Multi-scale buckling in thin sheets

E. Sharon, B. Roman, S. G. Shin, M. Marder and H. L. Swinney

We present an experimental study of the buckling process, which follows the tearing of plastic sheet. The
irreversible plastic flow around the crack tip creates a new, curved metric in the sheet. This metric is
smooth and all its components are either constant (in the propagation direction) or monotonically
increasing (towards the sheet's edge). Despite this monotonic behavior the sheet buckles and forms a
cascade of waves, superimposed on each other, with highly separated wavelengths. We show that all the
members of the cascade are similar and thus create a fractal, which spans over 2.5 decades. The problem has
two relevant length scales: The sheet thickness t and the plastic zone radius Rp, which provide the lower
and upper bounds for the possible wavelengths. Though t<<Rp, one can not take the limit t=0 keeping only Rp
as a single length scale. Instead, t remains the relevant parameter for the bending energy associated with
the buckling and sets an energy threshold for the buckling at all scales. This point is demonstrated by
showing that the wavelengths of each buckling cascade scale linearly with the sheet thickness t. We suggest
that this multi-scale buckling cascade is a general phenomenon that should appear whenever a thin sheet
posses a curved metric, which cannot be matched by buckling in a single wave number. Under these conditions
the sheet forms many "local solutions", each of them matches the metric over a different scale. Their
superposition constructs the global solution, which matches the metric at all scales and thus minimizes the
energy. We support this suggestion by performing molecular dynamics simulations of a thin elastic sheet
with a similar metric. The simulations show the formation of a buckling cascade similar to the one observed
in experiments. Finally we show that similar buckling patterns are very common in nature. We can find them
in flowers, leafs and living tissues. We suggest that these very complex surfaces are not necessarily the
results of highly complex growth mechanism. Instead, they might result from very simple processes, which
create non-flat metrics.