On the Occurrence of Elastic Singularities in Compressed Thin Sheets: Stress Focusing and Defocusing

Abstract

Compressing thin sheets usually yields the formation of singularities which focus curvature and stretch on points or lines. In particular, following the common experience of crumpled paper where a paper sheet is crushed in a paper ball, one might guess that elastic singularities should be the rule beyond some compression level. In contrast, we show here that, somewhat surprisingly, compressing a sheet between cylinders makes singularities spontaneously disappear at large compression. This “stress-defocusing” phenomenon is qualitatively explained from scale invariance and further linked to a criterion based on a balance between stretch and curvature energies on defocused states. This criterion is made quantitative using the scalings relevant to sheet elasticity and compared to experiment. These results are synthesized in a phase diagram completed with plastic transitions. They end up with a renewed vision of elastic singularities as a thermodynamic condensed phase where stress is focused, in competition with a regular diluted phase where stress is defocused. Different compression routes may be followed in this diagram by managing differently the two principal curvatures of a sheet, as experimentally achieved here. In practice, besides the famous Elastica and crumpled paper routes, this offers interesting alternatives for compressing a sheet with an amazing spontaneous regularization of geometry and stress that repels the occurrence of plastic damages.