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

TL;DR

This paper introduces an elastic multipole method to predict the linear deformation response of infinite 2D solid structures with holes and inclusions, leveraging electrostatic analogies to account for interactions and deformations.

Contribution

The paper presents a novel elastic multipole expansion technique that systematically models interactions in 2D structures with holes and inclusions, improving prediction accuracy.

Findings

Method agrees well with finite element simulations.

Method accurately predicts experimental results.

Captures complex interactions between inclusions and external stress.

Abstract

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.