Summary: Finite-Distance Singularities in the Tearing of Thin Sheets
E. Bayart, A. Boudaoud,* and M. Adda-Bedia
Laboratoire de Physique Statistique, Ecole Normale Supe´rieure, UPMC Paris 06, Universite´ Paris Diderot,
CNRS, 24 rue Lhomond, 75005 Paris, France
(Received 22 January 2011; published 11 May 2011)
We investigate the interaction between two cracks propagating quasistatically in a thin sheet. Two
different experimental geometries allow us to tear sheets by imposing an out-of-plane shear loading.
A single tear propagates in a straight line independently of its position in the sheet. In contrast, we find that
two tears converge along self-similar paths and annihilate each other. These finite-distance singularities
display geometry-dependent similarity exponents, which we retrieve using scaling arguments based on a
balance between the stretching and the bending of the sheet close to the tips of the cracks.
DOI: 10.1103/PhysRevLett.106.194301 PACS numbers: 46.50.+a, 46.70.De, 62.20.mt
Thin sheets are widespread in nature and techno-
logy. Examples include insect wings, leaves or tectonic
plates, and graphene, conducting layers, or metallic roofs.
Strikingly, these objects feature two types of energy focus-
ing, at a sharp fold as in crumpled paper  or at the tip of a
crack as in a torn sheet of paper. More generally, under-
standing and predicting the propagation of a crack in a
brittle material yield a central challenge in fracture me-