Stress Condensation in Crushed Elastic Manifolds

Abstract

We discuss an $M$-dimensional phantom elastic manifold of linear size $L$ crushed into a small sphere of radius $R\ensuremath{\ll}L$ in $N$-dimensional space. We investigate the low elastic energy states of 2-sheets $(M\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}2)$ and 3-sheets $(M\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}3)$ using analytic methods and lattice simulations. When $N\ensuremath{\ge}2M$ the curvature energy is uniformly distributed in the sheet and the strain energy is negligible. But when $N\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}M+1$ and $M>1$, both energies appear to be condensed into a network of narrow $M\ensuremath{-}1$ dimensional ridges. The ridges appear straight over distances comparable to the confining radius $R$.