Continuum micro-dilatation modeling of auxetic metamaterials

Abstract

Metamaterials with re-entrant type lattice structure with cubic symmetry are studied. The main new result is that the continuum theory of porous media generalized to the case of media with cubic symmetry can adequately describe the effective properties of a wide class of lattice structures as continuum media. Identification of the micro-dilatational elastic constants for the cellular metamaterials with negative Poisson's ratio is carried out. Methods for identifying the parameters of the micro-dilatation theory of media through the structural and rigidity parameters of lattice structures are developed. It is shown that material constants of the micro-dilatation theory can be directly linked with the geometrical and elastic parameters of the unit cell of the metamaterial. In particular, “void stiffness” modulus and voids diffusion parameter depend linearly on the stiffness of re-entrant structures in the unit cells and coupling modulus depends significantly only on the geometry of the unit cell. This result is obtained using micro-dilatational analytical solution of the homogeneous elasticity problem and FE modeling of the uniaxial tension, hydrostatic compression and simple shear of the beam lattice metamaterial models. The problem of non-homogeneous uniaxial deformation of the rod and the problem of the pure bending of the beam were used to determine the voids diffusion parameter. It is shown, in particular, that the coupled and scale effects are equivalent for the studied continuum and lattice models of the metamaterials in the considered range of parameters.