Mechanical Properties of Engineered Materials 2008 Part 10 doc

Note that this rather general definition of a composite applies to bothsynthetic man-made and natural existing in nature composite materials.Hence, concrete is a synthetic composite that

or heat treatment variations However, after a number of iterations, anasymptotic limit will soon be reached by this approach, as the propertiescome close to the intrinsic limits for any given system In contrast, an almostinfinite array of properties may be engineered by the second approach whichinvolves extrinsic modification by the introduction of additional (external)phases.

For example, the strength of a system may be improved by ment with a second phase that has higher strength than the intrinsic limit ofthe ‘‘host’’ material which is commonly known as the ‘‘matrix.’’ The result-ing system that is produced by the mixture of two or more phases is known as

reinforce-a composite mreinforce-aterireinforce-al

Note that this rather general definition of a composite applies to bothsynthetic (man-made) and natural (existing in nature) composite materials.Hence, concrete is a synthetic composite that consists of sand, cement andstone, and wood is a natural composite that consists primarily of hemi-cellulose fibers in a matrix of lignin More commonly, however, most of

us are familiar with polymer matrix composites that are often used in ern tennis racquets and pole vaults We also know, from watching athleticevents, that these so-called advanced composite materials promote signifi-cant improvements in performance.

mod-This chapter introduces the concepts that are required for a basicunderstanding of the effects of composite reinforcement on compositestrength and modulus Following a brief description of the different types

of composite materials, mixture rules are presented for composite systemsreinforced with continuous and discontinuous fibers This is followed by anintroduction to composite deformation, and a discussion on the effects offiber orientation on composite failure modes The effects of statistical var-iations in fiber properties on the composite properties are then examined atthe end of the chapter Further topics in composite deformation will bepresented inChap 10

9.2 TYPES OF COMPOSITE MATERIALS

Synthetic composites are often reinforced with high-strength fibers or kers (short fibers) Such reinforcements are obtained via special processingschemes that generally result in low flaw/defect contents Due to their lowflaw/defect contents, the strength levels of whiskers and fibers are generallymuch greater than those of conventional bulk materials in which highervolume fractions of defects are present This is shown in Table 9.1 inwhich the strengths of monolithic and fiber/whisker materials are compared.The higher strengths of the whisker/fiber materials allow for the develop-ment of composite materials with intermediate strength levels, i.e., betweenthose of the matrix and reinforcement materials Similarly, intermediatevalues of modulus and other mechanical/physical properties can be achieved

whis-by the use of composite materials

The actual balance of properties of a given composite system depends

on the combinations of materials that are actually used Since we are erally restricted to mixtures of metals, polymers, or ceramics, most syntheticcomposites consist of mixtures of the different classes of materials that areshown in Fig 9.1(a) However, during composite processing, interfacialreactions can occur between the matrix and reinforcement materials.These result in the formation of interfacial phases and interfaces (bound-aries), as shown schematically in Fig 9.1(b)

gen-One example of a composite that contains easily observed interfacialphases is presented inFig 9.2.This shows a transverse cross-section from

a titanium matrix (Ti–15V–3Cr–3Al–3Sn) composite reinforced with bon-coated SiC (SCS-6) fibers The interfacial phases in this composite

car-have been studied using a combination of scanning and transmissionelectron microscopy The multilayered interfacial phases in the Ti–15V–3Cr–3Al–3Sn/SCS-6 composite (Fig 9.2) have been identified to containpredominantly TiC However, some Ti2C and Ti5Si3 phases have alsobeen shown to be present in some of the interfacial layers (Shyue et al.,1995).

The properties of a composite can be tailored by the judicious control

of interfacial properties For example, this can be achieved in the Ti–15V–3Cr–3Al–3Sn/SCS-6 composite by the use of carbon coatings on the SiC/

TABLE9.1 Summary of Basic Mechanical/Physical Properties of

Selected Composite Constituents: Fiber Versus Bulk Properties

Young’smodulus(GPa)

Strengtha

(MPa)Alumina: fiber (Saffil RF) 300 2000

a Tensile and flexural strengths for fiber and monolithic, respectively.

FIGURE9.1 Schematic illustration of (a) the different types of composites and(b) interfaces and interfacial phases formed between the matrix and reinfor-cement materials

SCS-6 fibers The hexagonal graphite layers in the carbon coatings tend toalign with axial stress, thus making easy shear possible in the direction ofinterfacial shear stress Hence, the interfacial shear strengths of silicon car-bide fiber-reinforced composites can be controlled by the use of carboncoatings that make interfacial sliding relatively easy Such interfacial sliding

is critical in the accommodation of strain during mechanical loading orthermal cycling

Composite properties are also controlled by the selection of tuents with the appropriate mix of mechanical and physical properties(Tables 9.1 and 9.2) Since light weight is often of importance in a largenumber of structural applications, especially in transportation vehiclessuch as cars, boats, airplanes, etc., specific mechanical properties areoften considered in the selection of composite materials Specific propertiesare given by the ratio of a property (such as Young’s modulus andstrength) to the density For example, the specific modulus is the ratio

consti-of Young’s modulus to density, while specific strength is the ratio consti-ofabsolute strength to density

It is a useful exercise to compare the absolute and specific properties in

Table 9.2 This shows that ceramics and metals tend to have higher absoluteand specific moduli and strength, while polymers tend to have lower abso-lute properties and moderate specific properties In contrast, polymer matrixcomposites can be designed with attractive combinations of absolute specific

FIGURE 9.2 (a) Transverse cross-section of Ti–15V–3Cr–3Al–3Sn compositereinforced with 35 vol% carbon-coated SiC (SCS-6) fibers and (b) InterfacialPhases in Ti-15-3/SCS6 composite

TABLE9.2 Summary of Basic Mechanical/Physical Properties of Selected Composite Constituents: ConstituentProperties

Density(mg/m3)

Young’smodulus(GPa)

Strengtha(MPa)

Ductility(%)

Toughness,

KIC(M Pa m1=2Þ

Specificmodulus[(GPa)/

(mg/m3)]

Specificstrength[(MPa)/(mg/m3)]

strength and stiffness These are generally engineered by the judicious tion of polymer matrices (usually epoxy matrices) and strong and stiff(usually glass, carbon, or kevlar) fibers in engineering composites, whichare usually polymer composites.

selec-The specific properties of different materials can be easily comparedusing materials selection charts such as the plots of E versus , or f versus

in Figs 9.3 and9.4,respectively Note that the dashed lines in these figurescorrespond to different ‘‘merit’’ indices For example, the minimum weightdesign of stiff ties, for which the merit index is E=, could be achieved byselecting the materials with the highest E= from Fig 9.3 These are clearlythe materials that lie on dashed lines at the top left-hand corner of Fig 9.3

FIGURE9.3 Materials selection charts showing attractive combinations of cific modulus that can be obtained from engineering composites

spe-Similarly, the materials with the highest specific strengths, f=, are thematerials at the top left-hand corner of the strength materials selectionchart shown in Fig 9.4 In both charts (Figs 9.3and 9.4), polymer matrixcomposites such as carbon fiber-reinforced plastics (CFRPs), glass fiber-reinforced plastics (GFRPs), and kevlar fiber-reinforced plastics (KFRPs)emerge clearly as the materials of choice For this reason, polymer matrixcomposites are often attractive in the design of strong and stiff lightweightstructures.

A very wide range of synthetic and natural composite materials arepossible Conventional reinforcement morphologies include: particles (Fig.9.5), fibers [Fig 9.6(a)], whiskers [Figs 9.6(b) and 9.6(c)], and layers, Fig.9.6(d) However, instead of abrupt interfaces which may cause stress con-

FIGURE9.4 Materials selection charts showing attractive combinations of cific strength that can be obtained from engineering composites

spe-centrations, graded interfaces may be used in the design of coatings andinterfaces in which the properties of the system are varied continuously from100% A to 100% B, as shown schematically inFig 9.7.Such graded transi-tions in composition may be used to avoid abrupt changes in stress statesthat can occur at nongraded interfaces.

Furthermore, composite architectures can be tailored to support loads

in different directions Unidirectional fiber-reinforced architectures [Fig.9.6(a)] are, therefore, only suitable for structural applications in which theloading is applied primarily in one direction Of course, the composite fibermay be oriented to support axial loads in such cases Similarly, bidirectionalcomposite systems (with two orientations of fibers) can be oriented to sup-port loads in two directions

The fibers may also be discontinuous in nature [Figs 9.6(b) and 9.6(c)],

in which case they are known as whiskers Whiskers generally have highstrengths due to their low defect densities They may be aligned [Fig 9.6(b)],

FIGURE9.5 Schematic illustration of particulate reinforcement morphologies:(a) spherical; (b) irregular; (c) faceted

FIGURE9.6 Examples of possible composite architectures: (a) unidirectionalfiber reinforcement; (b) aligned whisker reinforcement; (c) randomly orientedwhisker reinforcement; (d) continuous layers.

FIGURE9.7 Schematic illustration of graded reinforcements

or randomly oriented,Fig 9.6(c) The reader may recognize intuitively thataligned orientations of whiskers or fibers will give rise to increased strength

in the direction of alignment, but overall, to anisotropic properties, i.e.,properties that vary significantly with changes in direction However, ran-dom orientations of whiskers will tend to result in lower average strengths inany given direction, but also to relatively isotropic properties, i.e., propertiesthat do not vary as much in any given direction

In addition to the synthetic composites discussed above, several posite systems have been observed in nature In fact, most materials innature are composite materials Some examples of natural compositesinclude wood and bone As discussed earlier, wood is a natural compositethat consists of a lignin matrix and spiral hemicellulose fibers Bone, on theother hand, is a composite that consists of organic fibers, inorganic crystals,water, and fats About 35% of bone consist of organic collagen proteinfibers with small rod-like (5 nm
5 nm
50 nm) hydroxyapatite crystals.Long cortical/cancellous bones typically have low fat content and compactstructures that consist of a network of beams and sheets that are known astrabeculae

com-It should be clear from the above discussion that an almost infinitearray of synthetic and artificial composite systems are possible However,the optimization of composite performance requires some knowledge ofbasic composite mechanics and materials concepts These will be introduced

in this chapter More advanced topics such as composite ply theory andshear lag theory will be presented inChap 10

9.3 RULE-OF-MIXTURE THEORY

The properties of composites may be estimated by the application of simplerule-of-mixture theories (Voigt, 1889) These rules can be used to estimateaverage composite mechanical and physical properties along different direc-tions They may also be used to estimate the bounds in mechanical/physicalproperties They are, therefore, extremely useful in assessment of the com-binations of basic mechanical/physical properties that can be engineered viacomposite reinforcement This section will present constant-strain and con-stant-stress rules of mixture

9.3.1 Constant-Strain and Constant-Stress Rules of

Mixtures

An understanding of constant-strain and constant-stress rules of mixturesmay be gained by a careful study ofFig 9.8 This shows schematics of thesame composite system with loads applied parallel [Fig 9.8(a)] or perpen-

dicular [Fig 9.8(b)] to the reinforcement layers In the case where the loadsare applied parallel to the reinforcement direction [Fig 9.8(a)], the strains inthe matrix and reinforcement layers must be equal, to avoid relative slidingbetween these layers.

In contrast, the strains in the individual matrix and reinforcementlayers are different when the loads are applied in a direction that is perpen-dicular to the fiber orientation, Fig 9.8(b) Since the same load is applied tothe same cross-sectional area in the reinforcement and matrix layers, thestresses in these layers must be constant and equal for a given load Theloading configuration shown in Fig 9.8(b), therefore, corresponds to aconstant stress condition

Let us now return to the constant strain condition shown cally in Fig 9.8(a) If the initial length of each of the layers, L, and appliedload, P, is partitioned between the load in the reinforcement, Pr, and the

schemati-FIGURE9.8 Schematic illustration of loading configurations for (a) strain rule of mixtures and (b) constant-stress rule of mixtures

constant-load in the matrix, Pm, then simple force balance gives

where E is Young’s modulus and " is the uniaxial strain

Substituting Eqs (9.2) and (9.3) into Eq (9.1), and using subscripts c,

m, and r to denote the composite, matrix, and reinforcement, respectively,gives

and

Pc¼ Ec"cAc¼ Em"mAmþ Er"rAr ð9:4bÞwhere Acis the area of composite, Am is the area of matrix, and Ar is thearea of the reinforcement Noting that the strains in the composite, matrix,and reinforcement are equal, i.e.,"c¼ "m¼ "r, we may simplify Eq (9.4b)

rule-of-posite properties are averaged according to the volume fraction of the posite constituents.

com-The fraction of the load supported by each of the constituents alsodepends on the ratio of the in moduli to the composite moduli Hence, formost reinforcements, which typically have higher moduli than those ofmatrix materials (Tables 9.1 and 9.2) most of the load is supported by thefibers, since:

Vr¼ ‘r

‘c

ð9:11bÞDividing both the left- and right-hand sides of Eq (9.10) by‘c, andnoting that Vm¼ ‘m=‘cand Vr¼ ‘r=‘c[from Eqs (9.11a) and (9.11b)] gives

The composite strain is, therefore, averaged between the matrix and cement for the constant stress condition The composite modulus for theconstant stress condition may be obtained by substituting " ¼ =E into

in Fig 9.9 For example, particulate composites will tend to have compositemoduli that are closer to the lower bound values, as shown in Fig 9.9.The expressions for modulus may be generalized for the wide range ofpossible composite materials between the constant strain (iso-strain) andconstant stress (iso-stress) conditions (Fig 9.9) This may be accomplished

by the use of an expression of the form:

where X is a property such as modulus, n is a number between þ1 and 1,and subscripts c, m, and r denote composite, matrix, and reinforcement,respectively Equation (9.15) reduces to the constant strain and constantstress expressions at the limits of n ¼ þ1 and n ¼ 1

Also, the wide range of possible composite properties may be mated for actual composites for which values of n are between 1 andþ1 Furthermore, the values of n for many composites are close to zero.However, there are no solutions for n ¼ 0, for which Eq (9.15) gives thetrivial solution 1 ¼ 1 Iterative methods are, therefore, required to obtainthe solutions for n ¼ 0

esti-Although the discussion so far has focused largely on iso-stress andiso-strain conditions, the above rule-of-mixture approach can be appliedgenerally to the estimation of physical properties such as density, thermalconductivity, and diffusivity The rule-of-mixture expressions can, therefore,

be used to estimate the effects of reinforcements on many important cal properties They may also be used to estimate the bounds in a wide range

physi-of physical properties physi-of composite materials Such rule-physi-of-mixture tions are particularly valuable because they can be used in simple ‘‘back-of-the-envelope’’ estimates to guide materials selection and design

calcula-Finally, in this section, it is important to note that the simple aging schemes derived above for two-phase composite systems can beextended to a more general case of any n-component system (where

During the initial stages of deformation, both the matrix and fibers deformelastically (Fig 9.10) Furthermore, since the axial strains in the matrix andfiber are the same (iso-strain condition), then the stresses for a given strain, ð"Þ, are given simply by Hooke’s law to be

If we now consider the specific case of a ductile matrix compositereinforced with strong brittle fibers, matrix yielding is most likely to precedefiber fracture In this case, the onset of composite yielding will correspond tothe matrix yield strain,"my, as shown schematically inFig 9.11 Also, thecomposite yield stress, and the stresses in the matrix and fiber are given,respectively, by

cð"myÞ ¼ Vm mð"myÞ þ Vf fð"myÞ ð9:18aÞ

FIGURE9.10 Stress–strain curves associated with uniaxial deformation of stiffelastic composite

mð"myÞ ¼ Vm mð"myÞ ð9:18bÞ

where the subscript ‘‘my’’ corresponds to the matrix yield stress and scripts c, m, and f denote composite, matrix, and fiber, respectively.Following the onset of matrix yielding, codeformation of the matrix andfibers continues until the fiber fracture strain, "cf, is reached The stressescorresponding to this strain are again given by constant-strain rule-of-mixtures to be

sub- mð"mfÞ  mð"cfÞ ð9:23bÞSimilarly, we may obtain a condition for the minimum fiber volumefraction, Vmin, at which the composite fracture strength, cð"mfÞ, exceeds thematrix fracture strength, mð"mfÞ:

mð"mfÞ ¼ ð1  VminÞ mð"cfÞ þ Vmin fð"cfÞ ð9:24aÞ

9.6 FAILURE OF OFF-AXIS COMPOSITES

So far, we have focused primarily on the deformation behavior of tional fiber-reinforced composites However, it is common in several appli-cations of composite materials to utilize fiber architectures that are inclined

unidirec-at an angle to the loading axis Such off-axis composites may give rise todifferent deformation and failure modes, depending on the orientation ofthe fibers with respect to the loading axis

To appreciate the possible failure modes, let us start by considering theloading of the arbitrary off-axis composite shown schematically inFig 9.14.The uniaxial force vector, F, may be resolved into two components: F cos and F sin The component F cos  results in loading of the fibers along

FIGURE 9.13 Loci of stress levels corresponding to matrix-dominated andcomposite failure modes

9.3 RULE -OF- MIXTURE THEORY

The properties of composites may be estimated by the application of simplerule -of- mixture theories (Voigt, 1889)... assessment of the com-binations of basic mechanical/ physical properties that can be engineered viacomposite reinforcement This section will present constant-strain and con-stant-stress rules of mixture... the effects of reinforcements on many important cal properties They may also be used to estimate the bounds in a wide range

physi -of physical properties physi -of composite materials Such