Twofold rotation between kinematically compatible martensites

In modelling phase-transforming solids, we employ an energy minimization procedure of finding the deformations as the minimizers of a free energy functional of the material occupying Ω ∈ ℝ
³
. In the case of an austenite phase coexisting with a laminate of two alternating martensite variants, a transition layer (phase boundary) is often present in the microstructure to accommodate the deformations on either side. The kinematic compatibility condition defining a rank-1 connection between the deformation gradients on either side of the interface leads to the elimination of the transition layer and the accompanying energy barrier to transformation. Hence, in alloys in which the austenite and martensite phases are kinematically compatible, the hysteresis is expected to be minimum. In the austenite–martensite laminate case, we are interested in finding the rank-1 connection between the identity matrix I (the deformation gradient of the undistorted austenite) and the average deformation gradient in the martensite laminate. Therefore, depending on the pair of martensite variants forming the laminates, the kinematic compatibility is either satisfied or not. This paper presents the proof of existence of a 2-fold (180° connected) rotation between two compatible variants of martensite which have a rank-1 connection between them. The application of this proof is in analyses like multiscale structure–property mathematical modelling which requires the knowledge of compatibility between the martensite variants.