Invariant solutions of the equation of motion for a sphere composed of Blatz-Ko material

The nonlinear partial differential equation governing the spherically symmetric dynamic deformations of Blatz-Ko materials is studied. The Lie groups and three invariant solutions of the governing equation are derived. Two of the invariant solutions are in separable form. The governing equation of the invariant solution in non-separable form is a singular second order non-autonomous ordinary differential equation (ODE). We transform this ODE to an autonomous ODE successfully and analyze its trajectories in the phase plane. By doing this we can capture the global behaviors of the solutions for the ODE. Especially, we can figure out what kind of problems will have singularity. The invariant solution in non-separable form studied in this paper is interesting because it shows that when localization or blowing up of some physical quantities occurs close to the outer boundary of a Blatz-Ko sphere the sphere may still look normal, that is, the distribution of the radial deformation as well as the boundary data still vary moderately. Therefore this special invariant solution gives us new insights about the behaviors of Blatz-Ko materials.