  The interaction of a thin viscous film with an elastic sheet results in
coupling of pressure and deformation, which can be utilized as an actuation
mechanism for surface deformations in a wide range of applications, including
microfluidics, optics, and soft robotics. Implementation of such configurations
inherently takes place over finite domains and often requires some
pre-stretching of the sheet. Under the assumptions of strong pre-stretching and
small deformations of the lubricated elastic sheet, we use the linearized
Reynolds and Foppl-von Karman equations to derive closed-form analytical
solutions describing the deformation in a finite domain due to external forces,
accounting for both bending and tension effects. We provide a closed-form
solution for the case of a square-shaped actuation region and present the
effect of pre-stretching on the dynamics of the deformation. We further present
the dependence of the deformation magnitude and timescale on the spatial
wavenumber, as well as the transition between stretching- and bending-dominant
regimes. We also demonstrate the effect of spatial discretization of the
forcing (representing practical actuation elements) on the achievable
resolution of the deformation. Extending the problem to an axisymmetric domain,
we investigate the effects arising from nonlinearity of the Reynolds and
Foppl-von Karman equations and present the deformation behavior as it becomes
comparable to the initial film thickness and dependent on the induced tension.
These results set the theoretical foundation for implementation of microfluidic
soft actuators based on elastohydrodynanmics.
