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Electronic Supplementary Material for
Soft network materials with isotropic negative Poisson’s ratios over large strains
Jianxing Liu1 and Yihui Zhang1,*
1 Center for Mechanics and Materials; Center for Flexible Electronics Technology; AML,
Department of Engineering Mechanics; Tsinghua University, Beijing 100084, China
Correspondence and requests for materials should be addressed to Y.Z. (email:
Keywords: Poisson’s ratio; Bioinspired designs; Lattice; Modeling; Shape memory effect
Electronic Supplementary Material (ESI) for Soft Matter.This journal is © The Royal Society of Chemistry 2017
2
Note I: Analytic solutions to the linear Poisson’s ratio
For a planar beam with an arbitrary curvy shape, as shown in Fig. S4a, a curvilinear
coordinate (S) can be adopted to describe the shape of its central axis in the Cartesian
coordinates (X, Y) with the origin at the left end of the microstructure, i.e.,
( ), ( )X X S Y Y S . (S1)
Consider the simply supported conditions in which the microstructure is subject to a
horizontal force N0 at the right end and moments MA and MB at the left and right ends
respectively, as shown in Fig. S4a. The axial force N and bending moment M at a cross
section S are then related to the internal force N0 and Q0 along X and Y directions as
0 0
0 0
( ) ( ) ( ) ,
( ) ( ) ( ) ,A
N S N X S Q Y S
M S M N Y S Q X S
(S2)
where d ( )
( )d
X SX S
S ,
d ( )( )
d
Y SY S
S and 0
0
A BM MQ
L
. For a microstructure with
uniform width, its strain energy is given by
0 02 2
0 0
1 1d d
2 2
S S
S S
U N S M SE A E I
, (S3)
where the membrane energy and bending energy are considered; S0 is the total arc length of
the microstructure curve; and ESA and ESI are the tensile and bending stiffnesses, respectively.
For a microstructure with a reduced width at various turning regions of the curved
microstructure, we number the beginning and ending sections of each segment from 1 to n, as
shown in Fig. S4b. Then the strain energy can be rewritten as
2 1 2 2
2 2 1
2 1 2 2
2 2 1
1 2
2 22 2
0 01 2
1 2
2 22 2
0 01 2
1 1d d
2 2
1 1d d ,
2 2
k k
k k
k k
k k
n n
S S
S Sk kS S
n n
S S
S Sk kS S
U N S N SE A E A
M S M SE I E I
(S4)
where A1 and E1 are area and inertia moment of section 1 (with initial width) respectively,
while A2 and E2 are used for section 2 with a reduced width. Substitution of Eq. (S2) into Eq.
3
(S4) gives the strain energy as
1 2 3 1
2 3 4 5
6 7 8
9
2 21 2 1 1 2 20 0
0 0 02
1 0 0 2
2 22 2 1 2 1 20
0 0 02
0 0 1
1 2 2 1 1
0 0
1
0 0
2( ) (
2 2
2) [ ( )
2
( 2 ) ( ) ( )
( )( )]
A B A B A B
S S
A B A B A BA
S
A B A B A A A B
A B
L LM M M M M MU N N N
E A L L E A
LM M M M M MN M N L
L L E I
M M M M M N L M M M
LN L M M
4 5 6
7 8 9
2 2 2 2 2 2 200 0
2
2 2 2
0 0 0 0
[ ( ) ( 2 )2
( ) ( ) ( )( )] ,
A A B A B
S
A A A B A B
M N L M M M ME I
M N L M M M N L M M
(S5)
where
2 1
12
2 1
22
2 1
32
2 1
42
2 1
52
6
1
21 2
00
1
21
00
1
21 2
00
1
21
00
1
21 2
300
1
1[ ( )] d
2( ) ( )d
1[ ( )] d
1d
1[ ( )] d
1
k
k
k
k
k
k
k
k
k
k
n
S
Sk
n
S
Sk
n
S
Sk
n
S
Sk
n
S
Sk
X S SL
X S Y S SL
Y S SL
SL
Y S SL
L
2 1
2
2 1
72
2 1
82
2 1
92
1
22
300
1
21
200
1
21
200
1
21
300
[ ( )] d
2( )d
2( )d
2( ) ( )d ,
k
k
k
k
k
k
k
k
n
S
Sk
n
S
Sk
n
S
Sk
n
S
Sk
X S S
Y S SL
X S SL
X S Y S SL
2 2
12 1
2 2
22 1
2 2
32 1
2 2
42 1
2 2
52 1
2
22 2
00
2
22
00
2
22 2
00
2
22
00
2
22 2
300
1[ ( )] d
2( ) ( )d
1[ ( )] d
1d
1[ ( )] d
k
k
k
k
k
k
k
k
k
k
n
S
Sk
n
S
Sk
n
S
Sk
n
S
Sk
n
S
Sk
X S SL
X S Y S SL
Y S SL
SL
Y S SL
2 2
62 1
2 2
72 1
2 2
82 1
2 2
92 1
2
22 2
300
2
22
200
2
22
200
2
22
300
1[ ( )] d
2( )d
2( )d
2( ) ( )d ,
k
k
k
k
k
k
k
k
n
S
Sk
n
S
Sk
n
S
Sk
n
S
Sk
X S SL
Y S SL
X S SL
X S Y S SL
(S6)
are dimensionless parameters depending only on the shape of microstructure; the superscript
number of β indicates the corresponding section; L0 is span between the two ends of
4
microstructure. In the condition of infinitesimal deformations, the relationship between the
normalized displacements and loads can be then obtained from Eq. (S5) as
1 1 2 2 7 2 2
5 5 9 9
2 2
7
9
2 2 2(2) (1) (2) (1) (2) (2) (1)2 2 2
3
(2) (1) (1) (2) (1) (2) (1)
7 9 93 3 3
2(2) (1)2
(2) (1)
73
(
3
1 1 1 1 1( ) ( ) ( ( )
12 24 2 24
1 1 1 1 1( ) ) ( ) ( )
2 2
1( )
24
1 1( )
2
1 1(
2
A
B
w w w
q q q q
q q q
w
qu
q
q
3 3 4
3 3 6
4 6 6
6 8
8
2 2 3 3
9
2(2) (1) (2)2
3 2(2) (1) (2)2
3(1) (2) (1)
3
(1) (2) (1)
832) (1) (2) (1)
9 83
2 2(2) (1) (2) (1)2 2
(2) (1)
93
1 1( ) (
12 1 1( ) (
121) ( )
1 1) ( )
21) ( )
1 1( ) ( ) (
24 12
1 1( )
2
w
q q w
q q
q
q
q
w w
q q
q
6 3 3
6 8 6 6
0
2(2) (2) (1)2
3
(1) (2) (1) (2) (1)
83 3
1 1( )
12
1 1 1) ( ) ( )
2
A
B
N
M
M
w
q q
q q
,(S7)
with
2
0 0 0 00
0 0 1 1 1
; ; ; ; A BA B
S S S
N L M L M Lu wu w N M M
L L E I E I E I , (S8)
where u is the displacement at the right end along the direction of N0; ωA and ωB are the
rotational angles at the left and right ends, respectively; w2 is the reduced width of
microstructure, and the width ratio is q = w2/w1.
For infinitesimal deformations, a triangular network material under horizontal stretching
is taken as an example to derive the Poisson’s ratio and elastic modulus. Two representative
unit cells with different configurations are analyzed, as shown in Fig. S4c. According to the
static equilibrium of the unit cell, the inner force iN (with i denoting the different
microstructures of a unit cell, and i = 1 for the horizontal microstructure), the bending
moments iAM and iBM , and the external loading satisfy the following relations:
3 2 3 3 2 23 3 ( ) ( ) 0A B A BN N M M M M , (S9)
1 1 2 2 22( ) ( ) 3 0A B A BM M M M N , (S10)
5
1 3 2 3 3 2 21 2
02
2 2
4 3( ) 3( )
2 3 ( )
A B A B
S
N N N M M M M
LA
w E I
, (S11)
1 1 2 2 3 3( ) ( ) ( ) 0A B A B A BM M M M M M , (S12)
where σx is the effective stress of the triangular lattice along horizontal direction. According
to the deformation compatibility in the periodical lattices, the angle between the tangent lines
of different microstructures keeps unchanged during the deformation. Then the unit cells
should also satisfy the geometric relations given by
2 1 3 2 1 3
3 1 3 2 2 1
1( )
3
1( )
3
A B A B
A B A B
u u
u u
and
2 1 1 3 2 3
3 1 2 1 2 3
1( )
3
1( )
3
B A A B
B A A B
u u
u u
, (S13)
1 2 3 1 2 3 0A A A B B B . (S14)
The effective strain of the lattice material defined as the percentage of elongation is equal to
the unit cell, and can be expressed as
1 1u . (S15)
The constitutive relation of the microstructure, Eq. (S7), can be re-written in the form of a
stiffness matrix [dij] for simplify, as given by
1 2 3 11 12 13 1 2 3
1 2 3 12 22 23 1 2 3
1 2 3 13 23 33 1 2 3
A A A A A A
B B B B B B
N N N d d d u u u
M M M d d d
M M M d d d
. (S16)
According to Eqs. (S9) - (S16), an analytic solution of the Poisson’s ratio and elastic modulus
for triangular network materials with arbitrarily shaped microstructures can be obtained as
2
12 13 11 22 23 33 22 23 33
2
12 13 22 23 33 11 22 23 33
2( ) ( 2 )( 2 )
2( ) ( 2 )(3 2 )
d d d d d d d d d
d d d d d d d d d
, (S17)
2
11 22 23 33 12 13 11 22 23 33 2 2
2 2
12 13 22 23 33 11 22 23 33 2 0
2 3( 2 )[ ( ) ( 2 )]
[ 2( ) ( 2 )(3 2 )]
sd d d d d d d d d d E I wE
d d d d d d d d d A L
. (S18)
These solutions are useful for the deterministic design of network materials to achieve desired
6
mechanical properties. Then we consider a specific zigzag microstructure whose shape can be
described as
0
0 0 0
00 0
(0 )4
3 ( )
2 4 4
3 ( )
4
Ly kx x
L L Ly kx k x
Ly kx kL x L
, (S19)
for a single unit. Then the dimensionless parameters of β can be derived as
21 1 1 1 2
1 2 3 42 2
2 2 21 3 1 3 1
5 6 7
21 2 1 2
8 9 1 22
2
3 42
1 (1 )= , 0, , 1 (1 ),
1 1
1 1(1 ) , (16 15 ), 0,
48 48
11 ( 1), (1 ) , = , 0,
16 1
, 11
r k rk r
k k
k k kr r r
k k rk r r
k
k rk
k
() () () ()
() () ()
() () (2) (2)
(2) (2)2 2
2 2 3
5
2 23 2 1 2
6 7 8 9
1, (3 3 ),
48
1 1(15 ), 0, 1 , (2 ).
48 16
k kr r r r
k k kr r k r r r
(2)
(2) (2) (2) ()
(S20)
Based on Eqs. (S7), (S16) and (S17), the analytic solution of Poisson’s ratio for the triangular
network materials with a single zigzag microstructure can be obtained. When the weakened
segments are much narrower than the other segments, i.e., 0q , the strain energy of the
wider segments can be neglected, leading to a simplified solution of Poisson’s ratio:
22 2 3 2
22 2 4 3 2
18 2 3 3 18 2 6 3
2 3 3 3 2 6 9 6 6
k r r r r r r
k r r r r r r
. (S21)
Another limiting condition involves the network design with uniform width, i.e., q = 1. In
this case, the solution of Poisson’s ratio becomes
22 3 2 4 6 2 2 4
22 3 2 2 4 2 2 4
1 2 8 4 2 32 1
15 2 8 1 12 2 3 32 1 3
k k k k k k w k w
k k k k k k w k w
. (S22)
where 2w w . For zigzag microstructures with multiple unit cells (i.e., periodicity nT), the
7
shape of microstructure can be described as
0 0 0 0
0 0 0 0 0 0
0 0 0 0 0
( ) ( )4
3( ) ( )2 4 4
3( ) (
4
T T T T
T T T T T T
T T T T T
L L L Ly kx k m m x m
n n n n
L L L L L Ly kx k m m x m
n n n n n n
L L L L L Ly kx k m m x m
n n n n n
0 )Tn
, (S23)
where [0, 1]Tm n . The solution of Poisson’s ratio is given by
23 2 2 2 2 4 2 2 2 4
23 2 2 2 2 4 2 2 2 4
9 2 8 8 1 2 4 32 1
9 2 24 8 1 2 3 12 32 1 3
T T
T T
k k k n k k k n w k w
k k k n k k k n w k w
. (S24)
Using a similar approach, the analytic solution of the Poisson’s ratio can be also obtained
for the honeycomb network material. For 0q , it is given by
2 2
2 2
(3 ( 3 ) ) 3
3 3 ( 3 ) ) 3(
k r r r
k r r r
. (S25)
For q = 1, the Poisson’s ratio is written as
22 4 2 2 2 2 4 2 2 2 2 4 4
22 4 2 2 2 2 4 2 2 2 2 4 4
1 192 9 16 16 1 4 1 3 64
3 1 64 3 16 48 1 4 1 192
T T T T T
T T T T T
k n k n k n k k n w k n w
k n k n k n k k n w k n w
. (S26)
8
Figure S1. Comparison between experimental images and FEA predictions for the
deformation sequences of the entire samples in Fig. 2a, b, and c. Scale bars, 5 cm.
9
Figure S2. FEA results on the distribution of maximum principal strain for the three examples
of triangular and honeycomb network materials in Fig. 2a, b, and c, under uniaxial stretching
(25% and 50%).
10
Figure S3. Results of the Poisson’s ratio based on two different definitions (ν' = − dεy/dεx and
ν = − εy/εx), for the three network materials in Fig. 2a, b, and c. The dashlines denote the
critical strain (εcr) of the zigzag microstructure of different lattice materials.
11
Figure S4. Schematic illustrations of theoretical model for an arbitrarily shaped curvy beam
and triangular unit cells. (a) A simply supported beam microstructure subject to an axial force
at the right end, and moments MA and MB at the two ends. The shape of microstructure is
described by curvilinear coordinate (S). (b) An arbitrarily shaped curvy beam with or without
a width reduction. (c) A pair of representative unit cells of triangular network materials
subject to a uniform tensile stress along horizontal direction, with the free-body diagram of
the horizontally aligned microstructure.
12
Figure S5. Geometric conditions of the normalized width ( w ) and slopes (tanθ0) that should
be satisfied to form a triangular network material without any self-overlay. Here, the zigzag
microstructure has no width reduction. The region below the solid line denotes the regime
where the geometric conditions are met.
13
Figure S6. Linear Poisson’s ratio (ν) of triangular network materials versus the microstructure
slope (tanθ0) for a range of unit number (nT) of the zigzag microstructure and the width-to-
length ratio ( w ), in the condition of w2/w1 = 1.
14
Figure S7. Theoretical and computational studies on the linear Poisson ratio of triangular
network materials under infinitesimal deformations. (a) Linear Poisson’s ratio (ν) versus the
width ratio (w2/w1) for a wide range of microstructure slopes (tanθ0) and a fixed length ratio
(L2/(L1+L2) = 0.1). (b) Linear Poisson’s ratio (ν) versus the length ratio (L2/(L1+L2)) for a
wide range of width ratio (w2/w1) and a fixed microstructure slopes (tanθ0 = 1.0).
15
Figure S8. Microstructure slopes (tanθ0) and width ratios (w2/w1) that yield zero linear
Poisson’s ratio (ν) for four different length ratio (L2/(L1+L2) = 0.1, 0.2, 0.3 and 0.4) of
triangular network materials (in Fig. 3a). The black solid line represents the curve fitted based
on all of the dashed curves.
16
Figure S9. FEA results of triangular lattice materials under uniaxial stretching along
different loading angles. (a) Poisson’s ratio (ν) versus the applied tensile strain (ε) along
different loading angles (0o, 10o, 20o, and 30o) for the triangular network materials with q =
w2/w1 = 1.0. (b)-(e) Similar results for network materials with w2/w1 = 0.6, 0.4, 0.2 and 0.1,
respectively.
17
Figure S10. Critical strain (εcr) of the zigzag microstructure as a function of the
microstructure slope (tanθ0).
18
Figure S11. FEA images of the deformed configurations of triangular network materials
under uniaxial stretching along vertical directions, with microstructure parameters: (a) tanθ0 =
1.2, w2/w1 = 0.1, L2/(L1+L2) = 0.1, and (b) tanθ0 = 1.2, w2/w1 = 0.6, L2/(L1+L2) = 0.1.
19
Figure S12. Maximum variation of Poisson’s ratio Δν for the applied strain εx increasing from
zero to the critical strain εcr for different length ratios (L2/(L1+L2) = 0.1, 0.2, 0.3 and 0.4). The
horizontal dashed line indicates Δν = 0.05.
20
Figure S13. Maximum variation of Poisson’s ratio Δν for the applied strain εy increasing from
0 to 0.5εcr.The horizontal dashed line indicates Δν = 0.06.
21
Figure S14. (a) Optical image for the as-fabricated artificial skin and schematic illustration of
a unit cell. (b) Optical image for the artificial skin at the onset of fracture (~ 82% applied
strain). Scale bar, 20 mm.
22
Figure S15. Theoretical and computational studies on the linear elastic modulus of triangular
network materials under infinitesimal deformations. (a) Normalized linear elastic modulus
(E/Es) versus the microstructure slopes (tanθ0) for a wide range of width ratio (w2/w1) and a
fixed length ratio (L2/(L1+L2) = 0.1). (b) Normalized linear elastic modulus (E/Es) versus the
length ratio (L2/(L1+L2)) for a wide range of width ratio (w2/w1) and a fixed microstructure
slopes (tanθ0 = 1.0).
23
Figure S16. FEA results of stress-strain curves and tangent modulus-strain curves for the
triangular network materials under horizontal stretching. (a) and (b) Stress-strain curves and
tangent modulus-strain curves for the triangular network materials in Fig. 5a. (c) and (d)
Similar results for the triangular network materials in Fig. 5b. The dashlines denote the
critical strain (εcr) of the zigzag microstructure.
24
Movie Captions
Movie S1. This video illustrates synchronized experimental and computational results on the
deformation processes of three network materials (with negative, ‘zero’ and positive
Poisson’s ratios) under uniaxial stretching (50%).
Movie S2. This video illustrates an architected cylindrical shell under axial compression and
tension. It consists of three segments that possess negative, ‘zero’ and positive Poisson’s
ratios.