U.
S.
Department
of
Commerce
National
Bureau
of
Standards
Research
Paper
BP
1745
Volume
37,
October
1946
Part
of
the
Journal
of
Research
of
the
National
Bureau
of
Standards
Thermal-Expansion Stresses
in
Reinforced Plastics 1
By
Philip
S.
Turner
2
Failure
of adhesive
bonds
is
attributed
to
boundary
stress
concentrations.
An
analysis
of
the
causes of internal-stress
concentrations
in rigid adhesive
layers
leads
to
the
conclusion
that
stress
concentrations
can
be
eliminated
in
many
cases
by
matching
the
coefficients of
thermal
expansion of
the
component
parts.
A stress-equilibrium formula for calculating
the
thermal-expansion coefficients of
mixtures
involves
the
density,
modulus
of elasticity,
coefficient of
thermal
expansion,
and
proportion
by
weight
of
the
ingredients.
Illustrations
of
the
application
of
the
derived formula include
lead-antimony
and
beryllium-aluminum
mixtures,
phenol-formaldehyde resin
and
glass-fiber mixtures,
and
plastic plywoods.
The
thermal-expansion
co
efficients of a
number
of
pure
and
reinforced
pl
ast
ics
are
reported.
Bonds
obtained
when
thermal
coefficients
are
matched
are
stable
over a wide te
mperature
range.
I.
Introduction
This
report describes a method of compounding
a plastic, or
other
mixture
or
compositions,
to
provide a
materia
l having a predetermined de-
sired coeffici
ent
of thermal expansion.
In
the
preparation of plastics heretofore
it
has
been
found practically impossible to produce
a satis-
factory bond between a plastic
and
a metal facing
or
other
metal reinforcement.
This
is largely
Cilue
to
the
difference in coefficients of expansion of
the
mater
ials;
the
plastics have relatively high coeffi-
cients of expansion as compared
with
those of
metals. Because of the differential expansion,
forces created upon changes in temperatures
have
been such as to
prevent
a satisfact
ory
bond being
obtained between metals
and
plastics except
where plastics of a flexible rubber-like, or
gummy
type
are used.
Even
in the bonding of foils of
tin, lead, aluminum,
etc.,
to
cardboard or
paper
for containers, only gummy
or
tacky
plastics
have been used.
The
problem of obtaining bonds between
pla
stics
or
other
adhesives and various materials
1
The
exper!m~nta
l
work
on
this project was sponsored by and conducted
with the financial assistance
o(
the National
Advi~ory
Committce (
or
Aero·
nautics, upon the recommendation
of
their Subcommittee on Wood and
Plastics
(or
Aircra(t.
• Deceased.
Thermal
Expansion
in
Plastics
Cont
ents
•
Page
I.
Introduction
____
_________________ ___
__ __
__
__
239
II.
Effect
of various
fac
tors on
bond
strengths
____ _ 240
1.
Thickness
of
materials
bonded
____
______ 240
2.
Thickness
of adhesive
__
________
_____
__
240
3.
Modulus
of
elasticity
____
___________
___
240
4.
Moisture
content
_______________ ______ 241
5.
Temperature
__
___________ ____________ 241
6.
Thermal
expansion
____
____________
____
241
7. Pressure of
application
___
_____________ 241
III.
Deve
lopment
of a
fo
rmu
la for
calculating
coe
ffi-
cients of
thermal
expansion of
mixtures
______ 241
IV.
Applications
to
plastic compositions ___________ 243
1.
Applica
tion
to
a
mixture
of
polystyrene
and
aluminum
oxide ____ _
___
_____ ___ 243
2. Application
to
compositions for filling
rivet depressions
_
______
___________
__
244
3. Application
to
mixtures
of
phenol-formal-
dehyde
resin
and
glass fibers ______
___
_ 245
V. Detlilrmination of
coefficient~
of
thermal
expan-
sion of plastics
___
__ __
___
__________________ 246
1.
Factors
affecting thermal-coefficient
measurements
of plastics _ _ _ _ _ _ _ _ _ _ _ _ 247
2.
Results
of thermal-expansion
measure-
ments
for plastics
____
___
____________ 247
VI. Conclusions _________________
___
___
___
______ 250
239
L
is considered as being substantially
the
same as
that
of removing
or
eliminating concentrated
stresses
at
the
boundaries.
If
internal-stress
concentrations can be removed,
the
full
strength
of
the
adhesive can be developed
to
resist ex-
ternal loads.
Bonds
produced
by
adhesives
can
be divided
into two general classes:
the
rupbery
or
yielding
bond
and
the
rigid bond.
In
the
first category
are found mo
st
th
ermopl
ast
ic cements,
rubber
cements,
and
combinations of
thin
rubber
layers
and
cements. Adhesives of
this
class have been
found
to
provide durable bonds between dissim-
ilar materials
at
moderate temperatures.
The
rigid
or
high modulus
bond
has
generally been
found unsatisfactory for such applications. A
possible exception is found
in
the
use of cold-
setting
cements of
the
phenol-formaldehyde
and
urea-formaldehyde types.
The
advantages of
the
first class over
the
second disappear
at
reduced
temperatures where
the
adhesive loses
its
ability
to
eliminate stress concentrations
by
yielding
with
the
dimensional changes of
the
materials
bonded.
The
rigid
bond
is superior for
many
purposes
to
the
yielding bond, if
it
can be ob-
tained, because
it
produces a stronger
and
less-
yielding product.
For
composite
structural
ma-
terials subjected
to
extreme temperature changes
a stable rigid bond is imperative.
In
attempting
to
bond
various materials to-
gether,
it
has
been found
that
materials having
widely different coefficients of expansion cannot
be bonded with
any
rigid cement available. Cer-
tain
thermoplastic adhesives produce satisfactory
bonds
at
ordinary temperatures,
but
fail
at
low
temperatures when
the
bond becomes rigid
and
is
no longer able
to
yield
to
changes in dimension of
the
materials bonded. Certain
rubber
cements
fall into
the
classification of bonds which are
soft enough to yield with changes in dimensions.
These yielding bonds, however,
do
not
produce
the
rigidity required for
structural
applications
and
fail
at
low temperatures for tbe same reason
that
rigid plastic bonds fail
at
ordinary tempera-
tures;
nam
ely, difference in
rates
of expansion
and
contmct,ion
with
changes in temperature.
These failures are caused
by
stresses resulting
from differential expansion or contraction. These
forces can be reduced
by
one
or
more of the follow-
ing factors. '
II. Effect
of
Various Factors on Bond Strengths
An
analysis of the factors responsible for
the
production of stress concentrations in a rigid bond
must
include
the
following:
1.
The
thickness of
the
materials bonded.
2.
The
thickness of the adhesive layer.
3.
The
modulus of elasticity of all components.
4.
Changes in dimensions
with
changes in
moisture
content
for all materials.
5.
Changes in temperature
to
be encountered
in
service.
6. Coefficients of linear thermal expansion of all
materials.
7. Pressure of application.
1.
Thickness
of
Materials
Bonded
It
is evident
that
if the thickness of
the
materials
bonded
approaches zero, stresses
set
up
by
any
other
combination of factors
must
also approach
zero. Accordingly,
it
has
been found possible
to
bond
thin
metal foils to
other
materials. A
method
of eliminating stresses
by
this means
would, however, have very limited practical appli-
240
cation.
This
stress-reducing factor is utilized
in
the production of mechanical mixtures of materials
of microscopic size or in the form of fine filaments,
which are incompatible on a larger scale.
2.
Thickne
ss
of
Adhesive
In
bonding two pieces of the same material
the
stresses
set
up in
the
bond as the result of
restrained dimensional changes in the bonding
material would be proportional to the thickness of
the
bond.
It
has
long been recognized
that
thin
adhesive layers produce stronger bonds
than
thicker ones.
3.
Modulus
of
Elasticity
If
the
modulus of elasticity of
anyone
of two
adjacent materials is zero, the stresses set
up
between
them
must
be zero. This is essentially
achieved
by
bonds which yield readily
and
offer
little resistance
to
static
loads whether produced
internally or externally.
Journal
of
Research
~----------------11--------------------------------------------~·~~~
4.
Moisture
Content
The
problem of bonding materials having dif-
ferent dimensional stability
with
changes in
relative humidity can be reduced, if
not
eliminated,
if
the
surface of the combination is impervious
to
moisture.
At
the time of bonding
or
at
the
end
of
the
bonding operation
the
material
or
materials
sensitive to changes
in
humidity
can
be
made
to
contain
the
normal
amount
of
water
present
under average conditions.
No
completely satis-
factory solution
to
this aspect of bonding has been
found.
5.
Telliperature
Changes in temperature
cannot
be eliminated
I
in
any
practical application.
For
cold-setting
phenol-formaldehyde
and
urea-formaldehyde ce-
ments
the effect of
temperature
changes is
minimized by bonding
at
temperatures close
to
those encountered in service.
6.
Therlllal
Expansion
From
the
standpoint
of the thermal stresses
produced, the problem of bonding two different
materials b
ec
omes identical with
the
problem of
bonding two pieces of
the
same material, provided
the
coefficients of thermal expansion of
the
two
materials are the same. A further reduction in
stresses can be achieved if the bonding agent has
the
same coefficient of
th
ermal expansion as
the
materials bonded. A
bond
between materials
having
different coefficients of expansion would
have
to
withstand the shear stresses produced
by
the
differential expansion or contraction expe-
rienced within the range of
temp
eratures en-
countered. Bonds can sometimes be maintained
if the materials bonded are
thin
enough to permit
the
relief of stresses
by
bending or warping, either
of which is
und
es
ira ble.
The
bonding
strength
exhibi ted
by
a particular adhesive is
the
addi-
tional load which, added to
the
internal concen-
trat
ed stresses, is sufficient to disrupt
the
material.
In
some cases this additional load is zero
and
the
materials are said
not
to
bond.
7.
Pressure
of
Application
When
two materials are
bond
ed together
at
high pressures, strains are produced
at
the
bond
when
the
pressure is released. This
strain
is
proportional to
the
ratio of the moduli of elasticity
of the two components.
The
low
est
pressure feas-
ible should be used to produce bonds
with
the
lowest internal strains.
The
logical conclusion to be
draW'}
from the fore-
going discussion is
that
stable rigid bonds are
possible over a wide range of temperatures if the
thermal-expansion coefficients of
the
component
parts
are matched. An example of
the
results to
be expected is furnished
by
the
combination of
concrete
and
steel which individuallj have practi-
cally equal coefficients of thermal expansion
and
contraction.
It
is noted
that
certain proportions
of ccment, sand, and stone produce
optimum
results.
Variolls methods
have
been used in
other
fields
to predetermine the thermal-expansion coefficients
of mixtures, such as alloys, glasses, etc.
Some of
these methods appear to work satisfactorily in
limited applications.
Notable
among
th
ese are
weighted averages based on the thermal-expan-
sion coefficients of the components
and
their pro-
portions either by weight or
by
volume.
Th
ese
methods do not, however, give satisfactory re-
sults for plastic compositions.
It
seems reasonable to expect
that
the
coefficient
of thermal expansion of
a mixture is a function of
the relative compressibilities, as well as
the
thermal expansion coefficients
and
the propor-
tions of the components.
In
the following sec-
tion a formula for the coefficient of thermal ex-
pansion
for a mixture is· obtained which, although
not
derived rigorously (according
to
the
theory
of elasticity), does
take
into account
the
relative
compressibilities of the components.
III. Development
of
a Formula for Calculating Coefficients
of
Thermal Expansion
of
Mixtures
If
it
is
considered
that
an
internal stress system
in
a mixture is such
that
the stresses are nowhere
sufficient to disrupt the material, the sum of
the
internal
forces can be equated
to
zero
and
an
Therm.al
Expansion
in
Plastics
expression for
the
thermal-expansion coefficient of
the
mixture is obtained. When small particles or
fine filaments are incorporated
into
a mixture,
the
small dimensions appear to
permit
combinations
241
r-
~--~-----------------------------------------------------------------~~~------------------------
of
materia
ls which would be incompatible on a
larger scale.
The
derivation of the resulting volume thermal
coefficient of a mixture follows.
The
symbols
used are:
a=coefficient of linear thermal expansion
{3=coefficient of cubical
therm
al expansion
K=
bulk
modulus = l /bulk compressibility
d=density
P=fraction
or
percent
by
weight
V=volume
I:J.T=temperature difference
S=stress.
Subscripts i
and
r refer respectively
to
the
property
of the
ith
component and of
the
resul
tant
mixture.
If
it
is assumed
that
each component in
the
mixture is constrained
to
change dimensions with
temperature changes
at
the same
rate
as
the
aggregate
and
that
shear deformation is negligible,
the
stresses acting on the particles of the various
components will be
(1)
as
the
stress is given
by
the
product
of volume
strain
and
bulk
modulus.
The
resultant
of
the
forces acting on
any
cross section of
the
mixture
must
vanish. Therefore,
SlA
1
+S
2
A
z
+
...
+SnAn=O,
(2)
where there are n components
and
the
A's
refer to
the
parts
of the cross-sectional area formed
by
the
various components. However, in a homo-
geneous mixture
the
relative areas formed in the
cross section
by
the
different components are
proportional
to
their
relative volumes. There-
fore,
it
follows from equation 2
that
SIV
1
+S
2
V
2
+
...
+SnVn=O.
(3)
Substitution for S from equation 1
in
equation 3
then
yields
({3r-
(31)I:J.TV
1
K
1
+ ({3r-
(32)
I:J.TV
2
K
2
+ +
((3r-
{3n)I:J.TVnKn=O.
(4)
As
which can
be
substituted
for V in equation 4.
Also, as
AT,
dr,
and
Vr
are common factors,
they
can be eliminated from each
term
of the expression.
Solving for
{3"
the
following expression is obtained:
242
{3,
..
+{3n
P
n
K
n
d
n
(5)
As
the coefficient of linear thermal expansion is
directly proportional to the cubical coefficient,
a
can
be
substituted
wherever
(3
appears
with
the
following
a1P1K
1
+ a
2
P
2
K
2
+
....
+ anPnKn
d
1
d
2
d
n
• (6)
P1K
1
+ P
2
K
2
+ .
...
+ PnKn
d
1
d
2
d
n
It
is
apparent
by
inspection
that
equation 6
based on stress equilibrium reduces
to
a percent-
age
by
volume calculation
if
the ingredients
have
u
o 3 0
~
el'
«0
w-
z .
:::d5
20
ll..CI)
O~
!--n.
zi'j
10
UJ
-...J
Uo(
iL!
u.,x
UlI1J
Ox
vI-
o
o M
EAS
URED
V A L U
ES
~
V
~
--::::::
~
-----
~
t:::-
~
~
~
r:---
o
20
40
60
80
100
PERCEI'-!TAGE
OF
LEAD
FIGURE
1.-Thermal-expansion
coefficients
of
lead-antimony
mixtures.
Curve
A.
Percentage
by
weight calculation.
Curve
B.
Percentage
by
volume or stress equilibrium calculation.
u
o
30
;0...
0(1
-4:0
w-
Z
~
_2
20
...JO
lLiIi
o~
1-0..
z~
W
10
-...J
u«
iL2:
LL<X
WUJ
o:r:
UI-
o
o
MEASURED
VALUES
~
~
N
~
~
t-=:~::
::::--
r---
o
20
40
60
80
100
PERCENTAGE
OF
BERYLLIUM
FJGURE
2.-Thermal-expansion
coefficients
of
beryllium-
aluminum
alloys
Curve
C.
Percentage
by
weight calculation.
Curve
D.
Percentage
by
volume calculation.
Curve
E.
Stress equilibrium calculation.
Journal
of
Research
the same bulk moduli.
If
the ingredients have
the same modulus
to
weight ratios, the calcula-
tion amounts to a percentage
by
weight
int
er-
polation.
Equation
6
has
been verified
with
experimental
values of several metallic mixtures.
The
thermal-
expansion coefficients calculated according
to
equation 6 for several mixtures of lead
and
anti-
mony
and
of beryllium 3 and aluminum are com-
pared
with the measured values
in
figures 1
ahd
2.
IV. Applications to Plastic Compositions
It
has been found
that
the size
and
shape of the
filler particles
in
plastic mixtures have
an
effect
on the
resultant
coefficient of expansion of the
mixture.
The
equations derived do
not
take
this phenomenon
into
consideration. Difficulty
is also encountered because
the
moduli of some
materials are
not
available.
To
solve these
problems, equ
at
ion 6 is modified
by
substituting
an
empirically determined constant 0 for
Kid
for
each material.
Constant
0 is
interpr
eted as p'ro-
portional to
the
modulus-density
ratio
ratherthan
being equal to it.
The
proportionality factor is
dependent on
the
shape
and
size of the particles
and
on the distribution of
the
material
in
the
matrix.
It
is assumed
that
th
e constant 0 for
each specific filler
and
each plastic material is
ind ependent of
the
other
components of a mixture
if the ingredients
are
evenly distributed.
Thi
s
modified equation
has
been used successfully
in
several investigations.
The
use of this technique
and
the effects of the shape and size of fillers have
been described
in
another
report.4
In
studies of the physical properties of rein-
forced plastics
it
has
be
en found
that
the
strongest
materials were produced
by
oriented fibers in
thermosetting resins.
The
material fabricated in
this manner is essentially non-isotropic in all of
its properties and
has
different coefficients of
expansion depending on
the
direction of the fibers
in the test specimen. Materials having isotropic
properties can be
obtained
by
random distribution
or
by
planned orientation.
The
so
lution for any
property
must
take
into
consideration the
OI
:ien-
tation
of the filler as well as .
the
percentage
composition.
For
a mixture whose components have nearly
•
Mi
xt
ur
es
of
Be
and
Al
apparently
form a mechanical mixture,
but
no
value
for
tbe bulk modulus or
bulk
compressibility was availabl
e.
:F'rom
the
thermal
coefficient,
17.8
X
1O-6jO
C,
ofaknown
mixture containing
32.7
percent
of Be and
66.3
percent
of
AI,
and
with repo
rt
ed densiti
es
and thermal co·
efficients
of
the lllgredients. the
bulk
modulus
of
Be was calculated from
equa·
tion 6 to be 1
5.2
X
10
6
Ib/
in
'.
From this value
the
thermal coefficients
of
other mixtures
of
Be and Al were calculated
with
the
results shown in figu
re
2 in comparison with tho re
ported
va
lu
es.
•
Philip
S.
'rurner.
Jewel
Doran,
and
Frank
W.
Reinhart, Fairing compo-
SItions
for
aircraft surfaces. N
ACA
Technical Note No.
958
(N ovem ber
1944)
TherIDal
Expansion
in
Plastics
equal values of Poisson's ratio
the
bulk
modu
li
are nearly proportional
to
the
corresponding
Young's moduli.
For
such mixtures Young's
modulus
may
therefore be
substitut
ed for bulk
modulus
in
equation 6
to
yield
the
following ex-
pression for the thermal expansion coefficient of
a mixture:
iX
1
P
1
E
1
+
iX
2
P
2
E
2
+ +
OlnPn
E
"
d
1
d
2
••
d"
(7)
iX
T
= P
1
E
1
+P
2
E
2
+
+Pn
E
n
'
d
1
d
2
• • d
n
Thus,
in
many
cases where
the
bulk moduli are
not available, equation 7
may
be used
instead
of
equation 6.
Th
e following experim
ents
on a
mixture
of polystyrene
and
aluminum oxide illus-
trate
the
reasonably successful application of
equation
7.
1.
Application
to
a
Mixture
of
Poly-
styrene
and
Alulllinulll
Oxide
The
applicability of
the
formula is illustrated
by
the behavior
of
brass inserts
in
a mixture of
styrene resin with fused aluminum-oxide powder.
Brass inserts in ordinary polystyrene cause the
polystyrene to crack because of
the
different
coefficients of thermal expansion.
Th
e coefficient
of linear thermal expansion of polystyrene is
approximately
70X
10-
6
1 deg C,
that
of brass is
approximately
17
X
10-
6
/
deg C.
Fused
aluminum oxide was chosen for use
in
the mixture because
it
has a low coefficient of
lineal' thermal expansion
(8.7XIO-
6
1 deg C)
and
a
high modulus of elasticity compared
to
its
density.
Its
choice for use with polystyrene was also deter-
mined
in
part
by
its desirable electrical properties.
Th
ere was no appreciable change in the exce
llent
electrical resistance of polystrene when
the
alumi-
num-oxide filler was added.
The
data
in table 1
for the coefficients of lineal' thermal expansion of
mixtures of polystyrene
and
fused aluminum oxide,
calculated according
to
the stress-equilibrium,
formula show
that
approximately 90 percent of
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