Archive for Rational Mechanics and Analysis ( IF 2.6 ) Pub Date : 2021-06-03 , DOI: 10.1007/s00205-021-01679-8 Theresa M. Simon
We analyze generic sequences for which the geometrically linear energy
$$\begin{aligned} E_\eta (u,\chi )\,{:}{=} \,\eta ^{-\frac{2}{3}}\int _{B_{1}\left( 0\right) } \left| e(u)- \sum _{i=1}^3 \chi _ie_i\right| ^2 \, \mathrm {d}x+\eta ^\frac{1}{3} \sum _{i=1}^3 |D\chi _i|({B_{1}\left( 0\right) }) \end{aligned}$$remains bounded in the limit \(\eta \rightarrow 0\). 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 \(\chi _i:{B_{1}\left( 0\right) } \rightarrow \{0,1\}\). 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
$$\begin{aligned} e(u) \in \bigcup _{1\le i\ne j\le 3} {\text {conv}} \{e_i,e_j\}, \end{aligned}$$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.
中文翻译：
形状记忆合金中分支微结构的刚性
我们分析了几何线性能量
$$\begin{aligned} E_\eta (u,\chi )\,{:}{=} \,\eta ^{-\frac{2}{3}}\int _{B_{1}\left ( 0\right) } \left| e(u)- \sum _{i=1}^3 \chi _ie_i\right| ^2 \, \mathrm {d}x+\eta ^\frac{1}{3} \sum _{i=1}^3 |D\chi _i|({B_{1}\left( 0\right) }) \end{对齐}$$保持在极限\(\eta \rightarrow 0\) 内。这里\( e(u) \,{:}{=}\,1/2(Du + Du^T)\)是位移u的（线性化）应变，应变\(e_i\)对应于经历立方到四方转变的形状记忆合金的马氏体应变和相分配由\(\chi _i:{B_{1}\left( 0\right) } \rightarrow \{0,1\} \). 众所周知，在这种情况下，除了简单的层压板外，还可能存在分支结构，如果存在奥氏体，则可以使合金形成习性平面。我们以无 ansatz 的方式证明了马氏体孪晶之间宏观界面的排列与众所周知的一级条件所预测的一样。我们的证明通过非凸、非离散值的微分包含进行
$$\begin{aligned} e(u) \in \bigcup _{1\le i\ne j\le 3} {\text {conv}} \{e_i,e_j\}, \end{aligned}$$满足有界能量序列的弱极限，我们对所有解进行分类。特别是几何结构复杂的夹杂物不存在凸积分解。
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