Chandler Thomas G J, Vella Dominic
Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, UK.
Proc Math Phys Eng Sci. 2020 Oct;476(2242):20200551. doi: 10.1098/rspa.2020.0551. Epub 2020 Oct 21.
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, which itself depends on the problem under consideration.
温克勒床垫模型常被用作简化模型，以理解诸如涂层之类的薄弹性层在承受分布法向载荷时如何变形：假定该层的变形与所施加的法向载荷成正比。这种简单性意味着温克勒模型在从软物质到地球物理学等广泛领域都有应用。然而，在不可压缩弹性层的极限情况下，该模型预测变形阻力无穷大，因此失效。由于应用中使用的许多薄层是弹性体，因此接近不可压缩，我们考虑温克勒模型何时适用于此类层的问题。我们正式推导了一个在薄弹性层的温克勒极限和不可压缩极限之间进行插值的模型，并通过详细考虑两个示例问题来说明该模型：涂层弹性体层的点压痕和软弹性流体动力润滑中的自持升力。我们发现，温克勒模型的适用性（或反之）并非仅由泊松比的值决定，而是由一个压缩性参数决定，该参数将泊松比与层的细长程度度量相结合，而细长程度本身又取决于所考虑的问题。
