Locally periodic homogenization of multiscale models for plant tissue biomechanics

Abstract

In this work, the derivation and multiscale analysis of a mathematical model for plant tissue biomechanics is considered. The microscopic model consists of a coupled systems of equations of linear elasticity, describing mechanical deformation, and reaction-diffusion and ordinary differential equations modelling the dynamics of load-bearing cross-links in cell walls and middle lamella, connecting individual cells in a plant tissue. It takes into account a two-way coupling between mechanical deformation and chemical processes in cell walls and middle lamella, where elastic properties depend on the density of cross-links and reactions terms, defining the stretching and breakage of the cross-links, depend on the deformation gradient. The nonlinear dependence of solutions of the ordinary differential equation on the displacement gradient and of the boundary conditions in reaction-diffusion problem on the displacement induces novel approaches in the derivation of a priori estimates. Using homogenization techniques of locally periodic (l-p) two-scale convergence, a macroscopic model for plant tissue biomechanics is derived, representing the first result in the multiscale analysis of a biomechanical model for tissues with non-periodic fibrous microstructure. For multiscale analysis and derivation of the macroscopic model, the non-periodic microstructure of plant cell walls, characterised by rotated planes of parallel aligned microfibrils, is approximated by the corresponding locally periodic microstructure. The two-way coupling between equations of linear elasticity and reaction-diffusion and ordinary differential equations requires the proof of the strong l-p two-scale convergence for the deformation gradient and for the density of stretched cross-links, in order to pass to the limit in the nonlinear functions.