Notes on superposed incremental deformations in the mechanics of lipid membranes

We present an analysis of the superposed incremental deformations of lipid membranes in contact with a circular substrate. A complete analytical solution describing the morphological transitions of lipid membranes is obtained via Monge parametric representation and admissible linearization. The corresponding solution demonstrates smooth and bounded behavior within the finite domain of interest (annular) and, more importantly, shows excellent stability as it approaches the boundary of the circular substrate with the radius of convergence compatible with a few nanometers. Under the prescription of the superposed incremental deformations, a complete analysis of the necessary boundary conditions, the anchoring condition of the lipid molecules on an edge, and other geometrical quantities of the membrane is illustrated for the case of the circular substrate-membrane system.