Volumetric growth of neo-Hookean incompressible material

We consider a mathematical model of an incompressible neo-Hookean material, which is widely used in the modeling of biological tissues. The derivation of the governing equations for the deformation field, pressure, and growth factor is given. The resulting model includes the steady-state moment balance equation, the mass balance equation, and the growth factor evolutionary equation. The problem of material growth under the action of hydrostatic pressure is considered. The solution is found using the Lyapunov-Schmidt method. A detailed analysis of the linearized equations is carried out. The existence of a strong solution to the nonlinear problem on an arbitrary time interval for small external load is proved.