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

The analogies between electrostatics and elasticity help us to solve complicated boundary value problems in linear elasticity of 2D solid structures. For example, a hole inside an elastic body is analogous to a perfect conductor in electrostatics in the sense that at the circumference of them the fields, namely the traction force and the parallel component of the electric field respectively, have to be zero. Similarly, an inclusion in an elastic material is equivalent to a dielectric since in both cases the fields have to be continuous at the edge of the inhomogeneity. Furthermore, just like external electric field induces polarization (dipoles, quadrupoles, etc.) of conductive objects, external stress induces elastic multipoles inside holes. Based on this idea, we present a method for obtaining deformations of 2D solid structures with circular holes and inclusions under applied load. We validated the method by comparing the results with table-top experiments. We also utilized the method of image charges from electrostatics to capture the effect of boundaries on the deformed shapes of the holes/inclusions. The excellent agreement with the experimental results shows the power of this intuitive method.