Engineering Mechanics of Composite Materials

Oxford University Press Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland Madrid and associated companies in Berlin Ibadan Copyright 1994 by Oxford University Press Inc Published by Oxford University Press Inc 200 Madison Avenue New York New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved No part of this publication may be reproduced stored in a retrieval system or transmitted in any form or by any means electronic mechanical photocopying recording or otherwise without the prior permission of Oxford University Press Library of Congress CataloginginPublication Data Daniel Isaac M Engineering mechanics of composite materialsIsaac M Daniel Ori Ishai p cm Includes bibliographical references and index ISBN 0195075064 1 Composite materialsMechanical properties 2 Composite materialsTesting I Ishai Ori II Title TA4189C6D28 1993 620118dc20 9311047 987654321 Printed in the United States of America on acidfree paper Engineering Mechanics of Composite Materials Engineering Mechanics of Composite Materials Isaac M Daniel Departments of Civil and Mechanical Engineering Northwestern University Evanston IL USA Ori Ishai Faculty of Mechanical Engineering TechnionIsrael Institute of Technology Haifa Israel New York Oxford OXFORD UNIVERSITY PRESS 1994 To my wife Elaine my children Belinda Rebecca and Max and the memory of my parents Mordochai and Bella Daniel ISAAC M DANIEL To my wife Yael ORI ISHAI Although the underlying concepts of composite materials go back to antiquity the technology was essentially developed and most of the progress occurred in the last three decades and this development was accompanied by a proliferation of literature in the form of reports conference proceedings journals and a few dozen books Despite this plethora of literature or because of it we are constantly faced with a dilemma when asked to recommend a single introductory text for beginning students and engineers This has convinced us that there is a definite need for a simple and up to date introductory textbook aimed at senior undergraduates graduate students and engineers entering the field of composite materials This book is designed to meet the above needs as a teaching textbook and as a selfstudy reference It only requires knowledge of undergraduate mechanics of materials although some knowledge of elasticity and especially anisotropic elasticity might be helpful The book starts with definitions and an overview of the current status of composites technology The basic concepts and characteristics including properties of constituents and typical composite materials of interest and in current use are discussed in Chapter 2 To keep the volume of material covered manageable we omitted any extensive discussion of micromechanics We felt that although relevant micromechanics is not essential in the analysis and design of composites In Chapter 3 we deal with the elastic macromechanical response of the unidirectional lamina including constitutive relations in terms of mathematical stiffnesses and compliances and in terms of engineering properties We also deal with transformation relations for these mechanical properties We conclude with a short discussion of micromechanical predictions of elastic properties In Chapter 4 we begin with a discussion of microscopic failure mechanisms which leads into the main treatment of failure from the macroscopic point of view Four basic macroscopic failure theories are discussed in detail Classical lamination theory including hygrothermal effects is developed in detail and then applied to stress and failure analyses of multidirectional laminates in Chapters 5 6 and 7 We conclude Chapter 7 with a design methodology for structural composites including a design example discussed in detail Experimental methods for characterization and testing of the constituents and the composite material are described in Chapter 8 Whenever applicable in every chapter example problems are solved and a list of unsolved problems is given Computational procedures are emphasized throughout and flow charts for computations are presented The material in this book which can be covered in one semester is based on lecture notes that we have developed over the last fifteen years in teaching formal courses and condensed short courses at our respective institutions and we have incorporated much of the feedback received from students We hope this book is received as a useful and clear guide for introducing students and professionals to the field of composite materials We acknowledge with deep gratitude the outstanding dedicated and enthusiastic support provided by two people in the preparation of this work Mrs Yolande Mallian typed and proofread the entire manuscript including equations and tables with painstaking exactitude Dr ChoLiang Tsai diligently and ably performed many computations and prepared all the illustrations 3 Elastic Behavior of Unidirectional Lamina 37 572 Symmetric Laminates with Specially Orthotropic Layers Symmetric Crossply Laminates 156 71 Introduction 234 Contents Engineering Mechanics of Composite Materials Blank Page Chapter 1 Engineering Mechanics of Composite Materials Introduction components sports equipment biomedical products and many other products designed to have high mechanical performance andor environmental stability coupled with low weight see Figs 1217 13 Overview of Advantages and Limitations of Composite Materials Composites have unique advantages over monolithic materials such as high strength high stiffness long fatigue life low density and adaptability to the Fig 14 Automobile leaf spring made of glasspolyester composite weighing 36 N 8 lb compared with original steel spring which weighed 285 N 64 lb Fig 15 Boeing 757 commercial aircraft with a large number of composite material components Courtesy of Boeing Commercial Airplane Group Fig 17 B2 stealth bomber made almost entirely of composite materials Courtesy of Dr R Ghetzler Northrop Corporation intended function of the structure Additional improvements can be realized in corrosion resistance wear resistance appearance temperaturedependent behavior thermal stability thermal insulation thermal conductivity and acoustic insulation The basis of the superior structural performance of composite materials lies in the high specific strength strength to density ratio and high specific stiffness modulus to density ratio and in the anisotropic and heterogeneous character of the material The latter provides the composite system with many degrees of freedom for optimum configuration of the material Composites also have some limitations that conventional monolithic materials do not have Below is a brief discussion of the advantages and limitations of composites and conventional structural materials mainly metals when compared on the basis of micromechanics macromechanics material characterization design and optimization fabrication technology maintenance and durability and cost effectiveness The fabrication process is one of the most important steps in the application of composite materials Structural parts rather than generic material form are fabricated with relatively simple tooling A variety of fabrication methods suitable for several applications are available They include autoclave molding filament winding pultrusion and resin transfer molding RTM Structural components consisting of different materials such as honeycomb sandwich structures can be manufactured in one step by the socalled cocuring process Thus the number of parts to be assembled and the number of required joints can be reduced On the negative side composite fabrication is still dependent on skilled hand labor with limited automation and standardization This requires more stringent and extensive quality control procedures The study of composites is a philosophy of material design that allows for the optimum material composition along with structural design and optimization in one concurrent and interactive process It is a science and technology requiring close interaction of various disciplines such as structural design and analysis materials science mechanics of materials and process engineering The scope of composite materials research and technology consists of the following tasks 1 Investigation of basic characteristics of the constituent and composite materials 2 Material optimization for given service conditions 3 Development of effective and efficient fabrication procedures and understanding of their effect on material properties 4 Development of analytical procedures for determining material properties and prediction of structural behavior 5 Development of effective experimental methods for material characterization stress analysis and failure analysis 6 Nondestructive evaluation of material integrity and structural reliability 7 Assessment of durability flaw criticality and life prediction important To these two requirements today added the need for quality assurance reproducibility and predictability of behavior over the lifetime of the structure New developments continue in all areas For example new types of carbon fibers are being introduced with higher ultimate strains Thermoplastic matrices are used under certain conditions because they are tough have low sensitivity to moisture effects and are more easily amenable to mass production and repair The use of woven fabric and short fiber reinforcement is receiving more attention The design of structures and systems capable of operating at elevated temperatures has spurred intensive research in high temperature composites such as metalmatrix ceramicmatrix and carboncarbon composites The utilization of conventional and new composite materials is intimately related to the development of fabrication methods The manufacturing process is one of the most important stages in controlling the properties and ensuring the quality of the finished product Quality control inspection and automation are being introduced in the manufacturing process The technology of composite materials although still developing has reached a state of maturity Prospects for the future are bright for a variety of reasons The cost of the basic constituents is decreasing due to market expansion The fabrication process is becoming less costly as more experience is accumulated techniques are improved and automation is introduced Newer high volume applications such as in the automotive industry will expand the use of composites greatly The need for energy conservation motivates more uses of lightweight materials and products Finally the availability of many good interactive computer programs make structural design and analysis simpler and more manageable for engineers who have a basic undergraduate education Chapter 2 Basic Concepts and Characteristics 21 Structural Performance of Conventional Materials Conventional monolithic materials can be divided into three broad categories metals ceramics and polymers Although there is considerable variability in properties within each category each group of materials has some characteristic properties that are more distinct for that group In the case of ceramics one must make a distinction between two forms bulk and fiber Table 21 presents a list of properties and a rating of the three groups of materials with regard to each property The advantage or desirability is ranked as superior good poor and variable v Thus for example metals are superior with regard to stiffness and hygroscopic sensitivity but they have high density and are subject to chemical corrosion Ceramics in bulk form have low tensile strength and toughness but good thermal stability high hardness low creep and high erosion resistance Ceramics in fibrous form behave very differently from those in bulk form and have some unique advantages They rank highest with regard to tensile strength stiffness creep and thermal stability The biggest advantage that polymers have is their low density but they rank poorly with respect to stiffness creep hardness thermal and dimensional stability and erosion resistance The observations above show that no single material possesses all the advantages for a given application property and that it would be highly desirable to combine materials in ways that utilize the best of each constituent in a synergistic way A good combination for example would be ceramic fibers in a polymeric matrix Basic Concepts and Characteristics Table 21 Structural Performance Ranking of Conventional Materials Property Metals Ceramics Polymers Bulk Fibers Tensile strength v Stiffness v Fracture toughness v Impact strength v Fatigue endurance v Creep v v Hardness Density Dimensional stability v Thermal stability v Hygroscopic sensitivity v v Weatherability v v v Erosion resistance Corrosion Resistance v v superior good poor v variable 22 Geometric and Physical Definitions 221 Type of Material Depending on the number of its constituents or phases a material is called single phase or monolithic bi phase or twophase three phase and multiphase The different phases of a structural composite have distinct physical and mechanical properties and characteristic dimensions much larger than molecular or grain dimensions 222 Homogeneity A material is called homogeneous if its properties are the same at every point or are independent of location The concept of homogeneity is associated with a scale or characteristic volume and the definition of the properties involved Depending on the scale or volume observed the material can be more homogeneous or less homogeneous If low variability exists from point to point on a macroscopic scale the material is referred to as quasi homogeneous 223 Heterogeneity or Inhomogeneity A material is heterogeneous or inhomogeneous if its properties vary from point to point or depend on location As in the case above the concept of heterogeneity is associated with a scale or characteristic volume As this scale decreases the same material can be regarded as homogeneous quasi homogeneous or heterogeneous In Figure 21 for example the material is considered homogeneous and anisotropic on a macroscopic scale because it has a similar composition at different locations A and B but properties varying with orientation On a microscopic scale the material is heterogeneous and isotropic having different but orientationindependent properties within characteristic volumes a and b 224 Isotropy Many material properties such as stiffness strength thermal expansion and thermal conductivity are associated with a direction or axis A material is isotropic when its properties are the same in all directions or are independent of the orientation of reference axes 225 AnisotropyOrthotropy A material is anisotropic when its properties at a point vary with direction or depend on the orientation of reference axes Basic Concepts and Characteristics along any direction are the same as those along a symmetric direction with respect to a plane then that plane is defined as a plane of material symmetry A material may have zero one two three or an infinite number of planes of material symmetry through a point A material without any planes of symmetry is called general anisotropic or aeolotropic At the other extreme an isotropic material has an infinite number of planes of symmetry Of special relevance to composite materials are orthotropic materials ie materials having at least three mutually perpendicular planes of symmetry The intersections of these planes define three mutually perpendicular axes called principal axes of material symmetry or simply principal material axes 23 Material Response Some of the intrinsic characteristics of the materials discussed before are revealed in their response to simple mechanical loading eg uniaxial normal stress and pure shear stress as illustrated in Figure 22 An isotropic material under uniaxial tensile loading undergoes an axial deformation strain εₓ in the loading direction a transverse deformation strain εᵧ and no shear deformation εₓ σₓ E εᵧ ν σₓ E γₓᵧ 0 where εₓ εᵧ γₓᵧ Axial transverse and shear strains respectively σₓ Axial stress E Youngs modulus v Poissons ratio Under pure shear loading τₓᵧ the material undergoes a pure shear deformation ie a square element deforms into a diamondshaped one with equal and unchanged side lengths The shearstrain change of angle γₓᵧ and the normal strains εₓ εᵧ are γₓᵧ τₓᵧ G γₓᵧ 2 τₓᵧ 1 ν E εₓ εᵧ 0 where τxy Shear stress G Shear modulus As indicated in Eq 22 the shear modulus is not an independent constant but is related to Youngs modulus and Poissons ratio An orthotropic material loaded in uniaxial tension along one of its principal material axes 1 undergoes deformations similar to those of an isotropic material and given by ε1 σ1 E1 ε2 ν12 σ1 E1 γ12 0 where ε1 ε2 γ12 Axial transverse and shear strains respectively σ1 Axial normal stress E1 Axial modulus in the 1direction ν12 Poissons ratio associated with loading in the 1direction and strain in the 2direction Under pure shear loading τ12 along the principal material axes the material undergoes pure shear deformation ie a square element deforms into a diamondshaped one with unchanged side lengths The strains are γ12 τ12 G12 ε1 ε2 0 Here the shear modulus G12 is an independent material constant and is not related to the Youngs moduli or Poissons ratios In both cases discussed before normal loading does not produce shear strain and pure shear loading does not produce normal strains Thus normal loading and shear deformation as well as pure shear loading and normal strains are independent or uncoupled A general anisotropic material under uniaxial tension or an orthotropic material under uniaxial tension along a direction other than a principal material axis undergoes axial transverse and shear deformations given by εx σx Ex ey νxy σx Ex γxy ηxx σx Ex where εx ey γxy Axial transverse and shear strains respectively σx Axial normal stress Ex Axial modulus in xdirection νxy Poissons ratio associated with loading in the xdirection and strain in the ydirection ηxy Shear coupling coefficient the first subscript denotes normal loading in the xdirection the second subscript denotes shear strain This mode of response characterized by ηxy is called shear coupling effect and will be discussed in detail in Chapter 3 Under pure shear loading τxy along the same axes the material undergoes both shear and normal deformations ie a square element deforms into a parallelogram with unequal sides The shear and normal strains are given by γxy τxy Gy εx ηsx τxy Gy ey ηsy τxy Gy where Gy Shear modulus referred to the x and yaxes ηsx ηsy Shear coupling coefficients to be discussed later The above discussion illustrates the increasing complexity of material response with increasing anisotropy and the need to introduce additional material constants to describe this response 24 Types and Classification of Composite Materials Twophase composite materials are classified into three broad categories depending on the type geometry and orientation of the reinforcement phase as illustrated in the chart of Figure 23 Matrix Particulate filler Discontinuous fibers or whiskers Continuous fibers Particulate composite Unidirectional discontinuous fiber composite Randomly oriented discontinuous fiber composite Unidirectional continuous fiber composite Crossply or fabric continuous fiber composite Multidirectional continuous fiber composite Fig 23 Classification of composite material systems Particulate composites consist of particles of various sizes and shapes randomly dispersed within the matrix Because of the randomness of particle distribution these composites can be regarded as quasi homogeneous on a scale larger than the particle size and spacing and quasiisotropic Particulate composites may consist of nonmetallic particles in a nonmetallic matrix concrete glass reinforced with mica flakes brittle polymers reinforced with rubberlike particles metallic particles in nonmetallic matrices aluminum particles in polyurethane rubber used in rocket propellants metallic particles in metallic matrices lead particles in copper alloys to improve machinability and nonmetallic particles in metallic matrices silicon carbide particles in aluminium SiCpAl Discontinuous or shortfiber composites contain short fibers or whiskers as the reinforcing phase These short fibers which can be fairly long compared with the diameter can be either all oriented along one direction or randomly oriented In the first instance the composite material tends to be markedly anisotropic or more specifically orthotropic whereas in the second it can be regarded as quasiisotropic Continuous fiber composites are reinforced by long continuous fibers and are the most efficient from the point of view of stiffness and strength The continuous fibers can be all parallel unidirectional continuous fiber composite can be oriented at right angles to each other crossply or woven fabric continuous fiber composite or can be oriented along several directions multidirectional continuous fiber composite In the latter case for a certain number of fiber directions and distribution of fibers the composite can be characterized as a quasiisotropic material Fiberreinforced composites can be classified into broad categories according to the matrix used polymer metal ceramic and carbon matrix composites Table 22 Polymer matrix composites include thermoset epoxy polyimide polyester or thermoplastic polyetheretherketone polysulfone resins reinforced with glass carbon graphite aramid Kevlar or boron fibers They are used primarily in relatively low temperature applications Metal matrix composites consist of metals or alloys aluminum magnesium titanium copper reinforced with boron carbon graphite or ceramic fibers Their maximum use temperature is limited by the softening or melting temperature of the metal matrix Ceramic matrix composites consist of ceramic matrices silicon carbide aluminum oxide glassceramic silicon nitride reinforced with ceramic fibers They are best suited for very high temperature applications Carbon carbon composites consist of carbon or graphite matrix reinforced with graphite yarn or fabric They have unique properties of relatively high strength at high temperatures coupled with low thermal expansion and low density In addition to the types discussed above there are laminated composites These consist of thin layers of different materials bonded together such as bimaterials clad metals plywood and formica Basic Concepts and Characteristics Table 22 Types of Composite Materials Matrix type Fiber Polymer Eglass Sglass Carbon graphite Aramid Kevlar Boron Metal Boron Borsic Carbon graphite Silicon carbide Alumina Ceramic Silicon carbide Alumina Silicon nitride Carbon Carbon Matrix E poxy Polyimide Polyester Thermoplastics PEEK polysulfone etc Aluminum Magnesium Titanium Copper Silicon carbide Silicon carbide Alumina Glassceramic Silicon nitride 25 Lamina Laminate Characteristics and Configurations A lamina or ply is a plane or curved layer of unidirectional fibers or woven fabric in a matrix In the case of unidirectional fibers it is also referred to as unidirectional lamina UD The lamina is an orthotropic material with principal material axes in the direction of the fibers longitudinal normal to the fibers in the plane of the lamina inplane transverse and normal to the plane of the lamina Fig 24 These principal axes are designated as 1 2 and 3 respectively In the case of a woven fabric composite the warp and fill directions are the inplane principal directions Fig 24 Unidirectional lamina and principal coordinate axes A laminate is made up of two or more unidirectional laminae or plies stacked together at various orientations Fig 25 The laminae or plies or layers can be of various thicknesses and consist of different materials Since the principal material axes differ from ply to ply it is more convenient to analyze laminates using a common fixed system of coordinates xyz as shown The orientation of a given ply is given by the angle between the reference xaxis and the major principal material axis fiber orientation of the ply measured in a counterclockwise direction on the xy plane Composite laminates containing plies of two or more different types of materials are called hybrid composites and more specifically interply hybrid composites For example a composite laminate may be made up of unidirectional glassepoxy carbonepoxy and aramidepoxy layers stacked together in a specified sequence In some cases it may be advantageous to intermingle different types of fibers such as glass and carbon or aramid and carbon within the same unidirectional ply Such composites are called intraply hybrid composites Of course one could combine intraply hybrid layers with other layers to form an intraplyinterply hybrid composite Comosite laminates are designated in a manner indicating the number type orientation and stacking sequence of the plies The configuration of the laminate indicating its ply composition is called layup The configuration indicating in addition to the ply composition the exact location or sequence of the various plies is called the stacking sequence Following are some examples of laminate designations 000000 0₆ Unidirectional 6ply 0909090 090ₛ Crossply symmetric 45454545 45ₛ Angleply symmetric 04545450 045ₛ Multi directional 00454500 0₂450₂ₛ 0151515150 015150ₜ Hybrid 0ₖ0ₖ45ₐ45ₖ90ₑ45ₖ45ₐ0ₖ0ₖ90ₑₛ where subscripts and symbols signify the following number subscript Multiple of plies or group of plies s Symmetric sequence T Total number of plies Overbar denotes that laminate is symmetric about the midplane of the ply Fig 26 Levels of observation and types of analysis for composite materials Micromechanics is particularly important in the study of properties such as strength fracture toughness and fatigue life which are strongly influenced by local characteristics that cannot be integrated or averaged Micromechanics also allows the prediction of average behavior at the lamina level as a function of constituent properties and local conditions Fig 27 Isochromatic fringe patterns in a model of transversely loaded unidirectional composite E1 E2 E3 Youngs moduli along the principal ply directions G12 G23 G13 Shear moduli in 12 23 and 13 planes respectively these are equal to G21 G32 and G31 respectively ν12 ν23 ν13 Poissons ratios the first subscript denotes the loading direction and the second subscript denotes the strain direction these Poissons ratios are different from ν21 ν32 and ν31 ie subscripts are not interchangeable F1p F2n F3t Tensile strengths along the principal ply directions F1c F2c F3c Compressive strengths along the principal ply directions F12 F23 F13 Shear strengths in 12 23 and 13 planes respectively these are equal to F21 F32 and F31 respectively α1 α2 α3 Coefficients of thermal expansion β1 β2 β3 Coefficients of moisture expansion κ1 κ2 κ3 Coefficients of thermal conductivity In addition to the above the composite lamina is characterized by the following properties Fiber volume ratio Vf volume of fibers volume of composite Fiber weight ratio Wf weight of fibers weight of composite Matrix volume ratio Vm volume of matrix volume of composite Matrix weight ratio Wm 1 Wf weight of matrix weight of composite Void volume ratio Vv 1 Vf Vm volume of voids volume of composite 28 Degrees of Anisotropy Some material properties such as density specific heat absorptivity and emitance have no directionality associated with them and are described by one scalar quantity for both isotropic and anisotropic materials On the other hand properties such as stiffness Poissons ratio strength thermal expansion moisture expansion thermal conductivity and electrical conductivity are associated with direction and are a function of orientation in anisotropic materials Fiber composite materials can exhibit various degrees of anisotropy in the various properties The largest differences occur between properties in the longitudinal fiber and transverse normal to the fiber directions Ratios of some properties along these two directions for some typical composite materials are listed in Table 23 29 Constituent Materials and Properties 291 Fibers A large variety of fibers are available as reinforcement for composites The desirable characteristics of most reinforcing fibers are high strength high stiffness and relatively low density Each type of fiber has its own advantages and disadvantages as listed in Table 24 Table 25 lists specific fibers with their manufacturer strength modulus and density Extensive discussions of fiber reinforcements for composite materials can be found elsewhere13 Glass fibers are the most commonly used ones in low to medium performance composites because of their high tensile strength and low cost They are somewhat limited in composite applications because of their relatively low stiffness low fatigue endurance and rapid property degradation with exposure to severe hygrothermal conditions Aramid or Kevlar fibers have higher stiffness and lower density but they are limited by very low compressive strength in the Basic Concepts and Characteristics 29 composite and high moisture absorption Boron fibers not widely used at present are useful in local stiffening applications because of their high stiffness Carbon graphite fibers come in many types with a range of stiffnesses and strengths depending on the processing temperatures High strength and high stiffness carbon fibers AS4 T300 C6000 are processed at temperatures between 1200 and 1500C 2200 and 2700F Ultrahigh stiffness graphite fibers GY70 Pitch are processed at temperatures between 2000 and 3000C 3600 and 5400F The increase in stiffness is achieved at the expense of strength as shown in Table 25 Ceramic fibers such as silicon carbide and aluminum oxide have high stiffness and moderate strength and are used in metalmatrix and ceramicmatrix composites for high temperature applications Most fibers behave linearly to failure as shown in Figure 28 Carbon fibers such as the AS4 fiber however display a nonlinear stiffening effect One important property of the fiber related to strength and stiffness is the ultimate strain or strain to failure because it influences greatly the strength of the composite laminate As mentioned previously the basis of the superior performance of composites lies in the high specific strength strength to density ratio and high specific stiffness modulus to density ratio These two properties are controlled by the fibers A twodimensional comparative representation of some typical fibers from the point of view of specific strength and specific modulus is shown in Figure 29 Engineering Mechanics of Composite Materials 30 Specific modulus 108 in 0 1 2 3 4 5 6 7 8 9 10 20 Sglass 18 Aramid Kevlar 16 Carbon high strength 14 Boron on tungsten 12 Eglass 10 Steel wires 8 Beryllium 6 Aluminum in bulk 4 Steel in bulk 2 0 0 2 4 6 8 10 12 14 16 18 20 22 24 26 Specific strength 106 cm Fig 29 Performance map of fibers used in structural composite materials Basic Concepts and Characteristics 31 292 Matrices As shown in Table 22 four types of matrices are used in composites polymeric metallic ceramic and carbon The most commonly used matrices are polymeric which can be thermosets epoxies polyimide polyester or thermoplastics polysulfone polyetheretherketone The most highly developed of these are epoxies of DGEBA type diglycidyl ether of bisphenol A They can be formulated with a range of stiffnesses as shown in Figure 210 There are two types of epoxies those cured at a low temperature 120C 250F and used in components exposed to low or moderate temperature variations eg sporting goods and those cured at a higher temperature 175C 350F and used in high performance components exposed to high temperature and moisture variations eg aircraft structures Polyimide matrices are used for high temperature applications up to 370C 700F Polyesters are used in quickcuring systems for commercial products Thermoplastics are more compatible with hot forming and injection molding fabrication methods and can be applied at temperatures up to 400C 750F Metal matrices are recommended for high temperature applications up to approximately 800C 1500F Commonly used metal matrices include aluminum magnesium and titanium alloys Their use temperature is limited by the melting point Stressstrain curves of three typical matrices Stressstrain curves of typical unidirectional composites in fiber direction Properties of Typical Unidirectional Composite Materials Fiber Volume ratio Vf Chapter 3 Elastic Behavior of Unidirectional Lamina 31 StressStrain Relations Fig 31 State of stress at a point of a continuum and ε11 S1111 S1122 S1133 S1111 S1112 S1113 S1121 σ11 ε22 S2211 S2222 S2233 S2221 S2222 S2221 σ22 ε33 S3311 S3322 S3333 S3311 S3312 S3311 σ33 ε23 S2311 S2322 S2333 S2312 S2321 S2321 σ23 ε31 S3111 S3122 S3133 S3112 S3131 S3111 σ31 ε12 S1211 S1222 S1223 S1231 S1212 S1231 σ12 ε32 S3211 S3222 S3233 S3212 S3221 S3221 σ32 ε13 S1311 S1322 S1333 S1312 S1311 S1311 σ13 ε21 S2111 S2122 S2123 S2131 S2112 S2113 σ21 or in indicial notation σij Cijkl εkl i j k l 1 2 3 εij Sijkl σkl where Cijkl Stiffness components Sijkl Compliance components Elastic Behavior of Unidirectional Lamina Repeated subscripts in the relations above imply summation for all values of that subscript The compliance matrix Sijkl is the inverse of the stiffness matrix Cijkl Thus in general it would require 81 elastic constants to characterize a material fully However the symmetry of the stress and strain tensors σij σji εij εji reduces the number of independent elastic constants to 36 It is customary in mechanics of composites to use a contracted notation for the stress strain stiffness and compliance tensors as follows σ11 σ1 σ22 σ2 σ33 σ3 σ23 τ23 σ4 τ4 σ31 τ31 σ5 τ5 σ12 τ12 σ6 τ6 ε11 ε1 ε22 ε2 ε33 ε3 2ε23 γ4 ε4 2ε31 γ5 ε5 2ε12 γ6 ε6 C1111C11 C1122C12 C1133C13 C111232C14 C11312C15 C111122C16 C2211C21 C2222C22 C2233C23 C22232C24 C22312C25 C22122C26 C3311C31 C3322C32 C3333C33 C33322C34 C33112C35 C33122C36 C2311C41 C2322C42 C2333C43 C23232C44 C23312C45 C23122C46 C3111C51 C3122C52 C3133C53 C31322C54 C13112C55 C31122C56 C1211C61 C1222C62 C1233C63 C12232C64 C12312C65 C12122C66 Thus the stressstrain relations for an anisotropic body can be written in the contracted notation as σ1 C11 C12 C13 C14 C15 C16 ε1 σ2 C21 C22 C23 C24 C25 C26 ε2 σ3 C31 C32 C33 C34 C35 C36 ε3 τ4 C41 C42 C43 C44 C45 C46 γ4 τ5 C51 C52 C53 C54 C55 C56 γ5 τ6 C61 C62 C63 C64 C65 C66 γ6 or in indicial notation σi Cij εj i j 1 2 3 6 εi Sij σj Energy considerations require additional symmetries The work per unit volume is expressed as W 12 Cij εi εj The stressstrain relation Eq 310 can be obtained by differentiating Eq 311 σi Wεi Cij εj By differentiating again we obtain Cij ²Wεiεj Elastic Behavior of Unidirectional Lamina In a similar manner by reversing the order of differentiation we obtain Cji ²W εjεi 314 Since the order of differentiation of W is immaterial Eqs 313 and 314 yield Cij Cji 315 In a similar manner we can show that Sij Sji 316 ie the stiffness and compliance matrices are symmetric Thus the state of stress or strain at a point can be described by six components of stress or strain and the stressstrain Eqs 38 and 39 are expressed in terms of 21 independent stiffness or compliance constants 312 Specially Orthotropic Material In the case of an orthotropic material which has three mutually perpendicular planes of material symmetry the stressstrain relations in general have the same form as Eqs 38 and 39 However the number of independent elastic constants is reduced to nine as various stiffness and compliance terms are interrelated This is clearly seen when the reference system of coordinates is selected along principal planes of material symmetry ie in the case of a specially orthotropic material Then σ₁ σ₂ σ₃ τ₄ τ₅ τ₆ C11 C12 C13 0 0 0 ε₁ C12 C22 C23 0 0 0 ε₂ C13 C23 C33 0 0 0 ε₃ 0 0 0 C44 0 0 γ₄ 0 0 0 0 C55 0 γ₅ 0 0 0 0 0 C66 γ₆ 317 ε₁ ε₂ ε₃ γ₄ γ₅ γ₆ S11 S12 S13 0 0 0 σ₁ S12 S22 S23 0 0 0 σ₂ S13 S23 S33 0 0 0 σ₃ 0 0 0 S44 0 0 τ₄ 0 0 0 0 S55 0 τ₅ 0 0 0 0 0 S66 τ₆ 318 It is clearly shown that an orthotropic material can be characterized by nine independent elastic constants This number does not change by changing the reference system of coordinates to one in which the stiffness and compliance matrices in Eqs 317 and 318 above are fully populated The terms of either the stiffness or compliance matrix can be obtained by inversion of the other Thus relationships can be obtained between Cij and Sij Three important observations can be made with respect to the stressstrain relations in Eqs 317 and 318 1 No interaction exists between normal stresses σ₁ σ₂ σ₃ and shear strains γ₄ γ₅ γ₆ ie normal stresses acting along principal material directions produce only normal strains 2 No interaction exists between shear stresses τ₄ τ₅ τ₆ and normal strains ε₁ ε₂ ε₃ ie shear stresses acting on principal material planes produce only shear strains 3 No interaction exists between shear stresses and shear strains on different planes ie a shear stress acting on a principal plane produces a shear strain only on that plane 313 Transversely Isotropic Material An orthotropic material is called transversely isotropic when one of its principal planes is a plane of isotropy ie at every point there is a plane on which the mechanical properties are the same in all directions Many unidirectional composites with fibers packed in a hexagonal array or close to it can be considered transversely isotropic with the 23 plane normal to the fibers as the plane of isotropy Fig 32 This is the case with unidirectional carbonepoxy aramidepoxy and glassepoxy composites with relatively high fiber volume ratios The stressstrain relations for a transversely isotropic material are simplified by noting that subscripts 2 and 3 for a 23 plane of isotropy in the material constants are interchangeable in Eqs 317 and 318 ie C12 C13 C22 C33 and S12 S13 S22 S33 319 Also subscripts 5 and 6 are interchangeable thus C55 C66 S55 S66 320 Furthermore the simple stress transformation illustrated in Figure 33 shows that stiffness C44 or compliance S44 is not independent Considering an element with sides parallel to the 2 and 3axes Fig 33a under pure shear stress τ₀ τ₂₃ and resulting shear strain γ₀ γ₂₃ we have from Eq 317 τ₄ τ₂₃ C₄₄ γ₂₃ C₄₄ γ₄ τ₀ 321 The state of stress shown in Figure 33a is equivalent to that of an element rotated by 45 and subjected to equal tensile and compressive normal stresses Fig 33b σ2 τ0 σ3 τ0 resulting in normal strains such that ε2 ε3 γ23 2 ε1 0 Then from Eq 317 σ2 C22 ε2 C23 ε3 C22 ε2 C23 ε2 or σ2 ε2C22 C23 γ23 2 C22 C23 since C22 C22 C23 C23 due to transverse isotropy C44 C22 C23 2 C44 C22 C23 2 Qij Cij Ci3 Cj3 C33 i j 1 2 6 332 32 Relations between Mathematical and Engineering Constants σ1 σ1 E1 ε2 ν12 E1 σ1 ε3 ν13 E1 σ1 γ4 γ5 γ6 0 Engineering Mechanics of Composite Materials Elastic Behavior of Unidirectional Lamina Engineering Mechanics of Composite Materials Elastic Behavior of Unidirectional Lamina and in general vij vij or vij Ei Ej i j 1 2 3 Note The above can also be deduced from Bettis reciprocal law according to which transverse deformation due to a stress applied in the longitudinal direction is equal to the longitudinal deformation due to an equal stress applied in the transverse direction As seen above the relations between compliances Sij and engineering constants are fairly simple This however is not the case for the relations between stiffnesses Cij and engineering constants To obtain such relationships we need first to invert the compliance matrix Sij and express the stiffnesses Cij as a function of the compliances Sij as follows C11 S22 S33 S232S C22 S33 S11 S132S C33 S11 S22 S122S C12 S13 S23 S12 S33S C23 S12 S13 S23 S11S C13 S12 S23 S13 S22S C44 1S44 C55 1S55 C66 1S66 where S S11 S12 S13 S12 S22 S23 S13 S23 S33 Substituting the relations between Sij and engineering constants in the above we obtain Engineering Mechanics of Composite Materials C11 1 nu23 nu32E2 E3 delta C22 1 nu13 nu31E1 E3 delta C33 1 nu12 nu21E1 E2 delta C12 nu21 nu31 nu23E2 E3 delta nu12 nu13 nu32E1 E3 delta C23 nu32 nu21 nu31E1 E3 delta nu23 nu21 nu13E1 E2 delta C13 nu13 nu12 nu23E1 E2 delta nu31 nu21 nu32E2 E3 delta C44 G23 C55 G13 C66 G12 where delta 1E1 E2 E3 1 nu21 nu31 nu12 1 nu32 nu13 nu23 1 It should be noted that in the case of a transversely isotropic material with the 23 plane as the plane of isotropy E2 E3 G12 G13 nu12 nu13 33 StressStrain Relations for Thin Lamina A thin unidirectional lamina is assumed to be under a state of plane stress therefore the stressstrain relations in Eqs 331 and 333 are applicable They relate the inplane stress components with the inplane strain components along the principal material axes sigma1 sigma2 tau6 Q11 Q12 0 epsilon1 Q12 Q22 0 epsilon2 0 0 Q66 gamma6 The relations above can be expressed in terms of engineering constants by noting that S11 1E1 S22 1E2 S12 nu12E1 nu21E2 S66 1G12 and Q11 E11 nu12 nu21 Q22 E21 nu12 nu21 Q12 nu21 E11 nu12 nu21 nu12 E21 nu12 nu21 Q66 G12 Thus as far as the inplane stressstrain relations are concerned the unidirectional lamina can be fully characterized by four independent constantsthe four reduced stiffnesses Q11 Q22 Q12 and Q66 or the four compliances S11 S22 S12 and S66 or four engineering constants E1 E2 G12 and nu12 Poissons ratio nu21 is not independent as it is related to nu12 E1 and E2 by Eq 349 34 Transformation of Stress and Strain Normally the lamina principal axes 1 2 do not coincide with the loading or reference axes x y Fig 36 Then the stress and strain components referred to the principal material axes 1 2 can be expressed in terms of those referred to the loading axes xy by the following transformation relations σ1 T σx σ2 τ6 or in brief σ12 T σxy and ε1 T εx ε2 εy or in brief ε12 T εxy where the transformation matrix T is given by