Propagation Of Shock Waves In One-dimensional Granular Chain

Based on the Hertz model,the shock wave in one dimensional bead chain is studied numerically.It is noted that the piston compression leads to the formation of a shock wave.Ahead of it,all the beads are at rest,while behind it all beads oscillate and move with an average velocity of the piston speed.The overlap of the beads behind the shock front is larger than that of ahead of the shock wave.It increases as the piston velocity increases as well as the initial prestress increases.It is also found that the velocity of the shock wave increases as both the piston velocity and the initial overlap increase.The limitation of the maximum bead velocity is two times the piston velocity.However,the larger the piston velocity is,the slower the maximum bead velocity is reached.It is found that the shock-wave front is composed by several solitary waves in the limiting case that the initial overlap is zero.Based on these results,we obtain the relation between shock velocity and the piston velocity and good agreements are obtained between the analytical results and the numerical ones.Furthermore,the dependence of the shock velocity on both the piston velocity and the initial prestress is obtained for the general case,which is in good agreement with the numerical simulation.The impact into a granular medium is common in many natural and human-made processes.The prestress is one of the most important parameters controlling their properties.It is a function of the parameters of the granular materials and the external force.The present results may be used to detect the material characters by measuring both the piston velocity and the shock velocity.It is also can be used to detect the piston velocity by measuring the shock velocity,which can be used to detect the velocity of the bullet if we use the granular medium to protect our human-made materials.