Compression induced folding of a sheet: an integrable system

Area of Science:

• Physics
• Materials Science
• Fluid Dynamics

Background:

• Elastic films on fluid substrates exhibit complex buckling behaviors.
• Understanding fold formation is crucial for predicting material failure and designing new structures.

Purpose of the Study:

• To find an exact mathematical solution for the shape of folds in compressed elastic films on fluid substrates.
• To describe the progression of buckling from initial sinusoidal waves to a self-contacting fold.

Main Methods:

• Analytical solution derived from principles of elasticity and fluid mechanics.
• Mathematical modeling of the buckling instability and fold evolution.

Main Results:

• An exact solution for the fold shape was determined.
• Buckling occurs at a specific wave vector determined by film stiffness and fluid density.
• The pressure decrease is quadratically related to lateral displacement.
• A complex wave vector with invariant magnitude under compression was identified.

Conclusions:

• The study provides a complete description of fold formation in compressed elastic films.
• The findings offer a fundamental understanding of elasto-capillary instabilities.
• The exact solution can inform the design of microstructures and soft materials.