TY - JOUR

T1 - Rigidity of Branching Microstructures in Shape Memory Alloys

AU - Simon, Theresa M.

N1 - Funding Information:
The author thanks her PhD adviser Felix Otto for suggesting the problem and the many helpful discussions. Partially funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - Project-ID 211504053 - SFB 1060.
Publisher Copyright:
© 2021, The Author(s).

PY - 2021/9

Y1 - 2021/9

N2 - We analyze generic sequences for which the geometrically linear energy Eη(u,χ):=η-23∫B1(0)|e(u)-∑i=13χiei|2dx+η13∑i=13|Dχi|(B1(0))remains bounded in the limit η→ 0. Here e(u):=1/2(Du+DuT) is the (linearized) strain of the displacement u, the strains ei correspond to the martensite strains of a shape memory alloy undergoing cubic-to-tetragonal transformations and the partition into phases is given by χi: B1(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)∈⋃1≤i≠j≤3conv{ei,ej},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.

AB - We analyze generic sequences for which the geometrically linear energy Eη(u,χ):=η-23∫B1(0)|e(u)-∑i=13χiei|2dx+η13∑i=13|Dχi|(B1(0))remains bounded in the limit η→ 0. Here e(u):=1/2(Du+DuT) is the (linearized) strain of the displacement u, the strains ei correspond to the martensite strains of a shape memory alloy undergoing cubic-to-tetragonal transformations and the partition into phases is given by χi: B1(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)∈⋃1≤i≠j≤3conv{ei,ej},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.

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U2 - 10.1007/s00205-021-01679-8

DO - 10.1007/s00205-021-01679-8

M3 - Article

AN - SCOPUS:85107441530

SN - 0003-9527

VL - 241

SP - 1707

EP - 1783

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

IS - 3

ER -