Comment on ‘‘Crack Street: The Cycloidal Wake of a Cylinder Tearing through a Thin Sheet’’ A new mode of oscillatory fracture in brittle thin films

A new mode of oscillatory fracture in brittle thin films has recently been reported in two independent experiments [1,2]. When an object is driven through a thin film clamped along its lateral boundaries, the film tears following a striking sawtooth pattern, as the crack tip propagates in an oscillatory manner. In their Letter, Ghatak and Mahadevan [1,3] make an analogy with the classical construction of a cycloid and claim that the crack path consists of series of arches of prolate cycloids (APC). This approach is based on the assumptions that (i) the crack tip moves along an arc of circle in the cutting tool’s frame of reference and (ii) it does so with an angular velocity that is constant with an alternating sign. Whereas these assumptions roughly mimic the oscillatory behavior, they bear some important limitations. First, they lack a physical justification, making their construction an ad hoc description of the experiments. For instance, this construction is justified by the fact that the crack tip moves ‘‘on the circumference of the tool’’ but a heuristic fitting parameter, > 1, is later introduced in the model, implying that the crack is ahead of the tool. Second, these assumptions are inconsistent with some of the experimental observations. In particular, (i) in our own experiments [2], we have measured fluctuations of the distance of the crack tip to the center of the cutting tool by as much as 40% of the tool radius and (ii) the velocity of the crack tip presented in their Fig. 4(b) is far from constant, even within the quasistatic regime (stick phase). In the absence of a solid physical basis, the claim that the morphology of fracture paths is cycloidal is supported solely by a superposition of an experimental path with an APC curve, shown in their Fig. 2(c). Although visually satisfactory, the similarity of the curves mostly reflects the ingredients that have been included in the construction by hand, namely, the periodicity, existence of angular points, and invariance under axial symmetry combined with translation by a half-period. With the aim of testing the APC construction on a more quantitative ground, we have performed an error analysis of this model: we compared its predictions to those of two reference models which were purposely chosen to be unphysical. To avoid any bias, the comparison was based on the experimental crack path presented by the authors in Fig. 2(c), and the reference models were chosen with the same number of adjustable parameters, two, as in the APC construction. The first reference model is a plain sinusoid y x sin x , with adjustable amplitude and wave number . Note that experimental patterns are far from sinusoidal [1]; hence, as intended, this model should provide a poor fit and establish an upper bound on the error estimation. The other reference model is based on repli-