Thinning instabilities in biological and electroactive membranes
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This thesis addresses several problems in the mathematical modelling of biological and electroactive membranes. We begin with a review of the necessary scientific background of biological and electroactive membranes. We then introduce the mathematical framework required for modelling such membranes, consisting of an overview of differential geometry and nonlinear elasticity. This is followed by a full derivation of the shape equations of a lipid membrane whose energy density depends on the mean and Gaussian curvatures, the stretch of the membrane midsurface, and the gradient of this stretch. We also show how these equations can be specialised to a specific geometry to obtain a set of ordinary differential equations, and thus how they can be used to predict the behaviour of membranes, and possibly to calibrate experimental results with the theory. We then turn our focus to the modelling of dielectric membranes, and develop a new mathematical model describing wrinkling and dielectric breakdown in thin dielectric elastomer devices. We compare our theory with experimental results reported in the literature, and find a good match between theory and experiment. Finally, we present some suggestions on how the theory developed for dielectric membranes might be extended to the case of biological membranes, in particular, to the modelling of pore formation in lipid bilayers interacting with peptide proteins.