Rigidity of Branching Microstructures in Shape Memory Alloys

Abstract We analyze generic sequences for which the geometrically linear energy $$\begin{aligned} E_\eta (u,\chi )\,{:}{=} \,\eta ^{-\frac{2}{3}}\int _{B_{1}\left( 0\right) } \left| e(u)- \sum _{i=1}^3 \chi _ie_i\right| ^2 \, \mathrm {d}x+\eta ^\frac{1}{3} |D\chi _i|({B_{1}\left( }) \end{aligned}$$ E η ( u , χ ) : = - 2 3 ∫ B 1 0 e ∑ i d x + | D remains bounded in limit $$\eta \rightarrow 0$$ → . Here $$ e(u) \,{:}{=}\,1/2(Du + Du^T)$$ / T is (linearized) strain of displacement u , strains $$e_i$$ correspond to martensite a shape memory alloy undergoing cubic-to-tetragonal transformations and partition into phases given by $$\chi _i:{B_{1}\left( \{0,1\}$$ { } In this regime it known that addition simple laminates, branched structures are also possible, if austenite was present would enable form habit planes. an ansatz-free manner we prove alignment macroscopic interfaces between twins as predicted well-known rank-one conditions. Our proof proceeds via non-convex, non-discrete-valued differential inclusion \in \bigcup _{1\le i\ne j\le 3} {\text {conv}} \{e_i,e_j\}, ∈ ⋃ ≤ ≠ j conv satisfied weak limits classify all solutions. particular, there exist no convex integration solutions with complicated geometric structures.