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**Random heterogeneous materials. Microstructure and macroscopic properties.**
*(English)*
Zbl 0988.74001

Interdisciplinary Applied Mathematics. 16. New York, NY: Springer. xxi, 701 p. DM 171.09; sFr 147.66; £59.00; $ 69.95 (2002).

The book consists of three parts and 23 chapters. The aim of the book is to give broad information about random heterogeneous materials, particularly based on microstructural and macrostructural properties of the constituent materials. The critiques of each chapter are briefly given below.

The first part consists of a single chapter and presents a general overview and definitions of some concepts like heterogeneity, effective properties and others. The second part of the book consists of eleven chapters. In chapter 2, the author discusses in detail the characterization of microstructure of the material. To describe a wide class of heterogeneity and randomness that the materials have, the author explores here various microstructural descriptors, i.e. microstructural correlation functions and related theorems. The statistical mechanics of many particle systems is discussed in chapter 3. Various concepts and techniques of statistical mechanics that will be used in the rest of the book are cultivated in this chapters. Chapter 4 is confined to the characterization of microstructure of heterogeneous materials consisting of a matrix and \(d\)-dimensional spherical inclusions. For that purpose, the author introduces \(n\)-point correlation function \(H_n\) and expresses it in terms of probability density function. By using these functions, various expressions for material properties are derived in this chapter. The discussion of material properties of composite materials consisting of the same spheres is given in chapter 5. In chapter 6, the author studies the properties of materials containing different sizes of spheres. Effective anisotropic properties of homogeneous but anisotropic media, like laminated or fiber-reinforced composites, are discussed in chapter 7. Chapter 8 is devoted to cell and random-field models by determining the \(n\)-point probability functions. The percolation and clustering of various material structures are discussed in chapter 9. In chapter 10, the author presents the derivation and discussion of some basic results of continuum percolation theory. Chapter 11 examines the influence of volume fraction fluctuations on effective moduli and other properties of materials. Computer simulations, image analysis and reconstructions of heterogeneous materials are presented in chapter 12.

Part III “Microstructure/property connection” consists of eleven chapters. In chapter 13, by homogenizing the heterogeneous material through averaging, the localized governing equations are obtained. Here the author also examines various effective moduli and symmetries of the material. The variational formulations for random media of arbitrary structure are introduced in chapter 14 to obtain approximate solutions. In chapter 15, the author derives expressions that link the effective properties of a two-phase heterogeneous material to the effective properties of some microstructures with interchanged phases. These expressions are given as phase interchanging relations, particularly for elastic and conductive properties. The exact results on effective moduli of some typical composite materials are given in chapter 16. In chapter 17, the single inclusion solutions are introduced and, by utilizing these solutions, effective medium approximations for various composites are given in chapter 18. The cluster expansion method for obtaining effective moduli is given in chapter 19, chapter 20 discusses exact contrast expansions for effective moduli of heterogeneous materials, and finally the effective moduli for rigorous bounding of species are derived in chapters 21, 22 and 23.

The book is well written and might be very useful to researchers working in the area of composite materials and porous media. I, therefore, strongly recommend to personal users and to the institute libraries to keep the book on their shelves.

The first part consists of a single chapter and presents a general overview and definitions of some concepts like heterogeneity, effective properties and others. The second part of the book consists of eleven chapters. In chapter 2, the author discusses in detail the characterization of microstructure of the material. To describe a wide class of heterogeneity and randomness that the materials have, the author explores here various microstructural descriptors, i.e. microstructural correlation functions and related theorems. The statistical mechanics of many particle systems is discussed in chapter 3. Various concepts and techniques of statistical mechanics that will be used in the rest of the book are cultivated in this chapters. Chapter 4 is confined to the characterization of microstructure of heterogeneous materials consisting of a matrix and \(d\)-dimensional spherical inclusions. For that purpose, the author introduces \(n\)-point correlation function \(H_n\) and expresses it in terms of probability density function. By using these functions, various expressions for material properties are derived in this chapter. The discussion of material properties of composite materials consisting of the same spheres is given in chapter 5. In chapter 6, the author studies the properties of materials containing different sizes of spheres. Effective anisotropic properties of homogeneous but anisotropic media, like laminated or fiber-reinforced composites, are discussed in chapter 7. Chapter 8 is devoted to cell and random-field models by determining the \(n\)-point probability functions. The percolation and clustering of various material structures are discussed in chapter 9. In chapter 10, the author presents the derivation and discussion of some basic results of continuum percolation theory. Chapter 11 examines the influence of volume fraction fluctuations on effective moduli and other properties of materials. Computer simulations, image analysis and reconstructions of heterogeneous materials are presented in chapter 12.

Part III “Microstructure/property connection” consists of eleven chapters. In chapter 13, by homogenizing the heterogeneous material through averaging, the localized governing equations are obtained. Here the author also examines various effective moduli and symmetries of the material. The variational formulations for random media of arbitrary structure are introduced in chapter 14 to obtain approximate solutions. In chapter 15, the author derives expressions that link the effective properties of a two-phase heterogeneous material to the effective properties of some microstructures with interchanged phases. These expressions are given as phase interchanging relations, particularly for elastic and conductive properties. The exact results on effective moduli of some typical composite materials are given in chapter 16. In chapter 17, the single inclusion solutions are introduced and, by utilizing these solutions, effective medium approximations for various composites are given in chapter 18. The cluster expansion method for obtaining effective moduli is given in chapter 19, chapter 20 discusses exact contrast expansions for effective moduli of heterogeneous materials, and finally the effective moduli for rigorous bounding of species are derived in chapters 21, 22 and 23.

The book is well written and might be very useful to researchers working in the area of composite materials and porous media. I, therefore, strongly recommend to personal users and to the institute libraries to keep the book on their shelves.

Reviewer: Hilmi Demiray (İstanbul)

### MSC:

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

74A40 | Random materials and composite materials |

74E05 | Inhomogeneity in solid mechanics |

74A60 | Micromechanical theories |

74E30 | Composite and mixture properties |

74Q15 | Effective constitutive equations in solid mechanics |