clear understanding of the key physical processes that are involved, and they provide methods of assessing whether, or not, candidate materials have good prospects of being used as improved engineering materials. Analytical models can also be used to develop exact solutions to relatively simple and amenable practical situations. These solutions can be used as special cases to validate the numerical methods which have much wider applicability.
The principal objectives of this book are to present, in a single publication, a description of the derivations of selected theoretical methods of predicting the effective properties of composite materials for situations where they are either undamaged or are subject to damage in the form of matrix cracking, in fibre-reinforced unidirectional composites, or in the plies of laminates, or to a lesser extent on the interfaces between neighbouring plies. The major focus of the book is on derivations of analytical formulae which can be the basis of software that is designed to predict composite behaviour, e.g. prediction of properties and growth of damage and its effect on composite properties. Software will be available from the John Wiley & Sons, Inc. website [1] including examples of software predictions associated with relevant chapters of this book.
The chapters of this book are grouped into three parts. The first group comprises Chapters 1 and 2, which provide the introduction and the fundamental relations for continuum models, and Chapters 3–7 that focus on preferred methods of estimating the thermoelastic properties of undamaged composites: particulate reinforced (both spheres and spheroids), fibre reinforced and laminates. The second group comprises Chapters 8–14 considering the fundamentals of ply cracking, and the predictions of ply crack formation in damaged composites, which are categorised into symmetric cross-plies and general symmetric laminates subject to general in-plane loading, and also nonsymmetric cross-ply laminates subject to combined biaxial bending and in-plane loading. A rigorous approach is developed that allows much theoretical development without having to know the detailed distributions of stress and strain within the laminates. Much effort is devoted to the development of very useful interrelationships between the effective properties of damaged laminates, and their use when using an energy balance approach to predict ply crack formation. Chapter 12 is concerned with an approach to the prediction of delaminations from preexisting ply cracks, whereas Chapters 13 and 14 consider ply crack formation under conditions of fatigue loading and under aggressive environmental conditions. The third and final group of chapters are more advanced texts where the mathematical details underpinning some of the earlier chapters are described in more detail. Spheroidal particle reinforcement for undamaged composites is considered in Chapter 15, and debonded fibre/matrix interfaces and crack bridging are described in Chapter 16, whereas crack bridging of ply cracks in laminates is described in Chapter 17. Stress transfer mechanics for ply cracks in general symmetric laminates is considered in Chapter 18, and Chapter 19 describes stress transfer mechanics for ply cracks in nonsymmetric cross-ply laminates subject to biaxial bending.
There is no attempt in this book to provide comprehensive accounts of relevant parts of the literature, although reference will be made to source publications related to the analytical methods described in the book. Some topics considered in this book, e.g. the chapters on particulate composites, delamination, fatigue damage and environmental damage, have been included to extend the range of applicability of the analytical methods described in the book. The content of these chapters is based essentially on specific publications by the author that are available in the literature.
Reference
1 1. John Wiley & Sons, Inc. website (www.wiley.com/go/mccartney/properties).
2 Fundamental Relations for Continuum Models
Overview: This chapter introduces the basic principles on which the mechanics of continua are based. Having defined the concepts of vectors and tensors, the physical quantities displacement and velocity are defined for continuous systems and then applied to the fundamental balance laws for mass, momentum (linear and angular) and energy. The principles of the thermodynamics of multicomponent fluid systems are first introduced. The strain tensor is then introduced so that the thermodynamic approach can be extended to solid systems for the single-component solids that will be considered in this book. The fundamental equations are then described for linear thermoelastic solids subject to infinitesimal deformations. The chapter then specifies the constitutive equations required for the analysis of anisotropic solids that will be encountered throughout the book, including the transformation of anisotropic properties following rotation about a given coordinate axis. The chapter concludes by considering bend formation applied to a homogeneous orthotropic plate.
2.1 Introduction
The principal objective of this book is to develop theoretical models and associated software that can predict the deformation behaviour of composite materials for situations where the system is first undamaged, but then develops progressively growing damage as the applied loading is gradually increased. Even though the composite systems are heterogeneous, continuum methods can be applied to each constituent of the composite, and to assist in the development of models for homogenised effectively continuous systems where the details of the reinforcement and damage distribution have been smoothed, and where effective properties may be defined.
The objective of this chapter is to describe the fundamental principles on which theoretical developments will be based. The topics to be covered are the nature of vectors and tensors, the definitions of displacement and velocity vectors, the balance laws for mass, momentum and energy, thermodynamics involving stress and strain state variables, and a thorough treatment of the linear elastic behaviour of anisotropic materials, including a contracted notation that is used widely in the composites field.
2.2 Vectors
A vector v is a mathematical entity that possesses both a magnitude and a direction. The vector is usually a physical quantity that is independent of the coordinate system that will be used to describe its properties. A very convenient approach is first to define an orthonormal set of coordinates (x1,x2,x3). Three axes are drawn in the x1,x2 and x3 directions, which are all at right angles to one another. Such a system is often described as a Cartesian set of coordinates. The positive directions of the x1, x1 and x3 axes are described by unit vectors i1 i2 and i3, respectively, where i1=(1,0,0),i2=(0,1,0),i3=(0,0,1). The unit vectors are such that
The three coordinates (x1,x2,x3) describe the location of a point x that is known as the position vector, which may be written as x=x1i1+x2i2+x3i3. In tensor theory
