Analytical Solutions of Shock Fitting Equations for the Multishock Compaction of Die-Contained Powder Media

In an earlier paper [4], two sets of equations which governed the processes of propagation of shock waves reflected from the punch and plug surfaces in a die-contained copper powder medium were presented. The pressure-specific volume relation included in the sets of equations was composed of three partial relations having different material constants. In the present paper the sets of equations are simplified by assuming that the pressure and specific volume at the front and back sides of the shock front are always related by the same material constants, and linear equations are obtained by introducing a further minor assumption into the simplified nonlinear equations included in the sets of equations. Two sorts of analytical solutions of the linear equations are obtained. One is a general-form solution, while the other is a closed-form solution. The general-form solution calculated is compared satisfactorily with the difference solution computed in the previous study, confirming that the assumption introduced into the simplified equations is minor. Furthermore, calculated characteristics of the general-form solution are revealed by the consideration of the simplified equations and the linear equations, giving greater insight into the compaction processes. The closed-form solution, which is obtained only for the propagation of the shock wave starting from the punch surface and returning from the plug surface, agrees well with the general-form solution.