Universal Shapes Formed by Two Interacting Cracks

We investigate the origins of the widely observed “en passant” crack pattern, which forms through interactions between two approaching cracks. A rectangular elastic plate is notched on each long side and then subjected to quasistatic uniaxial strain from the short side. The two cracks propagate along approximately straight paths until they pass each other, after which they curve and release a lenticular fragment. We find that, for materials with diverse mechanical properties, the shape of this fragment has an aspect ratio of $2:1$, with the length scale set by the initial crack offset $s$ and the time scale set by the ratio of $s$ to the pulling velocity. The cracks have a universal square root shape, which we understand by using a simple geometric model of the crack-crack interaction.

Figure 1

(a) Experiment schematic, with side 1 stationary and side 2 pulled at constant velocity $v$. The point at which the two crack paths first cross their mutual perpendicular sets $t=0$, and $[x,y]=0$. (b)–(f) Detail of gelatin sheet showing dynamics. Detail of photoelastic response of gelatin sheet (g) before and (h) during crack curvature.

Figure 2

(a) Measurement of aspect ratio $\Gamma$ of final fragments for constant-$v$ fracture of 5 materials; the dotted line is $L=2W$. (b) Average of edges $\ell (w)$ for 8 gelatin fragments, with both lengths scaled by $s$. The dashed line is Eq. (1) with $A=1$ and $\alpha =\frac{1}{2}$, which corresponds to the model [Eq. (4)]. Inset: Sample image of fragment showing length $L$, width $W\approx s$, and extracted edge $\ell (w)$. (c) Power-law exponent $\alpha$ as a function of $W$, obtained from fitting Eq. (1); the dashed line is mean $⟨\alpha ⟩=0.49$. (d) Power-law prefactor $A$ as a function of $W$, obtained from fitting Eq. (1) with fixed $\alpha =\frac{1}{2}$; the dashed line is mean $⟨A⟩=0.97$. In (c),(d) the paired points connected by vertical bars correspond to the two sides of a single fragment. All fits to Eq. (1) are performed over the portion of $\ell (w)$ with $w/s>0.1$.

Figure 3

(a) ${\tilde{x}}_i(\tilde{t})$ and (b) ${\tilde{y}}_i(\tilde{t})$ for 12 experiments on gelatin at $v=0.7$, 1.4, and $2.9\mathrm{mm}/\mathrm{s}$. (c) Preinteraction stage: $y_1(t)$ for a single crack ($v=0.7\mathrm{mm}/\mathrm{s}$) and a double crack ($v=1.4\mathrm{mm}/\mathrm{s}$). $y_0<0$ is the starting coordinate of each crack at time $t_0<0$. (d) Interaction stage: $⟨\tilde{y}(\tilde{t})⟩$.

Figure 4

(a) Model solution (thick lines) during interaction stage ($\tilde{t}>0$), with definitions of variables. Model in (b) the final state and (c) translated from the final state to show crack edges superimposed on the experimental image. Comparison to $\lambda =1,3$ shown as thin lines in (c).