Rigidity of Branching Microstructures in Shape Memory Alloys

We analyze generic sequences for which the geometrically linear energy E-eta(u, chi) :=eta(-2/3) integral(B1(0)) vertical bar e(u) - Sigma(3)(i=1)chi(i)e(i)vertical bar(2) dx + eta(1/3) Sigma(3)(i=1) vertical bar D chi(i)vertical bar(B-1 (0)) remains bounded in the limit eta -> 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 the partition into phases is given by chi(i) : B-1 (0) -> {0, 1}. In this regime it is known that in addition to simple laminates, branched structures are also 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) is an element of boolean OR(1 <= i not equal j <= 3) 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.

Automated summary

In this study, analysis was conducted on the properties of generic sequences in a particular scenario where the alignment of macroscopic interfaces between martensite twins was proven as predicted by rank-one conditions. It was found that in certain cases, branched structures were possible, while there were no convex integration solutions with complicated geometric structures.