  In the works of K.A. Pericak-Spector and S. Spector \cite{ps88, ps98} a class
of self-similar solutions are constructed for the equations of radial isotropic
elastodynamics that describe cavitating solutions. Cavitating solutions
decrease the total mechanical energy and provide a striking example of
non-uniqueness of entropy weak solutions (for polyconvex energies) due to
point-singularities at the cavity. To resolve this paradox, we introduce the
concept of singular limiting induced from continuum solution (or
slic-solution), according to which a discontinuous motion is a slic-solution if
its averages form a family of smooth approximate solutions to the problem. It
turns out that there is an energetic cost for creating the cavity, which is
captured by the notion of slic-solution but neglected by the usual entropic
weak solutions. Once this cost is accounted for, the total mechanical energy of
the cavitating solution is in fact larger than that of the homogeneously
deformed state. We also apply the notion of slic-solutions to a one-dimensional
example describing the onset of fracture, and to gas dynamics in Langrangean
coordinates with Riemann data inducing vacuum in the wave fan.
