Figure 3 illustrates the model problem considered in this
work.We focus on the formation of weak compaction waves
that are supported by a piston moving with constant speed
up; such waves are characteristic of most accidental impact
scenarios in practice. The initially stress-free granular solid
possesses an embedded interface across which the material
solid volume fraction abruptly changes from φL to φR, the
values of to the left and right of the interface, respectively.
For discussion purposes, a decrease in solid volume fraction
(increase in porosity) across the interface is depicted
in the figure. This simple spatial distribution was chosen so
that the time-dependent evolution of incident, transmitted,
and reflected compaction waves with respect to the interface,
and their dependence on φL, φR, and up, could be
cleanly investigated without the development of overly complex
wave interactions resulting from more spatially varying
distributions. Further, we believe that this distribution
represents in spirit a granular material of uniform solid volume
fraction φ0 = φR that has been quickly loaded and unloaded
resulting in a finite region of strain hardened material
having solid volume fraction φL adjacent to the loaded surface.
Subsequent impact by a constant speed piston supports
the propagation of an incident compaction wave through the
initially dense, precompacted material. For a piston speed