Abstract
In this paper we present a derivation and multiscale analysis of a mathematical model for plant cell wall biomechanics that takes into account both the microscopic structure of a cell wall coming from the cellulose microfibrils and the chemical reactions between the cell wall’s constituents. Particular attention is paid to the role of pectin and the impact of calcium-pectin cross-linking chemistry on the mechanical properties of the cell wall. We prove the existence and uniqueness of the strongly coupled microscopic problem consisting of the equations of linear elasticity and a system of reaction-diffusion and ordinary differential equations. Using homogenization techniques (two-scale convergence and periodic unfolding methods) we derive a macroscopic model for plant cell wall biomechanics.
Highlights
For a better understanding of plant growth and development it is important to analyse the influence of chemical processes on the mechanical properties of plant cells
The main novelty of this paper is twofold: (i) we derive a new model for plant cell wall biomechanics where the mechanical properties and biochemical processes in a cell wall are considered on the scale of its structural elements and (ii) using homogenization techniques we obtain a macroscopic model for plant cell wall biomechanics from a microscopic description of the mechanical and chemical processes
The elasticity tensor is defined as Eε(ξ, x) = E(ξ, x/ε), where the Y -periodic in y function E is given by E(ξ, y) = EM (ξ)χYM (y) + EF χYF (y), with constant elastic properties of the microfibrils and the elastic properties of cell wall matrix depending on the density of calcium-pectin cross-links
Summary
For a better understanding of plant growth and development it is important to analyse the influence of chemical processes on the mechanical properties (elasticity and extensibility) of plant cells. The main novelty of this paper is twofold: (i) we derive a new model for plant cell wall biomechanics where the mechanical properties and biochemical processes in a cell wall are considered on the scale of its structural elements (on the scale of the microfibrils) and (ii) using homogenization techniques we obtain a macroscopic model for plant cell wall biomechanics from a microscopic description of the mechanical and chemical processes. In the second case when all chemical substances diffuse, solutions of the reaction-diffusion equations have higher regularity with respect to the spatial variables and a point-wise dependence of the reaction terms on the displacement gradient can be considered In this situation in order to pass to the limit in the nonlinear reaction terms we prove the strong two-scale convergence for the displacement gradient.
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