Local Well-Posedness of a Model for Stress-Driven Growth in the Presence of Nutrients

Abstract

A model for morphoelastic growth, that is, growth influenced by elastic stress, driven by the absorption of nutrients is considered. The model features a multiplicative decomposition of the deformation gradient into an elastic contribution and a growth tensor. While the evolution of the system is governed by an ordinary differential equation for the growth tensor on a suitable Banach space, which depends on the elastic stresses and the concentration of a nutrient field, the total deformation is given by the solution of a quasi-static equilibrium equation arising from the formal Euler-Lagrange equations of a hyperelastic variational integral. The nutrient concentration is determined by a linear elliptic reaction-diffusion equation which is formulated in Lagrangian coordinates and whose coefficients depend on the growth tensor as well as the deformation gradient accounting for the change of material properties due to elastic deformation and growth. Existence and uniqueness of solutions of this fully coupled system of differential equations is proven via a fixed-point argument.