Rotation and dilation deformation mechanisms for auxetic behaviour in the ?-cristobalite tetrahedral framework structure

Abstract

Analytical expressions are derived for the Poisson's ratios associated with a three-dimensional network of regular, corner-sharing tetrahedra in which: (1) the tetrahedra are assumed to be rigid and free to rotate relative to each other; (2) the tetrahedra are assumed to maintain shape and orientation but are free to change size (dilate); (3) tetrahedral rotation and dilation are assumed to act concurrently. The structure has a primitive unit cell containing four tetrahedra and is analogous to the molecular structure of α-cristobalite. Strain-dependent variations in Poisson's ratio are also predicted by the models. For deformation due to tetrahedral rotation the network is found to exhibit negative Poisson's ratios in each of the three principal directions, with the magnitude of the Poisson's ratio being dependent on the angle of rotation of the tetrahedra. The behaviour of the Poisson's ratio is isotropic in the transverse plane, but anisotropic elsewhere. In the dilation model negative Poisson's ratios equal to −1 are observed for uniaxial loading in any of the principal directions, with the value being constant irrespective of tetrahedral size. The model for concurrent tetrahedral rotation and dilation allows positive as well as negative Poisson's ratios, with the values determined by the framework geometry and relative strengths of the two mechanisms. The concurrent model also offers a design route to materials and structures having ultrahigh Young's moduli.