Validity of Winkler's mattress model for thin elastomeric layers: Beyond Poisson's ratio

TL;DR

This paper investigates the limitations of Winkler's mattress model for thin elastomeric layers, deriving an interpolated model that accounts for compressibility and layer geometry, extending its applicability beyond the Poisson ratio constraint.

Contribution

The authors derive a new model that bridges Winkler's and incompressible limits for thin elastic layers, considering both Poisson ratio and layer slenderness.

Findings

Winkler's model fails for nearly incompressible layers unless modified.

The applicability depends on a combined compressibility parameter, not just Poisson's ratio.

The new model accurately predicts deformation in example problems.

Abstract

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.