Title: Rigidity of branching microstructures in shape memory alloys,
Abstract: We analyze generic sequences for which the geometrically linear energy
\[E_\eta(u,\chi):= \eta^{-\frac{2}{3}}\int_{B_{0}(1)} \left| e(u)-
\sum_{i=1}^3 \chi_ie_i\right|^2 d x+\eta^\frac{1}{3} \sum_{i=1}^3
|D\chi_i|(B_{0}(1))\] remains bounded in the limit $\eta \to 0$. Here $ e(u)
:=1/2(Du + Du^T)$ is the (linearized) strain of the displacement $u$, the
strains $e_i$ correspond to the martensite strains of a shape memory alloy
undergoing cubic-to-tetragonal transformations and $\chi_i:B_{0}(1) \to
\{0,1\}$ is the partition into phases. In this regime it is known that in
addition to simple laminates also branched structures are possible, which if
austenite was present would enable the alloy to form habit planes.
In an ansatz-free manner we prove that the alignment of macroscopic
interfaces between martensite twins is as predicted by well-known rank-one
conditions. Our proof proceeds via the non-convex, non-discrete-valued
differential inclusion \[e(u) \in \bigcup_{1\leq i\neq j\leq 3}
\operatorname{conv} \{e_i,e_j\}\] satisfied by the weak limits of bounded
energy sequences and of which we classify all solutions. In particular, there
exist no convex integration solutions of the inclusion with complicated
geometric structures.