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- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
PROPERTIES FOR DESIGN OF COMPOSITE STRUCTURES
A comprehensive guide to analytical methods and source code to predict the behavior of undamaged and damaged composite materials
In Properties for Design of Composite Structures: Theory and Implementation Using Software, distinguished researcher Dr. Neil McCartney delivers a unique and insightful approach to the development of predictive methods for the properties of undamaged and damaged laminated composite materials. The book focuses on presenting compact analytical formulae for several important effective properties—including mechanical, thermal, and electrical—that can be applied to a variety of reinforcement geometries.
The author introduces a compact notation that enables an explicit treatment of laminate property determination, including the out-of-plane shear properties required for three-dimensional numerical simulations of structural features using finite and boundary element analyses. There is also a detailed consideration of ply crack closure and a useful study of the interrelationships between the effective thermoelastic constants of damaged laminates.
The book also offers:
- A thorough introduction to the principles and formulae for homogenous materials and applications, including continuum and fracture concepts for homogeneous materials
- A comprehensive exploration of the properties of undamaged composites, including undamaged composite materials with multiple phases and the properties of a single undamaged lamina
- Practical discussions of the properties of damaged composites, including matrix cracking in UD composites and damaged laminates
- Consideration of effects of delamination, fatigue, and environmentally induced damage
- In-depth examinations of derivations of key results, including the analysis of bridged cracks and stress transfer mechanics for cross-ply and general symmetric laminates
Perfect for composite design engineers in all types of material-supplying industries and manufacturing companies, Properties for Design of Composite Structures: Theory and Implementation Using Software will also earn a place in the libraries of undergraduate and graduate students in engineering, aerospace, and materials departments.
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Properties for Design of Composite Structures
Theory and Implementation Using Software
Neil McCartney
National Physical Laboratory, Teddington, Middlesex, UK
This edition first published 2022
© 2022 John Wiley & Sons Ltd
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The right of Neil McCartney to be identified as the author of this work has been asserted in accordance with law.
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Library of Congress Cataloging-in-Publication DataNames: McCartney, Neil, author.Title: Properties for design of composite structures : theory and implementation using software / Neil McCartney.Description: Hoboken, NJ : John Wiley & Sons, 2022. | Includes bibliographical references and index.Identifiers: LCCN 2021034688 (print) | LCCN 2021034689 (ebook) | ISBN 9781118485286 (hardback) | ISBN 9781118789681 (pdf) | ISBN 9781118789780 (epub) | ISBN 9781118789797 (ebook)Subjects: LCSH: Composite materials--Design and construction.Classification: LCC TA418.9.C6 M3495 2022 (print) | LCC TA418.9.C6 (ebook) | DDC 620.1/18--dc23LC record available at https://lccn.loc.gov/2021034688LC ebook record available at https://lccn.loc.gov/2021034689
Cover image: © NPL Management Ltd
Cover design by Wiley
Set in 9.5/12.5pt STIXTwoText by Integra Software Services Pvt. Ltd, Pondicherry, India
Contents
Cover
Title page
Copyright
Preface
About the Companion Website
1 Introduction
2 Fundamental Relations for Continuum Models
3 Maxwell’s Far-field Methodology Applied to the Prediction of Effective Properties of Multiphase Isotropic Particulate Composites
4 Maxwell’s Methodology for the Prediction of Effective Properties of Unidirectional Multiphase Fibre-reinforced Composites
5 Reinforcement with Ellipsoidal Inclusions
6 Properties of an Undamaged Single Lamina
7 Effective Thermoelastic Properties of Undamaged Laminates
8 Energy Balance Approach to Fracture in Anisotropic Elastic Material
9 Ply Crack Formation in Symmetric Cross-ply Laminates
10 Theoretical Basis for a Model of Ply Cracking in General Symmetric Laminates
11 Ply Cracking in Cross-ply Laminates Subject to Biaxial Bending
12 Energy-based Delamination Theory for Biaxial Loading in the Presence of Thermal Stresses
13 Energy Methods for Fatigue Damage Modelling of Laminates
14 Model of Composite Degradation Due to Environmental Damage
15 Maxwell’s Far-field Methodology Predicting Elastic Properties of Multiphase Composites Reinforced with Aligned Transversely Isotropic Spheroids
16 Debonding Models and Application to Fibre Fractures and Matrix Cracks
17 Interacting Bridged Ply Cracks in a Cross-ply Laminate
18 Theoretical Basis for a Model of Ply Cracking in General Symmetric Laminates
19 Stress-transfer Mechanics for Biaxial Bending
Appendix A: Solution for Shear of Isolated Spherical Particle in an Infinite Matrix
Appendix B: Elasticity Analysis of Two Concentric Cylinders
Appendix C: Gibbs Energy per Unit Volume for a Cracked Laminate
Appendix D: Crack Closure Conditions for Laminates
Appendix E: Derivation of the Solution of Nonlinear Equations
Appendix F: Analysis for Transversely Isotropic Cylindrical Inclusions
Appendix G: Recurrence Relations, Differential Equations and Boundary Conditions
Appendix H: Solution of Differential Equations
Appendix I: Energy Balance Equation for Delamination Growth
Appendix J: Derivation of Energy-based Fracture Criterion for Bridged Cracks
Appendix K: Numerical Solution of Integral Equations for Bridged Cracks
Index
End User License Agreement
List of Figures
Chapter 2
Figure 2.1 Transformation of right-handed...
Figure 2.2 Schematic diagram of part...
Chapter 3
Figure 3.1 (a) Discrete particle model...
Figure 3.2 Dependence of ratio of effective...
Figure 3.3 Dependence of effective bulk...
Figure 3.4 Dependence of the effective...
Chapter 4
Figure 4.1 (a) Discrete fibre model and...
Figure 4.2 Comparison of results for...
Figure 4.3 Comparison of results for...
Figure 4.4 Comparison of results for...
Figure 4.5 Comparison of results for...
Chapter 6
Figure 6.1 Method of defining principal...
Figure 6.2 Method of defining principal...
Chapter 7
Figure 7.1 Schematic diagram of the...
Chapter 8
Figure 8.1 Geometry and loading of a...
Figure 8.2 A composite plate (a) in...
Chapter 9
Figure 9.1 Representative volume elements...
Figure 9.2 Predictions of the normalised...
Chapter 10
Figure 10.1 Schematic diagram illustrating..
Chapter 11
Figure 11.1 Geometry of a multilayered laminate...
Chapter 12
Figure 12.1 Schematic diagram of a...
Figure 12.2 Schematic diagram showing...
Chapter 13
Figure 13.1 Schematic diagram of...
Figure 13.2 Master curve for...
Figure 13.3 The continuum model...
Figure 13.4 Side view of a long...
Figure 13.5 Schematic diagram of...
Figure 13.6 Dependence of axial...
Figure 13.7 Dependence of axial...
Figure 13.8 Experimental fatigue [24] data...
Figure 13.9 Dependence of in-plane...
Figure 13.10 Dependence of through...
Chapter 14
Figure 14.1 Schematic diagram of...
Figure 14.2 The dependence of composite...
Figure 14.3 The dependence of the...
Figure 14.4 The dependence of the...
Figure 14.5 The dependence of the...
Figure 14.6 The dependence of the...
Figure 14.7 The dependence of the...
Figure 14.8 The dependence of the...
Figure 14.9 The dependence of the...
Figure 14.10 The dependence of the...
Figure 14.11 The dependence of the...
Figure 14.12 The dependence of composite...
Chapter 15
Figure 15.1 (a) Discrete particle model...
Figure 15.2 Comparison of results...
Figure 15.3 Comparison of results...
Figure 15.4 Comparison of results...
Figure 15.5 Comparison of results...
Figure 15.6 Comparison of results for...
Chapter 16
Figure 16.1 Schematic diagram of...
Figure 16.2 Schematic diagram showing...
Figure 16.3 Schematic diagram showing the...
Figure 16.4 Interfacial stress distributions...
Figure 16.5 Interfacial stress distributions...
Figure 16.6 Interfacial stress distributions...
Figure 16.7 Interfacial stress distributions...
Figure 16.8 Interfacial stress distributions...
Figure 16.9 Interfacial stress distributions...
Figure 16.10 Schematic diagram of...
Figure 16.11 For debonded interfaces, typical...
Figure 16.12 For debonded interfaces...
Chapter 17
Figure 17.1 Schematic diagram of a...
Figure 17.2 Dependence of the critical...
Figure 17.3 Geometry and loading of...
Figure 17.4 Geometry and loading of part...
Figure 17.5 Stress intensity factors for...
Chapter 18
Figure 18.1 Schematic diagram illustrating...
Figure 18.2 Distribution of ply cracking...
Figure 18.3 Illustration of the homogenisation...
Chapter 19
Figure 19.1 Geometry of a multilayered nonsymmetrical...
Appendix J:
Figure J1 Schematic diagram of a bridged...
List of Tables
Chapter 3
Table 3.1 Estimates of effective bulk...
Table 3.2 Estimates of thermal expansion...
Chapter 12
Table 12.1 Predictions of loading parameters...
Table 12.2 Limiting loading parameters...
Chapter 15
Table 15.1 Effective properties for random...
Table 15.2 Effective properties for...
Table 15.3 Predictions for
C
1111
...
Table 15.4 Predictions for
C
3333
...
Table 15.5 Comparison of effective properties...
Chapter 16
Table 16.1 Values of matrix cracking...
Chapter 17
Table 17.1 Elastic properties for...
Guide
Cover
Title page
Copyright
Table of Contents
Preface
About the Companion Website
Begin Reading
Appendix A: Solution for Shear of Isolated Spherical Particle in an Infinite Matrix
Appendix B: Elasticity Analysis of Two Concentric Cylinders
Appendix C: Gibbs Energy per Unit Volume for a Cracked Laminate
Appendix D: Crack Closure Conditions for Laminates
Appendix E: Derivation of the Solution of Nonlinear Equations
Appendix F: Analysis for Transversely Isotropic Cylindrical Inclusions
Appendix G: Recurrence Relations, Differential Equations and Boundary Conditions
Appendix H: Solution of Differential Equations
Appendix I: Energy Balance Equation for Delamination Growth
Appendix J: Derivation of Energy-based Fracture Criterion for Bridged Cracks
Appendix K: Numerical Solution of Integral Equations for Bridged Cracks
Index
End User License Agreement
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Preface
The effective properties of composite materials, such as thermoelastic and conductive, depend upon the nature of many materials’ characteristics, which are needed to account for the composite structure, e.g. particulate or fibre reinforcement, unidirectional plies or multi-layered laminates. Account also must be taken of the resultant anisotropies that are often required when designing three-dimensional engineering components. Indeed, the properties of some constituents can be anisotropic themselves, such as carbon fibres and the individual plies of laminates. The accurate experimental determination of these properties is rarely, if ever, achieved in practice, and this leads on to the need for reliable estimation methods using analytical and/or numerical methods.
It is emphasised here that this book is not intended to be a reference text. The principal objective is to provide readers with a single source for a wealth of analytical methods and explicit formulae that can be used to estimate the effective properties of a wide range of composite types for both undamaged and damaged states. Readers wishing to track in more detail the historical development of composite theory will need to consult the literature, possibly starting with the limited number of references included in this book.
This book has been written over a number of years following the encouragement of the late Professor Anthony Kelly, FRS, CBE, to whom the author owes an enormous debt of gratitude. Professor Kelly was the Chairman of the Panel at the National Physical Laboratory (NPL) that decided, in 1969, that the author, following six months’ probation, could be offered a permanent post. The author now has an Emeritus position at NPL (Senior NPL Fellow) that has enabled continued research on various aspects of material science, and the writing of this book. A much closer relationship developed when, in the early 1980s, we both independently solved a problem considered in Chapter 14. At that point and onwards Professor Kelly, who was then Vice-Chancellor of Surrey University, introduced the author to many theoretical aspects of composite technology, which are included in this book. Of particular note is the development of methods of analysing the effects of fibre fractures and matrix cracks in unidirectionally reinforced composites, extending the pioneering work of colleagues at NPL, which led to the very well-known ACK (Aveston, Cooper, Kelly) theory, developed in 1971, which continues to be widely cited in the literature.
Following the Introduction, Chapter 2 of this book introduces and describes in detail many continuum analyses that will be required in later chapters. Chapters 3–7 are concerned with a description of methods of estimating the effective properties of undamaged composite materials, including those reinforced with aligned spheroidal particles. Extensive use is made of the methodology developed by James Clerk Maxwell in his book on electromagnetism, where an ingenious analytical method is used to estimate the effective electrical conductivity of a uniformly distributed assembly of isotropic spherical particles. His result is thought to be the first expression ever developed for an effective property of a composite. His methodology is also extremely powerful and worthy of extensive use in this book.
During service, composite components are often exposed to loadings that are sufficient to cause the formation of microstructural damage such as fibre failures or matrix cracking, which are usually associated with the occurrence of fibre/matrix interface debonding. For laminates, delamination at ply interfaces is an additional form of load-induced damage. The initiation and growth of microstructural damage affects the effective properties of composite materials, which degrade. It is remarkable that the degradation of very many thermoelastic constants is characterised by a single damage dependent function, as will become clear in the book. Sometimes damage mechanisms lead to local stress-transfer within complex structures, which can delay the onset of structural failure during progressive loading. Chapters 8–14 of this book will focus on analytical and semi-analytical methods of predicting both the effect of damage on effective properties and, more importantly, the dependence of effective properties on local loading conditions when using an energy balance approach. Chapters 13 and 14 concern, respectively, a model of damage development during the fatigue loading of laminates and a model of composite degradation due to environmental damage (in fibres).
Chapters 15–19 comprise more advanced analyses, providing mathematical details of many complex derivations that arise when considering: i) the effective properties of aligned systems of spheroids, ii) interface debonding associated with a fibre fracture or matrix crack in a unidirectional composite, iii) the behaviour of bridged cracks of finite length in a unidirectional composite, iv) the behaviour of ply cracks of finite length in a cross-ply laminate, v) stress-transfer mechanics for general symmetric laminates subject to general in-plane loading and vi) stress-transfer mechanics associated with out-of-plane orthogonal bending of cross-ply laminates that do not need to be symmetric. The models considered can lead to analytical formulations that require the use of numerical methods to deal with the complexity arising from the use of analytical techniques. Software implementing the analysis described in this book is provided at the Wiley website link (www.wiley.com/go/mccartney/properties).
The author would like to take this opportunity to thank all colleagues at NPL, and in other institutions around the world, for all the technical interactions that have helped greatly to formulate and investigate the various aspects of composite materials that are described in this book. My thanks also extend to the staff at John Wiley, who have been very patient while the book has been completed, and for their considerable help in the final stages of book preparation.
This book is dedicated to my late wife, Irene, and all our family, who have encouraged the completion of this book despite the many years in preparation. The author cannot thank them enough for allowing him the time in their lives, first to undertake the research that is described, and then the writing of the book itself.
L N McCartney
April 2022
About the Companion Website
Properties for Design of Composite Structures: Theory and Implementation Using Software is accompanied by a companion website:
www.wiley.com/go/mccartney/properties
The website includes:
Software Notes, where relevant by chapter
1 Introduction
Improving the properties and performance of materials by reinforcing them with different types of stiffer and/or stronger phases, such as particles or fibres, leads to the class of material known as composites. This approach was first exploited during natural developments (both living plants and organisms) and later by mankind, e.g. the use by Egyptians when making clay building bricks reinforced with straw to improve handling and performance, and when using gravel to reinforce cement forming a much stronger concrete material. Over the centuries, increasingly sophisticated composites have been developed, responding especially to the advent of higher-performance fibres (for high stiffness, strength and/or high temperature resistance). Even greater benefits can arise by the manufacture of composites reinforced with hollow and/or multicoated inclusions, and nanotubes.
The simplest inclusion geometry for matrix reinforcement is a set of spherical particles which might exhibit a range of radii. When the matrix and reinforcement is homogeneously mixed, the resulting material has improved properties which are usually considered to be isotropic. Another simple geometry uses aligned continuous fibres to reinforce an isotropic matrix forming a material that is anisotropic such that the properties in the fibre (or axial) direction differ from transverse properties in the plane normal to the fibre direction. This type of material is known as unidirectionally fibre-reinforced composite where the transverse stiffnesses and strengths are usually much lower than the stiffness and strength in the fibre direction. To overcome this significant practical problem, composite laminates are considered where a stack of unidirectional composites known as plies are bonded together where the fibres in each ply in the laminate are aligned in a direction that varies from ply to ply. Laminates are often weak under compression because of a damage mode known as delamination where debonding occurs at or near the interfaces between the various plies. To overcome this practical problem, woven or stitched fibre architectures are used. Such composites can be analysed effectively only if numerical methods are used. This topic will not, therefore, be considered in this book, as the intention is to focus here on the development and use of analytical methods. Composites, often made where the reinforcement is a set of short fibres which are either aligned in a given direction or are randomly oriented, are also not considered in this book.
The engineering application of composite materials for equilibrium calculations requires a full understanding the key materials properties which can be thought of as being classified as elastic, thermal, electrical and magnetic. In addition, when nonequilibrium transport phenomena occur, such as heat flow and electrical conduction, an additional type of property known as conductivities must also be considered. All these properties are well known, and for isotropic materials often encountered in engineering, a limited number of material properties are sufficient to characterise, fully, the physical behaviour of such materials. However, when composite materials are considered, the properties are usually no longer isotropic, and many more properties describing the directional behaviour of the material, and even of the fibres themselves, need to be considered in the engineering design process. This complexity also leads to needs for suitable measurement methods capable of determining the values of the multitude of properties required to fully characterise an anisotropic composite material, an important topic that will not be considered in this book. It is noted that the numerical analysis of three-dimensional composite components requires a very large number of materials properties, associated with the anisotropic nature of composite materials, some of which are extremely difficult to measure (e.g. through-thickness shear moduli). Analytical methods provide a pragmatic way of providing such data, but this leads on to needs to have access to reliable data for fibre properties, which are frequently anisotropic, and some properties are seldom known reliably (e.g. Poisson’s ratios, shear moduli and transverse thermal expansion).
The effective properties of composite materials, which arise when samples are considered as homogeneous materials having anisotropic properties, depend in a complex way on the properties of the materials 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. John Wiley & Sons, Inc. website (
www.wiley.com/go/mccartney/properties
).
