Let $A$ be a ball of radius $R_0$ made of an elastic material. We compress the ball by applying an uniform force on the external surface till the radius of the ball is reduced to a radius $R_1<R_0$. We want to evaluate the the radial stress $p(r)$ in the ball $A$.
Clearly the solution has a spherical symmetry and therefore for an elementary volume $dV=dr d\theta d\phi$ to be in equilibrium it is enough that the force acting on the surfaces $dS_{(r+dr)}$ and $dS_{r}$ are the same ignoring the forces on the lateral surfaces of the volume which are in equilibrium such that the displacements in the material are only along $r$.
For the above to be true we have that the total force acting on any sphere of radius $r$ has to be constant which means $p(r)4\pi r^2=constant$.
and finally we find that $p(r)=C/r^2$, where $C$ is a constant.
However, this means that in the centre of the sphere the stress is infinite. Moreover, since the elastic energy is proportional to the square of the stress (or better the strains), and since $1/r^4$ is not integrable, it takes an infinite energy to compress the sphere even for $R_1$ very close to $R_0$.
Of course the above is impossible and there must be an error in my reasoning. Can anyone tell me where I have it wring?
