Analyzing Statistical Variability of Fracture Properties T.A. BISHOP, A.J. MARKWORTH, and A.R. ROSENFIELD A recently developed statistical analysis is applied to the estimation of lower-bound Charpy V-notch impact energies. Scatter of the impact-energy-absorption data, at any given temperature, is described using a two-parameter Weibull distribution, with the parameters allowed to vary with temperature. Values for the parameters are obtained using standard linear-regression techniques. As a result, the analysis naturally yields the temperature dependence of the p th percentile of the impact-energy distribution. Data with both large and small scatter are analyzed using this procedure.
I.
INTRODUCTION
THEextreme scatter in impact-energy-absorption data for steel tested within the ductile/brittle transition region is well known. Figure 1 shows an unusual, but not rare example (see Reference 1). These four Charpy V-notch samples were taken from the same plate and impacted at the same temperature. Yet the absorbed energy varied by more than a factor of 20. Such a large difference cannot be explained by experimental errors; it reflects a true material variability that needs to be taken into account in specifying load and operating temperatures. The Weibull distribution is a natural choice for dealing with variability in fracture data, because it long has been applied to brittle materials (for a review see Reference 2) and because it is quite flexible. Mathematically, if Y is an observation taken at random from a two-parameter Weibull distribution, then for any value y the probability that Y ~ y is given by the function F(y) = Prob[Y ~ y] =
- e -~/~)~
y ~ 0
where a > 0 and/3 > 0. The probability function F(') is referred to as the cumulative-probability-distribution function (CDF) for the Weibull distribution with parameters a and/3. Figure 2 shows that the Weibull distribution provides an adequate description of the data from Figure 1. In the more usual case, the transition is spread out over a wide temperature range so that the scatter at any given temperature is less than that illustrated by Figure 1. However, the problem still remains of incorporating the scatter into a safety analysis. The present paper uses Charpy V-notch impact-test results to illustrate a new statistical approach to the analysis of fracture-toughness data for steels in the ductile/brittle transition region. In the cases shown below, a full energy/ temperature curve is available to specify the temperature above which the impact energy exceeds some given value. While the engineer's judgment is usually used to draw a curve through the data points in such cases, in some applications statistical procedures 3-6 are used. If there are a large number of points (typically on the order of 10 to 20) and little scatter, there is no problem in defining a transition temperature. On the other hand, often there are few data points and/or large scatter. In those cases, significant ambiguities can arise. T.A. BISHOP, Associate Section Manager, A.J. MARKWORTH, Senior Research Scientist, and A. R. ROSENFIELD, Research Leader, are all at Battelle-Columbus, 505 King Avenue, Columbus, OH 43201. Manuscript submitted August 31, 1982. METALLURGICALTRANSACTIONS A
Determination of a full curve is not always mandated by codes and standards. For example, Paragraph NB2331 of the ASME Boiler and Pressure Vessel Code requires that only three samples be tested and that all three exceed a given energy at the test temperature. Alternatively, ASTM A370 for steels requires both that the energies absorbed by three samples all exceed some given toughness and that their average also exceeds some given (higher) toughness. It is not clear how the transition temperature determined by the above two methods relates to that determined by consideration of the full curve. The analysis in the present paper may provide a means of relating the two evaluations.
II.
RECENT METHODS OF ANALYSIS
Prior to the research described here, two techniques have been used to describe Charpy-impact data quantitatively, and both have been applied principally to nuclear-reactorpressure-vessel steels. The fundamental approach to data analysis, in these two cases, differed substantially from one another, and each is summarized below in order to provide perspective for the present work. One of these methods was developed by Oldfield, 3'4'5 and was applied, for example, by Wullaert et al.,6 to the analysis of fracture-toughness data for irradiated and unirradiated pressure-vessel steels. It should be noted, however, that a very similar approach to analyzing Charpy-impact data for certain metals was developed by Chun. 7 Application of Chun's method to specific examples was carried out by Koons, 8 who also demonstrated the manner in which variances of certain of the associated parameters could be calculated. The discussion presented below is, however, limited to the method as developed by Oldfield. The essence of the Oldfield model lies in the use of the hyperbolic tangent function to describe the generally sigmoidal variation of Charpy V-notch-test properties with temperature. One form of the regression equation used by Oldfield 3 and Chun 7 is Y = A + B tanhX
[2]
where X = (T - 1"o)/C
In Eq. [2], Y is a property obtained from the Charpy V-notch test ( e . g . , the impact energy absorption, percentage of shear fracture, or lateral expansion), T is the temperature, and A, B, To, and C are the regression parameters that are determined, via nonlinear analysis, by minimizing the sum of squares of deviations of experimental data about the Y-axis. Fracture toughness also has been analyzed by Eq. [2]. VOLUME 14A, APRIL 1983--687
Fig. 1--Charpy specimens of experimental steel 293 (0.10 C, 0.54 Mn, 0.76 Cr) broken at - 5 0 ~ in joules, are, left to right: 14, 16, 163, and 296, respectively.
Fig. 2--Impact energy (steel 293, as-rolled; tested at - 5 0 ~ as a Weibull distribution.
expressed
One characteristic of Eq. [2], which is applicable to the entire range of test data, is that both the upper- and lowershelf energies, A + B and A - B, respectively, are independent of temperature. Another is that it displays the symmetry property of remaining unchanged after two reflections, i.e., in the lines X = 0 and Y = A, which can be expressed mathematically as
~
[Y(X) + Y(-X)] = A
[3]
The symmetry condition expressed in Eq. [3] follows, of course, from the assumed form of the regression equation, Eq. [2]; there is no physically based reason for assuming that the experimental data, to which Eq. [2] is fitted, would themselves exhibit such symmetry. A generalization of Eq. [2] was introduced by Oldfield5 to account for possible variation with X of the upper- and lower-shelf values of Y; this was given by
+ [B+ 688--VOLUME 14A, APRIL 1983
( - ~ - - f f ) X ] tanhX
[4]
Impact energies,
where a and/3 represent additional regression parameters. With Eq. [4], the upper- and lower-shelf values of Y are given by A + B + /3X and A - B + aX, respectively. If a and/or/3 is taken to be non-zero, the symmetry condition, Eq. [2], is, of course, no longer valid. In his model, Oldfield4'5 assumed that the variance of the data had two values, i.e., one at the lower shelf, defined by him as data within the temperature range T < To - C, and another for the remaining data. Detailed examination of residuals was carried out by Oldfield4 for fracturetoughness-vs-temperature data, and he found that the distribution of "normalized" toughness values about the regression equation was appropriately described in terms of the Gaussian distribution except at the lower shelf, at which a more highly skewed distribution was found that reflects the fact that negative toughness values do not exist. The other approach, which differs considerably from that of Oldfield, was developed by Stallmann. 9 The Stallmann model, like that of Oldfield, 3'4 is based on a regressionanalysis approach. However, unlike that of Oldfield, his analysis is both linear and multivariate. To illustrate Stallmann's approach, let Yi be the value of the dependent variable (impact energy, lateral expansion, etc. ) for the ith specimen, and suppose that, for this same specimen, there are n independent variables Xik, k = 1, 2 . . . . . . . . n (test temperature, irradiation temperature, fluence, chemical composition, etc.). The various Xik variables could be taken as functions (e.g., logarithms) of the original variables, if appropriate. The variables were combined by Stallmann in a linear fashion, i.e., Yi=C0+
- x/)
~
[5]
k=l
where ~ and X/k are adjusted values (that is, true values, determined from the regression analysis) of Yi and X~k, respectively, X ~ is a nominal value of variable Xk, the Ck parameters (k = 1, 2 . . . . . n) are regression coefficients, with Co, an absolute constant, being the value of the dependent variable when all the independent variables are at their nominal values. It was pointed out by Stallmann9 that a linear approximation (such as Eq. [5]) of any nonlinear METALLURGICALTRANSACTIONSA
function is locally applicable if the data are restricted to a region within which the linearity approximation is valid. The method applied by Stallmann to determine Co, C1, . . . . . C, consisted of use of an iterative procedure to minimize the following sum of squares of residuals, denoted as E2:
O'y
k= 1
O'k
J
the summation over i being carried out over all the specimens, with ~ and ~ being the variances associated with variables Y~ and Xik, respectively (which are calculated as part of the computational procedure). It can be seen, from Eq. [6], that no distinction is made between dependent and independent variables in the minimization process. Insofar as data selection is concerned, Stallmann suggested that only those data be used that are pertinent to the parameters being evaluated. Thus, determination of the transition temperature should be based only on transitionregion data, and likewise, the upper-shelf energy should be based only on upper-shelf data. He indicated that a linear model (which, as also was noted above, may involve a nonlinear physical model) is appropriate for this purpose. He noted that, since any curve is close to being linear near the inflection point, the linear approximation of the Charpy impact-energy/temperature curve is appropriate, with random variations in the data being likely to be significantly greater than possible nonlinearities. Another approach to analyzing Charpy impact data is presented in the following section. As described therein, this method has a number of advantages including the facts that (1) it is used to obtain directly the p th percentile curve of the Charpy data as a function of temperature, with the value of p being chosen by the user, and (2) it is based on standard linear-regression techniques.
F(qp) = p
[7]
Using Eq. [1] to solve Eq. [7] for qp, it follows that qp = /3[-ln(l - p)]V~
[8]
In the case of Charpy data, there is a distinct Weibull distribution associated with each temperature, T, since the material properties are dependent upon temperature. Therefore, we are considering a family of Weibull distributions over the range of the temperature T. The Weibull distribution defined by Eq. [1] is easily generalized to account for the temperature dependence by allowing the parameters a and/3 to be functions of temperature. Thus, the Weibull model associated with the Charpy data is given by:
y<0 Fr(y) =
' - e-~/B(r~
y >- 0
where a(T) > 0 and/3(T) > 0. The p th percentile also becomes a function of temperature and is given by:
qp(T) = /3(T)[-ln(1 - p)]V,m
[10]
The function qp(T) is interpreted as follows: For any fixed temperature, T, the probability that a Charpy-energy measurement will fall below qp(T) is p. Given this formulation of the model describing the Charpy data, the lower-bound curve can be rigorously defined in terms of the pth percentile curve, qp(T). For example, if specifications were to require the establishment of a lowerbound curve that should be exceeded by 95 pct of the Charpy data at any fixed temperature, T, then the lowerbound curve would represent the 5th percentile curve associated with the Weibull distributions. The lower-bound curve would then be equivalent to Eq. [10] with p set equal to 0.05, and it would be denoted by
U*(T) = /3(T)[-ln(0.95)] v~m =/3(T)[0.0513] ':~(~ III. THE STATISTICAL DEFINITION AND ESTIMATION OF THE LOWER-BOUND CURVE
A.
Rigorous Definition of the Lower Bound Curve
The statistical procedure developed by Bishop 1~provides a method for analyzing temperature-dependent fracturetoughness data. It calculates the median and all percentiles as functions of temperature. In the analysis discussed below, attention is focused on the "lower-bound" Charpy-impact energy, arbitrarily defined as a 5 pct failure probability, which corresponds to the fifth percentile. The procedure is used to determine a transition temperature, arbitrarily defined by a lower-bound energy of 41 J (30 ft-lb). The statistical procedure described in this section was developed based on the assumption that the Charpy data obtained at a fixed temperature follow a two-parameter Weibull distribution. The two parameters of the distribution are assumed to depend upon the temperature, T, and the pth percentile is directly calculated as a function of temperature. The lower-bound curve then is defined rigorously in terms of the pth percentile (in this paper p = 5) of the Weibull distribution. The p th percentile for the Weibull distribution defined by Eq. [1] is the parameter qp that satisfies the equation: METALLURGICAL TRANSACTIONS A
[11]
where U* is the impact energy corresponding to the 5th percentile. In practice, the two functions a(T) and /3(T) are unknown and have been estimated using polynomial approximations and the Charpy data. The additional benefit of using the Weibull model and the polynomial approximations is that they lead to a simple method of estimating a (T) and /3(T) (and, hence U*(T)) using standard linear-regression techniques. Alternatively, p in Eq. [10] can be set equal to 0.5. If this is done, the present analysis can be made analogous to the Oldfield method by assigning a hyperbolic tangent function to /3(T) or to the Stallmann method by making /3(T) linear over the restricted range of interest. Such assumptions would increase the complexity of the analysis since nonlinear regression techniques would have to be employed. The general steps in the estimation procedure are outlined in the next section. Complete details and a simulation study establishing the statistical properties of the estimation technique can be found in Reference 10.
B. Statistical Estimation of U*(T) Consider a set of Charpy-impact-test data {Yi, Ti} where Yi is the Charpy measurement taken on the ith specimen at VOLUME 14A, APRIL 1983--689
temperature Ti, i = 1, 2 . . . . . n. The development of the statistical-estimation procedure begins by recognizing that Yi can be characterized in terms of simple exponential random variables. If Yi is a Weibull random variable with CDF
y -> 0
[12]
then Y~ can be expressed as ri -- f l ( r i ) z i l/a(ri)
[13]
where the Zi are simple exponential random variables. Eq. [13] can be recast in the form of a simple linear regression equation by taking natural logarithms. Letting Yi* -- ln(Y/), we have: Y~* = ln(fl(T~)) + ~ T ) l n ( Z ; )
= ln(fl(Ti)) + el*
data. The first set of data was obtained on an experimental high-strength, low-alloy (HSLA) steel* and is an example *Heat 283 (0.10 C, 0.49 Mn, 0.83 Cr, 0.91 Ni, 0.20 Mo, 0.089 Nb), as-rolled.
of well-behaved Charpy-energy measurements with little scatter. The second data set was obtained on ASTM A508 steel used in Oak Ridge National Laboratory Experiment TSE611 and illustrates the case where there is significant scatter in the data. For both data sets, the statistical-estimation procedure discussed in the previous section was used to establish the lower-bound curve (arbitrarily chosen as the 5th percentile of the Weibull distribution) as a function of temperature. For purposes of illustration, the procedure was allowed to iterate nine times after the initial fit for each data set.
A. Analysis of Data for Experimental Heat 283 [14]
where e~* = l/a(Ti) ln(Zz) is the random "error" term in the model. Information about the function fl(T) is contained in the deterministic portion of Eq. [14], and information about a(T) is contained in the random portion of Eq. [141. The estimation procedure involves two steps. It begins by fitting the {11,..}data to Eq. [ 14] using a polynomial approximation for ln(fl(T)) in order to obtain an estimate of ln(/3(T)) and, therefore,/3 (T). The polynomial approximation for In(/3 (T)) allows for the use of standard linear regression techniques in the estimation process. Since the predicted values from that fit are estimates of ln(fl(T)), the residuals R~ = (observed Y~* - predicted Yi*) are approximately equal to ln(Zi)/ a(T~). It follows that lnlR, I ~ - l n ( a ( T ~ ) ) + ln(llnZ, I). Therefore, the residuals can be used to obtain an estimate of a(T). These estimates then are combined using Eq. [11] to obtain an estimate of qp(T) and hence U*(T). It can be shown that the function a(T) affects the properties of the error term in Eq. [14]. In the general case of linear regression, the expected value of the error terms are assumed to be zero, that is,
E (ei*) = 0
Table I presents the results of Charpy V-notch tests made on 16 specimens over a temperature range from - 7 3 to 100 ~ The logarithms of the Charpy-energy measurements are plotted vs temperature in Figure 3. Based on the shape of the plot in Figure 3, a third-degree polynomial was chosen to approximate ln(/3(T)). The logarithms of the absolute values of the residuals from the initial fit were then plotted as a function of temperature. Based on this plot, a first-degree polynomial was used to approximate - I n (a(T)). Although the procedure was allowed to iterate nine times, convergence was achieved after only two iterations. The results of the initial fit and the first three iterations are presented in Table II. Figure 4 is a plot of the Charpy-energy data vs temperature with the estimated U*(T) lower-bound curve superimposed on the data. Table III gives the estimates of the Weibull parameters a(T) and fl(T). These are characteristic of the dispersion and central tendencies, respectively. Specifically, fl(T) is the 63rd percentile and a is inversely proportional to the coefficient of variation, provided ot -----5.12 Therefore, large values of a(T) are consistent with fl(T) being not very different from U*(T), as is the case in Table III.
However, for Eq. [ 14], it can be shown that
E(ei* ) ~
1
2a(L)
Table I.
Therefore, a(T) introduces a negative bias in the estimate of/3(T). (It also affects the error variances; see Reference 10 for a more complete discussion of this point.) However, once an estimate for ct(T) is available, it can be used to correct for the bias in order to obtain a better estimate of /3(T). Therefore, the procedure is iterated using previous estimates of a(T) to adjust for bias. The iteration scheme is then continued until convergence in the estimate U*(T) is obtained. Complete details of the statistical model and the required statistical algorithms can be found in Reference 10.
IV.
APPLICATIONS
In this section we illustrate the statistical-estimation procedure by applying it to two different sets of Charpy 690--VOLUME 14A, APRIL 1983
Impact Data for Experimental Steel No. 283
[15] Temperature (~ 73 73 46 46 18 18 0 0 24 24 52 52 71 71 100 100
Charpy V-Notch Impact Energy (J) 7 7 27 23 49 68 65 58 102 104 134 145 150 142 148 144
METALLURGICAL TRANSACTIONS A
16o
5.6o
I
i-~ 4.80
I
I
120 -
g
<
4.00
-,
3.20 --
u
fLU
0r
5 r Z
rU
2.40
40
o $~', -80
--
"6 0 .J
80
I -40
i
i
- 80
l
I 40
I
,I 80
t 120
Temperoture,C
I 1.60
I 0
I
,
- 40
I
i
I
,
0 40 Temperoture,C
I 80
Fig. 4 - - C h a r p y V-notch energy measurements and lower-bound-energy estimates v s temperature for experimental steel 283.
i 120
Fig. 3 - - Logarithm of Charpy V-notch energy-absorption measurements v s temperature for experimental heat 283.
Table IV.
Charpy V-Notch Impact Energy (J)
Temperature (~
Once the estimated U*(T) lower-bound curve is obtained, the transition temperature, which is arbitrarily defined by the lower-bound energy of 41 J (30 ft-lb), can be calculated. Using the estimated U*(T) values for iteration number 3 in Table II, or the graphical form for U(T) given in Figure 4, we find that the transition temperature is approximately - 18 ~ For these data the form of the Charpy-energy-absorption/ temperature curve and the breadth of the temperature range make it amenable to both the Oldfield and Stallmann analyses. We now turn to a case where applying either of these analyses would be much more difficult.
26.67 26.67 48.89 48.89 48.89 48.89 48.89 57.22 57.22 60.00 60.00 68.33 68.33 73.89 73.89 85.00 85.00 97.78 97.78
B. Analysis of Data for ASTM A508 Steel The data from the steel used in ORNL Experiment TSE6 were selected as an example of a data set with significant scatter." Table IV presents the results of Charpy V-notch tests made on 19 specimens over a temperature range from 27 to 98 ~ The same procedure was used on this steel that was used in the previous illustration. In this case, a firstdegree polynomial was chosen to approximate ln(/3(T)), and a first-degree polynomial also was used to approximate -ln(c~(T)). Table II.
Impact Data for ASTM A508 Steel
29.15 28.47 56.94 29.83 42.03 29.15 39.32 61.01 55.59 56.94 60.33 55.59 44.74 31.18 48.81 54.23 61.01 70.50 69.15
The extreme scatter associated with these data required more iterations before convergence was achieved. Table V presents the results of the initial fit and the first nine iterations. In this case, convergence has been achieved in the
Estimates of the Lower-Bound Charpy V-Notch Impact Energy by Iteration, Experimental Steel 283
Iteration
- 73
- 46
0 1 2 3
5.20 5.51 5.51 5.51
16.92 17.80 17.80 17.80
Table III.
Impact Energy (Joules) at Indicated Temperature in ~ - 18 0 24 52 40.05 41.84 41.83 41.83
59.22 61.62 61.62 61.62
85.07 88.10 88.10 88.10
108.97 112.32 112.32 112.32
71
100
119.84 123.17 123.17 123.17
130.99 134.10 134.10 134.10
Weibull Parameters ~(T) and a(T) for Experimental Steel 283 from the Final Iteration Parameter Values at Indicated Temperature in ~
Parameter
-73
-46
-18
0
24
52
71
100
[3(T), joules a(T)
7.78 8.59
24.03 9.90
54.20 11.47
78.00 12.60
108.45 14.30
134.38 16.56
144.88 18.30
154.16 21.31
METALLURGICAL TRANSACTIONS A
VOLUME 14A, APRIL 1983--691
!
7O O0
~
55
u 0 z
40
O el..)
__
9
9
00
9
9
/
z5
I0
/
,
I
,
I
40
,
I
60 80 Temperoture,C
,
I I00
I
120
Fig. 5--Charpy V-notch measurements and lower-bound-energy estimates vs temperature for ASTM A508 steel.
first decimal place for the rounded data after eight iterations. Figure 5 is a plot of the Charpy data vs temperature, with the final estimated U*(T) curve superimposed on the data. Table VI reports the values of a(T) and /3(T) for the ASTM A508 specimens. In contrast to Experimental Heat 283, there is a large difference between fl(T) and the lower-bound value. In addition, a is much lower for this steel because of the large scatter. Interpolating between the U*(T) values for the ninth iteration in Table V, the transition temperature associated with a lower-bound energy of 41 J is found to be approximately 83 ~
V.
The analysis described in this paper provides a means for obtaining lower-bound-toughness estimates for any singlevariable case. While the illustrative examples were Charpy
Iteration 0 1 2 3 4 5 6 7 8 9
Parameter
26.67 10.33 15.33 13.22 14.29 13.71 14.01 13.86 13.94 13.89 13.91
a (T)
26.67 37.05 3.03
692--VOLUME 14A, APRIL 1983
We are grateful to Mr. P.N. Mincer for performing the experiments. This research is being supported by the Army Research Office under Project P-18399-MS.
1. E B. Pickering: PhysicalMetallurgy and the Design of Steels, Applied Science Publishers, London, 1978, p. 95. 2. T.T. Shih: Engg. Fracture Mech., 1980, vol. 13, pp. 257-71. 3. W. Oldfield: ASTM Standardization News, 1975, vol. 3, no. 11, p. 24.
Estimates of Lower-Bound Charpy V-Notch Energy by Iteration, ASTM A508 Steel
Table VI.
fl(T), joules
ACKNOWLEDGMENTS
REFERENCES
DISCUSSION
Table V.
impact data, they also could have been crack-initiation (Ktc) or crack-arrest (KIo) data. In principle, the analysis also could be extended to other phenomena, such as creep and fatigue. Also, it is not limited to Weibull distributions, but can employ other mathematical formulations in cases where they may be more appropriate.12 The novel feature of the analysis is its use of the experimental scatter as an integral part of the iterative process. By this means, unrealistic predictions of nonzero probabilities of negative energies can be prevented. Allowing the scatter to vary with temperature also can prevent the problem of assigning a lower bound to one region of the curve that is influenced by data from another region. For example, there is no reason to expect that upper-shelf and lower-shelf data will have the same variability. The analysis is still being developed. It remains to establish a means of specifying confidence estimates for the percentile data. There is also a need to develop a means of relating these results to the three-specimen criteria described earlier. Following these developments, the model will be applied to more extensive data sets. Note Added in Proof: In the statistical literature the use of the notation a and/3 for the Weibull parameters is often the reverse of that used here.
48.89 19.58 23.23 22.38 22.81 22.57 22.70 22.63 22.67 22.65 22.65
Impact Energy (Joules) at Indicated Temperature in ~ 57.22 60.00 68.33 73.89 23.91 25.45 30.40 33.97 26.75 28.00 31.96 34.79 26.53 28.00 32.69 36.05 26.63 27.99 32.29 35.37 26.58 27.99 32.50 35.73 26.61 27.99 32.39 35.54 26.59 27.99 32.45 35.64 26.60 27.99 32.42 35.59 26.59 27.99 32.44 35.62 26.59 27.99 32.43 35.60
85.00 41.74 40.92 43.32 42.05 42.72 42.37 42.56 42.45 42.51 42.49
97.78 51.80 48.81 52.65 50.62 51.70 51.13 51.43 51.27 51.36 51.32
Weibull Parameters/3(T) and a(T) for A508B Steel from the Final Iteration
48.89 45.47 4.26
Temperature Parameter Values at Indicated Temperature in ~ 57.22 60.00 68.33 73.89 49.09 50.36 54.38 57.24 4.85 5.06 5.74 6.26
85.00 63.40 7.42
97.78 71.32 9.02
METALLURGICAL TRANSACTIONS A
4. W. Oldfield: J. Testing Evaluation, 1979, vol. 7, p. 326. 5. W. Oldfield: J. Eng. Mater Technol., 1980, vol. 102, p. 107. 6. R.A. Wullaert, W. Oldfield, and W.L. Server: J. Eng. Mater. Technol., 1980, vol. 102, p. 101. 7. D. Chun: J. Mater., 1972, vol. 7, p. 257. 8. G. E Koons: J. Testing Evaluation, 1979, vol. 7, p. 96. 9. F.W. Stallmann: U.S. NRC Report NUREG/CR-2408, ORNL/ TM-8081, January 1982. 10. T.A. Bishop: Battelle-Columbus, Columbus, OH, unpublished
METALLURGICALTRANSACTIONS A
research, 1982. 11. R.D. Cheverton, D.A. Canonico, S.K. Iskander, S.E. Bolt, P.P. Holz, R.K. Nanstad, and W.J. Stelzman: Aspects of Fracture Mechanics on Pressure Vessels and Piping, ASME Pub. PVP 58, 1982, pp. 1-15. 12. P. Stanley, A. D. Sivill, and H. Fessler: J. Strain Anal. Engg. Design, 1978, vol. 13, pp. 103-13. 13. P.G. Tracy, T.P. Rich, R. Bowser, and L.R. Tramontozzi: Int. J. Fracture, 1982, vol. 14, pp. 253-77.
VOLUME 14A, APRIL 1983--693
