6

6. Rules of Mixture

The mechanical and physical properties of composite materials depend in a complex way on the type, form, quantity and arrangement of the constituents. Rules of mixture are equations which attempt to express these dependencies in a predictable fashion. Several different approaches have been used (summarised in Daniel and Ishai, 1994 and discussed in more detail by Gibson, 1994), but they are all based on various degrees of simplification, and may be semi-empirical. It follows that the user should treat the predictions with caution, especially if properties are to be used for anything more than preliminary design, and should endeavour to understand the assumptions underlying the formulae.

The rule of mixtures for composite density was derived in Chapter 5. Here we concentrate on expressions for elastic and thermal properties. Models of composite strength are discussed in Chapter 9.

6.1 Unidirectional Ply longitudinal modulusThe sign and nomeclature convention used here is shown in Fig. 6.1. The orthogonal axes 1, 2 and 3 relate to the fibre direction, in-plane transverse and through-thickness transverse directions respectively.

We note that the unidirectional ply has two different in-plane tensile moduli (E1 and E2). To a first approximation, E3 ( E2. We also need two values of Poissons ratio to describe the lateral contraction resulting from in-plane tension, as shown in Fig. 6.2. The conventional notation is that ij denotes the contraction in the j-direction when stress is applied in the i-direction.

From Fig. 6.2, it can be deduced that the lateral strain (2) resulting from a stress applied in the fibre direction (1) is much larger than the longitudinal strain (1) resulting from a transverse (2) applied stress. Hence, 12 > 21. In fact the Poissons ratios and moduli are related by:

(6.1)

In deriving the rule of mixtures, the following assumptions are made:

Fibres are uniform, parallel and continuous.

Perfect bonding exists between fibres and matrix.

A longitudinal load produces equal strain in fibre and matrix.

We apply a load F1 in the longitudinal direction this is shared equally between reinforcement and matrix so that F1 = Ff + Fm.

Writing loads in terms of stresses and areas we obtain:

(6.2)

where A = Af + Am is the cross-sectional area of the ply.

Applying Hookes law we replace stress with the product of strain and modulus:

(6.3)

But we have assumed equal strain (1 = f = m), so:

(6.4)

or

(6.5)

The terms Af / A and Am / A are the area fractions of fibre and matrix respectively. In our unidirectional composite, these are clearly equivalent to the constituent volume fractions, so we can write:

(6.6)

A similar rule of mixtures is commonly used for Poissons ratio:

(6.7)

These two rules of mixture are generally accepted as corresponding well with experimental data Matthews and Rawlings (1999), for example, state that agreement is within 5%.

To a reasonable approximation, if Ef >> Em, then E1 = Ef Vf.

We should, however, take note of the fact that reinforcing fibres may not be isotropic. Carbon and aramid, for example, rely on a three-dimensional oriented microstructure for their exceptional mechanical properties, but quoted values almost always refer to the axial direction. There seem to be few data available on transverse properties (partly because they are obviously very difficult to measure). Shindo (2000) quotes values for the ratio of longitudinal to transverse modulus from about 16 for high strength carbon fibre, up to more than 80 for highly oriented high modulus carbon fibre. Hence both Ef and possibly f in Equations 6.6, 6.7 and the other rules of mixture presented here should be interpreted with care this applies to thermal as well as elastic properties.

6.2 Unidirectional Ply transverse modulusFor loading in the transverse (2) direction, the state of stress in the relatively flexible matrix is much more complex, and not surprisingly the transverse modulus E2 turns out to be a matrix-dominated property. Generally, the rules of mixture are based on simply assumptions of stress distribution, and are much less reliable than those for longitudinal properties.

The simplest approach is a series model of fibre and matrix in which any Poissons contraction is ignored, and the stress in each of the constituents is assumed to be the same. A detailed derivation can be found, for example, in Matthews and Rawlings (1999) or Hull and Clyne (1996). The result is:

or

(6.8)

Again, we note that Ef should properly be associated with the transverse rather than longitudinal value of modulus.

If Ef >> Em and Vf ( Vm then we may write

(6.9)

which is of course independent of the modulus of the reinforcement.

The literature abounds with alternative models for transverse modulus, which seek to improve on the rather poor experimental agreement observed with Equation 6.8. A simple modification (which allows for fibres restricting the Poisson contraction) is to replace the matrix modulus in Equation 6.8 with

(6.10)

where m is the Poissons ratio of the matrix.

A commonly-used alternative is the Halpin-Tsai model for transverse modulus:

(6.11)

where

The parameter ( is adjustable, but is usually close to unity. Equation (6.10) is generally considered to more reliable than the simpler alternative.

The various rules of mixture for longitudinal and transverse modulus in UD glass/epoxy are plotted in Fig. 6.3. Fibre and matrix moduli were taken as 70 GPa and 3 GPa respectively. Note that the simple constant stress model for E2 is a lower bound.

6.3 Unidirectional Ply shear modulusAs the tensile modulus relates tensile stress and strain, so the shear modulus is defined as the ratio of shear stress to shear strain:

Note that the subscripts ij indicate the plane in which the shear modulus is defined.

The rule of mixtures for shear modulus is based on the same assumptions as that for transverse tensile modulus, and hence should be treated with similar caution:

(6.12)

The Halpin-Tsai equations are applicable to shear modulus, giving the preferred expression:

(6.13)

with

(6.14)

Again, the parameter ( is approximately equal to 1.

Assuming transverse isotropy, we expect G13 = G12.

The third shear modulus is usually obtained from:

(6.15)

6.4 Unidirectional Ply Poissons ratioIf our unidirectional ply (Fig. 6.1) has transverse isotropy, we expect 13 = 12 (see Equation 6.7). Hull and Clyne (1996) give the following expression for the other out-of-plane Poissons ratio in terms of the bulk modulus (K):

(6.16)

where

with

and

Three dimensional elastic constants are rarely required for routine calculations, but they may well be needed in finite element or other numerical analyses. For reference, Tables 6.1 and 6.2 give theoretical values of all the elastic constants for UD glass and carbon/epoxy composites over a limited range of fibre volume fractions. The other 3 Poissons ratios may be obtained from Equation 6.1. The constituent properties used are listed in Table 6.3. There is a wide range of published values, and these data should be used for illustrative purposes only. Note that some of the properties in Table 6.3 have been obtained by back calculation i.e. deduced from empirical composite data rather than measured as primary data.

Table 6.1: Rules of mixtures values for elastic constants of UD E-glass fibre/epoxy (units of tensile and shear moduli are GPa).VfE1E2 = E312 = 13 23G12 = G13G23

0.429.86.50.330.654.32.0

0.4533.27.20.320.644.82.2

0.536.58.10.310.635.32.5

0.5539.99.10.300.626.02.8

0.643.210.40.290.606.73.2

0.6546.611.90.280.597.63.7

0.749.913.80.270.578.64.4

Table 6.2: Rules of mixtures values for elastic constants of UD high strength carbon fibre/epoxy (units of tensile and shear moduli are GPa).

VfE1E2 = E312 = 1323G12 = G13G23

0.489.86.80.320.703.52.0

0.45100.77.70.310.693.72.3

0.5111.58.70.300.684.02.6

0.55122.49.90.290.664.33.0

0.6133.211.40.280.654.63.5

0.65144.113.30.270.644.94.1

0.7154.915.80.260.635.34.9

Table 6.3: Constituent properties for Tables 6.1 and 6.2 (Gibson, 1994).E-glass fibreHS (T300) carbon fibre aramid fibreepoxy resin

longitudinal tensile modulus (GPa)732201523

transverse tensile modulus (GPa)7313.84.1-

longitudinal shear modulus (GPa)30.19.02.92.1

transverse shear modulus (GPa)30.14.81.5-

longitudinal Poissons ratio0.220.200.350.4

transverse Poissons ratio0.220.250.35-

6.5 Multidirectional Ply in-plane tensile modulusThe calculation of elastic properties for laminae or laminates is normally undertaken with numerical laminate analysis (Chapter 8). However, it is possible to modify Equation 6.6 to give an estimate of tensile modulus for composites in which the fibres are neither continuous or unidirectional.

A correction factor to allow for the loss of efficiency if fibres are not perfectly aligned in the load direction was given by Krenchel (1964) as:

(6.18)

where a proportion i of the fibres has orientation i, and the summation is carried out over the various angles present in the reinforcement. For example, in a biaxial reinforcement with fibres oriented at ( 45o to the load direction, we have

The orientation correction factor is then included in Equation 6.6, giving the semi-empirical:

(6.19)

Here L is a length correction factor for short reinforcing fibres. For fibres greater than a critical length (see section 6.6), and for continuous fibres, L = 1. Table 6.4 gives appropriate values of o for common forms of reinforcement.

Table 6.4: Orientation correction factors for non-UD reinforcement (Equations 6.18 and 6.19).

orientationo

unidirectional1

biaxial0.5

( 45o0.25

random (in-plane)0.375

random (3D)0.2

Equation 6.19 applies strictly to in-plane reinforcement. In woven fabrics, the fibres exhibit waviness in the through-thickness direction. This crimp is greatest in plain weave styles, but less in twill and satin forms (see Chapter 3). We can use Equation 6.18 to estimate an additional orientation factor to allow for the loss in reinforcing efficiency due to out-of-plane waviness.

Assuming that the path of a tow in a woven fabric is sinusoidal, the orientation factor can be obtained numerically as a function of weave crimp angle (Grove and Summerscales, 2000). The results are shown in Fig. 6.4. In a woven reinforcement, both orientation factors should be combined to allow for in-plane and out-of-plane deviation from the load direction.

6.6 Short Fibre ReinforcementWhen considering continuous fibres, we can safely assume conditions of uniform stress or strain in the longitudinal direction. This is not the case with discontinuous fibres, due to the nature of the load transfer at the fibre ends. When a stiff fibre is embedded in a relatively flexible matrix, shear stress and strain are a maximum at the fibre ends (Fig. 6.5). The tensile stress in the fibre, on the other hand, is zero at the fibre end and increases towards the centre (Fig. 6.6).

Schematically, the shear stress at the fibre/matrix interface and the tensile stress in the fibre are as shown in Fig. 6.7. The common theoretical derivation of these stress distributions relies on the so-called shear lag model, which describes the transfer of tensile stresses from matrix to fibre via shear stress at the interface. This and other models are described in detail in Hull and Clyne (1996).

For our purposes, a simplified model will suffice. We consider the balance of the tensile force in the fibre (diameter D) with the shear force at the interface:

(6.20)

Lc is the length of the fibre over which the interfacial shear forces act. Referring to Fig. 6.7, it is clear that the average tensile stress in the

fibre is less than its maximum value of Ef m, where m is the strain in the matrix. Moreover, if the fibre length L < Lc, then the tensile stress nowhere reaches its maximum value. Lc is thus referred to as the critical length defined as the minimum length of fibre required for the tensile stress to reach its failure value. At this point, from Equation 6.20 we have:

(6.21)

where f* is the fibre tensile strength. By inserting values of interfacial or matrix shear strength for in Equation 6.21, the critical aspect ratios (Lc / D) for various fibre/matrix combinations can be estimated. Table 6.5 gives some typical values.

Table 6.5: Critical length and aspect ratio for some composite systems (from Matthews and Rawlings, 1996).

matrixfibreLc (mm)Lc / D

aluminiumboron1.820

epoxyboron3.535

epoxycarbon0.235

polyesterglass0.540

aluminaSiC0.00510

The average stress in a fibre reaches 90% of the value in a continuous fibre when L ( 5Lc. For fibre lengths L > Lc, the average fibre stress is given approximately by

(6.22)

Although the length correction factor (L) in Equation 6.6 depends on fibre length as expected, the theoretical models also include elastic properties of fibre and matrix and the fibre distribution:

(6.23)

where

Here 2R is the interfibre spacing and Gm the matrix shear modulus.

Calculations in Hull and Clyne (1996) for carbon and glass reinforcement show that L = 0.2, 0.89 and 0.99 for fibre lengths of 0.1, 1.0 and 10.0 mm respectively. In Fig. 6.8, we have calculated the length correction factor for glass fibres in epoxy resin, assuming a mean interfibre spacing of 20 times the fibre diameter.

This would suggest that, so far as stiffness is concerned, the length correction factor can be ignored for fibre lengths over about 1 mm. It should be noted, however, that processes such as injection moulding of thermoplastics can cause considerable fibre damage. Data from Vetrotex examined glass fibre/nylon feedstock, containing 4 mm long fibres. After moulding, almost all the fibres had lengths ranging from 0.1 to 0.7 mm only 12% of all the fibres had a length greater than 0.8 mm. The lower stiffness of discontinuous fibre-reinforced composites is of course also due to the fact that typical volume fractions are much lower than with aligned, continuous reinforcements.

In short fibre-reinforced thermosetting polymer composites, it is reasonable to assume that the fibres are always well above their critical length, and that the elastic properties are determined primarily by orientation effects. The following equations give reasonably accurate estimates for the isotropic in-plane elastic constants:

(6.24)

Here, E1 and E2 are the tensile moduli of a unidirectional ply of the same fibre volume fraction, as given by Equations 6.6 and 6.11 (or its alternatives). Results from Equation 6.23 are plotted in Fig. 6.9 for a random glass / epoxy composite.

6.7 Thermal ExpansionCoefficients of thermal expansion are of considerable relevance, since many polymer composites experience temperature changes as an integral part of their processing cycle. Rules of mixture are discussed in Chapter 10.

6.8 Exercises1. Using an Excel spreadsheet, or otherwise, produce an equivalent set of curves to those in Fig. 6.3 for high strength carbon fibre reinforcement.

2. The following empirical data was published in an Owens Corning Design Data Book. Compare these data with the appropriate predictions from rules of mixtures.

3. An aligned short fibre composite is stressed to a mean interfacial shear stress of 5 MPa. If the tensile strength of the reinforcement is 3400 MPa, what fibre length is required for the composite to have its maximum load carrying capacity?

4. Using Fig. 6.8, plot an approximate graph of tensile modulus versus fibre length for a short fibre glass/epoxy composite.

5. Using Equation 6.22, estimate the fibre length at which you would expect a glass/polyester composite to have 95% of the strength of one reinforced with continuous fibres.

6. Sketch and label the variation of tensile stress along the length of a high strength carbon fibre of length L = Lc if it is aligned in the load direction and the applied strain is 0.1%.

7. A test specimen containing a single HS carbon fibre is subjected to a uniaxial tensile test. At maximum stress, the fibre breaks into pieces each of length 0.625 mm. The fibre is 10 m in diameter, has longitudinal tensile modulus 240 GPa and an ultimate tensile strength of 2500 MPa. (a) What is the interfacial shear strength of the composite? (b) If the composite has a longitudinal modulus of 80 GPa, what was the applied stress at fibre failure?(Gibson, 1994).

Fig. 6.1: Orthogonal directions in unidirectional ply.

1

3

2

1

3

2

fibre direction (1)

fibre direction (1)

1

2 = - 12 1

2

1 = - 21 2

Fig. 6.2: Definition of in-plane Poissons ratios in orthotropic material

EMBED Excel.Sheet.8

Fig. 6.3: Rules of mixtures modulus for UD glass/epoxy composite. Matrix Poissons ratio taken as 0.4.

EMBED Excel.Sheet.8

Fig. 6.4: Calculated orientation distribution factor for a plain weave tow with varying crimp angle.

Fig. 6.5: Finite element model of single glass fibre (blue) embedded in epoxy resin. The model is axisymmetric about the upper edge, and is subjected to a uniaxial tensile strain. Upper picture shows undeformed mesh note high shear strain at fibre end.

Fig. 6.6: Contours of tensile stress in the fibre direction. Note low stress at fibre end, increasing to maximum value about 7 radii towards the centre.

Fig. 6.6: Schematic variation of fibre tensile stress and interfacial shear stress in a short fibre. (Matthews & Rawlings, 1999)

L

Lc / 2

EMBED Excel.Sheet.8

Fig. 6.8: Theoretical length correction factor for glass fibre/epoxy, assuming inter-fibre separation of 20 D.

EMBED Excel.Sheet.8

Fig. 6.9: Elastic constants of 2D random glass fibre-reinforced epoxy. Poissons ratio is multiplied by 10 for clarity.

Fig. 6.7: Tensile and shear stres distribution for a single fibre embedded in a matrix (Daniel & Ishai, 1994).

16-15

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eta0.9730941704

Gf9

Gm2.1

nf0.2Kf122.2222222222

nm0.4Km5

Em3

Ef220

glass

VfE1E2 = E3E3n12n13Kn23etaG12G13G23

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0.6133.211.411.40.280.2811.77730192720.654.64.63.5

0.65144.113.313.30.270.2713.27700663850.644.94.94.1

0.7154.915.815.80.260.2615.2143845090.635.35.34.9

VfE1E2 (HS)E3n12n13n23G12G13G23

0.429.86.56.50.330.330.654.34.32.0

0.4533.27.27.20.320.320.644.84.82.2

0.536.58.18.10.310.310.635.35.32.5

0.5539.99.19.10.300.300.626.06.02.8

0.643.210.410.40.290.290.606.76.73.2

0.6546.611.911.90.280.280.597.67.63.7

0.749.913.813.80.270.270.578.68.64.4

VfE1E2 (HS)E3n12n13n23G12G13G23

0.489.86.86.80.320.320.713.54.94.3

0.45100.77.77.70.310.310.703.75.24.8

0.5111.58.78.70.300.300.704.05.65.3

0.55122.49.99.90.290.290.694.35.96.0

0.6133.211.411.40.280.280.694.66.26.7

0.65144.113.313.30.270.270.684.96.67.6

0.7154.915.815.80.260.260.675.36.98.6

Sheet1

E1 (equation 6.6)

E2 (equation 6.11)

E2 (equation 6.10)

E2 (equation 6.9)

E1

E2 simple

E2 mod

E2 HS

fibre volume fraction

modulus (GPa)

Sheet2

E

G

n x 10

E

G

v

fibre volume fraction

modulus (GPa) and Poisson's ratio x 10

Sheet3

_1119161403.unknown

_1100952767.xlsChart1

0.2421171961

0.5224260911

0.8021670925

0.9010754899

0.9505377448

fibre length (mm)

length correction factor

Sheet1

Vf0.1Ef7.00E+10Gm/Ef1.53E-02

D1.00E-05Gm1071428571.42857

a0.000028025

R9.01E-06

lengthbetaeta

1.00E-042.02E+042.42E-01

2.00E-042.02E+045.22E-01

0.00052.02E+048.02E-01

1.00E-032.02E+049.01E-01

2.00E-032.02E+049.51E-01

5.00E-032.02E+049.80E-01

1.00E-022.02E+049.90E-01

0.10.2421171961

0.20.5224260911

0.50.8021670925

10.9010754899

20.9505377448

Sheet1

fibre length (mm)

length correction factor

Sheet2

Sheet3

_1100598498.xlsChart2

333.57142857143

6.353.15078769693.74933047673.2885857861

9.73.3175355453.94588500563.6063348416

13.053.50291909924.16418798333.9579030977

16.43.71024734984.40806045344.3489932886

19.753.94366197184.68227424754.7866666667

23.14.20841683374.99286733245.2797731569

26.454.51127819555.34759358295.839556004

29.84.86111111115.75657894746.4805194805

33.155.2697616066.23330365097.2217036173

36.55.75342465756.79611650498.0886075949

39.856.3348416297.47065101399.1161825726

43.27.04697986588.293838862610.3536585366

46.557.93950850669.320905459411.8726655348

49.99.090909090910.638297872313.7816091954

53.2510.632911392412.38938053116.2527472527

56.612.804878048814.830508474619.5773195876

E1 (equation 6.6)

E2 (equation 6.11)

E2 (equation 6.10)

E2 (equation 6.9)

E1

E2 simple

E2 mod

E2 HS

fibre volume fraction

modulus (GPa)

Sheet1

fibre70Em'3.5714285714eta0.9178082192

matrix3

E1E2 simpleE2 modE2 HS

0333.57142857143

0.056.353.15078769693.74933047673.2885857861

0.19.73.3175355453.94588500563.6063348416

0.1513.053.50291909924.16418798333.9579030977

0.216.43.71024734984.40806045344.3489932886

0.2519.753.94366197184.68227424754.7866666667

0.323.14.20841683374.99286733245.2797731569

0.3526.454.51127819555.34759358295.839556004

0.429.84.86111111115.75657894746.4805194805

0.4533.155.2697616066.23330365097.2217036173

0.536.55.75342465756.79611650498.0886075949

0.5539.856.3348416297.47065101399.1161825726

0.643.27.04697986588.293838862610.3536585366

0.6546.557.93950850669.320905459411.8726655348

0.749.99.090909090910.638297872313.7816091954

0.7553.2510.632911392412.38938053116.2527472527

0.856.612.804878048814.830508474619.5773195876

Sheet1

E1 (equation 6.6)

E2 (equation 6.11)

E2 (equation 6.10)

E2 (equation 6.9)

E1

E2 simple

E2 mod

E2 HS

fibre volume fraction

modulus (GPa)

Sheet2

Sheet3