Thin elastic films and membranes under rectangular confinement

We address the deformations within a thin elastic film or membrane in a two-dimensional rectangular confinement. To this end, analytical considerations of the Navier-Cauchy equations describing linear elasticity are performed in the presence of a localized force center, that is, a corresponding Green's function is determined, under no-slip conditions at the clamped boundaries. Specifically, we find resulting displacement fields for different positions of the force center. It turns out that clamping regularizes the solution when compared to an infinitely extended system. Increasing compressibility renders the displacement field more homogeneous under the given confinement. Moreover, varying aspect ratios of the rectangular confining frame qualitatively affect the symmetry and appearance of the displacement field. Our results are confirmed by comparison with corresponding finite-element simulations.