Figure 1

Experimental setups for tests of [(a) and (b)] uniaxial and (c) radial compressions of [(a) and (c)] randomly folded paper and (b) aluminium foil.Reuse & Permissions

Figure 2

(a) Typical force $\left(F\right)$-compression $\left({\lambda }_{\parallel }\right)$ curve of randomly folded paper under the uniaxial compression: (1) loading, (2) unloading, and (3) relaxation shown in (d). (b) Typical stress (megapascals)–strain (arbitrary units) behavior of randomly folded aluminium foil under uniaxial compression: (1) loading and (2) unloading. (c) Compression force $F$ (newtons) versus time $t$ (hours) for fixed compression ratios ${\lambda }_{\parallel }=0.5$. (d) Lateral expansion ratio ${\lambda }_{\perp }$ versus time $t$ (hours) for fixed compression ratios ${\lambda }_{\parallel }=\left(1\right)$ 0.8, (2) 0.5, and (3) 0.3.Reuse & Permissions

Figure 3

(a) Perimeter $P$ (millimeters) versus height $H={\lambda }_{\parallel }{H}_0$ for axially compressed cylinders folded from sheets of paper of thickness $h=0.039\mathrm{mm}$ and size $L=500\mathrm{mm}$ with different contraction ratios $K=\left(1\right)$ 3.9, (2) 5.6, and (3) 8.3. (b) Lateral expansion ratio ${\lambda }_{\perp }$ versus axial compression ratio ${\lambda }_{\parallel }$ for all folded paper sheets tested under uniaxial compression; open symbols correspond to cylinders folded from sheets of size $250\mathrm{mm}$ and full symbols to cylinders folded from sheets of size $500\mathrm{mm}$ from papers of thickness: [(1) and (2)] 0.024, [(3) and (4)] 0.030, [(5) and (6)] 0.039, [(7) and (8)] 0.068, and [(9) and (10)] $0.087\mathrm{mm}$ with different contraction ratios; solid line—data fitting with the scaling relations [Eq. (2)] with $\nu =0.17$ (insert shows the same graph plotted in log-log coordinates). (d) Axial expansion ratio ${\lambda }_H$ versus radial compression ratio ${\lambda }_R$ for radially compressed cylinders folded from sheets of paper of size $L=500\mathrm{mm}$ and thickness of [(1) and (2)] 0.024 and [(3) and (4)] $0.068\mathrm{mm}$ with contraction ratios (1) and (3) and (2) and (4); symbols—experimental data and solid line—data fitting by the scaling relations [Eq. (4)] with $\alpha =0.2$.Reuse & Permissions

Figure 4

(a) Lateral expansion ratio ${\lambda }_{\perp }$ versus axial compression ratio ${\lambda }_{\parallel }$ for randomly folded aluminum foil with size of $500\times 500{\mathrm{mm}}^2$; circles—experimental data, doted line—data fitting with equations [Eq. (3)] with $\nu =0.22$ $(R^2=0.981)$, and solid line—data fitting with Eq. (1) with ${\nu }_e=0.34$ $(R^2=0.998)$. (b) Lateral strains $-{\epsilon }_{\perp }=[P-P\left(0\right)]∕P\left(0\right)$ versus axial strains ${\epsilon }_{\parallel }=(R_0-R)∕R_0$ for all aluminum foils tested in this work; solid line—data fitting with ${\nu }_e=1∕3$ $(R^2=0.974)$.Reuse & Permissions