A group of objects, arranged in a regular and periodic configuration, is forced to rearrange itself homogeneously into a different periodic structure. Out of the arbitrarily many ways of doing so, which is the one requiring least movement? This question is particularly relevant in displacive phase transitions in crystalline solids. There the objects could be atoms which are forced into a new crystalline structure due to changes in temperature. For complex lattices it is unclear how to derive such optimal transformations strains.
THE THREE-WELL PROBLEM
There are three principal ways of transforming a cubic to a tetragonal crystalline structure. Each possibility is called a tetragonal variant and locally the material needs to deform according to one of them. However, different regions may (and do) transform according to different variants - leading to complex patterns of different variants. The net deformation of these patterns is related to the quasiconvex hull of the variants and its determination is both mathematically interesting and of practical importance. The explicit knowledge would allow to predict the macroscopic response of the material. So far, it has only been possible to determine this hull in essentially two dimensional problems – but not in three.
A NEW THEORY FOR LATH MARTENSITE
All commonly accepted theories of lath martensite assume a large number of dislocations and the most influential papers on the morphology of lath martensite are by C.M. Wayman and date back to the 1980s with image resolutions on the scale of micrometers. Due to their impact, phenomenological theories explaining features of lath martensite, continued to look at steel on the same “macroscopic” scale. Based on the assumption of energy minimisation, we suggest a new model which suggests that, what appears to be dislocations, may in fact be very fine twins. The goal of this project is to use experimental techniques to look at lath martensite on a finer scale and to test our hypothesis.