Neck propagation as a shock wave

Abstract

Abstract." Neck propagation in the stretching of elastic solid filaments having a yield point was analyzed using the space one-dimensional thin filament governing equations developed previously by the authors and other researchers. Constitutive model for the filament was assumed to be expressible as engineering tensile stress r/(X) (tensile force) given as a function of elongational strain with the ~/(X) curve having a yield point maxima followed by a minima and a breaking point greater than the yield point maxima. Also incorporated into the model is the hysteresis of irreversible plastic deformation. When inertia is taken into consideration, the thin filament equations were found to reduce to the nonlinear wave equation 02/~(X)/02 2= CI02X/O'~ 2 where ~ is Lagrangean space coordinate, r is time, and C 1 is inertia coefficient. The above nonlinear wave equation yields a solution X(2, z) having a stepwise discontinuity in X which propagates along the 2 axis. The zero speed limit of the step wave solution was found to describe the above neck propagation occurring in solid filaments. Furthermore, it was recognized that the nonlinear wave equation was known for many years to also govern the plastic shock wave which propagates axially within a metal rod subjected to a very strong impact on its end. The one-dimensional atmospheric shock wave also was known to be governed by the nonlinear wave equation upon making certain simplifying assumptions. The above and other evidences lead to the conclusion that neck propagation occurring in the extension of solid filament obeying the above r/(X) function can be formally described as a shock wave.