  We investigate the crumpling of a sheet as it is repeatedly crushed onto
itself by rolling it into a cylinder and twisting it axially while allowing the
end-to-end length to evolve freely. As deduced from its plastic deformations,
the sheet creases and collapses into structures which repeat and sharpen over
hundreds of cycles to a remarkable degree before forming new configurations.
The observed metastablilty increases with applied cycles leading to recurrent
structures over a significant range of loading, but reconfigurations can
continue to occur for large enough loading as the creases develop tears. The
evolution of the sheet structure as measured by the mean curvature and the
total crease length is found to increase logarithmically with cycle number with
a rate which increases with degree of compression. We explain the overall
extent of creasing using flat folding models, and show the logarithmic growth
as being a consequence of individual creases becoming sharper with number of
folding cycles, and due to the bifurcation in the curvature field leading to
the formation of new creases and folding pathways. Thus, we show that
elastoplastic sheets can follow complex folding pathways to form convergent
structures after a sufficiently large number of training cycles provided
material fatigue remains unimportant.
