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A matrix formulation of the plane elastostatic inclusion problem via geometric function theory

Daehee Cho, Doosung Choi, Mikyoung Lim

TL;DR

This work tackles the planar elastostatic inclusion problem in an unbounded 2D medium with a general inclusion possessing arbitrary Lamé constants. It advances the state of the art by developing a matrix formulation that leverages an exterior conformal mapping and Faber polynomials within a geometric-function-theory framework, extending prior scalar and rigid-inclusion results to the full elasticity system. A key technical breakthrough is a cancellation mechanism that eliminates terms involving $1/\Psi'(w)$, enabling a finite 8×8 block-structured linear system that determines the density coefficients for the layer-potential representation. The resulting geometric-series solution framework incorporates the inclusion geometry directly into the matrix, paving the way for inverse problems, neutral inclusions, and effective-property analyses in composite and periodic structures.

Abstract

We investigate the two-dimensional elastostatic inclusion problem in an unbounded medium. Building on the recent developments for rigid inclusions \cite{Mattei:2021:EAS} and conductivity inclusions \cite{Jung:2021:SEL}, we extend these methodologies to the more general case of elastic inclusions with arbitrary Lamé constants. Our approach integrates layer potential techniques, geometric function theory, and the complex-variable formulation in plane elasticity. As a main result, we derive a matrix formulation of the elastostatic inclusion problem using basis functions defined via the exterior conformal mapping of the inclusion. This leads to a series solution framework that incorporates the geometry of the inclusion.

A matrix formulation of the plane elastostatic inclusion problem via geometric function theory

TL;DR

This work tackles the planar elastostatic inclusion problem in an unbounded 2D medium with a general inclusion possessing arbitrary Lamé constants. It advances the state of the art by developing a matrix formulation that leverages an exterior conformal mapping and Faber polynomials within a geometric-function-theory framework, extending prior scalar and rigid-inclusion results to the full elasticity system. A key technical breakthrough is a cancellation mechanism that eliminates terms involving , enabling a finite 8×8 block-structured linear system that determines the density coefficients for the layer-potential representation. The resulting geometric-series solution framework incorporates the inclusion geometry directly into the matrix, paving the way for inverse problems, neutral inclusions, and effective-property analyses in composite and periodic structures.

Abstract

We investigate the two-dimensional elastostatic inclusion problem in an unbounded medium. Building on the recent developments for rigid inclusions \cite{Mattei:2021:EAS} and conductivity inclusions \cite{Jung:2021:SEL}, we extend these methodologies to the more general case of elastic inclusions with arbitrary Lamé constants. Our approach integrates layer potential techniques, geometric function theory, and the complex-variable formulation in plane elasticity. As a main result, we derive a matrix formulation of the elastostatic inclusion problem using basis functions defined via the exterior conformal mapping of the inclusion. This leads to a series solution framework that incorporates the geometry of the inclusion.
Paper Structure (25 sections, 13 theorems, 134 equations)

This paper contains 25 sections, 13 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Layer potential formulation
  4. Complex-variable formulation
  5. Holomorphic function expressions for the single-layer potentials
  6. Exterior conformal mapping, Faber polynomials and their associated matrices
  7. Exterior conformal mapping
  8. Faber polynomials
  9. Matrices associated with the conformal mapping and Faber polynomials
  10. Matrices associated with $\Psi$.
  11. Grunsky coefficient matrix $C$.
  12. Expansion coefficient matrix $P$ for $F_m$.
  13. Expansion coefficient matrix $\widetilde{D}$ for $F'_m$.
  14. Series expansions for the background loading $H$
  15. Geometric series expansion for the single-layer potential
  16. ...and 10 more sections

Key Result

Lemma 2.1

Let $\mathbf{u}$ be the solution to (eqn:main:trans) and $u$ be its corresponding complex function as in complex:expression. Set $u_e:=u|_{\mathbb{C}\setminus \overline{\Omega}}$ and $u_i:=u|_{\Omega}$. Then there are functions $f_e$ and $g_e$ holomorphic in $\mathbb{C}\setminus\overline{\Omega}$ an where Moreover, for some constant $c$ and $z\in\partial\Omega$,

Theorems & Definitions (24)

  • Lemma 2.1: Ammari:2007:PMTMuskhelishvili:1953:SBP
  • Definition 1
  • Lemma 2.2: Ammari:2007:PMTAndo:2018:SPN
  • Lemma 2.3
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Lemma 5.1
  • proof
  • ...and 14 more