Establish a nontrivial upper bound for shear stiffness at zero bulk modulus

Establish whether a nontrivial upper bound strictly below the constituent shear modulus exists for the supremum S of normalized shear moduli of two-dimensional isotropic mixtures of an incompressible elastic solid and void with zero effective bulk modulus.

Background

For the class of two-dimensional isotropic solid–void mixtures considered in the paper, the authors define S as the supremum of the normalized effective shear modulus among materials with zero effective bulk modulus. Their construction proves only the lower bound S ≥ P/Q > 0.8528, while the uniform-strain argument supplies the trivial upper bound S ≤ 1.

The unresolved issue is whether structural or variational constraints force S to remain strictly below 1. Resolving it would determine whether the demonstrated improvement over 4/5 is close to optimal or whether substantially stiffer zero-bulk-modulus materials can be constructed.

References

On the other side, the only upper bound is the trivial $\mu{*}\le\mu$, so $P/Q\le S\le1$, and most of the remaining uncertainty lies on the upper side. Whether a nontrivial upper bound exists at zero bulk modulus and whether the construction has a three-dimensional analogue are the natural next questions.

— A hinged honeycomb with zero bulk modulus retaining more than four-fifths of its constituent's shear modulus  (2610.11280 - Chen, 8 Oct 2026) in Section Conclusions