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JOURNAL OF

C O MP O S I TE

MATER IALS Article

Study on static and fatigue behaviors of carbon fiber bundle and the statisticaldistribution by experiments

Jian Song, Weidong Wen, Haitao Cui and Sibo Zhao

Abstract

In this paper, the static, tension–tension fatigue, and residual strength behaviors of carbon fiber bundle have beeninvestigated using experimental methods. The corresponding mechanical models and statistical distribution regulationshave also been established based on the experimental data. The test results indicate that the test section length of fiberbundle had a marked effect on the static strength, which decreased with the increase of length. A two-parameter Weibulldistributional function was used to describe the distribution of static strength. Furthermore, the S-N curve was obtained

by a new segmented function to reflect the changing of fatigue life against stress levels. The experimental results of residual strength show that there was an abnormal tendency, which increased first and decreased later as the increase of cyclic number, for the residual strength of carbon fiber bundle in both tested stress levels 87% and 80%, respectively.Therefore, a new model for the residual strength was put forward and the good agreement between the fitting curvesand experimental was obtained.

Keywords

Carbon fiber bundle, static strength test and distribution law, fatigue life test and S-N, fatigue life distribution, residualstrength test and mechanical model

Introduction

Carbon fiber reinforced composites have been widely

used as reinforcement in polymer composites, owing to

their good mechanical properties and light weight. Most

of the early works focus on the certainty performance,1

such as the stiffness, strength, and fatigue characteristic

of composites, which causes that composites are often

overdesigned, such as heavier and more costly than

necessary. Therefore, probability design methods have

been gradually conducted into composites research.

Schaff and Davidson2 proposed a strength-based wear-

out model to predict the fatigue life and residual strength

of composite laminates and the related distributions

were investigated by a two parameter Weibull function.

Cheng and Hwu3 investigated the fatigue reliability of

composite laminates subjected to constant amplitude

loading and a residual analysis model was proposed to

predict the residual strength. Whitworth4–6 studied the

statistical distribution laws on graphite/epoxy compos-

ites based on a set of static and fatigue tests and put

forward to p-E-N and p-S-N models. Yang et al.7–9

proposed a three-parameter fatigue and residual

strength degradation model to predict the statistical fati-

gue behavior of angle-ply composites. Wu and Yao10

derived the fatigue life distribution of composite lamin-

ates based on their static strength distribution.

Although carbon fiber reinforced composites has

been studied in many aspects, the research on the per-

formance of carbon fiber bundle as main component is

quite limited. There are still some issues associated with

that left to be solved in the respects of non-uniform qual-

ityof filaments, length of limited fiber bundle and fatigue

characteristic, all of which have an impact on the static

and fatigue performance to some extent. Moreover, it is

inevitable to consider the statistical distribution

owing to the random question of self-defect and

College of Energy and Power Engineering, Nanjing University of

Aeronautics and Astronautics, China

Corresponding author:

Jian Song, College of Energy and Power Engineering, Nanjing University of

Aeronautics and Astronautics, No. 29 Street, Nanjing 210016, China.

Email: [email protected]

Journal of Composite Materials

2015, Vol. 49(25) 3157–3168

! The Author(s) 2014

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DOI: 10.1177/0021998314560385

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corresponding results are presented in Table 1, which

are also the residual strengths R(0) in the condition of

zero cycle.

U ¼ 4Pd 2 3000 ð1Þ

where P is the peak of static load; d is the nominaldiameter of carbon fiber bundle in which d is equal

to 7 mm.

From Table 1, we can see that the static strength of

fiber bundles was increased with the decrease of charac-

teristic lengths, which indicated a size effect on the static

strength of fiber bundles. In addition, the representative

load–displacement curves for different characteristic

lengths and the curve of influence of characteristic

length on average strength fitted by the model 2 were

shown in Figure 2, respectively. It is clearly found that

an obvious non-linear phenomenon happened as the

increase of load and also the average static strengths of

fiber bundle with different lengths were remarkable

reduced with the increase of characteristic length.

S 0 ¼ c0

L

1=mð2Þ

where S 0 is the average strength, L is the characteristiclength (30, 40, and 50 mm), c and m are the constants

determined by experimental data (m ¼ 1.632,c0 ¼ 3,625,272.57).

Damaged photographs subjected static loading are

given in Figure 3. From this figure, the location in

the surface of fiber bundles appears as the amountof fluffiness phenomena, which indicates that there

exists a certain quantity of monofilament taking

place failure gradually so that a de-bunching failure

mode finally happens in the middle of carbon fiber

bundle.

Statistical distribution of static strength. To research the

statistics distribution of fiber bundles, the two-para-

meter Weibull distribution function (3) is used to fit

the static strengths R(0) and thus we can plot the

related probability paper and distribution of static

strength in Figure 4.

F ðX Þ ¼ P U X f g ¼ 1 exp X

! ð3Þ

Table 1. Static strength test results of carbon fiber bundles.

Characteristiclength

Specimennumber

Peak load (N)

Strength(MPa)

Averagestrength (MPa)

Scaleparameter,

Shapeparameter,

50 mm L50-1 112.84 977.32 946.50 964.83 23.99L50-2 103.35 895.17

L50-3 104.05 901.23

L50-4 113.21 980.57

L50-5 106.95 926.31

L50-6 110.79 959.57

L50-7 113.81 985.73

40 mm L40-1 127.84 1107.29 1097.75 1111.23 37.61

L40-2 123.43 1069.05

L40-3 122.02 1056.91

L40-4 129.51 1121.73

L40-5 129.64 1122.84L40-6 128.00 1108.71

30 mm L30-1 130.72 1132.25 1172.73 1188.73 34.75

L30-2 131.74 1141.09

L30-3 132.65 1148.91

L30-4 133.48 1156.11

L30-5 134.92 1168.58

L30-6 137.84 1193.86

L30-7 140.68 1218.47

L30-8 141.15 1222.56

Song et al. 3159




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Figure 3. Static tensile test of carbon fiber bundle: (a) pre-test; (b) during the test; (c) photograph of failure specimen.

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0

20

40

60

80

100

120

140

160(a) (b) L30-1 : L=30mm

L40-3 : L=40mm

L50-4 : L=50mm

L o a d / N

displacement/%

25 30 35 40 45 50 55 60 65

800

900

1000

1100

1200

1300

1400

Fitting curve by Eq.(2)

Experimental data

A v e r a g e s t r e n g t h / M P a

Fiber length/mm

Figure 2. (a) Load–displacement curve of typical fiber bundle with different characteristic lengths; (b) influence of characteristic

length on average strength.

3160 Journal of Composite Materials 49(25)




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where F (X ) represents the statistical distribution of

carbon fiber bundles; U means the static strength;

and are scale parameter and shape parameter,

respectively.

From Figure 4, several important findings about the

static strength statistical distribution of carbon fiber

bundles were given as follows: First of all, it is logicalto consider that the statistical distribution for fiber

bundle strength can be estimated by using a two-

parameter Weibull distribution function (3) according

to the correlation coefficients (R ¼ 0.903, 0.884, and0.865 for L ¼ 50, 40, and 30mm, respectively) inFigure 4(a) to (c). Furthermore, although the strength

of fiber bundles varies with different characteristic

length, the value of shape parameters has a no signifi-

cant change (refer to Figure 4(d)) and the general

change trend is that with the increase of length, the

curve slope of Weibll distribution function in view of

one type of length decreases gradually.

In addition, the Kolmogorov–Smirnov hypothesis

tests were applied to verify quantitatively whether the

proposed two-parameter Weibull distribution function

is suitable for the statistical distribution of carbon fiber

bundles. The hypothesis test results are listed in Table 2

and the method is introduced briefly in Appendix 1.

According to Table 2, the values for all of the teststatistics (sqrt(n)*D) are outside the reject domain,

therefore, it is reasonable to accept the hypothesis

that the statistical distribution for static strength of

fiber bundles follows the Weibull distribution law.

Fatigue life test

Testing results of fatigue life. A set of 22 specimens sub-

jected to nine various constant amplitude stress levels

are listed in Table 3. The S-N curve of carbon fiber

bundles referred to equations (4) to (6) was then plotted

in Figure 5 and the related parameters fitted by the

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

6.78

6.80

6.82

6.84

6.86

6.88

6.90

(a) (b)

(c) (d)

L50: Experimental data

L50: Fitting curving

l n X

ln[-ln(1-F(X))]

Equation y = a + b

Adj. R-Squ 0.90297

Value Standard Er

B Intercep 6.871 0.00551

B Slope 0.041 0.00553

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

6.96

6.97

6.98

6.99

7.00

7.01

7.02

7.03

7.04

Equation y = a + b Adj. R-Squa 0.88385

Value Standard Err

D Intercept 7.0132 0.00409

D Slope 0.0265 0.00426



l n X

ln[-ln(1-F(X))]

-2.5 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

7.01

7.02

7.03

7.04

7.05

7.06

7.07

7.08

7.09

7.10

7.11

Equat ion y = a + b

Adj. R-Squa 0.86457

Value Standard Err

H Intercept 7.0806 0.00437

H Slope 0.0287 0.00426

l n X



ln[-ln(1-F(X))]

800 900 1000 1100 1200 1300 1400

0.0

0.2

0.4

0.6

0.8

1.0

L30:Fitting curving

L40:Fitting curving

L50:Fitting curving




F a i l u r e p r o b a b i l i t y

Static strength/MPa

Figure 4. Weibull distribution probability paper of static strength (a)–(c) and failure probability of carbon fiber bundles with different

characteristic lengths (d).

Song et al. 3161




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fatigue test data was directly given in equations (4)

to (6).

From the experimental results in Figure 5(a) (open

triangle ‘‘4’’), it is concluded that the normalized stress

basically underwent a linear trend between 0.85 and

0.94, while there was an obvious decay trend in the

range of 0.73 to 0.85. When the normalized stress was

less than 0.73, all of the fatigue life of fiber bundles was

more than 1e6, which was defined as the infinite life.

Therefore, it can be used a linear function to describe

the variation trend between 0.85 and 0.94 and an expo-

nential model to describe the change regularity of

0.73–0.85, respectively. The determined relationshipsbetween fatigue life and normalized stress were

expressed in the following equations

1. 0:85 max U

0:94 max

U ¼ 0:0506l g N f þ 1:059 ð4Þ

2. 0:75 max U

0:85

max

U ¼1

0:310

lg N f 4:014

lg N f 1:469 0:182

ð5

Þ

3. max U

0:75

lg N f 4 6 ð6Þ

From Figure 5(a), it is clearly found that good coin-

cidence between experiment and theory was obtained

for the fatigue issue of carbon fiber bundles. Moreover,

failure photograph under the fatigue load was shown in

Figure 5(b) and it can be seen that the failure model

mainly manifested de-bunching failure, which was simi-lar with the mode as that in static strength.

Nevertheless, the difference was that the failure regions

were concentrated in the middle of specimens compared

to that in the static tests.

Statistical distribution of fatigue life. Furthermore, in order

to investigate the distribution law for the fatigue of

fiber bundles, we assume that the statistical distribution

of the fatigue life also follows a two-parameter Weibull

distribution with reference to composite laminates3,6,18

F ðnÞ ¼ P N f n ¼ 1 exp n

l ! ð7Þ

where F (n) denotes the fatigue life distribution of

carbon fiber bundles, N is the fatigue life, and l are

the scale and shape parameter, respectively both of

which can be calculated by the same method as the

static strength.

The calculated results are presented in Table 4 and

corresponding distributional curves are plotted in

Figure 6. As for these results, the statistical distribution

Table 3. Tension–tension fatigue test results of carbon fiber

bundles.

Stress

level

Specimen

number

Fatigue

life, Nf lgNf Average lgNf

94% U 1 185 2.267 2.372

2 243 2.386

3 291 2.464

92% U 4 629 2.799 2.721

5 500 2.699

6 450 2.653

87% U 7 6984 3.844 3.836

8 7949 3.900

9 5923 3.773

10 6550 3.816

85% U 11 10,881 4.037 4.064

12 12,363 4.092

80% U 13 14,544 4.163 4.248

14 20,883 4.320

15 15,869 4.201

16 19,486 4.290

75% 17 235,079 5.371 5.397

18 239,358 5.379

19 274,297 5.438

73% 20 >1e6 >6 >670% 21 >1e6 >6 >6

60% 22 >1e6 >6 >6

Table 2. Hypothesis test results for static strength distribution

function.

Characteristic lengths L ¼ 30mm L ¼ 40mm L ¼ 50mmNumber of specimens 8 6 7

Kolmogorov D 0.2388 0.2974 0.2438

sqrt(n)*D 0.6755 0.7285 0.6450Reject domain at

significant level 0.05

[1.36,1)

D ¼ supjF n( x ) F ( x )j, where F ( x ) and F n( x ) are the assumed distributionalfunction and distributional function of samples, respectively; sqrt(n)*D

represents test statistics.





7/12

of fatigue life was fitted by the two-parameter Weibull

distribution fairly well, as shown in Figure 6(c).According to the statistical results mentioned above,

the excellent correlation implies that adoption of

Weibull distributional function in carbon fiber bundle

fatigue analysis is feasible. In addition, there was a

more discreteness in low stress level, which was also

reasonable because the lower stress level causes that

the fatigue damage takes place in more local positions

in the surface of fiber bundles and the propagation

paths are also more complex.

In addition, the similar approach as the static

strength was used to investigate whether the hypothesis

distribution regulation was reasonable for the fatigue

life of carbon fiber bundle at various stress levels. The

quantitative results are given in Table 5. It is clearly

seen that all the values of D were less than 1.36,

which was the lower limit of reject domain. Thus, the

hypothesis can be accepted.

Testing results for residual strength of carbon

fiber bundles

Testing results of residual strength. Fourteen specimens

were fatigued at two maximum stress levels 87% and

80%, respectively, up to 1/3 and 2/3 corresponding

average fatigue life, which were 7852 for 87% stress

level and 17696 for 80%. Their residual strength tests

were then performed and the results are listed in

Table 6. The failure photograph related to residual

strength tests is shown in Figure 7(a).

From Table 7, it is obviously found that the residual

strengths had a non-monotonic tendency with the

increase of cycle, which is similar to the notched

strength of woven composites.19,20 Figure 7(a) exhibits

several serious failure models with de-bunching and

fracture damages in the middle of fiber bundles. Thepossible reason for the phenomenon could be that the

monofilaments in the fiber bundle are gradually dis-

persed under the fatigue load, which may lead to

more uniform bearing in each of monofilaments.

Therefore, a higher load and further serious damage

are shown in the following static tensile test.

Meanwhile, a briefness ‘‘bang’’ sound could be listened

when the fracture occurred.

Furthermore, in order to investigate the difference

between static and residual strength, Figure 7(b) pre-

sents the load–displacement curves for three represen-

tative specimens used in static strength and residual

strength tests, respectively. It is found that the value

of residual strength experienced an increase firstly,

and then showed a decreased trend against cycle

number. The reasons of this abnormal phenomenon

may be that on the one hand, the stress state for each

of monofilaments is more uniform as mentioned above,

which could lead to the more bearing capacity for single

fiber bundle. On the other hand, the extent of accumu-

lative damage in monofilaments may increase subjected

cyclic loading, which could cause the decrease of the

mechanical performance to some degree. Therefore,

2 3 4 5 6

0.6

0.7

0.8

0.9

1.0

m a x

u l t

lgNf

(a) (b)

Figure 5. (a) The curve of fiber bundles normalized stress against lgNf ; (b) fatigue test specimen of carbon fiber bundle.

Table 4. Parameters in Weibull distributional function of fatiguelife.

Weibull distribution

parameters

87%

stress level

80%

stress level

G 7263.01 19,100.89

l 6.96 5.11

Song et al. 3163




8/12

the variation in residual strength versus cyclic number

may be the result of interaction of both factors

mentioned.

In order to describe the phenomenon quantitatively,

a new model was proposed. It is assumed that

dRðnÞdn

¼ gðS , rÞ f ðnÞ ð8Þ

-1.5 -1.0 -0.5 0.0 0.5

8.65

8.70

8.75

8.80

8.85

8.90

8.95

9.00

(a) (b)

(c)

Stress leve : 87% Ult Fitting curve

Equation y = a + b*

Adj. R-Squar 0.95498

Value Standard Erro

B Intercept 8.89055 0.01531

B Slope 0.14367 0.01787

ln[-ln(1-F(X))]

l n X

-1.5 -1.0 -0.5 0.0 0.5

9.55

9.60

9.65

9.70

9.75

9.80

9.85

9.90

9.95

10.00

Stress leve : 80% Ult Fitting curve

Equation y = a + b*

Adj. R-Squar 0.91796

Value Standard Err

H Intercept 9.8574 0.02849

H Slope 0.1955 0.03326

ln[-ln(1-F(X))]

l n X

4000 8000 12000 16000 20000 24000

0.0

0.2

0.4

0.6

0.8

1.0

Distribution function withmax

=80% U

Experimental data withmax

=80% U

Distribution function withmax

=87% U

Experimental data withmax

=87% U

Fatigue cycle Nf

F a i l u r e p r o b a b l i t y

Figure 6. Weibull distribution probability paper of fatigue life (a), (b) and failure probability (c).

Table 6. Residual strength tests results of carbon fiber bundles.

Stress

levels

Terminal

number

Peak

load (N)

Strength

(MPa)

Average

strength (MPa)

87% 1/3Nf * 169.41 1467.30 1391.34

150.69 1305.19

158.58 1373.58

163.86 1419.28

2/3Nf * 155.26 1344.81 1337.41

156.59 1356.34153.66 1330.93

152.12 1317.57

80% 1/3Nf ** 153.36 1328.33 1333.86

152.45 1320.43

156.19 1352.82

2/3Nf ** 110.35 955.81 1085.37

136.94 1186.06

128.64 1114.23

Nf * and Nf ** means the fatigue life of 87% and 80% stress levels,

respectively.

Table 5. Hypothesis test results for fatigue life distribution

function.

Stress levels 87% 80%

Number of specimens 4 4

Kolmogorov D 0.2465 0.2696

sqrt(n)*D 0.4931 0.5392

Reject domain at

significant level 0.05

[1.36,1)





9/12

Indicating that R(n) (the rate of residual strength at

the nth stress cycles) is equal to the product of the

effect, g(S , r), of the stress range, S , of the stress ratio,

r and the effect, f (n), of the number of cycle. The stress

range S is defined as the difference between the max-

imum and minimum cycle stress. So, the simplest

approximation to the functional form of g(S , r) and

f (n) is expressed as

gðS Þ ¼ k1S b, S ¼ ð1 rÞ max, k14 0 f

ðn

Þ ¼ Qn

þv, Q4 0, v4 0

ð9Þ

where k1, b, Q, and v are constants. In equation (9), it is

assumed for simplicity that the stress ratio is fixed so

that g(S, r) is a function of S alone and f (n) is the linear

function to ensure the variation of R(n) is in accordance

with the experimental data.

Substituting equation (9) into equation (8) and car-

rying out the integration from n1 to n2 cycles, we have

Rðn2Þ Rðn1Þ ¼ k1S bQ

2 n22 n21 þ k1vS bðn2 n1Þ

ð10Þ

For n2¼ n, n1¼ 0, equation (10) reduces to

RðnÞ Rð0Þ ¼ k1S bQ

2 n2 þ k1vS bn ð11Þ

Let a ¼ k1Q/2, c ¼ k1v, then

RðnÞ ¼ Rð0Þ S bðan2 cnÞ ð12Þ

Equation (12) is a theoretical residual model of

carbon fiber bundle involving three parameters a, b,

and c, which will be determined by amount of residual

strength tests in the following section.Based on the data of residual strength tests, the par-

ameters involved in equation (12) can be obtained. The

results for residual strength tests at high and low stress

amplitudes are shown in Table 7 and the related curves

are plotted in Figure 8. Additionally, the corresponding

residual strength under 87% and 80% stress levels for

1/3N f and 2/3N f cycles calculated by the equation (12)

are listed in Table 8. According to the results, the max-

imum error was only 8.14%, which took place under

80% stress levels for 2/3N f , therefore, it is reasonable to

accept the theoretical formula (12) to predict the resi-

dual strength of fiber bundle.

Furthermore, as equation (12) reflects the relation-

ship between the static strength and residual strength, it

is necessary to further validate whether the converted

static strengths R(0) calculated by equation (13) are

suitable to the statistical distribution of static strength.

Therefore, the residual strength data listed in Table 6

can be converted into the equivalent static strength

using the model given by equation (13). And the cor-

responding results are plotted in Figure 9 where the set

of converted equivalent static strength, R(0), using the

values of a, b, and c given in Table 7 was plotted as

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

0

30

60

90

120

150

180(b)(a)

Residual strength at 1/3Nf

Residual strength at 2/3Nf

Ultimate strength

L o a d / N

displacement/%

Figure 7. Residual strength failure photograph of carbon fiber bundle.

Table 7. Parameters in residual strength model.

Parameters

Low stress

amplitudes (80% U)

High stress

amplitudes (87% U)

a 1.55e-3 3.86e-3

b 0.532 1.009c 10.016 59.454

Song et al. 3165




10/12

solid squares and the solid curve was the same Weibull

distribution of the static strength presented in

Figure 4(d) with the 50 mm characteristic length.

Rð0Þ ¼ RðnÞ þ S bðan2 cnÞ ð13Þ

where R(0) represents the converted equivalent

strength.

It can be observed from Figure 9 that the correlation

between the Weibull distribution obtained directly by

static strength and converted data from residual

strength was comparatively well despite of stress

levels. Thus, it is reasonable to describe the residual

strength at various stress levels by equation (12).

Conclusions

The performances of static strength, fatigue life, and

residual strength for carbon fiber bundles of T300

fibers have been tested and the corresponding mechan-

ical models have also been built based on the testing

results. The following conclusions can be made:

1. The inner original defects derived from manufactur-

ing process existing in each of monofilaments caused

that the static strength of fiber bundle was correlated

with test section length. Therefore, the specimen

with a longer characteristic length has generally

lower strength under static tests.

2. Under T-T cyclic loading, the de-bunching phenom-

ena are more obviously and mainly concentrated in

the middle of fiber bundle, and less in the root part.

A segmented function was proposed to describe the

S-N curve of carbon fiber bundles and the related

parameters were then obtained based on the fatigue

0 1000 2000 3000 4000 5000 6000 7000

0

200

400

600

800

1000

1200

1400

(a) (b)

Residual strength curve obtained by Eq (12)

of 87% stress level

Average value of 87% stress level

Experimental data of 87% stress level

R e s i d u a l s t r e n g t h R ( n )

Cycle number n

0 4000 8000 12000 16000 20000

0

200

400

600

800

1000

1200

1400

Residual strength curve obtained by Eq (12)

of 80% stress level

Average value of 80% stress level

Experimental data of 80% stress level

Cycle number n

R e s i d u a l s t r e n

g t h R ( n )

Figure 8. Residual strength curves of carbon fiber bundle at high stress level (a) and low stress level (b).

840 860 880 900 920 940 960 980 1000 1020 1040

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Convered ultimate strength using equation (13)

Weibull distribution obtained form equation (6)

D i s t r i b u t i o n f u n c t i o n

Convered ultimate strength

Figure 9. Theoretical prediction of static strength and residual

strength data converted to static strength for carbon fiber bundle

under both stress levels.

Table 8. Comparison between the predicted values of residual

strength based on equation (12) and test data.

Stress

levels

Terminal

number

Test

results

Predicted

results Error

87% 1/3Nf * 1391.34 1386.26 0.37%

2/3Nf * 1337.41 1345.20 0.58%

80% 1/3Nf ** 1333.86 1245.92 6.59%

2/3Nf ** 1085.37 1173.69 8.14%

Nf * and Nf ** means the fatigue life of 87% and 80% stress levels, respect-

ively. All of the test data are from Table 6.





11/12

life data. In order to investigate the statistical distri-

bution of fatigue life, a two-parameter Weibull dis-

tribution function was well used to describe the

distribution.

3. In the end, the residual strength tests were conducted

and a theoretical model was derived based on the

assumption that the residual strength of fiber bun-dles is non-monotonic. Additionally, the failure

degree for residual strength tests was the most ser-

ious compared to others, which had not only obvi-

ous de-bunching damages, but also fracture

damages.

Funding

This work was supported by Jiangsu Innovation Program for

Graduate Education [grant number KYLX_0237].

Conflict of interest

None declared.

References

1. Degrieek J and Paepegem WV. Fatigue damage modeling

of fibre-reinforced composite materials: Review. Appl

Mech Rev 2001; 54(4): 279–300.

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Appendix

The Kolmogorov–Smirnov test.

This method21 proposed by Kolmogorov was

used to test quantitatively the difference degree

between the assumed distributional function F (x)

and the distributional function of samples F n(x)

(plotted in Figure 10). From the following

Figure 10. Schematic diagram of Dn.

Song et al. 3167




12/12

equation, it is defined as a distance Dn to describe

the difference

Dn ¼ sup F nðxÞ F ðxÞ ð14Þ

A reasonable test under a significant level is that if the hypothesis distribution is true, the value Dn men-

tioned above has a reduced trend, else the value

would be increased. Therefore, there exists a thresh-

old value k, i.e. when sqrt(n)*Dn< k, the hypothesis

could be accepted, else be rejected, where sqrt(n) is

used to reflect the effect of the number of test

specimens.

The limiting distribution of sqrt(n)*Dn was derived

by Kolmogorov as follows

limn!1

P ffiffiffi

np

Dn t ¼ F ðtÞ

F

ðt

Þ ¼1

2X

1

i ¼1 ð1

Þi 1e2i

2t2 ð15Þ

where F (t) was calculated and listed in the limiting dis-

tributional table of Dn.

Finally, the rejection domain can be obtained by

checking out the limiting distributional table under a

given significant levels , i.e. when ¼ 0.05, the rejectdomain U ¼ [1.36, 1).



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