  Discrete materials composed of masses connected by strongly nonlinear links
with anomalous behavior (reduction of elastic modulus with strain) have very
interesting wave dynamics. Such links may be composed of materials exhibiting
repeatable softening behavior under loading and unloading. These discrete
materials will not support strongly nonlinear compression pulses due to
nonlinear dispersion but may support stationary rarefaction pulses or
rarefaction shock-like waves. Here we investigate rarefaction waves in
nonlinear periodic systems with a general power-law relationship between force
and displacement $F \propto \delta^{n}$, where $0 < n < 1$. An exact solution
of the long-wave approximation is found for the special case of $n = 1/2$,
which agrees well with numerical results for the discrete chain. Theoretical
and numerical analysis of stationary solutions are discussed for different
values of $n$ in the interval $0 < n < 1$. The leading solitary rarefaction
wave followed by a dispersive tail was generated by impact in numerical
calculations.
