Transcript Composite Design - Plymouth University

Rules of Mixture
for 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
Ma  Mb  Mc  ... Ma Mb



 ...
V
V
V
In terms of the densities and volumes of the
constituents:
v a a v b b v c c



 ...
V
V
V
Density
But va / V = Va is the volume fraction of the
constituent a, hence:
  Va a  Vb b  Vc c  ...
For the special case of a fibre-reinforced matrix:
  Vf f  Vm m  Vf f  (1 Vf )m  Vf ( f  m )  m
since Vf + Vm = 1
Rule of mixtures density for
glass/epoxy composites
3000
f
2500
kg/m 3
2000
1500
m
1000
500
0
0
0.2
0.4
0.6
fibre volume fraction
0.8
1
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 ply
By 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.
3
1
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 crosssectional areas of fibre and matrix:
s1A = sfAf + smAm
where A (= Af + Am) is the total crosssectional area of the ply
Unidirectional ply - longitudinal
tensile modulus
• Applying Hooke’s law:
E1e1 A = Efef Af + Emem Am
where Poisson contraction has been ignored
• But the strain in fibre, matrix and
composite are the same, so
e1 = ef = em, 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
60
50
40
UD
30
biaxial
20
CSM
10
0
0
0.2
0.4
0.6
0.8
fibre volume fraction
Rule of mixtures tensile modulus
(T300 carbon fibre)
tensile modulus (GPa)
tensile modulus (GPa)
Rule of mixtures tensile modulus
(glass fibre/polyester)
200
150
UD
100
biaxial
quasi-isotropic
50
0
0
0.2
0.4
fibre volume fraction
0.6
0.8
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
Lm
Lf
Unidirectional ply transverse tensile modulus
s2
s2
It is assumed that the stress is the
same in each component (s2 = sf = sm).
Poisson contraction effects are ignored.
Unidirectional ply transverse tensile modulus
s2
s2
Lm
Lf
The total extension is d2 = df + dm, so
the strain is given by:
e2L2 = efLf + emLm
so that e2 = ef (Lf / L2) + em (Lm / L2)
Unidirectional ply transverse tensile modulus
s2
s2
Lm
Lf
But Lf / L2 = Vf and Lm / L2 = Vm = 1-Vf
So
e2 = ef Vf + em (1-Vf)
and
s2 / E2 = sf Vf / Ef + sm (1-Vf) / Em
Unidirectional ply transverse tensile modulus
s2
s2
Lm
Lf
But s2 = sf = sm, so that:
1 Vf (1  Vf )


E 2 Ef
Em
or
Ef E m
E2 
EmVf  Ef (1  Vf )
Rule of mixtures - transverse modulus
(glass/epoxy)
E2 (GPa)
16
14
12
10
8
6
If Ef >> Em,
4
2
0
E2  Em / (1-Vf)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
fibre volume fraction
Note that E2 is not particularly sensitive to Vf.
If Ef >> Em, E2 is almost independent of fibre
property:
Rule of mixtures - transverse modulus
16
carbon/epoxy
14
E2 (GPa)
12
10
8
6
glass/epoxy
4
2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
fibre volume fraction
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 semiempirical Halpin-Tsai equations).
A simple improvement for transverse
modulus is
Ef Em
E2 
Em Vf  Ef (1  Vf )
Em
where E m 
2
1m
Generalised rule of mixtures for
tensile modulus
E = hL ho Ef Vf + Em (1-Vf )
hL is a length correction factor. Typically, hL  1
for fibres longer than about 10 mm.
ho corrects for non-unidirectional reinforcement:
ho
unidirectional
biaxial
biaxial at 45o
random (in-plane)
random (3D)
1.0
0.5
0.25
0.375
0.2
Theoretical Orientation Correction Factor
ho = S ai cos4 qi
Where the summation is carried out over all the
different orientations present in the
reinforcement. ai is the proportion of all fibres
with orientation qi.
E.g. in a ±45o bias fabric,
ho = 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:
1
0.95
0.9
0.85
0.8
0.75
0.7
0
5
10
15
Crimp angle (degrees)
20
25
30
Theoretical length correction factor
tanhL / 2
hL  1 
L / 2

8Gm
E f D2 ln2R D
1
length correction factor
0.9
Theoretical length correction
factor for glass fibre/epoxy,
assuming inter-fibre
separation of 20 D.
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
fibre length (mm)
1.5
2
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:
E  ηLEfVf  Em(1 Vf )
The length correction factor (hL) can be derived
theoretically. Provided L > 1 mm, hL > 0.9
For composites in which fibres are not perfectly
aligned the full rule of mixtures expression is used,
incorporating both hL and ho.
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:
E  E1  E2
3
8
5
8
G  81 E1  41 E2
E

1
2G
where E1 and E2 are the ‘UD’ values
calculated earlier
60
50
40
UD
30
biaxial
20
CSM
10
0
0
0.2
0.4
0.6
0.8
fibre volume fraction
Rule of mixtures tensile modulus
(T300 carbon fibre)
tensile modulus (GPa)
tensile modulus (GPa)
Rule of mixtures tensile modulus
(glass fibre/polyester)
200
150
UD
100
biaxial
quasi-isotropic
50
0
0
0.2
0.4
fibre volume fraction
0.6
0.8
Rule of mixtures elastic modulus
glass fibre / epoxy resin
60
50
GPa
40
UD
biaxial
random
30
20
10
0
0.1
0.3
0.5
fibre volume fraction
0.7
GPa
Rule of mixtures elastic modulus
HS carbon / epoxy resin
180
160
140
120
100
80
60
40
20
0
UD
biaxial UD
quasi-isotropic UD
plain woven
0.4
0.5
0.6
fibre volume fraction
0.7
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:
1
Vf (1  Vf )


G12 Gf
Gm
• Poisson’s ratio:
12   fVf   m (1 Vf )
• Thermal expansion:
1
a1  (a f EfVf  a mEm 1  Vf )
E1
a 2  a fVf (1   f )  a m (1  Vf )(1   m )  a112