Intertwined Multiple Spiral Fracture in Perforated Sheets

We study multiple tearing of a thin, elastic, brittle sheet indented with a rigid cone. The $n$ cracks initially prepared symmetrically propagate radially for $n\ge 4$. However, if $n<4$ the radial symmetry is broken and fractures spontaneously intertwine along logarithmic spiral paths, respecting order $n$ rotational symmetry. In the limit of very thin sheets, we find that fracture mechanics is reduced to a geometrical model that correctly predicts the maximum number of spirals to be strictly 4, together with their growth rate and the perforation force. Similar spirals are also observed in a different tearing experiment (this time up to $n=4$, in agreement with the model), in which bending energy of the sheet is dominant.

Figure 1

Scanned crack paths obtained for $n=1$ to 5 initial radial notches (sheet thickness $30\mu \mathrm{m}$). Upper and lower rows correspond respectively to the pushing ($2\gamma$ is the opening angle of the cone) and pulling configurations (shown are $n=4$ initial radial notches). In the pulling case for $n=4$, spirals are obtained if notches are $S$ shaped. Horizontal bars in each subfigure are 5 cm long.

Figure 2

Semilog plot of the $n$ spiral crack paths in $30\mu \mathrm{m}$ thick sheets for (a) pushing and (b) pulling. The $n$ rotated branches of the $n$ spiral almost superpose in this graph. Straight lines evidence the exponential behavior.

Figure 3

Geometry of spiraling crack paths. (a) Perforation with a cone from a three-notch initial condition (inset), forms a three-spiral pattern (crack paths and “bending lines,” respectively, in color and dashed lines). (b),(c) Sequence of crack propagation ($A$ to $A^{\prime }$): contact point with the cone (arrowhead) moves from $P$ to $P^{\prime }$, and the tangency point from $T$ to $T^{\prime }$. (d) Elements of spiral tearing: the cone stretches the bending line $APT$ until the crack at $A$ starts to move. (e) Zero elasticity approximation: the bending line $\overline{AT}$ is straight, and the yellow crack path is the involute of the developed red crack path.

Figure 4

(a) Spiral growth rate for pushing (circle) and pulling (times) experiments ($30\mu \mathrm{m}$ sheets), and a comparison with Eq. (6) (lines). Values of $\epsilon$ are fitted separately for pushing and pulling. (Inset) Growth rate of a 2 spiral as a function of sheet thickness in a perforation experiment. (b) Perforation forces for spiral (red) and radial (black) patterns. Error bars, experiments with $t=30\mu \mathrm{m}$ thick sheets; open symbols, theoretical predictions after Eq. (7) with no adjustable parameters; dashed lines, theoretical predictions extended to noninteger values of $n$. The spiral solution (red, ${\alpha }_c=6°$, $\epsilon =0.035$, $G_{c}t=0.206\mathrm{N}$) is observed only for $n<4$, whereas the radial solution (black, ${\alpha }_c=7°$, $G_{c}t=0.288\mathrm{N}$) holds for $n\ge 4$.