  In the present work motivated by generalized forms of the Hertzian dynamics
associated with granular crystals, we consider the possibility of such models
to give rise to both shock and rarefaction waves. Depending on the value $p$ of
the nonlinearity exponent, we find that both of these possibilities are
realizable. We use a quasi-continuum approximation of a generalized inviscid
Burgers model in order to predict the solution profile up to times near the
shock formation, as well as to estimate when it will occur. Beyond that time
threshold, oscillations associated with the discrete nature of the underlying
model emerge that cannot be captured by the quasi-continuum approximation. Our
analytical characterization of the above features is complemented by systematic
numerical computations.
