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Statistical analysis of fracture toughness data
Engineering Frocrure Mechmics Printed in Great Britam.
00137944/87 s3.00 + 0.00 (3 1987 Pergamon Journals Ltd.
Vol. 27, No. 2, pp. 143-155, 1987
STATISTICAL
Brown
ANALYSIS OF FRACTURE DATA Boveri Research
D. J. NEVILLE Centre, CH-5405
Baden,
TOUGHNESS
Switzerland
Abstract-The underlying causes for the variations in fracture toughness, which are observed experimentally, are discussed. A theory covering the statistics of fracture is briefly described, and the distribution function based on this theory is shown to be applicable to metals, ceramics and powder-filled epoxy resins. The distribution function can be expected to work for all materials since the theory behind it involves no assumption concerning materials. Since large, consistent sets of data are rarely to hand in practice, it is not possible to apply the theoretical distribution function in an exact way. Instead approximate, conservative methods based on this new distribution function must be used and such methods are demonstrated in the present paper for a number of possible cases which may arise in practice.
NOTATION Weibull datum parameter Weibull scale parameter Scale parameter for the new distribution Weibull shape parameter Crack-opening displacement Shape parameter for the new distribution Cumulative probability Critical strain-energy release-rate Critical J-integral Critical stress-intensity factor Sampling (new distribution function)
function
function
1. INTRODUCTION FIRST basic requirement for an engineering analysis of the variations that occur in fracture toughness is an appreciation of the underlying causes of that variation. These causes are three. Firstly, there is always some experimental error involved in making the measurements from which toughness is calculated (crack-length, specimen-size, load at failure, etc.). This type of error is quite unavoidable, but fortunately in most cases reasonably small when testing is carefully carried out, and has been analysed in detail previously[l-31 and even measured[2, 33. In summary, the error in K,, from a carefully made test is about f 1% to f 2% of the mean value and the corresponding error in either crack-opening displacement or critical J-integral is about f 5% of the mean value, the error being greater in these two cases because of the more complicated and more numerous measurements required for each value of toughness. Secondly, inhomogeneity in materials causes a potentially very large amount of variation in toughness, the precise amount depending on the length of crack-front involved in each failure or test. Thirdly, different batches of nominally similar material must be expected to have different toughnesses and distributions thereof. If a rigorous analysis of the variations in toughness of a particular material is required, it is apparent from the above firstly that tests must be conducted on one batch or charge of material, secondly that they ought, for the sake of simplicity (to avoid the necessity to calculate the effect of specimen-size for each result), to be conducted on a set of specimens all of the same size and which must be cut in the same orientation, and thirdly that the parameters for the distribution of this one batch of material are not reliably applicable to any other batch of material. In the majority of instances no such large consistent sets of data will be available. Instead several small sets of data will have to be treated, and then one of several possible approximate approaches, the choice of which may be dependent somewhat on particular criteria, must be applied. A number of examples are dealt with in detail below. The theory is covered first in the present paper as background to the more practical methods which are described afterwards.
THE
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2. THEORETICAL
BACKGROUND
It is beyond the scope of the present paper to deal with the theoretical background in great detail, but the different types of theoretical analysis which deal with the distribution of toughness of materials are discussed briefly in order to throw some light on the problems that are encountered in consideration of the statistics of fracture. 2.1 Previous work It has been attempted to apply laws of mixture to the toughness of two-constituent metals (e.g. dual-phase ferrite-martensite steels)[49]. There are three problems with such an approach: firstly the effective toughness of the individual constituents may be different when the constituents are in situ in the compound material due to, for example, residual stress; secondly, if GIc, J,, and crack-opening displacement for a mixture are linearly related to the volume fractions of the constituents then the same cannot be said for K,, since G,,, J,, and crack-opening displacement are related to the square of K,, rather than linearly related to it; thirdly averaging G,, over a fracture surface or along a crack-front will be misleading since the failure of the weakest part will very probably result in total failure rendering such averaged properties meaningless. Indeed it is mistaken, as will be seen, to predict a single value of toughness for inhomogeneous materials because the presence of two or more constituents or other forms of inhomogeneity in materials does not generally lead to some average or extreme value of toughness, but rather to a distribution of values that must be described, modelled and explained in terms of statistics[3, lo]. Only in the extreme case of infinitely wide specimens will an inhomogeneous material exhibit single-valued toughness, and this is due to the infinitely long crack-front being certain to cut through a region containing material of minimum toughness. Even in the case of two distinct constituents, plastic interplay between the two constituents and variation of morphlogy from place to place ensure that a distribution of toughness results as opposed to two distinct values of toughness corresponding to the toughnesses of the two constituents[lO]. Several probabilistic theories of fracture have previously been published. Those of Curry and Knott[ 111, Evans[ 121 and Wallin, Saario and Tiirriinen[ 131 start out from a distribution of defectsizes to predict toughness, but all avoid the inevitable distribution of toughness by assuming, in effect, a particular value of cumulative probability to obtain a single value of toughness. Wallin et al. even go to the extent of using a temperature-dependent surface-energy term in order to achieve a reasonable prediction of the variation of toughness with temperature. Working on brittle materials in tension Jayatilaka and Trustrum[14] used a particular flawsize distribution to obtain a statistical distribution of failure stress equivalent to a two-parameter Weibull distribution, but found in practice that the effect of volume was not predicted precisely by means of the Weibull function; they extended this work to investigate other flaw-size distributions concluding that the distribution of failure stress is insensitive to the distribution of flaw size[lS]. Landes and Shaffer[16] explained the variation in the toughness of their rotor steel by means of a sampling argument, but applying the two-parameter Weibull function to their results does not, however, give a reliable fit[ 171. It is appropriate at this point to describe the Weibull function briefly since it is often mentioned in this paper. This function has three parameters each of which has a distinct role: the shape parameter, c, determines mainly the skewness of the function (whether the majority of values are above or below the mean); the scale parameter, b, has a determining influence on the spread (standard deviation) and mean of the function; and the datum parameter, a, which is the minimum value of the distributed variable, X: F@)
=
1
_e+(.Y--ni/hY.
(1)
Often the datum parameter, a, is ignored (set equal to zero) and then the function is in its familiar two-parameter form. For shape parameter, c, greater than 1, the mean of the distribution is always slightly less than the scale parameter, h, and when the shape parameter (sometimes called the “modulus”) is greater than 3 it has a very limited effect on the standard deviation of the function which is proportional to the scale parameter, h. It is therefore an error to treat the shape parameter (modulus)
Statistical analysis of fracture toughness data
145
as an inverse measure of standard deviation (spread) for the Weibull function as shown by the following example: Distribution 1: a = 0, b = 1, c = 3; standard deviation = 0.3, Distribution 2: u = 0, b = 4, c = 6; standard deviation = 0.75.
Wallin has also produced an analysis of variations in toughnesss[ 181 which serves to highlight many of the problems which can arise. He assumes the relevance of an “active volume” (in this case wedge-shaped and extending from the crack-tip to the edge of the plastic zone) to the determination of the probability of failure and yet invokes a weakest-link principle (meaning that only a very small part of the supposedly active volume is active in the failure). Further, application of this particular active volume implicitly assumes that fracture is unlikely to be initiated in the zone of very high stresses and strains next to the crack-tip. His mathematical derivation leads to a two-parameter Weibull function for toughness but is based on a mathematical procedure which is thought to be incorrect for the following reason. In considering infinitesimally thin layers of material within the active volume it is necessary to contemplate the failure of infinitesimally small parts of carbides which is difficult to justify in physical terms; if, however, the carbides within the active volume were considered individually as whole particles, then the mathematical approximation which Wallin uses ([ 1 -xl” _ e-““) t o reach a Weibull function would no longer be valid. (It can be shown that the failure of this approximation means that toughness is not distributed according to the two-parameter Weibull function for stress-conrolled failures of the type which Wallin considers, and that, if this approximation were correct, that toughness would indeed be distributed according to the two-parameter Weibull function for all materials). His assumption of an arbitrary value for minimum toughness (the third, datum parameter of the Weibull function) of between 5 and 15 MPafi brings the largest sets of data which he uses as evidence for his theory nearer to agreement with his assertion that the only appropriate shape parameter for the Weibull function when applied to toughness is four. However, the Weibull distribution does not fit these large sets of data at all well when the shape parameter is four[17]; the major reason for this is that toughness is (according to the new theory[3, 191 and to a method using the three-parameter Weibull function [3, lo]) distributed such that the distribution is alwaysposirively skewed towards low values (majority of values below the mean), whereas with the shape parameter equal to four the Weibull distribution is skewed in the wrong direction towards high values (majority of values above the mean). Other problems with Wallin’s analysis are that he associates the degree of spread of the distribution with the shape parameter of the Weibull function (which is mistaken, as shown above). He also states that inhomogeneity in the material affects only the mean value of the distribution; this is in contradiction of his own statement that the “slope” (shape parameter) describes the magnitude of “scatter”, since the spread (standard deviation) of the Weibull distribution is proportional to the mean value for a given shape parameter, both spread and mean being proportional to the scale parameter, b. Furthermore, without any effect of inhomogeneity, it is difficult to see any justification for his use of a weakest-link principle because the material would be equally weak everywhere. Inhomogeneity is rather to be seen as the principal underlying cause of the distribution of toughnessC2, 3, lo]. If the Weibull function is to be used for the description of distributions of toughness it is necessary that the three-parameter form be used in order to obtain a reasonable fit[3, 10, 171. In that case it is required to invoke a distance-criterion for failure at sharp cracks[20] in order to introduce the third (datum) parameter, a, the value for minimum toughness[lO]. However, the three-parameter Weibull distribution cannot fit data-sets from toughness properly (see Figs 1 and 2). The data in Fig. 1 are from a quenched and tempered low-alloyed structural steel (A533B) on the lower shelfC21] and were obtained using compact tension test-pieces. This misfit (which is to be seen for sets of data and materials other than those of Fig. 1) is due to the assumptions made by Weibull in his original derivation of his function[22]. The only merit[23] or Weibull’s choice for the material function, n(c) = ((x-a)/b)‘, is that it is the simplest possibility. Nonetheless, the Weibull function is extremely flexible and thus can be used in circumstances for which no specifically suited distribution function exists. That the Weibull function has an ever increasing hazard function for shape parameters greater than one makes it especially suitable for problems involving decay with time (e.g. creep, fatigue). A new statistical theory[3, 191 based on the conditions of stress and strain at crack-tips provides proper fits to toughness data with only two parameters (no flexibility) and is summarized
D. J. NEVILLE
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‘I
10 (K,, -K Ic
Fig. 1. Three-parameter
Weibull
100
rmnimum )Mh./iE
plot of K,, for a quenched
subsection derivation involve concerning or non-linearly elastic-plastic, etc.) materials. may for certain configurations.
and tempered
low-alloyed
steel[21].
193). and derivation of materials (they applicable of all stresses, awkward
191 sharp
passes The implication amount of material piece is at a fixed
a series
states[24] to the following region that, the stresses do not change, of material to any particular So the amount at or above given or strain to the square opening since (about displacements is of constant shape in two dimensions load. Since amount loaded given or strain would to calculate, to which be linearly to it. Sampling, as g, 4, or COD2, g(s), given
(S/W
(2)
where the scale parameter, B, normalizes S (and is actually the median value of s), and the shape parameter, D, allows the shape of the distribution to change with temperature as it is observed to do[17, 191 even for a single material. The shape parameter for a given material is also expected to be different for failure at microcracks in simple tensile tests than for failure at microcracks near a macrocrack because of a difference in the distributions of stress in these two cases. It can be shown[3, 193 that the cumulative probability of failure at the tip of a sharp crack is given by:
LOG (TOUGHNESS 1 Fig. 2. Systematic
misfit of the points
and the straight
line in Fig. 1.
Statistical 99.
1
analysis
of fracture
I I11111,
I
toughness
I I IIIII,
data I ,,,,I-
141
989590-
I
1
q‘?K&
1 I ,,,,I_ 100
(MPa+hii)4
Fig. 3. Plot using the new functionC3, 191 of the same data as in Fig. 1. Sampling is here K,, to the fourth power. The major deviation is no longer present. q is a constant which shifts the data uniformly parallel to the abscissa (a condition for using the data was not to reveal the values).
F(S)
go 1+g(S)
=
(3)
a function with only two parameters, B and D. The improvement in fit achieved by the use of the new distribution function can be seen by comparing Figs 1 and 3. A very good fit is also given in the case of ceramics. The data for this example are fracture toughness results from bend specimens with very fine slots (instead of fatigue precracks)[26]. Figure 4 shows the fit afforded by the new function for this set of data from RefelSic; the shape parameter, D, ( = slope of the plot = 5.34) is different from that of the steel (Fig. 3, slope, D = 2.20), and the distribution for the ceramic is more symmetrical but still somewhat skewed towards the lower values. Figure 5 shows the Weibull plot of the data in Fig. 4 using the Weibull function with only two parameters since for nearly symmetrical distributions a third parameter is superfluous; (a three-parameter plot shows the systematic misfit shown in Fig. 2). It should be noted in any event that three parameters make a function so flexible that “fits”, though perhaps acceptable in terms of statistical tests, might be quite meaningless from a theoretical, (and in this instance fracture-mechanical) point of view and are thus better regarded as non-theoretical descriptions than proper, theoretically based fits. It can be seen from comparison of Figs 4 and 5 that the new function (eqs 2 and 3, [3, 191) provides a proper fit, indicated by the coincidence 99s
100
I
I I,,,,,
(MPafi14
1000
KP, Fig. 4. Plot using the new function
of K,, for a ceramic material (Refel-Sic) the length of the plot.
[26]. The fit is good throughout
148
D. J. NEVILLE
!fd._J 100
Fig. 5. Weibull
1000
(MPa Jiif
plot of the same data as in Fig. 4. No fit exists, the points the plot.
lying on a curve throughout
of the points and the straight line over the whole range in Fig. 4, where the Weibull function cannot, as shown by the very distinct curvature of the plot in Fig. 5 (raising toughness to the fourth power makes no difference in the case of the two-parameter Weibull function). The new function[3, 193 may also be applied to the failure stresses from tests on pieces that contain many crack-like defects instead of one large crack, but in this case the load at failure to the fourth power rather than sampling, S, is plotted since the calculation of sampling would require knowledge of the size-distribution of the defects and their density in the material. Data[27] from the bursting pressure of 85 identical test-pieces made from filled epoxy resin are shown in Figs 6 and 7. The stress distribution in these tests was always of the same shape irrespective of load so that no integration over the volume of the test-pieces was necessary and the bursting pressures could be plotted directly. Figure 6 shows the better fit of the new function and Fig. 7 shows the curved plot of the Weibull function. Once again the general curve of the Weibull plot indicates that the Weibull function does not give a fit, whereas the new function gives a reasonable fit. There is one major deviation from the main trend in both of these plots which is probably due to an irregularity in the application of load (resulting from the complex loading arrangement), but the plot using the new function is clearly straighter overall. Rarely in practice will such good sets of data be available. It therefore remains to see how to apply the theoretical understanding obtained from large, consistent sets of data to the relatively 99 5
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-
5
2 -
l
4 l 05 105
I (BAR’)
406
2.406
(BURSTING PRESSUREI4 Fig. 6. Plot using the new function of the bursting pressures of powder-filled epoxy resin[27]. The plot is generally straight except for one excursion which is possibly due to the method of application or reading of load during the tests.
Statistical analysis of fracture toughness data
149
051 405
406
( t3AR4) (BURSTING
2 406
PRESSURE)’
Fig. 7. Weibull plot of the data in Fig. 6. In addition to the excursion observed in Fig. 6, the plot has a general curvature.
small, less consistent sets which are often encountered in practice. The next section deals with this problem. 3. PRACTICAL
APPLICATION
The following cases are dealt with since they represent the types of non-ideal situation which may arise: (1) Data from one charge of material is for some reason bimodal (split into two distinct populations with similar but not identical distributions). (2) Several data-sets are available; this is similar to (l), but the data-sets are not necessarily from one charge of material and the resulting distribution may be bimodal or even multimodal. (3) Too little data is available: no data-set of less than 7 elements is of any use whatsoever for statistical purposes because the uncertainty in the estimation of parameters is so great as to render any evaluation meaningless. Confidence improves strongly as the number of elements increases from 7 to 12, and the improvement carries on, although more slowly, as the number of elements is further increased. 3.1 Bimodality Figure 8 shows the appearance of a second, bimodal data-set from the Refel-SiC ceramic mentioned above[26], the bimodality being present presumably because of some inconsistency in the preparation and/or testing of the specimens since all the material came from one charge. If the specimens from which these results were obtained were available, it might be possible to separate the two populations with clarity and confidence. However, in the absence of that information an approximation must be made which offers another example of a possible procedure. Using the information from Fig. 4, (slope = 5.34, median value = 315 (MPafi)4) and the fact that the ratio of the specimen-widths for the two sets of specimens tested by Tradinik er al. was 1.75, a theoretical line can be drawn for the bimodal set of data. The ratio of the specimen-widths is the ratio of the lengths of the crack-fronts, and the volume subjected to greater than a given stress is proportional to crack-length. Thus, to compare the results from one set of specimens to those from a set of specimens n times wider, sampling (K& etc.) for the wider specimens must be multiplied by n.) (This method has been found to give an excellent prediction of the effect of specimensize[19], despite that the prediction is dependent on a fit which uses only two parameters.) Figure 9 shows this theoretical line as well as a tentative division of the data into two parts. The division
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D. J. NEVILLE
Fig. 8. Weibull
plot of further
data from Refel-SiC[26].
The bimodality
of the data is quite clear.
is tentative because it could be that the distribution of the consistent data-set is different from both those of the bimodal set; again, more information is required in order to know. The important points brought to light by these sets of data from Tradinik et al. are firstly, that data-sets must be checked to see whether they are consistent within themselves, and secondly that bimodal data-sets can only be split with confidence if more information is available than the simple data themselves. 3.2 Mixed data-sets The first possibility for mixed data-sets is that all the sets come from one charge of material. This example uses data from a martensitic rotor steel, 25 NiCrMo 14 5, [28]. Toughness (J,,) was measured at eight different distances from the centre of a forging 1.2 m in diameter, and there was no general trend to be seen in toughness between the outer five positions. The data from these positions are therefore treated as if from one charge of material. In Fig. 10 these fifteen points are plotted as sampling (in this case &), using the new function, and a good fit is obtained. The slope of the line (shape parameter) is 11.8 which is perhaps rather high for a steel, a slope of 1 to 2.5 being expected on the lower shelf of toughness, although no data-sets from the upper shelf have previously been treated with the new function. This might be explained in two different ways. 99 98
_
LINE FROM GOOD DATA
1 102
hlPav?ii)4
IO3
K?C of the data in Fig. 8, making use. of the information gained Fig. 9. A tentative separation specimens of different size but taken from the same material (see Fig. 4).
from a set of
Statistical
analysis
of fracture
toughness
data
151
2 (kJ/m2j2
Jh2 Fig. 10. Plot using the new function of J,, for rotor steel 25 NiCrMo 14 5[28]. The high slope indicates a rather symmetrical distribution which may be typical of toughness on the upper shelf, but could be due to errors influencing the distribution of toughness as well as material inhomogeneity.
(1) The mean of the results is 175.2 kJ/m2 and the errors in the experimentally obtained values of toughness will be about + 5% or about f 9 kJ/m2 in this case. The total range of the results is only 44 kJ/m* or about twice the range of error, and so the errors (which are symmetrically, normally distributed) are large enough that they tend to symmetrize the distribution of toughness which would otherwise be expected to be more skewed; that is a transitional distribution results. When two (or more) effects are working simultaneously (here errors and material variation) there is a tendency towards a symmetrical distribution regardless of the shapes of the individual contributing distributions (central limit theorem of statistics). It could also be that the five groups of three data are not from adequately similar material which would result in a distribution tending towards a normal distribution, and this would again be due to the central limit theorem. (2) It may be that toughness is indeed more symmetrically distributed on the upper shelf than on the lower shelf, the mechanisms of crack-propagation being totally different. The second possibility for mixed data-sets is that the individual sets come from similar material but not from the same charge. The data for the example are again from the martensitic rotor steel 25 NiCrMo 14 5[29], and are in two small sets, one of eight elements and the other of six. The set of eight is spread somewhat wider than would be expected from errors alone, and eight elements is just enough for a plot, see Fig. 11, the slope (shape parameter) of which is 8.4. The same considerations apply to this value of the shape parameter as to that of the previously considered data[28]. Either toughness is distributed more symmetrically on the upper shelf, or errors are playing a role. Were the smaller data-set sufficiently large for a plot it should not be plotted even 95
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00
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-
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I
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I
I
l
-
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/I
E
I
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5 109
,
I (MPa fi)4
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9
Fig. 1I. Plot using the new function of K, for rotor steel 25 NiCrMo 14 5[29]. Either toughness is fairly symmetrically distributed on the upper shelf or neither errors (symmetrically distributed) nor material inhomogeneity (heavily skewed) dominate, and an intermediate distribution of middling shape results.
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D. J. NEVILLE
2 109
Fig. 12. Mixing additional
(MPa4ii14
K;
40’0
data into that in Fig. 11 increases symmetry due to the central and thereby increases the slope for the new function.
limit theorem.
so, because the overwhelming source of the spread for this data-set is experimental error, and a plot would lead to a misleadingly symmetrical distribution. (It is also safer to assume a positively skewed distribution than a symmetrical one for reasons explained in section 3.4). Mixing the two sets of data together tends to increase symmetry due to the central limit theorem. For the new function as the slope (shape parameter, D) becomes larger so the distribution becomes more symmetrical. See Fig. 12, the slope is now about 10.2 as opposed to 8.4 from Fig. 11. Where errors outweight the effects of irregularity in the material or are at least large enough to affect the shape of the distribution, then it is possible to assume a shape parameter (for the new function), the choice of which is based on experience with results from similar material which were not seriously affected by error. In addition the mean value must be estimated in the normal way (sum divided by number of results). 3.3 Too little data Three possibilities exist when two little data is available, the available sets are all too small individually to be worth plotting or nothing is known from experience about the likely value of shape parameter. Firstly, all the data or data-sets may be mixed together to form one lot. The inevitable result of this treatment is that all the data together conform, in accordance with the central limit theorem of statistics, to a Gausssian or normal distribution. This would lead to a non-conservative estimation of the probability of failure (as will be explained in Section 3.4) and is therefore not a sensible procedure. Secondly, it is possible, when nothing better is available, to make plots for all the small (< 7 elements) data-sets to hand an take the averages (or even the minimum values) of the means and the shape parameters obtained as representative of the data generally. This will tend to result in a low (conservative) value for the overall mean, but the final shape parameter may not be appropriate. Thirdly, possibly the best procedure in such circumstances is to choose a conservatively skewed shape and calculate the mean of the available values. This mean will always tend to be low because a limited-number of values conforming to a positively skewed distribution will often tend to have a mean that is less than that of the distribution and it so happens that this tendency is particularly strong for the new function. 3.4 Sqfety in skewness When a limited number of values are taken from a population which is distributed according to a skewed distribution the mean and standard deviation will approximate those of the population. However, in some cases these approximations may systematically err on one side of the values for the population. For instance, for the new functionC3, 191 (eqs 1 and 2) with shape parameter equal to four, the mean for the function (or an infinite population) is 1.12 times the median value (scale parameter) but the mean as estimated by 10 values picked randomly from the population
Statistical
10
analysis
of fracture
toughness
data
NORMAL
CURVE
153
SELECTIONS
FUNDAMENTAL DISTRIBUTION
1000
0
FOR
_
SELECTIONS
1
2
VARIATE
Fig. 13. Demonstration of the conservatism of the new function compared to normal distributions for 10 and 1000 results. The new function gives a higher likelihood of failure having occurred for low values of toughness.
is expected to be only 1.07 times the median, and only 1.08 times for twenty results. Thus the new function affords a conservative estimation of mean. Similarly, the standard deviation is very much less for limited numbers of values than the standard deviation of the population from which the values were drawn and which is distributed according to the new function. The significance of these under-estimations of both mean and standard deviation is that if a normal distribution is assumed it will give serious under-estimations of the probabilities of failure at low values of load (see Fig. 13). Shown in Fig. 13 are a curve for the new function with a shape parameter of 4, and curves for assumed normal distributions based on the means and standard deviations of 10 and 1000 selections from the distribution represented by the curve for the new function. Thus there is safety in the skewness of the true distribution of toughness and danger in the assumption of a normal distribution. From the immediately preceeding considerations it is apparent that, when using the new distribution on a set of toughness-data, it is safer to err towards lower values of the shape parameter and towards lower values of the scale parameter (proportional to the mean). It is also noted that the mean of the Weibull distribution is rather accurately estimated by only a few values, meaning that the estimate is not conservative. Further, if a three parameter Weibull distribution is used to describe toughness data then the resulting estimates of probability of failure at low values will be non-conservative; indeed the estimate is zero below the datum parameter (which is, in any event, not easy to estimate for the best description of the data). This is ameliorated to some extent by the necessity to fulfill a criterion of distance before a metal can fail from a crack-tip[20] (the same effect adding to the conservatism resulting from the use of the new function). 4. SUMMARY Statistical analyses of toughness data should be done graphically using probability paper corresponding to the new function (in Fig. 14). The procedure is as follows. The values of sampling, S, are straightforwardly calculated. Then the N values for S to be plotted are each given a rankorder number, the smallest result having rank-order number 1, and the largest rank-order number N. Then the median rank for each result is calculated by means of an equation which gives a very good approximation to the actual median ranks (which would otherwise consume a great deal of time in calculation): Median rank = (n - 0.3)/(N+ 0.4) where N is the number of results to be plotted and n is rank-order number. are values of cumulative probability against which the values for sampling, probability paper (see Fig. 14). The slope of a plot on this probability paper parameter, D, and the median value (scale parameter, B) is read off probability = 50%.
(4) These median ranks S, are plotted on the is equal to the shape opposite cumulative
D. J. NEVILLE
154
SAMPLING Fig. 14. Probability
paper
/ LOAD
for the new function.
The new function is generally conservative in its estimates of failure-probability and is easy to use (no third parameter is required and thus need not be estimated), and graphical methods bring to light any problems involving irregularities in the data (e.g. bimodality, etc.). The analysis used must depend on the nature, quantity and quality of the data available, and ought to be chosen according to the purpose behind the analysis but always with a view to conservatism and safety. A~knoM,ledRement.s-Dr to the author. Thanks
H. Beer is gratefully acknowedged for making the data from the powder-filled also to Dr C. Wtithrich for helpful discussions.
epoxy resin available
REFERENCES [I] 123 [3] [4] [S] [6] [7] [S] [9] [IO] [ll] [12] [I31 1143 [15] [I67 [17] [18] [19] [20] 1211 [22] [23] [24]
W. J. Jackson and J. C. Wright, Metals Technol 4, 425 (1977). D. J. Neville and J. F. Knott, J. Mech. Phys. Solids 34, 243 (1986). D. J. Neville, Ph.D. thesis, University of Cambridge (1985). P. Stratman and E. Hornbogen, Proc. ICSMA 4 p. 607, Nancy (1967). E. Hornbogen and M. Graf, Acla MetaN 25, 877 (1977). M. Graf and E. Hornbogen, Acra Mefull 25, 883 (1977). K. Friedrich, Prog. Colloid Polym. Sci. 64, 104 (1978). E. Hornbogen and K. Friedrich, Proc. Inf. Conf. Pure appl. Chem., I.U.P.A.C. (vol. 3, p. 1422) (1979). E. Hornbogen and K. Friedrich, J. Mater Sci. 15, 2175 (1980). D. J. Neville and J. F. Knott, J. Mech. Phys. Solid.7 34, 256 (1986). D. A. Curry and J. F. Knott, Merail. Sci. 31, 341 (1971). A. G. Evans, Metal/. Truns. 14A, 1349 (1983). K. Wallin, T. Saario and K. Torriinen, Metall. Sci. 18, 13 (1984). A. de S. Jayatilaka and K. Trustrum, J. Maler. Sci. 12, 1426 (1977). K. Trustrum and A. de S. Jayatilaka, J. Mater. Sci. 18, 2765 (1983). J. D. Landes and D. H. Shaffer, ASTM STP 700, 368 (1980). D. J. Neville and J. F. Knott, J. Mech. Phys. Solids 34, 278 (1986). K. Walhn, Engng Fracture Mech. 19, 1085 (1984). D. J. Neville, to be published by the Royal Society (1987). R. 0. Ritchie, J. F. Knott and J. R. Rice, J. Mech. Phys. Sol& 321, 395 (1973). T. J. Williams, Rolls Royce and Associates, private communication. W. Weibull, A Statistical Theory of the Strength of Materials, Ingeniorswetenskapsakademiens Handhngar, Stockholm: Generalstabens Litografiska Anastalts Forlag, 1939. W. Weibull, J. crppl. Mech. 18, 293 (1951). R. M. McMeeking, J. Mech. Phxs. Solids 25. 357 (1977).
No. 151.
Statistical
analysis
of fracture
toughness
data
155
J. R. Rice and M. A. Johnson, in Inelastic Behaviour of Solids (Edited by Kanninen et al.), p, 641 McGraw-Hill, New York (1970). [26] W. Tradinik, G. Popp and R. F. Pabst, Z. Werkstofftech 13, 254 (1982). [27] H. R. Beer, Brown, Bovri & Cie., Central Laboratories, Baden, Switzerland, communication of unpublished results. [28] J. Albrecht and C. Wtithrich, Z. Werksfofftech, 13, 96 (1982). [29] J. Albrecht, Brown Boveri & Cie., Baden, Switzerland, COST 501, Project CH-11, final report: KLR 85-167 C. [25]
(Received 9 September
1986)
00137944/87 s3.00 + 0.00 (3 1987 Pergamon Journals Ltd.
Vol. 27, No. 2, pp. 143-155, 1987
STATISTICAL
Brown
ANALYSIS OF FRACTURE DATA Boveri Research
D. J. NEVILLE Centre, CH-5405
Baden,
TOUGHNESS
Switzerland
Abstract-The underlying causes for the variations in fracture toughness, which are observed experimentally, are discussed. A theory covering the statistics of fracture is briefly described, and the distribution function based on this theory is shown to be applicable to metals, ceramics and powder-filled epoxy resins. The distribution function can be expected to work for all materials since the theory behind it involves no assumption concerning materials. Since large, consistent sets of data are rarely to hand in practice, it is not possible to apply the theoretical distribution function in an exact way. Instead approximate, conservative methods based on this new distribution function must be used and such methods are demonstrated in the present paper for a number of possible cases which may arise in practice.
NOTATION Weibull datum parameter Weibull scale parameter Scale parameter for the new distribution Weibull shape parameter Crack-opening displacement Shape parameter for the new distribution Cumulative probability Critical strain-energy release-rate Critical J-integral Critical stress-intensity factor Sampling (new distribution function)
function
function
1. INTRODUCTION FIRST basic requirement for an engineering analysis of the variations that occur in fracture toughness is an appreciation of the underlying causes of that variation. These causes are three. Firstly, there is always some experimental error involved in making the measurements from which toughness is calculated (crack-length, specimen-size, load at failure, etc.). This type of error is quite unavoidable, but fortunately in most cases reasonably small when testing is carefully carried out, and has been analysed in detail previously[l-31 and even measured[2, 33. In summary, the error in K,, from a carefully made test is about f 1% to f 2% of the mean value and the corresponding error in either crack-opening displacement or critical J-integral is about f 5% of the mean value, the error being greater in these two cases because of the more complicated and more numerous measurements required for each value of toughness. Secondly, inhomogeneity in materials causes a potentially very large amount of variation in toughness, the precise amount depending on the length of crack-front involved in each failure or test. Thirdly, different batches of nominally similar material must be expected to have different toughnesses and distributions thereof. If a rigorous analysis of the variations in toughness of a particular material is required, it is apparent from the above firstly that tests must be conducted on one batch or charge of material, secondly that they ought, for the sake of simplicity (to avoid the necessity to calculate the effect of specimen-size for each result), to be conducted on a set of specimens all of the same size and which must be cut in the same orientation, and thirdly that the parameters for the distribution of this one batch of material are not reliably applicable to any other batch of material. In the majority of instances no such large consistent sets of data will be available. Instead several small sets of data will have to be treated, and then one of several possible approximate approaches, the choice of which may be dependent somewhat on particular criteria, must be applied. A number of examples are dealt with in detail below. The theory is covered first in the present paper as background to the more practical methods which are described afterwards.
THE
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D. J. NEVILLE
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2. THEORETICAL
BACKGROUND
It is beyond the scope of the present paper to deal with the theoretical background in great detail, but the different types of theoretical analysis which deal with the distribution of toughness of materials are discussed briefly in order to throw some light on the problems that are encountered in consideration of the statistics of fracture. 2.1 Previous work It has been attempted to apply laws of mixture to the toughness of two-constituent metals (e.g. dual-phase ferrite-martensite steels)[49]. There are three problems with such an approach: firstly the effective toughness of the individual constituents may be different when the constituents are in situ in the compound material due to, for example, residual stress; secondly, if GIc, J,, and crack-opening displacement for a mixture are linearly related to the volume fractions of the constituents then the same cannot be said for K,, since G,,, J,, and crack-opening displacement are related to the square of K,, rather than linearly related to it; thirdly averaging G,, over a fracture surface or along a crack-front will be misleading since the failure of the weakest part will very probably result in total failure rendering such averaged properties meaningless. Indeed it is mistaken, as will be seen, to predict a single value of toughness for inhomogeneous materials because the presence of two or more constituents or other forms of inhomogeneity in materials does not generally lead to some average or extreme value of toughness, but rather to a distribution of values that must be described, modelled and explained in terms of statistics[3, lo]. Only in the extreme case of infinitely wide specimens will an inhomogeneous material exhibit single-valued toughness, and this is due to the infinitely long crack-front being certain to cut through a region containing material of minimum toughness. Even in the case of two distinct constituents, plastic interplay between the two constituents and variation of morphlogy from place to place ensure that a distribution of toughness results as opposed to two distinct values of toughness corresponding to the toughnesses of the two constituents[lO]. Several probabilistic theories of fracture have previously been published. Those of Curry and Knott[ 111, Evans[ 121 and Wallin, Saario and Tiirriinen[ 131 start out from a distribution of defectsizes to predict toughness, but all avoid the inevitable distribution of toughness by assuming, in effect, a particular value of cumulative probability to obtain a single value of toughness. Wallin et al. even go to the extent of using a temperature-dependent surface-energy term in order to achieve a reasonable prediction of the variation of toughness with temperature. Working on brittle materials in tension Jayatilaka and Trustrum[14] used a particular flawsize distribution to obtain a statistical distribution of failure stress equivalent to a two-parameter Weibull distribution, but found in practice that the effect of volume was not predicted precisely by means of the Weibull function; they extended this work to investigate other flaw-size distributions concluding that the distribution of failure stress is insensitive to the distribution of flaw size[lS]. Landes and Shaffer[16] explained the variation in the toughness of their rotor steel by means of a sampling argument, but applying the two-parameter Weibull function to their results does not, however, give a reliable fit[ 171. It is appropriate at this point to describe the Weibull function briefly since it is often mentioned in this paper. This function has three parameters each of which has a distinct role: the shape parameter, c, determines mainly the skewness of the function (whether the majority of values are above or below the mean); the scale parameter, b, has a determining influence on the spread (standard deviation) and mean of the function; and the datum parameter, a, which is the minimum value of the distributed variable, X: F@)
=
1
_e+(.Y--ni/hY.
(1)
Often the datum parameter, a, is ignored (set equal to zero) and then the function is in its familiar two-parameter form. For shape parameter, c, greater than 1, the mean of the distribution is always slightly less than the scale parameter, h, and when the shape parameter (sometimes called the “modulus”) is greater than 3 it has a very limited effect on the standard deviation of the function which is proportional to the scale parameter, h. It is therefore an error to treat the shape parameter (modulus)
Statistical analysis of fracture toughness data
145
as an inverse measure of standard deviation (spread) for the Weibull function as shown by the following example: Distribution 1: a = 0, b = 1, c = 3; standard deviation = 0.3, Distribution 2: u = 0, b = 4, c = 6; standard deviation = 0.75.
Wallin has also produced an analysis of variations in toughnesss[ 181 which serves to highlight many of the problems which can arise. He assumes the relevance of an “active volume” (in this case wedge-shaped and extending from the crack-tip to the edge of the plastic zone) to the determination of the probability of failure and yet invokes a weakest-link principle (meaning that only a very small part of the supposedly active volume is active in the failure). Further, application of this particular active volume implicitly assumes that fracture is unlikely to be initiated in the zone of very high stresses and strains next to the crack-tip. His mathematical derivation leads to a two-parameter Weibull function for toughness but is based on a mathematical procedure which is thought to be incorrect for the following reason. In considering infinitesimally thin layers of material within the active volume it is necessary to contemplate the failure of infinitesimally small parts of carbides which is difficult to justify in physical terms; if, however, the carbides within the active volume were considered individually as whole particles, then the mathematical approximation which Wallin uses ([ 1 -xl” _ e-““) t o reach a Weibull function would no longer be valid. (It can be shown that the failure of this approximation means that toughness is not distributed according to the two-parameter Weibull function for stress-conrolled failures of the type which Wallin considers, and that, if this approximation were correct, that toughness would indeed be distributed according to the two-parameter Weibull function for all materials). His assumption of an arbitrary value for minimum toughness (the third, datum parameter of the Weibull function) of between 5 and 15 MPafi brings the largest sets of data which he uses as evidence for his theory nearer to agreement with his assertion that the only appropriate shape parameter for the Weibull function when applied to toughness is four. However, the Weibull distribution does not fit these large sets of data at all well when the shape parameter is four[17]; the major reason for this is that toughness is (according to the new theory[3, 191 and to a method using the three-parameter Weibull function [3, lo]) distributed such that the distribution is alwaysposirively skewed towards low values (majority of values below the mean), whereas with the shape parameter equal to four the Weibull distribution is skewed in the wrong direction towards high values (majority of values above the mean). Other problems with Wallin’s analysis are that he associates the degree of spread of the distribution with the shape parameter of the Weibull function (which is mistaken, as shown above). He also states that inhomogeneity in the material affects only the mean value of the distribution; this is in contradiction of his own statement that the “slope” (shape parameter) describes the magnitude of “scatter”, since the spread (standard deviation) of the Weibull distribution is proportional to the mean value for a given shape parameter, both spread and mean being proportional to the scale parameter, b. Furthermore, without any effect of inhomogeneity, it is difficult to see any justification for his use of a weakest-link principle because the material would be equally weak everywhere. Inhomogeneity is rather to be seen as the principal underlying cause of the distribution of toughnessC2, 3, lo]. If the Weibull function is to be used for the description of distributions of toughness it is necessary that the three-parameter form be used in order to obtain a reasonable fit[3, 10, 171. In that case it is required to invoke a distance-criterion for failure at sharp cracks[20] in order to introduce the third (datum) parameter, a, the value for minimum toughness[lO]. However, the three-parameter Weibull distribution cannot fit data-sets from toughness properly (see Figs 1 and 2). The data in Fig. 1 are from a quenched and tempered low-alloyed structural steel (A533B) on the lower shelfC21] and were obtained using compact tension test-pieces. This misfit (which is to be seen for sets of data and materials other than those of Fig. 1) is due to the assumptions made by Weibull in his original derivation of his function[22]. The only merit[23] or Weibull’s choice for the material function, n(c) = ((x-a)/b)‘, is that it is the simplest possibility. Nonetheless, the Weibull function is extremely flexible and thus can be used in circumstances for which no specifically suited distribution function exists. That the Weibull function has an ever increasing hazard function for shape parameters greater than one makes it especially suitable for problems involving decay with time (e.g. creep, fatigue). A new statistical theory[3, 191 based on the conditions of stress and strain at crack-tips provides proper fits to toughness data with only two parameters (no flexibility) and is summarized
D. J. NEVILLE
146
‘I
10 (K,, -K Ic
Fig. 1. Three-parameter
Weibull
100
rmnimum )Mh./iE
plot of K,, for a quenched
subsection derivation involve concerning or non-linearly elastic-plastic, etc.) materials. may for certain configurations.
and tempered
low-alloyed
steel[21].
193). and derivation of materials (they applicable of all stresses, awkward
191 sharp
passes The implication amount of material piece is at a fixed
a series
states[24] to the following region that, the stresses do not change, of material to any particular So the amount at or above given or strain to the square opening since (about displacements is of constant shape in two dimensions load. Since amount loaded given or strain would to calculate, to which be linearly to it. Sampling, as g, 4, or COD2, g(s), given
(S/W
(2)
where the scale parameter, B, normalizes S (and is actually the median value of s), and the shape parameter, D, allows the shape of the distribution to change with temperature as it is observed to do[17, 191 even for a single material. The shape parameter for a given material is also expected to be different for failure at microcracks in simple tensile tests than for failure at microcracks near a macrocrack because of a difference in the distributions of stress in these two cases. It can be shown[3, 193 that the cumulative probability of failure at the tip of a sharp crack is given by:
LOG (TOUGHNESS 1 Fig. 2. Systematic
misfit of the points
and the straight
line in Fig. 1.
Statistical 99.
1
analysis
of fracture
I I11111,
I
toughness
I I IIIII,
data I ,,,,I-
141
989590-
I
1
q‘?K&
1 I ,,,,I_ 100
(MPa+hii)4
Fig. 3. Plot using the new functionC3, 191 of the same data as in Fig. 1. Sampling is here K,, to the fourth power. The major deviation is no longer present. q is a constant which shifts the data uniformly parallel to the abscissa (a condition for using the data was not to reveal the values).
F(S)
go 1+g(S)
=
(3)
a function with only two parameters, B and D. The improvement in fit achieved by the use of the new distribution function can be seen by comparing Figs 1 and 3. A very good fit is also given in the case of ceramics. The data for this example are fracture toughness results from bend specimens with very fine slots (instead of fatigue precracks)[26]. Figure 4 shows the fit afforded by the new function for this set of data from RefelSic; the shape parameter, D, ( = slope of the plot = 5.34) is different from that of the steel (Fig. 3, slope, D = 2.20), and the distribution for the ceramic is more symmetrical but still somewhat skewed towards the lower values. Figure 5 shows the Weibull plot of the data in Fig. 4 using the Weibull function with only two parameters since for nearly symmetrical distributions a third parameter is superfluous; (a three-parameter plot shows the systematic misfit shown in Fig. 2). It should be noted in any event that three parameters make a function so flexible that “fits”, though perhaps acceptable in terms of statistical tests, might be quite meaningless from a theoretical, (and in this instance fracture-mechanical) point of view and are thus better regarded as non-theoretical descriptions than proper, theoretically based fits. It can be seen from comparison of Figs 4 and 5 that the new function (eqs 2 and 3, [3, 191) provides a proper fit, indicated by the coincidence 99s
100
I
I I,,,,,
(MPafi14
1000
KP, Fig. 4. Plot using the new function
of K,, for a ceramic material (Refel-Sic) the length of the plot.
[26]. The fit is good throughout
148
D. J. NEVILLE
!fd._J 100
Fig. 5. Weibull
1000
(MPa Jiif
plot of the same data as in Fig. 4. No fit exists, the points the plot.
lying on a curve throughout
of the points and the straight line over the whole range in Fig. 4, where the Weibull function cannot, as shown by the very distinct curvature of the plot in Fig. 5 (raising toughness to the fourth power makes no difference in the case of the two-parameter Weibull function). The new function[3, 193 may also be applied to the failure stresses from tests on pieces that contain many crack-like defects instead of one large crack, but in this case the load at failure to the fourth power rather than sampling, S, is plotted since the calculation of sampling would require knowledge of the size-distribution of the defects and their density in the material. Data[27] from the bursting pressure of 85 identical test-pieces made from filled epoxy resin are shown in Figs 6 and 7. The stress distribution in these tests was always of the same shape irrespective of load so that no integration over the volume of the test-pieces was necessary and the bursting pressures could be plotted directly. Figure 6 shows the better fit of the new function and Fig. 7 shows the curved plot of the Weibull function. Once again the general curve of the Weibull plot indicates that the Weibull function does not give a fit, whereas the new function gives a reasonable fit. There is one major deviation from the main trend in both of these plots which is probably due to an irregularity in the application of load (resulting from the complex loading arrangement), but the plot using the new function is clearly straighter overall. Rarely in practice will such good sets of data be available. It therefore remains to see how to apply the theoretical understanding obtained from large, consistent sets of data to the relatively 99 5
, I II,
l
99 l
98
to
2 95 -
2 80 c 90 2
70
--
*+
0
I'
F - 1: I' 4 "2: sro1 5 g 50
-
5
2 -
l
4 l 05 105
I (BAR’)
406
2.406
(BURSTING PRESSUREI4 Fig. 6. Plot using the new function of the bursting pressures of powder-filled epoxy resin[27]. The plot is generally straight except for one excursion which is possibly due to the method of application or reading of load during the tests.
Statistical analysis of fracture toughness data
149
051 405
406
( t3AR4) (BURSTING
2 406
PRESSURE)’
Fig. 7. Weibull plot of the data in Fig. 6. In addition to the excursion observed in Fig. 6, the plot has a general curvature.
small, less consistent sets which are often encountered in practice. The next section deals with this problem. 3. PRACTICAL
APPLICATION
The following cases are dealt with since they represent the types of non-ideal situation which may arise: (1) Data from one charge of material is for some reason bimodal (split into two distinct populations with similar but not identical distributions). (2) Several data-sets are available; this is similar to (l), but the data-sets are not necessarily from one charge of material and the resulting distribution may be bimodal or even multimodal. (3) Too little data is available: no data-set of less than 7 elements is of any use whatsoever for statistical purposes because the uncertainty in the estimation of parameters is so great as to render any evaluation meaningless. Confidence improves strongly as the number of elements increases from 7 to 12, and the improvement carries on, although more slowly, as the number of elements is further increased. 3.1 Bimodality Figure 8 shows the appearance of a second, bimodal data-set from the Refel-SiC ceramic mentioned above[26], the bimodality being present presumably because of some inconsistency in the preparation and/or testing of the specimens since all the material came from one charge. If the specimens from which these results were obtained were available, it might be possible to separate the two populations with clarity and confidence. However, in the absence of that information an approximation must be made which offers another example of a possible procedure. Using the information from Fig. 4, (slope = 5.34, median value = 315 (MPafi)4) and the fact that the ratio of the specimen-widths for the two sets of specimens tested by Tradinik er al. was 1.75, a theoretical line can be drawn for the bimodal set of data. The ratio of the specimen-widths is the ratio of the lengths of the crack-fronts, and the volume subjected to greater than a given stress is proportional to crack-length. Thus, to compare the results from one set of specimens to those from a set of specimens n times wider, sampling (K& etc.) for the wider specimens must be multiplied by n.) (This method has been found to give an excellent prediction of the effect of specimensize[19], despite that the prediction is dependent on a fit which uses only two parameters.) Figure 9 shows this theoretical line as well as a tentative division of the data into two parts. The division
150
D. J. NEVILLE
Fig. 8. Weibull
plot of further
data from Refel-SiC[26].
The bimodality
of the data is quite clear.
is tentative because it could be that the distribution of the consistent data-set is different from both those of the bimodal set; again, more information is required in order to know. The important points brought to light by these sets of data from Tradinik et al. are firstly, that data-sets must be checked to see whether they are consistent within themselves, and secondly that bimodal data-sets can only be split with confidence if more information is available than the simple data themselves. 3.2 Mixed data-sets The first possibility for mixed data-sets is that all the sets come from one charge of material. This example uses data from a martensitic rotor steel, 25 NiCrMo 14 5, [28]. Toughness (J,,) was measured at eight different distances from the centre of a forging 1.2 m in diameter, and there was no general trend to be seen in toughness between the outer five positions. The data from these positions are therefore treated as if from one charge of material. In Fig. 10 these fifteen points are plotted as sampling (in this case &), using the new function, and a good fit is obtained. The slope of the line (shape parameter) is 11.8 which is perhaps rather high for a steel, a slope of 1 to 2.5 being expected on the lower shelf of toughness, although no data-sets from the upper shelf have previously been treated with the new function. This might be explained in two different ways. 99 98
_
LINE FROM GOOD DATA
1 102
hlPav?ii)4
IO3
K?C of the data in Fig. 8, making use. of the information gained Fig. 9. A tentative separation specimens of different size but taken from the same material (see Fig. 4).
from a set of
Statistical
analysis
of fracture
toughness
data
151
2 (kJ/m2j2
Jh2 Fig. 10. Plot using the new function of J,, for rotor steel 25 NiCrMo 14 5[28]. The high slope indicates a rather symmetrical distribution which may be typical of toughness on the upper shelf, but could be due to errors influencing the distribution of toughness as well as material inhomogeneity.
(1) The mean of the results is 175.2 kJ/m2 and the errors in the experimentally obtained values of toughness will be about + 5% or about f 9 kJ/m2 in this case. The total range of the results is only 44 kJ/m* or about twice the range of error, and so the errors (which are symmetrically, normally distributed) are large enough that they tend to symmetrize the distribution of toughness which would otherwise be expected to be more skewed; that is a transitional distribution results. When two (or more) effects are working simultaneously (here errors and material variation) there is a tendency towards a symmetrical distribution regardless of the shapes of the individual contributing distributions (central limit theorem of statistics). It could also be that the five groups of three data are not from adequately similar material which would result in a distribution tending towards a normal distribution, and this would again be due to the central limit theorem. (2) It may be that toughness is indeed more symmetrically distributed on the upper shelf than on the lower shelf, the mechanisms of crack-propagation being totally different. The second possibility for mixed data-sets is that the individual sets come from similar material but not from the same charge. The data for the example are again from the martensitic rotor steel 25 NiCrMo 14 5[29], and are in two small sets, one of eight elements and the other of six. The set of eight is spread somewhat wider than would be expected from errors alone, and eight elements is just enough for a plot, see Fig. 11, the slope (shape parameter) of which is 8.4. The same considerations apply to this value of the shape parameter as to that of the previously considered data[28]. Either toughness is distributed more symmetrically on the upper shelf, or errors are playing a role. Were the smaller data-set sufficiently large for a plot it should not be plotted even 95
I
2
90
z g
00
-
$ m 70
-
/’
I
IIIII
I
I
l
-
0
/I
E
I
10 i
5 109
,
I (MPa fi)4
IIll 40’0
9
Fig. 1I. Plot using the new function of K, for rotor steel 25 NiCrMo 14 5[29]. Either toughness is fairly symmetrically distributed on the upper shelf or neither errors (symmetrically distributed) nor material inhomogeneity (heavily skewed) dominate, and an intermediate distribution of middling shape results.
152
D. J. NEVILLE
2 109
Fig. 12. Mixing additional
(MPa4ii14
K;
40’0
data into that in Fig. 11 increases symmetry due to the central and thereby increases the slope for the new function.
limit theorem.
so, because the overwhelming source of the spread for this data-set is experimental error, and a plot would lead to a misleadingly symmetrical distribution. (It is also safer to assume a positively skewed distribution than a symmetrical one for reasons explained in section 3.4). Mixing the two sets of data together tends to increase symmetry due to the central limit theorem. For the new function as the slope (shape parameter, D) becomes larger so the distribution becomes more symmetrical. See Fig. 12, the slope is now about 10.2 as opposed to 8.4 from Fig. 11. Where errors outweight the effects of irregularity in the material or are at least large enough to affect the shape of the distribution, then it is possible to assume a shape parameter (for the new function), the choice of which is based on experience with results from similar material which were not seriously affected by error. In addition the mean value must be estimated in the normal way (sum divided by number of results). 3.3 Too little data Three possibilities exist when two little data is available, the available sets are all too small individually to be worth plotting or nothing is known from experience about the likely value of shape parameter. Firstly, all the data or data-sets may be mixed together to form one lot. The inevitable result of this treatment is that all the data together conform, in accordance with the central limit theorem of statistics, to a Gausssian or normal distribution. This would lead to a non-conservative estimation of the probability of failure (as will be explained in Section 3.4) and is therefore not a sensible procedure. Secondly, it is possible, when nothing better is available, to make plots for all the small (< 7 elements) data-sets to hand an take the averages (or even the minimum values) of the means and the shape parameters obtained as representative of the data generally. This will tend to result in a low (conservative) value for the overall mean, but the final shape parameter may not be appropriate. Thirdly, possibly the best procedure in such circumstances is to choose a conservatively skewed shape and calculate the mean of the available values. This mean will always tend to be low because a limited-number of values conforming to a positively skewed distribution will often tend to have a mean that is less than that of the distribution and it so happens that this tendency is particularly strong for the new function. 3.4 Sqfety in skewness When a limited number of values are taken from a population which is distributed according to a skewed distribution the mean and standard deviation will approximate those of the population. However, in some cases these approximations may systematically err on one side of the values for the population. For instance, for the new functionC3, 191 (eqs 1 and 2) with shape parameter equal to four, the mean for the function (or an infinite population) is 1.12 times the median value (scale parameter) but the mean as estimated by 10 values picked randomly from the population
Statistical
10
analysis
of fracture
toughness
data
NORMAL
CURVE
153
SELECTIONS
FUNDAMENTAL DISTRIBUTION
1000
0
FOR
_
SELECTIONS
1
2
VARIATE
Fig. 13. Demonstration of the conservatism of the new function compared to normal distributions for 10 and 1000 results. The new function gives a higher likelihood of failure having occurred for low values of toughness.
is expected to be only 1.07 times the median, and only 1.08 times for twenty results. Thus the new function affords a conservative estimation of mean. Similarly, the standard deviation is very much less for limited numbers of values than the standard deviation of the population from which the values were drawn and which is distributed according to the new function. The significance of these under-estimations of both mean and standard deviation is that if a normal distribution is assumed it will give serious under-estimations of the probabilities of failure at low values of load (see Fig. 13). Shown in Fig. 13 are a curve for the new function with a shape parameter of 4, and curves for assumed normal distributions based on the means and standard deviations of 10 and 1000 selections from the distribution represented by the curve for the new function. Thus there is safety in the skewness of the true distribution of toughness and danger in the assumption of a normal distribution. From the immediately preceeding considerations it is apparent that, when using the new distribution on a set of toughness-data, it is safer to err towards lower values of the shape parameter and towards lower values of the scale parameter (proportional to the mean). It is also noted that the mean of the Weibull distribution is rather accurately estimated by only a few values, meaning that the estimate is not conservative. Further, if a three parameter Weibull distribution is used to describe toughness data then the resulting estimates of probability of failure at low values will be non-conservative; indeed the estimate is zero below the datum parameter (which is, in any event, not easy to estimate for the best description of the data). This is ameliorated to some extent by the necessity to fulfill a criterion of distance before a metal can fail from a crack-tip[20] (the same effect adding to the conservatism resulting from the use of the new function). 4. SUMMARY Statistical analyses of toughness data should be done graphically using probability paper corresponding to the new function (in Fig. 14). The procedure is as follows. The values of sampling, S, are straightforwardly calculated. Then the N values for S to be plotted are each given a rankorder number, the smallest result having rank-order number 1, and the largest rank-order number N. Then the median rank for each result is calculated by means of an equation which gives a very good approximation to the actual median ranks (which would otherwise consume a great deal of time in calculation): Median rank = (n - 0.3)/(N+ 0.4) where N is the number of results to be plotted and n is rank-order number. are values of cumulative probability against which the values for sampling, probability paper (see Fig. 14). The slope of a plot on this probability paper parameter, D, and the median value (scale parameter, B) is read off probability = 50%.
(4) These median ranks S, are plotted on the is equal to the shape opposite cumulative
D. J. NEVILLE
154
SAMPLING Fig. 14. Probability
paper
/ LOAD
for the new function.
The new function is generally conservative in its estimates of failure-probability and is easy to use (no third parameter is required and thus need not be estimated), and graphical methods bring to light any problems involving irregularities in the data (e.g. bimodality, etc.). The analysis used must depend on the nature, quantity and quality of the data available, and ought to be chosen according to the purpose behind the analysis but always with a view to conservatism and safety. A~knoM,ledRement.s-Dr to the author. Thanks
H. Beer is gratefully acknowedged for making the data from the powder-filled also to Dr C. Wtithrich for helpful discussions.
epoxy resin available
REFERENCES [I] 123 [3] [4] [S] [6] [7] [S] [9] [IO] [ll] [12] [I31 1143 [15] [I67 [17] [18] [19] [20] 1211 [22] [23] [24]
W. J. Jackson and J. C. Wright, Metals Technol 4, 425 (1977). D. J. Neville and J. F. Knott, J. Mech. Phys. Solids 34, 243 (1986). D. J. Neville, Ph.D. thesis, University of Cambridge (1985). P. Stratman and E. Hornbogen, Proc. ICSMA 4 p. 607, Nancy (1967). E. Hornbogen and M. Graf, Acla MetaN 25, 877 (1977). M. Graf and E. Hornbogen, Acra Mefull 25, 883 (1977). K. Friedrich, Prog. Colloid Polym. Sci. 64, 104 (1978). E. Hornbogen and K. Friedrich, Proc. Inf. Conf. Pure appl. Chem., I.U.P.A.C. (vol. 3, p. 1422) (1979). E. Hornbogen and K. Friedrich, J. Mater Sci. 15, 2175 (1980). D. J. Neville and J. F. Knott, J. Mech. Phys. Solid.7 34, 256 (1986). D. A. Curry and J. F. Knott, Merail. Sci. 31, 341 (1971). A. G. Evans, Metal/. Truns. 14A, 1349 (1983). K. Wallin, T. Saario and K. Torriinen, Metall. Sci. 18, 13 (1984). A. de S. Jayatilaka and K. Trustrum, J. Maler. Sci. 12, 1426 (1977). K. Trustrum and A. de S. Jayatilaka, J. Mater. Sci. 18, 2765 (1983). J. D. Landes and D. H. Shaffer, ASTM STP 700, 368 (1980). D. J. Neville and J. F. Knott, J. Mech. Phys. Solids 34, 278 (1986). K. Walhn, Engng Fracture Mech. 19, 1085 (1984). D. J. Neville, to be published by the Royal Society (1987). R. 0. Ritchie, J. F. Knott and J. R. Rice, J. Mech. Phys. Sol& 321, 395 (1973). T. J. Williams, Rolls Royce and Associates, private communication. W. Weibull, A Statistical Theory of the Strength of Materials, Ingeniorswetenskapsakademiens Handhngar, Stockholm: Generalstabens Litografiska Anastalts Forlag, 1939. W. Weibull, J. crppl. Mech. 18, 293 (1951). R. M. McMeeking, J. Mech. Phxs. Solids 25. 357 (1977).
No. 151.
Statistical
analysis
of fracture
toughness
data
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J. R. Rice and M. A. Johnson, in Inelastic Behaviour of Solids (Edited by Kanninen et al.), p, 641 McGraw-Hill, New York (1970). [26] W. Tradinik, G. Popp and R. F. Pabst, Z. Werkstofftech 13, 254 (1982). [27] H. R. Beer, Brown, Bovri & Cie., Central Laboratories, Baden, Switzerland, communication of unpublished results. [28] J. Albrecht and C. Wtithrich, Z. Werksfofftech, 13, 96 (1982). [29] J. Albrecht, Brown Boveri & Cie., Baden, Switzerland, COST 501, Project CH-11, final report: KLR 85-167 C. [25]
(Received 9 September
1986)