Fig. 5: Designing unit cells that leverage geometric nonlinearity to achieve DEM-identified desired effective constitutive laws, and experimental realization of the spring and chain.

a Example mechanisms which can be used to achieve various nonlinearities with geometry alone are shown, including the following nonlinearities: i) stiffening, ii) softening, iii) soft to stiff, iv) snap-through (but not bistable), and v) fully bistable. The plot on the right shows the nonlinear responses of these five example mechanisms (with the stiff to soft, snap-through, and bistable curves linearly scaled to be on the same scale as the thicker mechanisms). b–e Optimized spring design and chain realization for the case of minimizing peak transmitted kinetic energy aimed to match the nonlinear coefficients identified in Fig. 4a. b Results of the shape optimization, with the initial condition and optimized design shown. c The fabricated polycarbonate unit cell, consisting of four optimized springs (as seen in (b)) and a rigid frame. d The quasi-static test of the unit cell shown in (c), compared against the target the behavior simulated via COMSOL FEM. e The full chain of 20 unit cells, hung from a frame. The chain is clamped to the left of the leftmost unit cell, imposing a zero displacement boundary condition. The impact occurs at the right end of the chain. Source data are provided as a Source Data file.