Bifurcation Analysis of Periodic Kirigami Structure with ...

†

∗ • ∗∗ • ∗∗ • ∗∗∗

Bifurcation Analysis of Periodic Kirigami Structure with Out-Plane Deformation

by

Xiao-Wen LEI∗, Akihiro NAKATANI∗∗, Yusuke DOI∗∗ and Shintaro MATSUNAGA∗∗∗

Kirigami is a traditional art of papercraft, which makes 3 dimensional structure from 2 dimensional sheet. Recentyears, the mathematical foundations of kirigami and origami have been developed as well as the practical applications.In this research, we analyze the out-of-plane deformation of kirigami structure under tensile force, and investigate themechanism of the deformation based on the beam theory. We simulate the process from in-plane deformation to out-of-plane deformation using molecular dynamics, and discuss the effect of geometry parameters on stability.

Key words:Kirigami structure, Elasticity, Bifurcation, Beam theory, Strain energy

12

3

2 3

1)−3) kirigami origami

1970

4).

Silverberg 5) square

twist

2

6),7)

8)

† 29 7 12 Received July 12. 2017 c⃝2018 The Society of Materials Science, Japan∗ 910-8507∗ Department of Mechanical Engineering, Graduate School of Engineering, University of Fukui, Bunkyo, Fukui, 910-8507.∗∗ 565-0871∗∗∗ Department of Adaptive Machine Systems, Graduate School of Engineering, Osaka University, Yamadaoka, Suita, 565-0871.∗∗∗ 448-0029∗∗∗ DENSO Corporation, Showa-cho, Kariya, 448-0029.

22 ·1Fig.1(a)

Fig.1(a) 1

Fig.1(b)

x Lx y Ly z

h b l

λ = (Ly − 2b)/2

(a) schematics of simple kirigami structure

0.1Lx

0.4Lx

Lx

(l)

Thickness hl = Lx/2, λ = (Ly − 2b)/2x

y

z

0.5b

(λ)b(λ)

0.5b

Ly

(b) unit of analysis model

Fig. 1 Analysis model of kirigami structure.

2 ·2

I III Fig.2(a), (b)

Fig.1 1

4

y

I

I

III I III

Fig.2(a), (b)

uI, uIII

E

I III II IIII

EII EIIII

II IIII II = hλ3/12 IIII = λh3/12

E

II IIII

EII EIIII

x

y

z

uI

l

(a) mode I

x

z

y

uIII

l

(b) mode III

y

z

x

θ

λ

uI

uIIIFF

(c) mixed mode

Fig. 2 Deformation in-plane (mode I), out-of-plane (mode III),and mixed mode of mode I and mode III.

y F

I III

Fig.2(c) θ

1

y

δ Fig.2(c)

uI uIII θ δ (1)

(λ+ uI)2 + u2

III =(λ+

δ

2

)2, tan θ =

uIII

λ+ uI(1)

uI uIII (2)

uI =( δ2+ λ

)cos θ − λ, uIII =

( δ2+ λ

)sin θ (2)

W 4 δ

F

−Fδ Π

(3)

Π = W − Fδ =24EIIl3

u2I +

24EIIIIl3

u2III − Fδ (3)

δ θ

∂Π/∂δ = 0 ∂Π/∂θ = 0

∂Π

∂δ=

∂W

∂uI

∂uI

∂δ+

∂W

∂uIII

∂uIII

∂δ− F = 0 (4)

F =24EIIl3

{(λ+

δ

2

)cos2 θ − λcos θ

}

+24EIIII

l3(λ+

δ

2

)sin2 θ (5)

∂Π

∂θ=

∂W

∂uI

∂uI

∂θ+

∂W

∂uIII

∂uIII

∂θ= 0 (6)

｢材料｣ (Journal of the Society of Materials Science, Japan), Vol. 67, No. 2, pp. 202-207, Feb. 2018

論　　文

11-2017-0103-(p.202-207).indd 202 2018/01/10 20:34:26

†

∗ • ∗∗ • ∗∗ • ∗∗∗

Bifurcation Analysis of Periodic Kirigami Structure with Out-Plane Deformation

by

Xiao-Wen LEI∗, Akihiro NAKATANI∗∗, Yusuke DOI∗∗ and Shintaro MATSUNAGA∗∗∗

Kirigami is a traditional art of papercraft, which makes 3 dimensional structure from 2 dimensional sheet. Recentyears, the mathematical foundations of kirigami and origami have been developed as well as the practical applications.In this research, we analyze the out-of-plane deformation of kirigami structure under tensile force, and investigate themechanism of the deformation based on the beam theory. We simulate the process from in-plane deformation to out-of-plane deformation using molecular dynamics, and discuss the effect of geometry parameters on stability.

Key words:Kirigami structure, Elasticity, Bifurcation, Beam theory, Strain energy

12

3

2 3

1)−3) kirigami origami

1970

4).

Silverberg 5) square

twist

2

6),7)

8)

† 29 7 12 Received July 12. 2017 c⃝2018 The Society of Materials Science, Japan∗ 910-8507∗ Department of Mechanical Engineering, Graduate School of Engineering, University of Fukui, Bunkyo, Fukui, 910-8507.∗∗ 565-0871∗∗∗ Department of Adaptive Machine Systems, Graduate School of Engineering, Osaka University, Yamadaoka, Suita, 565-0871.∗∗∗ 448-0029∗∗∗ DENSO Corporation, Showa-cho, Kariya, 448-0029.

22 ·1Fig.1(a)

Fig.1(a) 1

Fig.1(b)

x Lx y Ly z

h b l

λ = (Ly − 2b)/2

(a) schematics of simple kirigami structure

0.1Lx

0.4Lx

Lx

(l)

Thickness hl = Lx/2, λ = (Ly − 2b)/2x

y

z

0.5b

(λ)b(λ)

0.5b

Ly

(b) unit of analysis model

Fig. 1 Analysis model of kirigami structure.

2 ·2

I III Fig.2(a), (b)

Fig.1 1

4

y

I

I

III I III

Fig.2(a), (b)

uI, uIII

E

I III II IIII

EII EIIII

II IIII II = hλ3/12 IIII = λh3/12

E

II IIII

EII EIIII

x

y

z

uI

l

(a) mode I

x

z

y

uIII

l

(b) mode III

y

z

x

θ

λ

uI

uIIIFF

(c) mixed mode

Fig. 2 Deformation in-plane (mode I), out-of-plane (mode III),and mixed mode of mode I and mode III.

y F

I III

Fig.2(c) θ

1

y

δ Fig.2(c)

uI uIII θ δ (1)

(λ+ uI)2 + u2

III =(λ+

δ

2

)2, tan θ =

uIII

λ+ uI(1)

uI uIII (2)

uI =( δ2+ λ

)cos θ − λ, uIII =

( δ2+ λ

)sin θ (2)

W 4 δ

F

−Fδ Π

(3)

Π = W − Fδ =24EIIl3

u2I +

24EIIIIl3

u2III − Fδ (3)

δ θ

∂Π/∂δ = 0 ∂Π/∂θ = 0

∂Π

∂δ=

∂W

∂uI

∂uI

∂δ+

∂W

∂uIII

∂uIII

∂δ− F = 0 (4)

F =24EIIl3

{(λ+

δ

2

)cos2 θ − λcos θ

}

+24EIIII

l3(λ+

δ

2

)sin2 θ (5)

∂Π

∂θ=

∂W

∂uI

∂uI

∂θ+

∂W

∂uIII

∂uIII

∂θ= 0 (6)

203面外変形を起こすキリガミ周期構造体の分岐解析

11-2017-0103-(p.202-207).indd 203 2018/01/10 20:34:26

θ = 0 (7)

cos θ =EIIλ

(EII − EIIII)(λ+ δ

2

) (8)

(2) (5) (7)

(8)

δ I I III

2 ·3Fig.1 Lx = 400.0A Ly = 69.3A

h = 9.80A b = 3.20A λ = 31.45A

Fig.3

[110] [112]

(111) x y xy

xy

x

y

z z

y

x

Fig. 3 Particle model.

y x z

(9) Morse

ϕ(r) = DMorseexp{−2A(r − r0)}

− 2DMorseexp{−A(r − r0)} (9)

DMorse = 1.0×10−21J

A = 6.5A−1r0 = 2.0A

a0 = 2.83A

4.036 × 10−25kg

y 109s−1

∆t = 0.75× 10−15s

0K

33 ·13 ·1 ·1

2.3 (3)

W Fig.4

W δ

θ W δ E

h W = W/(Eh3) δ = δ/h

(6) ∂W/∂θ = 09),10)

11)

−80 −60 −40 −20 0 20 40 60 800.0

0.5

1.0

1.5

2.0

2.5

0

0.01

0.02

0.03

0.04

W−

Energy landscapeEquilibrium path

Unstable path (mode I)

θ , deg

δ−

W−

Fig. 4 Strain energy W as a function of δ and θ.

Fc δc I,

III γ = EII/(EIIII)

Fc =24

l3EIIλ

γ − 1(10)

δc =2λ

γ − 1(11)

uI uIII F θ

δ

(i) 0 ≤ δ ≤ δc

uI(δ) =1

2δ (12)

uIII(δ) = 0 (13)

F (δ) =12EII

l3δ (14)

θ(δ) = 0. (15)

(ii) δ > δc

uI(δ) =λ

γ − 1(16)

uIII(δ) =γλ

γ − 1tan θ (17)

F (δ) =24EIIII

l3( δ2+ λ

)(18)

θ(δ) = cos−1 γ

(γ − 1)( δ2+ λ)

. (19)

3 ·1 ·2 Lx = 400.0A

b = 3.20A h = 9.80A

L0 = 69.3A Ly Ly = L0, 0.75L0

0.5L0 0.25L0

W δ Fig.5(a) W

δ E h

W = W/(Eh3) δ = δ/h

λ = (Ly − 2b)/2 Ly

λ = h γ = 1 (11)

δc

Ly = 0.375L0

δ < δc δ > δc

Fig.5(b) F δ F

E h F = F/(Eh2)

Ly δc

Ly

Ly = 0.25L0

λ < h

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

W−

δ−

Ly=L0, mode IL0, mode I, III

0.75L0, mode I0.75L0, mode I, III


0.375L0(λ=h), mode I0.25L0, mode I

(a) strain energy W

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

F−δ−





(b) tensile force F

Fig. 5 Strain energy W and tensile force F as functions of dis-placement δ with different size of specimen Ly .

II = hλ3/12 IIII = λh3/12

h λ

γ = EII/(EIIII) (11)

λ

δ = δc W F

Fig.6(a), 6(b) λ ≤ h

(11)

W F δ

λ = h

Fig.6(a)

Fig.6(b)

Fc δc

λ = h

δc λ = h

λ = h

λ < h

0.000

0.001

0.002

0.003

0.004

0.0 1.0 2.0 3.0 4.0 5.0 6.0

W−

δ−

λ = hBifurcation points

(a) bifurcation points on W – δ plane

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.0 1.0 2.0 3.0 4.0 5.0 6.0

F−

δ−


(b) bifurcation points on F – δ plane

Fig. 6 Bifurcation points on W – δ and F – δ planes.

3 ·2

Fig.7

x

z

xy

Fig.7(a)

Fig.7(b), (c)

204 雷　霄雯，中谷彰宏，土井祐介，松永慎太郎

11-2017-0103-(p.202-207).indd 204 2018/01/10 20:34:27

θ = 0 (7)

cos θ =EIIλ

(EII − EIIII)(λ+ δ

2

) (8)

(2) (5) (7)

(8)

δ I I III

2 ·3Fig.1 Lx = 400.0A Ly = 69.3A

h = 9.80A b = 3.20A λ = 31.45A

Fig.3

[110] [112]

(111) x y xy

xy

x

y

z z

y

x

Fig. 3 Particle model.

y x z

(9) Morse

ϕ(r) = DMorseexp{−2A(r − r0)}

− 2DMorseexp{−A(r − r0)} (9)

DMorse = 1.0×10−21J

A = 6.5A−1r0 = 2.0A

a0 = 2.83A

4.036 × 10−25kg

y 109s−1

∆t = 0.75× 10−15s

0K

33 ·13 ·1 ·1

2.3 (3)

W Fig.4

W δ

θ W δ E

h W = W/(Eh3) δ = δ/h

(6) ∂W/∂θ = 09),10)

11)

−80 −60 −40 −20 0 20 40 60 800.0

0.5

1.0

1.5

2.0

2.5

0

0.01

0.02

0.03

0.04

W−

Energy landscapeEquilibrium path

Unstable path (mode I)

θ , deg

δ−

W−

Fig. 4 Strain energy W as a function of δ and θ.

Fc δc I,

III γ = EII/(EIIII)

Fc =24

l3EIIλ

γ − 1(10)

δc =2λ

γ − 1(11)

uI uIII F θ

δ

(i) 0 ≤ δ ≤ δc

uI(δ) =1

2δ (12)

uIII(δ) = 0 (13)

F (δ) =12EII

l3δ (14)

θ(δ) = 0. (15)

(ii) δ > δc

uI(δ) =λ

γ − 1(16)

uIII(δ) =γλ

γ − 1tan θ (17)

F (δ) =24EIIII

l3( δ2+ λ

)(18)

θ(δ) = cos−1 γ

(γ − 1)( δ2+ λ)

. (19)

3 ·1 ·2 Lx = 400.0A

b = 3.20A h = 9.80A

L0 = 69.3A Ly Ly = L0, 0.75L0

0.5L0 0.25L0

W δ Fig.5(a) W

δ E h

W = W/(Eh3) δ = δ/h

λ = (Ly − 2b)/2 Ly

λ = h γ = 1 (11)

δc

Ly = 0.375L0

δ < δc δ > δc

Fig.5(b) F δ F

E h F = F/(Eh2)

Ly δc

Ly

Ly = 0.25L0

λ < h

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

W−

δ−





(a) strain energy W

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

F−

δ−





(b) tensile force F

Fig. 5 Strain energy W and tensile force F as functions of dis-placement δ with different size of specimen Ly .

II = hλ3/12 IIII = λh3/12

h λ

γ = EII/(EIIII) (11)

λ

δ = δc W F

Fig.6(a), 6(b) λ ≤ h

(11)

W F δ

λ = h

Fig.6(a)

Fig.6(b)

Fc δc

λ = h

δc λ = h

λ = h

λ < h

0.000

0.001

0.002

0.003

0.004

0.0 1.0 2.0 3.0 4.0 5.0 6.0

W−

δ−


(a) bifurcation points on W – δ plane

0.000

0.001

0.002

0.003

0.004

0.005

0.006

0.0 1.0 2.0 3.0 4.0 5.0 6.0

F−

δ−


(b) bifurcation points on F – δ plane

Fig. 6 Bifurcation points on W – δ and F – δ planes.

3 ·2

Fig.7

x

z

xy

Fig.7(a)

Fig.7(b), (c)


11-2017-0103-(p.202-207).indd 205 2018/01/10 20:34:28

x

y

zy

z

x

x

y

z

(a) δ = 0A

x

y

zy

z

x

x

y

z

(b) δ = 7.77A

x

y

z

y

z

xx

y

z

(c) δ = 25.95A

Fig. 7 Atomic configuration obtained by MD simulation. Grada-tion of color corresponds to the x coordinate of each atom.

3 ·3

I

III

EII = 1.067 × 10−25Pa ·m4,

EIIII = 2.533× 10−26Pa ·m4

Fig. 8(a) θ δ Fig. 8(b)

uI δ Fig. 8(c) uIII δ

Mode I θ = 0

δ

Mixed Mode

Geometrical

condition

−40

−30

−20

−10

0

10

20

30

40

50

0 5 10 15 20 25 30 35 40

θ , deg

δ , Å

Mode I

Mixed Mode

MD result

(a) θ − δ

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40

uI , Å

δ , Å

Mode I

Mixed Mode

MD result

Geometrical condition

(b) uI − δ

−30

−20

−10

0

10

20

30

40

0 5 10 15 20 25 30 35 40

uII

I , Å

δ , Å

Mode I

Mixed Mode

MD result


(c) uIII − δ

Fig. 8 Comparison between MD result and geometrical condi-tion.

MD

result

Mode I θ = 0

θ = 0

δ

Fig. 8(a)

10) δ

Fig. 8(b) (c)

δ

(2)

uI uIII MD result

δ θ

(2) uI

uIII Geometrical condition

Fig.8(b) 8(c)

uI uIII

δ θ

uI uIII

4

15K13835 B 17K14145

1) M. K. Bless, A. W. Barnard, P. A. Rose, S. P. Roberts,K. L. McGill, P. Y. Huang, A. R. Ruyack, J. W. Kevek,B. Kobrin, D. A. Muller and P. L. McEuen, “Graphenekirigami”, Nature, Vol. 524, pp. 204-207 (2015).

2) A. Rafsanjani and K. Bertoldi, “Buckling-inducedkirigami”, Physical Review Letters, Vol. 118, pp.084301-1-5 (2017).

3) A. Nakatani, X.-W. Lei, S. Matsunaga and Y. Doi,“Bifurcation analysis of anti-plane deformation inkirigami structure under tensile loading”, The Pro-ceedings of the Materials and Mechanics Conference,GS-51, pp.957-959 (2016).

4) URL http://www.miuraori.biz/hpgen/HPB/entries/11.html

5) J. L. Silverberg, J.-H. Na, A. A. Evans, B. Liu, T. C.Hull, C. D. Santangelo, R. J. Lang, R. C. Hayward andI. Cohen, “Origami structures with a critical transi-tion to bistability arising from hidden degrees of free-dom”, Nature Materials, Vol. 14, pp. 389-393 (2015).

6) T. C. Shyu, P. F. Damasceno, P. M. Dodd, A. Lam-oureux, L. Xu, M. Shlian, M. Shtein, S. C. Glotzer andN. A. Kotov, “A kirigami approach to engineering elas-ticity in nanocomposites through patterned defects”,Nature Materials, Vol. 14, pp. 785-789 (2015).

7) Y. Tang and J. Yin, “Design of cut unit geometryin hierarchical kirigami-based auxetic metamaterialsfor high stretchability and compressibility”, ExtremeMechanics Letters, Vol. 12, pp. 77-85 (2017).

8) X.-W Lei and A. Nakatani, “A deformation mech-anism for ridge-shaped kink structure in layeredsolids”, The American Society of Mechanical Engi-neers (ASME) Journal of Applied Mechanics, Vol. 82,pp. 071016-1-6 (2015).

9) D. Bigoni, “Nonlinear Solid Mechanics”, CambridgeUniversity Press (2012).

10) Y. Shibutani and A. Nakatani, “Mechanics of Materi-als”, Corona Publishing Co., Ltd. (2017).

11) D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni,“Structures buckling under tensile dead load”, Pro-ceedings of the Royal Society A, Vol. 467, pp. 1686-1700 (2011).

206 雷　霄雯，中谷彰宏，土井祐介，松永慎太郎

11-2017-0103-(p.202-207).indd 206 2018/01/10 20:34:28

x

y

zy

z

x

x

y

z

(a) δ = 0A

x

y

zy

z

x

x

y

z

(b) δ = 7.77A

x

y

z

y

z

xx

y

z

(c) δ = 25.95A

Fig. 7 Atomic configuration obtained by MD simulation. Grada-tion of color corresponds to the x coordinate of each atom.

3 ·3

I

III

EII = 1.067 × 10−25Pa ·m4,

EIIII = 2.533× 10−26Pa ·m4

Fig. 8(a) θ δ Fig. 8(b)

uI δ Fig. 8(c) uIII δ

Mode I θ = 0

δ

Mixed Mode

Geometrical

condition

−40

−30

−20

−10

0

10

20

30

40

50

0 5 10 15 20 25 30 35 40

θ , deg

δ , Å

Mode I

Mixed Mode

MD result

(a) θ − δ

0

2

4

6

8

10

12

14

16

18

20

0 5 10 15 20 25 30 35 40

uI , Å

δ , Å

Mode I

Mixed Mode

MD result


(b) uI − δ

−30

−20

−10

0

10

20

30

40

0 5 10 15 20 25 30 35 40

uII

I , Å

δ , Å

Mode I

Mixed Mode

MD result


(c) uIII − δ

Fig. 8 Comparison between MD result and geometrical condi-tion.

MD

result

Mode I θ = 0

θ = 0

δ

Fig. 8(a)

10) δ

Fig. 8(b) (c)

δ

(2)

uI uIII MD result

δ θ

(2) uI

uIII Geometrical condition

Fig.8(b) 8(c)

uI uIII

δ θ

uI uIII

4

15K13835 B 17K14145

1) M. K. Bless, A. W. Barnard, P. A. Rose, S. P. Roberts,K. L. McGill, P. Y. Huang, A. R. Ruyack, J. W. Kevek,B. Kobrin, D. A. Muller and P. L. McEuen, “Graphenekirigami”, Nature, Vol. 524, pp. 204-207 (2015).

2) A. Rafsanjani and K. Bertoldi, “Buckling-inducedkirigami”, Physical Review Letters, Vol. 118, pp.084301-1-5 (2017).

3) A. Nakatani, X.-W. Lei, S. Matsunaga and Y. Doi,“Bifurcation analysis of anti-plane deformation inkirigami structure under tensile loading”, The Pro-ceedings of the Materials and Mechanics Conference,GS-51, pp.957-959 (2016).

4) URL http://www.miuraori.biz/hpgen/HPB/entries/11.html

5) J. L. Silverberg, J.-H. Na, A. A. Evans, B. Liu, T. C.Hull, C. D. Santangelo, R. J. Lang, R. C. Hayward andI. Cohen, “Origami structures with a critical transi-tion to bistability arising from hidden degrees of free-dom”, Nature Materials, Vol. 14, pp. 389-393 (2015).

6) T. C. Shyu, P. F. Damasceno, P. M. Dodd, A. Lam-oureux, L. Xu, M. Shlian, M. Shtein, S. C. Glotzer andN. A. Kotov, “A kirigami approach to engineering elas-ticity in nanocomposites through patterned defects”,Nature Materials, Vol. 14, pp. 785-789 (2015).

7) Y. Tang and J. Yin, “Design of cut unit geometryin hierarchical kirigami-based auxetic metamaterialsfor high stretchability and compressibility”, ExtremeMechanics Letters, Vol. 12, pp. 77-85 (2017).

8) X.-W Lei and A. Nakatani, “A deformation mech-anism for ridge-shaped kink structure in layeredsolids”, The American Society of Mechanical Engi-neers (ASME) Journal of Applied Mechanics, Vol. 82,pp. 071016-1-6 (2015).

9) D. Bigoni, “Nonlinear Solid Mechanics”, CambridgeUniversity Press (2012).

10) Y. Shibutani and A. Nakatani, “Mechanics of Materi-als”, Corona Publishing Co., Ltd. (2017).

11) D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni,“Structures buckling under tensile dead load”, Pro-ceedings of the Royal Society A, Vol. 467, pp. 1686-1700 (2011).


11-2017-0103-(p.202-207).indd 207 2018/01/10 20:34:29

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