Homogenization of a Poroelasticity Model for Fibre-Reinforced Hydrogels

TL;DR

This paper develops a homogenized poroelastic model for fibre-reinforced hydrogels, capturing the effective macro-scale behavior resulting from micro-scale heterogeneities in the material structure.

Contribution

It introduces a rigorous homogenization approach for a complex fibre-hydrogel composite, deriving an effective model that accounts for micro-scale pore pressure effects.

Findings

Existence and uniqueness of solutions for the model

Derivation of a two-scale limit model as heterogeneity scale tends to zero

Identification of micro stress corrections at the macro scale

Abstract

In this paper, the analysis and homogenization of a poroelastic model for the hydro-mechanical response of fibre-reinforced hydrogels is considered. Here, the medium in question is considered to be a highly heterogeneous two-component media composed of a connected fibre-scaffold with periodically distributed inclusions of hydrogel. While the fibres are assumed to be elastic, the hydromechanical response of hydrogel is modeled via \emph{Biot's poroelasticity}. We show that the resulting mathematical problem admits a unique weak solution and investigate the limit behavior (in the sense of two-scale convergence) of the solutions with respect to a scale parameter, characterizing the heterogeneity of the medium. Letting this scale parameter tend to zero, we arrive at an effective model where the micro variations of the pore pressure give rise to a micro stress correction at the macro scale.