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Elastic materials with holes and inclusions are important in a large varietyof contexts ranging from construction material to biological membranes. Morerecently, they have also been exploited in mechanical metamaterials, where thegeometry of highly deformable structures is responsible for their unusualproperties, such as negative Poisson's ratio, mechanical cloaking, and tunablephononic band gaps. Understanding how such structures deform in response toapplied external loads is thus crucial for designing novel mechanicalmetamaterials. Here we present a method for predicting the linear response ofinfinite 2D solid structures with circular holes and inclusions by employinganalogies with electrostatics. Just like an external electric field inducespolarization (dipoles, quadrupoles and other multipoles) of conductive anddielectric objects, external stress induces elastic multipoles inside holes andinclusions. Stresses generated by these induced elastic multipoles then lead tointeractions between holes and inclusions, which induce additional polarizationand thus additional deformation of holes and inclusions. We present a methodthat expands the induced polarization in a series of elastic multipoles, whichsystematically takes into account the interactions of inclusions and holes withthe external stress field and also between them. The results of our method showgood agreement with both linear finite element simulations and experiments.

Elastic multipole method for describing linear deformation of infinite 2D solid structures with circular holes and inclusions