A self-consistent Weibull analysis of carbon fibre strength distributions

Fibre Scienceand Technology 16 (1982) 81 94

A SELF-CONSISTENT WEIBULL ANALYSIS OF CARBON FIBRE STRENGTH DISTRIBUTIONS

CHARLES P. BEETZ, JR Physics Department, General Motors Research Laboratories,

Warren, Michigan 48090 (USA)

SUMMARY

A new method is developed for analysing carbonfibre tensile strength distributions. The method uses a Weibull probabilistic model of fibre strength and treats the parameters of the probability distribution as an intrinsic fibre property, independent of fibre length. Self-consistency is built into the method by forcing a single set of parameters to give an optimum fit to experimental tensile strength data taken at several different gauge lengths. The modality of the probability distribution used to model the fibre strength is determined from a factor analysis of the experimental histograms. The method is applied to fibres from mesophase pitch ( Thornel-P) whose tensile properties have been measured at four different gauge lengths. The variation of the mean fibre strength with gauge length at very short gauge lengths predictedJrom the self consistent Weibull parameters is less than that predicted by non-self consistent methods.

1.

INTRODUCTION

The statistical or brittle nature of carbon fibre strength is due to the presence of strength-limiting defects randomly distributed along the fibre. A direct consequence of this defect-controlled failure mode is the gauge length dependence of fibre strength. ~ 7 The strength properties of such fibres are usually discussed in terms of a weakest-link statistical model, s The importance of a statistical model of fibre strength is its ability to predict fibre tensile strength behaviour in a region of gauge lengths (0.1-0.2 ram) inaccessible to experimental measurement. These distances are of the order of the critical length in fibre-reinforced composites. 9- 13 In a previous paper 7 a statistical model of fibre strength was developed which considered fibres having a bimodal distribution of tensile strengths due to a bimodal 81 Fibre Science and Technology 0015-0568/82/0016-0081/$02-75 4~ Applied Science Publishers Ltd,

England, 1982 Printed in Great Britain

82

CHARLES P. BEETZ, JR

distribution of strength-limiting defects. This model provides a more accurate description of actual experimental strength histograms which frequently show multiple modes. It was also pointed out that the use of a conventional (unimodal) Weibull distribution to model a multimodal strength distribution could lead to substantial overestimates of the fibre strength at short gauge lengths. In this paper a method is presented for obtaining a self-consistent set of Weibull (SCW) parameters from a set of fibre tensile strength distributions measured at several different gauge lengths. The major advantage of this approach is one of improved signal-to-noise ratio, in that all the data points measured at several different gauge lengths are used to calculate the Weibull parameters which are assumed to be independent of gauge length. The calculation is self-consistent because only a single set of Weibull parameters is needed to describe the fibre strength distributions at several different gauge lengths. 2.

THE SELF-CONSISTENT W E I B U L L ( S C W ) A P P R O A C H

In this section a Weibull probability model is adopted and a method described for obtaining a self-consistent set of Weibull parameters from experimental tensile strength data taken at several different gauge lengths. Consider a generalised fibre which contains n different types of defects, each randomly distributed over the fibre volume. It is assumed that each defect gives rise to a mode in the tensile strength distribution. The probability that such a fibre of length L will survive to a stress a is given by a mixture of the individual survival probabilities v for each of the n defects or n

P= ~'3xiP i

(1)

i-1

where

Pi =

exp ( -

L(a/Ooi)")

(2)

is the Weibull probability and the mixing parameters, xi, satisfy

~

xi =

1

(3)

i=1

The aoi and mi are respectively the Weibull scale and shape parameters associated with the ith failure mode. The probability density associated with eqn (1) is

f(t~,L)

=

~ Lx~mi~"' 1 exp ,,, Croi i-1

- L(~

~"~ ',yoi /

(4)

WEIBULL ANALYSIS OF CARBON FIBRE STRENGTH DISTRIBUTIONS

83

The Weibull parameters aol and mi are assumed to be empirical materials constants, independent of the dimensions of the fibre. The mixing parameters on the other hand are not materials constants and will depend on the length of the fibre. The x,s give the relative concentrations of defects and will change with gauge length. The self-consistent method consists of finding a single set of the m~ and tro~ parameters which will fit all the tensile strength data taken at several different gauge lengths. This is achieved by allowing the x~ to vary with gauge length during the calculation. The most probable set ofm~, Cro~and xi is determined by maximising the following likelihood function :7,14,15

l=~-](JPi) ~R' j=l

(5)

i=1

where ~R~ is the experimental frequency of failure for t h e j t h gauge length within a strength interval Aa about tensile strength a~, m is the number of different gauge lengths measured, k is the number of strength intervals in each experimental histogram and

JPi =

f(~i, aol . . . . ~o., rnl . . . . m., Jx I . . . . ~x., Lj) d~ i

(6)

1

is the probability of being in the ith strength interval at the j t h gauge length, Lj. Using the approximation for P~ given in Reference 15, eqn (5) can be rewritten as

ln(l)~,~JRiln[f(~i,~rol j-I

....

O'o., m 1. . . .

m.,JXl .... :xn, Lj)l~,A~r]

(7)

i--I

where Atr= try-cr~_l and cry,, is the midpoint of the ith strength interval. Maximising eqn (7) or equivalently minimising - l n ( l ) with respect to the parameters ao~,mi, Jxi gives rise to that set of parameters which maximises the probability of occurrence of the JR~ tensile results in each interval of the strength histogram at each gauge length. The maximisation of eqn (7) however cannot be implemented until the number n of independently contributing failure modes has been uniquely specified. 3.

FACTOR ANALYSIS OF STRENGTH DISTRIBUTIONS

In this section a method is developed for determining the number of independently contributing failure modes in carbon fibre strength distributions using the mathematical technique of factor analysis. Factor analysis is a technique, based on the theorems of linear algebra, 16-~9 for determining the minimum number of linearly independent vectors needed to span a given linear vector space. Consider the following N × M linear vector space generated by

84

CHARLES P. BEETZ, JR

+ Pal = ) ,

Poinki

~ = 1,..., N

i = 1,...,M

a,i>_ m

(8)

k=l

In the case of fibre tensile strengths, P~i represents the fibre frequency of failure for the ith gauge length, at tensile strength, a, Pok is the probability of a failure due to the kth defect in an interval Aa about a, and nk~ is the number of kth defect types contained in the ith gauge length. This analysis implicitly assumes that the physical system under investigation obeys a superposition principle. The problem is to find the minimum number m of linearly independent vectors P,k which span the entire space. The values P,i form a rectangular N × M matrix P. The number of linearly independent vectors in P is found by determining its rank. The rank of P is the order of the largest nonzero determinant contained in P. A frequently used method for finding m is to form a square symmetric covariance matrix17-22 (covariance about the origin) C = ppr

(9)

where pT is the transpose of P, and look for the m nonzero eigenvalues. Diagonalisation of C is accomplished by applying a similarity transformation X~CX = A

(10)

where X is a matrix of orthonormal eigenvectors, A is a diagonal matrix of eigenvalues, m of which are nonzero. In actual practice none of the eigenvalues is identically equal to zero due to fluctuations in the experimental data. Therefore a criterion must be set for identifying the nonzero eigenvalues. One method for identifying nonzero eigenvalues makes use of the fact that the eigenvalues of C, the covariance matrix, are themselves a measure of the variance accounted for by their corresponding eigenvector. 2° A plot of the logarithm of the eigenvalues (arranged in descending order) versus the eigenvalue number will yield a sudden step at the cutoff eigenvector. Another method 17'z~ defines a matrix Y as YTY=A

(11)

Substituting eqn (9) into eqn (10) and equating to eqn (11) yields yx = xrp

(12)

P = XY v

(13)

Solving for P yields where use has been made of the orthonormality of X. The P matrix can be approximated by using only the first m eigenvectors or

WEIBULL ANALYSIS OF CARBON FIBRE STRENGTH DISTRIBUTIONS

PI~'=?, XikYi k

85

(14)

k=l

It now becomes possible to use a X2 significance test to determine the value of m beyond which there is no significant contributions to P. The value of f2,,~ is given by 21

(PI~'')- Pii) 2

(15)

where Spu is the standard deviation of the element P~.

4.

RESULTS

Tensile strength distributions for Thornel-P (VSB-32-0) fibres from the Union Carbide Corporation were measured at four different gauge lengths using techniques described previously. 7 The normalised strength distributions and the cumulative probability plots are shown in Fig. 1 and Fig. 2 respectively. The tensile tests were carried out at an Instron crosshead speed of 0.125 mm/min. The elements of the corresponding P matrix (of dimension 22 × 4 resulting from 22 strength intervals at four gauge lengths) are given in Table 1. It appears from the histograms (and the cumulative probability plots) that the strength distributions are multimodal and change with gauge length. Constructing the C matrix (eqn (9)) and performing the diagonalisation (eqn (10)) yields the four eigenvalues shown in Table 2, two of which are an order-of-magnitude smaller than the others. A plot of the eigenvalue as a function of the number of the eigenvalue is shown in Fig. 3, illustrating the steplike behaviour at the cutoff eigenvalue (number 2). From this plot it can be seen that most of the variance in the four data sets can be accounted for by the eigenvectors associated with the first two eigenvalues of Table 2. This is further substantiated by performing the partial reconstruction of eqn (14). The reconstructed histograms are shown in Table 3 along with the )(2 values. The reconstruction is essentially complete after using only the first two eigenvectors of Table 2. It can be concluded from this analysis that the 22 × 4 vector space corresponding to the tensile histograms (Figs 1-4) can be spanned by two eigenvectors, implying that the tensile strength distributions are composed of a mixture of two independent failure mechanisms. This is in agreement with a previous analysis by the author 7 of carbon fibre strength distributions where a bimodal distribution was intuitively adopted and is in accord with experimental studies 2~'25 of the various defects found in carbon fibres.

0.3 a

Number Percent

0.2

of

Fibres 0.1

~

0.0 0.5

~ T ~

1,0 1.5 2.0 2.5 Tensile Strength (GPs)

3.0

0.3( b

Number Percent 0

"

2

t

~

of

Fibres 0,1

0.0 0.5

1.0 1.5 2.0 2.5 Tensile Strength (GPa)

0.3 0.2 Number Percent of Fibres 0,1

3.0

t I

I

o.o

0.5

1.0 1.5 2.0 2.5 Tensile Strength (GPa)

3.0

0,0 r~ 0.5 1.0 1.5 2.0 2.5 Tensile Strength (GPa)

3.0

0.3 d

Number Fibres

0,2 0.1

Fig. I. Tensile strength h i s t o g r a m for Thornel-P(VSB-32-0), crosshead speed 0 . 1 2 5 m m / m m : (a) G a u g e length = 2 5 . 4 m m , total n u m b e r of fibres = 104~ (b) gauge length = 19-05 m m , total n u m b e r of fibres = 103; (c) gauge length = 12-7 m m , t o t a l n u m b e r of fibres = 108; (d) gauge length = 6"35 mm, total n u m b e r of fibres = 113.

0

Cumulative Probability

-2

-4

I 2

-6

109

I 3

I 4

I S

I e

I 7

I I e 91010

i

Tensile S t r e n g t h (GPa)

2

/.

0 J

Cumulative Probability

)

/

,/

-2

-4

I

-6

10 g

I

I

I

I

i

4

s

e

7 e 91019

I

Tensile S t r e n g t h (GPa)

2

0 Cumulative Probability

/

-2

/

/

-4

I

-6

10s 2

0 Cumulative Probability

/

I

I

I

I

I

I

2 4 s e 7 8 91010 Tensile S t r e n g t h (GPa)

/

-2

-4 -S

i

10 s

i

i

i

i

i

i

i

2 3 • s • 7 e 9101o Tensile S t r e n g t h (GPa)

Fig. 2. Cumulative probability of failure for the tensile histograms of Fig. 1 : (a) Gauge length 25-4 m m ; (b) gauge length = 19.05 m m ; (c) gauge length = 12-7 m m ; (d) gauge length = 6.35 mm.

88

CHARLES P. BEETZ, JR TABLE 1

ELEMENTS OF THE P MATRIX PI,I

P1,2

PI,3

PI,a

0.0 0-0 0-009 709 0.009 709 0.0 0.029 126 0.0 0.029 126 0.048 544 0.116505 0-135 922 0-145631 0.155340 0.145 631 0.116505 0.048 544 0"009 709 0-0 0-0 0-0 0"0 0.0

0.0 0-0 0-0 0.009 615 0.0 0.009 615 0.048 077 0.057 692 0.019 231 0.134615 0.230 769 0-182692 0.211 538 0.067 308 0-0 0-0 0-009 615 0.009 615 0.0 0"0 0"0 0.009 615

P1,1

0"0

0"009 259

P2,1 P3.1

0"0 0"008 850

0"0 0"0

P4A PL~ P6,1 P~,I Ps,l Plo,l P~ 1,~ Pl2,1 P13,1 P~4.~ P~LI P~6,1

0-0 0"008 850 0"026 549 0"017 699 0"035 398 0'026 549 0'088 496 0"035 398 0"159292 0"221 239 0.230088 0.132743 0"008 850

0"0 0"0 0.0 0-0 0"037 037 0"009 259 0'064 815 0" 129 630 0'129630 0"222 222 0.231 481 0"111 111 0.046 296

P17,1

0"0

0'0

Pl s,l

0"0

0"009 259

P19,1 P20,1

0"0 0'0

0"0 0'0

P21.1

0"0

0"0

P22,1

0"0

0"0

Gauge length" (ram)

6.35

12.7

P9,1

19.05

25.4

a Each column corresponds to one of the normalised histograms of Fig. l(a~t). TABLE 2 ElGENVALUES AND EIGENVECTORSOF THE C MATRIX

Eigenvalues A Xll A1= A2 = A3 = A4 =

0.532 87 0.048 64 0"006 34 0'005 59

X~ 1 X21 X31 X41

0-109 68 0-127 23 -0.788 18 0.59208

Eigenvectors X X12 -0"283 79 0"877 48 -0"161 68 -0"351 22

Xl3 -0-817 51 - 0.110 11 0.267 50 0.50990

X~4 - 0"488 99 - 0.462 30 -0.530 16 -0.515 84

The relative magnitudes of the two principle eigenvalues also give a qualitative estimate of the relative concentration of the two defects. In this case the two defects would be expected to be present in approximately a 10:1 ratio. The SCW calculation can now be applied to the data of Fig. 1, by taking n = 2 in eqn (7). The minimisation of the negative ofeqn (7) was achieved through the use of a sequential simplex program. 24 The resultant fits of eqn (4), using the parameters in Table 4, to the experimental histograms are given by the curves in Fig. 1. The calculated strength distributions fit the experimental histograms reasonably well. The strength distribution at 25.4 m m gauge length has two components, major and minor, whose 1/exp points ( = ooiL - t/,,,; the value of stress at which 63 % of all

WEIBULL ANALYSIS OF CARBON FIBRE STRENGTH DISTRIBUTIONS

89

1.0

0.1 0

_= Q :) C 0

._= UJ

0.01

0.001

1

1 2

I 3

4

Eigenvalue N u m b e r

Fig. 3.

Plot of eigenvalue versus eigenvalue (eigenvector) number demonstrating that most of the variance in the data can be accounted for by only two of the eigenvectors.

fibres have failed) are close together (see Table 4). The major component has a large value of m (m~ =9.56) corresponding to a defect which is more frequently distributed along the fibre than the minor component. The minor component (m 2 = 4.94) occurs so infrequently that its contribution goes to zero at 6.35 m m gauge length. The variation of the fibre strength with gauge length is shown (Figs 1-4) by the movement of the histogram m a x i m u m to higher strengths with decreasing length. A plot of the mean strength for each of the four histograms versus gauge length is shown in Fig. 4. The error bars represent + one standard deviation from the mean. The solid line in Fig. 4 is a least squares fit to the data and has a slope of - 0.04 + 0.03 showing a weak dependence on gauge length. The dashed line in Fig. 4 is calculated from the m~ and a ol SCW parameters and corresponds to the case when the mixing parameter is 1-0 or when the fibre contains only type I defects. Since the mixing parameter is an unknown at short gauge lengths, this is the most reasonable way to extrapolate the results of the SCW calculation. This is also justified in that the mixing parameter seems to be tending to 1-0 (Table 4) at the shorter gauge lengths. The predicted mean strengths at very short gauge lengths, based on the m~ and a01 SCW parameters are given in Table 5. These predicted mean strengths are lower

90

CHARLES P. BEETZ, JR TABLE 3 RECONSTRUCTED P MATRICES

(a) Reconstruction with eigenvector corresponding to A 1 0-002 53 0-0 0.004 67 0-004 74 0.002 35 0.01643 0.016 83 0.041 04 0-026 02 0.103 01 0.13549 0-15864 0.21004 0-17623 0.093 49 0-026 59 0.004 74 0.004 95 0.0 0.0 0.0 0.002 42 Gauge length (mm) ;(2

6.35 2.39

0.002 60 0.0 0.004 79 0.004 87 0-002 42 0.01689 0.017 30 0.042 18 0.026 74 0-105 87 0.13925 0-16305 0.21587 0.181 13 0.096 08 0.027 33 0.004 87 0.005 09 0"0 0"0 0"0 0.002 49 12.7 1.99

0-002 26 0"0 0.004 18 0.004 24 0-002 11 0-01472 0-01508 0.036 78 0.023 32 0.092 32 0.121 43 0.142 18 0.18824 0.15794 0.083 78 0.023 83 0.004 24 0.004 44 0"0 0-0 0"0 0.002 17

0.002 40 0-0 0.004 42 0"004 49 0.002 23 0.015 58 0"015 96 0.038 91 0.024 66 0-097 65 0.12844 0.15039 0"199 11 0.16706 0'088 62 0.025 20 0-004 49 0.004 69 0-0 0"0 0"0 0"002 29

19"05 0.669

25.4 7-86

0-002 24 0.0 0-004 13 0'004 33 0-00206 0.01467 0.015 42 0.037 00 0.023 32 0.092 85 0.122 94 0.14256 0-188 27 0" 156 59 0.08272 0.023 65 0"004 33 0-004 50 0.0 0.0 0-0 0"002 26 19-05 0"662

0.000 37 0.0 0'000 82 0-011 00 - 0-00145 0-011 20 0"040 71 0"054 87 0'024 86 0" 137 60 0.240 78 0-17905 0-20106 0"066 82 0-01004 0.011 83 0.011 00 0'009 10 0.0 0.0 0.0 0-008 72 25.4 0-116

(b) Reconstruction with eigenvectors corresponding to A~ and A 2

Gauge length (mm) ;(2

0.003 79 0.0 0.006 91 0"000 67 0-00465 0.019 16 0.001 39 0"031 08 0.025 90 0.078 09 0.065 41 0.14077 0.208 82 0"238 75 0.14250 0.03493 0.000 67 0-002 21 0-0 0.0 0-0 -0-001 58 6.35 0"286

0-003 26 0"0 0"005 97 0"002 74 0"003 62 0-018 32 0.009 20 0"036 96 0.026 68 0.092 80 0-102 49 0.15367 0.215 23 0-213 93 0.121 80 0"031 70 0-002 74 0"003 65 0.0 0.0 0.0 0.000 39 12-7 0"555

91

WEIBULL ANALYSIS OF CARBON FIBRE STRENGTH DISTRIBUTIONS TABLE 3--contd.

(c) Reconstruction with eigenvectors corresponding to A~, A 2 and A 3 0.004 32 0.0 0.005 01 0-001 35 0.005 74 0-01442 0.008 37 0.034 33 0-016 65 0.070 20 0.058 25 0.141 11 0.221 94 0.242 11 0-12928 0.023 69 -0-001 35 0.003 69 0.0 0.0 0-0 - 0-000 62 -

Gauge length (ram) Z2

6.35 0.229

0.003 50 0.0 0.005 10 0.001 80 0.004 12 0,016 14 0.012 41 0,038 45 0.022 42 0.089 16 0.099 19 0.15383 0.221 27 0.215 47 0.11571 0.026 53 0.001 80 0-004 33 0.0 0-0 0-0 0.000 83 12.7 0.513

0.000 92 0,0 0,008 88 0.009 41 - 0,000 66 0-02652 - 0.002 00 0.028 89 0.046 41 0-112 57 0.140 83 0.141 72 0.15549 0.148 21 0.11576 0-051 73 0.00941 0.000 79 0.0 0-0 0-0 - 0.000 13 19.05 0.012

0-000 80 0-0 - 0.000 71 0-009 36 - 0.000 57 0-00736 0.046 34 0.057 49 0-017 39 0.131 22 0.235 00 0.17932 0-211 66 0.069 53 -0.00064 0.002 75 0.009 36 0.010 30 0.0 0-0 0-0 0.009 49 25.4 0.018

(d) Reconstruction with eigenvectors corresponding to A1, A2, A3 and A4 0"000 00 0-0 0-008 84 0.000 00 0-008 84 0"026 54 0'017 69 0.035 39 0-026 54 0.08849 0.035 39 0-15929 0.221 23 0'23008 0.13274 0.008 84 0'000 00 - 0'000 00 0.0 0.0 0.0 - 0.000 00 Gauge length (mm) X2

6.35 0.0

0.009 25 0.0 0"000 00 0-000 00 0.000 00 0.000 00 0.000 00 0.037 03 0-009 25 0.06481 0.129 62 0.12962 0.22222 0.231 48 0.111 11 0-046 29 0-000 00 0"009 25 0.0 0.0 0.0 0.0 12-7 0.0

0.000 00 0-0 0"009 70 0.009 70 0.000 00 0"029 12 0.000 00 0.029 12 0-048 54 0.11650 0.135 92 0.14563 0.15533 0.145 63 0.11650 0.048 54 0-009 70 0.000 00 0.0 0.0 0.0 0.000 00 19-05 0.0

0-0 0.0 - 0.000 00 0.009 61 - 0.000 00 0"009 61 0.048 07 0.057 69 0.019 23 0.13461 0.230 76 0.18269 0.211 53 0.067 30 - 0.00000 0.000 00 0.096 1 0.096 1 0.0 0.0 0.0 0.096 1 25.4 0.0

92

CHARLES P. BEETZ, JR

10.0,

Tensile Strength (GPa)

1.O

0.10

........

L

.O

.........

10.0

100.

Gauge Length (ram) Fig. 4. Plot of mean tensile strength versus gauge length for the data of Fig, 1. Solid line, least squares fit to data; dashed line, estimated variation of strength with gauge length using SCW parameters.

TABLE 4 SELF-CONSISTENT WEIBULL PARAMETERS

L (ram)

6'35 12"7 19'05 25"4

x1

ml

ao ~

6o~L l,m,

X2

m2

602

6o2L l,vn:

l'0 0"92 0"46 0"81

%56

1'23

2'08 1"94 1'86 1"80

0 0'08 0'54 0'19

4'94

0"91

2"54 2'20 2"03 1"92

TABLE 5 PREDICTED MEAN STRENGTHS

Gauge length (ram)

~r,ul, (GPa)

0"1 0'2 0"5 1"0

3"24 3"01 2.74 2"55

than a previous estimate, 7 m a d e by fitting a b i m o d al mixture o f Weibull distributions to a tensile strength h is t o g r a m measured at a single gauge length (L = 25.4 mm). T h e estimate from this previous m e a s u r e m e n t was high since at that time the variation o f the mixing p a r a m e t e r s with gauge length was not taken into c o n s i d e r a t i o n and the e x t r a p o l a t i o n to short gauge lengths was m a d e from the m o d e having the smallest m p a r a m e t e r . H o w e v e r as has been shown in this paper the c o n t r i b u t i o n f r o m this m o d e becomes negligible at shorter gauge lengths.

WEIBULL ANALYSIS OF CARBON FIBRE STRENGTH DISTRIBUTIONS

93

The predicted strengths in Table 5 are also appreciably smaller than earlier estimates for mesophase pitch based fibres. 4 These earlier predictions were arrived at using a conventional unimodal Weibull distribution (neglecting the bimodal character of the data), which led to overestimates of the fibre strength at short gauge lengths. 7 A more recent study of defects in 'clean' mesophase pitch based on fibres 25 which exhibited primarily unimodal strength distributions, found m values closer to those found by the SCW method. Hitchon and Phillips 6 made similar observations on PAN based fibres. They found that the conventional Weibull distribution led to overestimates of the fibre strength at short gauge length and ascribed the problem to neglect of multiple modes of failure.

5.

CONCLUSIONS

A method of finding a self-consistent set of Weibuli parameters from tensile strength data at several different gauge lengths has been presented. This method tacitly assumes that the Weibull parameters are an intrinsic materials property and hence that they do not vary with fibre length. A major advantage of the SCW method is that it makes use of data taken at many different gauge lengths and for this reason it forms a better basis for extrapolation to shorter gauge lengths. A factor analysis method was developed for determining the number of independently contributing modes in a set of experimental tensile strength histograms. This information is used to help to determine the modality of the probability distribution used to model the fibre strength. A factor analysis of tensile histograms taken at four different gauge lengths for Thornel-P (VSB-32-0) fibres yielded two independent failure modes and therefore the fibre strength was modelled by a bimodal mixture of Weibull distributions. The results of the SCW calculation are in good agreement with the observed data, but give rise to a weaker dependence of strength on length than previous measurements on mesophase pitch based fibres. These previous measurements used a simple unimodal Weibull probability distribution which has been shown to overestimate the strength at short gauge lengths. However, if another defect population, say the undetected flaws in Reference 25, becomes important for gauge lengths less than those considered so far, the SCW calculations will at best be a lower limit for the extrapolated strength; so a problem remains until accurate tensile strength measurements which incorporate scanning electron microscopy of the fracture cross-sections can be made at very short gauge lengths (L ~_ 0.1 mm).

ACKNOWLEDGEMENTS

The author thanks G. W. Budd for technical assistance in preparation and testing of the carbon fibres, G. W. Smith and J. F. Herbst for helpful discussions and J. G. Gay

94

CHARLES P. BEETZ, JR

for the use of the simplex program; also J. C. Price for assistance with the computer programming and R. L. Spencer for the use of the Instron tensile test apparatus.

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