Composite Structures Design, Mechanics, Analysis, Manufacturing, and Testing Manoj Kumar Buragohain

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Micromechanics of a Lamina

Publication details https://www.routledgehandbooks.com/doi/10.1201/9781315268057-3 Manoj Kumar Buragohain Published online on: 20 Sep 2017

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The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The publisher shall not be liable for an loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 E E d G F F E E E A A 3.2 briefly and touchparameters the upon elasticity-based models andmodels. semiempirical some models of mechanics materials-based for evaluation the thermoelastic of lamina focus Our isture. not areview of models; these we instead, dwell formulations on the of basic many models concepts.micromechanics micromechanics litera- the are in There considered be alternative can study an experimental as to of the lamina. lamina the context the Thus, in study of of lamina. product the design, of micromechanics the a experimental an subsequent from design obtained analysis and directly be laminate can analysis.and Alternatively, input for the data macro-level the analysis and of alamina level macro at analysis the the in used of lamina the subsequent and design laminate then functions are characteristics as of constituentestimated the lamina The properties. studymental of behavior its constituents, lamina and is reinforcements viz. matrix, and experi the from obtained input are data necessary the analysismechanical of a lamina, itsand behavior studied at two be can levels—micro level level. macro and For micro analysisite (and laminate design) at different levels. is amultiphase Alamina element point. Figure 3.1starting presents aschematic representation of process of the compos- ior is essential for design the of analysis acomposite is and the of alamina structure blocks building the acomposite are in knowledge structure; laminae behav of lamina overall the in aspect cial design of acomposite mentioned As Chapter structure. 1, in element, design composite is acru laminate and is structural alaminated A laminate 3.1 b

c Micromechanics of a Lamina a of Micromechanics 1 1 c c

f c f m f , , b

, A

c f

, E

In this chapter, this In we provide followed remark introductory an by abrief review of the , E F

f m , b f , A 2 2

f c

CHAPTER ROAD MAP CHAPTER

PRINCIPAL NOMENCLATURE m m

Shear moduluShear tively, of transversely isotropic fibers tively, of transversely isotropic composite element volume tive Area of crossArea sec Fiber diamet in a representativein volume element Young’s modu Areas of cross sectionAreas of composite, respectively, fibers,andmatrix, Widths of composite, respectively, fibers,andmatrix, in arepresenta Young’s modu Forces b shared Young’s longitudinal transverse the respec and directions, in moduli Young’s modu Total force on c Young’s longitudinal transverse the respec and directions, in moduli er s of isotropic fibers lus of isotropic composite lus of isotropic fibers lus of matrix omposite (representative volume element) y the fibers and matrix, respectivelyandmatrix, fibers y the tion of arepresentative volume element 79 3 ------Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 V s 80 γ β β α α w w W v t l G 3.1 FIGURE β β α α v ( l G , c c V

c f 12 1 1

, , c m , f 1 1 c c m 12 f

, m b

c f , f

c f f

, ,

l t

, , v c

) ,

V , f f w

f ,

, , W β ,

cri m β α

t m γ l G t m , 2

2 , m

2 m , m f 2 23 c

f 23 c

(

v c V

) min

Schematic representation of composite laminate analysis process. analysis laminate of composite representation Schematic sentative volume element volume sentative tative volume element Total volum Critical fiberCritical fibervolumeminimum fractionand fraction, volume Coefficien Mass fracti Fiber volume volume fraction, matrix voids fraction, and volume Lengths of composite,Lengths respectively, fibers,andmatrix, in arepresen Longitudinal and transverse and coefficientsLongitudinal expansion, thermal of Longitudinal and transverse and coefficientsLongitudinal expansion, thermal of Longitudinal (in alongitudinalLongitudinal plane) transverse and (in a transverse Coefficien Fiber spacin Fiber Longitudinal and transverse and coefficientsLongitudinal of moisture expansion, Coefficien Mass of fibers a plane) respectively, strains, shear composite in respectively, of transversely isotropic fibers Longitudi respectively, of transversely isotropic composite respectively, of transversely isotropic fibers respectively, of transversely isotropic composite respectively respectively fraction, element of transversely isotropic fibers Thicknesses ofThicknesses composite, respectively, fibers,andmatrix, in arepre Shear modulShear Coefficien Shear moduli in the longitudinal transverse the and planes, in moduli Shear respectively, Total weigh Length, widtLength, Volumesand ofvoids, fibers,matrix, respectively analysis oflamina Micromechanical nal and transverse and coefficientsnal of moisture expansion, t of thermal expansiont of thermal of matrix t of moisture expansion of isotropic composite t of thermal expansiont of thermal of isotropic composite t of moisture expansion of matrix on respectively offractionand fibersmass matrix, of t of composite e of composite g Macromechanical analysisoflamina us of matrix h, and thickness, respectively, thickness, h, and of arepresentative volume Analysis ofcompositestructure Analysis ofcompositelaminate nd mass of matrix, respectivelynd of mass matrix, Characterization of constituents Or study oflamina Experimental Composite Structures Composite - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 () σσ σσ ρ ξ ν ν η () () () () () () () Δ Δ Δ ( ( ( ν Δ Δ ( γ ( γ Micromechanics of a Lamina a of Micromechanics εε εε εε εε εε εε γ γ γ γ γ σ ε ε εε εε ε εε εε 12 12 12 12 12 12

12 12 12 T C C T T C

f c m 12 12 T C l l C T c

c c m m 12 12 12 12 12 12 , c m c m f f T C T C T T T f c c , c

f c T

)

c c ) f , , f m

, f f

) , , , , , ρ , c , , ult f c Δ

ult

, ) , ) Δ ult ult ult ν γ

ul ul f

ul ul ul ult , ult T C

f Δ γ C T

, T C

l C T tc tc 23 23 t c c tf tf

m m f f f

,( ,( c c , 23

, C

Δ , ,( ,(

f m

(

( Δ m m f γ γ

l C T 23 T C 23 Δ m f

) ) c C ) ) ) ) ult ult ult ul ult ult m t

Longitudinal (in alongitudinalLongitudinal plane) transverse and (in atransverse Ultimate longitudinal transverserespectively, and tensileUltimate strains, in Poisson’s r tively, transversely in isotropic fibers tively, transversely in isotropic composite Density of composite, respectively fibers,andmatrix, Ultimate longitudinal (inUltimate alongitudinal plane) transverse and (in a Longitudinal and transverse and tensileLongitudinal stresses, respectively, composite in Changes length in of composite, respectively, fibers,andmatrix, in a Longitudinal (in alongitudinalLongitudinal plane) transverse and (in atransverse Longitudinal and transverse and compressiveLongitudinal stresses, respectively, in Ultimate sh Ultimate Reinforci Fiber packi Longitudinal and transverse and compressiveLongitudinal respectively, strains, in Change len in Deformations composite, in respectively fibers,andmatrix, isotropic composite)isotropic composite respectively) of transversely isotropic fibers matrix transverse and compressiveLongitudinal respectively, strains, in longitudinal transverse and compressiveUltimate respec strains, transversely isotropic fibers fibers transversely isotropic composite composite transverserespectively, and tensileLongitudinal strains, composite in representative volume element respectively plane) respectively, strain, shear matrix in fibers transverse plane) respectively, strains, shear transversely in isotropic plane) respectively, strains, shear fibers in composite transverse plane) respectively, strains, shear transversely in isotropic Ultimate sh Ultimate Longitudinal and transverserespectively, and tensileLongitudinal strains, matrix in transverse and compressiveLongitudinal respectively, strains, in transverserespectively, and tensileLongitudinal strains, fibers in Ultimate sh Ultimate Change tem in Major Poisson’s ratios (inlongitudinal the plane transverse and plane, Ultimate longitudinal transverserespectively, and tensileUltimate strains, in Ultimate longitudinal (inUltimate alongitudinal plane) transverse and (in a Poisson’s r Changes moisture in content composite, in fibers,andmatrix, Ultimate longitudinal transverse and compressiveUltimate respec strains, Ultimate te Ultimate Ultimate te Ultimate te Ultimate te Ultimate ng factor (in Halpin–Tsai equations) atio of isotropic fibers atio of matrix nsile strain in matrix in nsile strain isotropic in nsile fibers strain isotropic in nsile composite strain nsile isotropic stress in composite (i.e., tensile of strength ng factor (in Halpin–Tsai equations) ear strain in matrix in strain ear ear strain in isotropic in composite strain ear ear strain in isotropic in fibers strain ear gth of arepresentativegth volume element perature 81 - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 τ has been donefield been the has in micromechanics. The of subject discussedalso differentis at () 82 terms of include constituentterms parameters These properties. Toward in several of lamina we the thermoelastic to this, determine parameters need is considered effect their and grossbehaviormatrix on the is determined. of lamina the Micromechanics study is the which in interaction the of reinforcements the the and we know, constituents these combine together asingle as unison act in and entity. A composite isup made of two lamina constituents—reinforcements As matrix. and 3.3 ( ( () σσ σσ () () () σσ σσ () ( ( τ ( τ τ τ τ τ τ σσ σσ σσ σ σσ σσ 12 12 12 12 12 12 12 C T C T 12 12 c m f m m 12 12 12 12 m C C T T T f f T ) f ) f m c Extensive work, reflected as by numerous papers research availableliterature, the in ◾ ◾ ◾ ◾ ) c c , f f f ult , , c , ult , , )

ult , )

◾ ◾ ◾ ◾ τ ult

ul ul τ ul ul ult

ul ul τ

Coefficients of moisture expansion (CMEs) Coefficients expansion thermal of (CTEs) parameters Strength moduli Elastic 23 t tm 23 C T tc tc , T C tf tf 23

, ,(

,( ,( m m f f ,( ,(

f ( INTRODUCTION c

(

τ m

τ 23

23 C C T C T f ) c ) ult ) ) ) ult ) ) ult ul ul ult ul

t t

Longitudinal (in alongitudinalLongitudinal plane) transverse and (in atransverse versely isotropic fibers (i.e.,transverseand strength)longitudinal shear compressive of strengths transversely isotropic fibers) tively, transversely in isotropic fibers (i.e.,transverseand longitudinal fibers)tropic verse compressive of strengths transversely isotropic composite) tively, transversely in isotropic composite (i.e., longitudinal trans and Ultimate tensile compressive and Ultimate stresses, respectively, (i.e., matrix in longitudinal transverse and tensileUltimate stresses, respectively, in Longitudinal and transverse and tensileLongitudinal stresses, respectively, matrix in transverse and tensileLongitudinal stressesfibers in Ultimate sh Ultimate Ultimate longitudinal transverse and compressiveUltimate stresses, respec sile of strengths transversely isotropic composite) transversely isotropic composite (i.e., longitudinal transverse and ten plane) stresses, shear respectively, matrix in plane) stresses, shear respectively, fibers in strengths)shear transversely isotropic composite (i.e., longitudinal transverse and longitudinal stresses, transverse and shear respectively,Ultimate in plane) stresses, shear respectively, composite in tensile compressive and of strengths matrix) matrix ofstrengths transversely isotropic fibers) transversely isotropic fibers (i.e.,transverseand tensile longitudinal longitudinal transverse and tensileUltimate stresses, respectively, in Ultimate sh Ultimate Longitudinal (in alongitudinalLongitudinal plane) transverse and (in atransverse Ultimate te Ultimate transverse and compressiveLongitudinal stressesfibers in Ultimate sh Ultimate Longitudinal (in alongitudinalLongitudinal plane) transverse and (in atransverse Longitudinal and transverse and compressiveLongitudinal stresses, respectively, in longitudinal transverse and compressiveUltimate stresses, respec Ultimate longitudinal stresses, transverse and shear respectively,Ultimate trans in nsile isotropic stress in fibers (i.e.,tensile of strength iso ear stress (i.e.,ear strength) shear of isotropic fibers ear stress (i.e.,ear strength) shear of isotropic composite ear stress in matrix (i.e., matrix stress in ofear strength shear matrix) Composite Structures Composite ------Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Chamis and Sendeckyj [ Sendeckyj and Chamis survey of A detailed various approaches lamina. isof provided aunidirectional by to develop models for prediction the especially elastic of various parameters, moduli, models have of been keen interest research several and approaches have adopted been for References instance, hexagonal, etc.),inclusion fiberalignment, of voids, etc. models relate of to different fibers array geometrical assumed rectangular, (square, models ofmechanics materials-based have past. the Several proposed in been of these (see micromechanics in Section 3.4.1) relaxed/modified suitably are and a number of Some between constituentsinterface the of common assumptions the is maintained. expressions for elastic Typically, parameters. continuity the of displacement across the arepresentative energyin and balance volume element (RVE) to derive desired the tions (see, for References instance, 8 ply design of preliminary pressure the vesselsin [ but due of simplicity used properties to their alamina the still underestimate they are Poisson’s effect provided to models assumed be These are typically by matrix. the vided completelyand transverse and by stiffness fibers.the shear hand, the other On pro it are strength longitudinaland is that assumed stiffness lamina; unidirectional is ignored of for strength and longitudinal a the matrix stiffness estimating the and into asimple classificationas follows: Thus, for of sake the convenience of discussion, models micromechanics put the be can ciated with rigorous complex treatmentand expressions. graphical and mathematical principles of elasticity models, they, semiempirical and asso typically the are barring involve grossly assumptions. simplifying rest of The approaches on based the the are models.semiempirical models netting modelsThe mechanics of and materials-based solutions,and statisticaltheories, methods, finite elementmicrostructure methods, self-consistentals, models, on based principles, techniques bounding variational exact levels of many texts treatmentin of mechanics on the composites [ Micromechanics of a Lamina a of Micromechanics discuss the mechanics of materials models in detail for all the parameters listed above. listed parameters for the modelsdiscuss of mechanics detail the all materials in overall aproduct in an design idea we required environment. shall mind, in With this 23 book;of for reviews, this in-depth refer can interested readers to References process-related factors into account empirical by taken variables [ incorporating are models, semiempirical complexityIn mathematical the effects the is and reduced of elasticbehavior influence theally lamina are unreliable to ignored, leading estimates. they design have the in utility of aproduct. limited Also, many variables actu that randomly with position.vary models, these All however, somewhat are complex and allowedare are relaxedthe and elastic array to parameters fibersofin regular aligned restrictions statistical methods, the the elastic bounds on the In to obtain parameters. of exact the self-consistent method is the model. Variational principles employed are techniques, including such methods numerical finite the element as A method. variation ofgeneral assumptions frame (see Section 3.4.1) is formulated solved and by various and the bibliographies the and providedchapter, this therein. In we attempt to provide shall Elasticity-based models involve more rigorous behavior treatmentof lamina (see, the models of mechanics The materials-based involve too grossly assump simplifying Netting models highly simplified are modelsin thebondwhich between the fibers ◾ ◾ ◾ ◾ An exhaustiveAn discussion of models the available is beyond literature the in scope the ◾ ◾ ◾ ◾ Semiempirical models Semiempirical Elasticity-based models models of materials-based Mechanics Netting models 6 11 ]; approachesanalysis, these netting are of mechanics materi – 20 ). exact elasticity an an In method, problem the within – 10 ). Averaged force in used are stresses strains and 7 ]. 1 – 5 ]. Micromechanics Micromechanics ]. 6 , 22 , and , and 21 83 ]. - - - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 stresses. is present interphase Also, betweenthematrix. and interface fibers at an the the ment is not Similarly,have perfect. can have can voids matrix the lamina initial the and (transversely isotropic) fibers.align are their andFibersgenerallyrandomly spaced modelsmechanics ofthe anisotropic materials-based discussed accommodate here can direction-dependent. next the elastic we As in are see strengths section, and shall moduli highlyanisotropic.are fibersTheybe considered transversely can as isotropic their and tions of various models. For example, are glassisotropic, fibers aramid and but carbon approach for elastic the moduli. A brief discussion is provided also elasticity on the approach semiempirical the and 84 and matrix. However, matrix. and of acomposite manufacture the deviations during laminate, do weAs know, two constituents—fibers acomposite is up made of primarily material 3.4.2.2 homogeneousness, fibers across the matrix. or uniform are of parameters these all parameters— stiffness we three the need E For isotropic fibers three— reducessuch to as glass,the number parameters of stiffness isotropic. For fibers, these we need parameters:five stiffness highlytheyanisotropicandbe considered are fibers transversely can as such as carbon ics would depend restriction on respect of the in behaviorsCertain ofand fibersmatrix. experimentally for micromechan use in determined to be number of parameters these experimentally.are determined The elastic ofThe strengths and and fibersmatrix moduli 3.4.2.1 follows: as variables are of anumber of terms in basic parameter variables. micromechanics desired the These general irrespective of procedure, modelThe micromechanics the is to used, express 3.4.2 assumptions follows:These as restrictions and are interface. itsthe and constituents, tions respect of in lamina, is, that and fibersmatrix, Micromechanics models on based anumber of are assumptions simplifying restric and 3.4.1 3.4 f

, ◾ ◾ ◾ ◾ ◾ Some of some not restrictions and relaxed realistic the deriva of are the are them in ◾ ◾ ◾ ◾ G

◾ ◾ ◾ ◾ ◾ ◾ ◾ ◾ ◾ f

Mass fractions ofand fibersmatrix Volumeand fractions ofvoids fibers,matrix, Densities ofand fibersmatrix ofStrengths and fibersmatrix Elastic of moduli and fibersmatrix interaction (iii) zone. is, nothat fiber–matrix interphase, has (i) between interface perfectbond,and fibersmatrix The (ii) no voids,and is (i) matrix homogeneous,The (ii) isotropic, (iii) elastic, linearly (iv) and void-free. (v)spaced, (vi) and perfectly aligned, void-free. are fibers(i) The homogeneous, (ii)linearly elastic, (iii) isotropic, (iv) regularly tropic, (iii) elastic, linearly (iv) and stress-free. initially is (i) lamina macroscopicallyThe homogeneous, (ii) macroscopically ortho

, and and ,

BASIC MICROMECHANICS BASIC Assumptions and Restrictions Micromechanics Variables Elastic Moduli and Strengths of Fibers and Matrix and ofFibers Strengths and Moduli Elastic Volume Fractions ν f

. On the other hand, all common matrix materials are isotropic are for materials which common matrix all other hand, the . On E m

, G m

, and and , ν m

. Further, under the restriction under. Further, the of E 1 f

, E 2 Composite Structures Composite f

, G 12 f

, ν 12 f

, and and , ν - 23 f

- - - - . Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Micromechanics of a Lamina a of Micromechanics FIGURE 3.2 FIGURE ofume voids volume total to the of composite. Thus, volumetotal of composite, voids and volumetheratio fraction isas defined the of vol Similarly, volume matrix theratio fraction isas defined the of the volume to matrix of of volume the of the composite fibersthein total to volumematerial of composite. and voids.matrix, Fiber parts—fibers, volumeoffraction three the ratio as is defined voids volume and occur total Thus, the introduced. are of a composite consists material in in fiber theoretical volume maximum the fraction,with as aviewshown to determining acompositeare In packedrandom fibersin a fashion. material, a lamina. However, Poisson’s ratio. fiber It is maximum to useful knowvolume theoretical the fraction of such properties as greatly influenceseter that longitudinal modulusandmajor lamina array. Fiber volume expressed be fractions can as where Figure 3.2 For ideal composite an v material, Dividing sides the both by v It is clear that v v v v V V We subsequent the in see shall sections fiber that volumefraction - is a key param V c m f v m f v

volume of matrix of volume total volumetotal of composite material volume of fibers of volume fiber volumefraction voids volume fraction matrix volume fraction matrix volume of voids of volume

, let us consider two regular arrays of fibers—square array and triangular triangular and, let array us consider of two arrays fibers—square regular Schematic representation of fiber packing. (a) Square array. (b) Triangular array. array. ofpacking. (b) fiber (a) representation Square Schematic Triangular (a) V s f == Fo v v c rs , we get c f d ,, quare vv VV fm V fm v VV ++ m

++ = fm array,

+= V (b) v v v

m = c vv

V 0 and we0 and get vc v = V an 1 = s f

d 1 =

π 4 V d s v 2 2 = d

v v c v

(3.5) (3.2) (3.3) (3.4) (3.1) 85 - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 tions withcross fibers are sectionof circular and 86 of mass composite. total to the of matrix Thus, composite Similarly,theratio material. fraction mass isas defined themass of matrix Fibertheratio fraction mass isas mass defined themass of of of the total to fibers 3.4.2.3 where where ρ are fiber diameter and fiber diameter fiber s are spacing,where respectively. dand volume. for Then, composite, the the write following:we can fibers,andmatrix, where ( It is clear that w w w W Substituting Equations 3.12 3.14 through Equation in 3.11, we get W For maximum fiber Forpacking, maximum Now, we product know the that of density volume and that in contained mass is the c m f m f

V c matrix mass fraction mass matrix fiber mass fraction mass fiber mass ofmass composite the material mass ofmass matrix mass ofmass fibers , ρ Mass Fractions Mass f ) max f , and ρ , and is the theoretical maximum fiber maximum volume theoretical is the fraction. m are densities are of composite, respectively. fibers,andmatrix, Fo rt Fo Fo riangular rs rt quare riangular d ρρ

= cc array, array, vv ww

s wv cf . Thus, theoretical maximum fiber maximum volume. Thus, theoretical frac wv =+ W W wv =+ mm ff m cc array, f = = = = = ff () () ρ ρ V ρ V w w w w fm fm m c c f c w m f ρ V

m mm f v

== == = π

4 23 23 π π d 07 2 s . 2

9 09 .

1 Composite Structures Composite

(3.15) (3.12) (3.10) (3.13) (3.14) (3.11) (3.6) (3.8) (3.7) (3.9) - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 responses of RVEs the to applied identical loads analysis thusthe are and of RVE an to each other, complete it Further, the we the is that presumed obtain can lamina. following: tion, we get expressions the foras fractions mass forand fibersmatrix FIGURE 3.3 FIGURE the Let and regularlyaligned. fiberas spacings taken straight be are fibers The strengths. RVEAn is considered expressions for obtaining of various elastic the and moduli 3.4.3 Materials (c) Idealized Kaw,ment. K. A. from volume permission element. (Adaptedwith t Micromechanics of a Lamina a of Micromechanics l c c

in the width and thickness directions, respectively. directions, widththickness and the in RVE we an Then, take of size × Now, substituting Equations 3.12 3.14 through Equations in 3.9 3.10, and we get the Equation 3.16 expression of mixtures rule is the for density of composite. Dividing sides the both by v Then, substitutingThen, Equation 3.16 Equations in 3.17 3.18, and with simple manipula Taking voids zero, fraction as V

× , CRC Press, Boca Raton, FL, 2006.) FL, Raton, Boca , CRC Press, Representative Volume Element Representative

t c Figure 3.3 as shown as Figure 3.3 in

(b ) (a) Schematic representation of a unidirectional lamina. (b) volume Representative ele lamina. (a) of aunidirectional representation Schematic (a) a shows schematic the representation lamina. of aunidirectional 3 0 l c 2 c W 1 and using and Equation 3.1, we get W f f b ρρ m = c cf = b such by that placing RVEs the next repeatedly

= WV =+ 11 WV

() 1 +− m t ρρ f c

− fm = () () = () VV / ρρ ρρ ρρ

V fm fm fm fm ρ ρ ρ ρ f (c) t , and we, and get following: the / / / m c c c f VV ρ b fm m f c V V +

b f f c m V

l t c c Mechanics of Composite Composite of Mechanics b m /2 b f b m /2 b (3.20) (3.16) (3.17) (3.19) (3.18) c and and 87 - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 major Poisson’s ratio ( analogy isanalogy shown Figure 3.4 in with a system springs-in-parallel of springs with This parallel. different in stiffnesses Thus, matrix. RVE loading condition under this is shown in An load fiberdirection. the in under uniaxial us considerLet lamina aunidirectional 3.5.1.1 further simplifiedas shownin Figure 3.3 further is sufficientThethe the complete characteristics of is RVE lamina. for determining 88 tic constants—longitudinal modulustic constants—longitudinal ( ( lamina A unidirectional 3.5.1 3.5 where ( given by given area of fibers, the area Notes: E 2 Now, cross-sectionalof composite total area the RVE, the in It is for easy that to see zero voids fraction, F F ◾ ◾ Now, force by total volume the taken the element bythe and fibers is shared the F c ) normal to the fiber direction, shear fibermodulusdirection, to the shear ( ) normal

◾ ◾ c m f

suffixes “ suffixes belongsto stress fact on parameter the the that composite. to the Similarly, modulus of fibers, E E the longitudinalthe Young’s modulus of composite, In the general the nomenclature,In composite elastic represented by are moduli We system have coordinate aCartesian used lamina. the to theplane of is normal 3-direction and the fibersin theplane lamina, of transversethealongthe direction, to is fibers,which the 2-direction is normal system. coordinate rial Here, 1-direction longitudinal is which the direction, is

2 MECHANICS OF MATERIALS-BASED MODELSMECHANICS force by fibers shared the force by matrix shared the total forcetotal representative on the volume element , Evaluation of Elastic Moduli Longitudinal Modulus ( Longitudinal G 12 , etc. However, chapter, this in additional “ we suffix an add shall f ” and “ ” and A f , and the cross-sectional area of the matrix, cross-sectionalof matrix, the , and the area ν 12 m c Figure 3.3 ” are used for fibers and matrix, respectively. for used ” are and fibersmatrix, Thus, ). m is the Young’s is the so on. modulus and of matrix, b. E ) is an orthotropic body characterized by four elas body orthotropic characterized a) is- an 1 c AA FF ) E cf cf Ab Ab Ab 1 =+ =+ mm c cc ff ) alongtransverse fiberdirection, the modulus c. = = = t t Figure 3.4 t c c F c A

m O

-123 mate usually known the as 12 c ) in the plane of the lamina, and and plane the of) in lamina, the E 2 . The RVEa. The compared be can f is the transverse is the Young’s A A Composite Structures Composite c , the cross-sectional, the m , are, respectively,, are, E 1 c (3.25) (3.24) (3.22) (3.23) (3.21) E is is c 1 ” - , Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 3.27, we obtain Composite Materials Composite of M. Jones, R. Mechanics from permission with analogy. parts (Adapted in Springs-in-parallel (b) Composite Materials Composite modulus as follows: as modulus volume corresponding the fractions. expression Thus, we the obtain for longitudinal the in Hooke’sin law Equation 3.26: write and area fractions by lengtharea the

3.4 FIGURE composite, fibers, and matrix are equal, that is, ε equal, are composite, fibers,andmatrix where is no void composite, the in it we write also as can longitudinalthe modulus composite. of aunidirectional Under restriction there the that Micromechanics of a Lamina a of Micromechanics Now, are restrictioncomposite, under the the that elastic, fibers,and matrix we bring σ σ From Equation 3.25, we obtain In the above the In equation, we multiply can numerator denominator the the the in and The fibers and matrix are perfectly bonded, and thus, the longitudinal strains in the in thus,perfectlyandtheare bonded, longitudinal strains and fibersmatrix The σ Equation 3.29 for is one it avery of and popular is “rule mixtures” referred the to as 1 1 1 c f m

longitudinal composite the stress in material longitudinal matrix the stress in longitudinal fibers the stress in

(a) Representative volume element under uniaxial stress in the fiber direction. direction. fiber the in stress (a) volume uniaxial Representative element under , CRC Press, Boca Raton, FL, 2006.) FL, Raton, Boca , CRC Press, , second edition, Taylor edition, , second New York, &Francis, 1999; Kaw, K. A. σ 1c (a) (b) σ 1c t EA c l 11 c cc of the RVE and see that the area fractions are equal to equal fractions of are RVE the area the that see and εε EE b c σσ EE 11 cf 11 EE 11 cc cf =+ 11 cf AA cf =+ =+ =+ =+ Matrix Matrix EA Fiber VE 11 l c fm A A ff VE c f ff fm E () σ 1 m 1 − EA V 1 mm c A A mm m

A V = m c ε

f 1

m 1 σ

f /2

1c m = b f σ

b 1c ε m 1 /2 m . Then, from Equation from . Then, Mechanics of of Mechanics (3.26) (3.27) (3.30) (3.28) (3.29) 89 Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 lamina is a fiber-dominated property. fiber-dominated isa lamina fibers.Thus, the viz. maywe concludethe that longitudinal modulus unidirectional of a springs is controlled by of resultantstiffer three stiffness the the the spring,seen that in-parallel. From the springs-in-parallel analogy ( we as Further, mentioned before, RVE the with asystem compared be can of springs- modulus do not matrix havethe any appreciable impact oncomposite the modulus. modulus.changesthesematerials, in In fiber thematrix modulushigherthan is far results.experimental For most composite advanced polymeric the matrix materials, predictions for well made tally longitudinal the modulus with of by mixtures rule the tions. It is widely design in used analysis; and it is not only simple but reliable also as givestures relation asimple of constituent the terms linear volume in and moduli frac fraction for given data the Example in 3.1.rulemix theof figure, the seenfrom As Boca Raton, FL, 2006.) FL, Raton, Boca Tayloredition, New York, &Francis, 1999; Kaw, K. A. Materials Composite of M. Jones, R. Mechanics from permission with ogy. parts (Adapted in 90 FIGURE 3.6 FIGURE verse to Owing Poisson’sdirection. effect, it undergoes contraction longitudinal the in Under load the shown as the figure, the in undergoes grossRVE extensiontrans the in RVE shownAn as transverse the direction stressedin in 3.5.1.2 3.5 FIGURE from Example 3.1). Example from Figure 3.5 shows variation of the longitudinal the modulus w.r.t. fiber the volume

Transverse Modulus ( Transverse

(b) σ (a) Representative volume element under transverse stress. (b) Springs-in-series anal (a) (b) stress. Springs-in-series volume transverse Representative element under Longitudinal modulus by mechanics of materials approach (constituent material data data (constituent material approach of materials by mechanics modulus Longitudinal Longitudinal modulus (E )

2c 1c 10 20 80 30 40 50 60 70 0 . . . . 0.8 0.6 0.4 0.2 0.0 arxFbrMatrix Fiber Matrix (a) E m E t E c 1c 2 c = E b ) c 1f Fiber volumefraction(V σ V 2c f +E m l ≈ V c m Mechanics of Composite Materials Composite of Mechanics Figure 3.4 σ 2c f ) Figure 3.6a is considered next. b m /2 b f E b) of RVE, the it be can b 1f m /2 Composite Structures Composite σ 2 1.0 c , CRC Press, , CRC Press, , second , second - - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 tive moduli as tive moduli fraction, respectively. Thus, butwe nothing width fiber the fractions that are see volumevolumefractionandmatrix where Thus, matrix. extension of transverse transverse sumtotal the is extensions the in and fibers the in springs-in-series series. isin analogy shown This in RVE The withdirection. asystem compared be can of springs with different stiffnesses hand sidehand of above the equation, by product the of of RVE, length thickness the and l where Micromechanics of a Lamina a of Micromechanics Further, transverse strains in composite, fibers, and matrix are related composite,the respec in to transverseFurther, strains fibers,andmatrix ε Dividing sides the both by b ε ε Equation definition the in Bringing strains, 3.31of normal as written be can Δ Δ Then, substitutingThen, Equations 3.34 3.38 through Equation in 3.33, we get Now, multiplying numerator the denominator, and right- width the fractions the in in Δ 2 2 2 c f m f m c

transverse extension matrix the in gross transverse extension composite the in transverse extension fibers the in transverse strain in the composite the in transverse strain transverse strain in the fibers the in transverse strain transverse strain in the matrix the in transverse strain c σ , Equation 3.32 as written be can EE εε εε 2 2 22 22 c c cf cc bb ∆∆ =+ =+ cf =+ ε ε ε σ =+ 2 2 b 2 b b b m 2 2 f c m c c f f f = = = b b ff = = V c f f σ σ E E σ V V E 2 2 2 2 2 m f m c m c f f ε ∆

σ ε 2 E

m 2 m 2 m mm m b

b b V m c Figure 3.6 gross transverse b. The m

(3.36) (3.37) (3.35) (3.34) (3.38) (3.39) (3.32) (3.33) (3.31) c 91 t c - , Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 or wholeThus, well as and fibersmatrix. the as transverse for to the stress same is the compositecross-sectional normal the area a as 92 FIGURE 3.7 FIGURE hand, other the On fiberhigh volumefractions. Suchhigh fiber fractions, volume however,unrealistic. are 3.43, is shown in modulus of a unidirectional lamina is a matrix-dominated property. is amatrix-dominated modulus lamina of aunidirectional deformations.ily dependent matrix on the Thus, we may conclude transverse the that composite under transverse stress, grossdeformation is lamina primar of lamina the of springs is the influenced heavily the by springs weak (matrix).unidirectional In a From springs-in-series the ( analogy resentative volume with under transverse be asystem stress can of springs-in-series. 0.8 it and is very modulus. close mentioned as Further, matrix to the earlier, rep the from Example 3.1). Example from Now, we look RVE at the under transverse Figure 3.6 stress in notice , and the that The variation ofThe E Taking void content zero, as Equation 3.42 as written be can Using Equation 3.40 Equation in 3.39, we get

Transverse modulus (E2c)

10 20 30 40 50 60 70 80 0 0. Transverse modulus by mechanics of materials approach (constituent material data data (constituent material approach of materials byTransverse mechanics modulus 00 Figure 3.7 E 2 2 c c E rises at avery rises low up rate to afiber volumefraction of about with V with m .2 . The variation in the transverse modulus is rather sharp at transverse the modulus variation in . The issharp rather E f for given data the Example in 3.1, on based Equation 2 c E = 2 E c σσ 1 Fiber Figure 3.6b), resultant stiffness we the that see can 22 = 22 EV c cf mf 0. == =+ EV E 40 vo mf 2 c E V lume fr = +− EE EE f f 2 2 E EV + fm fm m 2 V EV σ ff action f () V E 2 2 1 + E E m m fm m 2 .6 f

2 E

f (V (1–V m

f )

f ) 0. 81 E Composite Structures Composite 2 f .0 (3.40) (3.42) (3.43) (3.41) - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 E fibers and matrix. (Note that transverse(Notethat deformationsandare matrix. fibersnegative.) Thus, due transverse the direction to Poisson’sin effect. in lus we and consider force RVE shown longitudinal as an the direction in under uniaxial these twoshownthese cases are in matrix moduli are of similar order, of similar are and moduli matrix posite modulus to the is equal MMCs ofandCMCs, For fibers the matrix. or and fiber transversethe equal, modulusare the com of moduli fiberandmatrix the if lamina, this approachthis leadsof to underestimate transverse the modulus. ized form as follows: as form ized fiber volumefraction, Thus, fiber loadingdirection. the in under uniaxial strain longitudinal normal majorThe Poisson’sthe ratio negative isas defined to ratio transversestrain of normal 3.5.1.3 3.8 FIGURE hand, fiber-to-matrixhand, modulusare ratios in large PMCs. very Typical E Micromechanics of a Lamina a of Micromechanics 2 c Figure 3.9 Figure Now, transverse deformation total the sumof is the transverse deformations the in The modelThe for major the Poisson’s for to that longitudinal ratio the modu is similar Another wayAnother to express composite the transverse modulus nondimensional the is in From above the equation, we if that, see is asimple one, but it does not well compare results. with experimental general, In

Major Poisson’s ( Ratio

. The lamina deforms in the longitudinal direction due to direct stress and stress deforms and longitudinal due the direction in to direct lamina . The Variation of transverse modulus with different fiber-to-matrix modulus ratios. modulus fiber-to-matrix different with modulus Variation of transverse ν

12 E2c /Em 10 E c 20 40 60 80 0 0 2 =− 0. c 00 / E m ε ε

= 1 2 Figure 3.8 c E c EE

1 or 2 (Typical mm ν c .2 with 12 = ∆∆ c E ) Fiber 11 T c 2 +− c E MMCs σ

=+ 2f = E . The mechanics of materials-based model of mechanics . The materials-based for 1 0. vo () /E c 2

40 E f lume fr ≠ m / E T 2 f E = 5 and CMCs f / 0 (Typical m

EV 2 = 1 E values are typically small. On the other other the values On small. typically are f / 2 2f E ∆

action ff E /E an m T m .6 m m

= da . In other words, in a unidirectional words, other . In aunidirectional in

= 100 PMCs)

) (V 1 or 1 ll f ) others

0. E 81 2 f

= zer

E o m .0 , irrespective of the

2 c / E m plots for (3.46) (3.44) (3.45) 93 - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 gitudinal strain, Then, dividing both the sides dividingThen, the both of Equation 3.47 width with of the RVE, the tudinal strains in the composite, fibers, and matrix are all equal, that is, equal, all composite,are the in strains fibers,andmatrix tudinal strains and we can write Equation we and 3.46 write can strains as

94 where 3.9 FIGURE Composite Materials Composite major Poisson’s of Kaw, the K. A. from permission determination ratio. (Adaptedwith where Now,perfectlyare the bonded, longi and fibersmatrix restriction under the the that ε ε Deformations composite the in constituents the related and be respective to the can Δ Δ Δ Now, definition by ε 2 2 2 f m c T T T c m

transverse strain in the composite the in transverse strain transverse deformation matrix the in transverse deformation fibers the in transverse deformationtotal composite in transverse strain in the fibers the in transverse strain transverse strain in the matrix the in transverse strain

Representative volume element under uniaxial stress in the fiber direction for the the for direction fiber the in stress volumeRepresentative uniaxial element under ε , CRC Press, Boca Raton, FL, 2006.) FL, Raton, Boca , CRC Press, 1 c σ (or ε (or 1c 1 f or ε or b σ b c b 1c m m /2 t /2 b 1 c f m ), we get following: the b ε ε bb c 1 2 cc c c εε 22 =+ ν ν ν =+ 12 b b 12 m c f c f l c =− l ε + ∆ =− ε =− c ff 1 2 l f f c L ε ε ε ε ε ε 1 2 1 2 1 2 m m b c c b f f b m c

ε ε ε 2 1 2 m m b b

m c

+ ∆ /2 b f c T σ 1c b m /2 Composite Structures Composite σ 1 c ε 1 c

= b Mechanics of of Mechanics c

, and lon and , ε 1 f

= (3.50) (3.47) (3.48) (3.49) (3.51)

ε 1 m - - . Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Micromechanics of a Lamina a of Micromechanics where related to the shear stresses.related shear to the Thus, we express can deformations shear the follows: as volume corresponding to the fractions, we get FIGURE 3.10 FIGURE Thus, sumofis deformations the shear thematrix. and fibers the in stress shown as shear Figure 3.10 in For developing amodel for in-plane modulus, the shear RVE an is subjected to in-plane 3.5.1.4 fiber-dominated nor matrix-dominated. nor fiber-dominated not muchare different thus, each from composite and other Poisson’s ratio is neither Poisson’swhereas transverse the modulus Fiber matrix isand matrix-dominated. ratios ratio. We seen had before longitudinal the that modulus is a fiber-dominated property Mechanics of Composite Materials Composite of M. Jones, R. Mechanics from permission with parts in Francis, New York,Francis, 1999; Kaw, K. A. Δ Shear deformations are related to the shear strains and shear strains can in turn be be turn in can strains shear and deformationsShear strains related shear to the are Δ Δ Substituting above the Equation in 3.48 equal width notingfractions the are and that Equations 3.52 expressions 3.53 and of mixtures rule the for are major the Poisson’s For zero void content, c f m

shear deformation in the matrix the deformation in shear shear deformationshear composite the in shear deformationshear fibers the in In-Plane Shear Modulus ( Shear In-Plane

(a) Representative volume element under shear stress. (b) Shear deformation. (Adapted (Adapted (a) (b) deformation. stress. Shear volume shear Representative element under (b (a) ) t c Mechanics of Composite Materials Composite of Mechanics νν b 212 12 c ∆ ∆ νν τ cf 12c 212 12 ff =+ cc . The total shear deformation shear total volume the . The in element ∆∆ cf == G == l c =+ 12 cf γ γ l c =+ c 12 ) 12 VV b b fm VV c f fm ν G G ∆ τ τ ∆ ν () 12 12 c 12 12 1 m c c − f f

∆ b ∆ m b ∆ f b c m f m τ

f /2 12c

/2 /2 b

f , CRC Press, Boca Raton, FL, 2006.) FL, Raton, Boca , CRC Press, b m /2 , second edition, Taylor edition, , second & (3.56) (3.54) (3.55) (3.52) (3.53) 95 Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Equation 3.54, we get from Example 3.1). Example from are very similar to those for transversevery the similar modulus.are withAs approach equations These for in-plane modulus the shear lamina. of aunidirectional

96 FIGURE 3.11 FIGURE dominated property.dominated variation of Atypical G tional composite:tional equal, that is, that τ equal, b or m / Equations 3.60 3.61 and models the by are of mechanics the materials-based Under restriction is no there the that void, Dividing sides the both of above the equation with We all mayare stressescomposite, in shear note the here that fibers,andmatrix b c

V m , we get following the relation for in-plane modulus the shear of aunidirec

In-plane shear modulus (G12c) In-plane shear modulus by mechanics of materials approach (constituent material data data (constituent material approach of materials by mechanics modulus shear In-plane 10 15 20 25 30 35 40 12 0 5 0. c

00 =

τ 12 f G

= m

τ G 12 .2 12 m ∆ G . Then, substituting. Then, Equations 3.55 3.57 through in c 12 mm G G = b c G == 1 212 12 212 12 c Fiber = GV 12c c c γ mf = =+ =+ GV 12 0. G vo mf 40 G G m b V b +− lume fr GG GG V m f f f 12 12 GV 12 f f + + G G c 12 fm fm 12 with V with GV τ action 12f G ff f 12 12 G G G V () b 1 m (1 –V m m m m fm m m .6 b

(V m b f is shown Figure 3.11 in f f

) ) , and noting b , and that

0. E G 81 2 12 c , G f Composite Structures Composite 12 c is a matrix- also .0 f / b c

= .

V (3.57) (3.60) (3.58) (3.59) (3.61) f and and - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Micromechanics of a Lamina a of Micromechanics

dividing numerator the denominator and by ratio as ratio of fibersgiven is by only about takes 3%.whereas matrix the words,other about fiberstake 97% the theloadaxial composite,on the total of to each other. equal are Also, note that matrix Glass fiber is isotropicand replacewe can E Solution in-plane modulus.and shear sible composite longitudinal modulus, transverse modulus, major Poisson’s ratio, pos themaximum Consider crossand section the determine of fibersas circular forces ratio by the ofand shared axial matrix. fibers determine (c) and lamina Poisson’s ratio, in-plane modulus. and shear (b) Apply alongitudinal force on the G follows: as are E glass/epoxyFor constituent the properties aunidirectional material lamina, 3.1 EXAMPLE sharing takes place isfollows: takes as sharing modulus are obtained as modulus obtained are modulus,gitudinal transverse modulus, major Poisson’s ratio, in-plane and shear ν 12 m f For fibers of circular cross section, the maximum theoretical Forcrossvolume fibers section,fractionthemaximum of circular We fiberstake the that see In 31.67 taken thematrix. loadaxial by the times We longitudinaland fibers loads, in know longitudinal under uniaxial strains us applyLet a longitudinal force (a) composite the longitudinal Determine modulus, transverse modulus, major

with ν with =

1.4

GPa. Consider zero void content afiber and volumefraction of 0.6. f . Then, using. Then, Equations 3.30, 3.43, 3.53, 3.6), and respectively, lon the - G ν f E

E = 12 12 2 1 c c c c

76 = =× = =× F F

602 06 06 GPa, 1 636 06 614 06 1 m .. .( f .. .. () == ×+ ×+ V 76 fm EV EV ν mm f F F ff

+− 35 76 ax = 1 1 +− m f

== F × × 0.2, 0.2, (. 10 ((. (. = 10 10 10 1 14 c 36 −× 23 −× on the composite.on the ratio which in The load . EA . EA π 604 36 76 mm .. G .) ff 63 ε ε 60 63 67 × × f 1 1

A ). ) ) ×= = ×= 06 c 1 , we obtain the desired load sharing load desired , we the sharing obtain 9069 0

. f and E and 35 m f . .. A 64 5 66 30 =

f GPa, / = = A 167 31 c 33 84

2 70 . . . . f 24 with E with

0 V 0 E 4 f GP GP and and

m GP

= a a

a 3.6 f , A

G m

/ GPa, 12 A f c with G with

V ν m m . Then, . Then,

= f , and , and

0.3, 0.3, - 97 Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 98 obtained as obtained lus, transverse modulus, major Poisson’s ratio, in-plane modulusand shear are Using Equations 3.30, 3.43, 3.53, 3.61, and respectively, longitudinal the modu Solution whereas the matrix takes only about takes 1%.whereas matrix the words,other the loadaboutaxial composite, fiberstake on 99% the the total of place isfollows:takes calculatedas E

are givenare follows: as constituent the properties carbon/epoxyFor material aunidirectional lamina, 3.2 EXAMPLE m

Take afiber volumefraction of 0.6andzero void content. We see that the fibers take 100 times the axial load taken by the matrix. In Wetaken thematrix. loadaxial by the fiberstake times the that see 100 Under alongitudinal force composite, on the ratio which the in load sharing The corresponding elastic corresponding The for moduli fiber this volumegivenare fraction by d. b. a. c. =

Compare the elasticCompare the of with moduli carbon/epoxy those of the lamina Considerthemaximum cross section circular ofand fibers determine glass/epoxy lamina in Example 3.1. Example in glass/epoxy lamina Poisson’s ratio, in-plane modulus. and shear possible composite longitudinal modulus, transverse modulus, major forces byand shared axial fibersmatrix. major Poisson’s ratio, in-plane modulus. and shear Apply alongitudinal force ratio the of composite on the determine and composite the longitudinalDetermine modulus, transverse modulus, 3.6

GPa, G ν E ν E 12 12 m 2 1 G ν c c c c

E E = 12 12 =× = =× = 2 1

c c c c 0.3, 0.3, 9069 0 0976 9069 0 9069 0 9069 0 = =× = =× .( ...... E 603 06 06 636 06 614 06 1 F F G .. .( f .. ..

1 1 = m m f × ×+

×+ ×+ = == 240 240 02 14 36

1.4 EV EV 22 24 +−

1 ++− mm +− 35 76 GPa, +− +− ff

GPa. × × (. (. (. 10 10 × × 10 (. 10 (. ((. 10 10 10 14 10 36 − 14 36 . −× . −× . . E 240 604 36 .) 2 60 .) 62 6 9069 f 63 .. 9069

0935 9069 0976 9069 ). ) ) = ×= × ×× × ×= ×

24 24 06 2 ). ) 30 .. ) ×= .

4 44 145 6 ×= GPa, = = 36 302 03 = .. 32 73 . 100 . . 3 = = 0G 5 ν 926 69 082 10 12 646 26 GP

f .

Pa .G = . 0 GP a

99 0.3, 0.3, GP GP a Composite Structures Composite Pa

a a G

12 f

GPa, - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 and in the form the of in and composite well. as material Consequently, lamina aunidirectional in of load.nature to corresponds aspecific combination parameters strength andof direction loading composite of Each of of strength characterization these aunidirectional lamina. failure. There are five ( failure. are There parameters strength it subjected stress that be can to before maximum is the of strength amaterial The 3.5.2 Micromechanics of a Lamina a of Micromechanics The fibers and matrix haveand fibersmatrix individualtheir as The own characteristics entitiesfailure dominated, and the major the and Poisson’sdominated, ratio to is fibers neutral matrix. or matrix-are istransverse fiber-dominated,in-planeandthe moduli lamina shear largely In other the uninfluenced. words, longitudinal modulus unidirectional of a forcements only major marginally. the hand, other the Poisson’s On ratio remains transverse modulusby in-plane rein modulus the increased the shear and are modulusthegitudinal of glass fiber higher carbon than is fibermodulus. The is lon case moreof the increase the as prominent in carbon/epoxyThe lamina by longitudinalthe is reinforcements. modulusgreatly increased the of lamina the Note: 3.1 glass/epoxythe previous the in example lamina is given Table in of fibersgiven is by

A comparison ofA comparison elastic the ofwith those moduli carbon/epoxy of the lamina elastic corresponding The for moduli fiber this volumegivenare fraction by theoretical Forcrossvolume fibers section,fractionthemaximum of circular

Evaluation of Strengths From above, made comparison the wethat find w.r.t. modulus, thematrix E Modulus Elastic 3.2) (Example Moduli ofElastic Comparison TABLE 3.1 ν G E 12c 2c 1c 12c G ν E E 1 12 2 22 1 c c c c =× = = =× Absolute 9069 0 09240 9069 0 093 9069 0 9069 0 Value 47.0 .. .( .. .. UD Glass/Epoxy Lamina 0.24 3.3 8.4 ×+ × () V 03 14 fm Matrix Property As aRatiow.r.t. 661 24 22 +− +− +− ax × × (. (. (. 10 == 10 13.1 10 14 0.8 2.4 2.3 36 −× . . 23 0924 9069 0 π 9069 .) 0922 9069 9069 Table 3.2 Table 9069 0 ). ) ) . ×= × ×= Absolute 303 03 Value 36 145.4 UD Carbon/Epoxy Lamina .. 0.3 3.2 7.4 ) to be evaluated) to be for complete = =

571 15 92 1 99 217 .G . . 8 Matrix Property As aRatiow.r.t. Pa GP 40.4 1.0 2.3 2.1 a .

- - 99 Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 The corresponding stress in the fiber the stress corresponding in is tensile longitudinal The of strength fiber and the tensile strength of matrix, tensilethe of strength matrix, 100 follows: as strains limiting the it and wouldstrain denoted be as failure or matrix strain matrix place is maximum takes failure referred the to as matrix at which failure. strain The till linearly increases strain the tensile matrix, the stress in increasingSimilarly, under iselasticagradually and linearly failure till matrix the and it or strain fiberfailurefiber would be as denoted strain maximum the to as takes place at whichis referred fiberfracture failure. strain The till linearly increases failure. we Thus, as tic till apply increasing tensile gradually afiber, stress in its strain restriction,theperindividual linearly are As our equal fibersstrengths. fiberof is elas tion of elastic addition moduli. In to those assumptions restrictions, and we assume that Welamina. anumber made of assumptions simplifying restrictions and for evalua the - assumptions for made development are the of models for of strength the predicting a of characteristics quite be ure acomposite involved. can lamina However, simplifying have interface the and own their individual characteristics. failure aresult, As fail the with cracking. pullout, fiberfiberfracture ture, matrix, Fibers,matrix fiber debond, and under longitudinal tensile fiber load lamina frac are aunidirectional modesfailure in betweentwo.interface thethe and of The itspossible constituents—fibers andmatrix of characteristics failure acompositeThe depend characteristics failure on the lamina 3.5.2.1 more complex are ofstrengths those for alamina than moduli. aresult, As models concerned. the local imperfections as far are for evaluation the of extent. may Stiffness considered be with asmoothening aglobal as effect parameter as imperfections,These however, same to characteristics the do not stiffness the affect is highly sensitiveof alamina to local imperfections such voids, as etc. kink, fiber under different loading conditions, different found. be modes failure can failure The togetherand elastic with the fiber of moduli and fibersmatrix fraction. volume is governed (andrectional lamina by strains failure the strengths) ofand fibersmatrix The mechanics of materials-based modelmechanics ofThe materials-based for longitudinal the of strength a unidi Strength Parameters of a Unidirectional Lamina ofaUnidirectional Parameters Strength TABLE 3.2 In-plane shearstrength Transverse compressive strength Transverse tensilestrength Longitudinal compressive strength Longitudinal tensilestrength Strength Parameter

Longitudinal Tensile Strength Tensile Strength Longitudinal () σ () m T () σε () σε ε 11 T ul m m T f T Nature ofLoad Applied t . The strengths of strengths constituents the . The related to are Shear force Compressive force Tensile force Compressive force Tensile force ult ult ult and the corresponding stress in the matrix is matrix the stress corresponding in the and = = () σ () () 1 T c T m T f ul t ult ult E E m 1 f

In theplaneoflamina Normal tothefiberdirection Normal tothefiberdirection Along thefiberdirection Along thefiberdirection (in theplaneoflamina) (in theplaneoflamina) Loading Direction Composite Structures Composite () () σ ε (3.63) (3.62) 1 1 T T f f ult ult - - - - . .

Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Micromechanics of a Lamina a of Micromechanics matrix strain: (i) strain: matrix FIGURE 3.12 FIGURE ina, then giventhen by alone.load longitudinal ofby The strength compositeis matrix taken the is the lamina like holesare crossthe fibers in section the and total the composite of fractured lamina composite same the imply stress does not necessarily of failure composite. the The However, strain. at strain matrix in increase matrix in instantaneous increase an this effective implies in a Fiberdecrease fracture and cross-sectional of lamina the area tensileoccurs. load,and fiber fracture the tensile fiber exceeds strain its strain failure At avery low of strength lamina. to the the tribute fiber volumefraction, under small sion in parts from R. M. Jones, R. from sion parts in Case 1 Case separately. is given by longitudinal point, the At tensile of this strength composite of the strength matrix. the tensile to the is longitudinal equal the tensile and of strength lamina the but matrix pure fiber volumefractions.theWhen fiber fraction volume zero, is the composite is nothing under different us checkLet longitudinal the lamina tensile of strength aunidirectional New York, 1999; G. J. Kelly A. and Davies, Metallurgical Reviews () εε 1 T f Now, two possible are there relative fiber cases ofstrain maximum maximum to As we gradually increase the fiber we As the increase gradually volumefraction,initially, hardlythe con fibers V ul tm f

< < :

() ( () V εε T f (b (a) 1 T ) f min ult ) . (a) Strength of a unidirectional lamina. (b) Stress–strain curves for a unidirectional lam for aunidirectional curves (b) Stress–strain lamina. . (a) of aunidirectional Strength ul . (c) Stress–strain curves for a unidirectional lamina, lamina, for aunidirectional curves . (c) Stress–strain

s tm s V s T 1c s T 1f < f T Mechanics of materials model for the longitudinal strength of a unidirectional lamina, lamina, of a unidirectional strength longitudinal for model the of materials Mechanics m T 1c s T min ult m () () ult ult εε ult 1 T T ult f 0 V ult ul A f tm e ( T 1f < cri Figure 3.12 O () ult Mechanics of Composite Materials Composite of Mechanics V Fiber e T Composit f () ul σσ t and (ii) and Matrix 1 T c () e ult σσ ) m T e 1 T c = 1.0 ult ult () B () = εε m T s 1 T (c) T s () 1c f ult T 1c ul m T ult () tm 1 ult > s ult = − s s T 1f = T T m 1c ()

V s ult T s 1f ult ult f T s T m , 10(37), 1965, 1–77.)

ult ult ult , second edition, Taylor edition, , second & Francis, V . Let us consider. Let cases these V e f

f 1 –V 1f T > +

( ult V e f f e T 1f ) min Fiber Composite ult . (Adapted with permis . (Adaptedwith Matrix E m e

1 –V m T ult f (3.65) (3.64) 101 - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 OB from the matrix. Note that stress in the matrix at the point of maximum fiber strain is fiber point at the ofstrain maximum matrix Note the stress in that matrix. the from tensile contribution ofis whereasthe strength composite second the the term lamina, fractions above ( of composite is the entirely alone.volume contributed lamina At fiber by matrix the 3.65 region is valid. Note this in of that fiber volumefractions,the longitudinal strength 3.65 is applicable, for and volume fractions above ( stress in the composite. the stress in Beyond point, for this V same is at the readjustmentthere increases longitudinal and load strain the in sharing respectively. of Irrespective fiber the volumefraction, first.thethispoint, fails fiber At ( case case Mathematically, ( V very close are strength compositethe to each other. matrix for and hand, other the On the case,this of loadstrength composite In the and failure. due finally matrix to fails introduce the concept of minimum fiber concept introduce the ofvolume minimum fraction, ( volume fraction (i.e., at matrix)low and pure fiber volumefractions. point, this letAt us providedare for Thus, properties. Equation 3.65 better valid to be has for zero fiber obviously,Quite very contradicts principle the this of composites where reinforcements longitudinal tensile is given then strength by of composite well considered as the be strain can failure its the and as lamina strain to completeleads fiberfracture Thus, the the composite of failure failure fiber lamina. beyond is and strain matrix in its increase strain fractions, failure this fiber volume instantaneously. increases strain However, matrix the and occurs fracture at high 102

() valid for fiber all volumefractions; for volumefractions lower ( than above, indicated As lamina. not of equations these strength aunidirectional are dinal 3.66, expression the one obtains for fiber critical volumefractionas Equation in strength with by strength matrix Then, replacingof matrix. lamina the the we introduce another parameter referred to as the critical fiber critical referred the to as we parameter volume introduceanother fraction, ( for fibersbe the to aboveeffective strength increasingthethat in lamina thematrix, of V ε f 1

T f > Note that the lamina strength at ( strength Note lamina the that In Equationthe 3.66,contributionIn is term thefrom the first to the fibers longitudinal The model forThe for the longitudinal the lamina, tensile of strength aunidirectional Equation 3.65 addition implies the that strength! of reduces fibers lamina actually Equations 3.65 3.66 and represent micromechanics-based model the for longitu the f ) represent the lamina strength at fiber strength represent lamina volume the fractionsbelow and above ( cri

ul () ( is the fiber is the volumefraction abovethat is than morestrength thewhich lamina εε tm V E 1 T f f ) min ul . tm , the readjustment, the fiberincreases of after immediately failure load sharing < () T V V f ult ) f ) min , is pictorially explained in min too, once fiber the tensile exceeds fiber strain its strain, failure is obtained by is solving obtained Equations 3.65 3.66 and as () () σσ V 11 T fm c ult () in V = = fc () () ri σσ T V = m T f f ) () ult () ult min () σε σε σε VE 11 T m T + is lower Thus, of strength matrix. the the than m T f ff + () ult ult ult () 11 − T − − ε f 1 T () f () () Figure 3.12

ult < 1 T 1 T ult T f −

f f ( V V ult ult ult () f mf f ε ) ) E E E T () min min 1 f m m m − , Equation 3.66 is applicable. , the matrix continues to take matrix , the ult V

. The line segments line . The V f E ) min m

, below which Equation

Composite Structures Composite V f ) min , Equation AO V (3.66) (3.67) (3.68) V f and and ) f ) min cri - , . Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Micromechanics of a Lamina a of Micromechanics tinue to take loads till the strain reaches the fiber ultimate failure strain, thatthe is, reaches fiber the ultimate strain, failure strain the loadstinue till to take lamina is givenlamina by longitudinal the tensile and of strength the tensile to the of strength is matrix equal the longitudinalcomposite the tensile and of strength lamina the but is nothing matrix pure ferent fiber volumefirstfractions.case,the in As whenfiber fraction volume is zero, the Thus, we under dif check longitudinal the lamina tensile of strength aunidirectional for procedure developmentThe the first one. to the case ofis similar model the this in (c) Stress–strain curves for the lamina, V lamina, for the curves (c) Stress–strain 3.13 FIGURE 2 Case load. case, composite the of Also, that matrix. this the in is higher far than strength composite the and same at fails the beyond sharply strain failure strain the matrix the is given then by composite the and fails.longitudinalstrain of Thus, the strength composite the lamina is very steep; strain fiberin the exceedsincrease ultimate fraction,strain this its failure fiberAt a in fiber small increase strain. instantaneous volume an and of lamina the area effective the implies in cracking a decrease Matrix cross-sectional occurs. cracking exceeds matrix and tensile its tensile strain under strain failure small load, matrix the lamina, higher than a certain minimum value, ( minimum acertain higher than As we gradually increase the fiber the weAs increase gradually volumefraction, at a low fiber fraction, volume Note the similarity of Equation 3.70Note similarity the with Equation 3.66. At fiber volumefractions : () εε () 1 T εε f 1 T (b (a) ul f tm ) > σ σ σ σ ul

σ m m T 1c T tm T 1f T () 1c T > Mechanics of materials-based model for the longitudinal strength of a unidirectional of aunidirectional strength longitudinal for model the of materials-based Mechanics σ V T ult ult ult ult f () ult ul A t min 0 ε . (a) Strength of the lamina. (b) Stress–strain curves for the lamina, lamina, for the curves (b) Stress–strain . (a) lamina. of the Strength T m T ul ult Fiber t ( () O σε Figure 3.13 11 T c V Matrix ε f ult 1f T Composite = ult () f

> () m T ε σσ

) ( 1 1.0 T V ult c (c) f B ) EV ult min σ σ σ V 1f T m T . 1c T ff = σ σ f ) σ 1c T 1c T min ult ult () ult + ult , after matrix cracking, the fibers cracking, the con matrix , after ult m T () = = ε σ ult m T m Fiber T ε ε

m T 1f ult T ult () ult ult 1 Matrix − E ε E 1f T 1f 1f V V V ult f Composit f f + ε

σ m T ult e 1 –V f V f

(3.69) ( (3.70) V f 103 ) min - - . Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 and the composite the and for load. hand, other same at the fails the On beyond sharply strain the increases fiber failure the strain failure diately matrix after case case same stress in the composite. the stress in same For V point, isat the readjustment there increases longitudinal and loadstrain the in sharing respectively. of Irrespective fiberthis the first. At fails volumefraction,thematrix obtained by solvingobtained Equations 3.70 3.71 and as fiber volumefraction (i.e.,matrix), pure Equationit reduces to 3.69. Now, ( lamina is given then lamina by fiber stress ultimate thestress. exceedsThefiber strength the longitudinal of composite 104

OB continues load composite the to and take due finally to fails failure.fiber The model forThe for the longitudinal the lamina, tensile of strength aunidirectional First, we find the failure strains of fibers and matrix as follows:of weand matrix First, the fibers strains find failure Solution tions are readily calculatedas readily tions are

Equation 3.70 is applicable at fiber volumefractions lower ( than as follows: E follows: as are constituent the properties carbon/epoxyFor material aunidirectional lamina, 3.3 EXAMPLE represent the lamina strength at fiber strength represent lamina volume the fractionsbelow and above ( () Using Equations 3.67 fiber critical and 3.68, and volumefrac minimum the fiber is higherstrain. than failure strain failure We matrix the that see b. a. εε c. 1 T

f Study the stress, strain, and load-sharing characteristics at load-sharing characteristics afiber and Study stress, the strain, volume fraction of 0.6. of fraction 0.01. of fraction ume fraction. ume Study the stress, strain, and load-sharing characteristics at load-sharing characteristics afiber and Study stress, the strain, volume fiber volume minimum critical thefraction the and fiber Determine vol ul tm > () 1 f T

= ult

375 () , is pictorially explained in V () V fm

GPa, fm in in = = () () () σ 230 0 3600 008 0 3000 72 ε ε 1    1 T T () m T f f εε +− () 11 T 20083600 008 0 72 σσ ult ult ult f f 11 T

c < ult = == == −× ult

− ( 3000 3600 7 000 375 V . = 72 3000 () f ) min () () , m T ε . MPa m T , the readjustment, the of imme load sharing T ult f ult 00 Figure 3.13    ult EE × . E , V 2 E 008 0 m fm f . m +

= () ε

3.6 = T

. The line segments line . The 0142 0 ult

GPa, . m V

Composite Structures Composite ()

σ f

V > m T f )

ult ( min V = f . Note at zero ) min 72MPa , the fiber, the V AO f - V ) (3.72) (3.71) min f and and ) - . min is is - , Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Micromechanics of a Lamina a of Micromechanics change and the total load is shared by the matrix alone, is, load that bychange total matrix is shared the the and follows: as stresses cross-sectionaling corresponding the with the areas posite follows: calculatedas are placeFiber is when takes 0.008. failure longitudinal the strain fraction, However, for of sake the illustration, let us consider V composite Very strength. low fiber volumefraction suchas impractical.1% is fiber the in increase volumefraction further and the would increase of matrix the 50% around toally 60%. Composite is invariably strength much that higher than fiberin the compositevolume increase fraction tensile further wouldstrength. tensile any additional and strength increase compositethe matrix the is higher than strength. fiberadditional in increase volumefraction the compositewould increase tensile ofstrength composite the istensile lower However, strength. matrix the than any tensile strength. fiberadditional in increase volumefraction the composite reducewould actually ofstrength composite the istensile lower Also, strength. any matrix the than and Immediately after fiber failure, load sharing goes through an instantaneous instantaneous an fiber through after failure, goes Immediately load sharing and composite byare calculated fibers, shared the by matrix, Loads multiply Just before and com fiberthe the in longitudinal failure, matrix, stresses fibers, us applyLet atensile force RVE on its the increase magnitude. gradually and considerus first Let an Then, unitcross-sectionalat RVE of area. fiber volume carbon/epoxy aunidirectional composite,In fiber the volumefraction is gener At afiber volume fractionhigherthan 1.45%,the tensile longitudinal of strength At afiber volumefractionbetween 1.42%and 1.45%,the tensile longitudinal Thus, at afiber volumefraction than less 1.42%,the tensile longitudinal V σ f σ σ

= 1 T 1 T 1 T m f c

0.01, of cross-sectional the are composite, areas fibers,andmatrix =× =× =× 0 602 8 28 3600 008 0 0 37 008 0 008 0 . .( .. F F F () 1 1 1 V m f c fc ==×= =× =× 00 ri 3000 28 8512 58 ., 5 003000 55 000 = F F F , .. .. 7 000 375 1 1 80 1 1 m ×+ 000083600 008 0 3000 f c = 20083600 008 0 72 ======A A 00 A = 852100 512 58 100 00 512 58 −× m 92 1 873 48 512 28 99 .( f c 10 −× () = = = 13 .( .( MPa . == 1 09 00 == . . . mm % 852100 512 58 MPa N% N% 05 9 1 .( 09 mm N% mm .( 2 .) 605 512 58 3600 9 2 N% ×= 2 N%

12 = = . ) ) 7 0145 0 f

= . .) )

0.01. ) .

MPa

- - - 105 Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 106

posite follows: calculatedas are Fiber failure takes placeFiber is when takes 0.008. failure longitudinal the strain at fiberThen, volumefraction, composite the in of strength composite, is the the is, longitudinal that the strength follows: as stresses point, the are At this strain. failure loading,ther reachesmatrix finally, thewhen fails strain thematrix capacity additional the has loads. to take fur On lamina Thus, the of matrix. the load. same the However, lower is still strain failure strain the increased than this and the corresponding stresses corresponding are the and are fibers,andmatrix Equation 3.65 get and composite the as strength of composite the at afiber volumefraction of 1% is 71.28 Just before and com fiberthe the in longitudinal failure, matrix, stresses fibers, us applyLet atensile force RVE on its the increase magnitude. gradually and previous the in As case, let us consider RVE an of cross-sectional unit area. us nowLet consider afiber volumefraction of 0.6. loading load is possible at level. fails this No further matrix the as So, stress the Wethat takes find at whenincreases theplace, failure fiberstrain longitudinal stresses to these corresponding are strains The σ σ σ 1 T 1 T 1 T m f c =× =× =× σ σ σ 1 T 1 T 1 T m 0 375 008 0 0 602 8 28 3600 008 0 008 0 f c .. . .( =× = =× ε () ε 00 0 00 ε 1 σ T 1 T 1 . .( T m f c 1 T c 3600 2 20 = == = σ 06 σ σ ult ., 1 0 T 1 9103 59 00 T 1 T m 3600 f c ,, .. =× ×+ 0 3000 000 == = == . .. = 10 01 72 7 000 375 ×+ V 0 8512 58 512 58 A ×+ A f = A 09 =

= m . . f c 8512 58 . 00 1 772 MPa (.

= = = 10 9 0.6, of cross-sectional the composite, areas 0 . 09 MPa −= 1 .00164 04 06 . . mm 3600 9 930 128 71 3600 99 MPa 17 28 71 01 04 × 8512 58 9103 59 mm mm .) ×= 2 ). . . ×= 2 2 6011 52 1811 3600

MPa MPa = ). 0164 0 . MPa

MPa. WeMPa. als can

MPa . Composite Structures Composite MPa

o use - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 are [ are under longitudinal compression lamina modesfailure associated with aunidirectional different more and from complexare those under longitudinal tension. than Typical under longitudinal compression lamina of characteristics strength a unidirectional The 3.5.2.2 Micromechanics of a Lamina a of Micromechanics fiber failure and the corresponding thecomposite andcorresponding fiber failure the composite is stress strength, fiber Thus, at this limits. volumefraction,the composite after immediately fails instantaneously beyond far increase strain stress and their place, matrix takes the stressesthese are alone. is, That matrix the be by diatelyto fibershared the point, load total is required after this failure. At follows: as stresses cross-sectionaling corresponding the with the areas ◾ ◾ ◾ ◾ usealso Equation 3.66 get and composite the as strength is, longitudinalthat the tensile of strength composite the is 1811.52 stresses corresponding are the and ◾ ◾ ◾ ◾ 2 Note that the strain in the matrix is toohigh. We matrix the that in find when Note strain failure the fiber that to corresponding strains is too high. The matrix Note the stress in the that To loading checkis possible, whether further let us consider imme instant an and composite byare calculated fibers, shared the by matrix, Loads multiply Shear failure Shear Transverse and/or interface tensile of failure matrix Microbuckling of fibersin shear Microbuckling of fibersin extension , 25

Longitudinal Compressive Strength Compressive Strength Longitudinal – 27 () ] σ 1 T c ult =× 3000 FF F F 1 1 1 m f c =× =× =× ε ε ε 1 T 1 T 1 T σ m σ 8152 1811 σ 3000 28 f c 1 06 T 1 T 1 == = = T m F F F f c .. .. 1 1 1 80 == = == c m 0 f 06 588 4528 +× .. 3600 == .. == == 0 8152 1811 52 1811 0 3600 008 0 06 815 100 52 1811 ×+ 00 815 100 52 1811 41 .. 04 . 00 1 (% 8152 1811 . 10 == == . . . . 809 36 99 1800 == 15 12 . .. .8 3600 4 815 100 52 1811 N( ) N( × 20 8152 1811 5 588 4528 8 N( N( () ×− σ . 1 C (. . . c 10 ul = N( MPa %) t %) MPa 64 258 1 . 8152 1811 6 %)

). %) =

MPa

MPa. WeMPa. can

- - 107 Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 can be used for used be can longitudinal compressive [ strength is associated with low fiber volumethefraction and following approximate expression undergoes extension of compression and failure type transverse This the direction. in ( microbuckling fibermay buckling take Local place either out-of-phasein-phase. or Out-of-phase fiber longitudinal compression,theytinyand fibers like columnstend the act to buckle. schematicallyare shown in Materials Kaw, K. A. from permission (Adaptedwith (d) failure. Shear of matrix. ure shear. in of fibers (b)- in fail extension. ofMicrobuckling (a) Microbuckling tensile fibers (c) Transverse 108 compositeapproximated by be the mode can [ this in ofmicrobuckling fiber theis andmore common compressive longitudinal of strength mode ( phase shear or the 3.14 FIGURE is given by duetransverse tensile to Poisson’s strain effect under alongitudinal compressive stress interface. Now, fiber–matrix or the sion of matrix exceeds tensile the ultimate strain the when due transverse the tensile to Poisson’s strain effect under longitudinal compres tional lamina is giventional lamina by At fiber high volumefraction (typically, These basic failure modes in a unidirectional lamina under longitudinal compression lamina basicThese aunidirectional modes failure in Transverse and/or ( interface tensile of failure matrix the Thus, in this mode of longitudinal failure, this the Thus, in compressive of strength aunidirec (a) (c) , CRC Press, Boca Raton, FL, 2006.) FL, Raton, Boca , CRC Press,

Typical failure modes in a unidirectional lamina under longitudinal compression. compression. longitudinal under lamina aunidirectional in Typical modes failure Figure 3.14 extensionala) is the which mode buckling in matrix the Figure 3.14 Figure 3.14 () σ εν 1 C 21 T c c () == ult σ ) in which the matrix undergoes shear. which type b) in matrix the This 1 C c ≈ 21 ult . In a unidirectional composite under aunidirectional . In lamina 2 cc ε V ≈ C (d (b f ) ) V 1 VE 31 − f G

νσ > () ff m V 21 12 E

− 0.4),in-microbuckling fiber occurs f 1 cc 1 1 c V

E ]: C 1 f m ]

Figure 3.14 Mechanics of Composite Composite of Mechanics Composite Structures Composite ) takes place c) takes (3.75) (3.73) (3.74) - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 nal compressivenal shown be composite can as aunidirectional stress in lamina of by strength fibers.dictated shear the the By applyinglongitudirulemixtures, theof compositethe the of failure andcomposite fibers shear the due fails to direct is strength ( failure where Micromechanics of a Lamina a of Micromechanics given by pointat the of fiberthe compressivelongitudinal failure, shear is thematrix in stress of that fiber. the is higher than Then, strain failure we put matrix arestriction the that nal compressivenal fiber the stressto in corresponding failure is 2( fiber shear it stress on and a plane occurs at 45°dinal longitudi longitudinal toThus, the the axis. -

preliminary design calculations. preliminary have good match results. with experimental relations Thus, these only useful in are compressionmodes in complex rather are values theoreticallypredicted and do not different in compression lamina ofstrength aunidirectional modes. failure failure The Note: tion Equation from 3.83. longitudinal Thus, the composite expressed be can strength as discussed Section be ite in 3.5.2.3, will we rela and ofmechanics use the can - materials given Equations in 3.30 3.53. and of transverse compos tensile the ultimate The strain σ σ σ The fourth basic fourth shear is the compressionThe lamina aunidirectional mode failure in E ume fraction as 0.6. fraction as ume longitudinal the compressiveDetermine Take of strength lamina. fiber the the vol We stress under alongitudinal shear longitu load the is half maximum know the that are givenare follows: as constituent the properties carbon/epoxyFor material aunidirectional lamina, 3.4 EXAMPLE The compositeThe elastic given are moduli by formulae of mechanics the materials 1 1 1 C C C m c m f

We section models presented this in for longitudinal the predicting compressive

Figure 3.14 longitudinal compressive matrix the stress in longitudinal compressive fibers the stress in longitudinal compressive composite the stress in 3.6 στ 11 C m

GPa, = 2 () σ () G 1 C c ) that is associatedd) that with fiber high volume case,fractions.this In 21 m ult

fu = =

lt 1.4 () EE στ     E νν mf EV 11

C GPa, c 1 12 / 1 f

ff ult = ff VV σσ

=+ 240 . Thus, we get longitudinal the compressive as strength 11 C +− c () ν +− σ 21 m =+ 1 EV

C ()

= c GPa, mf mf ult

C 2 () () ., ( 0.3, 1 f 1 fu VV = fm lt E E     τ 2 V 12 12 f σ

cc f = ν f     () C )      ε 12 ult 11

() 22 T 1 c +−

E = E −

1 ult       m GPa,

f 36 E E () f

2 m

f MPa, and and MPa, −

G V 12 f       V f    

fm     

() 22 ε () T σ

GPa, m T ult ult

= ν τ 12 12 72MP f f

) = ult

. Here, 0.3, 0.3, (3.77) (3.76) (3.78) (3.79) a - . 109 - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 ure of a lamina. As we As of know,ure alamina. suitable reinforcements longitudinal the greatly enhance eter. “first ply The is generallytransverse failure” thedue tensileof a to laminate fail 110 The transverse composite param tensileThe is acritical of strength aunidirectional lamina 3.5.2.3 obtained as obtained as obtained compressive of strength composite the as is obtained nal compressivenal of strength composite the as is obtained failure modes. failure the We compressivelongitudinal determine first shall differentper as strength Solution with the horizontal axis. horizontal with the graph thealmost figure, mode coincident the is this seenin failure strength;for as example, longitudinal mode failure compressive gives shear direct minimum the of designer above the the data, mayexperimental minimum use the results. this In However, data. dependrather material on experimental absence the in of any well predictions. with theoretical design apractical Thus, in scenario, one would results experimental do also literature, not the match in modes.ure reported As anyamongtherefigure, is hardly the differentcomparison resultsper as fail shownally in carbon/epoxy composite Example in 3.4 w.r.t. fiber the volumefraction is pictori Note: of composite the as is obtained Shear failure mode usingThen, Equation 3.77, longitudinal compressive of strength composite is Transverse tensile failure matrix of Fiber mode microbuckling shear in Fiber microbuckling in extensional mode () σ

1 C The variation of The longitudinal the compressive of strength unidirectional the c Transverse Tensile Strength TensileTransverse Strength ult = = () 804 4830     σ () 4 000 240 σ 1 C Figure 3.15 c 1 C c ult 30 03 06 03 ,. .M ult .. =× =× ×+ 20 : Using Equation 3.79, longitudinal the compressive strength ×+ Pa 23 06 () . As found. As above the in calculations well as the in as σ . 6 60 1 C c × ×+ () .( 3600 ult ε     ×− m T 06 . = 6 ., ult 10 ×− 10 ×× 1400 ( == − σ 31 (. 4 000 240 10 4 0 3600 000 240 ×− 2 T 3600 .) : The ultimate tensile strain of matrix is of tensile matrix ultimate strain : The 3600 c 6 . : Using Equation 3.74, longitudinal the 72 ) (. 6 ul , t = 6 : Using Equation 3.73, longitudi the ) 06 3500     ×+ (. 00 10 ) −     1 2. MPa

     64 2000 22 ). =     3600 =

, 4942 24 ,M 36 − MPa Composite Structures Composite 10       × Pa

60     × 02

- - - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 transverse extensions we of show transverse can strains, terms in that ous failure modes (Example 3.4). (Example modes failure ous which gives us gives which transverse extensionsthat is, Δ fiberandmatrix, the in ( lus modulus. transverse the approach discuss model and with model of amechanics the line materials-based for in resultingthe models complex. also are Here, however, we would adopt asimplistic effectsThe of factors these complex rather transverse on the tensile are strength and voids,ing and etc. models Theoretical have developed been using parameters. these fiber-to-matrix interface/interphase, strength, fiberfiber fraction, volume fiber pack the of strength influencingtransverse thematrix, theare tensileof strength strength transverseinfluencesthe tensile Amongthe severalstrength. factors responsible for factor is about two. However, concentration stress or strain is not only the factor that analysis fibers.Theoretical the around shows matrix the concentrationthis stress that with dense fiberindicating packing place concentration matrix stress the from takes in fibers. been It has from found observations thatcrackinitiationexperimental usually discontinuitiesas stress/strain concentrations and develop the around matrix the in lower betensile considered fibers the of The strength can parent matrix! the than be can fact, transverse lamina In properties. the tensile of strength aunidirectional composite. ofproperties aunidirectional However, it is not case for the transverse the 3.15 FIGURE Micromechanics of a Lamina a of Micromechanics Under transverse load, the transverse stresses in the fiber and matrix are equal, equal, Underare transverse load, transverse fiberandmatrix the stresses the in Let us considerLet RVE the for used expression deriving the for transverse the modu Using Equation 3.81 Equation in 3.80, we get Figure 3.6

). grosstransverse extension The composite the in of sumtotal is the Variation of longitudinal compressive strength with fiber volume fraction as per per vari as fraction fiber with volume compressive strength Variation of longitudinal Longitudinal compressive strength 100,000 120,000 20,000 40,000 60,000 80,000 0 . . . . 0.8 0.6 0.4 0.2 0.0 εε εε 2 T 22 T c Fiber shearfailure Transverse tensilefailureofmatrix Fiber microbucklinginshear Extensional ﬁbermicrobuckling c =+ =+ εε      11 22 T f T EE f VV Fiber volumefraction       fm fm E E = 2 m f ε − 2 T 2 T ()       1 V m − fm

     f 2 T c

Δ f

Δ m . Expressing these 1.0 (3.80) (3.82) (3.81) 111 - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 where the terms l where terms the Then, we get the ultimate tensile strain of the unidirectional composite weThen, of unidirectional tensile get the as ultimate strain the Thus, is ineffectiveinterface alone. transverse by such stress is matrix total borne the the that ( section cross approximations.fying For example, let us consider of fiberswith array circular asquare models developed be Other also can of strength appropriate matrix. simpli the making composite transversethe tensile tensile of the strength unidirectional as the same is the of tensile other words, equation model, the strength. In but is nothing this matrix in cross section in a square array. asquare in section cross matrix becomes equal to the ultimate tensile strain of the matrix, that is, that of matrix, tensile the ultimate strain to the becomes equal matrix square array of fibers, array square i.e., b 112 FIGURE 3.16 FIGURE materials expressionmaterials for E using RVE an lamina showndirectional in Thus, tensile reaches ultimate stress the of matrix. the transverse tensile matrix the stress in And the transverse tensile strength of the unidirectional composite transverse the tensileAnd of strength unidirectional the is given by In this mode of place failure, this when the takes failure in In transverse the tensile strain We fiber the that see can volumefraction the for in RVE Equation 3.84 represents asimple model for transverse the tensile of strength auni

Figure 3.16 Representative volume element for transverse tensile strength with fibers of circular circular of fibers with strength volume tensile Representative element for transverse c , t c are given dare , and in t () c σε ). Here, approximation we an fiber-to-matrix the make that 2 () 22 T c b εε in Equation in 3.84, it shown be can right-hand the that side c c c

2 T = c ult ()

σ ult σσ t σσ =+ c T 2c ) is given by 2 T 22 T =+ c c E lt ult cc      11 V c l c =      f 11 =− =       () E E       π 4 Figure 3.6 T Figure 3.16 m T 2 m E m E f      lt 2 ult m t cc d − f c ()            − σ 2 td       c T 2c

V t −       c fm d V      . By substituting of mechanics the () fm . The composite. The when fails the

           () T

Dia. =d T ult ult

Figure 3.16 (considering a

Composite Structures Composite εε 2 T m = () (3.86) (3.87) (3.85) (3.84) (3.83) m T ult - - . Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 where Micromechanics of a Lamina a of Micromechanics arelation to Equation weadopted and 3.84 obtain can similar follows: as complex. Asimplistic for with approach that line in transverse be tensile can strength pendently combination or in with one another. Clearly, is mechanism failure final the fiber-to-matrix and interfaceThese crushing, maymodes inde failure. failure occur compression. compression fiber They are of failure shear matrix, of failure matrix, underA number transverse ofpossible modes failure are lamina aunidirectional in 3.5.2.4 composite is obtained as composite obtained is given follows: as E are constituent the properties carbon/epoxyFor material aunidirectional lamina, 3.5 EXAMPLE () compositeunidirectional as 3.5.2.5.) of Section end which (See is identically tensile the of as strength matrix. same note the the at the () Substituting Equation 3.87 Equation in 3.86, we get is obtained as is obtained of matrix. tensile the ultimate strain the and transversethe Wecompositemodulus unidirectional the of determine first shall Solution Take tensile the of as strength matrix. same the fiber the volumefractionas 0.6. σ ε m C m T Then, usingThen, Equation 3.84, transverse the tensile of strength unidirectional the as is obtained of matrix tensile the ultimate strain The We use also Equation 3.88 can transverse the tensile of strength obtain the and Using Equation 3.43, transverse the modulus composite of unidirectional the

ul ult t Transverse Compressive Strength Compressive Strength Transverse

= ultimate compressiveultimate of matrix the strain 72MPa () σ 2 T c . Verify is the transverse the tensile that of strength lamina the E 1 ult f

2 () = c σ =× () =

2 σε T 240 c 261 7226 22 C 636 06 c ult .. ()

ult σσ GPa, =× ×+ 2 T () =+ c ε 72     ult m T E +− E 22 ult = c 2            f

× 12      (. 11 () 36 = == 22 10 −× . 36 −×

m T 22 3600 .       72 ult E E

10 GPa, () 62 2 m σ            ) f 12 × 2 C 06 − − c π . ul 00 .. E 2 60 t              m     V = = ×=

2. V = fm π

     91 226 7 f ()

.M 3.6 .G        272 02 C

GPa, ult Pa

MPa ()

σ 2 T f

ult = 36 MPa, (3.88) (3.89) 113

- Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 relation to Equation 3.88 similar follows: as debond assuming fiber-to-matrix at and the interface, a array obtain square we can which is compressive. Alternatively, consideringcross fibers sectionin a of circular too, right-hand the side strength, of Equation 3.89 matrix the as same the is actually 114 shear is complex. also shear Here, we adopt asimplistic for with approach that line in the in-plane case of in the interface. mechanism failure in transverse Like the strengths, possible the the and and strength higher shear of failure shear matrix modes failure are fibers critical. The cometoo under are stress, strength shear butshear possessfibers far fiber-to-matrix the interface and strength Under shear in-plane stress, matrix shear the 3.5.2.5 tional composite as is obtained As in the case of the composite, transverse in As the tensile of strength unidirectional the here composite as tional Equation 3.90 transverse the compressive obtain and of strength unidirec the 3.5.2.5.) of Section end the which is identically compressive to the equal (See of strength matrix. note the at compressive ultimate the and of matrix. the strain transversethe Wecompositemodulus unidirectional the of determine first shall Solution 0.6. as tion compressive the as same is the Take of strength matrix. the fiber the volumefrac () given follows: as are constituent the properties carbon/epoxyFor material aunidirectional lamina, 3.6 EXAMPLE σ m C Then, usingThen, Equation 3.89, transverse the compressive of strength unidirec the compressiveas ultimate is obtained The of matrix the strain Assuming fibers of circular cross section in a square array,cross fibers Assuming sectionin a square of circular also use we can From Example 3.5, transverse the modulus composite of unidirectional the is

ult In-Plane Shear Strength Strength Shear In-Plane = 108MP () σ 2 C c a E ult . Verify transverse the compressive that of strength lamina the 1 () f

σ = =× 2 T

c 240 261 7226 ult ()

=× σσ GPa, 2 C 108 () c ε     ult m C E () +− E τ 2 ult = 2 12      c       f

12 36 cu = 22 () = == . −×

226 7 lt m C 22 3600 108 .G ult

10 GPa,            × 12 06 − π . Pa .. 00 E 60            m ×=

3. V = π 36 13

3.6 3108 03        .M

GPa, Pa

() MPa Composite Structures Composite σ 2 T f ult

= 36 MPa, (3.90) - - -

Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 shear strain in composite in follows: strain as shear with simple and, manipulation, strains relatedtions we shear to the express are can the strength ofstrength composite the as is obtained γ is, that of matrix, the strain shear ultimate to the becomes equal matrix the in strain or strength ofstrength matrix. the that is, Δ deformationsshear thematrix, fiberand the in shownas in in-plane modulus. shear us consider Let RVE the subjected to in-plane stress shear Micromechanics of a Lamina a of Micromechanics 12 Then, multiplyingThen, above the in-plane with modulus, the shear in-plane the shear Taking the mode of failure as the shear failure of the matrix, at failure, the shear at shear failure, the Taking of failure mode matrix, ofshear the the the as failure given follows: as are constituent the properties carbon/epoxyFor material aunidirectional lamina, 3.7 EXAMPLE shownIt be can right-hand the that side of Equation 3.96 shear the as same is the Using Equation 3.93 Equation in 3.92, simple after manipulations, we get ite and the ultimate in-plane shear strain of matrix. the in-plane strain ultimate ite shear the and in-planethe composWemodulus shear unidirectional the of determine first shall Solution Take of strength matrix. shear the fiber the the as volumefractionas 0.6. ( The shear stresses in fiber and matrix are equal, that is, τ equal, are stressesfiberandmatrix in shear The m τ

= m )

ult ( γ

m = ) ult

35 . Thus, . Figure 3.10

MPa. Verify that the in-plane shear strength of the lamina is the same same is the VerifyMPa. of in-plane strength lamina the shear the that E 1 f

= () . Total deformation shear volume the in element sumof is the τγ

240 12 () γγ cu 12 γγ

γγ cu GPa, lt 12 21 12 12 12 γγ =+ c cf 212 12 lt =+ =+ cf γγ =+ G G 21 12 12 12 =+ 12      11      12 11 ff c GG f

     11 =       VV       G

G fm G 22 G b b 12       = 12 m c f G m

f G GPa, f γ 12 − m − f γ mm − 12       ()       V G 1 m V − fm m       fm b      b

     = m c () fm 12 f     

1.4 () c

ult =

GPa, 12

ult Δ f

= f

τ () σ

12 Δ 2 T m m f , which gives us . Shear deforma. Shear ult = 36 MPa, (3.96) (3.95) (3.94) (3.92) (3.93) (3.91) - 115

- Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 now derive expression an coefficient for longitudinal the thermal of expansion based givesCTE arelativeLet fiber change direction.us the in dimension in of lamina the 116 For isotropic it an isas material, defined of is ameasure relative CTE The change dimensions in w.r.t. change temperature. in 3.5.3 where ite is obtained as ite is obtained composite is obtained as composite obtained is strengths are the same as the corresponding matrix strengths. matrix corresponding the as same the are strengths found,composite composite already the as series, and the is in by weakest the link springs-in-seriesThe is analogy applicable of failure the casesthe and these in assumption inherent on composite the the failure. that is failure due matrix to the transversestrength, compressive based are in-plane strength and strength, shear design calculations. models the very Further, preliminary in for transverse tensile models of mechanics The materials-based fordata. used be at strengths, best,can always strength advisable designer the experimentally determined that utilizes models not reliable are for any design use practical in analysis and exercise. It is highly simplified approximations, resultingin simple relations.As a result,these more far models oncomplicated based These stiffness. are characteristics are than using models. these lamina complex Failuredirectional modes are strength and We lamina. worked also a unidirectional out of strengths the ahypothetical uni tion, we discussed modelsmechanics of the materials-based for of strengths the For a unidirectional lamina, CTE is adirection-dependent parameter. CTE Longitudinal lamina, For aunidirectional Δ l Δ (See strength. shear note below.) matrix which the as same is the as is obtained of matrix the in-plane strain ultimate shear the and α

T l

Using Equation 3.61, in-plane modulusthe compos shear of unidirectional the Then, usingThen, Equation 3.96, of in-plane strength unidirectional the shear the Note on the mechanics of materials-based models for models strengths materials-based of mechanics Note the on

original length original length in change of material the CTE change temperature in Evaluation Coefficients of Thermal () τ cu lt G =× 12 151 3195 c = 614 06 ..     ×+ () γ +− mu      2000 22 22 1400 lt , α == × (. 10 = 14 1400 −× . 35 lT ∆ ∆ 62 10 l )       ×

025 0 . 2 .. 60 =     ×= 195 3

.G 2 35 025 Pa 5M Composite Structures Composite

Pa : In this sec this : In

(3.97) - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 schematic representation of longitudinal deformations due to change in temperature. in change due to deformations of longitudinal representation schematic Micromechanics of a Lamina a of Micromechanics composite, A or the fibers and matrix, we can write the write following:we can and fibersmatrix, the load. longitudinal the stresses Thus, adding in is isno there zero as structural lamina although thethe net in stress thematrix and generated fibers the stresses in are thermal deformations,ent thermal bond interface, due at the they to mismatchwould between CTEs, the undergo differ changea temperature of on simple approach. of mechanics us consider materials Let RVE an subject and it to 3.17 FIGURE where the same, that is, same, that Δ the differential deformation from them between net restrains them the deformation and is Dividing sides the both of above the equation cross-sectional of with the the area A We can bring in the thermal deformationsWe equation rewrite and the thermal above the in bring can follows: as Now, deformations change related follows: temperature as to the are thermal A f m

cross-sectional of matrix the area cross-sectional of fibers the area c , we get

Representative volume element for longitudinal coefficient of thermal expansion— thermal of coefficient volumeRepresentative element for longitudinal E El l 1 c 1 , which is the thermal deformation, which of thermal is composite. the the aresult, As f fc      () ∆∆ ∆∆∆ ∆∆ Δ ll cf T Δ − l . If we consider the fibers and matrix to be free, that wefree,is, . If be no to considerand fibersmatrix the −+ Change intemperature= c l f and Δ and σσ lV 11 ff σσ ff       Matrix Matrix VV Fiber VE 11 ΔΔ ΔΔ ff ΔΔ fm l AA l m +− c + lT lT lT , respectively ( mm cc ff += El = = = mc mf α α α () ()      10 1 1 mm ∆∆ ll cm ∆ −− T − l l l l c c c c

= 0 lV mf

Figure 3.17 ∆l       ()

∆l 10 () f 10 c −= ∆l m V f = ). However, bond the

(3.100) (3.104) (3.102) (3.103) (3.101) (3.99) (3.98) 117 - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 ficient of expansionas ( expansion expansionsverse thermal sumof to thermal the the ofand fibersmatrix the - trans fied relation themechanics total equatingmaterials-based of by be obtained can given below [ approach; of mechanics the from materials that as for it stated be can transverse CTE, expression The verse CTE. for rigorous from longitudinal CTE analysis as same is the

118 or temperature. in change due to deformations of transverse representation schematic 3.18 FIGURE Figure 3.18 In the above the In relation, for ν In the case of the transverse CTE, ignoring the Poisson’s ignoring the case of the transverse the CTE, In effect, ahighly simpli Dividing sides the both by Rigorous have methods employed been for well as longitudinal both CTE trans as is given longitudinalThe CTE by Equation 3.106 follows: as Solution longitudinal Take transversethe and CTEs. fiber the volumefractionas 0.6. 6 Substituting above the Equation in 3.101 sides dividing and the both by Rearranging the terms, we show terms, can the is given longitudinal the CTE that Rearranging by given follows: as constituentare the properties carbon/epoxyFor material a unidirectional lamina, 3.8 EXAMPLE

× α

10 1 c − = 6

m/m/°C, m/m/°C, ), is, that Δb −× 28

05 .. ]: b Representative volume element for transverse coefficient of thermal expansion— thermal volume of coefficient Representative element for transverse c b b 240 m m 240 E /2 /2 αν b EV α 1 f 22 f ×+ 11

cf m = ×+ fc

=+ c 636 06 () =

ααα αα

06 = .. 240

() αα α 60 11 Change intemperature=

Δb 22 1 −+ 12 c cc

GPa, αα ∆∆ × Δ c = f 60

, the mechanics of materials expression of mechanics , the materials used. be can Tb 22 +

bT Tb Tb Tb 11 cf 10 ff αα ×−

×× =+ αν Δb c 11 , we get expression an coef for transverse thermal − (. E ff 36 10 6 =+ Matrix Fiber Matrix EV EV

ff .( m m/m/°C, and and m/m/°C, m VV 1

= ff VV EV ++

ff 6 3.6 fm fm 10 mc +− ) +− () − ()

α EV GPa, mf .) ∆ 6 −− T () EV () mm 1 1 mf ×= − α α αα ν () mf 00 10 1 12 1 mm f

∆ b f () = f

− 10 f = +∆b mf 6 (b (b

−

− m m 0.28, 0.28, 0.5 +∆b +∆b f

11 0 9 10 .0099

× = ν m m ν )/2 )/2

m 2 10 b ×° c

Composite Structures Composite c

= +∆b

− 6

0.3. Determine 0.3.Determine m/m/°C, m/m/°C, − c 6 m/m/ Δ Tl c α C , we get (3.106) (3.107) (3.105) (3.108) (3.109) 2 f

= - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 mechanics of materials approach. ofmechanics us consider materials Let RVE an ( is of m/m/kg/kg. unit transverse The CME the direction. in gives transverse the CME hand, other the of us ameasure relative change dimension in length changeunit unit per of mass in moisture content of mass unit body. per the On dimension directionthe per thein linear isfiber as change definedlongitudinal in CME where FIGURE 3.19 FIGURE it isas material, defined change dimension in w.r.t. change moisture in content body. the in For isotropic an abody absorbsWhen moisture, it expands of is ameasure size. relative CME in The 3.5.4 Micromechanics of a Lamina a of Micromechanics schematic representation of longitudinal deformations due to change in moisture content. moisture in change due to deformations of longitudinal representation schematic follows: by Equation 3.109. Towardthemajor wePoisson’s this, determine first ratio as ics rigorous per approach. as of transverse analysis CTE materials The is given Let us nowLet derive expression an for simple on based the longitudinal the CME is a direction-dependent CME parameter. the Thus, the lamina, For a unidirectional Δ l Δ β

C l

α Transverse follows: as is obtained then CTE rigorous per as longitudinal mechan The analysis CTE the per as same is the is given transverse CTE The by Equation 3.108 follows: as

2 c change moisture content in of mass unit body per (kg/kg) the CME of the material (m/m/kg/kg) of material the CME original length (m) length original change length in (m) Evaluation of Coefficients Moisture =+ = [( 3 5 10 335 952 10 α

. 2 Representative volume element for longitudinal coefficient of moisture expansion— of moisture coefficient volumeRepresentative element for longitudinal c =× .) 28 ×° Chan [. 60 ×× ν − ge 12 60 6 c m/m/ in moisturecontent 66 =× +× .( 02 61 .. ++ 01 C 80 (. Matrix Matrix Fiber −× l 60 03 c β 06 .) +× = per unitmassofcomposite= ×× )] .( lC 31 ∆ 60 ∆ 027 10 l −−

(. −= 10 66 −− =× 06 ∆l .) ∆l f ∆l c 60 . )( m 61 288 0 . −× 0 .) 50 ∆C

Figure 3.19 m/m/ c .] 8 10 288 ° C

× ). the Let − (3.110) 6

- 119 Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 tion above follows: as can write the following: the write can load.we longitudinaland fibersmatrix, the stressesthe Thus, adding in no structural there isas zerois although the lamina netin stress thematrix and fibers the in ated mation of composite the due to moisture absorption. aresult, As gener stresses are deformation net the deformation and is, same, that is the composite, A

CMEs, as shown as CMEs, Figure 3.19 in is, that no free, bond to be interface, due at the to mismatch betweenmatrix the volume element quantity of moisture. we absorb If consider acertain and fibers the 120

related change to the moisture in content follows: as However, CTE. gitudinal deformations the due to moisture are expansion they and are where Δ or l m We deformations the in bring can due to moisture absorption rewrite equa and the Substituting of above the Equation in 3.114 sides dividing and the both by l Rearranging the terms, we terms, get the Rearranging Dividing sides the both of above the equation cross-sectional of with the the area A Note that till this point, the procedure for procedure point, the derivation this for to that Note till that lon is very similar A , respectively. However, differential them from bond the between restrains them f m

cross-sectional of fibers the area cross-sectional of matrix the area EC f 11 , we get fc () βββ ββ E ∆∆ El 1 1 f fc      cf β () ∆∆ ∆∆∆ ∆∆ 1 −+ c ll cf = − 11 l −+ c ββ 11 ff CV σσ EC lV , they would undergo different deformations, 11 ff ff σσ ff       VV VE 11 ΔΔ ∆∆ ∆∆ ∆∆ ff fm AA lC lC +− lC + mm ff cc ff = = = VE += El EC mc mf mc β β β EC +− () () 1      1 () 1 10 mm ∆∆ cc Δ ll mm ∆∆ cm −− − mc cc l fc l c l l cm =

−− lV CV mf

mf       ()

10 () () 1 10 −= CV Δ V l mf c , which net is the defor f () = 10

Composite Structures Composite

c , we get Δ (3.115) (3.116) (3.117) (3.112) (3.113) (3.119) (3.118) (3.114) (3.111) l f and and - - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 the expansionsthe ( ofand fibersmatrix the transversefor wethat expansion and total transverse the equate the CTE sum of to the relation ofmechanics materials-based is derived here. methodology to The is similar where Micromechanics of a Lamina a of Micromechanics deformations we and related transverse get to the CMEs are following: the contents in the fibers and matrix. Thus, contentsand fibersmatrix. the in or schematic representation of transverse deformations due to change in moisture content. moisture in change due to deformations of transverse representation schematic FIGURE 3.20 FIGURE dinal CME as CME dinal Dividing sides the both by ρ In the case of transverse CME, ignoring the Poisson’s ignoring the case of the transverseIn CME, effect, ahighly simplified Dividing sides the both by Δ Now, moisture content total composite the the in sumof to the moisture is equal Substituting Equation 3.122 Equation in 3.119, we get expression the for longitu the β Δ Δ Δ β β 2 2 m C C C f c

c f m

transverse CME of fibertransverse the CME change moisture in content of mass unit matrix per the of matrix the CME change moisture in content of mass unit fibers per the transverse CME of compositetransverse the CME change moisture in content of mass unit composite per the β 1 c

= b Representative volume element for transverse coefficient of moisture expansion— volumeof moisture coefficient Representative element for transverse c     Change b b ββ m m 11 /2 /2 ff b EC f ∆∆ ∆∆ ∆∆ CV Cl in moisture content per unitmassofcomposite = in moisturecontent ∆∆ ∆ ββ cc CC ff β ρρ ρρ 22 2 c C cc c ∆∆ ff =+ cc c VE = bt fm Cb = c ρ C 1 +− l ββ +− c cf c c ∆∆ 2 b b [] =+ cf ff c c ∆∆ CV t =+ Matrix Matrix , we get c mm CV Fibe , we get ff CV ρρ ρρ Cl ff r ff ρρ VV mf Figure 3.20 fm VC () fm CV 1 fc +− +− Cb +− fm ∆ mf bt ff () C 1 fc mf ∆ CV c () 1 CV β mf ∆ mf mm ), is, that Δb ()     b () mf 1 Cl ∆ 1 (b (b     f +∆b () EV 1 mm m m ρ Cb 1 +∆b +∆b ρ − f ff VVV f V fm cm m m m bt +− +−

)/2 )/2 ∆C ρ EV

c c b c mf

c () = +∆b 1 () 1

Δb c f f

+    

Δb

m . These . These (3.125) (3.124) (3.120) (3.122) (3.123) (3.121) 121 - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3

122

lows [ lows approach; of mechanics folthe as materials it stated for be can transverse the CME, sion for more from longitudinal rigorous the CME analysis from that as same is the effect ignored. been has More rigorous have methods employed. been also expres The onbased simplistic assumptions. case of the transverse the coefficient, In the Poisson’s CME as CME given by Equation 3.126 follows: as sis is given by Equation 3.127. density The of composite the is given by rigorous the per approach. as ofmechanics analy transverse CTE materials The

Substituting that carbon fiber carbon that does not absorb moisture, Δ is given CME Longitudinal by Equation 3.123. Now, given under the assumption Solution changed the dimensionsmoisture, of determine laminate. the moisture. CMEs. Take fiber the volumefractionas 0.6. Assume carbon doesfiber not absorb kg/kg, given follows: as constituentare the properties carbon/epoxyFor material a unidirectional lamina, 3.9 EXAMPLE We have developed expressions for longitudinal well both as transverse as CMEs The transverse CME as per the mechanics of materials-based approach of mechanics the per materials-based as is transverse CME The And the longitudinal the Poisson’sAnd ratio of composite the is given by rigorous the per as longitudinal analysisThe the CME per as same is the If a unidirectional laminate of size 400 laminate aunidirectional If 4 ]: β β 1 ν c 2 12 β c = = f

2 = = c 012 0     β Δ 00 = = .

2     0.28, and and 0.28, βν C c +× 484 0 21     = c 01 00 E from Equation from 3.122, expression an we obtain can for transverse ff . 01 ∆∆ +× 1 +× .. m/m/kg/kg     35 f +× ββ

∆∆ ν CV = ∆ 2 12 ∆ ff CV .(

m/m/kg/kg ρ 35 240 ∆∆ C c () C 11 36 c ff ∆∆ =× m CV ρρ ++ =× m ν CV

02 m ×× GPa, ∆ 80 11 06 18 ff ..

fm = .. ρρ ∆ fm .( C 2 11 80 +− .( +− ff

11 ×− m C 0.3. Determine the longitudinal0.3. transverse the and Determine ×− E ×− fm m m +− CV

60 10 = +× (. +× 10 06 CV

βν 3.6 06 CV − .) mf mm mf .) .( () .) .( () 1 6

31 1 GPa, CV

6 mf mm     ×× ) () 10 1     C () −= × [. 806 18 −= f ρ

+− ×     = f     240 06

18 .) [ = 61

.) ρρ .. 300 β mf

ff 1 1.8, 1.8, × V ×+ f

. () =

1 0 636 06 mm + +

288 0 . .. ρ β 661 52 . m 1 2 +× f

.. mmf =

= × 11 () ×−

    1.1, 1.1,

11 0. Then, 0. 8 ρβ − (. Composite Structures Composite

cc ×− mm absorbs50 mm V − β (. 10 m 06 −

]1 11 =

06 )] ν

.) 0.35 6

2 c )

   

m/m/ (3.126) (3.127)

g - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 bounds for Young’s the modulus of isotropic an derived composite be can [ material energy have for used been lower upper bound respectively. bound, and following The potential elastic complementary moduli.energy minimum Principles and of minimum exact solutions, self-consistent and models [ subcategories—models intothem three on based techniques, bounding models with nience of discussion. way One to classify elasticity-based the models is to categorize elasticity as ered models. Also, classification the these of models matter is a of conve dimensions, conditions, equilibrium compatibility and conditions. approach is on based more rigorous relations including treatment, three in stress–strain RVE. elasticity the In approach, too, concept the of RVE is adopted. However, this relationsCompatibility 3D and may stress–strain not satisfied be theeachat point in conditions. Simplifying assumptions suchstress distribution made. uniform as are derived chapter are presentedeters this simple by in utilizing forms of equilibrium equations of mechanics forThe hygrothermal materials param and strength, stiffness, 3.6 Lamina a of Micromechanics als approach)als follows: as obtained are

Bounding techniques are associated with finding the upper andthe associated withupper finding lowerBounding are techniques bounds the for weAs mentioned had earlier, several of are there models consid be types can that

Thus, the changed dimensions of the unidirectional laminate are changedThus, the dimensions laminate of unidirectional the changes dimensions(fromThe the in of mechanics of laminate the the materi is of mass matrix The the as is obtained of mass laminate The the rigorous the per as transverse approach as the CME Then is obtained ELASTICITY-BASED MODELS β 2 c = = ∆ ∆ 625 0     ∆ 00 .m h b l +× w = = =× m 484 0 012 0 484 0 .( = 35 01 .. . .. /m/kg/kg +× w (. 0 53059 310 35 400 030 40 c ∆ ∆ ××× ××× = ×× C .. C 1000 1000 1000 030 40 300 400 m m 8 E mm mm mm mm ×+ ×× lowe ××× .( 10 08 11 ×× r 1000 1000 1000 ×− 1000 1000 = 50 50 50 )( ×− 815 08 .) VE .. 31 dm 10 × ×− 06 2 5210 059 10 1000 4592 1 521 4592 1 521000 4592 1 .) EE ]. (. .. md + .) 61 2 VE 06 × = md ×= × ×= 4592 1 82 ) . .k .. 1     0000 ×

8 = 1 . 24kg 4224 0 5 1 288 0 012 0 552 = . g 02 03 −×

.m .

8 5 .. mm mm m

(3.128) 1 , 14 - 123 ]: - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 FL, 2006.) FL, Kaw, K. A. from permission (Adapted with analyzing the response the analyzing of composite the conditions. cylinder applied to the boundary applied composite onare the cylinder elastic desired the and modulus by is obtained outer an cylinder ( and for matrix the RVEThe is acombination cylinder of inner represents two that fiber cylinders—an the or at random. array either aregular in cross section in arranged and circular as taken solutions. composite the In assemblage cylindrical (CCA) approach [ it is verified the whether governingdifferentialequationsare satisfied the assumed by solution assumed The assumed. involves or displacement stress, strain, components and the isingIn not geometry exact specified. solutions approach,possible a solution is volume fiber is that pack apart far fractions. bounds are of One reasonsthe that the composite.unidirectional However, for most apart far too fiber practical bounds are the 124 in a designin environment. Halpin–Tsai equations [ design variables. simple models, are Empirical easy hand, other to on use and the some in range applicabilityand cases, of narrow their to arather is restricted also match results. with experimental Elasticity-based models generally complicated, are modulus in-plane modulus the not shear and very are reliable they as do not have good We notedhad before modelsmechanics of the that materials-based for transverse the 3.7 and 3.21 FIGURE where are called semiempirical. called are curve fitting have the in used parameters physical these thus models and significance were developed byand curve fitting of elasticity-basedexperimental The modeldata. modelsempirical models these briefly and section.Thesethis discussed are in models These bounds can also be interpreted as boundsfor as transverse the modulus interpreted be also of bounds can These a V V E E E E

d m upper lower d m

SEMIEMPIRICAL MODELS SEMIEMPIRICAL

upper bound for Young’s modulus matrix volume fraction matrix lower bound for Young’s modulus volume fraction matter of dispersed the Young’s modulus material of dispersed the Young’s modulus of matrix the

Representative volume element in the composite cylinder assemblage (CCA) assemblage cylinder composite volumeRepresentative elementthe in approach. EE upper Mechanics of Composite Materials Composite of Mechanics =+ Figure 3.21 dd VE mm ). Appropriate boundary conditions conditions boundary ). Appropriate 1 V , 21

] are the most the commonly] are used Matrix Fibe r , CRC Press, Boca Raton, Raton, Boca , CRC Press, Composite Structures Composite 15 ],are fibers the (3.129) - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 In-plane shearmodulus,G Transverse modulus,E Longitudinal modulus,E Thus, from EquationThus, from 3.131 elastic moduli. elastic tions or values reliable recommended The of data. experimental ξ Equations ofand comparing curvea procedure fitting 3.130and 3.131with elasticity solu- apply; however, depends accuracy choice on the their of parameter the geometry,packing loading the condition. and Halpin–Tsai very equations simple are to 3.7.2.1 3.7.2 Source: Major Poisson’sratio,ν Desired Modulus Factors Reinforcing Recommended TABLE 3.3 expressedbe follows: as general formThe of Halpin–Tsai the equations for elastic can the of moduli alamina 3.7.1 and Micromechanics of a Lamina a of Micromechanics in which,in coefficient the ηis given by Using values the for parameter The V M M M f

f m

Lancaster, 1980; B.Paul,Transactions oftheMetallurgical Society ofAIME,100,1960,36–41. W. Tsai and H. ThomasHahn,Introduction toCompositeMaterials,Technomic Publishing, S. R. M.Jones, General Form of Halpin–Tsai Equations Halpin–Tsai Equations for Elastic Moduli corresponding fibercorresponding modulus corresponding matrix modulus matrix corresponding fiber volumefraction desired compositedesired modulus, is, that Longitudinal Modulus ξ Mechanics ofCompositeMaterials 2 is called the reinforcing the is called factor it and depends on fiber geometry, fiber 12 1 12 ξ given in 1 ∞ 2 ∞ 2 b a 3l ξ n M Table 3.3 Table η      b a = =             () () 1 MM MM ξ 1 η + − =∞ fm fm = , we can obtain the expressions the , we obtain can for different / ξη / η For fibersof rectangular cross sectioninatriangular For fibersofcircular cross sectioninasquare array For fibersof rectangular cross sectioninatriangular For fibersofcircular cross sectioninasquare array , secondedition,Taylor 1999; &Francis,New York, E array (Figure 3.22b) (Figure 3.22a) array (Figure 3.22b) (Figure 3.22a) 0 V 1 V c

f , E

f       + − 2 M c 1 ξ , ν m

c , or G 12 Remarks Table 3.3 given are Table in c – – ξ . It is determined by . It is determined (3.130) (3.132) (3.133) (3.131) 125 . Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 where as that by approach that of mechanics as the with materials zero void content. 126 we show can that case of the longitudinal in As the modulus, 3.7.2.3 Halphin–TsaiThe equation for transverse the modulus is given of alamina by 3.7.2.2 and 2006.) Kaw, K. A. from permission with array. (Adapted atriangular in section cross array. (b) of rectangular Fibers asquare in section cross 3.22 FIGURE as that by approach that of mechanics as the with materials zero void content. And ξ And Thus, wethe Halpin–Tsaithat find equation themajor for Poisson’sratio the same is On substitutionOn Equation in 3.130, we get Thus, wethe Halpin–Tsaithat find equation the for longitudinal modulus the same is

Transverse Modulus Transverse Major Poisson’s Ratio

2 or

(a) Fiber cross section and fiber packing in Halpin–Tsai equations. Halpin–Tsaiin equations. (a) packing of fiber Fibers and section cross Fiber ξ

2 a / Table 3.3 Table per b as Mechanics of Composite Materials Composite of Mechanics νν EE 212 12 11 cf cf E η =+ =+ 2 c = ξη = () ()       EE EE =− VE 1 VV 1 2 2 fm + . fm fm − fm E E (b) b / 1 ξη m η ξ f

ν V = V f f

() + − 1 () 0, following and procedure, a similar 1 a 1      

− 1/ E ξ − m V

f f

, CRC Press, Boca Raton, FL, FL, Raton, Boca , CRC Press, Composite Structures Composite (3.136) (3.137) (3.135) (3.134) (3.138) circular circular Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 whereas Halpin–Tsai equations yield closer match. approachof gives materials of underestimates transverse in-plane moduli, and shear approach well. isals as agood there and match data with experimental mechanics The ratio, Halpin–Tsai equations provide identical results mechanics ofwith as the materi range of fiber volumefractions. the For longitudinal themajormodulus and Poisson’s for to that transverse the modulussimilar it and is given by Halphin–TsaiThe equation for is in-plane modulus the shear lamina of aunidirectional 3.7.2.4

where Micromechanics of a Lamina a of Micromechanics The constituent material properties are as follows: as are constituent properties The material E Solution Exampleresults in 3.2. with those obtained ofarray. 0.6. fiberin a square the cross Also,the section take Compare as circular plane modulus shear by using Halpin–Tsai equations. Take afiber volumefraction posite longitudinal modulus, transverse modulus, major Poisson’s ratio, in- and com Example the in carbon/epoxy 3.2,For determine unidirectional lamina the 3.10EXAMPLE Halpin–Tsai equations for simple elastic the are to of apply moduli a lamina a wide in ν using Equation 3.137, as ηis obtained And ξ And 12 f Using Equation 3.138, major the Poisson’s as ratio is obtained usingThen, Equation 3.136, we get transverse the modulus as For transverse the modulus, ξ Using Equation 3.135, as longitudinal the is modulusobtained of lamina the

In-Plane Shear Modulus Shear In-Plane

0.3, 0.3, =

1 or G 12 f ξ

= =

E E 22 2 1 c 3l c

GPa, =× = ν n( 12      06 ab 12 c .( / +× =× E 10 η −× ) m 603 06 Table 3.3 Table per as 240

= = .. G η ..

12 (. 6538 0 6538 (. 3.6 24 24 =

.. +− c = = /

/ ()

() GPa, 36 10 36 2 for circular fibers in a square array. 2 for fibersin a square Then, circular GG GG       +− 1 12 12 1 × 06 ) ) + (. fm − fm + 10 − 06 .) / ν 63 ξη η 1 2 m V

×= V      = = . f ×= 60 f + −

6538 0 0.3, and       ). .. 36 4 44 145 6 ×= G . 1/ ξ .. m

30 057 10

G 1 . m f 3

= GP = GP

240 1.4 a a

GPa. GPa, E 2 f

= (3.140) (3.139)

24, 24, - 127 - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 128 and easy adesign toand use in environment. they have applicability. limited simple models, Semiempirical are hand, other on the on more rigorous behavior. treatmentof lamina generally They complicated are and siderations involving averaged Elasticity-based stresses strains. and models based are modelsof simple materials-based are on based simple tools con are that equilibrium assumptions made, also of which are relaxed some some are in models. mechanics The models. pirical models, models, of mechanics materials-based elasticity-based models, semiem and of they including different parameters; are types, of netting lamina the determination A number fractions mass ofandof and fibersmatrix. are models the available the for basic variables, elastic viz. certain densities, strengths, moduli, volume and fractions coefficientsand and strengths moisture expansion knowledgethermal of fromthe of elastic viz. of composite moduli, parameters, the determination the around lamina level. micro at the lamina level, At micro the study the of acomposite revolves material chapter, this In we reviewed basic the concepts tools and available for analysis the of a 3.8 approach for evaluation the of transverse in-plane moduli. and shear Halpin–Tsaiseen that mechanics of the materials suited than better equations are transverse in-plane moduli. the and Thus, it underestimates shear is data, mental before approach, of mechanics the when that materials with experi compared those by approach. of mechanics larger the than materials We mentioned had verse modulus in-plane modulus the shear and Halpin–Tsai per as equations are expressionsthe trans the hand, other identical methods. the the On both are in major the and Poisson’s identical. ratio is astraightforward are outcome This as Note: below: byina Halpin–Tsai equations approach of mechanics with the is materials given Then, usingThen, Equation 3.140, as η is obtained

Typically, RVE an is considered models. micromechanics the in Several basic

For in-plane modulus, the shear The comparison of comparison resultsThe the for elastic the of moduli carbon/epoxy the lam usingThen, Equation 3.139, we get in-plane modulus the shear as SUMMARY From above, made comparison the wethe that find longitudinal modulus E Modulus Elastic ν G E G 12c 1c 2c 12c 12 c =      11 +× 10 η −× = Materials Approach (. (. .. 8803 0 22 22 8803 .. Mechanics of / / 14 14 145.4 ξ

0.3 3.2 7.4 = ) ) × 06 + −

1 for circular fibers in a square array. 1 for fibersin a square circular 06 1 1 =      ×= 8803 0 . 445 14 .. Halpin–Tsai

Equations 145.4 3 10.57 0.3 4.53 GP a Composite Structures Composite

- - - - - Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 EXERCISE PROBLEMS mostthe commonly models. used models mechanics of the respect, materials-based well as Halpin–Tsai as models are design alternative ofan work to experimental a product. this In preliminary only the in models for prediction the of at elastic bestconsidered be can strengths and moduli as studywhile the of gives micromechanics agood insight into how composites work, the is generally a gap between micromechanics predictions results. experimental and Thus, Micromechanics of a Lamina a of Micromechanics

aresult, As Many ofunrealistic. there assumptions the are micromechanics in made 3.5 3.4 3.3 3.2 3.1

and 2.0.and Plot v the fiber type/diameter? fiber fraction forume composites? each of three the they dependent Are on the dent on fiber type/diameter?(b) fiber maximum theoretical theWhat is vol Consider glass/epoxy carbon/epoxy aunidirectional aunidirectional and Consider given data the Exercise in 3.3. density the If of sample the is exper digestion amatrix testIn (see Chapter 11 for details) of carbon/epoxy sam Consider Exercise in data the 3.1. (a) fiberfraction the in mass Determine Consider fiber the cross sectionas circular. the (a)fiber Determine volume Consider given data the volume fraction, (c) volume matrix (d) fraction, and voids volume fraction. two materials. w.r.t. fiber volumefractions.the in trend in Commentthe loadon sharing Tabulate plot and percentage the of share load by fiberseachin the lamina fibersmall volumefraction of 0.1, increase to gradually it 0.9in steps of 0.1. E tension. each subjected to auniaxial lamina, foundimentally 1.48 as Hint: Plot W the for fiber spacing-to-fiber ratios diameter ( array of (i) ofpacking fiber and fiber array packing (ii) square triangular forpacking fiber spacing-to-fiber ratios diameter ( array of (i) offractionfiber in and fiber array packing (ii) square triangular ple, following the were recorded: Assume zero void content. (a) the Determine (b) fiberfractionand mass fiber fraction. volume 1.1Density of matrix Density of fiber Mass removal of crucible matrix with sample after Mass of crucible with sample removal before matrix Mass of empty crucible the three composites? three the they dependent Are fiber ontype? the type? fiberfractionmass (b) maximum foreach theoretical is the What of f

76 Use the rule of mixtures for composite Useof mixtures rule the density.

GPa (glass fiber),and f Epoxy matrix Kevlar fiber Glass fiber Carbon fiber versus s/

= f

versus 1.80 curves. Are the fiberfractions the mass d curves. dependent Are on fiber

g/cm

g/cm g/cm = below:

s/ 30.1525 3 curves. Are the fiber the d curves. Are volumefractions depen )Density(g/cm Diameter (µm) 3 3 , determine the (a) the , determine fiberfraction,(b)mass fiber E m 12 16 –

7 =

3.5

GPa (epoxy with matrix). Starting s/ d ) of 1.0, 1.25, 1.5, 1.75, 2.0. and E 1 f

= s/

= d 240 1.45 2.58 1.80 1.1

) of 1.0, 1.25, 1.5, 1.75, =

30.4590

30.5903

GPa (carbon fiber),(carbonGPa 3 )

g - - - - 129 Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 130

3.10 3.9 3.8 3.7 3.6

Determine the (a) the Determine longitudinal modulus, (b) transverse modulus, (c) in-plane Consider a hybrid carbon–glass/epoxy fol with the unidirectional lamina (a) (b) longitudinal the tensile ofat an strength lamina. If the Determine Consider following the for carbon/epoxy data aunidirectional lamina: Consider glass/epoxy aunidirectional following with the lamina constitu glass/epoxy fraction mass of the unidirectional fiberIf in a is 0.67, lamina Assume zero void content afiber and volumefractionthe of 0.6.Determine Consider glass/epoxy aunidirectional following with the lamina constituent determine the void the determine content. following Assume the data: composite longitudinal modulus, transverse modulus, major Poisson’s ratio, properties: material example, Hint: modulus,shear (d) and major Poisson’s ratio of lamina. the respectively, change longitudinal the of the strength lamina. in the determine reduceelevated moduli by fiberandmatrix the and temperature, 20%, 10% lus, transverse modulus, major Poisson’s ratio, in-plane modulus. and shear possible themaximum compositeandfibers determine longitudinal modu- forces by shared and fibersmatrix. (c) Considercross sectionthe circular of ratioApply the of axial determine alongitudinal and force lamina on the transverse modulus, major Poisson’s ratio, in-plane modulus. and shear (b) volume fraction is 60%. (a) composite the longitudinal Determine modulus, ν ρ lowing constituent characteristics: properties:ent E material EG voids composite on the properties. mechanical m m ν Ca Glas Epoxy ff

12 = = == Compare the resultsCompare the withExample those in 3.1. Discuss effect the of rbon f

Replace two the reinforcement equivalent phases with an one. For 0.3, and 1.1 76 = sf )() () EE iber ma 02

GP 11 g/cm fibe ., ff EE ttrix 5 equivalent aG 11 : ,. rG fm EG an νν G 3 :, = == , and ρ , and :. ff EE m d EG ==

12 240 4500 = V mm ff =× 636 76 ff 02

== == 1.4 = × GP c 4 24 240 GP 51 03 14 35

,, MPa, =

05 )() () GPa. If the voidsGPa. the If volume fraction is 2% fiber the and VV aG .

aG f ,. 1.75 fm GP

ff = ,, carbon == abnglas carbon Pa

an aG 76 () 35

V ,. g/cm d

f GPa, () 505 35 + σ glas )() () Pa VV m T 3 ff s . ult ν EG Pa abnglas carbon Pa () f

V == = ,. GP f V ν s

72MPa Pa 0.2, 0.2, + ff carbon ff 60 14 03 36 aG = == ,. ., ,, ν G G GP 201 02 mf 12 ., f ==

= 5 f aG , an =

s () 35 σ ++ d mm 20 T ,.

V GPa, () an == E ult Composite Structures Composite d f Pa ρ .,

V glas f

E = . m s

2.54 = 04

3.6

g/cm .

GPa, Pa 3 - - , Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 Micromechanics of a Lamina a of Micromechanics SUGGESTED AND READINGREFERENCES

11. 10. 9. 8. 7. 6. 5. 4. 3. 2. 1. 3.15 3.14 3.13 3.12 3.11

FL, 2011. Composites, thirdedition,John Wiley &Sons,New York, 2006. ments, AIAAJournal, 2,1964,348. Lancaster, 1980. model, Journal ofMechanics andPhysicsofSolids , 13,1965,189–198. Aviation Conference, Los Angeles, California,March12–16,1961. degradation factors, Journal ofPressure Vessel Technology, 117(4),1995,390–394. fibrous composites,Journal ofCompositeMaterials , 2(3),1968,332–358. Administration, 1972. R. Hill, Theory ofmechanicalpropertiesfiber-strengthened materials—III.Self-consistent Z. Hashin, B. W. Shaffer, Stress–strain relationsofreinforcedplasticsparallelandnormalto the internalfila J. C. Ekvall, Elastic properties of orthotropic monofilament laminates, ASME Paper 61-AV-56, B. W. Tew, Preliminary design of tubular composite structures using netting theory and composite C. ChamisandG.P. Sendeckyj, Critiqueontheoriespredictingthermoelasticpropertiesof E. J.Barbero, S. W. Tsai andH. Thomas Hahn, A. K.Kaw, B. D. Agarwal, L. J. Broutman, and K. Chandrashekhara, R. M.Jones,

Given the material data in Exercise in Given data 3.14, material the plot percentage the change vol in constituent the proper carbon/epoxyFor material lamina, a unidirectional Is it possible such its that longitudinal lamina to design aunidirectional For a unidirectional carbon/epoxy lamina, the constituent the properties carbon/epoxyFor material lamina, aunidirectional (a) the Determine longitudinal tensile (b) strength, longitudinal compressive V of size 400 laminate ofume a unidirectional not absorb moisture. results Compare the Example with those in 3.9. fiber longitudinal Assume transverse carbon the and CMEs. does determine β change? along fiberdirection) the should not change when subjectedtemperature to tion Exercise in of carbon/epoxy the 3.12 lamina its if length (dimension dimension invariant? should is fiber What temperature the be volumefrac 300 for change the dimensions in w.r.t.Determine dimensions room temperature ties are giventies are follows: as of 25°Cperature above dimensions original are room The temperature. given follows: as E are data: additional ing (e)and Exercise in 3.7. of in-plane lamina strength the shear Use follow the (c)strength, transverse tensile (d) strength, transverse compressive strength, resultsthe Example with those in 3.1. in-plane modulusand shear using Halpin–Tsai the formulations. Compare m/°C, m/°C, m f if it if absorbs 50

V mm

f 0.35 Theory ofFiber Reinforced Materials , NASA-CR-1974, National Aeronautics andSpace

= Mechanics ofComposite Materials,CRCPress,BocaRaton,FL,2006. Mechanics ofComposite Materials,secondedition, Taylor &Francis,New York, 1999. α Introduction toCompositeMaterialsDesign

× 0.5 and and 0.5 2 () () f

m/m/kg/kg, m/m/kg/kg, σ σσ

= 300 1 m T T

f 6 ult

ult

× mm. Comment effect on the mm. of =

carbon V 10 g moisture. Take void content zero. as f 72

− = 6

= m/m/°C, m/m/°C, 0.6 if the lamina is subjected elevated0.6 to an lamina the if tem ν 1 PPa f 4500 12

E = Introduction toCompositeMaterials,TechnomicPublishing, f

1 an =

240 =

d MP 0.28, 0.28,

240 ()

τ α GPa, aM 12 m ,,

fu () GPa, = ν m

lt T f 60 E

= carbon m ult

E × = 0.3. If If 0.3. glas m

, secondedition,CRCPress,BocaRaton,

3.6 = = mm s = −

V 36 6

3.6 Analysis and Performance of Fiber GPa,

f m/m/°C, and and m/m/°C, . 3500

× MPa V

GPa,

300 = α

0.6 and and 0.6 1

f Pa mm

ρ = f

−

× 1.8, 1.8, 0.5

8 ν V

12 × mm w.r.t. mm ρ v f

m = = 10

0.02, 0.02, 0.28. 0.28. −

6 1.1, 1.1,

m/ - - - - - 131

- Downloaded By: 10.3.98.104 At: 07:34 26 Sep 2021; For: 9781315268057, chapter3, 10.1201/9781315268057-3 132

28. 27. 25. 24. 23. 22. 21. 20. 19. 18. 17. 16. 15. 14. 13. 12. 26.

2489–2504. ites, AIAAJournal, 4,102,1966. ites withperiodicmicrostructure, phase materials,Journal oftheMechanics andPhysicsofSolids,100,1963,127–140. Reviews, 10(37),1965,1–77. Journal, 4(9),1966,1537–1542. 1975, 1311–1318. Society of AIME,100,1960,36–41. Society Journal ofCompositeMaterials,2(3),1968,380–404. Applied Mechanics Reviews , 63,2010. Composite Materials,48(19),2014,2349–2362. for theeffective tetragonalelasticmodulioftwo-phase fiber-reinforced composites, International Journal ofSolidsandStructures, 31(21),1994,2933–2944. unidirectional composites,Journal ofCompositeMaterials,3,1969,368–381. 481–505. Applied Mechanics, 100,1964,223–232. Structural Applications,NASA CR-207, April 1965. Composite Materials,1,1967,188–193. 1969. J. M. Whitney andM.B.Riley, Elasticpropertiesoffiberreinforcedcompositematerials, R. A. Schapery, Thermal expansion co-efficients ofcompositematerialsbasedonenergy principles, H. Schuerch,Predictionofcompressive strengthinunidirectionalboronfibermetalmatrixcompos L. B.Greszczuk,Microbuckling failure ofcircular fiber-reinforced composites, A. Kelly andG.J.Davies, The principlesofthefiberreinforcementsmetals, N. Charalambakis,Homogenizationtechniquesandmicromechanics. A survey andperspectives, Z. Hashin, Analysis ofcompositematerials—Asurvey, J. C.Halpin, S. G. Mogilevskaya, V. I. Kushch, and D. Nikolskiy, Evaluation of some approximate estimates E. J.Barbero, T. M.Damiani,andJ. Trovillion, Micromechanicsoffabric reinforcedcompos R. Lucianoand E. J.Barbero, Formulas for the stiffness of compositeswithperiodicmicrostructure, D. F. Adams and S. W. Tsai, The influence of random filament packing on the transverse stiffness of Z. HashinandB. W. Rosen, The elasticmodulioffibrereinforcedmaterials, Z. HashinandS.Shtrikman, A variational approachtothetheoryofelasticbehavior ofmulti B. Paul, Predictionofelasticconstantsmultiphasematerials, J. M. Whitney, Elasticmoduliofunidirectionalcompositeswithanisotropicfilaments, N. F. Dow andB. W. Rosen, Effects ofEnvironmental Factors onCompositeMaterials Evaluations ofFilament Reinforced Compositesfor Aerospace International Journal ofSolidsandStructures Journal of Applied Mechanics Transactions oftheMetallurgical , AFML-TR-67-423, June , AFML-TR-67-423, Composite Structures Composite AIAA Journal, 100, ASME Journal of , 42(9–10),2005, Metallurgical , 50(3),1983, Journal of Journal of AIAA - - -