Eq. (27) and ignoring the terms higher than A4, the
energy then becomes
Ustrain
u00 ðsÞ2 ds;
C4 ¼
ð1
ð1
u0 ðsÞ2 /0 ðsÞ2 ds;
(29)
s¼1
/ðsÞ
u0 ðsÞds;
C6 ¼
ð1
/0 ðsÞ2 ds
s¼1
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1 u
aðC1 þ C5 Þ
u
pffiffiffiffiffiffiffiffiffiffiffi
A ¼ pffiffiffi tpffiffiffi
2C
2
7C
b
4
L
2 452 L2 þ 12að1þ
1þ
Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
16C6 þ a2 C3
a2 L
pffiffiffiffiffiffiffiffiffiffiffi
v0 pffiffiffi
2
12 2ðC1 þ C5 Þa b 1 þ 4
0
ð1
/0 ðsÞ4 ds;
Minimization of the energy yields
The postbuckling behavior is studied by energy minimization
of the potential energy of the beam. The displacements due to
buckling at neutral axis are uðzÞ; vðzÞ; and wðzÞ. Then, the general
point at a section has the following displacements by ignoring the
warping as [17,18]
>
:
ð1
s¼1
s¼1
2
8
>
<
C2 ¼
s¼1
a ð 1 pffiffiffi
a 9
>
a
>
1 n4 /p n2
c cJ1=4 c2 dc>
=
8
8
16
0
dndf (24)
ð
2 1
>
a
a
>
>
1
c4 /p c2 dc
;
8
16 0
s
u0 ðsÞ/0 ðsÞds;
s¼1
C5 ¼
8
>
ðs
ðf >
< pffiffiffi
a uðsÞ ¼ a
n nJ1=4 n2
8
f¼1 n¼0 >
>
:
ð1
"
rffiffiffiffiffiffiffiffiffiffiffi
Ea3 b
24ab2 v0 1 þ ¼A
ðC1 þ C5 Þ þ 16C6
96ð1 þ ÞL
2
a2 L
#
Eab5 7
a2 L2
2
C
(28)
þ a C3 þ A 4
þ
C
2
4
8L3 45
12ð1 þ Þb2
(30)
Finite Element Analysis
The finite element method is also used to study the lateral buckling of the beam in order to validate the analytical solutions. The
element B33 in ABAQUS is used to discretize the beam with left end
fixed and right end subjected to a vertical displacement. We first
determine the eigenvalues and eigenmodes from the buckling
analysis. The eigenmode is then used as initial small geometrical
imperfection to trigger the buckling in the postbuckling analysis.
It should be noted that the imperfections are always small enough
to ensure that the results are independent of the imperfections.
5
Results and Discussion
We take a beam with L ¼ 20, a ¼ 1, b ¼ 0.1, and ¼ 0.3 as an
example to show our results. Equations (12) and (15) give the critical displacement for the symmetrical buckling modes. The first
two roots of Eq. (12) are obtained as L2 bcr =8 ¼ 3:491 and
L2 bcr =8 ¼ 6:653, which yield the critical displacements
vcr
0 ¼ 0:5773 and 1.1002 for the first and second symmetrical
buckling modes, respectively. Similarly, Eqs. (15) and (22) give
the critical displacement for the antisymmetric buckling modes.
The first two roots of Eq. (22) are L2 bcr =8 ¼ 4:6146 and
L2 bcr =8 ¼ 6:0476, which yield the critical displacements
vcr
0 ¼ 0:7632 and 1.0001. According to the magnitude of critical
displacement, mode 1 for lateral buckling is symmetric, mode 2
is antisymmetric, mode 3 is antisymmetric, and mode 4 is symmetric. The buckling shapes can then be obtained by Eq. (17) for
symmetric mode and Eqs. (23) and (24) for antisymmetric mode.
Table 1 shows the critical displacements for these four modes of
lateral buckling, which are compared to those from eigenvalue
buckling analysis in the finite element software ABAQUS [19]. The
analytical predictions agree well with numerical simulations with
an error less than 3%. The buckling shapes of rotation of the
cross-section and displacement of center in x direction, /ðsÞ and
uðsÞ, are shown in Figs. 3 and 4, respectively. The results from
finite element simulations are also shown for comparison and
validation. The analytical predictions (solid line) have good agreements with the finite element simulations (dot).
2
where C1–C6 are constants, which only depend on the buckling
shapes and are given by
041031-4 / Vol. 80, JULY 2013
Table 1 Critical displacements for lateral buckling of a beam
with a 5 0.1, b 5 1, L 5 20
cr v
0cr analytical
v0 FEM
Mode 1
Mode 2
Mode 3
Mode 4
0.5773
0.7632
1.0001
1.1002
0.5613
0.7420
0.9732
1.0708
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vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1:3336 u
1
A ¼ pffiffiffi u
u
b2 0:6584 pffiffiffiffiffiffiffiffiffiffiffi
L t
3:4001 2 þ
1þ
1þ
L
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
a2 L
v0 3:2905 2 pffiffiffiffiffiffiffiffiffiffiffi
b 1þ
Let us consider the postbuckling for mode 1 with L2 bcr =8
¼ 3:491. a ¼ bcr L2 is then obtained as 27.9280. The constants
C1–C6 can be calculated from Eq. (29) as given by C1
¼ 6:323e 2, C2 ¼ 2:186e 1, C3 ¼ 8:997e 2, C4 ¼ 1:013e 2,
C5 ¼ 2:682e 2, and C6 ¼ 4:390. The amplitude A in Eq. (30)
then becomes
For b L, if b L, A can be simplified as
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
p
ffiffiffiffiffiffiffiffiffiffiffi
v0
1 a2
4
3:2905 pffiffiffiffiffiffiffiffiffiffiffi 2
A ¼ 1:6435 1 þ L
1þb
Fig. 3
The buckling shape of rotation of the cross-section /ðs Þ
(31)
(32)
Figure 5 shows the amplitude A versus the applied displacement
v0 for the buckling mode 1 of a beam with L ¼ 20, a ¼ 1, b ¼ 0.1,
and ¼ 0.3. The amplitude A remainspzero
v0 is smaller
ffiffiffiffiffiffiffiffiffiffiffiwhen
2
1 þ b2 ¼ 0:5773, and
than a critical value, vcr
0 ¼ 3:2905a L=
buckling does not occur. This critical value is consistent with that
(Eq. (15)) from initial buckling analysis. Once v0 exceeds the critical value, lateral buckling occurs, and the amplitude increases as
v0 increases. To validate the analytical solution, a postbuckling
analysis is performed in ABAQUS. The buckling modes are first
obtained through the eigenvalue buckling analysis and then used
as initial small geometrical imperfections to trigger the buckling
of the interconnect. It should be noted that the imperfection should
be small enough to ensure that the solution is accurate. As shown
in Fig. 5, the finite element results agree well with analytical
solutions.
6
Concluding Remarks
We have derived analytical solutions for lateral buckling of
interconnects under shear in a noncoplanar mesh design for
stretchable electronics. The critical buckling load and mode are
obtained analytically by solving the equilibrium equations, and
they agree well with finite element simulations. The postbuckling
behavior is studied by energy minimization of the potential
energy, including up to fourth power of the displacement. A simple expression of the amplitude to characterize the magnitude of
deformation is obtained, and it agrees well with the finite element
simulations without any parameter fitting. The models in this paper may provide a route to study complex buckling modes of
interconnects, such as diagonal compression/stretching involving
both compression and shear.
Fig. 4 The buckling shape of displacement of center in x direction u ðs Þ
Acknowledgment
This work was supported by the Provost Award (University of
Miami) and NSFC (Grant No.10972194).
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