Winkler’s mattress model is often used as a simplified
model to understand how a thin elastic layer,
such as a coating, deforms when subject to a
distributed normal load: the deformation of the
layer is assumed proportional to the applied normal
load. This simplicity means that the Winkler model
has found a wide range of applications from soft
matter to geophysics. However, in the limit of an
incompressible elastic layer the model predicts infinite
resistance to deformation, and hence breaks down.
Since many of the thin layers used in applications
are elastomeric, and hence close to incompressible,
we consider the question of when the Winkler model
is appropriate for such layers. We formally derive
a model that interpolates between the Winkler and
incompressible limits for thin elastic layers, and
illustrate this model by detailed consideration of
two example problems: the point indentation of a
coated elastomeric layer and self-sustained lift in
soft elastohydrodynamic lubrication. We find that the
applicability (or otherwise) of the Winkler model is
not determined by the value of the Poisson ratio alone,
but by a compressibility parameter that combines the
Poisson ratio with a measure of the layer’s slenderness
that depends on the problem under consideration.
