Stress-modulated growth in the presence of nutrients -- existence and uniqueness in one spatial dimension

TL;DR

This paper proves the existence and uniqueness of solutions for a one-dimensional stress-modulated growth model that incorporates nutrient effects and elastic deformation, providing a rigorous mathematical foundation for such biological growth processes.

Contribution

It establishes the mathematical existence and uniqueness of solutions for a coupled stress-growth model in one dimension, considering nutrient influence and elastic deformation.

Findings

Existence of solutions is proven for the model.

Uniqueness of solutions is established under given conditions.

The model captures stress and nutrient effects on growth in a rigorous framework.

Abstract

Existence and uniqueness of solutions for a class of models for stress-modulated growth is proven in one spatial dimension. The model features the multiplicative decomposition of the deformation gradient $F$ into an elastic part $F_e$ and a growth-related part $G$. After the transformation due to the growth process, governed by $G$, an elastic deformation described by $F_e$ is applied in order to restore the Dirichlet boundary conditions and therefore the current configuration might be stressed with a stress tensor $S$. The growth of the material at each point in the reference configuration is given by an ordinary differential equation for which the right-hand side may depend on the stress $S$ and the pull-back of a nutrient concentration in the current configuration, leading to a coupled system of ordinary differential equations.