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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:

yihuizhang@tsinghua.edu.cn).

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

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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

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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)

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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

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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

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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)

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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.

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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.

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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.

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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).

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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.

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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.