Shear Deformation in Auxetic Solids

Abstract

This chapter establishes the effect of auxeticity on shear deformation in laterally-loaded thick beams, laterally-loaded thick circular, polygonal and rectangular plates, buckling of thick columns and plates , and vibration of thick plates. Results show that shear deformation reduces as the Poisson’s ratio becomes more negative, thereby implying that geometrically thick beams and plates are mechanically thin beams and plates, respectively, if the Poisson’s ratio is sufficiently negative. In other words, results of deflections in Timoshenko beam and Mindlin plate approximate those by Euler-Bernoulli beam and Kirchhoff plate, respectively, as the Poisson’s ratio approaches −1. In the study of buckling of isotropic columns, it was found that auxeticity increases the buckling load such that the buckling loads of Timoshenko columns approximate those of Euler-Bernoulli columns as $v \to - 1$. In the case of vibration of thick isotropic plates, it was shown that as a plate’s Poisson’s ratio becomes more negative, the Mindlin-to-Kirchhoff natural frequency ratio increases with diminishing rate. Furthermore, simplifying assumptions such as constant shear correction factor and exclusion of rotary inertia is valid for plates with positive Poisson’s ratio, and that the assumptions of constant shear correction factor and no rotary inertia for auxetic plates give overestimated natural frequency.