Effect of initial geometry on crumpled thin shells

Why do milk cartons and aluminum cans crumple differently from paper? Besides distinct elasticity and plasticity, their unique geometric structures also play a role. Over the past few decades, crumpled sheets have been widely studied for their fascinating mechanical, energetic, and statistical properties. These include the power-law relationship between the volume density of crumpled balls and pressure, the ratio of bending to stretching energies, and statistics on the number and length of deformations. Since these conclusions were primarily derived from studies on flat sheets, verifying whether they can be safely applied to the 3D hollow objects we encounter daily is important. A cubical box, with its sides forming flat sheets, is an ideal subject for comparison. The crumpling properties of such boxes have been examined through molecular dynamics simulations and experiments. By systematically dismantling the sides of the box, we clarified the roles of sides, corners, and open/closed boundaries. This knowledge allowed us to identify corners as the key factor causing deviations from the crumpling properties of flat sheets. One significant difference is that the power-law relationship between compaction and pressure is missing, replaced by a bump characteristic of the buckling transition. Different types of corners are found to be equally important, whether they are sharp at a singular point (as in boxes), along a curve (as in aluminum cans), or rounded (as in balls and vehicles). Special attention has been given to explaining why the vertices on the surface of a crumpled ball do not buckle, allowing them to retain the properties of a flat sheet despite their 3D appearance.