rules of mixture for elastic properties. 'rules of mixtures' are mathematical expressions...

Rules of Mixturefor Elastic Properties

'Rules of Mixtures' are mathematical expressions which give some property of the composite in terms of the properties, quantity and arrangement of its constituents.

They may be based on a number of simplifying assumptions, and their use in design should tempered with extreme caution!

Density

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

......

V

M

V

M

V

MMM bacba

In terms of the densities and volumes of the constituents:

...V

v

V

v

V

v ccbbaa

Density

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

For the special case of a fibre-reinforced matrix:

... ccbbaa VVV

mmffmfffmmff VVVVV )()1(

since Vf + Vm = 1

Rule of mixtures density for glass/epoxy composites

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

Micromechanical models for stiffness

Unidirectional ply

Unidirectional fibres are the simplest arrangement of fibres to analyse.

They provide maximum properties in the fibre direction, but minimum properties in the transverse direction.

fibre direction

transverse direction

Unidirectional ply

We expect the unidirectional composite to have different tensile moduli in different directions.

These properties may be labelled in several different ways:

E1, E||

E2, E

Unidirectional plyBy convention, the principal axes of the ply are labelled ‘1, 2, 3’. This is used to denote the fact that ply may be aligned differently from the cartesian axes x, y, z.

1

3

2

Unidirectional ply - longitudinal tensile modulus

We make the following assumptions in developing a rule of mixtures:

• Fibres are uniform, parallel and continuous.

• Perfect bonding between fibre and matrix.

• Longitudinal load produces equal strain in fibre and matrix.


• A load applied in the fibre direction is shared 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: E11 A = Eff Af + Emm Am

where Poisson contraction has been ignored

• But the strain in fibre, matrix and composite 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 fractions Vf and Vm = 1-Vf. Hence:

E1 = Ef Vf + Em (1-Vf)


E1 = Ef Vf + Em ( 1-Vf )

Note the similarity to the rules of mixture expression for density.

In polymer composites, Ef >> Em, so

E1 Ef Vf

Rule of mixtures tensile modulus (glass fibre/polyester)

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8


ten

sile

mo

du

lus

(GP

a)

UD

biaxial

CSM

Rule of mixtures tensile modulus (T300 carbon fibre)

0

50

100

150

200

0 0.2 0.4 0.6 0.8


ten

sile

mo

du

lus

(GP

a)

UD

biaxial

quasi-isotropic

This rule of mixtures is a good fit to experimental data

(source: Hull, Introduction to Composite Materials, CUP)

Unidirectional ply - transverse tensile modulus

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

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

L2

matrix

fibre

LfLm


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

Poisson contraction effects are ignored.

22


The total extension is 2 = f + m, so the strain is given by:

2L2 = fLf + mLm

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

22

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

22

LfLm


But 2 = f = m, so that:

22

LfLm

m

f

f

f

E

V

E

V

E

)1(1

2

)1(2fffm

mf

VEVE

EEE

or

Rule of mixtures - transverse modulus (glass/epoxy)

0

2

4

6

8

10

12

14

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8


E2

(GP

a)

If Ef >> Em,

E2 Em / (1-Vf)

Note that E2 is not particularly sensitive to Vf.

If Ef >> Em, E2 is almost independent of fibre property:

Rule of mixtures - transverse modulus

0

2

4

6

8

10

12

14

16

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8


E2

(GP

a)

carbon/epoxy

glass/epoxy

The transverse modulus is dominated by the matrix, and is virtually independent of the reinforcement.

The transverse rule of mixtures is not particularly accurate, due to the simplifications made - Poisson effects are not negligible, and the strain distribution is not uniform:

(source: Hull, Introduction to Composite Materials, CUP)


Many theoretical studies have been undertaken to develop better micromechanical models (eg the semi-empirical Halpin-Tsai equations).

A simple improvement for transverse modulus is

)1(2fffm

mf

VEVE

EEE

21 m

mm

EE

where

Generalised rule of mixtures for tensile modulus

E = L o Ef Vf + Em (1-Vf )

L is a length correction factor. Typically, L 1 for 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

Theoretical Orientation Correction Factor

o = icos4 i

Where the summation is carried out over all the different orientations present in the reinforcement. i is the proportion of all fibres with orientation i.

E.g. in a ±45o bias fabric,

o = 0.5 cos4 (45o) + 0.5 cos4 (-45o)

Assuming that the fibre path in a plain woven fabric is sinusoidal, a further correction factor can be derived for non-straight fibres:

0.7

0.75

0.8

0.85

0.9

0.95

1

0 5 10 15 20 25 30

Crimp angle (degrees)

Theoretical length correction factor

2/L

2/Ltanh1L

DR2lnDE

G82

f

m

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.5 1 1.5 2

fibre length (mm)

len

gth

co

rrec

tio

n f

acto

r

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

Stiffness of short fibre composites

For aligned short fibre composites (difficult to achieve in polymers!), the rule of mixtures for modulus in the fibre direction is:

)V(EVEηE fmffL 1

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

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

28

518

3 EEE

24

118

1 EEG

1G2

E

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:

where E1 and E2 are the ‘UD’ values calculated earlier

Rule of mixtures tensile modulus (glass fibre/polyester)

0

10

20

30

40

50

60

0 0.2 0.4 0.6 0.8


ten

sile

mo

du

lus

(GP

a)

UD

biaxial

CSM

Rule of mixtures tensile modulus (T300 carbon fibre)

0

50

100

150

200

0 0.2 0.4 0.6 0.8


ten

sile

mo

du

lus

(GP

a)

UD

biaxial

quasi-isotropic

Rule of mixtures elastic modulusglass fibre / epoxy resin

0

10

20

30

40

50

60

0.1 0.3 0.5 0.7


GP

a

UD

biaxial

random

Rule of mixtures elastic modulusHS carbon / epoxy resin

020406080

100120140160180

0.4 0.5 0.6 0.7


GP

a

UD

biaxial UD

quasi-isotropic UD

plain woven

Rules of mixture properties for CSM-polyester laminates

Larsson & Eliasson, Principles of Yacht Design

Rules of mixture properties for glass woven roving-polyester laminates

Larsson & Eliasson, Principles of Yacht Design

Other rules of mixtures

• Shear modulus:

• Poisson’s ratio:

• Thermal expansion:

m

f

f

f

G

V

G

V

G

)1(1

12

)1(12 fmff VV

1212

11

)1)(1()1(

)1(1

mfmfff

fmmfff

VV

VEVEE

rules of mixture for elastic properties. 'rules of mixtures' are mathematical expressions...

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