CONCERNING DISTRIBUTION
THE
GLASS
FIBER
STRENGTH
FUNCTION
S. L . R o g i n s k i i , V. S. S t r e l y a e v , and L. L. Sachkovskaya
UDC 678.5.06:539.4.014
The statistical s t r e n g t h distribution functions of gIass monofilaments of various composition a r e cor.Lsidered. On the basis of the experimental data it is shown that the fiber strength d i s tributions can be d e s c r i b e d by a t h r e e - p a r a m e t e r function of the WeibulI type.
In a s e r i e s of studies [1-3] it has been shown that in g l a s s - r e i n f o r c e d plastics (GRP) the breaking s t r e s s is prol~)rtional to the m e a n strength of the r e i n f o r c i n g f i b e r s and depends on its variance. On the other hand, these p a r a m e t e r s must be d e t e r m i n e d to p e r m i t statistical control of the f i b e r manufacturing p r o c e s s . The fullest information can be obtained by constructing integral strength distribution c u r v e s and approximating them analytically by means of theoretical distribution flmctions. An analysis of the published data [4, 5] shows that depending on the g e o m e t r y (diameter, length), the composition of the glass, and the fabrication technology glass f i b e r s have various types of distributions, as may be seen f r o m Fig. 1. Thus, it may not always be possible to use a norms1 distribution to d e s c r i b e the test data. Despite the obvious importance of determining the theoretical distribution function that c o r r e sponds m o s t Closely to the t e s t data, no such statistical analyses have been published. Most authors confine t h e m s e l v e s to constructing h i s t o g r a m s [6] o r to the graphic interpretation of the experimental data [2]. We have analyzed the r e s u l t s of m a s s testing of the strength of 273-A and 112 glass f i b e r s and comm e r c i a l aluminoborosilicate glass f i b e r s 9-11 p in d i a m e t e r and 10 m m long. The volume of test data was l a r g e enough to yield quite reliable e s t i m a t e s . The samples were numbered 814,509, and518 for the c o m m e r c i a l fiber and for the 273-A and 112 fibers, r e s p e c t i v e l y . In o r d e r to determine the theoretical distribution functions we found the statistical c h a r a c t e r i s t i c s of the samples [7]: the mean n
XOut~ %t n -
(1)
where Cruti is the tensile strength of the i-th fiber, and n is the sample volume, and the standard deviation 2- (Z~ut~)2)' s = ~ t-~_~-(~Cud 1 n
(2)
F o r the c o m m e r c i a l aluminoborosilicate f i b e r s and glasses 273-A and 112 we obtained the foI1o~lng means and standard deviations, k g f / m m 2 : 2 2 5 and 55.4; 295 and 37; 358 and 32.2, r e s p e c t i v e l y . The e x p e r i m e n t a l distributions a r e p r e s e n t e d in the f o r m of h i s t o g r a m s (Fig. 2), f o r constructing which we d e t e r m i n e d the class interval f r o m the f o r m u l a [8] Erm a x - - ( ~ m l n
=
12
(3)
and calculated the class f r e q u e n c y 5 v ~ - -
nc
(4)
All-Union S c i e n t i f i c - R e s e a r c h Institute of G l a s s - R e i n f o r c e d Plastics and Glass F i b e r s , Moscow Regions. T r a n s l a t e d f r o m Mekhanika P o l i m e r o v , No. 1, pp. 131-136, J a n u a r y - F e b r u a r y , 1970. Original article submitted F e b r u a r y 10, 1969. © 1972 Consultants Bureau, a division of Plenum Publishing Corporation, 227 West 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.
113
J~
6"
mm2 z~5
~s
~
akgf/mm2
~
Fig. 1. H i s t o g r a m s of the distribution of the tensile s t r e n g t h of E - g l a s s f i b e r s , a) n = 178, ~ut = 340.2 k g f / m m 2 ; s = 31.5 k g f / m m 2 [4]; the thick line r e p r e s e n t s the theoretical n o r m a l distribution; b) n = 206, ~ut= 1 9 8 . 6 k g f / m m 2, f i b e r d i a m e t e r 125 # [5].
t
b
-~
-~o
o
2.0
Fig. 2. H i s t o g r a m s of the f i b e r tensile s t r e n g t h distribution and t h e o r e t i c a l f r e q u e n c y function c u r v e s for a n o r m a l law (1} and a Weibull d i s t r i b u t i o n (2): a) c o m m e r c i a l a l u m i n o b o r o s i l i c a t e g l a s s ; b) 273-A glass; e) 112 g l a s s . If in the f i r s t a p p r o x i m a t i o n it is a s s u m e d that the mean and v a r i a n c e of the s a m p l e a r e equal to the m e a n and v a r i a n c e of the m a i n population, then the frequency function f o r a n o r m a l law is calculated f r o m the f o r m u l a [7] t
(~) ~ - - - - e ]/2~ s
( a t l t - a u~ 2 2s2
(5)
The c o r r e s p o n d i n g distribution function o
/.
P(~)
! e
~ s _:
_
(qJt-aUt)~
~s,
dour.
(6)
The i n t e g r a l distribution c u r v e s f o r these types of f i b e r s a r e p r e s e n t e d in Fig. 3. The fine lines r e p r e s e n t the 95% confidence i n t e r v a l s f o r the t h e o r e t i c a l distribution curve.
114
99,99r 99~t ~ 9~,5
991
95 = 90
-
q
, / / ( - - ~ 3 ~
~ i. . . . . . .
:-
80 7O 5O O0 3O 20 tO
-
~/3
I:
/~j
-! !
I/.~!
f OA O,Z
50
t5o
zso
35o
zoo
3oo
4oo
6oo
F i g . 3. I n t e g r a l t e n s i l e s t r e n g t h d i s t r i b u t i o n c u r v e s of v a r i o u s t y p e s of g l a s s f i b e r s f o r a n o r m a l law. F o r n o t a t i o n s e e F i g , 2.
5o
too
¢5o
2oo
zso
3oo ~5o ~oo 45o
F i g . 4. W e i b u l l - t y p e i n t e g r a l f i b e r s t r e n g t h d i s t r i b u t i o n c u r v e s . F o r n o t a t i o n s e e F i g . 2,
115
T A B L E 1. E s t i m a t e of the C o n v e r g e n c e o f t h e F i b e r S t r e n g t h D i s t r i b u t i o n H i s t o g r a m to t h e N o r m a l and W e i b u l l D i s t r i b u t i o n n=518, ~ut=358 kgf/mrr~, s = 32.2 kgf]mmz
~i=Pn~Ut)
(ni-n~)~
nc
tZ i
Weibull fistribution
(ni--ni)~
208
~7
~8
248 268 288 308 328 348 368 388 408 428 448
7 31 5 63 133 13 7 1
5
4,9
11,9
5,95 22,8 1136 ~'9 i28,5 I06 60,9 22,8 7
1,05
8,2 47'9
2~'5 12,1 7,8 2
$
24
2,02
6,08
1.1 67.2 62,5 18~5~02 625 146 60,8 4
0,135 2,94
9,66 24,4 50,5 88.0
1.03
I7.4 0,157 5,9 2.4 2,67 0,572
130 69,2 9,01 3
tl i - - n +.
(n -n'i)'-
0,92
~,846
2,66 7,0~ 6,6 4,86 22,5 626,25 127,5 3,8 5,99 2
],5
14,4 35,3 4
(n ~-n'i)~ .......
0,189 0.732 0,178 0,124 7.t 9,157 0,007 0,2(~8 3.87 1,33
Y 13,84
35,274
The WeibulI fiber strength distribution is written in the form [2]
+ where G(crut) is the probability of the fibers breaking at a s t r e s s equal to or l e s s than ~ut; m, ~0, and ~u are parameters of the distribution curve, the exponent m characterizing the degree of damage and ~u being the minimum strength. In order to determine the values of the parameters of the Weibull distribution the experimental fiber strength data were plotted on probability paper in the coordinates 1/[1 - G(~ut)] along the ordinate axis and In Crut along the axis of a b s c i s s a s (Fig. 4) in a c c o r d a n c e with the equation 1 In In "1- G(crul~ =m In (~It -ou) - m in (r0.
(8)
It is c l e a r f r o m Fig. 4 that in the f i r s t approximation it m a y be a s s u m e d that the m i n i m u m s t r e n g t h ~u ~- 0 for all three types of fibers. The parameter m is equal to the slope of the straight line and hence is given by the equation 1
lain l _ G ( ~ u t l ) m
1
lain 1--G(cut~)
(9)
In Outl-ln Cut~
w h e r e ~utl and ~ut~ a r e a r b i t r a r i l y selected values of the f i b e r strength. F o r a m i n i m u m strength ~u = 0 the p a r a m e t e r ~0 is found as the value of the f i b e r s t r e n g t h at a p r o b e bility level ~63%. F o r the types of f i b e r s investigated the values of m and % a r e equal to 4.3 and 247 k g f / m m 2 for the c o m m e r c i a l f i b e r s ; 10.5 and 310 kgf/inm 2 f o r the 273-A f i b e r s ; and 13.4 and 370 k g f / m m 2 f o r the f i b e r s m a d e of g l a s s 112. The Weibull f r e q u e n c y functions w e r e calculated f r o m the e x p r e s s i o n
( ~ut t" g(~ut)=e
m ( ~ u t ) ~-'
(10)
~o ", ~o
I t s h o u l d b e k e p t i n m i n d t h a t the n o r m a l i z e d W e i b u l l f r e q u e n c y f u n c t i o n c a n b e o b t a i n e d b y m u l t i p l y i n g e x p r e s s i o n (10) b y t h e s a n d a r d d e v i a t i o n o f t h e c o r r e s p o n d i n g s a m p l e . Since the sample volumes were large, the normal and Weibull distributions were compared with the e x p e r i m e n t a l h i s t o g r a m by m e a n s o f t h e P e a r s o n c r i t e r i o n p{X2). T h e s e q u e n c e o f c a l c u l a t i o n s f o r c o m m e r c i a l f i b e r s i s i l l u s t r a t e d in the t a b l e . A s t h e b o u n d a r y b e t w e e n t h e r a n d o m and t h e s i g n i f i c a n t i t i s u s u a l to t a k e a 5% s i g n i f i c a n c e l e v e l [8]. F o r the n o r m a l d i s t r i b u t i o n ×2 = 35.274, w h i c h i s g r e a t e r t h a n 14.065, w h i c h
116
corresponds to the 5% significance level. Consequently, the discrepancy between the experimental and fitted normal frequencies wilt be significant. For the Weibull distribution •2 = 13.845 (which is less than 14.065) and the discrepancy between the experimental and fitted frequencies may be assumed random. This indicates that the strength distribution for the commercial fibers is governed by a Weibull distribution law. The authors thank A. !. Ivanova and S. Z. Vol'sk~ya for supplying the experimental data on the strength of 273-A and 112 glass fibers. LITERATURE I. 2.
3. 4.
5. 6. 7. 8.
CITED
J. A. Kies, Trans. J. Plast. Inst., 30, 85 (1962). H. T. Corten, Mechanics of Failure of Glass-Fiber Reinforced PIastics [Russian translation], Khimiya (1967). S. V. Serensen and V. S. Strelyaev, Proc. Georgian Rep. Conf. on Engineering Mechanics of Polymers [in Russian] (1968). A. J. Patrick, 20th Ann. Teclm. Conf. SPI, Sec. 8D (1966). E. H. Jsffe, 21st Ann. Techn. Conf. SPI, Sec. 8C (1966) M. S. A,~lanova, Modern Views on the Strength of Glass Fibers [in Russian], Moscow (1965). N. V. Smirnov and I. V. Dunin-Barkovskii, Course of Theory of Probability and Mathematical Statistics [in Russian], Moscow (1965). A. K. Mitropol'skil, Statistical Calculation Techniques [in Russian], Moscow (1961).
117
