Twin plane

A twin plane is a compatible planar interface between differently oriented crystal variants. In a piecewise homogeneous elastic deformation its reference normal is the $N$ in a rank-one connection between the two variant deformation gradients. Continuity of the deformation holds along this plane even though its derivative changes. Other material planes crossing the interface can kink after deformation.

Let $B=\mathrm{diag}(b,a,a)$, with $a,b>0$, $a\ne b$, and $A=R\mathrm{diag}(a,b,a)$ be a twin-related gradient with $\cos \theta =2ab/(a^2+b^2)$, $\sin \theta =\kappa (b^2-a^2)/(a^2+b^2)$, $\kappa =\pm 1$. Their jump is $(b^2-a^2)(-b,\kappa a,0)\otimes (1,\kappa ,0)/(a^2+b^2)$.

A plane through the origin with normal $p=(p_1,p_2,0)$ is spanned by $e_3$ and $t=(-p_2,p_1,0)$. Both gradients map $e_3$ to the same direction, so the two image half-planes are coplanar exactly when $At$ and $Bt$ are parallel in their transverse section. Direct calculation givesIts nonzero prefactor makes planarity equivalent to $p\parallel (1,\kappa ,0)$, the twin normal. On that plane the two affine maps agree, proving sufficiency as well. The restriction on the third normal component matters; this statement is not a classification of every possible material plane.