Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Analytic asymptotic formulas for effective parameters of planar elastic composites

Daehee Cho, Doosung Choi, Mikyoung Lim

Abstract

We investigate the effective elastic properties of periodic dilute two-phase composites consisting of an homogeneous isotropic matrix and a periodic array of rigid inclusions. We assume the rigid inclusion in a unit cell is a simply connected, bounded domain so that there exists an exterior conformal mapping corresponding the inclusion. Recently, an analytical series solution method for the elastic problem with a rigid inclusion was developed based on the layer potential technique and the geometric function theory \cite{Mattei:2021:EAS}. In this paper, by using the series solution method, we derive expression formulas for the elastic moment tensors--the coefficients of the multipole expansion associated with an elastic inclusion--of an inclusion of arbitrary shape. These formulas for the elastic moment tensors lead us to analytic asymptotic formulas for the effective parameters of the periodic elastic composites with rigid inclusions in terms of the associated exterior conformal mapping.

Analytic asymptotic formulas for effective parameters of planar elastic composites

Abstract

We investigate the effective elastic properties of periodic dilute two-phase composites consisting of an homogeneous isotropic matrix and a periodic array of rigid inclusions. We assume the rigid inclusion in a unit cell is a simply connected, bounded domain so that there exists an exterior conformal mapping corresponding the inclusion. Recently, an analytical series solution method for the elastic problem with a rigid inclusion was developed based on the layer potential technique and the geometric function theory \cite{Mattei:2021:EAS}. In this paper, by using the series solution method, we derive expression formulas for the elastic moment tensors--the coefficients of the multipole expansion associated with an elastic inclusion--of an inclusion of arbitrary shape. These formulas for the elastic moment tensors lead us to analytic asymptotic formulas for the effective parameters of the periodic elastic composites with rigid inclusions in terms of the associated exterior conformal mapping.
Paper Structure (10 sections, 5 theorems, 65 equations)

This paper contains 10 sections, 5 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Elastic moment tensors
  4. Conformal mapping and Faber polynomial
  5. Explicit series expansion for the single-layer potential in complex form
  6. Asymptotic formula for elastic moment tensors
  7. EMTs with general conformal mapping
  8. EMTs with rigid inclusion
  9. Solvability of the series solution method in Mattei:2021:EAS for the uniform loading
  10. Inclusions with conformal mappings having upto negative third order

Key Result

Theorem 1.1

Let $D$ be a bounded, simply connected, and planar Lipschitz domain. We use def:Psi for the exterior conformal mapping associated with $D$. Then, the elastic moment tensors of $D$, $(m^{ij}_{kl})$, have the form where $\mathbb{E}^{(1,i)}$ and $\mathbb{E}^{(2,i)}$ are defined by

Theorems & Definitions (11)

  • Theorem 1.1
  • Definition 1
  • Theorem 2.1: Mattei:2021:EAS
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Example 1
  • Example 2
  • ...and 1 more