A Continuum Macro-Model for Bistable Periodic Auxetic Surfaces

Abstract

A macro-constitutive model for the deformation response of periodic rotating bistable auxetic surfaces is developed. Focus is placed on isotropic surfaces made of bistable hexagonal cells composed of six triangular units with two stable equilibrium states. Adopting a variational formulation, the effective stress-strain response is derived from a free energy function expressed in terms of the invariants of the logarithmic strain. To address the mathematical ill-posedness and numerical artifacts--such as mesh sensitivity--arising from the double-well nature of the free energy, two regularization approaches are introduced: (i) a gradient-enhanced first invariant of the logarithmic strain, and (ii) an artificial material rate dependency. Although neither regularization guarantees solution uniqueness, the former mitigates mesh sensitivity, while the latter improves the convergence behavior of the nonlinear numerical scheme by promoting smooth temporal evolution of transition localization and enabling the system to overcome snap-backs induced by local non-proportional loading near transition fronts. The model is implemented using membrane/shell structural elements and plane stress continuum ones within the ABAQUS finite element suite. Numerical simulations demonstrate the efficacy of the proposed formulation and its implementation.