Optimality of General Lattice Transformations with Applications to the Bain Strain in Steel

Abstract

This article provides a rigorous proof of a conjecture by E.C. Bain in 1924 on the optimality of the so-called "Bain strain" based on a criterion of least atomic movement. A general framework that explores several such optimality criteria is introduced and employed to show the existence of optimal transformations between any two Bravais lattices. A precise algorithm and a GUI to determine this optimal transformation is provided. Apart from the Bain conjecture concerning the transformation from face-centred cubic to body-centred cubic, applications include the face-centred cubic to body-centred tetragonal transition as well as the transformation between two triclinic phases of Terephthalic Acid. The GUI can be accessed under http://uk.mathworks.com/matlabcentral/fileexchange/55554-optlat