1. 1
Thermal
distortion
in
composites
Author:
Roland
Papp
Experiment
performed
on
3
and
6
November
2015
in
the
Department
of
Materials
Science
and
Metallurgy
of
the
University
of
Cambridge.
Abstract
This
experiment
is
the
investigation
of
the
distortion
of
carbon
fibre
reinforced
Nylon
6
composites
due
to
temperature
change.
We
produced
laminates
of
different
stacking
sequences
(uniaxial,
simple
cross-­‐ply
and
symmetrical
cross-­‐ply)
by
hot
pressing.
After
cooling
down
the
simple
cross-­‐ply
had
a
saddle
shape,
while
the
others
with
symmetrical
stacking
were
(mostly)
flat.
We
performed
4-­‐point
bending
to
get
the
stiffness
of
the
uniaxial
laminate:
𝐸!"#!$ = 86.3 GPa
𝐸!"#$%&'"%' = 8.57 GPa
From
this
we
worked
out
the
stiffness
of
the
carbon
fibres
only:
𝐸!!"#$% = 168 GPa
𝐸!!"#$%&'$& = 16.1 GPa
In
the
next
part
of
the
experiment
we
put
a
cross-­‐ply
strip
into
liquid
nitrogen
to
measure
the
curvature
after
bending.
It
bent
just
as
expected
from
a
bi-­‐material
strip.
From
the
curvature
and
previous
data
we
calculated
the
thermal
expansivity
of
the
Nylon
6
matrix
to
be:
𝛼! = 30 ∙ 10!!
K!!
Unlike
the
stiffness
data
this
did
not
happen
to
be
very
accurate,
however
this
experiment
was
not
for
simply
measuring
the
thermal
expansion
coefficient
of
Nylon
6
in
a
rather
complicated
way,
but
to
understand
composites’
reactions
to
temperature
changes
and
these
reactions’
dependence
on
the
stacking
sequence.
1 Introduction
This
report
is
about
an
experiment
performed
in
the
Department
of
Materials
Science
and
Metallurgy
of
the
University
of
Cambridge.
We
produced
carbon
fibre
reinforced
composite
laminates
with
different
stacking
sequences
(uniaxial,
simple
cross-­‐ply
and
symmetrical
cross-­‐ply)
and
investigated
their
properties.
We
took
precise
computer
controlled
4-­‐point
bending
measurements
to
work
out
the
stiffness
of
the
composite
in
axial
and
transverse
directions.
Then
by
the
Voigt
and
Halpin-­‐Tsai
expressions
we
calculated
the
Young
modulus
of
carbon
fibres
in
both
axial
and
transverse
directions.

2. 2
In
the
next
part
of
the
experiment
we
measured
the
bending
of
a
simple
cross-­‐ply
composite
due
to
great
temperature
change
by
putting
it
in
liquid
nitrogen.
We
noted
the
curvature
and
by
using
data
from
the
previous
part
we
calculated
the
thermal
expansion
coefficient
of
the
Nylon
6
matrix.
At
the
end
we
investigated
the
composite
samples
under
microscope
to
search
for
porosities
and
to
get
a
clear
view
of
the
distribution
of
carbon
fibres.
2 Theoretical
Background1
2.1
Stiffness
The
second
moment
of
area
of
a
section
of
a
beam
is
𝐼 =
𝑤ℎ!
12
For
a
4-­‐point
bending
specified
in
Figure
2
the
central
deflection
is
𝛿 =
𝐹𝑠 3𝐿!
− 4𝑠!
48𝐸𝐼
Therefore
𝐸 =
𝑠 3𝐿!
− 4𝑠!
48𝐼 ∙
d𝛿
d𝐹
By
knowing
the
Young
modulus
of
the
matrix
𝐸!
and
the
fibres
𝐸!
and
the
fibre
volume
fraction
𝑓
we
can
work
out
the
stiffness
of
a
strip
in
axial
(1)
and
perpendicular
(2)
directions
by
the
following
equations:
Voigt
equation:
𝐸! = 𝑓𝐸! + 1 − 𝑓 𝐸!
Halpin-­‐Tsai
expression:
𝐸! =
𝐸! 1 + 𝜂𝑓
1 − 𝜂𝑓
, where 𝜂 =
𝐸!
𝐸!
− 1
𝐸!
𝐸!
+ 1
These
equations
can
be
rearranged
for
the
stiffness
of
the
fibres
in
two
directions:
𝐸!! = 𝑓!!
𝐸! − 1 − 𝑓 𝐸!
𝐸!! = 𝐸! ∙
1 + 𝜂
1 − 𝜂
, where 𝜂 = 𝑓!!
∙
𝐸! − 𝐸!
𝐸! + 𝐸!
2.2
Thermal
Contraction
The
curvature
due
to
a
misfit
strain
Δ 𝜀
in
a
bi-­‐material
is
𝜅 =
6𝐸! 𝐸! ℎ! + ℎ! ℎ!ℎ!
𝐸!
!
ℎ!
!
+ 4𝐸! 𝐸!ℎ!
!
ℎ! + 6𝐸! 𝐸!ℎ!
!
ℎ!
!
+ 4𝐸! 𝐸!ℎ!ℎ!
!
+ 𝐸!
!
ℎ!
! Δ𝜀
where
𝐸
and
ℎ
are
the
Young
modulus
and
thickness
of
the
strips.

3. 3
Figure
1
—
Geometry
and
equation
for
the
relationship
between
the
curvature,
𝜿,
and
the
consequent
lateral
displacement,
𝜹
for
any
curved
piece
of
material.
The
Poisson
ratio
of
the
composite
is
𝜈!"! = 𝑓𝜈! + 1 − 𝑓 𝜈!
where
𝜈!
and
𝜈!
are
that
of
the
fibre
and
the
matrix.
The
predicted
values
for
the
axial
and
transverse
thermal
expansivities
for
a
single
uniaxial
ply
of
composite
are
𝛼!
!"
=
𝛼! 1 − 𝑓 𝐸! + 𝛼! 𝑓𝐸!
1 − 𝑓 𝐸! + 𝑓𝐸!
𝛼!
!"
= 𝛼! 1 − 𝑓 1 + 𝜈! + 𝛼! 𝑓 1 + 𝜈! − 𝛼!
!"
𝜈!"!
By
definition
Δ𝛼 =
Δ𝜀
Δ𝑇

4. 4
3 Method
3.1 Production
We
used
a
unidirectional
pre-­‐preg
tape
made
of
carbon
fibre
reinforced
Nylon
6
to
cut
out
12
plies
for
each
of
the
three
laminate
samples.
We
stacked
the
12
plies
onto
each
other
creating
the
following
sequences:
i. Unidirectional
—
0°!"
ii. Simple
cross-­‐ply
—
0°!/90°!
iii. Symmetrical
cross-­‐ply
—
0°!/90°!/0°!
We
put
these
stacks
into
a
hot
press
to
heat
them
up
to
280℃
under
a
pressure
of
50
atmospheres
for
about
30
minutes
and
cool
them
down
slowly.
3.2
After
cooling
We
examined
the
laminates
qualitatively.
The
difference
in
thermal
distortion
between
the
three
stacking
sequences
was
obvious
to
the
eye.
3.3
Stiffness
measurement
We
cut
out
two
strips
from
the
unidirectional
(i)
sample:
One
parallel
and
one
perpendicular
to
the
fibres.
The
dimensions
of
the
strips
are
in
Table
1.
We
used
a
scanning
laser
extensometer
to
perform
displacement
controlled
4-­‐point
bending
tests
on
these.
The
data
was
captured
by
a
computer
and
put
in
an
Excel
table
format.
The
loading
configuration
is
shown
on
Figure
2.
Figure
2
—
Loading
configuration
during
4-­‐point
bending,
showing
the
forces,
dimensions
and
central
deflection
3.4
Thermal
contraction
We
also
cut
out
a
strip
from
the
simple
cross-­‐ply
laminate
(ii)
to
perform
a
thermal
extension
experiment.
After
fixing
it
securely
in
a
tray
we
poured
liquid
nitrogen
into
the
tray
to
cool
the
strip
down.
The
configuration
is
shown
below
on
Figure
3.

5. 5
Figure
3
Generation
of
a
misfit
strain
and
hence
a
curvature
in
a
bi-­‐material
strip.
We
used
this
cooling
configuration
for
the
thermal
contraction
measurement.
We
measured
the
deflection
𝛿
and
calculated
the
curvature
𝜅
in
the
way
explained
in
Figure
1.
3.5
Microscopic
examination
As
the
microscopic
examination
destroyed
the
samples,
we
performed
it
at
the
end
of
the
experiment.
We
mounted
and
polished
pieces
of
the
three
strips
to
look
for
porosities
and
check
the
alignment
and
distribution
of
the
fibers
under
optical
microscope.
4 Results
Some
data:
𝐿 = 100 mm
𝑠 = 25 mm
Table
1
—
Book
values
of
a
few
physical
properties
𝐸 GPa
𝜌 Mg/m!
𝛼 K!!
∙ 10!!
𝜈
Carbon
fibre
(axial)
-­‐
1.75
-­‐1
0.2
Carbon
fibre
(transverse)
-­‐
1.75
10
0.2
Nylon
6
5
1.1
-­‐
0.4

6. 6
Table
2
—
Measured
properties
of
the
strips
𝑚 (g)
𝑙 (mm)
𝑤 (mm) ℎ (mm) 𝜌 (kg/m!
) 𝐼 (mm!
)
Uniaxial
parallel
(1)
5.64
116.2
21.9
1.7
1304
8.97
Uniaxial
transverse
(2)
5.34
119.8
18.14
1.72
1429
7.69
Simple
cross-­‐ply
6.2
121.1
21.1
1.58
1536
6.94
4.1
After
cooling
When
the
laminates
have
cooled
down
we
looked
at
them
and
found
clear
differences:
i. The
unidirectional
was
flat.
It
was
much
easier
to
bend
it
in
one
direction
than
in
the
other
parallel
to
it.
ii. The
simple
cross-­‐ply
was
saddle-­‐shaped
and
stiff
in
all
directions.
iii. The
symmetrical
cross-­‐ply
was
also
slightly
saddle
shaped,
however
by
a
gently
force
it
could
be
clicked
to
the
other
side,
giving
two
symmetrical
stable
phases.
4.2
Fibre
volume
fraction
The
average
density
of
the
three
strips
is:
𝜌 = 1423 kg/m!
The
fibre
volume
fraction
of
the
composite
is
therefore:
𝑓 =
𝜌 − 𝜌!"!"#
𝜌!"#$%&'($#) − 𝜌!"#$%
= 49.7% ≈ 50%
The
above
equation
holds
if
and
only
if
the
composite
is
free
of
porosities.
To
check
this
we
took
micrographs
of
the
three
strips
(see
Micrograph
1-­‐3).
I
do
not
see
porosity
anywhere
in
the
samples,
not
even
at
the
interface
of
differently
oriented
plies.
The
fibre
volume
fraction
is
indeed
50%.

7. 7
Micrograph
1
—
Unidirectional
composite,
axial
view
Micrograph
2
—
Unidirectional
composite,
transverse
view

8. 8
Micrograph
3
—
Cross-­‐ply
composite
4.3
Stiffness
measurement
The
gradients
of
the
plots
for
the
axial
(1)
and
transverse
(2)
loadings
(Figure
4
and
Figure
5)
give
the
stiffness
in
these
directions:
𝐸! =
𝑠 3𝐿!
− 4𝑠!
48 ∙ 𝐼! ∙
d𝛿
d𝐹 !
=
𝑠 3𝐿!
− 4𝑠!
48𝐼! ∙ −gradient!"#$%
= 86.3 GPa
𝐸! =
𝑠 3𝐿!
− 4𝑠!
48 ∙ 𝐼! ∙
d𝛿
d𝐹 !
=
𝑠 3𝐿!
− 4𝑠!
48𝐼! ∙ −gradient!"#$%&'"%'
= 8.57 GPa

9. 9
Figure
4
—
Axial
loading
data
of
the
uniaxial
composite
strip
Figure
5
—
Transverse
loading
data
of
the
uniaxial
composite
strip
y
=
-­‐0.0185x
+
22.501
R²
=
0.99994
2.05E+01
2.10E+01
2.15E+01
2.20E+01
2.25E+01
2.30E+01
0.00E+00
2.00E+01
4.00E+01
6.00E+01
8.00E+01
1.00E+02
1.20E+02
δ0-­‐δ
(mm)
F(N)
Axial
y
=
-­‐0.2174x
+
21.618
R²
=
0.99989
1.92E+01
1.94E+01
1.96E+01
1.98E+01
2.00E+01
2.02E+01
2.04E+01
2.06E+01
2.08E+01
2.10E+01
2.12E+01
2.14E+01
0.00E+00
2.00E+00
4.00E+00
6.00E+00
8.00E+00
1.00E+01
1.20E+01
δ0-­‐δ
(mm)
F(N)
Transverse

10. 10
Using
the
rearranged
Voigt
and
Halpin-­‐Tsai
expressions
(see
Theoretical
Background)
the
axial
and
transverse
stiffness
of
the
carbon
fibres
can
be
obtained:
𝐸!!"#$% = 𝑓!!
𝐸! − 1 − 𝑓 𝐸! = 168 GPa
𝜂 = 𝑓!!
∙
𝐸! − 𝐸!
𝐸! + 𝐸!
= 0.526
𝐸!!"#$%&'$& = 𝐸! ∙
1 + 𝜂
1 − 𝜂
= 16.1 GPa
4.4
Thermal
contraction
4.4.1
Measurement
Table
3
—
Directly
measured
data
after
cooling
𝑥 mm
𝛿 mm
Δ𝑇 (K)
85
10
-­‐220
The
curvature
follows
as
𝜅 =
2 sin tan!! 𝛿
𝑥
𝑥! + 𝛿!
= 2.73 m!!
The
thickness
of
each
of
the
two
layers
(one
layer
is
made
of
six
plies)
is
ℎ = ℎ! = ℎ! =
1
2
ℎ!"#$%& !"#$$!!"# = 0.79 mm
The
equation
in
section
2.2
giving
the
curvature
from
the
misfit
strain
can
be
transformed
algebraically
to
give
the
misfit
strain
𝛥 𝜀
from
𝜅.
After
executing
the
simplification
ℎ = ℎ! = ℎ!
and
the
substitutions
𝐸! = 𝐸!
and
𝐸! = 𝐸!
it
follows
as:
Δ𝜀 = 𝜅
12 ∙ 𝐸! 𝐸!
𝐸!
!
+ 14𝐸! 𝐸! + 𝐸!
!
∙ ℎ
!!
= 4.34 ∙ 10!!
Therefore
the
difference
in
thermal
expansivities
between
the
layers
is
Δ𝛼 =
Δ𝜀
Δ𝑇
= 2.0 ∙ 10!!
K!!
4.4.2
Calculation
Using
the
expressions
in
section
2.2
and
data
given
in
Table
1
we
get
the
following
results:
𝜈!"! = 0.3

11. 11
Δ𝛼 = 𝛼!
!"
− 𝛼!
!"
= 𝑎! 1 − 𝑓 1 + 𝜈! + 𝛼!
!"
∙ 𝑓 1 + 𝜈! − 𝜈!"! + 1 𝛼!
!"
= 𝛼! 1 − 𝑓 1 + 𝜈! − 𝜈!"! + 1
1 − 𝑓 𝐸!
1 − 𝑓 𝐸! + 𝑓𝐸!
!"
+ 𝛼!
!"
𝑓 1 + 𝜈! − 𝜈!"! + 1
𝛼!
!"
∙ 𝑓 ∙ 𝐸!
!"
1 − 𝑓 𝐸! + 𝑓𝐸!
!" ≡ 𝛼! 𝑋 + 𝑌
By
rearranging
this
equation
we
can
work
out
the
expansivity
of
the
Nylon
matrix:
𝛼! =
Δ𝛼 − 𝑌
𝑋
= 3.0 ∙ 10!!
K!!
5 Discussion
5.1
After
cooling
The
qualitative
results
were
everywhere
in
line
with
our
expectations:
i. The
unidirectional
laminate
does
not
have
a
misfit
strain
due
to
thermal
contraction
and
is
therefore
flat.
As
the
fibres
are
much
stiffer
than
the
matrix,
it
is
much
easier
to
bend
the
laminate
parallel
to
than
perpendicular
to
the
fibres.
ii. The
saddle
shape
of
the
simple
cross-­‐ply
laminate
is
due
to
different
thermal
expansivity
in
axial
and
transverse
direction
in
every
layer.
As
the
stacking
sequence
is
asymmetric,
this
creates
a
saddle
shape.
iii. As
the
stacking
sequence
is
symmetric
in
this
laminate,
in
theory
it
should
have
been
flat.
As
all
plies
are
under
stress
due
to
each
other,
and
the
preparation
was
not
fully
precise
is
reasonable
to
have
two
phases.
The
symmetry
of
the
equilibrium
phases
show
the
symmetry
of
the
stacking.
5.2
Stiffness
measurement
The
micrographs
do
not
show
porosities,
so
we
can
use
the
𝑓 = 50%
value
for
the
fibre
volume
fraction.
We
can
also
notice
that
the
fibres
are
very
well
aligned
and
fairly
homogenously
distributed.
This
means
that
the
Voigt
and
Halpin-­‐Tsai
expressions
give
a
good
accuracy
for
this
calculation.
We
checked
the
samples
after
the
measurements
were
taken
and
found
no
bentness.
This
proved
that
the
strips
were
elastic
throughout
the
whole
experiment.
According
to
Vircon2
,
the
approximate
stiffness
of
standard
modulus
carbon
fibres
is
220 − 241 GPa,
which
is
only
slightly
higher
than
the
168 GPa
we
got
for
the
axial
stiffness.
Considering
that
it
depends
a
lot
on
the
certain
material
choice
we
can
take
it
as
a
plausible
value.
According
to
H.
Miyagawa’s
article,
Comparison
of
experimental
and
theoretical
transverse
elastic
modulus
of
carbon
fibers,3
different
experimental
and
theoretical
techniques
give
a
broad
range
of
values
for
the
transverse
Young
modulus
of
the
fibres
illustrated
by
Figure
6.
Our
obtained
value
of
16.1 GPa
is
very
close
to
the
median
of
these
values,
therefore
our
value
is
reasonable.

12. 12
Figure
6
—
Comparison
of
the
transverse
elastic
modulus
of
carbon
fibers,
experimentally
measured
by
Raman
spectroscopy
and
nanoindentation,
numerically
analyzed
by
FEM,
and
calculated
from
Mori–Tanaka,
Halpin–Tsai,
and
Uemura
equations.3
5.3
Thermal
contraction
The
very
high
error
(>20%)
in
this
experiment
built
up
even
more
during
the
following
calculation,
which
used
many
previously
determined
values
without
known
exact
errors.
Perhaps
this
is
why
the
thermal
expansivity
of
Nylon
6
was
measured
to
be
𝛼! = 30 ∙ 10!!
K!!
,
about
the
third
of
the
book
value4
of
80 − 85 ×10!!
K!!
.
In
the
context
of
this
experiment
this
is
a
quite
good
value
as
it
gives
the
right
order
of
magnitude.
This
confirms
that
the
other
values
measured
and
calculated
throughout
the
experiment
were
right
as
well
or
at
least
they
were
in
the
right
order
of
magnitude.
To
measure
the
thermal
expansion
coefficient
of
Nylon
6
there
are
many
ways,
which
are
obviously
more
precise
and
accurate
than
cooling
a
cross-­‐ply
carbon
fibre
reinforced
Nylon
strip.
Without
the
carbon
fibre
and
by
the
use
of
a
material
with
a
known
𝛼
the
same
bi-­‐material
experiment
could
be
performed
for
example.

13. 13
6 Conclusion
Although
the
only
difference
was
in
their
stacking
sequences,
the
three
laminates
looked
and
behaved
very
differently
after
having
been
cooled
down
to
room
temperature.
The
symmetric
stackings
had
symmetric
(basically
flat)
shapes,
while
the
simple
cross-­‐ply
laminate
ha
a
saddle
shape
due
to
the
different
thermal
expansivity
in
axial
and
transverse
direction
in
every
layer.
From
the
data
obtained
by
the
4-­‐point
bending
tests
we
calculated
the
stiffness
of
the
uniaxial
laminate
in
the
two
directions:
𝐸!"#!$ = 86.3 GPa
𝐸!"#$%&'"%' = 8.57 GPa
Using
these
values,
the
Halpin-­‐Tsai
and
Voigt
equations
gave
the
stiffness
of
the
carbon
fibres
themselves:
𝐸!!"#$% = 168 GPa
𝐸!!"#$%&'$& = 16.1 GPa
We
put
a
strip
cut
from
the
cross-­‐ply
laminate
into
liquid
nitrogen.
As
expected
it
bended
due
to
the
bi-­‐material
effect.
From
the
curvature
and
previous
data
we
calculated
the
thermal
expansivity
of
the
Nylon
matrix
to
be:
𝛼! = 30 ∙ 10!!
K!!
This
is
not
a
precise
technique,
however
the
obtained
value
is
still
within
a
plausible
error
range.
This
experiment
showed
and
quantified
the
bending
problems
of
composites
in
real
world
applications
due
to
temperature
change
and
showed
that
these
are
mostly
avoidable
by
using
symmetrical
stacking
sequences.
As
we
saw
for
the
symmetrical
cross-­‐ply,
this
implies
stresses
in
the
material,
which
might
cause
a
problem.
Therefore
it
may
not
be
desirable
to
use
composites
for
applications
where
the
temperature
change
can
be
extreme.

14. 14
7 Appendices
Further
readings:
Hiroaki
Miyagawa,
Thomas
Mase,
Chiaki
Sato,
Edward
Drown,
Lawrence
T.
Drzal,
Kozo
Ikegami
Comparison
of
experimental
and
theoretical
transverse
elastic
modulus
of
carbon
fibers
Carbon
44
(2006)
2002–2008
http://www.sciencedirect.com/science/article/pii/S0008622306000819
A.
Margossian,
S.
Bel,
R.
Hinterhoelzl
Bending
characterisation
of
a
molten
unidirectional
carbon
fibre
reinforced
thermoplastic
composite
using
a
Dynamic
Mechanical
Analysis
system
Composites
Part
A:
Applied
Science
and
Manufacturing
Volume
77,
October
2015,
Pages
154–163
http://www.sciencedirect.com/science/article/pii/S1359835X15002146
Chensong
Dong
,
Heshan
A.
Ranaweera-­‐Jayawardena,
Ian
J.
Davies
Flexural
properties
of
hybrid
composites
reinforced
by
S-­‐2
glass
and
T700S
carbon
fibres
Composites
Part
B:
Engineering
Volume
43,
Issue
2,
March
2012,
Pages
573–581
http://www.sciencedirect.com/science/article/pii/S1359836811004161
Wolfgang
Grellmann,
Sabine
Seidler
Testing
of
Composite
Materials
Polymer
Testing
(Second
Edition)
2013,
Pages
513–563
http://www.sciencedirect.com/science/article/pii/B9781569905487500116
L.J.
Hart-­‐Smith
Comparison
between
theories
and
test
data
concerning
the
strength
of
various
fibre–polymer
composites
Composites
Science
and
Technology
Volume
62,
Issues
12–13,
September–October
2002,
Pages
1591–1618
http://www.sciencedirect.com/science/article/pii/S0266353801002123
Innocent
Kafodya,
Guijun
Xian,
Hui
Li
Durability
study
of
pultruded
CFRP
plates
immersed
in
water
and
seawater
under
sustained
bending:
Water
uptake
and
effects
on
the
mechanical
properties
Composites
Part
B:
Engineering
Volume
70,
1
March
2015,
Pages
138–148
http://www.sciencedirect.com/science/article/pii/S1359836814004909
J.
Deng,
J.
Eisenhauer
Tanner,
D.
Mukai,
H.
R.
Hamilton,
and
C.
W.
Dolan

15. 15
Durability
Performance
of
Carbon
Fiber-­‐Reinforced
Polymer
in
Repair/Strengthening
of
Concrete
Beams
Materials
Journal
Volume:
112
Issue:
2
Pages(s):
247-­‐258
https://www.concrete.org/publications/internationalconcreteabstractsportal.aspx?
m=details&i=51687104
8 References
1
Part
II
Practical
Book
—
P1
2
http://www.vircon-­‐composites.com/3_1_2.asp
3
Hiroaki
Miyagawa,
Thomas
Mase,
Chiaki
Sato,
Edward
Drown,
Lawrence
T.
Drzal,
Kozo
Ikegami
Comparison
of
experimental
and
theoretical
transverse
elastic
modulus
of
carbon
fibers
Carbon
44
(2006)
2002–2008
http://www.sciencedirect.com/science/article/pii/S0008622306000819
4
http://www.engineeringtoolbox.com/linear-­‐expansion-­‐coefficients-­‐d_95.html