  We present an analytical 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. Specifically, 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.
