  The general exact solution describing the dynamics of anisotropic elastic
spheres supported only by tangential stresses is reduced to a quadrature using
Ori's mass-area coordinates. This leads to the explicit construction of the
root equation governing the nature of the central singularity. Using this
equation, we formulate and motivate on physical grounds a conjecture on the
nature of this singularity. The conjecture covers a large sector of the space
of initial data; roughly speaking, it asserts that addition of a tangential
stress cannot undress a covered dust singularity. The root equation also allows
us to analyze the case of self-similar spacetimes and to get some insight on
the role of stresses in deciding the nature of the singularities in this case.
