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

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 <= nu(yx) , nu(yz) <= 1) and one negative Poisson's ratios (-1 >= nu(xz) >= 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.