# Boundary integral equations for isotropic linear elasticity

## TL;DR

This paper develops boundary integral equations for 3D isotropic linear elasticity in biphasic models, analyzes spectra for spherical inclusions, and proposes a Galerkin discretization for multiple inclusions with numerical validation.

## Contribution

It introduces a new integral equation approach and Galerkin discretization for multiple spherical inclusions with different Lamé parameters in isotropic elasticity.

## Key findings

- Eigenfunctions of boundary operators for spherical inclusions identified
- Spectra of boundary operators explicitly computed for spherical cases
- Numerical tests demonstrate effectiveness of the proposed discretization

## Abstract

This articles first investigates boundary integral operators for the three-dimensional isotropic linear elasticity of a biphasic model with piecewise constant Lam\'e coefficients in the form of a bounded domain of arbitrary shape surrounded by a background material. In the simple case of a spherical inclusion, the vector spherical harmonics consist of eigenfunctions of the single and double layer boundary operators and we provide their spectra. Further, in the case of many spherical inclusions with isotropic materials, each with its own set of Lam\'e parameters, we propose an integral equation and a subsequent Galerkin discretization using the vector spherical harmonics and apply the discretization to several numerical test cases.