Modeling the Fourth Power Law Response of Viscoplastic Materials
A standard way to parametrize a steady structured shock going through a viscoplastic material, is by plotting the stress jump across the wave (Ds) as function of its maximum strain rate () plotted on a logarithmic scale. Doing this we usually get a straight line, which means that . Grady and coworkers noticed in the 1980s that for many materials b=4. Since then, several workers in the field tried to understand this unusual behavior, but they were not able to arrive at any plausible explanation. Recently, Grady revisited his fourth power law with an extensive review, including an outline of various mechanisms proposed as possible explanations of this law. But still, none of these mechanisms is able to explain the fourth power law response. In view of this background, we use here the fourth power law to calibrate the strain rate component of the constitutive response of Al 6061-T6. To this end, we integrate the steady viscoplastic equations (developed long ago) describing the propagation of a plane wave from the elastic precursor level to the stress plateau level. We find that in order to reproduce the fourth power law, the plastic deformation rate has to depend on the deviator overstress by: , with α=2.38. But for high shock levels, when the elastic precursor is overdriven, using this value of α leads to a higher value of the slope b, close to b=6. This result has yet to be verified experimentally.
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