Popping Balloons: A Case Study of Dynamical Fragmentation

Understanding the physics of fragmentation is important in a wide range of industrial and geophysical applications. Fragmentation processes involve large strain rates and short time scales that take place during crack nucleation and propagation. Using rubber membranes, we develop an experimental analysis that enables us to track the fragmentation process in situ in both time and space. We find that bursting a highly stretched membrane yields a treelike fragmentation network that originates at a single seed crack, followed by successive crack tip-splitting events. We show that a dynamic instability drives this branching mechanism. Fragmentation occurs when the crack tip speed attains a critical velocity for which tip splitting becomes the sole available mechanism of releasing the stored elastic energy. Given the general character of the fragmentation processes, this framework should be applicable to other crack networks in brittle materials.

Figure 1

Cross section of the experimental setup. The dotted line represents the initial flat state of the latex sheet before its inflation. Notations for the inflated membrane: $s(z)$ represents the arc length and $r(z)$ the distance to the axis of symmetry.

Figure 2

The two different regimes of explosion are visualized using fast imaging. (a) Opening regime. A single fracture splits the membrane in two. The vertical line appearing on the images is the rod that supports the blade used to trigger the explosion. (b) Fragmentation regime. From a single initial fracture, a network of cracks develops, eventually turning the membrane into numerous shreds (presently 15). In this example, the explosion is spontaneous. In (a) and (b), the dark disk in the center of the images is the circular frame, its inner diameter (52 mm) sets the scale. (c) Postmortem sample ($e=350\mu \mathrm{m}$). (d) Close-up extracted from the movie corresponding to the sample shown in (c). The time interval between two successive images is $1/60000\mathrm{s}$. Two successive tip splittings has turned an initial single crack into three cracks. The colored circles and arrows indicate the corresponding events on the postmortem sample.

Figure 3

Final number of cracks $N_f$ as a function of the ratio $T/e$, where $T$ is the membrane tension measured at the pole of the balloon and $e$ the initial membrane thickness. The results for four different thicknesses are displayed. Filled (open) symbols represent triggered (spontaneous) explosions. Since each measurement of $T/e$ has a relative uncertainty of the order of 2%, the dispersion of the data points illustrates the stochastic aspect of the fragmentation.

Figure 4

(a) Distance $s(t)$ covered by the fractures since their creation for membranes with initial thickness $e=800\mu \mathrm{m}$. Black symbols represent the progression of cracks in the opening regime. The velocity is constant and increases with the tension of the membrane. Colored symbols represent the progression of the extremity of a crack network in the fragmentation regime. The cracks propagate at a constant velocity, now independent of the membrane tension. The limit velocity is $v_l=570\pm 15\mathrm{m}/\mathrm{s}$. (b) Number $N$ of cracks as a function of the coordinate $r$ for experiments with $e=800\mu \mathrm{m}$. In the opening regime (black symbols), the number of cracks is constant. In the fragmentation regime (colored symbols), $N$ starts to increase linearly and eventually saturates. Panels (a) and (b) display data from the same experiments and share the same symbols. (c) Values of $G$ as defined by Eq. (3) as a function of $T/e$. As the calculation holds for the fragmentation regime, no data point is displayed in the opening regime (colored rectangle). The dashed line represents the behavior of $G$ that is consistent with the data of Fig. 3. The red dotted line indicates the estimate of $\mathrm{\Gamma }$ as the asymptotic value of $G$.