A generalization of director tensors for odd matter in classical continuum mechanics

Abstract A general tensorial framework for odd and hyperelastic contributions in linear elasticity is developed within classical continuum mechanics. The formulation is based on complementary decompositions of the second-order identity tensor and an off-diagonal selector tensor, together with their associated director tensors. By forming suitable dyadic combinations of these decompositions, the fourth-order elastic tensor can be represented compactly while preserving the required minor symmetries. The resulting construction provides a systematic classification of normal-normal, shear-shear, and shear-normal couplings, and accommodates both reciprocal and non-reciprocal constitutive responses in anisotropic media. Within this framework, classical Beltrami stress-potential representations are revisited, while the Airy, Maxwell, and Morea specializations are summarized. Special attention is given to the Airy stress function in planar elasticity, where the class of odd constitutive responses compatible with the classical biharmonic structure is identified. This analysis is then applied to the problem of a pressurized hole in an infinite medium. It is shown that, for the class of odd responses compatible with the Airy formulation under classical continuum assumptions, the stress field coincides with the classical solution, while odd effects appear at the level of the displacement field. The results clarify which odd elastic constitutive responses can be represented consistently within classical continuum mechanics and provide a structured basis for future studies of anisotropic and non-reciprocal elastic media.