Quasiconvexity and hysteresis

Hysteresis is a universal feature of first order phase transformations. During observation of the transformation, this may present itself as different transformation temperatures on heating and cooling, or different transformation stresses on loading and unloading. One way to approach this mathematically is through a study of local minimizers. A natural concept for local minimizer emerges from the constraint of geometric compatibility, and has links to the concept of quasiconvexity (joint work with J. M. Ball). This concept seems to explain, at least qualitatively, the hysteresis observed in some biaxial tests the author did with C. Chu. However, it clearly does not explain the hysteresis observed in many other experiments, for example, the simple experiment of measuring transformation temperatures upon heating and cooling. By surveying the hysteresis in lots of systems, we arrive at a related but different concept. At this time of the writing of this abstract, it is not clear whether this new concept also has links to quasiconvexity, but in any case it relates in some way to the idea that, if a material has certain special distortions, then the phases fit together unusually well.