Mathematical modeling of neo-Hookean material growth

A mathematical model of the volumetric growth of an incompressible neo-Hookean material is derived. Models of this type are used to describe the evolution of the human brain under the action of an external load. In the paper, we show that the space of deformation fields in a homeostatic state coincides with the Möbius group of conformal transforms in $\mathbb{R}^3$. We prove the well-posedness of the linear boundary value problem obtained by linearizing the governing equations around a homeostatic state. The behavior of solutions when the time variable tends to infinity is studied. The main conclusion is that changes in the material, caused by a temporary increase in pressure (hydrocephalus) are irreversible.