Modeling the Behavior of an Extensible Sheet in a Pressurized Chamber

Packing of slender objects into receptacles that are filled with a fluid is a ubiquitous process in biological systems and technological advances. Motivated by these applications, we study the quasi-static evolution of an extensible thin sheet that is confined between the two sides of a rectangular closed chamber. The two sides of the chamber, above and below the sheet, are filled with an ideal fluid. We derive an analytical model that accounts for the mutual interaction between the elastic deformation of the sheet and the pressure difference that it induces in the chamber. Our model reveals that the evolution of the system is governed by three dimensionless parameters: the normalized lateral displacement of the sheet, the slenderness of the sheet, and the bendo-gases parameter, that accounts for the ratio between the energy of the fluid and the bending energy of the sheet. We derive the state diagram of the system on this three-dimensional parameter space, and show that in addition to the flat configuration, the sheet exhibits three different branches of buckled solutions. We extract the details of the flat-to-buckle instability and derive approximated solutions to the height functions, and therefore the pressure drops, in each branch. Overall, our analysis sheds light on mechanical instabilities that emerge from the interaction between thin elastic bodies and a compressible fluid medium.