  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.
