Macroscale Simulation of Post-Critical Behavior in Flexible Metamaterials

Numerical simulation of periodic metamaterials that exploit unit-cell instabilities is challenging when approached through fully resolved finite element (FE) simulations or brute-force reduced-order models that explicitly represent every unit-cell member. Reduced-order effective continuum descriptions offer a promising route to lower computational cost while retaining modeling rigor. However, the resulting macroscale boundary-value problem becomes ill-posed due to loss of ellipticity under quasi-static conditions, resulting in nonunique solutions and numerical artifacts such as mesh sensitivity. This work shows that, when the post-critical material instability underlying the effective constitutive model governs the macroscale response–;i.e., when the applied loading activates the same instability pattern on which the constitutive model was calibrated–;two regularization strategies enable robust macroscale FE simulation of instability-driven metamaterials: artificial material rate dependence, which smooths the temporal evolution of instabilities, and, where scale separation is assumed, gradient enhancement, which introduces a material length scale and reduces mesh dependence.