'Rules of Mixtures' are mathematicalexpressions which give some property ofthe composite in terms of the properties,quantity and arrangement of itsconstituents.

They may be based on a number ofsimplifying assumptions, and their use indesign should tempered with extremecaution!

For a general composite, total volume V,containing masses of constituents Ma, Mb, Mc,...the composite density is

......���

����

VM

VM

VMMM bacba

�

In terms of the densities and volumes of theconstituents:

...����

Vv

Vv

Vv ccbbaa ���

�

But va / V = Va is the volume fraction of theconstituent a, hence:

For the special case of a fibre-reinforced matrix:

...���� ccbbaa VVV ����

mmffmfffmmff VVVVV �������� �������� )()1(

since Vf + Vm = 1

0

500

1000

1500

2000

2500

3000

0 0.2 0.4 0.6 0.8 1

fibre volume fraction

kg/m

3

�f

�m

Unidirectional fibres are the simplestarrangement of fibres to analyse.They provide maximum properties in thefibre direction, but minimum properties inthe transverse direction.

fibre direction

transversedirection

We expect the unidirectional composite tohave different tensile moduli in differentdirections.These properties may be labelled inseveral different ways:

E1, E||

E2, E�

By convention, the principal axes of the ply arelabelled ‘1, 2, 3’. This is used to denote the factthat ply may be aligned differently from thecartesian axes x, y, z.

1

3

2

We make the following assumptions indeveloping a rule of mixtures:

• Fibres are uniform, parallel andcontinuous.

• Perfect bonding between fibreand matrix.

• Longitudinal load produces equalstrain in fibre and matrix.

• A load applied in the fibre direction isshared between fibre and matrix:

F1 = Ff + Fm

• The stresses depend on the cross-sectional areas of fibre and matrix:

�1A = �fAf + �mAm

where A (= Af + Am) is the total cross-sectional area of the ply

• Applying Hooke’s law:E1�1 A = Ef�f Af + Em�m Amwhere Poisson contraction has been ignored

• But the strain in fibre, matrix andcomposite are the same, so�1 = �f = �m, and:

E1 A = Ef Af + Em Am

Dividing through by area A:E1 = Ef (Af / A) + Em (Am / A)

But for the unidirectional ply, (Af / A) and(Am / A) are the same as volume fractionsVf and Vm = 1-Vf. Hence:

E1 = Ef Vf + Em (1-Vf)

E1 = Ef Vf + Em ( 1-Vf )

Note the similarity to the rules of mixtureexpression for density.

In polymer composites, Ef >> Em, soE1 � Ef Vf

Rule of mixtures tensile modulus (glass fibre/polyester)

010

2030

4050

60

0 0.2 0.4 0.6 0.8

fibre volume fraction

tens

ile m

odul

us (G

Pa)

UDbiaxialCSM

Rule of mixtures tensile modulus (T300 carbon fibre)

0

50

100

150

200

0 0.2 0.4 0.6 0.8

fibre volume fraction

tens

ile m

odul

us (G

Pa)

UDbiaxialquasi-isotropic

(source: Hull, Introductionto Composite Materials,CUP)

For the transverse stiffness, a load isapplied at right angles to the fibres.

The model is very much simplified, andthe fibres are lumped together:

L2matrix

fibreLfLm

It is assumed that the stress is thesame in each component (�2 = �f = �m).

Poisson contraction effects are ignored.

�2 �2

The total extension is �2 = �f + �m, sothe strain is given by:

�2L2 = �fLf + �mLm

so that �2 = �f (Lf / L2) + �m (Lm / L2)

�2 �2

LfLm

But Lf / L2 = Vf and Lm / L2 = Vm = 1-Vf

So �2 = �f Vf + �m (1-Vf)

and �2 / E2 = �f Vf / Ef + �m (1-Vf) / Em

�2 �2

LfLm

But �2 = �f = �m, so that:

�2 �2

LfLm

m

f

f

f

EV

EV

E)1(1

2

��� )1(2

fffm

mf

VEVEEEE

���or

Rule of mixtures - transverse modulus (glass/epoxy)

02468

10121416

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

fibre volume fraction

E 2 (G

Pa)

If Ef >> Em,

E2 � Em / (1-Vf)

Note that E2 is not particularly sensitive to Vf.

If Ef >> Em, E2 is almost independent of fibreproperty:

Rule of mixtures - transverse modulus

02468

10121416

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

fibre volume fraction

E 2 (G

Pa)

carbon/epoxy

glass/epoxy

The transverse modulus is dominated bythe matrix, and is virtually independentof the reinforcement.

The transverserule of mixtures isnot particularlyaccurate, due tothe simplificationsmade - Poissoneffects are notnegligible, and thestrain distributionis not uniform:

(source: Hull, Introduction toComposite Materials, CUP)

Many theoretical studies have beenundertaken to develop bettermicromechanical models (eg the semi-empirical Halpin-Tsai equations).A simple improvement for transversemodulus is

)1(2fffm

mf

VEVEEEE

���

�� 21 m

mm

EE��

��where

E = �L �o Ef Vf + Em (1-Vf )�L is a length correction factor. Typically, �L � 1for fibres longer than about 10 mm.

�o corrects for non-unidirectional reinforcement:�o

unidirectional 1.0biaxial 0.5biaxial at �45o 0.25random (in-plane) 0.375random (3D) 0.2

� �� �2/L

2/Ltanh1L�

����

� �DR2lnDEG8

2f

m��

00.10.20.30.40.50.60.70.80.9

1

0 0.5 1 1.5 2

fibre length (mm)

leng

th c

orre

ctio

n fa

ctor

Theoretical length correctionfactor for glass fibre/epoxy,assuming inter-fibreseparation of 20 D.

For aligned short fibre composites (difficult toachieve in polymers!), the rule of mixtures formodulus in the fibre direction is:

)V(EVEηE fmffL ��� 1

The length correction factor (�L) can be derivedtheoretically. Provided L > 1 mm, �L > 0.9

For composites in which fibres are not perfectlyaligned the full rule of mixtures expression is used,incorporating both �L and �o.

285

183 EEE ��

241

181 EEG ��

���

In short fibre-reinforced thermosetting polymer composites, itis reasonable to assume that the fibres are always well abovetheir critical length, and that the elastic properties aredetermined primarily by orientation effects.

The following equations give reasonably accurate estimatesfor the isotropic in-plane elastic constants:

where E1 and E2 are the ‘UD’ valuescalculated earlier

Rule of mixtures tensile modulus (glass fibre/polyester)

010

2030

4050

60

0 0.2 0.4 0.6 0.8

fibre volume fraction

tens

ile m

odul

us (G

Pa)

UDbiaxialCSM

Rule of mixtures tensile modulus (T300 carbon fibre)

0

50

100

150

200

0 0.2 0.4 0.6 0.8

fibre volume fraction

tens

ile m

odul

us (G

Pa)

UDbiaxialquasi-isotropic

Larsson & Eliasson,Principles of Yacht Design

Larsson & Eliasson,Principles of Yacht Design

• Shear modulus:

• Poisson’s ratio:

• Thermal expansion:

m

f

f

f

GV

GV

G)1(1

12

���

)1(12 fmff VV ��� ���

� �

1212

11

)1)(1()1(

)1(1

�������

���

������

���

mfmfff

fmmfff

VV

VEVEE