A 3D extension of pantographic geometries to obtain metamaterial with semi-auxetic properties

Abstract

In this work we present a three-dimensional extension of pantographic structures and describe its properties after homogenization of the unit cell. Here we rely on a description involving only the first gradient of displacement, as the semi-auxetic property is effectively described by first-order stiffness terms. For a homogenization technique, discrete asymptotic expansion is used. The material shows two positive ( $0\le {\nu }_{\mathrm{yx}},{\nu }_{\mathrm{yz}}\le 1$ ) and one negative Poisson’s ratios ( $-1\ge {\nu }_{\mathrm{xz}}\ge 0$ ). If, on the other hand, we assume inextensible Bernoulli beams and perfect pivots, we find a vanishing stiffness matrix, suggesting a purely higher gradient material.