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 () and one negative Poisson’s ratios (). 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.