The implications of the fiber truss concept for creep properties of laminated composites

Composite Structures 11 (1989) 85-100

The Implications of the Fiber Truss Concept for Creep Properties of Laminated Composites

D. A. Dillard, K. C. Gramoll & H. F. Brinson Engineering Science and Mechanics Department, Virginia Polytechnic Institute, Blacksburg, VA 24061, USA

ABSTRACT A numerical procedure based on classical lamination theory has accurately predicted the time dependent creep in laminated composites consisting of only two fiber directions. Predictions for laminates with more than two fiber orientations have been accurate at short times, but have fallen well below the longer time experimental results obtained on narrow (12 ram) specimens. Further testing has helped establish that relaxation of the fiber trusses along the free edges of narrow specimens is much more significant for time dependent behavior than under static conditions. Insights into the design of laminated composites for long term loading and certain precautions for experimental evaluation of time dependent properties are also given. The paper also highlights the non-conservative errors that can be made by using static test results to indicate long term behavior.

1 INTRODUCTION A l t h o u g h design engineers are often aware of the viscoelastic response of polymeric materials, it is easy to lose sight of this time dependence when these plastics are reinforced with advanced fibers. Indeed, some fibers such as graphite exhibit very little viscoelastic behavior. Nonetheless, even t h o u g h the fibers act in an elastic m a n n e r and the resulting laminate is loaded to serve in a fiber dominated mode, the time dependence may not be negligible. Experimental results indicate that a substantial amount of creep and eventual delayed failures can result when typical laminates are uniaxially loaded with the principal load direction slightly off-axis from a 85 Composite Structures 0263-8223/89/$03.50 O 1989Elsevier SciencePublishers Ltd, England. Printed in Great Britain

86

D. ,-t. Dillard, K. ( . Gramol[, H. t~: Brinson

dominant fiber direction. This is particularly significant because the principal stress directions do not coincide with the principal material directions at all locations in most composite structures. Creep is not a universal problem with all composite applications, but it must be given consideration in many usages where residual stresses, applied mechanical loads of long duration, and environmental factors combine to create a scenario leading to time dependent deformation and the possibility of delayed failures. Although composites made with the higher cure temperature resin systems are often quite stable at lower temperatures, the use of lower cure temperature systems can result in laminated composites which exhibit considerable time dependence even at room temperature. It is further known that the effect of moisture and other solvents serves to plasticize the matrix and can result in a substantial decrease in the glass transition temperature and acceleration in the viscoelastic processes.' In order to understand and predict the viscoelastic response of a general laminate subjected to a given environment and loading history, it is important to develop a methodology for analytically modelling and predicting time dependent response. An accelerated characterization procedure e advocated by the authors is based on conducting short term tests on unidirectional laminae and using this information to estimate the long term response of general laminates. The procedure is designed to utilize only a minimal amount of tests in order to reduce the cost of the viscoelastic analysis. Figure 1 outlines the procedure used to predict the long term response. Relatively short term uniaxial compliance tests are conducted on the unidirectional coupons of material at 0°, 10° and 90° from the fiber direction. This information is collected at several temperatures and stress levels and, by utilizing techniques such as the time-temperature superposition principle (TTSP), these short term data can be used to predict the long term response of the unidirectional material. Compliance data from these laminae are sufficient to characterize the orthotropic nonlinear viscoelastic properties of a lamina? Delayed failure data are also collected for the three unidirectional coupons and are extended to predict delayed failures over long time periods by the use of techniques such as that proposed by Z h u r k o v ? With the predicted long term properties of the unidirectional material, one can then estimate the long term response of general laminated composites subjected to arbitrary loading, assuming the failure mechanisms (including damage development) are the same in the test coupon and the predicted structure. The current procedure used to predict the laminate behavior is an incremental numerical procedure 5'~'based on classical lamination theory (CLT). The program calculates the current stress state within each ply and then allows each ply to creep during the next time step. Because the plies creep

Fiber truss concept for creep properties of laminated composites

Shor, Lamina

1%(t,{=~ ,T ,M,-'

•oo,,,m

Lo mlno

Response Model

Constitutive Model

)

l

[

]

87

'

M,-.,

i.oo0,,

o.o]

Failure Model

\_ic / Long Term Laminate Predictions

Long Term Creep Compliance and Deioyed Failure Predictions for General Laminated Composites

Fig. 1. Schematic diagram of accelerated characterization procedure for laminated composites. different amounts in various directions, some mechanism is needed at the end of each time step to force the deformations to be consistent. An appropriate algorithm is similar to the equivalent load approach used to account for the mismatch due to thermal expansion of the plies. 7

2 DISCUSSION Numerical predictions of a fiber truss effect

The compliance predictions of the program have been quite good for laminates which consist of only two fiber directions. Figure 2 illustrates the degree of fit which has been obtained for these types of layups. This plot is for a T300/934 Gr/Ep crossply coupon which is loaded with a uniaxial load applied 15° off-axis from the 0° direction. The degree of fit is quite good and one should note that the predictions for the compliance at the longer times indicate a very strong dependence on the applied stress level although this effect is quite small at the short times. This serves to caution the use of static results to estimate the long term behavior unless the viscoelastic processes are adequately recognized. One should further note that the creep is quite l a r g e - - m o r e than 4 times the static strain after only 1 week of loading (at an

88

D. A. Dillard, K. C. Grarnoll, H. F. Brinson 1.5

CREEP COMPLIANCE COMPARISON FOR J [(15/-75)4,] AT :~.0" F (f60° C)

A

~

!

0 -X

1.2

----o. "~

2.0

m

!

0 X

EXPERIMENTAL PREDICTED

/ /-

1.6

%

/i 11/%

0.9

i.,I (.3 Z
Z (1.

o"i • 5 0 0 kli (34.5 MPo) o'. = I 0 0 0 {69.0)

0 u

o".= 20.00 (137.9) % = zs. lo (173.1)

06

n l.i.J LU n~

[ _/

~ , Js.oo (Io3.4)

//

o'4 i //

//

/P" ./" ~:11_~,,~o-''~

~ I

o', ~.;~

0.3

.

I

0

%%

o.

0 (.3 0.4

=4

0.0

'o.B

...--" ~,~.i "~-

I0

rl

£I. hl hl OC

=,

I I

I 2

I 3

I 4

3.0

LOG TIME (rain) Fig. 2. Experimental and predicted nonlinear compliance for a laminate with two fiber directions.

elevated temperature). These large strains are typical of laminates consisting of only two fiber directions because the fibers tend to scissor and result in very large deformations. Because these laminates are highly susceptible to large viscoelastic deformations, they should be avoided for applications where it is possible for the load directions to deviate from the predominant fiber direction. The program predictions obtained for laminates consisting of three (or more) fiber directions have been less satisfactory. Because of the assumptions of classical lamination theory, the program assumes that normals through the laminate remain straight and normal. When the fibers are present in at least three directions, the CLT assumptions imply that the fibers act as pinned trusses which form a vast network of triangular truss elements. Because triangular trusses are rigid, the program predictions tend asymptotically to approach an upper limit which we have referred to as the fiber truss limit. This theoretical limit can be obtained by assuming that the matrix properties completely relax, thereby forcing the elastic fibers to carry all of the load. Actual experimental results and predictions are shown in Fig. 3 for a T300/934 laminate with three fiber directions. While the program predicts the fiber truss effect, narrow ( 1 3 m m ) specimens with three fiber directions may exhibit creep which is considerably greater than that predicted by the numerical procedure. This concept is

Fiber truss concept for creep properties of laminated composites

89

FIBER TRUSS COMPLIANCE .0921X I0 "s psi -I

oo: V .........

l

lo,

CREEP COMPLIANCE COMPARISON FOR

~- 0.088 tk '-"

EE,,o,,,,-~,,o,,.]

o.o.e I-

-

-

AT ~'O'F ,,~O'C>

EXPERIMENTAL PREi~CTEO

-

-t O.,'>, X

...,~:7.-/~3.;/."

" ~ ~'~ ] ~ " "~

o.o,,

_

"10.IZ5

4o.,= /

~

~

"

~

0 . 0 " 0 t, -I

1

"i "is.o l i tel.6 l l ~ l l

_~ _ \ _

I 0

I I

~" I0

I 2

I 3

(179.31

-t 0.119

I 5

/ 0.116

I 4

I

t~J

LOG TIME(rain) Fig. 3. Experimental and predicted nonlinear compliance for a laminate with three fiber directions.

,_,8_~ ~_R_~_~_~. ~_0,_~_~,~.

/___ ¢_____7

....

/ Z-:N~-.~ ss~

s ~ ~'S S ~ s j'

%

(..) ~ W nt.J

/

-

~

SCHEMATIC REPRESENTATION OF LAMINATION THEORY LIMITATIONS AS MATRIX STRESSES RELAX SIGNIFICANTLY

~°'l

~-

ACTUAL COMPLIANCE LAMINATION THEORY PREDICATION

o. I ']

lrll I DIFFERENT STRESS LEVELS

% LOG TIME Fig. 4. Schematic of fiber truss limiting compliance.

90

D. A. Dillard, K. ( . (;ramoll, tt. k. Brinson

illustrated schematically in Fig. 4. Much of the discrepancy apparently arises because the assumptions of lamination theory are not valid near the free edges of these specimens. While the CLT over-prediction of static modulus of narrow specimens has been observed, the time dependent deviation can be much larger. Since the interlaminar shear strains are controlled by the viscoelastic resin, this edge effect can increase significantly at longer times. Experimental data have now been collected to illustrate the effect of specimen width on the measured viscoelastic response of laminated composites. As specimen width is increased, the ratio of the area near the free edge to that at the specimen interior decreases, and one approaches the predictions obtained from the numerical procedure. The implications are twofold: (1) the program predictions appear to be valid for wider specimens and structural components: (2) care must be used in selecting specimen size for laminates with three or more fiber directions for creep validation measurements in the laboratory'. This fiber truss effec! has several consequences for the design of composite structures for long term durabilitv. Specifically, the model can provide insights for designing laminates to provide better resistance to long term deformations. It is well known that the stiffness and strength estimates predicted by classical lamination theory and measured experimentally can be significantly higher than the corresponding predictions from a rule of mixtures or netting analysis because of the constraint imposed bv the plies on one another. From a viscoelastic standpoint, much of this increased stiffness and strength can be attributed to the presence of the fiber trusses imposed by three or more fibers and the assumptions of CLT. These assumptions that normals remain straight and normal can be rigorously demonstrated for regions away from free edges and where the applied stress field is relatively uniform and the individual plies have uniform properties in plan form. At the free edges these assumptions are not valid, yet for many practical structures the rcgion affected by the free edges is relatively small. Obvious examples are aircraft wing skins which are bolted at the edges, and tubular structures. This ignores the localized behavior around holes and cut-outs but addresses the overall properties of the structure. Obviously these localized effects are very important for failure analysis, and damage can occur as a function of time at those points. More elaborate techniques would be necessary to predict these localized effects. To demonstrate the significance of the fiber truss effect in regions well away from free edges, several example layups are now considered. Because the matrix properties are often much more viscoelastic than the fibers, one would expect the laminates to be much more time dependent in directions not dominated by fibers. Figure 5 illustrates the predicted nonlinear response of a ( 4 5 / - 4 5 / 9 0 , ), laminate as x is varied from 0% to 31V~f of the

91

Fiber truss conceptfor creep properties of laminated composites 160 o O_

Creep Compliance Predictions of T300/934 Graphite/Epoxy Laminates

140

[45/-45/90xls

120 Q~ 0

o

*here x is percent of Iominote

/

x percent o~

/ /

160 Deg C and 34.5 MPo

100 2~,

80

Q.

E

0 C)

n Q) L.

(D

60 20N 30~

40 20 0 0

Log Time

(sec)

week

Yeor

Fig. 5. Numerical example of creep compliancefor [45/-45/90x]S with percent of 90' plys varying. total laminate ply content. These predictions are based on the properties obtained for the load applied in the 0O direction. The compliance at short times is 40% greater for the laminate with no 90° plies than for the case where x = 30%. While this difference is not small, one notes that the response after 10 weeks is estimated to be more than 200% greater than for the laminate with 30% 90° plies, One should also note that only 3% 90° plies is sufficient to reduce this difference by a factor of 2. The caution one should see from this type of prediction is that observations based on short term tests could be very much in error. The wide divergence appears quite slowly, but it results in long term behavior which is grossly different from what initial observations would imply. Another example is illustrated by predictions in Figs 6-8 for a more fiber dominated type of laminate. The (0/90),~ laminate may be typical of certain laminates which have been used widely for pressure vessel applications; it is similar to many of the minimum thickness skins which are fabricated from woven material. Consider the case where a uniaxial load is applied at small angles away from the principal material directions. Although the actual loading is often biaxial, this loading situation will illustrate the point. In Fig. 6 we observe that even small angles between the load and the fiber direction can give rise to significant creep and rapid delayed failures. Figure 7 shows that the addition of only 4% (45/-45) plies can significantly reduce the time dependence of the laminate. Figure 8 suggests that the time dependence is

D. A. Dillard, K. C. Gramoll, H. F. Brinson

92

~121 cl I--

2O

~-

16

J

J 4°

J qJ C)

2° 0o

12

c

0

Predicted Failure

EL

E

8

Q_ a) Q)

4

o (D

Creep C o m p l i a n c e Predictions of T 3 0 0 / 9 3 4 Graphite/Epoxy Laminate Load Applied at Various Off-Axis Angles [ 9 0 / 0 / / ~ 4 5 x ]s where x ie O~ 160 Deg C and 138 MPa

0

Log Time (sec)

I

Week

Year

Fig. 6. Numerical example of predicted creep compliance for [90/0]s with load applied at various angles.

" O- " cl

2O

'--

16

10o

6o ~

4° ~

(.9 c-

2o

13°

12

0

Q_

E o

8

Creep Compliance P r e d i c t i o n s of T 3 0 0 / 9 3 4 Graphite/Epoxy Laminate

Q_

~

I1) L_

Load Applied at Various Off-Axis Angles [ 9 0 / O / ~ 4 5 x ] S where x ie 4~ 160 Deg C and 138 MPo

4

f.D 0 0

Log Time

(sec)

Week

1 8 Year

Fig. 7. Numerical example of predicted creep compliance for [90/0/-45/45]s with load applied at various angles.

Fiber truss concept for creep properties of laminated composites

93

20 EL J 16 I

c) co °--

10 ° 8° 6° 4° 2o 0o

12

E).._

E

0 (..)

8

Compliance

Graphite/Epoxy

cl

~

Creep

4

Predictions

of T300/934

Laminate

Load Applied at Various Off-Axis Angles [ 9 0 f O / + 4 5 x ] s where x is 10~ 160 De9 C and 1.38 MPa

CD O

I

I 8

Log Time

(sec)

Week

Year

Fig. 8. Numerical example of predicted creep compliance for [90/0/-45/45]s with load applied at various angles.

minimal when the percentage of (45/-45) plies is increased to 10%. The addition of at least a third fiber direction induces the fiber truss effect which significantly reduces the time dependence. Again one should note that the observations at short times could lead to grossly non-conservative estimates of long term behavior. Experimental verification of fiber truss effect To illustrate the validity of these predictions, tests have been run on several specimens. Figure 9 shows the significant effect of the third fiber direction on 25 mm wide specimens of Kevlar/epoxy at 82°C. The specimens were loaded to produce the same initial strain in them. The load on the (45/-45)2s specimen was 3.0MPa, and the load on the (45/-45/902)s specimen was 20.0 MPa. Note the extreme divergence of the laminate with only two fiber directions. The initial compliances were different by a factor of nearly 7. This occurred because, at the higher temperature, the short time response was shifted far to the right on the time vs. compliance curve (Fig. 6). In an evaluation of the width effect, data were collected for specimens of different widths with layups of (45/-45)2s and (45/-45/902)~; the results are given in Figs 10 and 11. These specimens ranged from 12-38 mm wide and were tested at various stress levels. There was a trend for the compliance of

94

D. A. Dillard, K. C. Gramoll, H. F. Brinson

Creep C o m p l i a n c e of K e v l a r / E p o x y

~. 2000 O Q_ p-

L

a

m

i

n

a

t

e

s

,-- 1 600 (O

~- 1200 CD

Q_

E o

°

800

o

÷ [902/45/-45]s

at 20 ~lPo

Cl_

~o ~9

400

L

C)

I

0

I

0.5

~

I

I

|

)

I

I

{'

I

I

I

I

I

I

I

:

,

I

, ~

I

I

1.

I

I

I

:

:

~

,

a,

I

3.s

Log Time

(Sec)

Doy

(hour)

Fig. 9. Comparisonof actualcreeptest resultsof [45/-45], and [902/45/-45]~Kevlar/epoxy laminates.

1.18 (O C o ..--

1.16 1.14

~3-11 2

E

o (D

-(3

1.1

N o r m a l i z e d E x p e r i m e n t a l Creep C o m p l i a n c e of [ . 4 5 / - 4 5 ] s K e v l a r / E p o x y Laminates

,/~

25 C and 21 ~lPa Specimen Width

1~/'~ pjff"

a 12.4 enm × 19.1 men

/J ,/f

÷ 25.4 men 4, 2514 m m

1.08

N

-o

1.06

E

~- 1.04 O Z 1.02

2

Log Time

3

'I

(see)

,oor

4

0oy

Fig. 10. Creep complianceof [45/-45]~ Kevlar/epoxylaminates(note lack of fiber truss effect).

95

Fiber truss concept for creep properties of laminated composites

1.35 Normalized Experimental Creep Compliance of [ 9 0 2 / 4 5 / - 4 5 ] s Kevlar/Epoxy Laminates

1.3 E ,--

2 5 ° C end

1.25

Specimen n

o_

E o

21 MPa

f

Width

/

12"4mm

+

/

÷ 2s I, mm

1.2

,~=

7

O

-o q~ 1.15 . IN

o

E o

z

1.1

1.05 1

•w-

1

i

i

i

3

Log Time

(sec)

i

5

I

(ooy)

week

Fig. 11. Truss effect on [902/45/-45]s Kevlar/epoxy laminates.

both laminates to be somewhat larger for narrow specimens than for wider specimens. However, this width effect is more pronounced for the laminate with three fiber directions, as would be predicted by an understanding of the interaction of the free edges and the fibre truss effect. While this effect was fairly small for these laminates, the width effect should be greater as the laminate compliance reaches the fiber truss limit. These tests have not yet been performed. In addition to examining the global deformation predictions, further insights may also be gained by considering the stress redistribution predictions within a laminate. This matrix stress relaxation and load shifting to fibers will generally take place over a long period of time. As an example, the relaxation of the matrix octahedral shear stress, which is a function of the lamina transverse (o'22) and shear (o12) stresses, for the laminate used in Fig. 3 takes place over 18-19 decades of time (see Fig. 12). It should not be assumed that the individual lamina stresses that are used to calculate the matrix octahedral stress (o'22 and o-,2) monotonically approach zero stress. Figures 13 and 14 show the transverse and shear stresses in each ply. Notice that the transverse stress in the individual plies can actually change from tension to compression, as it approaches zero stress. This fluctuation of stresses can be attributed to the different relaxation rates that each of the directions (transverse, shear, fiber) and ply orientation exhibit. The shear

96

D. A. Dillard, K. C. Gramoll, H. F. Brinson 40 35

Predicted

o

Ply

30 25

to X

m 09 (1)

L

o

Graphite/Epoxy (T300/934) 160° C Load= 179MPa

20

03

X "~"

Matrix Stress Relaxation

[10/55/-35/10]"

15

(1) 10 03 5 0

I

'i'~'~';';'1'1

i

13

I

|

I

15

i

17

i

(9

Log Time (sec) Fig. 12. Numerical prediction of stress relaxation in matrix for graphite/epoxy laminate.

20 o

15

O_ .._i

Predicted Matrix Sheer Stress Relaxation

[io/55/-35/1 o]=

10

5

C4

~-~

0

-5 03 L_ o

-10

t03

-15 -20

Graphite/Epoxy ( T 3 0 0 / 9 3 4 )

~

~

Load 179 MPo t

10 ° Ply

';'~'~'-;'~'11

i

i

1'3

I

l's

I

1'7

i

1'9

Log Time (see) Fig. 13. Numerical prediction of shear stress relaxation in matrix for graphite/epoxy laminate.

97

Fiber truss concept for creep properties of laminated composites

10 C3

c1_

9! 8

ss* P,y

7" 6

03 O9

5 4 3 2

k_

1

t"q ¢N

tD

q)

(13 Q) (n k_

> O3 E C3 k..

I--

Predicted Matrix Transverse Stress Relaxation Load179&MPII [to/ss/-3s/lo], Graphite/Epoxy (T300/934) ,__.J.,.,..,

N

--35 ==Ply

~],

0 -1 -2 -2 -4

10 °

Ply

-5 -6

i

i

-1 Log T i m e

i

i

1'5 1'7

i

;9

(sec)

Fig. 14. Numerical prediction of transverse stress relaxation in matrix for graphite/epoxy laminate. will shed its stress faster than the transverse direction, thus causing the transverse direction actually to pick up load in the first few decades and then shed this load later after the shear has totally relaxed. Further comments o n the power law

In attempting to understand and model long term creep compliance, there has been some concern about how accurate the simpler power law is for modeling the long term behavior of a laminate. The simple form of the power law gives the linear viscoelastic compliance as S(t) = Sg + mt n

where Sg is the glassy or initial compliance, rn is the coefficient for the power law term, and n is the exponent. This form suggests that the compliance is u n b o u n d e d for large values of time. Polymer experience suggests that crosslinked resins such as epoxy should have a limiting rubbery compliance, St. This suggests an alternative form of the power law given by (St - Sg)

S(t) = Sg + (1 + r/t)"

where r is the characteristic time of the power law. Figure 15 illustrates the

98

D. A. Dillard, K. C. Grarnoll, H. F. Brinson

1.2 Simple and Generalized Power Law Comparison (D

°_ 60 u-

P a r a m e t e r Values

(D

(,.) c" o

'~ o

[

0.8

n = 0.4 m 0 001583

0.6

=

=

~

// ///

s, = 0.o sr = ,.o

o.--

#_a E

/ f -,~.~-~

st--

Simple ...-~/ Power Low /~-,..

' lo,ooo,ooo

Generalized

/////

~g +mt. n

._

Power L o w

~/ S. + --(sL- s,)

0.4 oC) Z

~g

'

(1 + r / O n

0.2 0

i

-4

t

i

-2

i

0

i

i

2

i

i

i

i

4

'1

i

6

t

8

10

Log Time Fig. 15. Simple and generalized power law compliance comparison.

Simple and Generalized • Power Law Comparison c"

._~

Simple ~ / / sg + mtn . / Z

Power Law ~ /

0

g

P a r a m e t e r Values

0

--

Sg = 0 . 0

c- Q_

S

2E F-- 0

-~ (..)

S r --

r -

1 0 .

- - --'/'/'~....r ~ - - ~

/r

/

-1

\

/

-.

Generalized Power Law

_

8= -ee

0n

Eo I,....

o Z

o

-2

_..J

-3 -4

10

Log Time Fig. 16. Simple and generalized power law log compliance comparis(m.

Fiber truss concept for creep properties of laminated composites

99

degree of fit between these two forms in log compliance vs. log time space. Figure 16 shows the same results for compliance vs. log time. A numerical experiment revealed that, if one assumes the general power law to be the correct form, the simpler power law is in very slight error when fitted to the initial portion of the curve. In fact, if the simple power law is fitted to the extreme left of the figure, which corresponds to a maximum transient compliance of only 4% of (St - Sg), the values at 50% of (St - Sg) are only 8% in error. This gives confidence that the simple power law is adequate for characterizing the response up through the glass transition temperature. Because the simple power law is conservative in nature, and because most composite structures are used well below the glass transition temperature, this simple form is adequate for most situations. If the temperature is above Tg or if environmental effects significantly lower Tg, the more general form of the power law should be used. Also, if predictions are to be made at a very long time, the more general form may be needed. Within this region, however, it appears that failure will be imminent. Other difficulties in using the power law have been discussed previously. ~-'° 3 CONCLUSIONS The effect of the free edge has been shown to be more significant for long term compliance of laminates than might be predicted by short term observations. This free edge effect can cause narrow specimens to be significantly more compliant than CLT would predict. For long term durability, it has been demonstrated that specimens with three or more fiber directions have dramatically less creep than laminates consisting of only two fiber directions, even though the difference for short term loading is much smaller. A discussion of the simple and more general form of the power law for modeling the viscoelastic response of composite materials suggests that the simple form of the power law should be adequate for predictions below the glass transition temperature (at the service life), but that the more general form is needed when large scale molecular rearrangement is taking place. ACKNOWLEDGEMENTS The authors would like to acknowledge the continued support of NASAAmes for contract NASA-NSG-2038 which has been monitored by Dr Howard Nelson. We are also grateful to Dr Hal Loken and the Dupont Corporation for providing support and furnishing many of the specimens used in this study. We would also like to thank Shelia Lucas for typing the manuscript.

100

D. A. Dillard, K. C. Gramoll, H. F. Brinson REFERENCES

1. Myhre, S. H., Labor, J. D. & Aker, S. C., Composites, 13 (1982) 289. 2. Brinson, H. F., Morris, D. H. & Yeow, Y. T., A new experimental method for the accelerated characterization of composite materials. Paper presented at Sixth International Conference on Experimental Stress Analysis, Munich, September 1978. 3. Lou, Y. C. & Schapery, R. A., Viscoelastic characterization of nonlinear fiber-reinforced plastic. J. Compos. Mater., 5 (1971) 208-34. 4. Zhurkov, S. N., Kinetic concept of the strength of solids. Int. J. Fracture Mech., 1 (1965). 5. Dillard, D. A. & Brinson, H. F., A numerical procedure for predicting creep and delayed failures in laminated composites. In Long Term Behavior of Composites, ed. T. K. O'Brien, ASTM STP 813, 1983, pp. 23-37. 6. Dillard, D. A., Morris, D. A. & Brinson, H. F., Predicting viscoelastic response and delayed failures in general laminated composites. In Composite Materials: Testing and Design (Sixth Conference), ed. I. M. Daniel, ASTM STP 787, 1982, pp. 357-70. 7. Jones, R. M., Mechanics of Composite Materials. McGraw-Hill, New York. 1975. 8. Dillard, D. A. & Hiel, C., Singularity problems of the power law for modeling creep compliance. Proc. SEM Spring Conference on Experimental Mechanics, Las Vegas, June 1985. 9. Hiel, C., Cardon, A. H. & Brinson, H. F., The nonlinear viscoelastic response of resin matrix composite laminates. NASA Contractor Report 3772, July 1984. 10. Tuttle, M. E. & Brinson, H. F., Accelerated viscoelastic characterization of T300/5208 graphite/epoxy laminates. NASA Contractor Report 3871, March 1985.