Compatibilities and supercompatibility conditions in shape memory alloys determined from correspondence, metrics and symmetries

The phenomenological theory of martensite crystallography (PTMC) explains the habit planes of bi-variant laminate martensite product and the transformation twins between the variants. It also permits to determine whether or not the austenite and martensite lattice parameters satisfy supercompatibility by three mathematical equations called “cofactor conditions”. The PTMC calculations are in great part based on mathematical tools of continuum mechanics (polar decompositions and stretch tensors). Here, we propose to use an alternative approach called correspondence theory (CT). Supercompatibility is obtained when three types of compatibility are satisfied: a) austenite/martensite (A/M) compatibility by invariant plane strain (IPS) lattice distortion, b) martensite/martensite (M/M) compatibility by transformation twins, and c) their intercompatibility by a parallelism between the shear direction of the IPS and the shear direction of the twin. The satisfaction of this “shear/shear” condition implies that of the cofactor conditions. These conditions can be written with pure crystallographic parameters (metrics, group of symmetries and correspondence). The twin equations are obtained from double-coset decomposition of the group of austenite symmetries. The A/M compatibility condition appears as a degeneracy condition of a quadratic form of a symmetric matrix called “compatibility by metric correspondence” (CMC). The IPS shear vector in the shear/shear equation is obtained from another matrix called “shear by metric correspondence” (SMC). As example, it is shown that B19’ martensite is supercompatible if its lattice parameters follow specific analytical functions of the monoclinic angle. The CT offers a promising framework to develop new shape memory alloys with supercompatible martensite and high fatigue resistance.