Supplementary text and figures from Functionally graded energy-absorbing metamaterials with negative stiffness curved beams: sequentially-adaptive modal multi-stability encoding

Fascinating properties like negative stiffness and multiple stable states can be achieved in lattice metamaterials with constituting elements as curved beams. We introduce spatially-varying and graded beam-level architectures for the most optimum material utilization in programming the stability behaviour and negative stiffness of bi-level architected lattice metamaterials. As a direct derivative, the specific energy absorption capability can be enhanced and modulated significantly based on novel modal insights concerning the beam-level constitutive relations. A minimum potential energy-based sequentially-adaptive mode-dependent analytical formulation is presented, showing a significant improvement in the prediction accuracy compared to available literature. In this context, we have developed a highly efficient matrix-based formulation for dealing with higher-order solutions, which are further validated by separate large-deformation finite element simulations and experimental investigations. Subsequently, we extend the elementary beam-level concepts towards developing programmable lightweight lattices with tailorable constitutive behaviour and enhanced specific energy absorption capability based on the bi-level design space of spatially graded beam profile and layer-wise varying unit cell architectures. The proposed optimum sustainable metamaterials with shape recoverability will lead to a range of critical engineering applications including energy-absorbing armours and protecting structures, stiffness modulation, vibration control, soft robotics, morphing, microelectromechanical devices, and various mechanical systems across the length scales.