## Abstract

The use of cellular elastomer substrates not only reduces its restriction on natural diffusion or convection of biofluids in the realm of stretchable electronics but also enhances the stretchability of the electronic systems. An analytical model of “zigzag” cellular substrates under finite deformation is established and validated in this paper. The deformed shape, nonlinear stress–strain curve, and Poisson’s ratio–strain curve of the cellular elastomer substrate calculated using the reported analytical model agree well with those from finite element analysis (FEA). Results show that lower restriction on the natural motion of human skin could be achieved by the proposed zigzag cellular substrates compared with the previously reported hexagonal cellular substrates, manifesting another leap toward mechanically “invisible” wearable, stretchable electronic systems.

## 1 Introduction

Significant progress has been achieved in materials [13] and mechanics [416] in offering the capability for stretchable electronics to be deformed into complex shapes without failure in functionality or structure. A recent direction of devising mechanically “invisible” skin-mounted stretchable electronics [1719], which are ultrasoft and hardly detectable by skin through tactile sensation, demands a new class of compliant elastomer substrates. Like many biological materials, e.g., skin [20,21] and viscid spider silk [22,23], which show “J-shaped” stress–strain behaviors as a result of collagen microstructures, cellular elastomer substrates offer similar behavior [24,25]. The integration of stretchable electronics onto cellular substrates could achieve a similar “J-shaped” stress–strain response as in the biological materials, with a compliant mechanical behavior at low stretching strains and an increasingly stiff response at larger stretching strains to balance wear comfort and structural integrity, shows promising application potential in bio-integrated electronic microsystems and tissue engineering [2631].

Apart from “J-shaped” stress–strain behavior and high permeability of biofluids [25,32], cellular substrates could achieve much larger stretchability compared with counterpart uniform solid substrates [33]. Some analytical models have been established to investigate the equilibrium and deformation compatibility of hexagonal cellular substrates under finite stretching [3437]. The stretching behaviors of substrates affect the interfacial shear stress between substrate and human skin, which is one of the key considerations when improving the wear comfort of stretchable, epidermal electronics. The shear stress between hexagonal cellular substrates and human skin is usually higher than the skin sensitivity threshold under finite deformation [37]. In this paper, “zigzag” cellular substrates are investigated and engineered to achieve higher compliance than the previously reported hexagonal cellular substrates, inducing lower interfacial shear stress under finite deformation.

An analytical model to investigate the zigzag cellular substrates with finite deformation is presented in this paper. Results show that the stress–strain relationship calculated by the analytical model matches that from finite element analysis (FEA). The interfacial shear stress between zigzag cellular substrates and human skin is well below the skin sensitivity threshold (20 kPa) [38] in any stretching direction and is noticeably lower than that for the hexagonal cellular substrates [34,35].

## 2 Analytical Model

The walls of zigzag cellular substrate shown in Fig. 1(a) can be modeled as beams when the length of cellular beam segment l is much larger than its width δ. All beam segments in the zigzag substrate are assumed to have the same length l in this study. The thickness of zigzag cellular substrates is assumed to be larger than the width of walls. For uniaxial stretching along x-/y-axis, the basic unit is denoted in the dashed rectangle in Fig. 1(a). 1/8 of the basic unit is chosen for analysis on account of symmetry as presented in Fig. 1(b). A local coordinate system ${x¯i,y¯i}$ is constructed as shown in Fig. 1(b). The $x¯i$-axis is parallel to the beams of zigzag cellular substrate, which is also illustrated in Fig. 2. The stretching force per unit thickness acting on the 1/8 basic unit can be denoted as P. The equivalent nominal stress is normalized by the initial modulus of substrate material E0 as
$σ¯=P(2sinα1+sinα2)E0l$
(1)
where αi is the rotation angle between the local coordinate system and the global coordinate system (i = 1 or 2 denotes beam AB and BC, respectively). The incompressible Mooney–Rivlin constitutive relation [34,39] gives the uniaxial stress–strain relationship $σ=E0f(ε)$, where f(ɛ) is
$f(ε)=115(4+11+ε)[1+ε−1(1+ε)2]$
(2)
Fig. 1
Fig. 1
Close modal
Fig. 2
Fig. 2
Close modal
The equilibrium equations of the beams in Fig. 2 give
$dQidsi−Nidφidsi=0$
(3a)
$dNidsi+Qidφidsi=0$
(3b)
where Qi is the shear force, Ni is the axial force, and φi is the angle between the beam axial direction and $x¯i$-direction. Let S and s represent the axial length of the beam segment in zigzag cellular substrate before and after stretching, respectively. The relationship between dsi and dSi is
$dsidSi=1+f−1(NiE0δ)$
(4)
The moment balance of the beam gives
$dMidsi=Qi,ordMidSi=[1+f−1(NiE0δ)]Qi$
(5)
where Mi is the moment. The axial force Ni and shear force Qi are related to P and φi as
$Ni=Pcos(αi−φi)$
(6a)
$Qi=−Psin(αi−φi)$
(6b)
The moment–curvature relation gives
$Mi=E0Idφi/dSi$
(7)
where E0I is the in-plane bending stiffness.
Substitution of Eqs. (6) and (7) into the moment equilibrium Eq. (5) gives
$d2φidSi2=−PE0Isin(αi−φi){1+f−1[Pcos(αi−φi)E0δ]}$
(8)
For beam AB, the moment at point B is denoted by M0, and the moment condition at point B is
$(dφ1/dS1)|S1=l=M0E0I$
(9)
Substitution of Eq. (9) into Eq. (8) gives
$(dφ1dS1)2=−2PE0I∫α1−β1φ1sin(α1−x){1+f−1[Pcos(α1−x)E0δ]}dx+(M0E0I)2$
(10)
where $β1=α1−φ1|S1=l$ is determined from the following equation by integrating Eq. (10) from S1 = 0 to S1 = l
$l=∫0α1−β1dφ1−2PE0I∫α1−β1φ1sin(α1−x){1+f−1[Pcos(α1−x)E0δ]}dx+(M0E0I)2$
(11)
For beam BC, the moment condition at point C is
$(dφ2/dS2)|S2=l/2=0$
(12)
Substitution of Eq. (12) into Eq. (8) gives
$(dφ2dS2)2=−2PE0I∫α2−β2φ2sin(α2−x){1+f−1[Pcos(α2−x)E0δ]}dx$
(13)
where $β2=α2−φ2|S2=l/2$ is determined from the following equation by integrating Eq. (13) from S2 = 0 to S2 = l/2
$l2=∫α1−β1α2−β2dφ2−2PE0I∫α2−β2φ2sin(α2−x){1+f−1[Pcos(α2−x)E0δ]}dx$
(14)
where the lower limit of integration is the rotation angle, at point B, of α1β1 as shown in Fig. 2(b). The local coordinates of points B and C can be obtained from $dx¯/dsi=cosφi$ and $dy¯/dsi=sinφi$, as
${x¯By¯B}=∫0α1−β1{1+f−1[Pcos(α1−φ1)E0δ]}{cosφ1sinφ1}dφ1−2PE0I∫α1−β1φ1sin(α1−x){1+f−1[Pcos(α1−x)E0δ]}dx+(M0E0I)2$
(15a)
${x¯Cy¯C}=∫α1−β1α2−β2{1+f−1[Pcos(α2−φ2)E0δ]}{cosφ2sinφ2}dφ2−2PE0I∫α2−β2φ2sin(α2−x){1+f−1[Pcos(α2−x)E0δ]}dx$
(15b)
where $x¯$, $y¯$ represent the local coordinates. The global coordinates of points B and C after deformation are related to their local coordinates by
${xC−xB=x¯Ccosα2−y¯Csinα2yC−yB=x¯Csinα2+y¯Ccosα2$
(16a)
${xB=x¯^Bcosα1−y¯^Bsinα1yB=x¯^Bsinα1+y¯^Bcosα1$
(16b)
The moment at point B can be obtained based on the global coordinates of points B and C as
$M0=(yC−yB)P$
(17)
Based on Eqs. (11), (14), (15), (16), and (17), the global coordinates of points B and C can be obtained. The global coordinates of point C before deformation are
$XC=(cosα1+12cosα2)l$
(18a)
$YC=(sinα1+12sinα2)l$
(18b)
The nominal strain can be obtained as
$εx_celluar=xCXC−1=x¯Ccosα2−y¯Csinα2+x¯Bcosα1−y¯Bsinα1(cosα1+12cosα2)l−1$
(19a)
$εy_celluar=yCYC−1=x¯Csinα2+y¯Ccosα2+x¯Bsinα1+y¯Bcosα1(sinα1+12sinα2)l−1$
(19b)
Based on the increments in $εx_celluar$ and $εy_celluar$, the equivalent Poisson’s ratio is obtained as
$νcelluar=−dεy_celluardεx_celluar$
(20)
The above process can also be used to calculate the deformed shape. Similar to Eq. 15(a), for a specific location along beam AB with angle β3 (the angle between x-direction and the beam axial direction along beam AB after deformation) ranging between β1 and α1, the local coordinates can be obtained as
${x¯β3y¯β3}=∫0α1−β3{1+f−1[Pcos(α1−φ1)E0δ]}{cosφ1sinφ1}dφ1−2PE0I∫α1−β1φ1sin(α1−x){1+f−1[Pcos(α1−x)E0δ]}dx+(M0E0I)2$
(21)
The global coordinates after deformation can be obtained based on Eq. 16(b)
${xβ3=x¯β3cosα1−y¯β3sinα1yβ3=x¯β3sinα1+y¯β3cosα1$
(22)
Similar to Eq. 15(b), for a specific location along beam BC with angle β4 (the angle between x-direction and the beam axial direction along beam BC after deformation) ranging between β2 and $α2−α1+β1$, the local coordinates can be obtained as
${x¯β4y¯β4}=∫α1−β1α2−β4{1+f−1[Pcos(α2−φ2)E0δ]}{cosφ2sinφ2}dφ2−2PE0I∫α2−β2φ2sin(α2−x){1+f−1[Pcos(α2−x)E0δ]}dx$
(23)
The global coordinates after deformation can be obtained based on Eq. 16(a)
${xβ4=x¯Bcosα1−y¯Bsinα1+x¯β4cosα2−y¯β4sinα2yβ4=x¯Bsinα1+y¯Bcosα1+x¯β4sinα2+y¯β4cosα2$
(24)

## 3 Results

Figure 3 compares the deformed shape predicted by the proposed analytical model and that by FEA, with α1 = 30 deg, α2 = 120 deg, δ = 0.1 l, and l = 0.3 mm at different stretching strains. Four-node shell elements are chosen for the cellular substrate in the commercial FEA software suite abaqus. The deformed shape calculated by the analytical model agrees well with that extracted directly from FEA.

Fig. 3
Fig. 3
Close modal

The stress–strain curves predicted by FEA and the analytical model for the zigzag cellular substrates are demonstrated in Fig. 4. Figure 4(a) manifests the stress–strain curves with α1 = 30 deg, α2 = 120 deg for different widths. Figure 4(b) shows the stress–strain curves with δ = 0.1 l, α2α1 = 90 deg for different angles α1. Figure 4(c) illustrates the stress–strain curves with δ = 0.1 l, α1 = 30 deg for different angle differences α2α1. The results of parameter analysis show that a lower equivalent modulus can be obtained with a smaller width and a slenderer (larger lateral-parallel aspect ratio when uniaxially stretched; described by larger α1) and wavier (longer curve length within the same span; described by larger (α2α1)) zigzag shape. The stress–strain responses calculated by the analytical model agree well with the FEA results.

Fig. 4
Fig. 4
Close modal

Figure 5 presents the Poisson’s ratio–strain curves calculated by the analytical model and predicted by FEA for the zigzag cellular substrates. Poisson’s ratio versus stretching strain with α1 = 30 deg, α2 = 120 deg for different widths is shown in Fig. 5(a). The width has a noticeable effect on the Poisson’s ratio. Figure 5(b) illustrates Poisson’s ratio versus stretching strain with δ = 0.1 l, α2α1 = 90 deg for different angles α1. The response of Poisson’s ratio versus stretching strain with δ = 0.1 l, α1 = 30 deg for different angle differences α2α1 is presented in Fig. 5(c). Results show that a slenderer (described by larger α1) and wavier (described by larger (α2α1)) zigzag shape yields a lower Poisson’s ratio. The Poisson’s ratio–strain relations calculated by the analytical model perfectly match the FEA results.

Fig. 5
Fig. 5
Close modal
The zigzag cellular substrates possess an advantageous feature over the previously reported hexagonal cellular substrates in terms of reducing their restriction on the natural motion of human skin, by inducing a lower interfacial shear stress between the substrate and skin. Figures 6(a) and 6(b) manifests two representative cellular geometries (thickness 0.3 mm, porosity 80%, dimensions 3.4 mm × 3.4 mm, elastic modulus of elastomer substrate material 500 kPa) bonded to the skin (thickness 1 mm, elastic modulus 130 kPa [41]). The porosity ψ of the zigzag cellular substrate is calculated as [35,40]
$ψ=1−3δ(2cosα1+cosα2)(2sinα1+sinα2)l$
(25)
Fig. 6
Fig. 6
Close modal

The shear stress between the cellular substrates and the skin is computed using FEA where the cellular substrates are modeled with four-node shell elements and the skin with eight-node solid elements. Figures 6(c) and 6(d) gives the shear stress between the cellular substrate and the skin under 50% uniaxial stretching of the skin along x-direction. The maximum interfacial shear stress along the stretching direction between the skin and the zigzag cellular substrate (Fig. 6(d)) is 14.8 kPa, which is smaller than the human skin sensitivity of 20 kPa [38], suggesting smaller constraints on the skin than the hexagonal cellular substrate (Fig. 6(c)). The same phenomenon also holds when stretching along y-direction, as shown in Figs. 6(e) and 6(f). The interfacial shear stress distribution is insensitive to a moderate increase of cellular substrate thickness.

Figure 7 shows the dependence of the interfacial shear stress (on the skin) on the stretching direction (which is denoted by the angle between the stretching direction and the positive x-direction (in Fig. 6)), for hexagonal and zigzag cellular substrates. S13 and S23 are the interfacial shear stress components along and perpendicular to the stretching direction, respectively. Whatever the stretching direction, both S13 and S23 with the zigzag cellular substrate are lower than those with the hexagonal cellular substrate. The maximum shear stress between the zigzag cellular substrate and the skin is noticeably lower than the human skin sensitivity of 20 kPa for any stretching direction. Lower constraints on the natural movement of skin could be achieved using the zigzag cellular substrates.

Fig. 7
Fig. 7
Close modal

## 4 Conclusions

An analytical model of zigzag cellular substrates under finite deformation is proposed. The deformed shape, nonlinear stress–strain curves and Poisson’s ratio–strain curves can be conveniently calculated by the reported analytical model and agree well with FEA predictions. The results of parameter analysis show that a smaller width and a slenderer (larger aspect ratio) and wavier (longer curve length within the same span) zigzag shape yield a lower equivalent modulus and a lower Poisson’s ratio for the zigzag cellular substrates. Whatever the stretching direction of skin, the skin-mounted zigzag cellular substrates could achieve lower constraints on the natural motion of human skin compared with those hexagonal cellular substrates, suggesting promising application opportunities toward mechanically “invisible” epidermal electronics.

## Acknowledgment

The authors gratefully acknowledge the support from the National Natural Science Foundation of China under Grant Nos. 11972059 and 11572023. Y. H. acknowledges the support from the National Science Foundation, USA (Grant No. CMMI1635443).

## References

1.
Kim
,
H.-J.
,
Thukral
,
A.
, and
Yu
,
C.
,
2018
, “
Highly Sensitive and Very Stretchable Strain Sensor Based on a Rubbery Semiconductor
,”
ACS Appl. Mater. Interfaces
,
10
(
5
), pp.
5000
5006
. 10.1021/acsami.7b17709
2.
Tao
,
L.-Q.
,
Tian
,
H.
,
Liu
,
Y.
,
Ju
,
Z.-Y.
,
Pang
,
Y.
,
Chen
,
Y.-Q.
,
Wang
,
D.-Y.
,
Tian
,
X.-G.
,
Yan
,
J.-C.
,
Deng
,
N.-Q.
,
Yang
,
Y.
, and
Ren
,
T.-L.
,
2017
, “
An Intelligent Artificial Throat With Sound-Sensing Ability Based on Laser Induced Graphene
,”
Nat. Commun.
,
8
, p.
14579
. 10.1038/ncomms14579
3.
Kim
,
Y.
,
Chortos
,
A.
,
Xu
,
W.
,
Liu
,
Y.
,
Oh
,
J. Y.
,
Son
,
D.
,
Kang
,
J.
,
Foudeh
,
A. M.
,
Zhu
,
C.
,
Lee
,
Y.
,
Niu
,
S.
,
Liu
,
J.
,
Pfattner
,
R.
,
Bao
,
Z.
, and
Lee
,
T.-W.
,
2018
, “
A Bioinspired Flexible Organic Artificial Afferent Nerve
,”
Science
,
360
(
6392
), pp.
998
1003
. 10.1126/science.aao0098
4.
Khang
,
D.-Y.
,
Jiang
,
H.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2006
, “
A Stretchable Form of Single-Crystal Silicon for High-Performance Electronics on Rubber Substrates
,”
Science
,
311
(
5758
), pp.
208
212
. 10.1126/science.1121401
5.
Avila
,
R.
, and
Xue
,
Y.
,
2017
, “
Torsional Buckling by Joining Prestrained and Unstrained Elastomeric Strips With Application as Bilinear Elastic Spring
,”
ASME J. Appl. Mech.
,
84
(
10
), p.
104502
. 10.1115/1.4037347
6.
Zhang
,
P.
, and
Parnell
,
W. J.
,
2017
, “
Band Gap Formation and Tunability in Stretchable Serpentine Interconnects
,”
ASME J. Appl. Mech.
,
84
(
9
), p.
091007
. 10.1115/1.4037314
7.
Li
,
Y.
,
Zhang
,
J.
,
Xing
,
Y.
, and
Song
,
J.
,
2017
, “
Thermomechanical Analysis of Epidermal Electronic Devices Integrated With Human Skin
,”
ASME J. Appl. Mech.
,
84
(
11
), p.
111004
. 10.1115/1.4037704
8.
Wang
,
A.
,
Avila
,
R.
, and
Ma
,
Y.
,
2017
, “
Mechanics Design for Buckling of Thin Ribbons on an Elastomeric Substrate Without Material Failure
,”
ASME J. Appl. Mech.
,
84
(
9
), p.
094501
. 10.1115/1.4037149
9.
Zhang
,
M.
,
Liu
,
H.
,
Cao
,
P.
,
Chen
,
B.
,
Hu
,
J.
,
Chen
,
Y.
,
Pan
,
B.
,
Fan
,
J. A.
,
Li
,
R.
,
Zhang
,
L.
, and
Su
,
Y.
,
2017
, “
Strain-Limiting Substrates Based on Nonbuckling, Prestrain-Free Mechanics for Robust Stretchable Electronics
,”
ASME J. Appl. Mech.
,
84
(
12
), p.
121010
.
10.
Xu
,
Z.
,
Fan
,
Z.
,
Zi
,
Y.
,
Zhang
,
Y.
, and
Huang
,
Y.
,
2019
, “
An Inverse Design Method of Buckling-Guided Assembly for Ribbon-Type 3D Structures
,”
ASME J. Appl. Mech.
,
87
(
3
), p.
031004
. 10.1115/1.4045367
11.
Ma
,
Y.
,
Choi
,
J.
,
Hourlier-Fargette
,
A.
,
Xue
,
Y.
,
Chung
,
H. U.
,
Lee
,
J. Y.
,
Wang
,
X.
,
Xie
,
Z.
,
Kang
,
D.
,
Wang
,
H.
,
Han
,
S.
,
Kang
,
S.-K.
,
Kang
,
Y.
,
Yu
,
X.
,
Slepian
,
M. J.
,
Raj
,
M. S.
,
Model
,
J. B.
,
Feng
,
X.
,
Ghaffari
,
R.
,
Rogers
,
J. A.
, and
Huang
,
Y.
,
2018
, “
Relation Between Blood Pressure and Pulse Wave Velocity for Human Arteries
,”
Proc. Natl. Acad. Sci. U. S. A.
,
115
(
44
), pp.
11144
11149
. 10.1073/pnas.1814392115
12.
Yang
,
S.
,
Qiao
,
S.
, and
Lu
,
N.
,
2016
, “
Elasticity Solutions to Nonbuckling Serpentine Ribbons
,”
ASME J. Appl. Mech.
,
84
(
2
), p.
021004
. 10.1115/1.4035118
13.
Yan
,
Z.
,
Wang
,
B.
, and
Wang
,
K.
,
2019
, “
Stretchability and Compressibility of a Novel Layout Design for Flexible Electronics Based on Bended Wrinkle Geometries
,”
Composites, Part B
,
166
, pp.
65
73
. 10.1016/j.compositesb.2018.11.123
14.
Yan
,
Z.
,
Wang
,
B.
,
Wang
,
K.
, and
Zhang
,
C.
,
2019
, “
A Novel Cellular Substrate for Flexible Electronics With Negative Poisson Ratios Under Large Stretching
,”
Int. J. Mech. Sci.
,
151
, pp.
314
321
. 10.1016/j.ijmecsci.2018.11.026
15.
Zhang
,
Y.
,
Jiao
,
Y.
,
Wu
,
J.
,
Ma
,
Y.
, and
Feng
,
X.
,
2020
, “
,”
Extreme Mech. Lett.
,
34
, p.
100604
. 10.1016/j.eml.2019.100604
16.
Yan
,
Z.
,
Wang
,
B.
,
Wang
,
K.
,
Zhao
,
S.
,
Li
,
S.
,
Huang
,
Y.
, and
Wang
,
H.
,
2019
, “
Cellular Substrate to Facilitate Global Buckling of Serpentine Structures
,”
ASME J. Appl. Mech.
,
87
(
2
), p.
024501
. 10.1115/1.4045282
17.
Yu
,
X.
,
Xie
,
Z.
,
Yu
,
Y.
,
Lee
,
J.
,
,
A.
,
Luan
,
H.
,
Ruban
,
J.
,
Ning
,
X.
,
Akhtar
,
A.
,
Li
,
D.
,
Ji
,
B.
,
Liu
,
Y.
,
Sun
,
R.
,
Cao
,
J.
,
Huo
,
Q.
,
Zhong
,
Y.
,
Lee
,
C. M.
,
Kim
,
S. Y.
,
Gutruf
,
P.
,
Zhang
,
C.
,
Xue
,
Y.
,
Guo
,
Q.
,
Chempakasseril
,
A.
,
Tian
,
P.
,
Lu
,
W.
,
Jeong
,
J. Y.
,
Yu
,
Y. J.
,
Cornman
,
J.
,
Tan
,
C. S.
,
Kim
,
B. H.
,
Lee
,
K. H.
,
Feng
,
X.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2019
, “
Skin-Integrated Wireless Haptic Interfaces for Virtual and Augmented Reality
,”
Nature
,
575
(
7783
), pp.
473
479
. 10.1038/s41586-019-1687-0
18.
Li
,
J.
,
Zhao
,
J.
, and
Rogers
,
J. A.
,
2019
, “
Materials and Designs for Power Supply Systems in Skin-Interfaced Electronics
,”
Acc. Chem. Res.
,
52
(
1
), pp.
53
62
. 10.1021/acs.accounts.8b00486
19.
Kim
,
N.
,
Lim
,
T.
,
Song
,
K.
,
Yang
,
S.
, and
Lee
,
J.
,
2016
, “
Stretchable Multichannel Electromyography Sensor Array Covering Large Area for Controlling Home Electronics With Distinguishable Signals From Multiple Muscles
,”
ACS Appl. Mater. Interaces
,
8
(
32
), pp.
21070
21076
. 10.1021/acsami.6b05025
20.
Yang
,
W.
,
Sherman
,
V. R.
,
Gludovatz
,
B.
,
Schaible
,
E.
,
Stewart
,
P.
,
Ritchie
,
R. O.
, and
Meyers
,
M. A.
,
2015
, “
On the Tear Resistance of Skin
,”
Nat. Commun.
,
6
, p.
6649
. 10.1038/ncomms7649
21.
Ling
,
S.
,
Zhang
,
Q.
,
Kaplan
,
D. L.
,
Omenetto
,
F.
,
Buehler
,
M. J.
, and
Qin
,
Z.
,
2016
, “
Printing of Stretchable Silk Membranes for Strain Measurements
,”
Lab Chip
,
16
(
13
), pp.
2459
2466
. 10.1039/C6LC00519E
22.
Meyers
,
M. A.
,
McKittrick
,
J.
, and
Chen
,
P. Y.
,
2013
, “
Structural Biological Materials: Critical Mechanics-Materials Connections
,”
Science
,
339
(
6121
), pp.
773
779
. 10.1126/science.1220854
23.
Cranford
,
S. W.
,
Tarakanova
,
A.
,
Pugno
,
N. M.
, and
Buehler
,
M. J.
,
2012
, “
Nonlinear Material Behaviour of Spider Silk Yields Robust Webs
,”
Nature
,
482
(
7383
), pp.
72
76
. 10.1038/nature10739
24.
Ma
,
Q.
,
Cheng
,
H.
,
Jang
,
K. I.
,
Luan
,
H.
,
Hwang
,
K. C.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Zhang
,
Y.
,
2016
, “
A Nonlinear Mechanics Model of Bio-Inspired Hierarchical Lattice Materials Consisting of Horseshoe Microstructures
,”
J. Mech. Phys. Solids
,
90
, pp.
179
202
. 10.1016/j.jmps.2016.02.012
25.
Lee
,
Y. K.
,
Jang
,
K.-I.
,
Ma
,
Y.
,
Koh
,
A.
,
Chen
,
H.
,
Jung
,
H. N.
,
Kim
,
Y.
,
Kwak
,
J. W.
,
Wang
,
L.
,
Xue
,
Y.
,
Yang
,
Y.
,
Tian
,
W.
,
Jiang
,
Y.
,
Zhang
,
Y.
,
Feng
,
X.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2017
, “
Chemical Sensing Systems That Utilize Soft Electronics on Thin Elastomeric Substrates With Open Cellular Designs
,”
,
27
(
9
), p.
1605476
26.
Ma
,
Y.
,
Feng
,
X.
,
Rogers
,
J. A.
,
Huang
,
Y.
, and
Zhang
,
Y.
,
2017
, “
Design and Application of ‘J-Shaped’ Stress–Strain Behavior in Stretchable Electronics: A Review
,”
Lab Chip
,
17
(
10
), pp.
1689
1704
. 10.1039/C7LC00289K
27.
Lee
,
C. H.
,
Ma
,
Y.
,
Jang
,
K.-I.
,
Banks
,
A.
,
Pan
,
T.
,
Feng
,
X.
,
Kim
,
J. S.
,
Kang
,
D.
,
Raj
,
M. S.
,
McGrane
,
B. L.
,
Morey
,
B.
,
Wang
,
X.
,
Ghaffari
,
R.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2015
, “
Soft Core/Shell Packages for Stretchable Electronics
,”
,
25
(
24
), pp.
3698
3704
28.
Liu
,
J.
, and
Zhang
,
Y.
,
2018
, “
A Mechanics Model of Soft Network Materials With Periodic Lattices of Arbitrarily Shaped Filamentary Microstructures for Tunable Poisson's Ratios
,”
ASME J. Appl. Mech.
,
85
(
5
), p.
051003
. 10.1115/1.4039374
29.
Zhang
,
E.
,
Liu
,
Y.
, and
Zhang
,
Y.
,
2018
, “
A Computational Model of Bio-Inspired Soft Network Materials for Analyzing Their Anisotropic Mechanical Properties
,”
ASME J. Appl. Mech.
,
85
(
7
), p.
071002
. 10.1115/1.4039815
30.
Liu
,
J.
, and
Zhang
,
Y.
,
2018
, “
Soft Network Materials With Isotropic Negative Poisson’s Ratios Over Large Strains
,”
Soft Matter
,
14
(
5
), pp.
693
703
. 10.1039/C7SM02052J
31.
Ma
,
Y.
,
Zhang
,
Y.
,
Cai
,
S.
,
Han
,
Z.
,
Liu
,
X.
,
Wang
,
F.
,
Cao
,
Y.
,
Wang
,
Z.
,
Li
,
H.
,
Chen
,
Y.
, and
Feng
,
X.
,
2019
, “
Flexible Hybrid Electronics for Digital Healthcare
,”
, p.
1902062
32.
Dou
,
Y.
,
Jin
,
M.
,
Zhou
,
G.
, and
Shui
,
L.
,
2015
, “
Breath Figure Method for Construction of Honeycomb Films
,”
Membranes
,
5
(
3
), pp.
399
424
. 10.3390/membranes5030399
33.
Jang
,
K.-I.
,
Chung
,
H. U.
,
Xu
,
S.
,
Lee
,
C. H.
,
Luan
,
H.
,
Jeong
,
J.
,
Cheng
,
H.
,
Kim
,
G.-T.
,
Han
,
S. Y.
,
Lee
,
J. W.
,
Kim
,
J.
,
Cho
,
M.
,
Miao
,
F.
,
Yang
,
Y.
,
Jung
,
H. N.
,
Flavin
,
M.
,
Liu
,
H.
,
Kong
,
G. W.
,
Yu
,
K. J.
,
Rhee
,
S. I.
,
Chung
,
J.
,
Kim
,
B.
,
Kwak
,
J. W.
,
Yun
,
M. H.
,
Kim
,
J. Y.
,
Song
,
Y. M.
,
Paik
,
U.
,
Zhang
,
Y.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2015
, “
Soft Network Composite Materials With Deterministic and Bio-Inspired Designs
,”
Nat. Commun.
,
6
, p.
6566
. 10.1038/ncomms7566
34.
Chen
,
H.
,
Zhu
,
F.
,
Jang
,
K.-I.
,
Feng
,
X.
,
Rogers
,
J. A.
,
Zhang
,
Y.
,
Huang
,
Y.
, and
Ma
,
Y.
,
2018
, “
The Equivalent Medium of Cellular Substrate Under Large Stretching, With Applications to Stretchable Electronics
,”
J. Mech. Phys. Solids
,
120
, pp.
199
207
. 10.1016/j.jmps.2017.11.002
35.
Zhu
,
F.
,
Xiao
,
H.
,
Li
,
H.
,
Huang
,
Y.
, and
Ma
,
Y.
,
2019
, “
Irregular Hexagonal Cellular Substrate for Stretchable Electronics
,”
ASME J. Appl. Mech.
,
86
(
3
), p.
034501
. 10.1115/1.4042288
36.
Tancogne-Dejean
,
T.
,
Karathanasopoulos
,
N.
, and
Mohr
,
D.
,
2019
, “
Stiffness and Strength of Hexachiral Honeycomb-Like Metamaterials
,”
ASME J. Appl. Mech.
,
86
(
11
), p.
111010
. 10.1115/1.4044494
37.
Zhu
,
F.
,
Xiao
,
H.
,
Xue
,
Y.
,
Feng
,
X.
,
Huang
,
Y.
, and
Ma
,
Y.
,
2018
, “
Anisotropic Mechanics of Cellular Substrate Under Finite Deformation
,”
ASME J. Appl. Mech.
,
85
(
7
), p.
071007
. 10.1115/1.4039964
38.
Wang
,
S.
,
Li
,
M.
,
Wu
,
J.
,
Kim
,
D.-H.
,
Lu
,
N.
,
Su
,
Y.
,
Kang
,
Z.
,
Huang
,
Y.
, and
Rogers
,
J. A.
,
2012
, “
Mechanics of Epidermal Electronics
,”
ASME J. Appl. Mech.
,
79
(
3
), p.
031022
. 10.1115/1.4005963
39.
Mooney
,
M.
,
1940
, “
A Theory of Large Elastic Deformation
,”
J. Appl. Phys.
,
11
(
9
), pp.
582
592
. 10.1063/1.1712836
40.
Gibson
,
L. J.
, and
Ashby
,
M. F.
,
1999
,
Cellular Solids: Structure and Properties
,
Cambridge University Press
,
Cambridge, UK
.
41.
Schwindt
,
D. A.
,
Wilhelm
,
K.-P.
,
Miller
,
D. L.
, and
Maibach
,
H. I.
,
1998
, “
Cumulative Irritation in Older and Younger Skin: A Comparison
,”
Acta. Derm.-Venereol.
,
78
(
4
), pp.
279
283
. 10.1080/000155598441864