Coupled Mechanical–Electrical modeling of pantographic metamaterials with simplified flexoelectric bending

Pantographic metamaterials display distinctive mechanical behaviors arising from their mesostructure, which consists of two orthogonal families of fibers interconnected by hinges. Owing to this architecture, such lattices are naturally described within the framework of second-gradient continuum mechanics, since the bending of fibers contributes to the strain energy through the second gradient of the placement field. In this context, flexoelectricity, namely, the generation of electric polarization induced by strain gradients, represents a natural multiphysics coupling for this class of metamaterials. Numerical simulations are presented to illustrate the resulting electric field distribution in pantographic specimens subjected to bias-extension, compression, and shear loading conditions.