Time-dependent modelling of thin poroelastic films drying on deformable plates

TL;DR

This paper develops simplified, asymptotic models for the time-dependent stress and deformation in drying, particle-laden thin films on deformable plates, validated against finite element solutions and experiments.

Contribution

It introduces novel asymptotic models for coupled poroelastic film and elastic plate dynamics, accounting for non-uniform thickness and stress distributions.

Findings

Models agree with finite element simulations.

Plate deflection scales as t^{1/2} at high Péclet numbers.

Models match experimental observations.

Abstract

Understanding the generation of mechanical stress in drying, particle-laden films is important for a wide range of industrial processes. The cantilever experiment allows the stress in a drying film that has been deposited onto a thin plate to be quantified. Mechanical stresses in the film are transmitted to the plate and drive bending. Mathematical modelling enables the film stress to be inferred from measurements of the plate deflection. The aim of this paper is to present simplified models of the cantilever experiment that have been derived from the time-dependent equations of continuum mechanics using asymptotic methods. The film is described using nonlinear poroelasticity and the plate using nonlinear elasticity. In contrast to Stoney-like formulae, the simplified models account for films with non-uniform thickness and stress. The film model reduces to a single differential equation that can be solved independently of the plate equations. The plate model reduces to an extended form of the Foppl-von Karman (FvK) equations that accounts for gradients in the longitudinal traction acting on the plate surface. Consistent boundary conditions for the FvK equations are derived by resolving the Saint-Venant boundary layers at the free edges of the plate. The asymptotically reduced models are in excellent agreement with finite element solutions of the full governing equations. As the P\'eclet number increases, the time evolution of the plate deflection changes from $t$ to $t^{1/2}$, in agreement with experiments.