Porous mechanical metamaterials as aggregates of elastic charges

We present a new framework to describe the complex nonlinear response of two-dimensional porous mechanical metamaterials. We adopt a geometric approach to elasticity in which pores are represented by elastic charges, and show that this method captures with high level of accuracy both the shape deformation of individual pores as well as collective deformation patterns resulting from interactions between neighboring pores. The response of the metamaterials, both in the linear and nonlinear regime, is obtained by minimizing the nonlinear energy of the system written in terms of interacting elastic charges located at each hole. In particular, we show that quadrupoles and hexadecapoles - the two lowest order multipoles available in elasticity - are capable of matching the experimentally observed evolution in shape and relative orientation of the holes in a variety of periodic porous mechanical metamaterials. Our work demonstrates the ability of elastic charges to capture the physics of highly deformable porous solids and paves the way to the development of new theoretical frameworks for the description and rational design of porous mechanical metamaterials.