Determine the true shear modulus of the hinged honeycomb

Determine the true effective shear modulus of the hinged honeycomb, including whether the certified lower bound at the selected interface angle is sharp.

Background

The paper constructs a regular-hexagonal hinged honeycomb with zero effective bulk modulus and proves that, at a particular interface angle, its normalized shear modulus is at least P/Q > 0.8528, thereby exceeding the previously established value of 4/5. The authors note that this bound is based on one angle, one tiling, and one polynomial degree, so it may not equal the actual shear modulus of the construction.

Determining the true shear modulus would clarify how much shear stiffness the hinged honeycomb can retain and whether improvements to the stress-field ansatz, interface angle, polynomial degree, or geometry can substantially strengthen the result.

References

The bound is unlikely to be sharp, because it certifies one structure at one angle with one polynomial degree. A higher degree, another angle, or another tiling may yield more, and the true shear modulus of the honeycomb is unknown.

— 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

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