A Continuum Macro-Model for Bistable Periodic Auxetic Surfaces

TL;DR

This work develops a variational macro-model for isotropic periodic rotating bistable auxetic surfaces, expressing the free energy in terms of invariants of the logarithmic strain to derive an effective in-plane constitutive law. To cope with the double-well energy and associated ill-posedness, it introduces gradient-enhanced regularization on the first invariant and an artificial viscosity, enabling robust finite element simulations in ABAQUS. Calibrated against unit-cell simulations, the model captures transition-front localization, front propagation, and energy-release characteristics across tension and expansion scenarios, with qualitative agreement to experiments. Limitations include isotropy, absence of rate- and temperature-dependent effects, and the restriction to displacement-controlled loading; the framework nonetheless offers a practical tool for design and optimization of programmable bistable metamaterials.

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.

Figures (11)

• Figure 1: Schematic of the strip geometry with the finite element mesh.
• Figure 2:
• Figure 3: a) Axial, volume-averaged, normalized stress vs strain responses for the strip tension test for mesh densities of $L_e=l/3$, $L_e=l/5$, and $L_e=l/7$ for the gradient-enhanced and rate-dependent model; b) & c) band transition zone for the different mesh densities.
• Figure 4: Uniaxial stress vs strain response for the rate-independent model and the rate-dependent one with the maximum viscosity parameter, $\eta=1.5$ MPa . s, used in the numerical simulations.
• Figure 5: Axial, volume-averaged, normalized stress vs strain responses for the strip tension test for mesh densities of $L_e=l/3$, $L_e=l/5$, and $L_e=l/7$ using the local (non–gradient-enhanced), rate-dependent model.