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

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

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

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)

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

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

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

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

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)

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

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

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

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

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)

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

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

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

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

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

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

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