Chiral materials do not have a center of inversion symmetry. Left and right handed versions of the material are distinct. Chiral materials are also called hemitropic or noncentrosymmetric. Some material properties such as piezoelectricity and pyroelectricity can only occur in chiral materials. Classical elasticity makes no distinction between chiral and non-chiral materials because the elastic modulus tensor is fourth rank. By contrast, generalized continuum theories such as Cosserat elasticity do make such a distinction.

Curiously the subject of chiral elasticity has recently become popular. Our research in this area is summarized below: the first prediction of stretch twist or squeeze twist coupling in an elastic solid represented as a continuum. We have performed the first experimental demonstrations of chiral elastic behavior and the first production of chiral elastic materials in 2D and in 3D.

**Rod Lakes and Robert Benedict,
Noncentrosymmetry in micropolar elasticity,
International Journal of Engineering Science, 20 (10), 1161-1167, (1982).**

A solid which is isotropic with respect to coordinate rotations but not with respect to inversions is called noncentrosymmetric, acentric, hemitropic, or chiral. Chirality has no effect upon the classical elastic modulus tensor. In Cosserat elasticity, chirality (hemitropy) has an effect. A chiral Cosserat solid has three new elastic constants in addition to the six considered in the fully isotropic micropolar solid. The chiral micropolar solid is predicted to undergo torsional deformation when subjected to tensile load. Thus chiral solids have different mechanical behavior from solids with a center of symmetry, as allowed by the more general Cosserat elastic theory. The theory predicts coupling between stretching and twisting.

A theoretical and experimental investigation is conducted of a two-dimensionally chiral honeycomb. The honeycomb exhibits a Poisson's ratio of -1 for deformations in-plane. This Poisson's ratio is maintained over a significant range of strain, in contrast to the variation with strain seen in known negative Poisson's ratio materials.

This is the first two-dimensional metamaterial that is chiral; we did not call it by such a name.

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Chiral three-dimensional isotropic cubic lattices with rigid cubical nodules and multiple deformable ribs are developed and analyzed via finite element analysis. The lattices exhibit geometry dependent Poisson's ratio that can be tuned to negative values. Poisson's ratio decreases from positive to negative values as the number of cells increases. Isotropy is obtained by adjustment of aspect ratio. The lattices exhibit significant size effects. Such a phenomenon cannot occur in a classical elastic continuum but it can occur in a Cosserat solid. The material exhibits squeeze-twist coupling and stretch-twist coupling. Coupling of this type cannot occur in classical elasticity. Lattices with this property are made by 3D printing.

This is the first three-dimensional metamaterial that is chiral; we did not call it by such a name.

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Cubic 3D lattices were designed, made by 3D printing, and studied experimentally. One lattice was simple cubic with ribs of diameter 1/5 the cell size. The second lattice was chiral, with spiral ribs. The chiral lattice, but not the achiral lattice, exhibited squeeze-twist coupling with size effects. Squeeze-twist coupling cannot occur in a classically elastic solid but is anticipated by theory in a Cosserat solid. Both lattices exhibited size effects in bending and torsion.

https://doi.org/10.1115/1.4044047 https://doi.org/10.1115/1.4044047

A chiral 3D lattice was designed, made by 3D printing, and studied experimentally. The lattice exhibited squeeze-twist coupling and a Poisson's ratio near zero. Squeeze-twist coupling does not occur in classical elasticity which makes no provision for chirality. By contrast, chiral effects are allowed in Cosserat elasticity. An experimental squeeze-twist coupling strain ratio on the order of unity and a Poisson's ratio near zero are in reasonable agreement with prior finite element analysis of a lattice with similar structure, for which negative Poisson's ratio is anticipated for a sufficient number of cells.

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We report experiments that show a stiff surface lattice exhibits elastic chirality.

**D. R. Reasa and R. S. Lakes, Nonclassical Chiral Elasticity of the Gyroid Lattice, Phys. Rev. Lett. 125, 205502, 13 November (2020). **

The gyroid lattice is a metamaterial which allows chirality that is tunable by geometry. Gyroid lattices were made in chiral and non-chiral form by 3D printing. The chiral lattices exhibited nonclassical elastic effects including coupling between compressive stress and torsional deformation. Gyroid lattices can approach upper bounds on elastic modulus. Effective modulus is increased by distributed moments but is, for gyroid cylinders of sufficiently small radius, softened by a surface layer of incomplete cells. Such size dependence is similar to that in foams is but unlike most lattices.

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Poisson's ratio in chiral Cosserat elastic solids is considered. Chirality allows the Poisson's ratio to exceed classical bounds, even if the material is directionally isotropic and all elastic moduli are within thermodynamic limits based on strain energy density. Poisson's ratio in chiral rods depends on the chiral elastic constants as well as on the shear and bulk moduli, assumed positive. Poisson's ratio can be greater than 0.5 or smaller than -1 for slender chiral specimens.

preprint pdf doi link

doi link https://doi.org/10.1002/pssb.202200338

doi link https://doi.org/10.1016/j.compstruct.2023.117068

Poisson's ratio in chiral isotropic elastic solids can be larger or smaller than the classical thermodynamic bounds. The effect of Cosserat coupling constant k is studied. Analysis shows that solids with weak coupling exhibit Poisson's ratio anomalies for larger specimen sizes than corresponding solids with strong coupling. Experiments on a quasi-isotropic composite with chiral inclusions show Poisson's ratio greater than 0.5.

First published: 14 November (2023). pdf available via agreement between publisher and the university. doi link

See also Poisson main page. These lattices are also known to be Cosserat solids; see Cosserat page. The 3D lattices are made by 3D printing.

Remark: Some "odd" elastic materials have been described with internal moments. Such materials are not classically elastic. They could be magnetic materials or they could be generalized Cosserat solids.