  Elastic materials with holes and inclusions are important in a large variety
of contexts ranging from construction material to biological membranes. More
recently, they have also been exploited in mechanical metamaterials, where the
geometry of highly deformable structures is responsible for their unusual
properties, such as negative Poisson's ratio, mechanical cloaking, and tunable
phononic band gaps. Understanding how such structures deform in response to
applied external loads is thus crucial for designing novel mechanical
metamaterials. Here we present a method for predicting the linear response of
infinite 2D solid structures with circular holes and inclusions by employing
analogies with electrostatics. Just like an external electric field induces
polarization (dipoles, quadrupoles and other multipoles) of conductive and
dielectric objects, external stress induces elastic multipoles inside holes and
inclusions. Stresses generated by these induced elastic multipoles then lead to
interactions between holes and inclusions, which induce additional polarization
and thus additional deformation of holes and inclusions. We present a method
that expands the induced polarization in a series of elastic multipoles, which
systematically takes into account the interactions of inclusions and holes with
the external stress field and also between them. The results of our method show
good agreement with both linear finite element simulations and experiments.
