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Fibre composite fatigue-life determination
Composites Part A 28A (1997) 5- 15
Copyright ci 1996 Published by Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-835X/97/$17.00
Sl359-835X(96)00088-7
ELSEVIEK
Fibre composite
J. Lee, B. Harris”,
fatigue-life
D. P. Almond
determination
and F. Hammett
School of Materials Science, University of Bath, Bath BA2 7AY, UK (Received 10 November 1995; revised 13 May 1996)
A methodology is described for the statistical characterization of fibre composite fatigue-life data. The approach determines the most appropriate probability distribution function and then estimates the associated descriptive parameters. The methodology includes the determination of the most accurate minimum-life estimation method. Two sets of carbon fibre composite fatigue-life data are found to be best characterized by Weibull distributions, from which minimum lives may be estimated. The results of this study highlight a large difference between estimated median and minimum fatigue lives of composite materials. One set of the samples exhibited a median life of 37 022 cycles and a minimum life of 0 cycles, and the other a median life of 85 250 cycles and a minimum life of 16 000 cycles. It is concluded that minimumlife S-N plots are more appropriate for use in design engineering activities than S-N plots based, in the conventional
way, on median
lives. Copyright
‘c 1996 Published
by Elsevier Science Limited
(Keywords: fibre composite; fatigue-life determination; methodology: statistical characterization)
INTRODUCTION If composite structures are to be used effectively and efficiently in design applications then the statistical characteristics of the monotonic and fatigue strengths of the material must be understood. This can be achieved through the accurate estimation of the probability distributions and their associated descriptive parameters. There are several methods of carrying out this estimation process and it is the purpose of this paper to derive the most accurate procedure. It is normally assumed that fatigue lives for carbon fibre-reinforced plastic materials follow a Weibull distribution“2. Moreover, earlier work on fatigue life prediction3 for a 5245T800 carbon fibre epoxy laminate, and research within this department on three other aerospace composite systems, suggested that the Weibull modulus, the parameter defining the shape of the distribution, was very close to unity. A similar result was reported by Whitney’. In this case the exponential distribution provides the best fit, implying that failure is a random event and independent of life. This is an alarming prospect for designers, since it casts doubt on methods that have been employed in the past to estimate safe fatigue life. As a result of this ambiguity it was decided to carry out an exercise focusing exclusively on the distribution estimation method. Another factor which impacts on the accuracy of the distribution characterization, and in particular provides *To
whom correspondence
should
be addressed
a confident prediction of minimum life, is the approach employed for estimating the distribution parameters. There are numerous techniques available for this task and because of this the second objective of this study was to determine the most appropriate method of parameter estimation. This was achieved by an exercise similar to that described in ref. 4 and is described in detail in a later section.
MATERIALS
AND TESTING
PROCEDURES
E.uperimental materials
The fatigue test specimens were manufactured from the ICI Fiberite HTA/982A composite system. This material is reported to have similar characteristics to the HTA/913 and T300/9 13 laminates which have been used in many other research programmes and design applications. Here, 16-ply laminates of dimensions 400 mm x 300mm were laid up with unidirectional (UD) and [(*45, 02)2]s configurations. The panels were then cured by hot-pressing according to the approved processing schedule of the manufacturer. The test specimens had nominal dimensions of 200mm x 20mm x 2 mm. A water-cooled diamond saw was used to cut the test pieces to size from the cured laminates. Any visible damage zones and areas close to the laminate edges were discarded to reduce the risk of producing specimens of inconsistent quality. To minimize the risk of the presence of fatigue crack initiators caused by cutting, the specimen edges were wet polished with 220 grade abrasive paper. The test specimens were then lightly abraded at
Fibre composite
fatigue-life
determination:
J. lee
et al.
the tails to key the surface and therefore aid the bonding, with Ciba Geigy Redux 403 epoxy resin paste, of 1.6mm thick aluminium end tabs to the samples. The end tabs were positioned to support the test pieces against crushing by the jaws of the test machines, whilst maintaining a 100 mm free length, allowing a 50 mm gauge length to be accommodated.
SAMPLE
FATIGUE
1 GRAPHICAL
Testing procedures
SIZE DETERMINATION
- 50 TEST PIECES
EXPERIMENTS
ANALYSIS
-DISTRIBUTION
I
The series of fatigue trials were carried out, for a single peak stress level, on two 1300 series servohydraulic Instron machines rated at f 100 kN. For the UD test pieces an R ratio (minimum stress to peak stress) of 0.1 with an associated peak stress of 1.6 GPa was selected (73% of tensile strength). The specimens manufactured with a [(&45, OZ)& lay-up were fatigue tested at the same R ratio but the peak stress for these trials was reduced to 1.1 GPa (87% of tensile strength). Both peak stresses were selected to provide fatigue test results with short and meaningful life cycles. All the values for fatigue lives were taken from samples exhibiting separation failures. Moreover, samples which failed in the proximity of the end tabs were not included.
4 SELECTION OF PARAMETER ESTIMATION METHOD TESTS BASED ON EDF STATISTICS f
1ESTIMATE
MINIMUM
LIFE 1
Figure 1 Flow chart
METHODOLOGY The methodology is summarized described in Figure 1.
in the flow chart
Sample size selection
Time constraints dictate that S-N curves are typically constructed from five to seven fatigue samples per stress level, but to increase confidence in the distribution estimation process a larger sample size was used here. As the fatigue life distribution was assumed to be unknown for this study, non-parametric statistical techniques were employed to determine an appropriate sample size for a given confidence level. From the tables in ref. 5 a sample size of 22 was selected, giving a 90% certainty that, for these conditions, 90% of the fatigue specimen population will have a fatigue life greater than the smallest value in a random sample taken from the population. This sample size is similar to that suggested in ref. 6. For the unidirectional case the sample size was increased to 50 to accommodate the parameter estimation exercise described later. Establishment of the distribution
The distribution estimation consisted of two main steps: a graphical goodness-of-fit procedure, ‘followed by a test based on empirical distribution function (EDF) statistics. Both are means of testing the null hypothesis, Ho, which states that a given random variable-in this study it is fatigue life-follows a stated probability law, F(x), e.g. the Weibull model. The tests measure in some way the conformity, or lack of it, of the set of fatigue lives
6
to the hypothesized distribution. Graphical goodness-offit techniques offer an exploratory test method for reducing the number of potential distributions. The results of this first step must be reinforced with a more formal numerical study using EDF statistics if any level of confidence can be placed in the selection of the postulated distribution. Graphical goodness-of-jit. Graphical goodness-of-fit tests involve drawing probability plots to establish the adherence or deviation from the hypothesized distribution. A probability plot is achieved by plotting the linearized transformation of the cumulative distribution function (cdf) against a function of the probability. The plotting parameters for several potential models are described in Table 17. It should be noted that the cdfs for the normal and log-normal distributions are not included as the pdfs for these distributions cannot be integrated in their simple form. Instead the pdf is integrated numerically, usually for the standard normal distribution cdf where the mean is 0 and the standard deviation is 1. Since the Weibull distribution is widely used in engineering practice* it was the main contender for providing an appropriate model for the fatigue-life data and will be discussed in detail. The Weibull pdf is: f(x)
= [~~~)~-‘]{““p[-~~)IlJ x>xo;t9>xo
where: x0 is the minimum value of x, also called the
Fibre composite
fatigue-life
determination:
J. Lee et al.
Table 1 Plotting formulae for various distributions” __ Distribution _____
Cdf F(x)
Horizontal axis
Vertical axis
Normal Log-normal Exponential Weibull (two-parameter) Weibull (three-parameter)
Tables (see text) Tables (see text) 1 - exp(-(x/B)) 1 - exp( -(.z/8)m) 1 - exp[-((.v - .vs)/@)“]
-v* Ink) X, fn(.u,) In(.x, - so)
-, 2, - fn(l -A) In(-ln(l -pi)) tn(-In(l -P,))
-
’ Where: X, is the random variable, in this case fatigue life; pi is the probability plotting position given by the formula (i- 0.5)/n; n is the total number of samples; 2, is a transformation of p, to accommodate the conversion of arithmetic paper to normal probability paper. given by the formulae: z = sign( F(x)- 0.5)( 1.238r( I + 0.0262r)) t = { - ln[4(F(.x)( 1 - F(x))]}‘/’ sign(F(.y) - 0.5) = 1 if F(I) -- 0.5 > 0 or - 1 if F(x)- 0.5 < 0 0 is the characteristic value or scale parameter; M is the shape parameter; and x0 is the minimum life
location parameter; m is the shape parameter or Weibull slope; and 0 is the characteristic value, or scale parameter. As stated earlier the cdf is obtained by the integration of the pdf: F(s)
=
J
Taking natural logarithms, In
1
,Y Ii’ 1 - F(x) = 0; 1
_‘lf(x)dr
= /‘.f(r)dx 50
Inhi 1 - F(x)
= m(ln X) - (m In 0)
Thus, in the linear equation, Y=mX+C Y = In{- ln[l - F(x)]}
Let
___ m=
(on Table 1, F(x) is replaced by pI for plotting purposes)
( >Y x-x0
L9-- 5’)
X = 1n.u
Therefore:
C=mlnB m-1
Then plotting ln{ - ln[l - F(x)]} versus 1n.u should give a straight line if the hypothesis that the Weibull twoparameter distribution models the fatigue-life data is valid. Any deviation from a straight line suggests that a different model is required.
&dx=dy 0
or F(x) =
J
e-?‘dy
_ _ e-?
? ?‘o
= -exprs)m
[,
Therefore: F(x) = 1 - exp -
[ (E$m]
This is referred to as the three-parameter Weibull model in ref. 9. If the location parameter x0 is assumed to be zero, the cdf reduces to the two-parameter form: F(x) = 1 - exp[-(i)nl]
To linearize the function for probability plotting the twoparameter cdf is rewritten as: 1 1 - F(x)
=exp
Xm
00
Before the second stage of the Parameter estimation. goodness-of-fit process could be started the parameters which define the pdf had to be estimated. There are numerous methods for estimating the parameters, and as a result it was decided to carry out an exercise similar to that undertaken by Talreja4 where a group of 50 fatigue tests is treated as five sets of 10 samples. Parameter estimation methods were then applied to the five sub-groups and the results compared between the groups and with the estimations determined for the complete 50 specimen data set. Any of the parameter estimation techniques which produced widely different values for the distribution descriptors in relation to the other methods were discounted. The most accurate estimation method was selected out of the remaining techniques by considering the degree of variability, (range/&), where n is the number of sub-groups, for the parameters across the five sets. The chosen approach was then compared with the estimations of the distribution descriptors determined for the complete data set thereby reinforcing, or otherwise, the accuracy of the selection. It should be noted that the above comparisons were
7
Fibre composite fatigue-life
determination: J. Lee et al.
undertaken by visual examination of the data contained in Table 3. To obtain a more rigorous comparison statistical techniques should be used. To aid this analysis an expert system” was employed to calculate the parameters for the following estimation procedures:
where Xi < X, < . . . < X,. X has a distribution The EDF is then number of observations < x F,(x) =
F(x).
n
and F,(x) =O, x < X,
(9 Least-squares methods. This family of techniques
(ii)
(iii)
(iv)
(v)
aims to minimize the sum of squares of the differences between the predicted and the empirical values. The various versions of this method use different scales to measure the errors. In this study three forms were employed, namely probability least squares, return period least squares and standard least squares. The problem with least-squares methods is deciding on which errors should be minimized. Moreover, these techniques are sensitive to the plotting position formulae used. Percentiles is an approach which aims to give as many percentiles in the sample and the hypothetical distribution as the number of parameters to be estimated. The method of moments consists of equating as many of the theoretical distribution moments to the sample moments as the number of unknown parameters. By assuming the sample comes from a population with a distribution belonging to a parametric family, the maximum likelihood method aims to determine values for the parameters that maximize the probability of occurrence of the sample data. This method is favoured by statisticians owing to its statistical properties which allow confidence intervals in the parameters to be easily obtained’. However for minimal Weibull families, as is the case in this work, this estimation process can lead to problems. For some values of the shape parameter (1 < rn < 2) there are some regularity problems’. Moreover, Antle and Klimko” do not recommend maximum likelihood estimation of all three parameters for a sample size d 10. Both the above concerns are relevant to the estimation of the parameters of the distribution of fatigue lives for fibre-reinforced composite materials. The standard variable estimation method as suggested in ref. 4 was also used. To obtain an indication of the effect of sampling, the 50 UD fatigue samples were grouped into five sets of as-produced test pieces and five sub-groups of randomly selected specimens, and the parameter estimation exercise undertaken on both.
The findings of the graphical goodness-of-fit exercise were reinforced by a more accurate numerical techniquetests based on empirical distribution function (EDF) statistics. The EDF is a step function, F,(x), and is calculated from the experimental data to provide an estimate of the pdf. A definition of the EDF follows. Consider a random sample of size 12with Xi,. . . , X,, Test based on EDF statistics: Cramer-von
Mises.
Xi<=X
F,(~)=i/n,
n-1
F,(x) = 1, X, <= x
As x increases it takes a step up of height l/n with every sample observation. F,(x) displays the proportion of observations less than or equal to x, for any x. It is this measure of the difference, or more accurately the vertical difference, between F,(x) and F(x) that is the EDF statistic. There are two classes of EDF statistics: supremum and quadratic. Kolmogorov is an example of the first class but in this work we use a quadratic class member-Cramer-von Mises-because this approach is easier to compute. The test for the null hypothesis, Ho, that the random sample x1, x2, . . . , x, comes from the proposed distribution is established through the goodness-of-fit statistics W2, U2 and A2 (as defined below). The methodology7 for testing the null hypothesis, Ho, that the random sample comes from the Weibull distribution will now be discussed. The fatigue life values, Xi, are arranged in ascending order. Then z(i) = F(x(i)) is calculated by using the formula for either the two- or three-parameter Weibull distribution. By using the following formulae the EDF test statistics are determined:
w2= C i
1 z(i)
-
(2i (2n)
- l) 2+ l/( 12n) >
u2 = w2 - n(Z - 0.5)2 (2i - 1) log Z(i) + (2~2+ 1 - 2i) log{ 1 - ,Qz}
1
The above formulae calculate the general test statistics and must be modified depending on the knowledge of the distribution. This work considers case 0, where the distribution is completely specified. Consequently, the above values have to be adjusted in the following manner before a comparison can be made with the tabulated figures: W2 becomes ( W*)2 = ( W2 - 0.4/n + 0.6/n2)(1 + l/n); U2 becomes (Ut)2 = (U2 -0.1/n+ 0.1/n2)(1 +03/n); and A2 becomes (A*)2 and is calculated in the same manner. If the statistic exceeds the tabulated value7 at level o, the null hypothesis, Ho, is rejected at significance level Q.
APPLICATION OF METHODOLOGY CONSTANT-STRESS FATIGUE-LIFE Constant-stress UD fatigue
fatigue-life l$e results,
TO DATA
data Figure 2 shows a histogram
Fibre composite
of the 50 fatigue-life results obtained for the UD specimens. The range extends from 1392 to 190 860 cycles. The lowest observed value for fatigue life suggests that the location parameter, or minimum life, is likely to be 0. The sample set has an arithmetic mean of 44682 cycles.
16
fatigue-life
determination:
J. Lee et al.
[(f45, 02)2]s lay-upfatigue life data. The fatigue life data for the 22 [(f45, OZ)JS test specimens are summarized in Figure 3. The graph displays similar characteristics to that for the UD samples, the plot being skewed to the right. The fatigue-life values have an arithmetic mean of 130906 cycles and a range of 19 099-340 738
i
FATIGUE LIVES Figure 2
Histogram
of fatigue
o-4!xw9
lives for HTA-982A
50000-
UD
looooO14!w!J
15oooo199999
2OIMOO249999
FATIGUE LIVES Figure 3
Histogram
of fatigue
lives for HTA-982A
[(f45,
O?)&
2!wtOO299999
3wOl~0349999
3SOOOO399999
determination: J. lee et al.
Fibre composite fatigue-life
are shown in Figures 4-9. As can be seen, the graphs for the normal and log-normal distributions clearly do not fall on a straight line. Since the remaining linearized plots are all approximately linear, the most likely distributions of the fatigue lives from the UD samples are exponential, two-parameter Weibull or three-parameter Weibull. Several values for the location parameter in the three-parameter Weibull model were applied. The values quoted, x0 = 10 and 1000 cycles, are two examples but none of the x0 selections provided as good a fit as the
cycles. This higher minimum observed value for the fatigue life suggests that a location parameter may exist for these conditions. Graphical goodness-of-fit UD fatigue-l&e data graphical goodness-of-j2 tests. All five ‘contender’ distributions-normal, log-normal, exponential, two-parameter Weibull and three-parameter Weibull-were plotted for the UD fatigue-life data and
0
100000
50000
200000
150000
X (CYCLES TO FAILURE) Figure 4
HTA-982A UD graphical analysis: normal distribution hypothesis
5
6
12 Li
Figure 5
X
(CiCLE;
HTA-982A UD graphical analysis: log-normal distribution hypothesis
0
5oooo
loo000
150000
X (CYCLES TO FAILURE) Figure 6
10
13
TO F:ILU&
HTA-982A UD graphical analysis: exponential distribution hypothesis
2ooooo
Fibre composite
-5
HTA-982A
determination:
J. Lee et al.
I 5
Figure 7
fatigue-life
6
UD graphical analysis: two-parameter
12
13
12
13
Weibull distribution hypothesis
-5 5
6
L&-IO)
F&J&-IO)
(&CLE:
TO Figure 8
HTA-982A
UD graphical analysis: three-parameter Weibull distribution (10) hypothesis
J-y,,,,,;,
-5 5
: / 13
6
Ln(X-hO0)(&CLE; Figure 9
HTA-982A UD graphical analysis: three-parameter
TO F:~LURhOO;2
Weibull distribution (1000) hypothesis
two-parameter Weibull distribution. This is consistent with the observed data points which suggest either no minimum life or a very small minimum value. The above observations are reinforced by linear regression analysis which gives the highest correlation coefficient for the closest fit (Table 2).
Graphical goodness-of-fit tests for ((4x45, Oz)& lay-up fatigue-life results. The same candidate distributions as
used above were applied to the 22 results from the [(f45, O& lay-up specimen fatigue data. The plots for the Weibull distributions are described in Figures 10-12. Although not detailed here, the plots for the other
11
Fibre composite fatigue-life Table 2
determination: J. Lee et al.
Correlation coefficients for goodness-of-fit tests’ Correlation coefficient, r
Distribution
UD
Kf45,
Normal Log-normal Exponential Two-parameter Weibull Three-parameter Weibull (x1) Three-parameter Weibull (x2)
0.93 0.96 0.98 0.99 0.99 0.98
0.86 0.98 0.97 0.99 0.98 0.99
02h1s
‘For the UD case: x, = 10, .x2 = 1000; and for the [(f45, Oz)& samples: x, = 5000, x1 = 16 000
distributions gave deviations away from linearity similar to those in the corresponding graphs for the UD fatiguelife data. Unlike the UD case where the two-parameter model provided the best fit for the fatigue-life data, the most appropriate model for the [(f45, Oz)& lay-up fatigue-life data is provided by the three-parameter Weibull distribution. After several iterations a location parameter of 16 000 cycles was found to give the best fit (Figure 12). Again the correlation coefficients are shown in Table 2 and reinforce the hypothesis that the three-parameter model with a minimum life of 16 000 cycles is the most appropriate for this data set.
2
1
1
0
Figure 10
HTA-982A [(f45, 02)& graphical analysis: two-parameter Weibull distribution hypothesis
Figure 11
HTA-982A [(f45, Oz)& graphical analysis: three-parameter Weibull distribution (5000) hypothesis
12
Fibre composite
fatigue-life
determination:
J. Lee
et al.
/ -4 13
5 &&OO)(Ck’CLE~ Figure 12
Parameter
HTA-982A
[(f45.
01)2]S graphical
analysis:
three-parameter
estimation
By using the expert system” and the standard variable estimation procedure described above, the parameters for the Weibull distribution were estimated for the UD fatigue-life data and are tabulated in Table 3. To obtain an indication of the influence of sampling, the parameter exercise was carried out twice, firstly with the sample population divided into five as-produced sub-groups, and then with the five sub-groups containing a random selection of samples. By using the selection procedure as discussed above, the maximum likelihood approach was found to be the most appropriate, and this technique was applied to the complete data set for the UD fatigue life and the 22 results obtained for the [(f45, OZ)Z]slay-up specimens. The values for the Weibull parameters are shown in Table 4. The application of the selection procedure is described in a later section.
Tests based on EDF statistics By following the procedure for a statistical analysis with the goodness-of-fit tests based on the EDF statistics, as described above, the null hypothesis, Ho, that the fatigue lives for the two sets of samples belong to a Weibull distribution, was accepted at the 25% level of significance’.
DISCUSSION As can be seen, the graphs in Figures 2 and 3 are clearly skewed to the right, suggesting that the data belong to a distribution other than normal, necessitating further analysis of the fatigue-life data. This was achieved through the graphical goodness-of-fit exercise, which indicated that the fatigue data for the fibre-reinforced composite material in both ply configurations was most accurately modelled by a Weibull distribution, as these
TO F:UJRjFf-16~ Weibull
distribution
(16000)
hypothesis
linearized plots were the straightest. In the case of the UD sample fatigue-life data, the two-parameter form of the Weibull was the most appropriate. The fatigue-life data for the [(f45, 02)& lay-up specimens was most accurately modelled by the three-parameter version of the Weibull distribution. Statistical tests based on empirical distribution functions reinforce the hypothesis that the fatigue-life data belong to a Weibull distribution. For the UD test pieces an R ratio of 0.1 with an associated peak stress of 1.6GPa was selected (73% of tensile strength). The specimens manufactured with a [(f45, OZ)& lay-up arrangement were fatigue tested at the same R ratio but the peak stress for these trials was reduced to 1.1 GPa (87% of tensile strength). Both peak stresses were selected to provide fatigue-test results with short and meaningful life cycles. However, the higher percentage of tensile strength used as the peak stress in the fatigue trials of the [(f45, OZ)& lay-up test pieces results in a prediction of the three-parameter model and a minimum life, whereas the lower percentage used in the UD specimen fatigue tests produces a two-parameter Weibull model prediction. This reversal of peak stress level and number of minimum life cycles between the layup configurations could be explained by the [(f45, O& arrangement having a lower susceptibility to the many random factors affecting fatigue life. This work only considered one R ratio and peak stress for each lay-up configuration. Further research needs to be carried out to assess the impact of different R ratios and peak stresses on the Weibull parameters. The results of the parameter estimation method selection .procedure, as described earlier, are summarized in Table 3. The table is divided into three portions, the first of which (on the left-hand side) is sub-divided into two sections, one detailing the results from the as-produced fatigue-life data. the second containing data on the randomly grouped fatigue-life data. The adjacent segment averages the results from the as-produced and random group analyses. The final portion (on the extreme right) describes the results of the application of the parameter
13
Fibre composite fatigue-life
determination: J. Lee et al.
estimation techniques to the complete data set of 50 fatigue lives. The first stage of the procedure to select the most appropriate parameter estimation method, as described earlier, considered which of the parameter estimation techniques produced widely different values for the distribution descriptors in relation to the other approaches. Two options were rejected at this point, namely the standard variable and return period methods. By considering the actual values for all three sections in Table 3 it can be seen that both these approaches produce consistently low estimates when compared to the other techniques. The most accurate estimation method was selected out of the remaining techniques by considering the degree of variability, (range/&), where 12is the number of subgroups, for the parameters across the five sets. Again by considering the results in both the sub-groups and average for sub-groups sections it can be seen that the maximum likelihood method always provides estimates for m with the lowest variability and estimates for 8 with one of the lowest variabilities when compared to the other parameter estimation techniques. The selection of the maximum likelihood estimation technique is reinforced when the average result for the sub-groups is compared with the maximum likelihood estimation for the complete data set, and the values for m and 0 are found to be comparable. It should be noted that, for the data used in this work, five of the estimation techniques give similar results. From a practical engineering viewpoint there appears to be little to choose between the probability least squares, standard least squares, moments, percentiles and maximum likelihood method. In certain instances where a computer package, or similar, was not available to carry out the complicated iterations involved with some of the techniques (e.g. the maximum likelihood method), estimations from the simpler approaches may give results of sufficient level of accuracy. The estimated values for the Weibull parameters, as determined by using the maximum likelihood method, are described in Table 4. From the ‘2 x standard error on m’column it can be seen that it is unlikely that the shape parameter for either lay-up configuration is equal to unity, and as a result there is a higher confidence that the fatigue lives belong to the Weibull distribution and do not follow the exponential model. By using the ranked version of the Mann-Whitney U test the samples from the two sub-groups were found to come from a population with the same mean. This suggests that sampling does not affect the estimation of the distribution parameters. The above methodology is similar to that initially carried out by Weibull12 where an empirical analysis suggested that the Weibull distribution provided a suitable model for several groups of experimental data. To understand why the Weibull shape parameter is close to unity for these materials a different approach may be required. This is provided by Freudenthal and Gumbel699 where the derivation of the Weibull distribution has been achieved from an extremal phenomenon viewpoint, in
Fibre composite fatigue-life
Table 4
determination: J. Lee et al.
Weibull distribution parameters
Lay-up configuration
Shape parameter, m
2 x standard error on M
UD [(+45? 02111s
1.34 1.1
f0.04 f0.08
Characteristic value, 0 (cycles) _~___ ~
which the fatigue failures are considered to form a group of the weakest specimens and the theories of extreme values can be applied. In the first of these papers6 the authors developed a statistical scheme for evaluating fatigue failures. The concept of minimum life and the derivation of the three-parameter model was introduced in the second report’. Essentially the model is derived from the log-normal distribution which belongs to the exponential class. As a result the model is strongly influenced by the random nature of failure, characterized by the exponential distribution. However, Freudenthal and Gumbel introduce the concept of progressive damage where the damage resulting from any particular loading cycle is dependent on damage induced by previous loading cycles, leading to the development of the Weibull distribution. Consequently, it is suggested that the shape parameters for these fibre-reinforced composite materials are close to unity because the various stages of the manufacturing and testing route introduce randomness into the material characteristics. This is compounded by the random nature of the damage accumulation process. However, the fatigue failures result from the influence of progressive damage on top of the initial random properties, resulting in the requirement for a modified exponential model, the Weibull distribution. It is widely recognized that there is large scatter associated with fatigue-life values for composite materials. The practical extent of this scatter is highlighted in this study through a comparison of the difference in median and minimum life values. From the usual expression for the median,
1
~~
48 669 118960
k = 0 (ln2); [ the medians for both sets of materials were calculated as: UD samples (set of 49), 1 = 37022 cycles; and [(f45, Oz)& samples, R = 85 250 cycles. The corresponding estimates of minimum life values are 0 and 16 000 cycles, respectively. This gives a difference between the median and minimum life values of over 10 000 cycles for the UD case and over 33 000 cycles for the [(f45, 02)& sample data set. The disparity in the two parameters for both lay-up configurations reinforces the necessity for accurate parameter estimation and the need for careful consideration of the design engineering requirements.
2 x standard error on 0
Median value, .r
Location parameter, .vs (cycles)
f1527 &IO300
37022 85 250
0 I6000
composite material the results indicated that the most appropriate model was provided by a very broad Weibull distribution, with a shape parameter close to unity. The most accurate parameter estimation method was found to be the maximum likelihood technique. However, from a practical engineering viewpoint, estimations from any one of five techniques (probability least squares, standard least squares, moments, percentiles and maximum likelihood method) give satisfactory results. The large difference in median and minimum life values estimated in this study reinforces the requirement for accurate distribution parameter estimation and questions whether median life S-N curves are at all appropriate for design engineering applications. ACKNOWLEDGEMENTS The authors are grateful to the Engineering and Physical Sciences Research Council, UK for financial support of this work. The authors would like to thank Professor Alfonso Fernandez Canteli and Dr Maria Jesus Lamela Rey of the University of Oviedo for their guidance and useful suggestions on the use of the expert system for the analysis of extreme value problems.
REFERENCES 1
2 3
Whitney, J.M. Fatigue characterisation of composite materials. In ‘Fatigue of Fibrous Composite Materials’, ASTM STP 723, American Society for Testing and Materials, Philadelphia. PA, 1981, pp. 1333151 Hahn, H.T. and Kim, R.Y. Fatigue Behaviour of composite laminate. J. Compos. Muter. 1976, 10. 156-180 Gathercole, N.. Reiter, H., Adam, T. and Harris. B. Life prediction for fatigue of T800/5245 carbon-fibre composites: I. Constant-amplitude loading. Inf. J. Fatigue 1994, 16(8), 523532
4
8 9 IO
CONCLUSIONS
11
When the methodology illustrated in the flow chart (Figure 1) was applied to fatigue-life data for a fibre
12
Talreja, R. ‘Fatigue of Fibrous Composite Materials’, ASTM STP 723, American Society for Testing and Materials, Philadelphia, PA, 1981 Owen, D.B. ‘Handbook of Statistical Tables’, Addison-Welsey Publishing Co., Inc., Massachusetts, 1962, p. 318 Freudenthal, A.M. and Gumbel, E.J. On the statistical interpretation of fatigue tests. Proc. Rqval Sot. A 1953, 216, 230 D’Agostino, R.B. and Stephens, M.A. (Ed) ‘Goodness-of-Fit Techniques’, Marcel Dekker. Inc., New York. 1986, pp. 1 ~-62, 97-100, 145-150 Castillo. E. ‘Extreme Value Theory in Engineering’, Academic Press, Inc., London, 1988 Freudenthal, A.M. and Gumbel, E.J. Minimum life in fatigue. J. Am. Statistical Assoc. 1954. 49, 5155597 Alavarez. A.. Castillo. E., Cobo. A. and Herrero, T. An expert system for the analysis of extreme value problems, technical report, University of Cantabria. Santander, Spain, 1993 Ante, C.E. and Khmko, L.A. Choice of model for reliability studies and related topics II, Report ARL-73-0121. AD 772775, US Army Research Lab., NASA-Lewis Research Group, Ohio, 1973 Weibull, W. A statistical distribution function of wide apphcabihty. J. Appl.Mechanics 1951. 18. 193 --297
15
Copyright ci 1996 Published by Elsevier Science Limited Printed in Great Britain. All rights reserved 1359-835X/97/$17.00
Sl359-835X(96)00088-7
ELSEVIEK
Fibre composite
J. Lee, B. Harris”,
fatigue-life
D. P. Almond
determination
and F. Hammett
School of Materials Science, University of Bath, Bath BA2 7AY, UK (Received 10 November 1995; revised 13 May 1996)
A methodology is described for the statistical characterization of fibre composite fatigue-life data. The approach determines the most appropriate probability distribution function and then estimates the associated descriptive parameters. The methodology includes the determination of the most accurate minimum-life estimation method. Two sets of carbon fibre composite fatigue-life data are found to be best characterized by Weibull distributions, from which minimum lives may be estimated. The results of this study highlight a large difference between estimated median and minimum fatigue lives of composite materials. One set of the samples exhibited a median life of 37 022 cycles and a minimum life of 0 cycles, and the other a median life of 85 250 cycles and a minimum life of 16 000 cycles. It is concluded that minimumlife S-N plots are more appropriate for use in design engineering activities than S-N plots based, in the conventional
way, on median
lives. Copyright
‘c 1996 Published
by Elsevier Science Limited
(Keywords: fibre composite; fatigue-life determination; methodology: statistical characterization)
INTRODUCTION If composite structures are to be used effectively and efficiently in design applications then the statistical characteristics of the monotonic and fatigue strengths of the material must be understood. This can be achieved through the accurate estimation of the probability distributions and their associated descriptive parameters. There are several methods of carrying out this estimation process and it is the purpose of this paper to derive the most accurate procedure. It is normally assumed that fatigue lives for carbon fibre-reinforced plastic materials follow a Weibull distribution“2. Moreover, earlier work on fatigue life prediction3 for a 5245T800 carbon fibre epoxy laminate, and research within this department on three other aerospace composite systems, suggested that the Weibull modulus, the parameter defining the shape of the distribution, was very close to unity. A similar result was reported by Whitney’. In this case the exponential distribution provides the best fit, implying that failure is a random event and independent of life. This is an alarming prospect for designers, since it casts doubt on methods that have been employed in the past to estimate safe fatigue life. As a result of this ambiguity it was decided to carry out an exercise focusing exclusively on the distribution estimation method. Another factor which impacts on the accuracy of the distribution characterization, and in particular provides *To
whom correspondence
should
be addressed
a confident prediction of minimum life, is the approach employed for estimating the distribution parameters. There are numerous techniques available for this task and because of this the second objective of this study was to determine the most appropriate method of parameter estimation. This was achieved by an exercise similar to that described in ref. 4 and is described in detail in a later section.
MATERIALS
AND TESTING
PROCEDURES
E.uperimental materials
The fatigue test specimens were manufactured from the ICI Fiberite HTA/982A composite system. This material is reported to have similar characteristics to the HTA/913 and T300/9 13 laminates which have been used in many other research programmes and design applications. Here, 16-ply laminates of dimensions 400 mm x 300mm were laid up with unidirectional (UD) and [(*45, 02)2]s configurations. The panels were then cured by hot-pressing according to the approved processing schedule of the manufacturer. The test specimens had nominal dimensions of 200mm x 20mm x 2 mm. A water-cooled diamond saw was used to cut the test pieces to size from the cured laminates. Any visible damage zones and areas close to the laminate edges were discarded to reduce the risk of producing specimens of inconsistent quality. To minimize the risk of the presence of fatigue crack initiators caused by cutting, the specimen edges were wet polished with 220 grade abrasive paper. The test specimens were then lightly abraded at
Fibre composite
fatigue-life
determination:
J. lee
et al.
the tails to key the surface and therefore aid the bonding, with Ciba Geigy Redux 403 epoxy resin paste, of 1.6mm thick aluminium end tabs to the samples. The end tabs were positioned to support the test pieces against crushing by the jaws of the test machines, whilst maintaining a 100 mm free length, allowing a 50 mm gauge length to be accommodated.
SAMPLE
FATIGUE
1 GRAPHICAL
Testing procedures
SIZE DETERMINATION
- 50 TEST PIECES
EXPERIMENTS
ANALYSIS
-DISTRIBUTION
I
The series of fatigue trials were carried out, for a single peak stress level, on two 1300 series servohydraulic Instron machines rated at f 100 kN. For the UD test pieces an R ratio (minimum stress to peak stress) of 0.1 with an associated peak stress of 1.6 GPa was selected (73% of tensile strength). The specimens manufactured with a [(&45, OZ)& lay-up were fatigue tested at the same R ratio but the peak stress for these trials was reduced to 1.1 GPa (87% of tensile strength). Both peak stresses were selected to provide fatigue test results with short and meaningful life cycles. All the values for fatigue lives were taken from samples exhibiting separation failures. Moreover, samples which failed in the proximity of the end tabs were not included.
4 SELECTION OF PARAMETER ESTIMATION METHOD TESTS BASED ON EDF STATISTICS f
1ESTIMATE
MINIMUM
LIFE 1
Figure 1 Flow chart
METHODOLOGY The methodology is summarized described in Figure 1.
in the flow chart
Sample size selection
Time constraints dictate that S-N curves are typically constructed from five to seven fatigue samples per stress level, but to increase confidence in the distribution estimation process a larger sample size was used here. As the fatigue life distribution was assumed to be unknown for this study, non-parametric statistical techniques were employed to determine an appropriate sample size for a given confidence level. From the tables in ref. 5 a sample size of 22 was selected, giving a 90% certainty that, for these conditions, 90% of the fatigue specimen population will have a fatigue life greater than the smallest value in a random sample taken from the population. This sample size is similar to that suggested in ref. 6. For the unidirectional case the sample size was increased to 50 to accommodate the parameter estimation exercise described later. Establishment of the distribution
The distribution estimation consisted of two main steps: a graphical goodness-of-fit procedure, ‘followed by a test based on empirical distribution function (EDF) statistics. Both are means of testing the null hypothesis, Ho, which states that a given random variable-in this study it is fatigue life-follows a stated probability law, F(x), e.g. the Weibull model. The tests measure in some way the conformity, or lack of it, of the set of fatigue lives
6
to the hypothesized distribution. Graphical goodness-offit techniques offer an exploratory test method for reducing the number of potential distributions. The results of this first step must be reinforced with a more formal numerical study using EDF statistics if any level of confidence can be placed in the selection of the postulated distribution. Graphical goodness-of-jit. Graphical goodness-of-fit tests involve drawing probability plots to establish the adherence or deviation from the hypothesized distribution. A probability plot is achieved by plotting the linearized transformation of the cumulative distribution function (cdf) against a function of the probability. The plotting parameters for several potential models are described in Table 17. It should be noted that the cdfs for the normal and log-normal distributions are not included as the pdfs for these distributions cannot be integrated in their simple form. Instead the pdf is integrated numerically, usually for the standard normal distribution cdf where the mean is 0 and the standard deviation is 1. Since the Weibull distribution is widely used in engineering practice* it was the main contender for providing an appropriate model for the fatigue-life data and will be discussed in detail. The Weibull pdf is: f(x)
= [~~~)~-‘]{““p[-~~)IlJ x>xo;t9>xo
where: x0 is the minimum value of x, also called the
Fibre composite
fatigue-life
determination:
J. Lee et al.
Table 1 Plotting formulae for various distributions” __ Distribution _____
Cdf F(x)
Horizontal axis
Vertical axis
Normal Log-normal Exponential Weibull (two-parameter) Weibull (three-parameter)
Tables (see text) Tables (see text) 1 - exp(-(x/B)) 1 - exp( -(.z/8)m) 1 - exp[-((.v - .vs)/@)“]
-v* Ink) X, fn(.u,) In(.x, - so)
-, 2, - fn(l -A) In(-ln(l -pi)) tn(-In(l -P,))
-
’ Where: X, is the random variable, in this case fatigue life; pi is the probability plotting position given by the formula (i- 0.5)/n; n is the total number of samples; 2, is a transformation of p, to accommodate the conversion of arithmetic paper to normal probability paper. given by the formulae: z = sign( F(x)- 0.5)( 1.238r( I + 0.0262r)) t = { - ln[4(F(.x)( 1 - F(x))]}‘/’ sign(F(.y) - 0.5) = 1 if F(I) -- 0.5 > 0 or - 1 if F(x)- 0.5 < 0 0 is the characteristic value or scale parameter; M is the shape parameter; and x0 is the minimum life
location parameter; m is the shape parameter or Weibull slope; and 0 is the characteristic value, or scale parameter. As stated earlier the cdf is obtained by the integration of the pdf: F(s)
=
J
Taking natural logarithms, In
1
,Y Ii’ 1 - F(x) = 0; 1
_‘lf(x)dr
= /‘.f(r)dx 50
Inhi 1 - F(x)
= m(ln X) - (m In 0)
Thus, in the linear equation, Y=mX+C Y = In{- ln[l - F(x)]}
Let
___ m=
(on Table 1, F(x) is replaced by pI for plotting purposes)
( >Y x-x0
L9-- 5’)
X = 1n.u
Therefore:
C=mlnB m-1
Then plotting ln{ - ln[l - F(x)]} versus 1n.u should give a straight line if the hypothesis that the Weibull twoparameter distribution models the fatigue-life data is valid. Any deviation from a straight line suggests that a different model is required.
&dx=dy 0
or F(x) =
J
e-?‘dy
_ _ e-?
? ?‘o
= -exprs)m
[,
Therefore: F(x) = 1 - exp -
[ (E$m]
This is referred to as the three-parameter Weibull model in ref. 9. If the location parameter x0 is assumed to be zero, the cdf reduces to the two-parameter form: F(x) = 1 - exp[-(i)nl]
To linearize the function for probability plotting the twoparameter cdf is rewritten as: 1 1 - F(x)
=exp
Xm
00
Before the second stage of the Parameter estimation. goodness-of-fit process could be started the parameters which define the pdf had to be estimated. There are numerous methods for estimating the parameters, and as a result it was decided to carry out an exercise similar to that undertaken by Talreja4 where a group of 50 fatigue tests is treated as five sets of 10 samples. Parameter estimation methods were then applied to the five sub-groups and the results compared between the groups and with the estimations determined for the complete 50 specimen data set. Any of the parameter estimation techniques which produced widely different values for the distribution descriptors in relation to the other methods were discounted. The most accurate estimation method was selected out of the remaining techniques by considering the degree of variability, (range/&), where n is the number of sub-groups, for the parameters across the five sets. The chosen approach was then compared with the estimations of the distribution descriptors determined for the complete data set thereby reinforcing, or otherwise, the accuracy of the selection. It should be noted that the above comparisons were
7
Fibre composite fatigue-life
determination: J. Lee et al.
undertaken by visual examination of the data contained in Table 3. To obtain a more rigorous comparison statistical techniques should be used. To aid this analysis an expert system” was employed to calculate the parameters for the following estimation procedures:
where Xi < X, < . . . < X,. X has a distribution The EDF is then number of observations < x F,(x) =
F(x).
n
and F,(x) =O, x < X,
(9 Least-squares methods. This family of techniques
(ii)
(iii)
(iv)
(v)
aims to minimize the sum of squares of the differences between the predicted and the empirical values. The various versions of this method use different scales to measure the errors. In this study three forms were employed, namely probability least squares, return period least squares and standard least squares. The problem with least-squares methods is deciding on which errors should be minimized. Moreover, these techniques are sensitive to the plotting position formulae used. Percentiles is an approach which aims to give as many percentiles in the sample and the hypothetical distribution as the number of parameters to be estimated. The method of moments consists of equating as many of the theoretical distribution moments to the sample moments as the number of unknown parameters. By assuming the sample comes from a population with a distribution belonging to a parametric family, the maximum likelihood method aims to determine values for the parameters that maximize the probability of occurrence of the sample data. This method is favoured by statisticians owing to its statistical properties which allow confidence intervals in the parameters to be easily obtained’. However for minimal Weibull families, as is the case in this work, this estimation process can lead to problems. For some values of the shape parameter (1 < rn < 2) there are some regularity problems’. Moreover, Antle and Klimko” do not recommend maximum likelihood estimation of all three parameters for a sample size d 10. Both the above concerns are relevant to the estimation of the parameters of the distribution of fatigue lives for fibre-reinforced composite materials. The standard variable estimation method as suggested in ref. 4 was also used. To obtain an indication of the effect of sampling, the 50 UD fatigue samples were grouped into five sets of as-produced test pieces and five sub-groups of randomly selected specimens, and the parameter estimation exercise undertaken on both.
The findings of the graphical goodness-of-fit exercise were reinforced by a more accurate numerical techniquetests based on empirical distribution function (EDF) statistics. The EDF is a step function, F,(x), and is calculated from the experimental data to provide an estimate of the pdf. A definition of the EDF follows. Consider a random sample of size 12with Xi,. . . , X,, Test based on EDF statistics: Cramer-von
Mises.
Xi<=X
F,(~)=i/n,
n-1
F,(x) = 1, X, <= x
As x increases it takes a step up of height l/n with every sample observation. F,(x) displays the proportion of observations less than or equal to x, for any x. It is this measure of the difference, or more accurately the vertical difference, between F,(x) and F(x) that is the EDF statistic. There are two classes of EDF statistics: supremum and quadratic. Kolmogorov is an example of the first class but in this work we use a quadratic class member-Cramer-von Mises-because this approach is easier to compute. The test for the null hypothesis, Ho, that the random sample x1, x2, . . . , x, comes from the proposed distribution is established through the goodness-of-fit statistics W2, U2 and A2 (as defined below). The methodology7 for testing the null hypothesis, Ho, that the random sample comes from the Weibull distribution will now be discussed. The fatigue life values, Xi, are arranged in ascending order. Then z(i) = F(x(i)) is calculated by using the formula for either the two- or three-parameter Weibull distribution. By using the following formulae the EDF test statistics are determined:
w2= C i
1 z(i)
-
(2i (2n)
- l) 2+ l/( 12n) >
u2 = w2 - n(Z - 0.5)2 (2i - 1) log Z(i) + (2~2+ 1 - 2i) log{ 1 - ,Qz}
1
The above formulae calculate the general test statistics and must be modified depending on the knowledge of the distribution. This work considers case 0, where the distribution is completely specified. Consequently, the above values have to be adjusted in the following manner before a comparison can be made with the tabulated figures: W2 becomes ( W*)2 = ( W2 - 0.4/n + 0.6/n2)(1 + l/n); U2 becomes (Ut)2 = (U2 -0.1/n+ 0.1/n2)(1 +03/n); and A2 becomes (A*)2 and is calculated in the same manner. If the statistic exceeds the tabulated value7 at level o, the null hypothesis, Ho, is rejected at significance level Q.
APPLICATION OF METHODOLOGY CONSTANT-STRESS FATIGUE-LIFE Constant-stress UD fatigue
fatigue-life l$e results,
TO DATA
data Figure 2 shows a histogram
Fibre composite
of the 50 fatigue-life results obtained for the UD specimens. The range extends from 1392 to 190 860 cycles. The lowest observed value for fatigue life suggests that the location parameter, or minimum life, is likely to be 0. The sample set has an arithmetic mean of 44682 cycles.
16
fatigue-life
determination:
J. Lee et al.
[(f45, 02)2]s lay-upfatigue life data. The fatigue life data for the 22 [(f45, OZ)JS test specimens are summarized in Figure 3. The graph displays similar characteristics to that for the UD samples, the plot being skewed to the right. The fatigue-life values have an arithmetic mean of 130906 cycles and a range of 19 099-340 738
i
FATIGUE LIVES Figure 2
Histogram
of fatigue
o-4!xw9
lives for HTA-982A
50000-
UD
looooO14!w!J
15oooo199999
2OIMOO249999
FATIGUE LIVES Figure 3
Histogram
of fatigue
lives for HTA-982A
[(f45,
O?)&
2!wtOO299999
3wOl~0349999
3SOOOO399999
determination: J. lee et al.
Fibre composite fatigue-life
are shown in Figures 4-9. As can be seen, the graphs for the normal and log-normal distributions clearly do not fall on a straight line. Since the remaining linearized plots are all approximately linear, the most likely distributions of the fatigue lives from the UD samples are exponential, two-parameter Weibull or three-parameter Weibull. Several values for the location parameter in the three-parameter Weibull model were applied. The values quoted, x0 = 10 and 1000 cycles, are two examples but none of the x0 selections provided as good a fit as the
cycles. This higher minimum observed value for the fatigue life suggests that a location parameter may exist for these conditions. Graphical goodness-of-fit UD fatigue-l&e data graphical goodness-of-j2 tests. All five ‘contender’ distributions-normal, log-normal, exponential, two-parameter Weibull and three-parameter Weibull-were plotted for the UD fatigue-life data and
0
100000
50000
200000
150000
X (CYCLES TO FAILURE) Figure 4
HTA-982A UD graphical analysis: normal distribution hypothesis
5
6
12 Li
Figure 5
X
(CiCLE;
HTA-982A UD graphical analysis: log-normal distribution hypothesis
0
5oooo
loo000
150000
X (CYCLES TO FAILURE) Figure 6
10
13
TO F:ILU&
HTA-982A UD graphical analysis: exponential distribution hypothesis
2ooooo
Fibre composite
-5
HTA-982A
determination:
J. Lee et al.
I 5
Figure 7
fatigue-life
6
UD graphical analysis: two-parameter
12
13
12
13
Weibull distribution hypothesis
-5 5
6
L&-IO)
F&J&-IO)
(&CLE:
TO Figure 8
HTA-982A
UD graphical analysis: three-parameter Weibull distribution (10) hypothesis
J-y,,,,,;,
-5 5
: / 13
6
Ln(X-hO0)(&CLE; Figure 9
HTA-982A UD graphical analysis: three-parameter
TO F:~LURhOO;2
Weibull distribution (1000) hypothesis
two-parameter Weibull distribution. This is consistent with the observed data points which suggest either no minimum life or a very small minimum value. The above observations are reinforced by linear regression analysis which gives the highest correlation coefficient for the closest fit (Table 2).
Graphical goodness-of-fit tests for ((4x45, Oz)& lay-up fatigue-life results. The same candidate distributions as
used above were applied to the 22 results from the [(f45, O& lay-up specimen fatigue data. The plots for the Weibull distributions are described in Figures 10-12. Although not detailed here, the plots for the other
11
Fibre composite fatigue-life Table 2
determination: J. Lee et al.
Correlation coefficients for goodness-of-fit tests’ Correlation coefficient, r
Distribution
UD
Kf45,
Normal Log-normal Exponential Two-parameter Weibull Three-parameter Weibull (x1) Three-parameter Weibull (x2)
0.93 0.96 0.98 0.99 0.99 0.98
0.86 0.98 0.97 0.99 0.98 0.99
02h1s
‘For the UD case: x, = 10, .x2 = 1000; and for the [(f45, Oz)& samples: x, = 5000, x1 = 16 000
distributions gave deviations away from linearity similar to those in the corresponding graphs for the UD fatiguelife data. Unlike the UD case where the two-parameter model provided the best fit for the fatigue-life data, the most appropriate model for the [(f45, Oz)& lay-up fatigue-life data is provided by the three-parameter Weibull distribution. After several iterations a location parameter of 16 000 cycles was found to give the best fit (Figure 12). Again the correlation coefficients are shown in Table 2 and reinforce the hypothesis that the three-parameter model with a minimum life of 16 000 cycles is the most appropriate for this data set.
2
1
1
0
Figure 10
HTA-982A [(f45, 02)& graphical analysis: two-parameter Weibull distribution hypothesis
Figure 11
HTA-982A [(f45, Oz)& graphical analysis: three-parameter Weibull distribution (5000) hypothesis
12
Fibre composite
fatigue-life
determination:
J. Lee
et al.
/ -4 13
5 &&OO)(Ck’CLE~ Figure 12
Parameter
HTA-982A
[(f45.
01)2]S graphical
analysis:
three-parameter
estimation
By using the expert system” and the standard variable estimation procedure described above, the parameters for the Weibull distribution were estimated for the UD fatigue-life data and are tabulated in Table 3. To obtain an indication of the influence of sampling, the parameter exercise was carried out twice, firstly with the sample population divided into five as-produced sub-groups, and then with the five sub-groups containing a random selection of samples. By using the selection procedure as discussed above, the maximum likelihood approach was found to be the most appropriate, and this technique was applied to the complete data set for the UD fatigue life and the 22 results obtained for the [(f45, OZ)Z]slay-up specimens. The values for the Weibull parameters are shown in Table 4. The application of the selection procedure is described in a later section.
Tests based on EDF statistics By following the procedure for a statistical analysis with the goodness-of-fit tests based on the EDF statistics, as described above, the null hypothesis, Ho, that the fatigue lives for the two sets of samples belong to a Weibull distribution, was accepted at the 25% level of significance’.
DISCUSSION As can be seen, the graphs in Figures 2 and 3 are clearly skewed to the right, suggesting that the data belong to a distribution other than normal, necessitating further analysis of the fatigue-life data. This was achieved through the graphical goodness-of-fit exercise, which indicated that the fatigue data for the fibre-reinforced composite material in both ply configurations was most accurately modelled by a Weibull distribution, as these
TO F:UJRjFf-16~ Weibull
distribution
(16000)
hypothesis
linearized plots were the straightest. In the case of the UD sample fatigue-life data, the two-parameter form of the Weibull was the most appropriate. The fatigue-life data for the [(f45, 02)& lay-up specimens was most accurately modelled by the three-parameter version of the Weibull distribution. Statistical tests based on empirical distribution functions reinforce the hypothesis that the fatigue-life data belong to a Weibull distribution. For the UD test pieces an R ratio of 0.1 with an associated peak stress of 1.6GPa was selected (73% of tensile strength). The specimens manufactured with a [(f45, OZ)& lay-up arrangement were fatigue tested at the same R ratio but the peak stress for these trials was reduced to 1.1 GPa (87% of tensile strength). Both peak stresses were selected to provide fatigue-test results with short and meaningful life cycles. However, the higher percentage of tensile strength used as the peak stress in the fatigue trials of the [(f45, OZ)& lay-up test pieces results in a prediction of the three-parameter model and a minimum life, whereas the lower percentage used in the UD specimen fatigue tests produces a two-parameter Weibull model prediction. This reversal of peak stress level and number of minimum life cycles between the layup configurations could be explained by the [(f45, O& arrangement having a lower susceptibility to the many random factors affecting fatigue life. This work only considered one R ratio and peak stress for each lay-up configuration. Further research needs to be carried out to assess the impact of different R ratios and peak stresses on the Weibull parameters. The results of the parameter estimation method selection .procedure, as described earlier, are summarized in Table 3. The table is divided into three portions, the first of which (on the left-hand side) is sub-divided into two sections, one detailing the results from the as-produced fatigue-life data. the second containing data on the randomly grouped fatigue-life data. The adjacent segment averages the results from the as-produced and random group analyses. The final portion (on the extreme right) describes the results of the application of the parameter
13
Fibre composite fatigue-life
determination: J. Lee et al.
estimation techniques to the complete data set of 50 fatigue lives. The first stage of the procedure to select the most appropriate parameter estimation method, as described earlier, considered which of the parameter estimation techniques produced widely different values for the distribution descriptors in relation to the other approaches. Two options were rejected at this point, namely the standard variable and return period methods. By considering the actual values for all three sections in Table 3 it can be seen that both these approaches produce consistently low estimates when compared to the other techniques. The most accurate estimation method was selected out of the remaining techniques by considering the degree of variability, (range/&), where 12is the number of subgroups, for the parameters across the five sets. Again by considering the results in both the sub-groups and average for sub-groups sections it can be seen that the maximum likelihood method always provides estimates for m with the lowest variability and estimates for 8 with one of the lowest variabilities when compared to the other parameter estimation techniques. The selection of the maximum likelihood estimation technique is reinforced when the average result for the sub-groups is compared with the maximum likelihood estimation for the complete data set, and the values for m and 0 are found to be comparable. It should be noted that, for the data used in this work, five of the estimation techniques give similar results. From a practical engineering viewpoint there appears to be little to choose between the probability least squares, standard least squares, moments, percentiles and maximum likelihood method. In certain instances where a computer package, or similar, was not available to carry out the complicated iterations involved with some of the techniques (e.g. the maximum likelihood method), estimations from the simpler approaches may give results of sufficient level of accuracy. The estimated values for the Weibull parameters, as determined by using the maximum likelihood method, are described in Table 4. From the ‘2 x standard error on m’column it can be seen that it is unlikely that the shape parameter for either lay-up configuration is equal to unity, and as a result there is a higher confidence that the fatigue lives belong to the Weibull distribution and do not follow the exponential model. By using the ranked version of the Mann-Whitney U test the samples from the two sub-groups were found to come from a population with the same mean. This suggests that sampling does not affect the estimation of the distribution parameters. The above methodology is similar to that initially carried out by Weibull12 where an empirical analysis suggested that the Weibull distribution provided a suitable model for several groups of experimental data. To understand why the Weibull shape parameter is close to unity for these materials a different approach may be required. This is provided by Freudenthal and Gumbel699 where the derivation of the Weibull distribution has been achieved from an extremal phenomenon viewpoint, in
Fibre composite fatigue-life
Table 4
determination: J. Lee et al.
Weibull distribution parameters
Lay-up configuration
Shape parameter, m
2 x standard error on M
UD [(+45? 02111s
1.34 1.1
f0.04 f0.08
Characteristic value, 0 (cycles) _~___ ~
which the fatigue failures are considered to form a group of the weakest specimens and the theories of extreme values can be applied. In the first of these papers6 the authors developed a statistical scheme for evaluating fatigue failures. The concept of minimum life and the derivation of the three-parameter model was introduced in the second report’. Essentially the model is derived from the log-normal distribution which belongs to the exponential class. As a result the model is strongly influenced by the random nature of failure, characterized by the exponential distribution. However, Freudenthal and Gumbel introduce the concept of progressive damage where the damage resulting from any particular loading cycle is dependent on damage induced by previous loading cycles, leading to the development of the Weibull distribution. Consequently, it is suggested that the shape parameters for these fibre-reinforced composite materials are close to unity because the various stages of the manufacturing and testing route introduce randomness into the material characteristics. This is compounded by the random nature of the damage accumulation process. However, the fatigue failures result from the influence of progressive damage on top of the initial random properties, resulting in the requirement for a modified exponential model, the Weibull distribution. It is widely recognized that there is large scatter associated with fatigue-life values for composite materials. The practical extent of this scatter is highlighted in this study through a comparison of the difference in median and minimum life values. From the usual expression for the median,
1
~~
48 669 118960
k = 0 (ln2); [ the medians for both sets of materials were calculated as: UD samples (set of 49), 1 = 37022 cycles; and [(f45, Oz)& samples, R = 85 250 cycles. The corresponding estimates of minimum life values are 0 and 16 000 cycles, respectively. This gives a difference between the median and minimum life values of over 10 000 cycles for the UD case and over 33 000 cycles for the [(f45, 02)& sample data set. The disparity in the two parameters for both lay-up configurations reinforces the necessity for accurate parameter estimation and the need for careful consideration of the design engineering requirements.
2 x standard error on 0
Median value, .r
Location parameter, .vs (cycles)
f1527 &IO300
37022 85 250
0 I6000
composite material the results indicated that the most appropriate model was provided by a very broad Weibull distribution, with a shape parameter close to unity. The most accurate parameter estimation method was found to be the maximum likelihood technique. However, from a practical engineering viewpoint, estimations from any one of five techniques (probability least squares, standard least squares, moments, percentiles and maximum likelihood method) give satisfactory results. The large difference in median and minimum life values estimated in this study reinforces the requirement for accurate distribution parameter estimation and questions whether median life S-N curves are at all appropriate for design engineering applications. ACKNOWLEDGEMENTS The authors are grateful to the Engineering and Physical Sciences Research Council, UK for financial support of this work. The authors would like to thank Professor Alfonso Fernandez Canteli and Dr Maria Jesus Lamela Rey of the University of Oviedo for their guidance and useful suggestions on the use of the expert system for the analysis of extreme value problems.
REFERENCES 1
2 3
Whitney, J.M. Fatigue characterisation of composite materials. In ‘Fatigue of Fibrous Composite Materials’, ASTM STP 723, American Society for Testing and Materials, Philadelphia. PA, 1981, pp. 1333151 Hahn, H.T. and Kim, R.Y. Fatigue Behaviour of composite laminate. J. Compos. Muter. 1976, 10. 156-180 Gathercole, N.. Reiter, H., Adam, T. and Harris. B. Life prediction for fatigue of T800/5245 carbon-fibre composites: I. Constant-amplitude loading. Inf. J. Fatigue 1994, 16(8), 523532
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8 9 IO
CONCLUSIONS
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When the methodology illustrated in the flow chart (Figure 1) was applied to fatigue-life data for a fibre
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Talreja, R. ‘Fatigue of Fibrous Composite Materials’, ASTM STP 723, American Society for Testing and Materials, Philadelphia, PA, 1981 Owen, D.B. ‘Handbook of Statistical Tables’, Addison-Welsey Publishing Co., Inc., Massachusetts, 1962, p. 318 Freudenthal, A.M. and Gumbel, E.J. On the statistical interpretation of fatigue tests. Proc. Rqval Sot. A 1953, 216, 230 D’Agostino, R.B. and Stephens, M.A. (Ed) ‘Goodness-of-Fit Techniques’, Marcel Dekker. Inc., New York. 1986, pp. 1 ~-62, 97-100, 145-150 Castillo. E. ‘Extreme Value Theory in Engineering’, Academic Press, Inc., London, 1988 Freudenthal, A.M. and Gumbel, E.J. Minimum life in fatigue. J. Am. Statistical Assoc. 1954. 49, 5155597 Alavarez. A.. Castillo. E., Cobo. A. and Herrero, T. An expert system for the analysis of extreme value problems, technical report, University of Cantabria. Santander, Spain, 1993 Ante, C.E. and Khmko, L.A. Choice of model for reliability studies and related topics II, Report ARL-73-0121. AD 772775, US Army Research Lab., NASA-Lewis Research Group, Ohio, 1973 Weibull, W. A statistical distribution function of wide apphcabihty. J. Appl.Mechanics 1951. 18. 193 --297
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