In modelling electrostatically actuated micro-and nano-electromechanical systems, researchers have typically relied on a small-aspect ratio to form a leading-order theory. In doing so, small gradient terms are dropped. Although this approximation has been fruitful, its consequences have not been investigated. Here, this approximation is re-examined, and a new theory which includes often neglected small curvature terms is presented. Furthermore, the solution set of the new theory is explored for the unit disk domain and compared to the standard theory. Also, the analytical results are compared to experimental data.

We develop a three-dimensional model for capillary origami systems in which a rectangular plate has finite thickness, is allowed to stretch and undergoes small deflections. This latter constraint limits our description of the encapsulation process to its initial folding phase. We first simplify the resulting system of equations to two dimensions by assuming that the plate has infinite aspect ratio, which allows us to compare our approach to known two-dimensional capillary origami models for inextensible plates. Moreover, as this two-dimensional model is exactly solvable, we give an expression for its solution in terms of its parameters. We then turn to the full three-dimensional model in the limit of small drop volume and provide numerical simulations showing how the plate and the drop deform due to the effect of capillary forces.

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Abstract. In this paper, the classical solution set (λ, u) of the one-dimensional prescribed mean curvature equationfor λ > 0 and L > 0, is analyzed via a time map. It is shown that the solution set depends on both parameters λ and L and undergoes two bifurcations. The first is a standard saddle node bifurcation, which happens for all L at λ = λ * (L). The second is a splitting bifurcation, namely, there exists a value L * such that as L transitions from greater than or equal to L * to less than L * the upper branch of the bifurcation diagram of problem ( ) splits into two parts. In contrast, the solution set of the semilinear version of problem ( ) is independent of L and exhibits only a saddle node bifurcation. Therefore, as this analysis suggests, the splitting bifurcation is a byproduct of the mean curvature operator coupled with the singular forcing.