Micromechanics of Composites

1st Edition

Multipole Expansion Approach

Authors: Volodymyr Kushch
Hardcover ISBN: 9780124076839
eBook ISBN: 9780124076600
Imprint: Butterworth-Heinemann
Published Date: 26th June 2013
Page Count: 512
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Micromechanics of Composites: Multipole Expansion Approach is the first book to introduce micromechanics researchers to a more efficient and accurate alternative to computational micromechanics, which requires heavy computational effort and the need to extract meaningful data from a multitude of numbers produced by finite element software code. In this book Dr. Kushch demonstrates the development of the multipole expansion method, including recent new results in the theory of special functions and rigorous convergence proof of the obtained series solutions. The complete analytical solutions and accurate numerical data contained in the book have been obtained in a unified manner for a number of the multiple inclusion models of finite, semi- and infinite heterogeneous solids. Contemporary topics of micromechanics covered in the book include composites with imperfect and partially debonded interface, nanocomposites, cracked solids, statistics of the local fields, and brittle strength of disordered composites.

Key Features

  • Contains detailed analytical and numerical analyses of a variety of micromechanical multiple inclusion models, providing clear insight into the physical nature of the problems under study
  • Provides researchers with a reliable theoretical framework for developing the micromechanical theories of a composite’s strength, brittle/fatigue damage development and other properties
  • Includes a large amount of highly accurate numerical data and plots for a variety of model problems, serving as a benchmark for testing the applicability of existing approximate models and accuracy of numerical solutions


A multidisciplinary audience consisting of researchers, professionals and graduate students in materials science, mechanics, engineering, applied mathematics, physics and related areas dealing with heterogeneous solids.

Table of Contents


Chapter 1. Introduction

1.1 Motivation for the Work

1.2 Geometry Models

1.3 Method of Solution

1.4 Homogenization Problem: Volume vs. Surface Averaging

1.5 Scope and Structure of the Book


Part I: Particulate Composites

Chapter 2. Potential Fields of Interacting Spherical Inclusions

2.1 Background Theory

2.2 General Solution for a Single Inclusion

2.3 Particle Coating vs. Imperfect Interface

2.4 Re-Expansion Formulas for the Solid Spherical Harmonics

2.5 Finite Cluster Model (FCM)

2.6 Composite Sphere

2.7 Half-Space FCM


Chapter 3. Periodic Multipoles: Application to Composites

3.1 Composite Layer

3.2 Periodic Composite as a Sandwich of Composite Layers

3.3 Representative Unit Cell Model

3.4 3P Scalar Solid Harmonics

3.5 Local Temperature Field

3.6 Effective Conductivity of Composite


Chapter 4. Elastic Solids with Spherical Inclusions

4.1 Vector Spherical Harmonics

4.2 Scalar and Vector Solid Spherical Biharmonics

4.3 Partial Solutions of Lame Equation

4.4 Single Inclusion in Unbounded Solid

4.5 Application to Nanocomposite: Gurtin & Murdoch Theory

4.6 Re-Expansion Formulas for the Vector Harmonics and Biharmonics

4.7 Finite Array of Inclusions (FCM)

4.8 Isotropic Solid with Anisotropic Inclusion

4.9 Effective Stiffness of Composite: Modified Maxwell Approach

4.10 Elastic Composite Sphere

4.11 RSV and Effective Elastic Moduli


Chapter 5. Elasticity of Composite Half-Space, Layer, and Bulk

5.1 Vector Harmonics and Biharmonics for Half-Space

5.2 Vector Lame Solutions for Half-Space

5.3 FCM for Elastic Half-Space

5.4 Doubly Periodic Models

5.5 Triply Periodic Vector Mu


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About the Author

Volodymyr Kushch

Dr. Volodymyr Kushch has thirty years of successful experience in Micromechanics of Composite Materials and Computational Mechanics, including – Advanced analytical and numerical methods in composite mechanics; – Particulate and fiber reinforced solid composites; – Suspensions and bubbly liquids; – Multiscale analysis of composite materials; – Computational solid mechanics and materials science; – Methods of mathematical physics and theory of special functions; – FEA simulation of technological processes; – Applied software development.

Affiliations and Expertise

Head of Laboratory, Institute for Superhard Materials, The National Academy of Sciences of Ukraine


"Kushch presents the multipole expansion method as an alternative to computational micromechanics for analyzing heterogeneous materials on the level of individual constituents. Being mostly analytical in nature, he says, it constitutes a theoretical basis for high-performance computational algorithms and has found applications in astronomy, physics, chemistry, engineering, statistics, and other fields."--Reference & Research Book News, October 2013