  This paper aims to quantitatively relate the energy dissipated at a shock
wave in a nonlinearly elastic bar to the energy in the oscillations in two
related dissipationless, dispersive systems. In contrast to a phase boundary,
there is no kinetic relation associated with a shock wave. Three
one-dimensional dynamic impact problems are studied: Problem 1 concerns a
nonlinearly elastic bar, Problem 2 a discrete chain of particles, and Problem 3
a continuum with a strain gradient term in the constitutive relation. In the
impact problem considered, the free boundary of each initially quiescent body
is subjected to a sudden velocity that is then held constant for all subsequent
time. There is energy dissipation at the shock in Problem 1 but Problems 2 and
3 are conservative. Problem 1 is solved analytically, Problem 2 numerically and
an approximate solution to Problem 3 is constructed using modulation theory.
The rate of increase of the oscillatory energy in Problems 2 and 3 are
calculated and compared with the dissipation rate at the shock in Problem 1.
The results indicate that the former is a good measure of the latter.
