Emergence of dispersive shocks and rarefaction waves in power-law contact models

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 dispersive 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 quasicontinuum approximation of a generalized inviscid Burgers model in order to predict the solution profile up to times near the formation of the dispersive shock, as well as to estimate when it will occur. Beyond that time threshold, oscillations associated with the highly dispersive nature of the underlying model emerge, which cannot be captured by the quasicontinuum approximation. Our analytical characterization of the above features is complemented by systematic numerical computations.

Figure 1

Strain wave propagation for $p=1.5$ and temporal plots of strain waves for (a) ${\delta }_0=0$, (b) ${\delta }_0=0.001$, and (c) ${\delta }_0=0.01$. The black solid line is obtained from the direct simulation of the discrete model and the red dashed line is the prediction based on Eq. (15) (i.e., the continuum model). Strain curves are offset to ease visualization (the tick interval in the vertical axis indicates 1.0). Insets (i) and (ii) show the magnified view of the leading part at and after the predicted shock time, respectively. Space-time contour plots of strain wave propagation for (d) ${\delta }_0=0$, (e) ${\delta }_0=0.001$, and (f) ${\delta }_0=0.01$. The gray solid line indicates the predicted analytical shock wave time.

Figure 2

Strain wave propagation for $p=0.5$ and temporal plots of strain waves for (a) ${\delta }_0=0$, (b) ${\delta }_0=0.001$, and (c) ${\delta }_0=0.01$. The black solid line is obtained from the direct simulation of the discrete model and the red dashed line is the prediction based on Eq. (15) (i.e., the continuum model). Strain curves are offset to ease visualization (the tick interval in the vertical axis indicates 1.0). Insets (i) and (ii) show the magnified view of the tail part at and after the predicted shock time, respectively. Space-time contour plots of strain wave propagation for (d) ${\delta }_0=0$, (e) ${\delta }_0=0.001$, and (f) ${\delta }_0=0.01$. The black solid line indicates the predicted analytical shock wave time.

Figure 3

Effect of the exponent value $p$ on the shock wave time for (a) ${\delta }_0=0.001$ and (b) ${\delta }_0=0.01$. The red circles indicate the analytically predicted shock wave time ($T_s$), and the black squares denote the shock wave time from numerical simulations ($T_s^{\prime }$).

Figure 4

Effect of the precompression ${\delta }_0$ on the shock wave time for (a) $p=0.5$ and (b) $p=1.5$. The red circles indicate the analytically predicted shock wave time ($T_s$), and the black squares denote the shock wave time from numerical simulations ($T_s^{\prime }$). Data are plotted on a log-log scale.