Rules of Mixture for Elastic Properties. 'Rules of Mixtures' are mathematical expressions which give...
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Transcript of Rules of Mixture for Elastic Properties. 'Rules of Mixtures' are mathematical expressions which give...
'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
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.
Unidirectional ply - longitudinal tensile modulus
• 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
Unidirectional ply - longitudinal tensile modulus
• 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
Unidirectional ply - longitudinal tensile modulus
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)
Unidirectional ply - longitudinal tensile modulus
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
fibre volume fraction
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
fibre volume fraction
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
Unidirectional ply - transverse tensile modulus
It is assumed that the stress is the same in each component (2 = f = m).
Poisson contraction effects are ignored.
22
Unidirectional ply - transverse tensile modulus
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
Unidirectional ply - transverse tensile modulus
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
Unidirectional ply - transverse tensile modulus
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
fibre volume fraction
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
fibre volume fraction
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)
Unidirectional ply - transverse tensile modulus
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
fibre volume fraction
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
fibre volume fraction
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
fibre volume fraction
GP
a
UD
biaxial
random
Rule of mixtures elastic modulusHS carbon / epoxy resin
020406080
100120140160180
0.4 0.5 0.6 0.7
fibre volume fraction
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