bifurcation analysis of periodic kirigami structure with
TRANSCRIPT
†
∗ • ∗∗ • ∗∗ • ∗∗∗
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
0.50L0, mode I0.50L0, 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−
Ly=L0, mode IL0, mode I, III
0.75L0, mode I0.75L0, mode I, III
0.50L0, mode I0.50L0, mode I, III
0.375L0(λ=h), mode I0.25L0, mode I
(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−
λ = hBifurcation points
(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−
Ly=L0, mode IL0, mode I, III
0.75L0, mode I0.75L0, mode I, III
0.50L0, mode I0.50L0, 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−
Ly=L0, mode IL0, mode I, III
0.75L0, mode I0.75L0, mode I, III
0.50L0, mode I0.50L0, mode I, III
0.375L0(λ=h), mode I0.25L0, mode I
(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−
λ = hBifurcation points
(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)
205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 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
Geometrical condition
(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
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
Geometrical condition
(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).
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