Shear deformation in beams with negative Poisson’s ratio

Abstract

This paper evaluates the effect of auxeticity on the extent of shear deformation in isotropic beams. It is herein shown that the ratio of maximum deflection by Timoshenko theory to the maximum deflection by the Euler–Bernoulli theory reduces as the Poisson’s ratio of the beam material becomes more negative. The obtained results suggest that the ratio of shear deformation to bending deformation is lower in auxetic beams than in conventional beams, and that the use of Euler–Bernoulli beam theory for quantifying the deflection in auxetic beams gives lesser deviation from the Timoshenko beam theory than in conventional beams. Specifically, the shear deformation becomes insignificant for square beams with Poisson’s ratio of –1. A set of design equations that allows the use of simpler Euler–Bernoulli beam theory instead of the Timoshenko beam theory for the considered cases are proposed herein to justify the use of the former theory on beam analysis on the basis of reduced shear deformation as the beam’s Poisson’s ratio becomes more negative. The findings obtained herein also suggest that in comparison with conventional beams (of equal Young’s modulus, load distributions, and boundary conditions) the lower shear deformation in auxetic beams translates into smaller transverse deflections and hence better dimensional stability.