Kirigami Meta-Sheet for Enhanced Impact Absorption

1 Introduction

• Reusable absorption requires large deformation, energy dissipation, and recovery without sacrificial failure, while common vertical-stack absorbers are less suited to thin, broad protection.A planar sheet can spread impact into lateral deformation and be held mainly along its edge, reducing reliance on a ground or backing surface.Source: §1 · pp. 1–2

2.1 Kirigami Meta-sheet Design

• Edge-constrained kirigami units spread local out-of-plane impact laterally, and compliant low-hinge-ratio geometry enables negative stiffness; the planar pattern can be enlarged by tiling.The design converts impact into hinge buckling, unit rotation, and progressive lateral spreading; sheet-level manufacturing routes include cutting, molding, and lamination.Source: §2.1 · pp. 2–3, §1 · p. 2

2.2 Effect of Stiffness Profiles on Impact Mitigation

• Under the same modeled pulse, NSR dissipated normalized energy W = 0.46, versus 0.13 for ZSR and 0.02 for PSR.The model attributes this advantage to a positive–negative–positive stiffness sequence: negative stiffness reduces transmitted force, then hardening limits displacement and restores load capacity.Source: §2.2 · pp. 3–4

2.3 Quasistatic Indentation for 3D-printed Kirigami Meta-sheets

• LHR showed the clearest negative-stiffness transition, while HHR stiffened monotonically and MHR was intermediate.FE simulations localized stress near hinges and hinge–unit junctions; maximum von Mises stresses were 47.49 MPa for LHR, 40.60 MPa for MHR, and 32.95 MPa for HHR. LHR also provided the largest normalized dissipation despite HHR's larger absolute dissipated energy.Source: §2.3 · p. 4

2.4 Drop-tower Impact Test and Dynamic Dissipation

• LHR absorbed-energy fraction was 0.584 ± 0.003 (MHR 0.527 ± 0.006; HHR 0.444 ± 0.004), and LHR vibration-mediated dissipation was 10.36 ± 0.70 mJ.First impact force followed LHR < MHR < HHR; the accelerometer-derived energy is a reference indicator, not a full-field energy balance.Source: §2.4 · p. 5

2.5 Egg Drop Test

• The kirigami meta-sheet protected both a jumbo egg and the absorber, while polyethylene mesh and styrofoam failed to protect the egg.The egg remained intact, and the meta-sheet showed no visible tearing, hinge fracture, or permanent crushing; the mesh was too stiff, and the styrofoam fractured.Source: §2.5 · pp. 5–6

3 Conclusion

• The positive-negative-positive stiffness transition produced the lowest rebound coefficient, highest absorbed energy fraction, smallest first impact force, and largest dissipation in drop-tower experiments.The meta-sheet achieves these effects without sacrificial crushing.Source: §3 · p. 6
• Local out-of-plane impact drives distributed hinge buckling, unit rotation, stiffness transitions, and vibration, allowing softening followed by hardening without sacrificial crushing.The planar architecture avoids the vertical stacking and backing surface limitations of many metamaterial-based absorbers.Source: §3 · p. 6
• The planar meta-sheet can be enlarged by geometric scaling, tiled over broad areas, and manufactured using laser cutting, casting, molding, or lamination.These sheet-based processes support scalable impact attenuation.Source: §3 · p. 6
• Further work for repeated use should address fatigue-resistant hinges, multilayer assemblies, standardized peak-force measurements, and scalable manufacturing.Hinge reinforcement and elastomeric lamination may suppress hinge stress concentration while preserving the negative stiffness transition.Source: §3 · p. 6

4 Experimental Section

• Printed LHR, MHR, and HHR specimens had calibrated hinge ratios of 0.08045, 0.09739, and 0.15682, respectively, and were frame-fixed to reproduce modeled edge constraints.The specimens were designed to deploy into dome-shaped configurations and tested as flat printed sheets before deployment.Source: §4 · p. 6
• Quasistatic indentation used an 80 mm min−1 crosshead speed and 30 mm chuck; loading–unloading hysteresis energy was calculated by numerical integration.The FE indentation models used calibrated hinge geometry and material properties, experimental indenter geometry, and matching edge constraints to inspect buckling, inversion, lateral spreading, and localized hinge stress.Source: §4 · p. 7
• Drop-tower tests used a 0.203 kg assembly, synchronized imaging and force/acceleration recording, and camera-derived impact and rebound velocities.A free-fall trajectory fit supplied a camera time correction factor of K = 1.4359; sensor data retained the hardware time base.Source: §4 · p. 7
• Impact signals were calibrated and filtered, and damping analysis excluded trials with impact-induced velocity changes outside the 1–8 m s−1 acceptance range.Impact start and end were identified using force thresholds below 1 N around the first force value exceeding 3 N and the post-peak force decline.Source: §4 · pp. 7–8

Egg Drop Test:

• Egg-drop tests compared a larger kirigami meta-sheet with 3.2 mm PE mesh and 10 mm styrofoam using jumbo eggs released 1383.8 mm above the absorber.A clamp assembly maintained identical egg positioning across tests.Source: Egg Drop Test · p. 8

setup

• The planar pattern followed a previously reported computational design procedure, was printed flat, and was fixed to a reinforced frame to reproduce the simulation boundary condition.Source: setup · p. 21
• 0.06250, 0.09000, and 0.15000 were the nominal hinge ratios for the low-, medium-, and high-HR designs, respectively.One grid cell represented 1 cm, and fabricated hinge dimensions were measured because printing can deviate, particularly at narrow hinges.Source: setup · p. 21
• 14.25 mm was used as the effective unit-cell length to normalize measured hinge thicknesses, and the resulting as-fabricated hinge ratios—not nominal CAD values—were used in ABAQUS.The note reports that representative hinge thicknesses were measured near the sheet center and that calibrated values are summarized in Table S2.Source: setup · pp. 21–22
• The mesh used hybrid eight-node brick elements, with five elements through the thickness and eight seeded in the hinge region, to resolve large-deformation bending and localized hinge stress.The note also describes using wedge and tetrahedral elements where required by geometry.Source: setup · p. 23
• 80 mm/min was the indentation speed, and all three hinge-ratio models were indented 25 mm from contact for the stress-field comparison.The quasistatic simulations used Dynamic Implicit displacement control, HHT α = −0.05, and geometric nonlinearity.Source: setup · p. 23

Supplementary Note 3. Reinforced PLA frame and specimen assembly

• A printed PLA frame constrained the meta-sheet perimeter while exposing the central kirigami region; PLA balanced printability, stiffness, and resistance to parasitic bending.Source: Supplementary Note 3. Reinforced PLA… · p. 23
• Because an outer-boundary-only frame bent noticeably during testing, side-region ribs reduced parasitic bending while leaving accessible fishing-line tying points.Source: Supplementary Note 3. Reinforced PLA… · p. 24
• After printing, fishing lines fixed the meta-sheets to frames and adhesive locked the boundaries; Figure S2 shows LHR, MHR, and HHR specimens.Source: Supplementary Note 3. Reinforced PLA… · p. 24, Figure S2 · p. 24

Supplementary Note 4. Indenter and drop-tower chuck fabrication

• Both PLA chucks used a 30 mm contact diameter to keep contact geometry consistent across indentation and drop-impact tests.The shared diameter was used for both experimental protocols.Source: Supplementary Note 4. Indenter and… · p. 25
• The indentation chuck was a single cylindrical part for the tensile machine, while the drop-test chuck comprised male and female parts screwed together for drop-tower attachment.Figure S3 shows the printing layouts and mounted chucks.Source: Supplementary Note 4. Indenter and… · pp. 25–26

Supplementary Note 5. Experimental setup and instrumentation

• The indentation specimen rested on two leveled laboratory lifts, and a 30 mm PLA chuck on a universal tensile machine contacted its central region vertically.The frame was leveled before each test.Source: Supplementary Note 5. Experimental… · p. 26
• A Bruel & Kjaer Type 4393 accelerometer measured local surface acceleration one unit cell from impact, on a connection between triangular units, without obstructing direct contact.Its signal was amplified and recorded using an oscilloscope.Source: Supplementary Note 5. Experimental… · p. 26
• The accelerometer reference sensitivity was 0.3206 pC/(m.s−2), equivalent to 3.144 pC/g at 159.2 Hz, with voltage converted at 10 mV/(m.s−2).Source: Supplementary Note 5. Experimental… · p. 26

Supplementary Note 6. High speed video time calibration

• Although the tracking software used a 0.00033 s time step (3000 fps), the free-fall segment showed acceleration substantially above gravity, indicating compressed video timing.Source: Supplementary Note 6. High speed video… · p. 27
• The video time axis was calibrated with a free-fall constraint, yielding K=1.4359 for correcting measured timestamps.The correction factor was derived from the relationship between measured and true acceleration.Source: Supplementary Note 6. High speed video… · pp. 27–28
• For a fall distance of h = 0.428m, the physical fall time was ttrue ≈0.295s, compared with tmeas ≈0.179s in raw tracking data.Source: Supplementary Note 6. High speed video… · p. 27
• The time correction applied only to video-derived tracking data; accelerometer data retained the hardware-clock sampling interval of ∆t = 0.001s.Source: Supplementary Note 6. High speed video… · p. 28

Supplementary Note 7. Video-derived velocity and impact velocity

• A 51-frame Savitzky–Golay filter smoothed corrected displacement before differentiation, reducing pixel-tracking noise without suppressing the sharp pre-impact velocity peak.The resulting velocity was used to estimate impact velocity before contact.Source: Supplementary Note 7. Video-derived… · p. 28
• For each trial, impact velocity v0 was defined as the maximum downward velocity immediately before contact.Source: Supplementary Note 7. Video-derived… · p. 28
• Average calibrated impact velocities for the three specimen groups are reported in Table S4, including rail-friction and small aerodynamic-loss effects.These values were used for rebound and damping calibration.Source: Supplementary Note 7. Video-derived… · pp. 28–29, Table S4 · p. 29

• The same accelerometer protocol measured local vertical acceleration of the kirigami surface in the LHR, MHR, and HHR groups.Source: Supplementary Note 8. Accelerometer… · p. 29
• The raw accelerometer signal was recorded as amplified voltage V(t), with a hardware baseline offset of approximately −50mV that shifted uncorrected traces away from zero.Source: Supplementary Note 8. Accelerometer… · p. 29
• A pre-impact baseline computed independently for each trial corrected the bias, establishing a common zero-acceleration reference and preventing amplifier offsets from being read as physical acceleration.Source: Supplementary Note 8. Accelerometer… · p. 29

Supplementary Note 9. Filtering and time alignment of acceleration data

• A first-order Butterworth high-pass filter (fc = 2.0 Hz) suppressed DC offsets and low-frequency drift that can cause nonphysical velocity drift during integration.The filter order was n = 1.Source: Supplementary Note 9. Filtering and… · p. 30
• The filtered acceleration was integrated to obtain velocity and displacement changes.Source: Supplementary Note 9. Filtering and… · p. 30
• For five repeated trials, impact time was the sharp acceleration rise from baseline; traces began 0.24 s before impact and spanned 3.0 s to include post-impact vibration and settling.Source: Supplementary Note 9. Filtering and… · p. 30

Supplementary Movies

• Movie S1 shows front-view indentation experiments for LHR, MHR, and HHR kirigami meta-sheets.For each specimen group, the front-view recording is followed by a corresponding bottom-view recording.Source: Supplementary Movies · p. 31
• Movie S2 shows drop-tower impact experiments of LHR, MHR, and HHR kirigami meta-sheets.Each drop-impact recording is followed by a high-speed surface recording.Source: Supplementary Movies · p. 31
• Movie S3 compares egg-drop protection by the larger kirigami meta-sheet, polyethylene (PE) mesh, and styrofoam control absorbers.Source: Supplementary Movies · p. 31