Mathematical modelling and homogenization of thin fiber-reinforced hydrogels

TL;DR

This paper develops a mathematical model and homogenization approach for thin fiber-reinforced hydrogels, combining elasticity and poroelasticity to derive a simplified macroscopic behavior of the composite material.

Contribution

It introduces a novel homogenization dimension reduction technique for a coupled poroelastic-fiber system with simultaneous limits on periodicity and thickness.

Findings

Derivation of a Kirchhoff-Love-type limit displacement.

Establishment of the asymptotic behavior of the coupled system.

Proof of the uniqueness of the macroscopic solution.

Abstract

This work considers simultaneous homogenization dimension reduction of a poroelastic model for thin fiber-reinforced hydrogels. The analysed medium is defined as a two-component system consisting of a continuous fiber framework with hydrogel inclusions arranged periodically throughout. The fibers are assumed to operate under quasi-stationary linear elasticity, whereas the hydrogel's hydromechanical behavior is represented using Biot's linear poroelasticity model. The asymptotic limit of the coupled system is established when the periodicity and thickness parameters are of the same order and tend to zero simultaneously, utilizing the re-scaling unfolding operator. It is demonstrated that the limit displacement exhibits Kirchhoff-Love-type behavior using the decomposition of plate displacements. Towards the end, a unique solution for the macroscopic problem has been demonstrated.