Shock wave in a one-dimensional granular chain under Hertz contact

The shock wave in one-dimensional bead chain is studied numerically. When the shock wave arrives, the bead velocity oscillates around the piston velocity. It is found that the shock front is composed of several solitary waves and the limitation of the maximum bead velocity is 2 times the piston velocity in the limiting case where the initial overlap is zero. If the initial overlap is not zero, then the maximum bead velocity is less than 2 times the piston velocity but larger than the piston velocity. As the initial overlap increases from zero to the finite value, the shock velocity depends on not only the piston velocity but also the initial overlap. The crossover of the dependence of the shock velocity on the piston velocity from the zero initial prestress to the finite value is obtained in the present manuscript. It is an improvement of the results presented in Phys. Rev. Lett. 108, 058001 (2012)10.1103/PhysRevLett.108.058001. In other words, the dependence of the shock velocity on the parameters of the granular materials is given.