Shock–induced martensitic phase transitions: critical stresses, Riemann problems and applications

Abstract

In this paper we develop a theory for phase transitions in solids under shock loading. This theory applies, in particular, to a class of experiments which, as a result of solid–to–solid phase transitions, give rise to certain characteristic patterns consisting of two shock–like waves. We show that the single assumption that stresses in a phase cannot lie beyond its transition boundaries leads to a complete model for the physical systems at hand. This model is different from others proposed in the literature: it does not make use of kinetic relations and it accounts for the observed wave histories without parameter fitting. The first part of this paper focuses on the basic mathematical description of our model and it presents solutions to the complete set of Riemann problems which could arise as a result of dynamic interactions, including the basic two–wave structures mentioned above. In the second part of the paper we use our Riemann solver to construct general solutions for the piecewise constant initial–value problems usually arising in experiment, and we specialize our solutions to two widely studied polymorphic phase changes: the graphite–diamond transition and the α–∈ transition in iron. We show that, in the presence of well–accepted quations of state for the pure phases, our model leads to close quantitative agreement with a wide range of experimental results; possible sources of disagreement in certain fine features are also discussed. Interestingly, in some cases our theory predicts sequences of events which differ from those generally accepted. In particular, our model predicts a variety of regimes for the iron experiments which previous theoretical investigations had not surmised, and it indicates the existence of certain unexpected transformation domains in the graphite systems.

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