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Article Dans Une Revue Physical Review Letters Année : 2020

The nature of crack path instabilities in thin sheets cut by blunt objects

Résumé

Cutting a brittle soft sheet with a blunt object leaves an oscillating crack that seemingly violates the principle of local symmetry for fracture. We experimentally find that at a critical value of a well chosen control parameter the straight propagation is unstable and leads to an oscillatory pattern whose amplitude and wavelength grow by increasing the control parameter. We propose a simple model that unifies this instability with a related problem, namely that of a perforated sheet, where through a similar bifurcation a series of radial cracks spontaneously spiral around each other. We argue that both patterns originate from the same instability. A problem under active development in fracture theory concerns the prediction of the crack path and the associated instabilities: when a piece of material breaks, what determines the shape of the resulting pieces? In this respect, an oscillatory instability occurring in quasi-static propagation of cracks in thermally quenched strips of glass [1] has played an important role in the development of theories for unstable fracture path. Such a simple and clear situation was indeed useful as a test case for theoretical approaches, and has stimulated a number of studies over the past years [2-4]. Similar instabili-ties have been observed in oscillatory cracks in stretched rubber [5], drying colloidal films [6], and in the failure of coatings [7], also triggering theoretical developments [8]. In this Letter we analyze two seemingly different crack path trajectories in brittle thin elastic sheets (an oscilla-tory and a spiral path), and show that they both result from the same instability mechanism, by identifying the common control parameter. When a thin elastic film, clamped along its edges, is cut by a blunt tool displaced parallel to the sheet (config-uration S, for Straight, in Fig.1a), the expected straight cut is not observed [9-11], but instead an oscillatory path develops along the tool trajectory, breaking the left-right symmetry (Fig. 1c). In a different situation (configu-ration C, for Conical), when a conical tool perforates a brittle sheet (Fig. 1d), N cracks may propagate with a radial straight trajectory when N ≥ 4. But when N ≤ 3, intertwined spiraling trajectories [12] are observed. Both experiments suggest that the straight path is unstable despite the symmetry of both systems. Previous works focused on the developed patterns (oscillatory and spiral), with both geometries correctly captured by a simplified theory for tearing [9, 12, 13], but fail in explaining why the straight path is not observed. In this article we derive a more general framework that captures this feature , and compare its predictions with an experimental setup dedicated to study the instability conditions. We start by reporting a disregarded experimental fact in previous experiments with configuration S (Fig. 1a). A rectangular sheet (bi-oriented polypropylene with thick-a) c) 100mm Radially stable (N=4) Radially unstable (N=3) 100mm f) d) W Stable Unstable Clamps Tool Crack tip w e) g) h) Configuration S Configuration C Tool b) FIG. 1. (a-c) Configuration S and oscillatory crack instability: a) Setup: a rigid tool of width w with rectangular section is driven along a clamped sheet (width W w); b) upper view: the white region is the convex hull H of the cut and the lower edge of the sheet, while the clear grey region represents the material that is stretched due to the pushing tool; c) scanned crack path for w = 15mm, W = 155mm with a long straight path before oscillatory instability appears at ψ ≈ 60 •. (d-h) Configuration C and spiral crack instability: d) a rigid cone is driven across a clamped sheet with N = 4 initial radial cuts; e-f) ongoing perforations with N = 4 (e) and N = 3 (f); g-h) corresponding scanned crack paths: stable (g) radial path for N = 4 and unstable (h) radial path for N = 3 leads to three intertwined spiral paths. ness t = 30µm, length 900mm and width W = 148mm) is clamped along its two long edges and prepared with a centered notch (5 to 10mm long) on its lower (short) edge. The tool has a rectangular section, whose width w = 15mm (the only relevant dimension), is displaced at
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Dates et versions

hal-03015219 , version 1 (19-11-2020)

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Eugenio Hamm, Iryna Sivak, Benoît Roman. The nature of crack path instabilities in thin sheets cut by blunt objects. Physical Review Letters, 2020, 124 (17), ⟨10.1103/PhysRevLett.124.174101⟩. ⟨hal-03015219⟩
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