1. Cold Gas Pressure Folding of Miura-ori Sheets
Mark Schenk, Julian M. Allwood, Simon D. Guest

2. Miura-ori sheets - description
Introduction
Miura-ori sheet (literally: Miura-fold) named after Koryo Miura, who introduced it to engineering applications
Miura-ori sheet

3. Miura-ori sheets - description
developable textured sheet
can be folded from flat sheet material
1DOF mechanism
Description
can be folded from flat sheet material, only bending along folds
when regarded as hinged-panels, it is a 1DOF system

4. Miura-ori sheets - applications
folded sandwich panel cores
deployable structures
flexible surfaces
Applications
folded sandwich panel cores
changing fold pattern gives curved core
open ventilation channels (no moisture build-up due to condensation)
can fold high-performance but brittle materials such as resin-impregnated fibre paper (Nomex/Kevlar)
deployable structures : 1 Degree of Freedom
deployable solar array
flexible surfaces (CUED Advanced Structures Group)
in-plane flexibility
double-curvature
Heimbs (2007)

5. Manufacturing - folding
manufacturing solution : folding
deep texture with only low-energy bending operations
challenges
large reduction in surface area
kinematic coupling (1DOF)
Manufacturing by Folding
deep texture with only low energy bending operations
Challenges
large reduction in area
can’t use fixed tooling
biaxial slip between material and tooling
kinematic coupling
difficult to have both folded and unfolded regions in the sheet

6. Manufacturing – folding methods
synchronous
folding takes place along all folds simultaneously
gradual folding
sheet gradually transitions from flat to fully folded
pre-gathering
material is corrugated before adding double-corrugation
Existing Folding Methods
overview in PhD thesis - Schenk (2011)
Synchronous
folding takes place along all folds simultaneously
Gradual Folding
sheet gradually transitions from flat to fully folded
e.g. using series of rollers, or by embossing pattern on sheet material and buckling into shape
Pre-Gathering
material is corrugated before adding double-corrugation

7. Manufacturing – folding methods
synchronous
folding takes place along all folds simultaneously
deformations: only bending along fold lines
Synchronous
folding takes place along all folds simultaneously
e.g. vacuum actuated
deformations: only folding along fold lines

8. Manufacturing – folding methods
gradual folding
sheet gradually transitions from flat to fully folded
e.g. embossing of pattern, followed by local buckling
deformations: bending along folds and facets
Gewiss (1959)
Gradual Folding
sheet gradually transitions from flat to fully folded
e.g. using series of rollers, or by embossing pattern on sheet material and buckling into shape
deformations: necessarily involves bending of facets

9. Manufacturing – folding methods
pre-gathering
material is corrugated before adding double-corrugation (reduces to successive uni-axial contractions)
deformations: inversion of fold lines, shifting of folds and vertices through material, in-plane strain
Pre-Gathering
material is corrugated before adding double-corrugation
reduces to successive uni-axial contractions
Deformations : may include inversion of fold lines, shifting of fold and vertices through the material, in-plane strain.
Although literature suggests that the deformations are surprisingly minor, and folding is very closely approximated.
Ichikawa (1995)

10. Cold Gas Pressure Folding
pre-weakening along fold lines
hinged spacer plates
Cold Gas Pressure Folding
minimal initial tooling
pre-weakening of fold lines (CNC Milling / waterjetting)
hinged spacer plates (waterjetting)
sandwich of two sheets with spacer plates

11. Cold Gas Pressure Folding
folding process :
vacuum + external pressure (autoclave)
biaxial contraction
Cold Gas Pressure Folding
actuation
internal volume is strictly decreasing function
vacuum + external pressure enforces folding
bi-axial contraction enabled through moving spacer plates

12. Plastic Analysis – work balance
internal work : plastic hinge with Mp along fold lines
external work : applied pressure p
kinematic relationship between dV and dα in θ :
Plastic Analysis – Work Balance
internal work : plastic hinge with Mp along fold lines
external work : applied pressure p
quasi-static equilibrium
kinematics relates dV and dα in single parameter θ

13. Plastic Analysis – kinematics
unit cell kinematics
geometry : a, b, 
fold angle : q
two coupled fold angles (q and )
Unit Cell Kinematics
parallelogram geometry
side lengths, a,b
included angle, 
unit cell configuration
fold angle θ
fold angles along two fold lines
side length a, fold angle (π/2-)
side length b, fold angle θ

14. Plastic Analysis – internal work
internal work : plastic hinges
k : pre-weakening along fold lines
σy : yield stress
t : sheet thickness
Internal Work
plastic hinge along all fold lines
fold lines are pre-weakened by (1-κ)

15. Plastic Analysis – external work
external work : displaced volume V
initially dVtop dominant, later dVside
Vtop reaches maximum at θ=45°
External Work
displaced volume V combination of volume Vtop top surface, and Vside around perimeter
Vtop reaches maximum (and thereby dVtop=0) at θ=45°
after θ=45° the external pressure works to unfold the sheet, but exceeded by effect of pressure around perimeter
NB: the height of the spacers will affect the volume calculations; we choose the height to be twice the desired fold depth
Vtop
Vside

16. Plastic Analysis – work balance
external pressure p at every fold depth
more interesting: max pressure for desired fold depth
Work Balance
required external pressure p can be calculated at every fold depth (quasi-statically)
more interesting is the maximum pressure required to attain desired fold depth
normalised maximum pressure
A0 takes into account a,b,
teq is the equivalent weakened thickness; teq=t √κ
pmax : maximum pressure
A0 : surface area of flat unit cell
teq : equivalent weakened thickness
sy : yield stress

17. Plastic Analysis – Maximum Pressure
contour plot of maximum normalised pressure for varying  with a/b=1
for geometry chosen in our case: p = 9.3 [-]
geometry
a=b
fold depth

18. Trial Results Al 5241-H22 22 gauge (0.71mm) a = b = 25mm g = 60° = 0.4
experimentally
determined
σy=270MPa

19. Trial Results freely formed ridge ridge supported by spacers
Observations:
cracking at folds
fold lines avoid the holes drilled at the intended vertices
ridge supported by spacers

20. Trial Results - comparison
discrepancies:
elastic deformation
springback
vertices
strain hardening
Comparison trial results with analysis
this graph shows the quasi-static required pressure during folding
previous analysis considered unit cells; for 4x4 sheet the lack of folding around perimeter is taken into account
Discrepancies:
initially no plastic hinges (elasticity for low fold angles)
fold depth was measured after springback
more complex plastic deformations at vertices
strain hardening

21. Discussion Conclusion Further analysis
successfully manufactured Miura-ori sheet
existing plastic analysis underestimates required pressure
Further analysis
more accurate measurement of Mp
more experimental validation
Finite Element Analysis of vertex deformation
analysis of springback
Conclusions
successfully manufactured Miura-ori sheet from aluminium sheet
plastic analysis underestimates required forming pressure
Further Analysis
more accurate measurement of Mp(θ)
e.g. Shanley column
more experimental validation
necessary pre-weakening κ of fold lines
(some trials with κ>0.4 did not fold as intended)
effect of sheet thickness and pattern geometry
Finite Element Analysis of deformations at vertex
springback