(corresponding to $v=e_1$) is a simple solution in the first case, but a degenerate solution in the second case. Correspondingly, $$\text{$|A_1v|-|B_1v|=O(\alpha|v-e_1|)$ but $|A_2v|-|B_2v|=O(\alpha|v-e_1|^2)$.  }$$ Therefore, small deviations of the interfacial direction from $v=e_1$ generate less elastic energy in the case $K_2$ than in the case $K_1$, leading to a lower global energy. If rotations are neglected, which is the case if one treats the scalar case, then the two terms are of the same order, $$A_1v-B_1v=-2\alpha v_2 e_1=O(\alpha |v-e_1|)$$ and $$A_2v-B_2v=-2\alpha v_2 e_2=O(\alpha |v-e_1|)\,.$$ Therefore, the difference between $K_1$ and $K_2$ is not present in scalar models. The detailed exploitation of this effect in the construction requires an appropriate interaction between the two components and will be discussed in Section \[secUBtheo2\], see in particular Lemma \[eqconstronerecttheo2\].

From a mechanical viewpoint, in most materials the phases are symmetry-related and in particular one expects the eigenstrains to have the same determinant [@B9_BallJames87; @B9_Bhatta; @OtsukaWayman1999], which corresponds to our case $j=1$. The two-dimensional geometry we discuss corresponds to a three-dimensional setting in which both eigendeformations coincide in the third dimension. This would be, for example, the case for the three-dimensional eigenstrains $$A_1'= 
 \begin{pmatrix}
   1 & -\alpha & 0\\
 0 & 1 & 0 