Inverse-designed Disordered Nonlinear Metamaterial Chains for Robust Impact Mitigation

  • Tamur, Caglar (University of California, San Diego)
  • Brandy, Maya (University of California, San Diego)
  • Boechler, Nicholas (University of California, San Diego)
  • Kim, H Alicia (University of California, San Diego)

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Impact mitigating metamaterials can exploit elastic nonlinearity to redirect and dissipate energy, often relying on bistable energy locking mechanisms. Yet in the wave-dominated regimes their performance can be fragile where the small variations in loading or constitutive parameters may strongly degrade attenuation [1,2]. We investigate whether robustness can be improved by introducing engineered spatial disorder in the nonlinear part of the elastic response. We study a one-dimensional idealization of a metamaterial as a chain of identical masses connected by uniform linear dashpots, in parallel with nonlinear springs that model the effective elastic response of each unit cell. The linear stiffness is fixed and spatially uniform, while the nonlinear coefficients are allowed to vary along the chain through a number of spring families, yielding realizable heterogeneous designs. For a prescribed point-mass impact, we solve a constrained optimization problem to minimize the peak kinetic energy transmitted to a protected boundary with nonlinear coefficients chosen as the design variables. Following the bilevel inverse design approach of [2], the unit cell level geometries to achieve the target elastic response are realized via geometrically nonlinear level-set topology optimization. Allowing spatial heterogeneity in the nonlinear coefficient yields three key outcomes. First, optimized heterogeneous chains achieve equal or lower transmitted peak kinetic energy than the homogeneous designs, while doing so without requiring bistability. Second, these high-performing heterogeneous chains are substantially more robust to perturbations in nonlinear constitutive parameters and input energy than their homogeneous counterparts. Third, when bistability is explicitly excluded via admissibility constraints, the resulting heterogeneous designs remain high-performing and exhibit improved robustness relative to designs that allow bistability. These findings suggest a practical design principle for impact mitigating metamaterials: rather than enforcing perfect periodicity and relying on sharp bistable transitions, intentionally distributing nonlinear elastic response can deliver high attenuation with improved tolerance to manufacturing variability and uncertainty. [1] Fancher, R., et al. Extreme Mech. Lett. (2023) [2] MacNider, B. et al. Nat. Commun, (2025)