Homogenization of Biomechanical Models for Plant Tissues

Andrey Piatnitski,Mariya Ptashnyk

doi:10.1137/15m1046198

Abstract

In this paper homogenization of a mathematical model for plant tissue biomechanics is presented. The microscopic model constitutes a strongly coupled system of reaction-diffusion-convection equations for chemical processes in plant cells, the equations of poroelasticity for elastic deformations of plant cell walls and middle lamella, and Stokes equations for fluid flow inside the cells. The chemical process in cells and the elastic properties of cell walls and middle lamella are coupled because elastic moduli depend on densities involved in chemical reactions, whereas chemical reactions depend on mechanical stresses. Using homogenization techniques we derive rigorously a macroscopic model for plant biomechanics. To pass to the limit in the nonlinear reaction terms, which depend on elastic strain, we prove the strong two-scale convergence of the displacement gradient and velocity field.

Highlights

Analysis of interactions between mechanical properties and chemical processes, which influence the elasticity and extensibility of plant cell tissues, is important for better understanding of plant growth and development, as well as their response to environmental changes
It has been shown that chemical properties of pectin and the control of the density of calcium-pectin cross-links greatly influence the mechanical deformations of plant cell walls [34], and the interference with pectin methylesterase (PME) activity causes dramatic changes in growth behavior of plant tissues [50]
In order to pass to the limit in the nonlinear reaction terms, we prove the strong twoscale convergence for the displacement gradient and fluid flow velocity, essential for the homogenization of the coupled problem considered here

Summary

Introduction

Analysis of interactions between mechanical properties and chemical processes, which influence the elasticity and extensibility of plant cell tissues, is important for better understanding of plant growth and development, as well as their response to environmental changes. To describe the coupling between the mechanics and chemistry, we consider the dynamics of pectins, calcium, and calcium-pectin cross-links, water flow in a plant tissue, and the poroelastic nature of cell walls and middle lamella. ∂tcf − div(Df ∇cf − G(∂tuf )cf ) = gf ∂tce − div(De∇ce) = ge in Ωf , in Ωe, where the chemical reaction term gf = gf (cf ) in Ωf describes the decay and/or buffering of calcium inside the plant cells (see, e.g., [52]), ge models the interactions between calcium and demethylesterified pectin in cell walls and middle lamella and the creation and breakage of calcium-pectin cross-links, and G is a bounded function of the intracellular flow velocity ∂tuf. The Gagliardo–Nirenberg and trace inequalities, together with the extension properties of bεe and cε (see Lemma 3.1), yield (26)

Cδ cεe ε bεe

The definition

Πτ v

Thus applying