Rigidity of Branching Microstructures in Shape Memory Alloys

<jats:p>We analyze generic sequences for which the geometrically linear energy <jats:disp-formula><jats:alternatives><jats:tex-math>$$\begin{aligned} E_\eta (u,\chi )\,{:}{=} \,\eta ^{-\frac{2}{3}}\int _{B_{1}\left( 0\right) } \left| e(u)- \sum _{i=1}^3 \chi _ie_i\right| ^2 \, \mathrm {d}x+\eta ^\frac{1}{3} \sum _{i=1}^3 |D\chi _i|({B_{1}\left( 0\right) }) \end{aligned}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>η</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace /> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mspace /> <mml:msup> <mml:mi>η</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:msup> <mml:msub> <mml:mo>∫</mml:mo> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mfenced> <mml:mn>0</mml:mn> </mml:mfenced> </mml:mrow> </mml:msub> <mml:msup> <mml:mfenced> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>-</mml:mo> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>3</mml:mn> </mml:munderover> <mml:msub> <mml:mi>χ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:mfenced> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace /> <mml:mi>d</mml:mi> <mml:mi>x</mml:mi> <mml:mo>+</mml:mo> <mml:msup> <mml:mi>η</mml:mi> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msup> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>i</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>3</mml:mn> </mml:munderover> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>D</mml:mi> <mml:msub> <mml:mi>χ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mfenced> <mml:mn>0</mml:mn> </mml:mfenced> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math></jats:alternatives></jats:disp-formula>remains bounded in the limit <jats:inline-formula><jats:alternatives><jats:tex-math>$$\eta \rightarrow 0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. Here <jats:inline-formula><jats:alternatives><jats:tex-math>$$ e(u) \,{:}{=}\,1/2(Du + Du^T)$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mspace /> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:mspace /> <mml:mn>1</mml:mn> <mml:mo>/</mml:mo> <mml:mn>2</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>D</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>D</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mi>T</mml:mi> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> is the (linearized) strain of the displacement <jats:italic>u</jats:italic>, the strains <jats:inline-formula><jats:alternatives><jats:tex-math>$$e_i$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>e</mml:mi> <mml:mi>i</mml:mi> </mml:msub> </mml:math></jats:alternatives></jats:inline-formula> correspond to the martensite strains of a shape memory alloy undergoing cubic-to-tetragonal transformations and the partition into phases is given by <jats:inline-formula><jats:alternatives><jats:tex-math>$$\chi _i:{B_{1}\left( 0\right) } \rightarrow \{0,1\}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>χ</mml:mi> <mml:mi>i</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mrow> <mml:msub> <mml:mi>B</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mfenced> <mml:mn>0</mml:mn> </mml:mfenced> </mml:mrow> <mml:mo>→</mml:mo> <mml:mrow> <mml:mo>{</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>. 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 <jats:disp-formula><jats:alternatives><jats:tex-math>$$\begin{aligned} e(u) \in \bigcup _{1\le i\ne j\le 3} {\text {conv}} \{e_i,e_j\}, \end{aligned}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable>

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