Internal sites of actuation and activation in thin elastic films and membranes of finite thickness

TL;DR

This paper provides a theoretical analysis of the internal force and stress sites in thin elastic films and membranes of finite thickness, deriving fundamental solutions for localized singularities and exploring stability conditions.

Contribution

It derives three-dimensional Green's functions for force and stress singularities in finite-thickness elastic films, extending previous two-dimensional models and analyzing stability under different boundary conditions.

Findings

Fully free-standing membranes lead to diverging solutions.

Support at one surface stabilizes the solutions.

Differences between free-standing and supported membranes are explained.

Abstract

Functionalized thin elastic films and membranes frequently feature internal sites of net forces or stresses. These are, for instance, active sites of actuation, or rigid inclusions in a strained membrane that induce counterstress upon externally imposed deformations. We theoretically analyze the geometry of isotropic, flat, thin, linearly elastic films or membranes of finite thickness, laterally extended to infinity. At the mathematical core of such characterizations are the fundamental solutions for localized force and stress singularities associated with corresponding Green's functions. We derive such solutions in three dimensions and place them into the context of previous two-dimensional calculations. To this end, we consider both no-slip and stress-free conditions at the top and/or bottom surfaces. We provide an understanding for why the fully free-standing thin elastic membrane leads to diverging solutions in most geometries and compare these situations to the truly two-dimensional case. A no-slip support of at least one of the surfaces stabilizes the solution, which illustrates that the divergences in the fully free-standing case are connected to global deformations. Within the mentioned framework, our results are important for associated theoretical characterizations of thin elastic films, whether supported or free-standing, and of membranes subject to internal or external forces or stresses.