Dynamic bifurcation during high-rate planar extension of a thin rectangular block

Abstract

Dynamic bifurcations in deformation from a uniform state of rapid extension are studied for a rectangular block. The block may be viewed as a segment of a thin shell which is one period of a deformation which is periodic along the circumference. The material is taken to be incompressible and characterized by an incrementally linear, rate-independent elastic-plastic constitutive law. The stress-state in the block prior to bifurcation is approximately uniform and the effect of bifurcation mode inertia is taken into account. The bifurcation problem is formulated as a two-variable problem in terms of a constant stress and a variable representing the rate of growth of a bifurcation mode, here called the localization speed. A qualitative lower limit on growth of a diffuse mode in terms of the localization speed and a limit on the background motion due to the requirement of subsonic deformation are suggested. Bifurcation stress-localization speed distributions are computed and important changes from the quasistatic bifurcation behavior responsible for the phenomenon of multiple necking are shown to exist. In particular, it is found that the long wavelength modes are suppressed because their rate of formation is too slow compared to the background deformation. The main focus is on diffuse bifurcations; however, other bifurcation phenomena are touched upon by way of incremental wave propagation arguments.