  Investigations of the elastic behavior of bulk elastomers have traditionally
proceeded on the basis of classical rubber elasticity, which regards chains as
thermally fluctuating but disregards the thermal fluctuations of the
cross-links. Here, we consider an incompressible and flat elastomer film of an
axisymmetric shape confined between two large hard co-planar substrates, with
the axis of the film perpendicular to the substrates. We address the impact
that thermal fluctuations of the cross-links have on the free energy of elastic
deformation of the system, subject to the requirement that the fluctuating
elastomer cannot detach from the substrates. We examine the behavior of the
deformation free energy for one case where a rigid pinning boundary condition
is applied to a class of elastic fluctuations at the confining surfaces, and
another case where the same elastic fluctuations are subjected to soft "gluing"
potentials. We find that there can be significant departures (both quantitative
and qualitative) from the prediction of classical rubber elasticity theory when
elastic fluctuations are included. Finally, we compare the character of the
attractive part of the elastic fluctuation-induced, or pseudo-Casimir, stress
with the standard thermal Casimir stress in confined but non-elastomeric
systems, finding the same power law decay behavior when a rigid pinning
boundary condition is applied, for the case of the gluing potential, we find
that the leading order correction to the attractive part of the fluctuation
stress decays inversely with the inter-substrate separation.
