The full set of zero-energy cubic-to-tetragonal strains
Fix $\alpha, \beta > 0$ with $\alpha \ne \beta$. Let $U_1 = \operatorname{diag}(\alpha, \beta, \beta)$, $U_2 = \operatorname{diag}(\beta, \alpha, \beta)$ and $U_3 = \operatorname{diag}(\beta, \beta, \alpha)$, and let $K = \bigcup_{i=1}^3 SO(3) U_i$, the energy wells of a cubic-to-tetragonal martensitic transformation. Let $K^{qc}$ be its quasiconvex hull: $F \in K^{qc}$ when on $Q = (0,1)^3$ there are maps $y_j \in W^{1,\infty}(Q; \mathbb{R}^3)$ with $y_j = Fx$ on $\partial Q$ and uniformly bounded gradients such that $$\int_Q \operatorname{dist}(Dy_j(x), K)^2 \, dx \longrightarrow 0.$$ Every $F \in K^{qc}$ has $\det F = \alpha\beta^2$. Ball and James computed the quasiconvex hull of two rank-one connected wells of equal determinant, among them $SO(3)U_1 \cup SO(3)U_2$. The positive diagonal matrices in $K^{qc}$ are exactly the $\operatorname{diag}(v_1, v_2, v_3)$ with $v_1 v_2 v_3 = \alpha\beta^2$ and $\min(\alpha, \beta) \le v_i \le \max(\alpha, \beta)$ for each $i$, a result of Peigney to which Ball and Galanopoulou give a short proof. Dolzmann and Kirchheim proved that when $\alpha\beta^2 = 1$ the identity is an interior point of $K^{qc}$ relative to the matrices of determinant one.
Problem. Determine $K^{qc}$ for every such pair $\alpha, \beta$: give necessary and sufficient conditions for a matrix $F \in \mathbb{R}^{3 \times 3}$ to belong to $K^{qc}$, deciding general matrices, including non-diagonal ones.
Source: Ball (2002), Problem 16, for the energy wells of martensitic transformations in general, with the cubic-to-tetragonal case singled out there; stated for that case by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, martensitic phase transformations, quasiconvex hulls
References
@incollection{ball2002,
author = {Ball, John M.},
title = {Some open problems in elasticity},
booktitle = {Geometry, Mechanics, and Dynamics},
editor = {Newton, Paul and Holmes, Philip and Weinstein, Alan},
publisher = {Springer},
address = {New York},
year = {2002},
pages = {3--59},
doi = {10.1007/0-387-21791-6_1},
}
@article{ball1992,
author = {Ball, J. M. and James, R. D.},
title = {Proposed experimental tests of a theory of fine microstructure and the two-well problem},
journal = {Philos. Trans. Roy. Soc. London Ser. A},
volume = {338},
number = {1650},
year = {1992},
pages = {389--450},
doi = {10.1098/rsta.1992.0013},
}
@article{peigney2013,
author = {Peigney, Micha{\"e}l},
title = {On the energy-minimizing strains in martensitic microstructures---{P}art 1: {G}eometrically nonlinear theory},
journal = {J. Mech. Phys. Solids},
volume = {61},
number = {6},
year = {2013},
pages = {1489--1510},
doi = {10.1016/j.jmps.2012.12.009},
}
@misc{ball2026,
author = {Ball, John M. and Galanopoulou, Myrto},
title = {Compatibility of martensitic microstructures in polycrystals},
year = {2026},
eprint = {2607.12572},
archivePrefix = {arXiv},
}
@article{dolzmann2003,
author = {Dolzmann, Georg and Kirchheim, Bernd},
title = {Liquid-like behavior of shape memory alloys},
journal = {C. R. Math. Acad. Sci. Paris},
volume = {336},
number = {5},
year = {2003},
pages = {441--446},
doi = {10.1016/S1631-073X(03)00082-7},
}
@misc{aim-tetragonal-quasiconvex-hull,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The full set of zero-energy cubic-to-tetragonal strains}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/155-tetragonal-quasiconvex-hull.md},
note = {Entry 155 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00907, "The full set of zero-energy cubic-to-tetragonal strains", clopen.solutions.