# Linear models for thin plates of polymer gels

## TL;DR

This paper derives two simplified plate models for thin polymer gel structures by analyzing the asymptotic behavior of three-dimensional gel theories under different diffusivity scalings, revealing how fluid pressure distribution affects plate deformation.

## Contribution

It introduces two novel plate models for polymer gels based on asymptotic analysis, considering different diffusivity regimes and their impact on fluid pressure distribution.

## Key findings

- First model: fluid pressure affine across thickness, determined by boundary traces.
- Second model: fluid pressure governed by 3D diffusion equation.
- Both models incorporate fluid pressure effects into plate deformation analysis.

## Abstract

Within the linearized three-dimensional theory of polymer gels, we consider a sequence of problems formulated on a family of cylindrical domains whose height tends to zero. We assume that the fluid pressure is controlled at the top and bottom faces of the cylinder, and we consider two different scaling regimes for the diffusivity tensor. Through asymptotic-analysis techniques we obtain two plate models where the transverse displacement is governed by a plate equation with an extra contribution from the fluid pressure. In the limit obtained within the first scaling regime the fluid pressure is affine across the thickness and hence it is determined by its instantaneous trace on the top and bottom faces. In the second model, instead, the value of the fluid pressure is governed by a three-dimensional diffusion equation.