2D repetitive pattern includes kerfing,dukta® and Lamina emergent mechanisms (LEM). This content was presented in IASS in August 2015

1. Analysis and design of elastic material
formed using 2D repetitive slit pattern
Taisuke Ohshima[1], Tomohiro Tachi[1], Hiroya Tanaka[2],
Yasushi Yamaguchi[1]
!
[1]The University of Tokyo , [2] Keio University

2. ・Kerfing / Dukta® [1]
・Zigzag spring / Serpentine spring [2]
・Lamina Emergent Mechanisms(LEM) [3]
2D repetitive slit pattern
[3]An Introduction to Multilayer Lamina Emergent Mechanisms L. Delimont et.al
[2]from the web
[1] [2]
[3]
2

3. Applications of 2D repetitive slit pattern
Spring stool
by Carolien Laro
[1] US-Patent by Apple in 2013
Elastic buffer
Elastic hinges
[1]“Interlocking flexible segments formed from a rigid material” US 2013/0216740 A1
Bending
Kerf Pavilion @ MIT
Actuator or Deployable structure
3
[2] LEM
[2]”Fundamental Components for Lamina Emergent Mechanisms"

4. Research questions
High stiffness Processed flexible
Repetitive Pattern Material
(RPM)
-3D printing
-CNC cutting
・How does this pattern enable materials to be flexible ?
・How do we utilize this patten for designing flexibility ?
4
fig from the web (*1)
(*1) http://www.pontrilasmerchants.co.uk/products/mdf.php

5. Table of contents
1. Modeled relationship
between pattern and resulting flexibility
2. Experiment to evaluate this model
5
3. Dimensional analysis that explains characteristics
of this pattern

6. f (a,b,l,n,E,G) = stiffness of RPM
pattern parameter :(a,b,l,n)
material parameter :(E,G)
E :Yoiung's modulus
G :shear modulus
Local beam
Stiffness function & pattern parameter
We define stiffness function f
RPM
6

7. ・We view RPM as 1D elastic rod
Concept of our model
MBIP = EBIP IBIPφ MBOP = EBOP IBOPφ MT = GT JTφTPs = Ksds
< Stiffness function in each deformation >
fBIP (a,b,l,n,E,G)
Stiffness function in BIP-mode
Stretching Bending in plane Bending out of plane Twisting
S-mode BIP-mode BOP-mode T-mode
7
φT =
dθT
dx
θT

8. Stiffness
functionLocal beamGlobal elastic rod
∝ E
n3
a4
b
l
Ks = E
12na3
b
l3
∝ E
n3
ab4
l
∝G
na3.4
b1.8
l
Overview of our contribution
S-mode
BOP-mode
BIP-mode
T-mode
Equation of
deformation
Ps = Ksds
MBOP = EBOP IBOPφ
MT = GT JTφ
MBIP = EBIP IBIPφ
pattern parameter
8

9. Stiffness function in stretching (S-mode)
Global elastic rod Local beam
Ps = Ksds
fs (a,b,l,n) = E
12na3
b
l3
< Parameter >
PS
P = PS
ds
Stiffness function
9

10. Stiffness function in bending out of plane (BOP)
Global elastic rod Local beam
∵J is torsion constant
∵φBOP =
θBOP
a + g
MBOP = EBOP IBOPφBOP
MBOP
M = MBOP
Stiffness function
θBOP
< Parameter >
fBOP (a,b,l,n)
= G(a + g)J(a,b) (pure torsion)
∝
Gna3.5
b1.6
l
⎧
⎨
⎪
⎩⎪
∵G is material parameter (shear modulus)
10

11. Stiffness
function
EBIP IBIP ∝ E
n3
a4
b
l
Ks = E
12na3
b
l3
GT JT ∝ E
n3
ab4
l
EBOP IBOP ∝G
na3.4
b1.8
l
Overview of our contribution
S-mode
BOP-mode
BIP-mode
T-mode
Equation of
deformation
Ps = Ksds
MBOP = EBOP IBOPφ
MT = GT JTφ
MBIP = EBIP IBIPφ
pattern parameter
(*1) using warping torsion model
Local beamGlobal elastic rod
11

12. Dimensional Analysis
< Parameter >
pattern parameter :
(a,b,l,n)
material parameter :
(E,G)
a lb n
4 1 -1 3
(3.7)
32 -1 3
3 -3 11
1-1.11.83.4
a = 4mm,b = 5mm,l = 50mm, n = 2,
1≤ a ≤ 5, 4 ≤ b ≤ 8,40 ≤ b ≤ 80,1≤ n ≤ 8
⎧
⎨
⎪
⎩⎪
(3.2)
(1.7)
S-mode
T-mode
BOP-mode
12
EBIP IBIP ∝ E
n3
a4
b
l
Ks = E
12na3
b
l3
GT JT ∝ E
n3
ab4
l
EBOP IBOP ∝G
na3.4
b1.8
l

13. Suitable pattern for elastic hinge
・S-mode has high sensitivity about l
・BIP- and T-mode have high sensitivity about n
Decreasing l and increasing n realize compliant in
BOP -mode but stiff in the other modes
Sensitive parameter
Elastic hinges
13

14. Experiment result in BOP-mode
Physical testComputer simulation
・Used medium density fiber broad (MDF)
・Measured load and displacement with three-point bending
・Tested multiple samples by scaling pattern parameter.
14

15. Laminated material (MDF)(*1)
fiber !
(stiff)
glue!
(compliant)
G ≠
E
2(1+υ)
Shear modulus G of laminated materials (MDF)
E = 1261MPA Giso = 934 MPA (isotropic)
Glm = 126 MPA (laminated)
Measured shear modulus
Measured shear modulus
Measured G is ten times lower than isotropic G
G =
E
2(1+υ)
(*1) 構造用複合材料 影山和郎著
15

16. Dimensional Analysis
< Parameter >
pattern parameter :
(a,b,l,n)
material parameter :
(E,G)
a lb n
4 1 -1 3
(3.7)
32 -1 3
3 -3 11
1-1.11.83.4
a = 4mm,b = 5mm,l = 50mm, n = 2,
1≤ a ≤ 5, 4 ≤ b ≤ 8,40 ≤ b ≤ 80,1≤ n ≤ 8
⎧
⎨
⎪
⎩⎪
(3.2)
(1.7)
S-mode
T-mode
BOP-mode
16
EBIP IBIP ∝ E
n3
a4
b
l
Ks = E
12na3
b
l3
GT JT ∝ E
n3
ab4
l
EBOP IBOP ∝G
na3.4
b1.8
l

17. Experiment result in BOP-mode(1)
x: l (mm) y: stiffness = Load/Dsiplacement (N/mm)
Physical results
Simulation results (Warping torsion)
Simulation results (Pure torsion)
< Parameter >
17

18. Experiment result in BOP-mode(2)
x: a (mm) y: stiffness = Load/Dsiplacement (N/mm)
< Parameter >
18
Physical results
Simulation results (Warping torsion)
Simulation results (Pure torsion)

19. Conclusion
・Proposed model explains local beam deformation
determines stiffness of RPM
・Experiment result indicates this model is valid
in BOP-mode
・Dimensional analysis explains how stiffness of RPM
scales with changing pattern parameter
・We propose design guideline for elastic hinge with
dimensional analysis and experiment
19

20. Future work
・Implementing system to simulate and design
elastic bending(hinge)
・Modeling buckling condition of local beam
・Utilizing this pattern for deployable structure
・Finishing experiment for the other deformation cases
20

21. Thank You For Listening