Thin Sheet Mechanics

Whenever one dimension of a solid becomes small compared to the others, we say that the object is thin.  Thin objects (films or sheets) deform in unique ways when confined, or in other words, stuffed into a space that is smaller than the sheets two longer sides.  If confinement is small, a sheet can simply bend to deal with the constraints.  This is mainly because bending (though rooted in stretching) is a much lower energy deformation than compressing the sheet.  However, if the confinement is increased further the sheet will eventually need to bend in an orthogonal direction to the first bend.  When this happens, a big energetic cost is incurred because the sheet must now stretch (its Gaussian curvature is forced to change).  This is a big energetic cost for the sheet, and it may cause the sheet to focus the stretching to a sharp point. 

While still continuum mechanics, the focusing is quite hard to understand and represents a weakness in our ability to model thin films.  What is more, the sharp point discussed above occurs in many applications involving thin sheets like origami, thin film laminating processes, encapsulation, and crumpling (the inevitable end point of highly confining a sheet).  It is an open problem understanding how sharp points and folds affect the strength of thin sheet structures.