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

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 Fe$F_e$ and a growth-related part G. After the transformation due to the growth process, governed by G, an elastic deformation described by Fe$F_e$ 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.