Shock Waves in Nonlinear Granular Chain with Viscosity under Hertz Contact

Abstract

By considering the dependence of the viscosity on the relative velocity between adjacent beads, the shock wave structure in a one-dimensional nonlinear granular system is studied in the present paper. Two different distributions of the shock wave are found. In one distribution, the shock front is a discontinuous surface; in the other, there is an oscillation in the shock front. Furthermore, the wave structure depends on the magnitude of the viscosity and the external impact. There is a critical value of viscosity beta(c). It is found that in the shock front the beads oscillate and the maximum bead velocity is larger than the piston speed if the magnitude of the viscosity is less than beta(c). In contrast, the shock front is a discontinuous surface and the bead velocities are all equal to the piston speed, if the viscosity exceeds beta(c). All numerical results are verified by analytical studies. In the long-wavelength approximation, we obtain a Korteweg-de Vries-Burgers (KdV Burgers) equation that can completely explain the numerical results, and both the numerical and analytical results are in good agreement. This KdV-Burgers equation can be used to study many previous works on both solitary waves and shock waves.