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

・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]

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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

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[2] LEM

[2]”Fundamental Components for Lamina Emergent Mechanisms"

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 ?

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fig from the web (*1)

(*1) http://www.pontrilasmerchants.co.uk/products/mdf.php

Table of contents

1. Modeled relationship between pattern and resulting flexibility

2. Experiment to evaluate this model

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3. Dimensional analysis that explains characteristics of this pattern

f (a,b,l,n,E,G) = stiffness of RPM

pattern parameter : (a,b,l,n)material parameter : (E,G)

E :Yoiung 's modulusG : shear modulus

Local beam

Stiffness function & pattern parameter

We define stiffness function f

RPM

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・We view RPM as 1D elastic rod

Concept of our model

MBIP = EBIPIBIPφ MBOP = EBOPIBOPφ 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

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φT =dθTdx

θT

Stiffness functionLocal beamGlobal elastic rod

∝E n3a4bl

Ks = E12na3bl 3

∝E n3ab4

l

∝G na3.4b1.8

l

Overview of our contribution

S-mode

BOP-mode

BIP-mode

T-mode

Equation of deformation

Ps = Ksds

MBOP = EBOPIBOPφ

MT = GT JTφ

MBIP = EBIPIBIPφ

pattern parameter8

Stiffness function in stretching (S-mode)

Global elastic rod Local beam

Ps = Ksds

fs (a,b,l,n) = E12na3bl 3

< Parameter >PS

P = PS

ds

Stiffness function

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Stiffness function in bending out of plane (BOP)Global elastic rod Local beam

∵ J is torsion constant

∵φBOP =θBOP

a + gMBOP = EBOPIBOPφBOP

MBOP

M = MBOP

Stiffness function

θBOP

< Parameter >

fBOP (a,b,l,n)= G(a + g)J(a,b) (pure torsion)

∝ Gna3.5b1.6

l

⎧⎨⎪

⎩⎪∵G is material parameter (shear modulus)

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Stiffness function

EBIPIBIP ∝E n3a4bl

Ks = E12na3bl 3

GT JT ∝E n3ab4

l

EBOPIBOP ∝Gna3.4b1.8

l

Overview of our contribution

S-mode

BOP-mode

BIP-mode

T-mode

Equation of deformation

Ps = Ksds

MBOP = EBOPIBOPφ

MT = GT JTφ

MBIP = EBIPIBIPφ

pattern parameter

(*1) using warping torsion model

Local beamGlobal elastic rod

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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

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EBIPIBIP ∝E n3a4bl

Ks = E12na3bl 3

GT JT ∝E n3ab4

l

EBOPIBOP ∝Gna3.4b1.8

l

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

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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.

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Laminated material (MDF)(*1)

fiber !(stiff)

glue!(compliant)

G ≠ E2(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 = E2(1+υ)

(*1) 構造用複合材料 影山和郎著

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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

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EBIPIBIP ∝E n3a4bl

Ks = E12na3bl 3

GT JT ∝E n3ab4

l

EBOPIBOP ∝Gna3.4b1.8

l

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 >

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Experiment result in BOP-mode(2)

x: a (mm) y: stiffness = Load/Dsiplacement (N/mm)

< Parameter >

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Physical results Simulation results (Warping torsion)Simulation results (Pure torsion)

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

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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

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Thank You For Listening