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

: Stilz, Maximilian; Plappert, David; Gutmann, Florian; Hiermaier, Stefan


Mathematics and mechanics of solids: MMS (2021), Online First, 14 S.
ISSN: 1081-2865 (Print)
ISSN: 1741-3028 (Online)
Fraunhofer EMI ()
auxetic; homogenization; metamaterial; pantographic structure

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≤νyx,νyz≤1) and one negative Poisson’s ratios (-1≥νxz≥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.