Homogenization of a viscoelastic model for plant cell wall biomechanics

Mariya Ptashnyk,Brian Seguin

doi:10.1051/cocv/2016060

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 governing the viscoelastic deformations of plant cell walls contain memory terms. The derivation of the macroscopic problem for the degenerate viscoelastic equations is conducted using a perturbation argument.

Highlights

To obtain a better understanding of the mechanical properties and development of plant tissues it is important to model and analyse the interactions between the chemical processes and mechanical deformations of plant cells
The biomechanics of plant cell walls is determined by the cell wall microstructure, given by microfibrils, and the physical properties of the cell wall matrix
Pectin can be modified by the enzyme pectin methylesterase (PME), which removes methyl groups by breaking ester bonds

Summary

Introduction

To obtain a better understanding of the mechanical properties and development of plant tissues it is important to model and analyse the interactions between the chemical processes and mechanical deformations of plant cells. Due to the coupling between the viscoelastic properties and the biochemistry of a plant cell wall, the elastic and viscous tensors depend on space and time variables This fact introduces additional complexity in the derivation and in the structure of the macroscopic equations, compered to classical viscoelastic equations. The main novelty of this paper is the multiscale analysis and derivation of the macroscopic problem from a microscopic description of the mechanical and chemical processes This approach allows us to take into account the complex microscopic structure of a plant cell wall and to analyse the impact of the heterogeneous distribution of cell wall structural elements on the mechanical properties of plants.

Microscopic model for viscoelastic deformations of plant cell walls

Findings

Main results