We study multiple tearing of a thin, elastic, brittle sheet indented with a rigid cone. The n cracks initially prepared symmetrically propagate radially for n≥4. However, if n<4 the radial symmetry is broken and fractures spontaneously intertwine along logarithmic spiral paths, respecting order n rotational symmetry. In the limit of very thin sheets, we find that fracture mechanics is reduced to a geometrical model that correctly predicts the maximum number of spirals to be strictly 4, together with their growth rate and the perforation force. Similar spirals are also observed in a different tearing experiment (this time up to n=4, in agreement with the model), in which bending energy of the sheet is dominant.
Isometric deformations in thin elastic films easily form ridges to connect large flat regions or facets. Depending on the forces applied or the boundary conditions imposed, these ridges can be isometric, with no stretching or "stretching ridges" when bending and stretching are required to relax the elastic energy. Here we study a simple configuration to observe the transition between an isometric ridge to the well-known stretching ridge observed in crumpled films, and obtain the parameters that determine the ridge type. Specifically, we show that the transversal size of a stretching ridge acts as a cutoff length scale: a ridge is isometric if its width is greater than this characteristic length.
This study investigates the tearing of a thin notched sheet when two points on the sheet are pulled apart. The concepts that determine the crack trajectory are reviewed in the general anisotropic case, in which the energy of the fracture depends on the fracture direction. When observed as a flat sheet a purely geometric "tearing vector" is defined through the location of the crack tip and the pulling points. Both the Griffiths's criterion and the "maximum energy release rate criterion" (MERR) predict a fracture path that is parallel to the tearing vector in the isotropic case. However, for the anisotropic case, the application of the MERR leads to a crack path that deviates from the tearing vector, following a propagation direction that tends to minimize the fracture energy. In the case of strong anisotropy, it is more difficult to obtain an analytical prediction of the tearing trajectory. Thus, simple geometrical arguments are provided to give a derivation of a differential equation accounting for crack trajectory, according to the natural coordinates of the pulling, and in the case that the anisotropy is sufficiently weak. The solution derived from this analysis is in good agreement with previous experimental observations.
We report a new oscillatory propagation of cracks in thin films where three cracks interact mediated by two delamination fronts. Experimental observations indicate that delamination fronts joining the middle crack to the lateral crack tips swap contact periodically with the crack tip of the middle crack. A model based on a variational approach analytically predicts the condition of propagation and geometrical features of three parallel cracks. The stability conditions and oscillating propagation are found numerically and the predictions are in favorable agreement with experiments. We found that the physical mechanism selecting the wavelength structure is a relaxation process in which the middle crack produces a regular oscillatory path.
The physical rules governing the tearing of a packaging or of a piece of paper are not completely elucidated despite being common phenomena in daily life. Here, we investigate how the presence of a straight rigid object, a ruler, guides the fracture of a thin sheet within a wide range of pulling directions. In the case of thin isotropic brittle sheets, a simple geometrical analysis shows that fracture follows the direction of a tearing vector, which differs significantly from the pulling direction. In addition to geometry, bending energy or material anisotropy has to be implemented in the case of thicker or anisotropic sheets, to predict the direction of propagation. A generalization of the Wulff's type construction introduced by Takei et al. [1] accounts successfully for our experimental results.
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