To blister or not to blister: the importance of finite size effects

We consider the distinction between wrinkling and blistering in a compressed elastic sheet of size L floating on a liquid. We prescribe a blister of size B which is delaminated at the center of the sheet, and in contact with the liquid in the left and right regions. We solve the linearized force balance in those three regions, the central one being characterized by the absence of buoyancy. There are four important length scales: L, B, the characteristic wavelength of wrinkles (arising from the competition between bending and buoyancy) and the bendogravity length (arising from the competition between bending and the weight of the sheet). If the bendogravity length is infinite — equivalent to neglecting the weight of sheet where it is delaminated — we nondimensionlize the problem with the wavelength and obtain an eigenvalue problem for the compressive load, τ, and the deflection of the sheet. We distinguish symmetric and antisymmetric modes, and find τ as a function of B and L for each mode. Next, we minimize the energy with respect to B and determine the conditions under which the sheet is blistered or simply wrinkled (namely B=0). Interestingly, antisymmetric wrinkles can sometimes be more energetically favorable than a symmetric blister. Finally, we explore the consequences of a finite bendogravity length, noting that the mathematical nature of the problem changes dramatically.