Cracks in Thin Sheets: When Geometry Rules the Fracture Path

We study the propagation of brittle fractures coupled to large out-of-plane bending, as when a brittle elastic thin sheet is cut by a moving object. Taking into account the separation of the film’s bending and stretching energies and using fracture theory we show that such cracks propagate according to a simple set of geometrical rules in the limit of small thickness. In particular, this provides some insight into the geometrical origin of the oscillatory fracture patterns reported in two recent experiments. Numerical integration of our geometrical rules accurately reproduces both the shape of the fracture pattern and the detailed time evolution of the propagation of the crack tip, for various geometries of the cutting object.

Figure 1

(a) Schematic of an oscillatory path $P$, obtained when a cutting tool $I$, is driven at constant velocity $v$, through a thin polymer film clamped at two of its lateral boundaries [6]. $\lambda$ and $A$ are the wavelength and amplitude of the pattern, respectively. (b) Our 2D model divides the film into three regions: the soft region (white) where the presence of the tool I is accommodated by mere bending of the film, the active region $(TVU)$ (dark gray) where the elastic energy is stored and the outer region (light gray).

Figure 2

Typical frames from numerical simulations for various dimensionless times $\overline{t}=vt/\lambda$: (a) during quasistatic propagation; (b) just before and (c) just after a kink followed by a burst of dynamic propagation. The soft, stretched, and outer regions are shown in white, dark gray, and light gray, respectively. The circle in gray represents the cutting tool. In frame (b), the active region has two components $S$ and $S^{\prime }$ which define two nonzero angles $\varphi$ and ${\varphi }^{\prime }$ at the tip.

Figure 3

Comparison of crack tip motion from the model and the experiments for a cylindrical cutting tool ($w=20.4\mathrm{mm}$, $h=50\mu \mathrm{m}$, $v=1.2\mathrm{mm}·{\mathrm{s}}^{-1}$). The model’s parameters were $\alpha =0.20$ and $\beta =0.03$. (a) Time series for the crack tip coordinates $x(t)/w$ and $y(t)/w$ as a function of nondimensionalized tool advance, $vt/w$. Note the periodic gaps in crack position, corresponding to dynamic propagation. (b) Final pattern in the film plane ($x$, $y$).

Figure 4

(a) Comparison of experimental (data points) and numerical (solid curves) crack patterns, with a cutting tool made of a dimer (two parallel cylinders with diameter $d=3.95\mathrm{mm}$, with axes separated by $l=12\mathrm{mm}$, overall width being $w=d+l$) tilted at variable angle $\theta$, as in (b). Rescaled amplitude, $A/w$, and wavelength, $\lambda /w$ are given as a function of tilt angle $\theta$. Although this tool is considerably different from that of Fig. 3, no parameter was further adjusted. Both the experimental and numerical curves deviate significantly from the naive prediction, shown in dotted line, that the pattern amplitude is given by the dimer’s frontal width, $d+l\cos \theta$.