Martensite

A product of a displacive transformation. In carbon steel, rapid cooling can trap carbon in a distorted iron lattice, producing high hardness and low as-quenched toughness.

A martensitic variant is one of the lattice deformations corresponding to the low-temperature crystal phase. The parent crystal's discrete symmetries generate different variants. In a nonlinear elasticity model each representative $U_i$ supplies a hyperelastic energy well $SO(3)U_i$. A representative need not itself be a symmetric stretch tensor; multiplication by a proper rotation leaves the same well.

Deformation twinning joins differently oriented crystal variants across a coherent interface. In a piecewise affine nonlinear elasticity model the two deformation gradients must differ by $a\otimes N$, where $N$ is the reference interface normal. The Hadamard jump condition gives this requirement from continuity along every interface tangent. Compatible alternating layers can then make a mixture of variants whose macroscopic shape differs from either variant alone.

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.