Micromechanics of Composites

Micromechanics of Composites

Multipole Expansion Approach

1st Edition - May 13, 2013

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  • Author: Volodymyr Kushch
  • Hardcover ISBN: 9780124076839
  • eBook ISBN: 9780124076600

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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

  • Preface

    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 Multipoles

    5.6 RUC Model of Elastic Spherical Particle Composite

    5.7 Numerical Study


    Chapter 6. Conductivity of a Solid with Spheroidal Inclusions

    6.1 Scalar Spheroidal Solid Harmonics

    6.2 Single Inclusion: Conductivity Problem

    6.3 Re-Expansion Formulas for Spheroidal Solid Harmonics

    6.4 Finite Cluster Model of Spheroidal Particle Composite

    6.5 Double Fourier Integral Transform of Spheroidal Harmonics

    6.6 Doubly Periodic Harmonics

    6.7 Triply Periodic Harmonics

    6.8 Heat Conduction in Periodic Composite

    6.9 Numerical Examples


    Chapter 7. Elastic Solid with Spheroidal Inclusions

    7.1 Background Theory

    7.2 Single-Inclusion Problem

    7.3 Re-Expansion Formulas for the Spheroidal Lame Solutions

    7.4 Finite Cluster Model of Composite with Spheroidal Inclusions

    7.5 Half-Space Problem

    7.6 RUC Model of Elastic Spheroidal Particle Composite


    Chapter 8. Composites with Transversely Isotropic Constituents

    8.1 Transversely Isotropic Conductivity

    8.2 Transversely Isotropic Elastic Solid with Spherical Inclusions

    8.3 RUC Model

    8.4 Numerical Examples


    Part II: Fibrous Composites: Two-Dimensional Models

    Chapter 9. Circular Fiber Composite with Perfect Interfaces

    9.1 In-Plane Conductivity and Out-of-Plane Shear: The Governing Equations

    9.2 Finite Array of Circular Inclusions

    9.3 Half Plane with Circular Inclusions

    9.4 Infinite Arrays of Circular Inclusions

    9.5 Representative Unit Cell Model

    9.6 Finite Array of Circular Inclusions: In-Plane Elasticity Problem

    9.7 Circular Inclusions in Half-Plane

    9.8 RUC Model of Fibrous Composite: Elasticity

    9.9 Statistics of Microstructure, Peak Stress and Interface Damage in Fibrous Composite


    Chapter 10. Fibrous Composite with Interface Cracks

    10.1 General Solution for a Single, Partially Debonded Inclusion

    10.2 Finite Array of Partially Debonded Inclusions

    10.3 Conductivity of Fibrous Composite with Interface Damage

    10.4 In-Plane Elasticity: General Form of the Displacement Solution

    10.5 Displacement Solution for the Partially Debonded Inclusion

    10.6 A Finite Number of Interacting Inclusions withInterface Cracks

    10.7 RUC Model of Fibrous Composite with Interface Cracks


    Chapter 11. Solids with Elliptic Inclusions

    11.1 Single Elliptic Inclusion in an Inhomogeneous Far Field

    11.2 Re-Expansion Formulas for the Elliptic Solid Harmonics

    11.3 Finite Array of Inclusions

    11.4 Half-Space Containing a Finite Array of Elliptic Fibers

    11.5 Periodic Complex Potentials

    11.6 Micromechanical Model of Cracked Solid

    11.7 Numerical Examples


    Chapter 12. Fibrous Composite with Anisotropic Constituents

    12.1 Out-of-Plane Shear

    12.2 Periodic Complex Potentials

    12.3 Plane Strain

    12.4 Effective Stiffness Tensor


    Appendix A. Sample Fortran Codes

    A.1 FCM Conductivity Problem (Chapter 2)

    A.2 RUC Conductivity Problem for Spherical Particle Composite (Chapter 3)

    A.3 FCM Elasticity Problem (Chapter 4)

    A.4 RUC Elasticity Problem for Spherical Particle Composite (Chapter 5)

    A.5 RUC Conductivity and Elasticity Problems for Fibrous Composite (Chapter 9)

    A.6 Standard Lattice Sums




Product details

  • No. of pages: 512
  • Language: English
  • Copyright: © Butterworth-Heinemann 2013
  • Published: May 13, 2013
  • Imprint: Butterworth-Heinemann
  • Hardcover ISBN: 9780124076839
  • eBook ISBN: 9780124076600

About the Author

Volodymyr Kushch

Prof. Volodymyr Kushch is Leading Research Scientist, Institute for Superhard Materials of the National Academy of Sciences of Ukraine. He is the author of Micromechanics of Composites as well as dozens of peer-reviewed papers. He has over thirty years of research and industry experience in micromechanics of composites and computational mechanics, with research interests that include advanced analytical and numerical methods in mechanics of heterogeneous solids, suspensions and bubbly liquids, computational materials science, mathematical physics, and more.

Affiliations and Expertise

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

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