Statistical study of strength and fatigue life of composite materials

Statistical study of strength and fatigue life of composite materials W. HWANG and KS. HAN (State University of New York, USA) Fatigue life scattering of glass fibre-reinforced epoxy composites has been studied at 3 0 0 K and 77 K. The static strength and the fatigue life distributions were studied using normal, log-normal and twoparameter Weibull distribution functions; the value of mean fatigue life was analysed using mean fatigue life, mean log fatigue life and expected value of the Weibull distribution function. The effect on fatigue life of two stress levels (low-high and high-low) was investigated using the distribution functions; a modification of Miner's rule is made to predict the resulting scattering.

Key words: composite materials; fatigue testing; compressive strength; fatigue life; statistical analysis; two-stress level fatigue; glass fibres; epoxy resin

Glass fibre-reinforced composites are prospective materials not only for room temperature but also for cryogenic temperature applications, where high strength-to-density and strength-to-thermal conductivity ratios are required, t'2 Although there have been many investigations into the mechanical properties of composites at room temperature, there are only a few studies at cryogenic temperatures? -I° The results showed that the strength at 77 K is much higher than at 300 ICt,3-6 while fatigue life is slightly longer at 77 K than at 300 K. t,z.7-9 Due to the wide scatter of composite material properties, especially the fatigue life, statistical methods are necessary to analyse experimental data and predict properties. There have been some studies 6-8, .t-.4 on this topic. The Weibull distribution function t5 has been most frequently used for the analysis of fatigue scattering problems because of its excellent applicability and accuracy. However, some authors ~' claim that the log-normal distribution is better at analysing fatigue data than the two-parameter Weibull distribution. Although normal, log-normal and Weibull distribution functions are used widely for scattering studies, the characteristics of each function are not clearly known. It is the purpose of this study to compare these functions with experimental data and to determine the applica.bility of each function to the analysis of mechanical properties, such as static strength, onestress level and two-stress level fatigue life, of composite materials at room and cryogenic temperatu res.

EXPERIMENTAL DETAILS Glass fibre cloth epoxy composite, G-10 CR, manufactured by Spaulding Fiberglass Corp, was studied. The matrix of G-10 CR comprises an aminecatalysed, heat-activated, epichlorohydrin-bisphenol A type solid epoxy resin: the reinforcement is a silanefinished fabric, woven from continuous filament E-glass fibres having a diameter of 0.009 mm. Test specimens were cut from moulded rods, made by wrapping prepreg on a small mandrel, removing the mandrel, heat pressing in cylindrical moulds and grinding to size. The cylindrical specimens were 15.7 mm in diameter and 41.1 mm long. To prevent premature failure by end splitting and brooming, end caps made of heat-treated and quenched tool steel were used. The end caps were 50.8 mm in diameter and 15.7 mm thick, and contained a hole 15.7 mm in diameter and 5.1 mm deep. An MTS machine was used for static testing. Load was monitored by the load cell (Interface 1330-AF 100K), and load and displacement were recorded by an X - Y recorder. Compressive strength measurements were conducted at a strain rate of 1.3 mm min -t. Loadcontrolled fatigue tests were carried out on a closedloop. electro-hydraulic test machine, using sinusoidal wavelbrm to simulate the loading of superconductive magnetic energy storages (SMES). The fatigue loading cycle frequency was 1 Hz, which is believed to give a negligible temperature rise during tests. The ratios of peak stress to ultimate strength (stress ratio, R) used were 0.9, 0.85, 0.8, 0.75, 0.7, 0.65 and 0.6: a constant minimum stress level of 12.9 MPa was maintained.

0010-4361/87/010047-07 $3.00 © 1987 Butterworth 8" Co (Publishers) Ltd COMPOSITES. VOLUME 18. NO 1 . JANUARY 1987

47

Testing was conducted at 300 K in air and at 77 K with liquid nitrogen. For testing at 77 K, the specimens were vapour cooled slowly after liquid nitrogen for 15 min and then immersed into the liquid nitrogen for 15 min to ensure that they had reached an equilibrium temperature of 77 K. Two series of tests were conducted to examine the cumulative damage caused by two-level fatigue tests. To study low-high stress levels, the specimens were first subjected to fatigue at the lower stress level (266 MPa) to 10 000 cycles. Those that survived were then fatigued at the higher stress level (310 MPa) up to failure. In the high-low test, specimens were cycled at 310 MPa for 300 cycles and then fatigued at 266 MPa up to failure. Two-stress level tests were conducted at 300 K only.

Mean (expected value) and variance of the three distributions are as follows.

1) Normal distribution Mean: ttl

1 ~xi

.~_

(5)

171

i~l

Variance:

(6) i=1

2) Log-normal distribution RESULTS AND DISCUSSION

Mean:

Static strength and constant amplitude fatigue life lnx =

The experimental probability of failure for static strength and constant amplitude fatigue life data was determined by the median rank:

.-~

lnx i

(7)

l(inxi)=_(]-~)=l

(8)

i=1

Variance:

(1)

P = (m i - 0.3)/(m + 0.4)

where P is the probability of failure, m is the total number of samples and m i is the failure order. The experimental data, determined for seven stress levels (R = 0.9. 0.85, 0.8, 0.75, 0.7, 0.65 and 0.6) at 300 K and for four stress levels (R = 0.85, 0.8, 0.75 and 0.7) at 77 K in the case of fatigue data, are compared with the predictions of normal, log-normal and two-parameter Weibull distributions.

o,==

l ~" .-7 i=1

3) Weibull distribution E x p e c t e d value:

E(x) =

/

.ff(x)dx = / 3 F(I + l/at)

(9)

The cumulative distribution functions (CDF) are given by the following equations. Variance:

1) For normal distribution:

V(x) = E(x ~) -- [E(x)] 2

_(x-.V)lo

F(.v) = (l/x/~-~o')

I

= (t'l(x -.V)/(r]

(2)

2) For log-normal distribution: In.v

x

(hv-

I

(I/lnf) e x p l - ( l / 2 d 2)

Ira-) 21 tit.

= (1)l(lnx - lrtx)/dl

(3)

where hlv is the mean of In.v and a' is the standard deviation of lwc.

3) For two-parameter Weibull distribution: F(x) = I -- e x p l - ( . v / O ) " I

where a is the shape parameter and/3 is the scale parameter.

48

whereflx) is the probability density function of the Weibull distribution and 3, is the g a m m a function. Comparisons between experimental data and predicted values are made in Fig. 1 for static strength distribution and in Figs 2-6 for fatigue life distribution. Results of experiments at 300 K and at R ----0.6, 0.7 and 0.8 are shown in Figs 2-4 respectively; tests at R --- 0.7 and 0.8 yielded results showing similar trends. Likewise, the results of tests performed at 77 K and at R = 0.7 and 0.8 are shown in Figs 5 and 6; similar trends were observed in data obtained at R = 0.75. The least squares method was used to determine the parameters of the Weibull distribution function.

where.~ is the data mean and tr is the standard deviation.

F(.v) = ( l / ~ ' e ' )

(lO)

= fl2 [F(I + 2 / a t ) - l-~(l + l/a)]

exp(--?/2) dt

(4)

The results show that the three distributions predict static strength well. However, general trends show that the log-normal and two-parameter Weibull distributions predict fatigue life better than the normal distribution. To determine the appropriate mean fatigue life, the mean values are compared using the mean of the normal, log-normal and Weibull distributions and the scale parameter of the Weibull distribution function. The S/N curves based on these values are presented in Figs 7 and 8 for 300 K and 77 K, respectively. The

COMPOSITES . JANUARY

1987

99

shape parameters and the mean values are presented in Table I for 300 K and in Table 2 for 77 K. Of the four prediction methods, the mean log-normal distribution predicts the lowest fatigue life while the expected value of the Weibull distribution function predicts the highest fatigue life at both 300 K and 77 K. The fatigue life at 77 K is slightly longer than that at 300 K. Standard deviations are presented in Tables 3 and 4 for 300 K and 77 K. respectively.

I!

90

99

70

/

A

It.

50

E3 CJ 30

Normal /// Log-normal ,~//i Weibull . ~

. . . . • ~

90

d; 7O A

z~ U. a

10

-./

(j

...

1 /~/}i

I-"-i

.Norma, i

50 30 ~..._...-- ° o ~ ' ~ • / / / ° ~ ° j /

, 10

wiegibn~i~al

~

350 375

400

425 450

475

1

/lI I I I /

0.5

I

500

Static strength (MPa)

I

I

2

3

Fig. 1 Static strength distribution

Fatigue life, log N Fig. 3 Fatigue life distribution, R = 0.7. 300 K

99

99

I

Normal I,

90

9O

70

-

Log-normal

/e

-

Weibull

I ,;

'2

70

A

LL

A

/

50

50

£3 ¢.J

30

•/is"

10

Normal Log-normal Weibull

is S s

1 L

3

S':

30

10/ /

m

~

I

J

4 Fatigue life, log N

Fig. 2 Fatigue life distribution, R -- 0.6,300 K

COMPOSITES . JANUARY 1 987

5

6

1

0

1

2

3

Fatigue life, log N Fig, 4 Fatigue life distribution, R = 0.8, 300 K

49

99

nt is the fatigue cycles tested at stress S~ and #l~ is the equivalent cycles at stress $2 which produces the same damage as n~ cycles at S~.

I

-

.....

Normal

/•

......

Log-normal

/

1) For normal distribution:

90 Weibull

,'i~

[

x < ,,

@l(x - ~ ) / o ' , l

70

/~,/

F(x) =

iI

(13)

Lcill(x

n~ + n , 2 - E d c r 2 ]

x >

n~

A

50 LL E3 O

where

30

(14)

, , : = y: + o':l(n~ - -f,)/o-,I

-'£'t is the mean fatigue life at stress S~ and-'7"2 is the mean fatigue life at stress Sz.

10

/ ."

0

I

I

[

1

2

3

4

1.0

Fatigue life, log N Fig. 5

Fatigue life distribution, R = 0.7, 77 K

0.9

99 Normal 90 =

I

........

Log-normal

- - - - - -

Weibull

0.8

l'~'/

o"

~Li

--

/ t~'j •

0.7

j,/

-.9../

70 _

o

--

-

":'~.~'~

~

A

/,//

50 ii E3 ¢J

•

..

0.6

-

, n x

o ........

- .__J...l-

30

-

E(x)

A 0.5

10

I

I

I

1

1

2

3

4

5

Number of cycles, log N

0

I

I

I

1

2

3

Fig. 7

S/N

curve based on the mean values, 3 0 0 K

4 O

0.9

Fatigue life, log N

Inx Fig. 6

Fatigue life distribution. R = 0.8, 77 K o

Two-stress level fatigue life prediction by percent failure rule

•\

Two-strcss level ( l o w - h i g h and h i g h - l o w ) fatigue life

has buell analysed by the percent failure damage model, 12 using normal, log-normal and Weibull distribution functions. The total distribution function can bc expressed according to the following equations by the percent failure damage theory,

F(.v) =

[

LF2(.v - n, + nlz)

.

.

.

.

.

E(x)

-

-

0.75

~

.v < nl ( I 1) .~, > nl

0.65

whcrc

50

.

\.

o"

0

Ft(n,) = Fz(n,,)

.

"" ~ ~ o o

0.8

0.7 Fi(.v)

.

0.85

I 1

I 2

I 3

I 4

I 5

Number of cycles, log N (12)

Fig. 8

S/N

curve based on the mean values, 77 K

COMPOSITES . J A N U A R Y 1 9 8 7

Table 1.

Comparison of mean values and parameters of Weibull distribution function, 3 0 0 K

Stress ratio, R

x"

exp ~

E(x)

[3

0.6 0.65 0.7 0.75 0.8 0.85 0.9 Static

42400 18800 725 1500 205 65 6 63.7

20970 13780 350 944 129 15 2 63.5

44100 20200 790 1600 340 93 10 63.3

40240 21170 680 1600 270 41 5 66

Table 2.

0.84 1.14 0.81 0.98 0.70 0.47 0.53 12

Comparison of mean values and parameters ofWeibulldistributionfunction, 77 K

Stressratio, R

~

exp0"~')

E(x)

~

a

0.7 0.75 0.8 0.85

1337 1375 875 108

318 703 138 9

2843 1385 1430 118

1090 1300 490 33

0.44 0.88 0.42 0.39

Table 3.

Standard devi~ion, 3 0 0 K

where

Stress ratio, R

Normal

Log-normal

Weibull

0.6 0.65 0.7 0.75 0.8 0.85 0.9 Static

44658 16888 896 1400 133 101 11.1 5.26

1.3418 0.7394 1.3497 1.1020 1.3603 1.7981 1.3055 0.085

81463 2003 930 1657 499 230 21.3 6.34

Table 4.

,in =/~,(,1,//~) "'/~'

The experimental data and predicted values are presented in Figs 9 and 10 for the low-high and highlow test, respectively. The lower stress level is 0.6 and the normalized higher stress level is 0.7. The results show that all the distribution functions predict a higher probability of failure for the low-high test and a lower probability of failure for the high-low test than the experimental data.

Modification of Miner's rule to two-stress level fatigue life distribution

Standard deviation, 77 K

Stress ratio, R

Normal

Log-normal

Weibull

0.7 0.75 0.8 0.85

1674 1588 1145 220

2.4792 1.2557 2.5870 2.2229

7668 1575 4234 385

Most cumulative d a m a g e theories are not based on statistical analysis, therefore they cannot be directly applied to analyse multi-stress level fatigue life scattering: some special modifications are needed such as the Monte Carlo simulation, t~ In this study, the strength-life equal rank assumption '3,'4 is considered and expanded to modify non-statistical cumulative damage theories for the analysis of multi-stress level fatigue life distribution. The modification can be made in the following manner.

2) For log-normal distribution:

1)

[Ol(IrL~-

1--~:~',)/o',I

x

< ,,, 2)

F(x) =

tOllln(x

-.,

+ ,,,=) - Ta-~,l/o',}

x >.,

(15) where n, 2 = exp{Irtx-2 + o¢, [(Inn1 - Irtv,)/~d}

(16)

lrt,., is the mean log fatigue life at stress S, and lnx2 is the mean log latigue life at stress $2.

3) For Weibull distribution: = F(x)

[ I -- exp[-- (.v//3,)",]

x < nt

[

x > n1

I - exp {-[(x - nl + .,,)//321%}

(17) COMPOSITES . JANUARY 1987

(18)

3)

Choose a distribution function which is accurate at predicting one-stress level fatigue life. The strength-life equal rank assumption is expanded so that a specimen has a unique and the same rank in static strength, one-stress and multi-stress level fatigue life. In other words, if a specimen has a probability o f failure 0.5 in static strength, the probability of failure will be 0.5 not only for onestress level fatigue life but also for the multi-stress level case. For a probability o f failure 0.5 the fatigue life at each applied stress level can be determined. Therefore the non-statistical cumulative d a m a g e role can be applied and the remahfing life can be found. It could .be said that the predicted remaining life implies a probability of failure of 0.5 in the multi-stress level' fatigue test by the above assumption. In this way, the other remaining life can be predicted.

51

1?1

99

(19a)

Z ni/g i = 1 i=l

l/

and 90

where/h, ' 7 2 . . . . . rim_ I are constants, tl m and Ni are random variables.

70

For two-stress level fatigue:

A

~R It.

50

n2 = N2(I

a (.9

-

(20a)

11,/N,)

and

30 °°

_.

"'"-"

----"-" " " /

F I ( N , ) = F2(Na) = P

10 / /

/

/

Weibull

x

=

fl{

Inll

-

-

F(x)l}

Ni

(low-high

=

fli[

--

ln(l -- p)lt/,,i

,1~ = / ~ , 1

.y Q

90

t, a

/.-.... ~

50 30

Normal Log-normal Weibull

1

1

i 4

I 2

3

×

{l

-

-

ln(l

,1,/I/3,(

P)] '/'~,

-

-

ln(I

Two-stress

level f a t i g u e

life d i s t r i b u t i o n

(high-low

5

test)

function is chosen for the two-stress level fatigue life and only Miner's rule is investigated, because there may bc some difficulties and calculation errors in modifying the other damage theories. Following the above interpretation, Miner's rule

52

(23)

(24)

The predictions of Equations (23) and (24) are compared with experimental data in Tables 5 and 6 for the low-high and high-low test. respectively. (The discrepancies between experimental and predicted results arise due to the wide scatter of fatigue results for composite materials.) The result shows that the remaining fatigue lives predicted by the percent failure

Table 5. Comparison of predicted remaining fatigue life with experimental data (low-high test) Probability of failure, P

Experimental data

Percent failure rule

Modified Miner's rule

(number of cycles)

In this study, the two-parameter Weibull distribution

becomes:

P))'/'~']}

At a probability of failure, P, the percent failure damage rule (Equation (17)) predicts the following remaining life:

Fatigue life, log N Fig. 1 0

-

n 2 = fl~[ -- In(l -- p ) l t / a , _ n,~

Ill

10

(22)

Substitution of Equation (22) into Equation (20b) provides:

test)

99

70

(21)

'/"

For fatigue life, Equation (21) can be expressed as follows:

Fatigue life, log N Two-stress level f a t i g u e life d i s t r i b u t i o n

(20b)

The two-parameter Weibull distribution function (Equation (4)) can be rewritten:

Normal Log-normal

/

I 4

Fig. 9

(19b)

Fi(Ni) = p

0.3936 0.4562 0.5187 0.5813 0.6439 0.7064 0.7690 0.8316 0.8941 0.9565

315 400 500 1101 2535 3165 3725 6500 9325 15041

128 208 301 412 546 713 929 1226 1685 2631

125 203 295 405 538 703 918 1213 1670 2614

COMPOSITES. JANUARY 1987

Table 6. Comparison of predicted remaining fatigue life with experimental data (high-low test) Probability of failure, P

Experimental data

Percent failure rule

Modified Miner's rule

(number of cycles) 0.4881 0.5409 0.5937 0.6464 0.6992 0.7520 0.8048 0.8575 0.9103 0.9631

6478 9852 10932 13341 19000 28833 29757 39000 49000 50900

6684 11587 17248 23864 31779 41495 53905 70746 96460 148393

6894 11917 17692 24420 32449 42280 54811 71787 97663 149830

rule a n d m o d i f i e d M i n e r ' s rule are a l m o s t same. It is very interesting that E q u a t i o n s (23) a n d (24) have s i m i l a r forms to each other, the difference b e i n g that the last term o f E q u a t i o n (23) varies with p r o b a b i l i t y o f failure while the last term o f E q u a t i o n (24) is constant.

CONCLUDING REMARKS A m o n g the d i s t r i b u t i o n functions used here, the W e i b u l l d i s t r i b u t i o n function is the best to use for the p r e d i c t i o n o f fatigue life scattering. It is found that if there is only a s m a l l difference between the variables, as in the static strength case, the scatter is followed by the n o r m a l distribution. O n the o t h e r h a n d , it is expected that the l o g - n o r m a l distribution will be the most a p p l i c a b l e in the case o f extreme scattering. C o m p a r i s o n o f m e a n values shows that it is most desirable to take m e a n log fatigue life as a m e a n value of fatigue life. T h e m e a n fatigue life a n d the expected value o f the W e i b u l l d i s t r i b u t i o n s o m e t i m e s exceed the value o f the scale p a r a m e t e r o f the W e i b u l l d i s t r i b u t i o n function; therefore, for safety c o n s i d e r a t i o n s , these s h o u l d not be taken as a m e a n value of fatigue life. All the d i s t r i b u t i o n functions predict a h i g h e r p r o b a b i l i t y o f failure for the l o w - h i g h test a n d a lower p r o b a b i l i t y o f failure for the h i g h - l o w test t h a n the e x p e r i m e n t a l data. Non-statistical c u m u l a t i v e d a m a g e theories can be m o d i f i e d for the p r e d i c t i o n o f multi-stress fatigue life d i s t r i b u t i o n by the s t r e n g t h - l i f e equal r a n k a s s u m p t i o n .

REFERENCES Hen, K.S. "Compressive fatigue behaviour of a glass fibre-

reinforced polyester composite at 300 K and 77 K" Composi, o.s 14 No 2 (April 1983) pp 145-150

COMPOSITES. JANUARY 1987

2

Han, K.S. and HamS, M. "Fatigue life scattering of RP/C" 38th Ann RP/CI Conf 1983 (Reinforced Plastics/Composites

Division, SPI) paper 12-G "~ Hen, K.S. et al "Compressive tests of composite tubes at 300 K and 77 K" Advances hi C~. ogenic Engng Mawr 28 (ICMC. Plenum Pt'ess, 1982) pp 253-260 4 Khalil, A. and Han, K.S. "Mechanical and thermal properties of glass-fiber-reinforced composites at cryogenic temperatures" Advances in Cryogenic Engng Mater 28 (ICMC, Plenum Press, 1982) pp 243-252 4 Kasen, M.B. et al "Mechanical, electrical and thermal characterization of G-10CR and G-I ICR glass-cloth/epoxy laminates between room temperature and 4 K" Advances in Cr)'ogenic Engng Mawr 1'6 (ICMC, Plenum Press, 1980) pp 235-244 5 Dahlerup-Petersen, K, "Test of composite materials at cryogenic temperatures: facilities and results" Advances ill Co,ogenic Engng Mater 26 (ICMC, Plenum Press, 1980) pp 268-279 7 MeLoughlin, J.R. "Low temperature properties of high performance composites" 30th Ann RP/CI Col~ 1975 (Reinforced Plastics/Composites Division, SP1) paper 18-D 8 Abdhel Mohsen, M.H., Hart, K.S. and Rowlands, R.E. "Fatigue of glass-epoxy composite at 77 K and 300 K: observation and prediction" Advances i, Co'ogenic E,gng Mater 30 (ICMC, Plenum Press, 1984) pp 17-24 9 Kasem, M.B, Sehramm, R.E. and Read, D.T. "Fatigue of composites at cryogenic temperatures" Fatigue of Filamentart, Composiw Materials. ASTM STP 636. edited by K.L Reifsnider and K.N. Lauraitis (American Society for Testing and Materials, 1977) pp 141-151 10 Soltysiak, D.J. and Toth Jr, J.M, "Static fatigue of fiber glass pressure vessels from ambient to cryogenic temperatures" 22nd Ann RP/CI Cot~ 1967 (Reinforced Plastics/Composites Division. SPI) paper 14-E I I Shimokawa, T. and Hamaguchi, Y. "Distributions of fatigue life and fatigue strength in notched specimens of a carbon eight-harness-strain laminate" J Composiw Mater 17 (1983) pp 64-76 12 Chou, P.C. "A cumulative damage rule for fatigue of composite materials' in 'Modern Developments in Composite Materials and Structures' edited by J.R. Vinson (ASME. 1979). pp 343-355 13 Hahn, H.T. and Kim, R.Y. 'Proof testing of composite materials" J Composite Mawr 9 (1975) pp 297-311 14 Chou, P.C. and Croman, R. "Residual strength in fatigue based on the strength-life equal rank assumption" J Composite Mater 12 (1978) pp 177-194 15 Weibull, W. "A statistical distribution function of wide applicability"J Appl Mech ( 195 I) pp 293-297 16 Miner, M.A. 'Cumulative damage in fatigue" J Appl Meeh (1945) pp A-159-164 17 Johnsen, S.E.J. and Doner, M.A. "A statistical simulation model of Miner's rule" J Engng Mater and Tech 103 (1981) pp 113-117

AUTHORS T h e a u t h o r s are at T h e State University of New York at Buffalo, D e p a r t m e n t o f M e c h a n i c a l a n d A e r o s p a c e Engineering, Buffalo, NY 14260, USA. Inquiries should be directed to Professor H a n in the first instance.

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