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7/27/2019 Stress Relaxation Behavior of 3501-6 Epoxy Resin
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St ress Re laxa t ion Behav io r of 3501-6 Epoxy Res i n
Dur ing Cure
YEONG K. KIM* and SCOTT R.WHITE
Department o Aeronautical and Astronautical EngineeringUniversity o Illinois at Urbana-Champaign
Urbana, Illinois 61801
Epoxy resins and other thermosetting polymers change from liquids to solids
during cure. A precise process model of these materials requires a constitutive
model tha t is able to describe th is transformation in its entirety. In this study the
viscoelastic properties of a commercial epoxy resin were characterized using a
dynamic mechanical analyzer (DMA). Specimens were tested at several different
cure s tat es to develop master curves of s tre ss relaxation behavior during cure.Using this experimental data, the relaxation modulus was then modeled in athermorheologically complex manner. A Prony (exponential) series was used to
describe the relaxation modulus. A n original model was developed for the stress
relaxation times based on similar work by Scherer (16) on the relaxation of glass.
Shift functions used to obtain reduced times are empirically derived based on curve
fits to the data. The dat a show that the cure s tate ha s a profound effect on the s tres s
relaxation of epoxy. More important, the relaxation behavior above gelation is
shown to be quite sensitive to degree of cure.
INTRODUCTION In order to accurately model the development of
ne of the most significant problems in the manu-0 acturing of polymer composites is the develop-
ment of residual st res s and warpage. Residual
stresses have detrimental effects on many issues from
dimensional stability to durability. If composites are
to be utilized in greater number and in new applica-
tions, then the ability to predict processing-induced
residual str ess (an d its effects)is critical. The focus of
this paper is on one aspect of the residual s tre ss prob-
lem: modeling the development of mechanical proper-
ties during cure. This work represents the first sys-
tematic analysis of the effect of cure state on Tg,
relaxation modulus, a nd relaxation spectrum.
Modeling of the development of mechanical proper-ties during cure is not simple. During the cure cycle
the matrix changes from a liquid-like uncrosslinked
material in the early stages of cure to a viscoelastic
solid at the end of curing. The residual stresses that
arise during cure a re influenced by this complex con-
stitutive behavior. For example, chemical shrinkage
strains, which occur early in the cure cycle, usually do
not contribute to the residual stresses at the end of
the cure cycle since stress relaxation occurs quickly
when the matrix is uncrosslinked (1).
* Currently at Korea Institute of Aeronautical Technology, Korean Air, Seoul.Korea.
mechanical properties, a constitutive law must ac-
count for both the time- and cure-dependent n ature of
the material behavior. Recently, Kim and White (2 , 3)
have presented stress relaxation test results for
3501-6 epoxy resin during cure. In this paper the
results of s tre ss relaxation testing an d the develop-
ment of a practical constitutive model for both time-
and cure-dependent effects are reported.
Stress relaxation testing was accomplished using a
dynamic mechanical analyzer (DMA) in str ess relax-
ation mode for specimens cured to different degrees of
cure. The master curves and shift functions for each
degree of cure case were obtained by time-tempera-
ture superposition. The Tgwas also obtained for eachdegree of cure using the DMA in fixed frequency mode
at 1 Hz.
The stress relaxation data were modeled in a
chemo-thermo-rheologicallycomplex manner. Nor-
mally, st res s relaxation characterization is performed
by conversion of either creep da ta in the Laplace do-
main or dynamic mechanical data in the frequency
domain. For example, recent work by Dilman and
Seferis (4)presents experimental d ata for the dynamic
mechanical properties of a reacting polymer system.
However, since direct testing of st ress relaxation in the
time domain was performed in this research, the cor-
ruption of the data by errors associated with conver-
sion calculations was precluded.
2852 POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1- Vol.36,NO.23
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In the following sections the experimental proce-
dure and specimen manufacturing are presented first.
Next, the re sul ts from str ess relaxation testing at sev-
eral different cure states and temperatures are pre-
sented. Subsequently, material models are developed
for the relaxation modulus, shift function, an d relax-
ation time spectrum . Finally, the correlation to exper-
iments is presented along with some practical simpli-
fications to the material models.
EXPERIMENTAL PROCEDURE
SpecimenPreparation
The material used in this study was 3501-6 epoxy
resin (Hercules, Inc. ),a commercial resin widely used
in the aerospace industry. The detailed chemical
structure is proprietary; however, the res in is known
to be a diglycidyl ether 'of bisphenol A (DGEBA) type
resin cured with a multifunctional amine. Small beam
specimens (60.0 X 12.6 X 3.55 mm) were manufac-
tured for the stress relaxation tests on a TA Instru-ments DMA 983. Silicone rubber molds were made,
and epoxy was poured into the mold to produce neat
resin beam specimens.
Before the specimens were cured, the epoxy was
debulked by removing all entrapped air bubbles using
a hot plate. The temperature of the hot plate was kept
-105-120°C to achieve minimum viscosity of the
res in. The low viscosity of the resin assisted in migrat-
ing the bubbles to the upper surface, where they could
be removed. Once the air bubbles were removed, the
specimens were transfer red to a Tetrahedron MTP- 14
hot press for curing. The cure cycle used was a 1 h
dwell at 116°C followed by a 2 h dwell at the curetemperature. The cure temperature was chosen be-
tween 120°C an d 177°C to achieve several different
final cure states . A total of six beam specimens were
cured in each run. After the cure cycle was finished,
the residual exothermic heat was measured to deter-
mine the final degree of cure using a DuPont DSC 10
Differential Scanning Calorimeter (DSC). The DSC
sample weights ranged from 4.8 to 9.5 mg and sam-
ples were heated from room temperature to 310°C at
10"C/min in a nitrogen atmosphere. In addition, the
total heat of reaction of unc ured epoxy was measured.
In every case a min imum of three DSC test s was per-
formed. The total heat of reaction of 3501 6 epoxy was
found to be H , = 498.7 ? 1.6 J/g. his compares
favorably with previously reported values in the range
of 473.6 t 5. 4 J/g y Let: and Springer (5) nd 502 ?
21 J/g y Hou and Bal( 6).The final degree of cure for
eac h specimen was determined from
where HR s the residual lieat of reaction and af s the
final degree of cure. Five different cure temperatures
(177, 160, 150, 135, 130'C) were used to obtain spec-
imens with final degrees of cure of 0.98, 0.89, 0.80,
0.69 and 0.57, respectively.
St r e s s R e l axa t ion B e hav ior of 3501 -6 Epoxy Res in During C ure
POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1996, Vol.36,NO.23 2853
Stress Relaxation Testing
Figure 1 shows the test setup used for the stress
relaxation tests. After the geometric dimensions were
measured , the specim ens were clamped to the parallel
ar ms of the DMA. The test s were performed using the
stress relaxation mode at several different test tem-
pera tures . After clamping, the specimens were equil-
ibrated for 20 min a t 30°C. The specimens were then
deformed for 30-40 min and stress relaxation data
were captured. Afterwards, the temperature was in-
creased 5-15°C an d the procedure was repeated. Tem-
perature was increased in uniform steps until the
specimen was fully relaxed. Once the data were ac-
quired, the relaxation m odulus was calculated using
E( ) = 2( + v ) G ( ) ( 2 )
Here, v is Poisson's ratio, and G ( t ) s the sample shear
relaxation modulus obtained in the experiment. Bo-
getti a nd Gillespie (7 )assumed a constant Poisson's
ratio during cure in their model and investigated the
effect of the cons tan t Poisson's ratio on the modulusdevelopment of a polyester resin during isothermal
cure . Their resu lts were compared with the resu lts of
Levitsky an d Shaffer (8 ,9). in which bulk modulus is
assum ed to remain constan t during cure. It was found
that both models give nearly identical predictions of
modulus development during cure . In addition, Beck-
with (10 , 11) stud ied th e viscoelastic creep of unidi-
rectional and angle-plied S-901 glass /Shel l 58-68R
epoxy composite and Shell 58-68R epoxy neat resin.
During creep testing, Poisson's ratio of the epoxy was
measured at several different temperature and s tre ss
clampspecimen
I J
A
vFlg. 1 . Experimental setupfor stress relaxation test showingspecimen clamping arrangement and mode of deformation.
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Yeong K. Kim an d Scott R . White
levels. No appreciable time-dependence of the Pois-
son's ratio was observed. On the basis of these stud-
ies, it was assumed that the Poisson's ratio of 3501-6
epoxy resin was a constant 0.35 during cure.
STRESSRELAXATIONTESTRESULTS
Two tests were performed for each degree of cure. All
data presented are the mean values of these two tests .A complete test for each degree of cure case required
between 11and 14 h depending upon the temperature
range and step increment. Figure 2 shows the raw
data for the a, = 0.89 case. Figure 3 shows the same
data plotted vs. the logarithmic time axis. Subse-
quently, the data a t each temperature were shifted to
obtain the master curve by the time-temperature su-
perposition principle using the reduced time, 6 , given
by
6 = I,' & dt (3 )
where a, s the shift function. The reference tempera-
ture for all curves was chosen to be 30°C.
Figure 4 shows the master curve for a, = 0.89 after
superposit ion. This type of analysis was performed for
each degree of cure case. The master curves for the
remaining cases are shown in Fig. 5. It is remarkable
that the initial (elastic)modulus is nearly independent
of degree of cure. The shift functions used to obtain
the master curves are shown in Fig. 6. Obviously, the
sta te of cure has a significant effect on the relaxation
behavior. The logarithm of the shift function wasfound to be linearly dependent on temperature, and
the slope of this line decreases in magnitude as the
cure state advances.
In addition to the relaxation modulus, the relax-
ation spectra are plotted in Figs. 4 and 5. To under-
stan d the physical relevance of the relaxation spec-
trum, consider the relaxation modulus of a
generalized Maxwell-type material composed of N
* E
- TemDerature-
150
p
100 3
G
ve
p
50
elements,
N
W = l
(4)
In Eq 4, Em s the fully relaxed modulus (equilibrium
modulus) and E, is the rigidity of the element associ-
ated with the stress relaxation time, 7,. If N is in-
creased without bound, then a n integral form of theequation is obtained,
The term Hd(1n T ) is defined as the relaxation spec-
trum. Once the equilibrium modulus and spectra l in-
formation for a material is obtained, the relaxation
modulus E(t) can be calculated using this equation
(12). Theoretically, relaxation (or retardation) time
spectra can be calculated using the Laplace or Fourier
transform from dynamic mechanical measurementsof stress relaxation (or creep) n the frequency domain.
However, since the functional form of the relaxation is
usually very complex, analytical closed-form solutions
for the relaxation spectrum are rarely attempted. To
calculate the relaxation spectrum in the present
study, the Alfrey approximation (12, 13) s used, such
that
Equation 6 shows that the relaxation spectrum ca n be
determined directly from the slope of the relaxationmodulus master curve. These results are plotted to-
gether with the master curve for each degree of cure
case in Fig. 5. The spectral peak increases as cure
advances. In addition, the relaxation spectrum is
broad at high degree of cure and it narrows a s the
degree of cure is reduced.
Unfortunately, no experimental data could be ob-
tained at very low degrees of cure in this study. This is
because the strength of the epoxy at a < 0.57 is so low
that specimens break during clamping or during ini-
tial deformation. Other types of experimental tech-
niques such a s parallel plate (DMA) stress relaxation
or oscillatory rheometry could be used to obtain data
for low degrees of cure. Such data would be particu-
larly useful in providing the full range characteriza-
tion necessary to develop constitutive models over the
full range of cure states. However, if the application of
such a material model is in prediction of residual
stresses, material behavior before gelation ( a E= 0.5)ha s little importance since stresses developed in this
stage of curing are almost immediately relaxed.
CONSTITUTIVE MODELING DURING CURE
0 100 20 0 300 400 500 600 700 800
Time (min)
Fg. . Raw data o a stress relaxation test or LY, = 0.89.
Stress Relaxation Modulus
The stress relaxation curve for a thermorheologi-cally simple material is usually modeled either by a
2854 POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1 M , Vol.36,NO.23
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Str ess Relaxation Behavior of 3501 -6Epoxy Resin During Cure
3.5-
3
2 .5%-
-2--
k! 2 :w 1.5--
1
0 5 -
3
2.5
2
n&0 1.5
W
W
1
0.5
0
1
4- Relaxation Modulus--
--
O *
-1 -0 .5 0 0.5 1 1.5
log(t) (min)
1%. 3. 30 min stress relaxation profles at various temperatures or a, = 0.89.
Temp ("C)
- 0
- 5
- 0
- 5
-0
- 05
- 20
- 35
- 50
-t- 65
- 80
- 95
power law
or by a discrete exponential series
(7)
where b is a material constant, Em s the fully relaxed
modulus, E" is the unrelaxed modulus, W, are weight
factors, 7, are discrete stress relaxation times, and 5 isthe reduced time. It wa:s found that a single value of b
cannot sufficiently describe the stress relaxation of
3501-6 over a wide temperature range. In addition,
the exponential series model provides for computa-
tional convenience in viscoelastic solution techniques
such as the recursive formulation for time superposi-
tion integration calculations (14, 15). Equation 8 can
be expanded to include degree of cure dependence a s
E (a . 5) = E m(a )
Equation 9 describes a thermorheologically complex
material undergoing cure since the relaxation behav-
ior can no longer be obtained by simple horizontal
shifting along the time axis. From the experimental
results it was found that Eq 9 could be simplified
greatly by taking the weight parameters, W,, as cure-independent. The str ess relaxation data in Fig.5 can
be used to obtain E" and E", and W,. However, the
reduced time, 6, and the stress relaxation times, 7,
must now be developed for a curing epoxy.
Stress Relaxation Time
The approach that is taken is based on the work of
Scherer (16) on the relaxation of glass. Scherer pro-
posed a model of volume relaxation for thermorheo-
logically complex material behavior based on the
Adam-Gibbs equation
POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1- Vol. 3s,NO.23 2855
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Yeong K . Kim and S c o t t R . White
3 --
2.5 --
2 --
1.5
1
0.5
0
v--
--
--
--
w
2
0--
-2--
-4--2 -6-1
-8--
-lo--
-12--
-14 ~ ~ ~ ~ ~ " " ~ " " ~ " " ~ " "
hv -M .03
0
3.5I 1
rearrangement. The discrete relaxation times can be
expressed similarly as
T,(T)= TOexp(&T) (1 2 )
in which P, are now discrete potential barriers. Thespectral response describing the distribution of relax-
ation time scales can be defined as
TJT ' )A, =
7 , ( T 0 )
Combining E q s 1 -13 yields
1
0.8
0.6
-d
0.4
0.2
0
-2 0 2 4 6 8 10 12
log(5) (min)
Fg. . Relaxation modulus master curves and stress relaxation time spectrums for all degree o cure cases .
(13)
where T~ and 9 re constants, and S, is the conforma-
tional entropy. E q u a t i o n 10 is based on the assump-
tion developed by Gibbs and DiMarzio ( 17) that flow
involves the cooperative rearrangement of increas-
ingly large numbers of molecules as temperature de-
creases. Using E q 10 ,Scherer suggested that the sin-
gle stress relaxation time in Eq 7 can be expressed as
PT p U ) = ToexP(m) ( 1 1 )
where To is a reference temperature, and P is a term
that depends on the potential barrier to molecular
Crete energy barriersbf thermo-rheologically complex
materials with discrete relaxation times. Assuming for
the moment that the effects of cure on molecular re-
laxation are similar to those induced by thermal acti-
vation, then a similar expression can be obtained for
T,((Y). In this case the stress relaxation time is ex-
pressed in a more general form as
P , is proposed to have the same functional form as P ,
in E q 14 such that
P , = f ' ( ( ~ ) ( a - aO)log ( 1 6 )
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Str ess Relaxat ion Behavior o 3501-6 poxy R esin During Cure
I Ir u - / I
2f0
0 0.2 0.4 0.6 0.8 1
Degree of Cure
Q. 7 . Peak value o the stress relaxatfon time us. degree ocure. [Solid line is the modei predictionfrom E q 20.1
where
(17 )
Here, a is the reference degree of cure where the
stress relaxation behavior is known, a n d f ( a ) is a
model to describe the unique energy barrier of chemo-
thermo-rheologically simple materials. Equation 14
implies that the stress relaxation time decreases as
temperature increases. However, in epoxy curing,
stress relaxation time irzcreases with the degree of
3 T
2.5-
2
ncd
8 1.5-1-
wW
1
0.5---
cure. Equation 16was modified to reflect this behavior
by transposing the reference and current degree of
cure.
The spectral peak in stress relaxation time is plotted
vs. degree of cure in Fig. 7. At low degree of cure, the
peak relaxation time is short, and it increases as cure
advances. This effect is quite profound for 3501-6
epoxy. The peak relaxation time increases by -6 or-
ders of magnitude from a degree of cure of 0.57 to0.98.
This trend was compared with the change in Tgwith
degree of cure. To measure Tg . the beam specimens
were mechanically cycled at 1 Hz to obtain the storage
and loss moduli as temperature was increased at 2"C/
min from room temperature to 240°C.To minimize the
temperature error, a heat shield was used to insulate
the thermocouple from radiant heating effects from
the heaters. Figure 8 shows the storage and loss mod-
uli vs. temperature for off 0.98 case. The glass tran-
sition is defined as the peak of the loss modulus curve,
194.8"C n this case. Figure 9 shows the results of alltests including that of the uncured epoxy, which was
assumed to be 10°C (18) .A second order polynomial
was fit to the data , yielding
T& a )= 10.344 + 1 1 . 8 5 9 ~ ~178.04a2("C) (18)
This curve is somewhat dependent on the assumed
value for the uncured material (18);however, it has
little influence over the tested range of degree of cure
(a> 0.5).NormalizingEq 18with respect to the Tg t a
= aJ = 0.98 yields the chemical hardening function,
I 0.3
--0.25
--0.2
--0.15
--0.1
--0.05--
7
0 50 100 150 200 250
Temperature ("C)Fig.8. Storage and loss modulus at 1 Hz u s . temperaturefo r a, = 0.98.
POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1 Vol. 36, No. 23 2857
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Yeong K. Kim and Scott R. White
3 .5
3
2.5--
h 2-T
k9 .5-;
W
h
eP
--
c
/r/50- odel
10%-
50--
(
0 " ' ~ " ' ~ " ' / ' ' ' ~ ' ' '
0 0. 2 0 .4 0.6 0. 8
Degree of Cure
Fig. 9. G l a s s transition temperature measured by DMA us .degree of cure. (Solid line is the model prediction from E q 19.1
Table 1. Results of Nonlinear Curve Fitting for Equation 9for ao= 0.98 Case.
w r, (min) w,
1
2
3
4
5
7
8
9
6 ( T P )
2.92 E+ l
2.92 E+3
1.82 E+51.10 E+7
2.83 E+8
7.94 E+9
1.95 E+113.32 E+12
4.92 E+14
0.0590.0660.0830.112
0.1 54
0.262
0.1 84
0.049
0.025
Em 0.031 GPa E" = 3.2 GPa
f (a):
= 0.0536 + 0.0615a + 0.9227a2 (19)
If the glass transition temperature and stress relax-
ation time are influenced by the same mechanisms
during cure, the nf( a) should also describe the changein normalized stress relaxation time; i.e.,
(20)
From Fig.5, min was found a s the peak stress
relaxation time for a, = 0.98. Using Eq 20, peak stres s
relaxation times were calculated an d are shown in Fig.
7 together with the experimental results. An excellent
correlation is obtained over the experimental range of
degree of cure. This result supports the assumption
that the same effective mechanisms that govern the
change in glass transition also describe the change in
stress relaxation time during curing.
'i0 .5 {
0 02 0 2 4 6 8 1 0 1 24
Fig. 10. Relaxation modulus master curve prediction using Eq9 overlaid with experimental data or a, = 0.98.
"
v =
2 -0.4--
0 0. 2 0 .4 0.6 0.8 1
Degree of Cure
Fig. 1 1 . Slope of the shqt function us. degree of cure. (Solid
line Is the model prediction using Eq 24.)
If a, = 0.98 is chosen a s reference degree of cure, a',
then the potential functionf'(a) in E q 16 can be ob-
tained from
f ' ( a )=f a1- 1 (21)
since Pa=
0 when a = 0.98. Finally, after substitutingE q 16 into 15, and converting to log scale, the st ress
relaxation times are
Once the discrete relaxation times a t a reference de-
gree of cure are obtained (a 0.98 in this case), the
discrete relaxation times at any degree of cure can be
found using E q 22.
To find the discrete relaxation times for the refer-
ence degree of cure, nonlinear curve-fitting of the data
was performed using the Levenberg-Marquardt
method ( 19).Nine exponential terms an d one constan t
were obtained from the curve fit; the results are listed
in Table 1. The peak stress relaxation time is the sixth
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Stress Relaxation Behavior of 3501 6 E p o x y Resin During Cure
hc$W
bD03
0 50 100 150 200 250
Temperature ("C)0 7 . 12. Shiftfunctions predicted using E q 23 overlaid with the experimental data.
term in the exponential.series. These results are sub-
stituted into E q 9, and the model is compared with
data of a = 0.98 in Fig. 10. In this case E" and E" are
31 MPa and 3.2 GPa, respectively. An excellent corre-
lation was obtained for the reference degree of cure
case.
Shift Function and Reduced Times
To use E q s 9 and 22 to model the development of
relaxation modulus during cure, what remains is to
describe the shift func1:ion and its dependence on de-
gree of cure. Here, a simple cure fit to the experimental
data is applied. In Fig . 6 the shift functions for the
master curves in Fig . 5 are presented. All the shift
functions were linearly fit with respect to temperature,
and it was found that the slope of the shift function
increases as the degree of cure decreases. Observing
these results, the shift function is modeled as a linear
function of temperature with cure-dependent coeM-
cients such that
log(ad = c l ( a ) T+ c2(a) (23)
where c l (a)s the slope of the log(+) vs. T curve, and
c2[a)s the intercept. Since the shift function for a <0.57 (outside the range of the experiments) is not
available, extrapolation below a = 0.57 is arbitrary. As
a first approximation the slope of the shift function is
assumed to be exponentially dependent on degree of
cure in the form,
c l ( a )= -
A least squares fit of the da ta yields the constants a,
1.4/"C and a2 = O.O712/"C.Figure 11 shows model
predictions for the slope of the shift function together
with the experimental data. Once c , (a) s obtained,
c2(a)s calculated from
log(1)= 0 = cl(cr)To+ c z ( a )
or
In this case, the reference temperature, 'I0 = 30°C.The
shift function model from Eqs 23,24, nd 25 s shown
together with the data in Fig. 12.Now the stress relaxation behavior a t any degree of
cure can be modeled using E q s 3, 9, and 22-25. For
3501-6 epoxy the model predictions and experimental
data are shown in Fig. 13.The equilibrium and unre-
laxed moduli used to produce these plots are given inTable 2. Since the peak stress relaxation model of E q
20 does not predict the exact experimental value for
each degree of cure case (see FQ. 7). ome of the
relaxation curves are shifted (along the time axis)from
the data. This is particularly evident in the case of
a, = 0.80. Nevertheless, the profiles of the relaxationsare well correlated with the data for all cases.
From Table 2, " and E" are shown to be relatively
constant over the experimental degree of cure range.
The assumption of cure-independent initial modulus
has been previously proposed for polymers in the
glassy state (20, 21). For the case of E m , akahama
an d Geil (22) investigated the dependence of equilib-
rium modulus on crosslink density by dynamic me-
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nEc3
wv
J.J
Model3
2 .5
2
1.5
1
0 . 5
0
-2 0 2 4 6 8 10 12
1% (5)@in)Q. 13. Relaxation modulus master curve model predictions using Eq 9 and data.
chanical analysis in the frequency domain. They con-
trolled the crosslink density of diglycidyl ether of
bisphenol A by varying the amounts of dibasic acid
anhydride and monobasic acid anhydride curing
agents. The equilibrium modulus was found to in-
crease by about two orders of magnitude over the
range of crosslink density in their tests.
In the present experiments it is difficult to find adiscernible relationship between the equilibrium mod-
ulus and degree of cure. Moreover, the measured
modulus in this range ( G O MPa) is beyond the DMA
equipment sensitivity. If E“ and E are assumed to be
constant with degree of cure , then E q 9 can be reduced
to a simpler form
Figure 14 shows the data and the model predictions
using this simplified equation. In the Figure, the
model predictions for 0.1 an d 0.3 degree of cure cases
are also presented.
Overall, the correlation to the experimental data is
quite good. There is a slight shift of the relaxation
Table 2. E” and E“ Used in Equation 9 for Different Degree
of Cure Cases.
0 EmGPa) Eu (GPa)
0.98 0.031
0.89 0.029
0.80 0.032
0.69 0.032
0.57 0.032Model 0.032
3.20
2.90
3.17
3.22
3.223.20
4
curve for (Y = 0.57 to shorter times, notably above
reduced times of lo4 min. This effect is more pro-
nounced for the (Y = 0.80 case. The utility of E q 26 is
in its practical application in process models to pre-
dict residual stresses in polymers and polymer com-
posites.
CONCLUSIONS
The development of stress relaxation behavior of
3501-6 epoxy during cure was presented in this pa-
per. S tress relaxation tests a t five different degrees of
cure from 0.57 to 0.98were performed using a DMA in
stress relaxation mode. Master curves and shift func-
tions a t each degree of cure were obtained by time-
temperature superposition. The glass transition tem-
perature at each degree of cure was also measured by
dynamic mechanical testing at 1Hz. t was found tha t
the st ress relaxation is significantly influenced by the
cure sta te of an epoxy. The Tg nd the peak relaxation
time were shown to develop in the same functionalmanner during cure, indicating that similar micro-
structural mechanisms are involved in their develop-
ment.
The stress relaxation data was modeled in a chemo-
thermo-rheologically complex manner. An exponen-
tial (Prony) series with cure-dependent terms was
used to describe the development of relaxation mod-
ulus. Relaxation times were modeled using a semi-
empirical approach based on the work of Scherer ( 16).
Excellent correlation to the data was found over the
range of experimental degrees of cure (0.57 to 0.98).
For the specimens cured less than ar = 0.57, many
experimental problems occurred, such as specimencracking during clamping or failure during initial de-
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Stress Relaxation Behavior of 3501-6 E p o x y Resin During Cure
FUJ. 14 . Relaxation modulus master curve model predictions using Eq 26 and data. (Constant unrelaxed and equilibrium moduli.)
formation. However, since residual stresses develop
primarily after the e p c q ha s gelled ( a = 0.51, he
present results for relaxation modulus, shift function,
relaxation time, and glass transition temperature barrier to molecular rearrangement.
should prove to be extremely valuable in the analysis
of processing-induced residual stress. rearrangement.
P = Potential energy barrier to molecular
rearrangement.
Pa = Cure-dependent discrete potential
P, = Discrete potential barrier to molecular
S, = Conformational entropy.
ACKNOWLEDGMENTS t = Time.
The authors wish to gratefully acknowledge the
support of the Office of Naval Research for this re-
search (Grant No. NOOO14-93-1-0535] nd Dr.
Roshdy Barsoum (Program Monitor).
NOMELNCLATURE
a , = Constant.
a2 = Constant.
aT = Shift function.
b = Constant.
c , (a ) = Slope of the log(%) vs. T curve.c,(a) = Intercept of the log(%)vs. T curve.
E ( t = Relaxation modulus.
E , = Discrete rigidity of a n element associated
with T, relaxation time.
E" = Unrelaxed (.elastic)modulus.
E x = Relaxed (equilibrium)modulus.
f = Unique energy barrier model for chemo-
thermo-rhe ologically simple materials.
T = Temperature.
T = Glass transition temperature.
T = Reference temperature.
W, = Weighting factors.
a = Degree of cure.
a = Reference degree of cure.
a, = Final degree of cure.
A m = Ratio between unique and discrete stre ssrelaxation times.
A = Ratio between cure-dependent unique
an d discrete stress relaxation times.v = Poisson's ratio.
T = Relaxation time.
7, = Discrete relaxation times.
T~ = Peak relaxation time.
T~ = Constant in Adam-Gibbs equation.
= Constant in Adam-Gibbs equation.
5 = Reduced time.
o = Index.
f(a)= Chemical hardening function.
G(t) = Shear relaxation modulus.
Hd(ln 7) = Relaxation spectrum.
H R = Residual hlzat of reaction.
H , = Total heat of reaction.
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Revised September 1996