Homogenization of a viscoelastic model for plant cell wall biomechanics

TL;DR

This paper derives macroscopic equations for plant cell wall biomechanics by homogenizing a microscopic viscoelastic model that includes chemical processes, resulting in coupled equations with memory effects.

Contribution

It introduces a homogenization approach for a coupled viscoelastic and chemical model of plant cell walls, deriving macroscopic equations with memory terms.

Findings

Derived macroscopic equations with memory effects

Coupled viscoelastic and chemical processes modeled

Homogenization techniques applied to complex biological materials

Abstract

The microscopic structure of a plant cell wall is given by cellulose microfibrils embedded in a cell wall matrix. In this paper we consider a microscopic model for interactions between viscoelastic deformations of a plant cell wall and chemical processes in the cell wall matrix. We consider elastic deformations of the cell wall microfibrils and viscoelastic Kelvin--Voigt type deformations of the cell wall matrix. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive macroscopic equations from the microscopic model for cell wall biomechanics consisting of strongly coupled equations of linear viscoelasticity and a system of reaction-diffusion and ordinary differential equations. As is typical for microscopic viscoelastic problems, the macroscopic equations for viscoelastic deformations of plant cell walls contain memory terms. The derivation of the macroscopic problem for degenerate viscoelastic equations is conducted using a perturbation argument.