The attainable stiffness tensors of two-phase elastic composites
Fix isotropic elasticity tensors $C_i E = 2\mu_i E + \lambda_i \operatorname{tr}(E) I$ ($i = 1, 2$) acting on symmetric $3 \times 3$ strains, with $\mu_i > 0$, $3\lambda_i + 2\mu_i > 0$ and $C_2 - C_1$ positive definite, and fix a volume fraction $\theta \in (0,1)$. For every measurable periodic indicator $\chi : (\mathbb{R}/\mathbb{Z})^3 \to \{0, 1\}$ with $\int_{[0,1]^3} \chi = \theta$ let $C_\chi = \chi C_1 + (1 - \chi) C_2$, and define its effective tensor by $$E : C_\chi^{\mathrm{hom}} E = \inf_{\varphi \in H^1_{\mathrm{per}}([0,1]^3; \mathbb{R}^3)} \int_{[0,1]^3} \bigl(E + e(\varphi)\bigr) : C_\chi \bigl(E + e(\varphi)\bigr) \, dx,$$ where $e(\varphi) = (D\varphi + D\varphi^T)/2$. Let $$G_\theta(C_1, C_2) = \operatorname{cl} \{ C_\chi^{\mathrm{hom}} : \chi \text{ as above} \},$$ the closure in the finite-dimensional space of elasticity tensors, which includes limits of attainable effective tensors.
For the analogous conductivity problem, the G-closure of two isotropic phases at a fixed volume fraction is characterized by explicit bounds on the eigenvalues of the effective conductivity tensor, which extend the Hashin–Shtrikman bounds to anisotropic composites (Murat and Tartar; Lurie and Cherkaev). For elasticity, Hashin–Shtrikman bounds on the effective energy were computed explicitly for two isotropic phases by Allaire and Kohn in two dimensions and, for the primal energy from above and the complementary energy from below, by Burazin, Crnjac and Vrdoljak in three dimensions.
Problem. Give explicit necessary and sufficient conditions on a tensor $C$ for $C \in G_\theta(C_1, C_2)$. The characterization must cover arbitrary microstructures and the full tensor, not only the energy under one selected loading.
Source: The G-closure problem of Lurie and Cherkaev (1981); for two well-ordered isotropic elastic phases, see Milton (2002), Chapter 30, and Burazin, Crnjac and Vrdoljak (2024), §1. Stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- OC
- Keywords
- AIM Open Applied Problems, composite materials, G-closures, homogenization
References
@article{lurie1981,
author = {Lurie, K. A. and Cherkaev, A. V.},
title = {{$G$}-closure of a set of anisotropic conducting media in the case of two dimensions},
journal = {Dokl. Akad. Nauk SSSR},
volume = {259},
year = {1981},
pages = {328--331},
note = {English translation: Soviet Phys. Dokl. 26 (1981), 657--659},
}
@incollection{milton2002,
author = {Milton, Graeme W.},
title = {Properties of the {$G$}-closure and extremal families of composites},
booktitle = {The Theory of Composites},
chapter = {30},
publisher = {Cambridge University Press},
address = {Cambridge},
year = {2002},
pages = {643--670},
doi = {10.1017/CBO9780511613357.031},
}
@article{burazin2024,
author = {Burazin, Kre{\v s}imir and Crnjac, Ivana and Vrdoljak, Marko},
title = {Explicit {H}ashin--{S}htrikman bounds in {3D} linearized elasticity},
journal = {Filomat},
volume = {38},
number = {17},
year = {2024},
pages = {6033--6048},
doi = {10.2298/FIL2417033B},
}
@incollection{murat1985,
author = {Murat, Fran{\c c}ois and Tartar, Luc},
title = {Calcul des variations et homog{\'e}n{\'e}isation},
booktitle = {Les M{\'e}thodes de l'Homog{\'e}n{\'e}isation: Th{\'e}orie et Applications en Physique ({B}r{\'e}au-sans-{N}appe, 1983)},
series = {Collection de la Direction des {\'E}tudes et Recherches d'{\'E}lectricit{\'e} de France},
volume = {57},
publisher = {Eyrolles},
address = {Paris},
year = {1985},
pages = {319--369},
}
@article{lurie1986,
author = {Lurie, K. A. and Cherkaev, A. V.},
title = {Exact estimates of the conductivity of a binary mixture of isotropic materials},
journal = {Proc. Roy. Soc. Edinburgh Sect. A},
volume = {104},
number = {1-2},
year = {1986},
pages = {21--38},
doi = {10.1017/S0308210500019041},
}
@article{allaire1993,
author = {Allaire, Gr{\'e}goire and Kohn, Robert V.},
title = {Explicit optimal bounds on the elastic energy of a two-phase composite in two space dimensions},
journal = {Quart. Appl. Math.},
volume = {51},
number = {4},
year = {1993},
pages = {675--699},
doi = {10.1090/qam/1247434},
}
@misc{aim-elastic-two-phase-g-closure,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{The attainable stiffness tensors of two-phase elastic composites}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/233-elastic-two-phase-g-closure.md},
note = {Entry 233 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-00588, "The attainable stiffness tensors of two-phase elastic composites", clopen.solutions.