Micromechanics of Composites - 1st Edition - ISBN: 9780124076839, 9780124076600

Micromechanics of Composites

1st Edition

Multipole Expansion Approach

Authors: Volodymyr Kushch
eBook ISBN: 9780124076600
Hardcover ISBN: 9780124076839
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 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





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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