Programming nonlinear mechanical responses by topology optimization of limit-point instabilities

Abstract

Limit-point instabilities provide a powerful but underexplored mechanism for programming nonlinear mechanical responses. However, integrating such instabilities into topology optimization under large deformation remains challenging due to path-tracking and geometric regularity issues. Here, we develop an explicit design framework that programs prescribed force–displacement curves by deliberately exploiting limit-point instabilities over large deformation. We extend the Moving Morphable Void (MMV) method with finite-deformation equilibrium tracing and a curvature regularization that suppresses nonphysical sharp features while retaining design freedom. The framework synthesizes architectures that realize strain-hardening-type, yield-plateau-type, and strain-softening-type structural responses under compressive strains up to 20%, with typical curve-matching errors on the order of 1–2%. Mechanistically, the macroscopic response is governed not only by instability onset but by the spatial distribution, sequential activation, and coupling of buckling and snap-through modes embedded in the topology. We demonstrate overload protection, low-frequency vibration isolation, and instability-driven energy dissipation, and extract low-dimensional parametric templates that preserve the deformation mechanisms while enabling efficient retuning. The results establish controlled limit-point instability as a systematic design resource for programming constitutive-like nonlinear behavior at the structural level.