Tearing a thin sheet by forcing a rigid object through it leads to complex crack morphologies; a single
oscillatory crack arises when a tool is driven laterally through a brittle sheet, while two diverging cracks
and a series of concertinalike folds forms when a tool is forced laterally through a ductile sheet. On the
other hand, forcing an object perpendicularly through the sheet leads to radial petallike tears in both
ductile and brittle materials. To understand these different regimes we use a combination of experiments,
simulations, and simple theories. In particular, we describe the transition from brittle oscillatory tearing
via a single crack to ductile concertina tearing with two tears by deriving laws that describe the crack paths
and wavelength of the concertina folds and provide a simple phase diagram for the morphologies in terms
of the material properties of the sheet and the relative size of the tool.