Properties for Design of Composite Structures. Neil McCartney

used in their manufacture (e.g. reinforcements and matrix) and on the geometrical arrangement of these materials. It is plainly not feasible to undertake an experimental programme designed to use measurement methods to determine the relationships between effective properties of composite materials and the constituent properties and structure. Instead, theoretical methods are used based on the well-established principles of continuum thermodynamics defined in its most general form so that both continuum mechanics and electrodynamics are considered in a thermodynamic context. It is indeed of interest to know that James Clerk Maxwell, developer of the famous Maxwell equations of electrodynamics, is believed to be the first scientist to develop a formula for an effective property of a composite material. He considered a cluster of spherical particles, all having the same isotropic permittivity value, embedded in an infinite matrix, having a different value for isotropic permittivity, and developed an elegant method of estimating the effective properties of the particle cluster. Although Maxwell argued that his neglect of particle interactions would limit the validity of his effective property to low volume fractions, it is known that results obtained using his methodology are in fact valid for much larger volume fractions. This important scientific contribution appeared in 1873 as part of Chapter 9 in his book entitled A Treatise on Electricity and Magnetism, published by Clarendon Press, Oxford. Maxwell’s methodology will be used in this book to help understand the relationship of many effective composite properties to the properties of the reinforcements and their geometrical arrangements within a matrix.

In recent years, although many developments of materials and structures in the composites field have used a make-and-test philosophy, the scientific understanding that has now developed means that predictive methods of assessing composite performance are being used more widely. There is a wide spectrum of predictive techniques that can be used ranging from analytical models, which is the theme of this book, through to numerical simulations of engineering components, having complex geometries and loadings, which are based on numerical techniques such as the finite element and boundary element methods. There is a need to employ both analytical and numerical techniques. The former are models where predictions are possible through use of mathematical formulae that relate the important parameters that might be varied when designing a new material, whether a unidirectional ply or a complex laminate. The parameters normally encountered are the fibre volume fractions, the thermoelastic constants of both fibre and matrix, or of a ply or a laminate. When assessing damage resistance, other types of property are encountered such as fracture energies. Analytical methods provide a 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