Reconfigurable structures are of great technological importance, e.g.,, in soft robotics, deployable systems, shape-morphing technologies, and programmable metamaterials. These architectures are based on structural units whose tessellation induces multi-stability, enabling the system to transition between equilibrium states. As these systems grow in size and complexity, modeling them as discrete structures becomes increasingly impractical and computationally demanding, especially when it comes to design exploration and optimization.
Here, an “effective” macro-hyperelastic description of periodic bistable auxetic surfaces is developed. Focus is placed on surfaces made of bistable hexagonal cells composed of six triangular units with two stable equilibrium states: a closed (undeformed) configuration and an open (volumetrically strained) configuration. Such an auxetic bistable tiling has been proposed in Rafsanjani and Pasini, Extr. Mech. Let. [2016] and Chen et al., ACM Trans. Graph. [2022]. The hyperelastic free energy function is based on invariants of the logarithmic strain, which allows for the Kirchhoff stress tensor to be expressed as the sum of response terms (one of them being the pressure) that are mutually orthogonal and, in turn, facilitates its calibration against numerical simulations of the deformation response of the unit cell under periodic boundary conditions. The model is regularized by a non-local description of the pressure in order to avoid the pathological mesh dependence and ill-posed mathematical description, which may result from the double-well nature of the free-energy potential. Membrane and shell structural elements are developed in Abaqus-suite and numerical simulations depict the robustness of the model and verify its numerical implementation.
