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



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.



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



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



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 )

2856 POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1 VOl.36,NO.23



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



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

2858 POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1 Vol. 3Ei NO. 23



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-

POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1W6, V d . 36,NO.23 2859




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-

2860 POLYMER ENGINEERING AND SCIENCE, MID-DECEMBER 1- Vol. 36,NO. 23



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.

REFERENCES

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Revised September 1996

Stress Relaxation Behavior of 3501-6 Epoxy Resin

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