Proceedings of the ASME 2018 37th International Conference on Ocean, Offshore and Arctic Engineering OMAE2018 June 17-22, 2018, Madrid, Spain
OMAE2018-77038
COMPARISON OF METHODS TO FIND THE WEIBULL STRESS PARAMETERS
O.J. Coppejans TNO, Structural Dynamics
Leeghwaterstraat 44 NL-2628 CA Delft, The Netherlands
C.L. Walters*
TNO, Structural Dynamics Leeghwaterstraat 44 NL-2628 CA Delft, The Netherlands
Delft University of Technology Mekelweg 2
2628 CD Delft, The Netherlands
ABSTRACT
The Weibull stress (Beremin, 1983) relates the local first principal stress and plastically strained volume to the probability of fracture. It requires two constants (m, σu) as input, which are generally regarded as material parameters. As the Beremin approach is used in the background to structural analysis rules, the Beremin parameters are now being used in other situations, including engineering analysis. However, the currently accepted way to find the Beremin constants requires twenty tests, which is considered to be unacceptably expensive for industrial application. Less expensive ways of finding the Beremin parameters have been published in the literature, but they have never been compared to the de facto standard. In this paper, the Beremin parameters were found by the de facto standard method and two other ways, and a comparison is made. It was found that the Beremin parameters can be estimated with reasonable accuracy with a method that uses on just two sets (six specimens) of fracture specimens by careful application of the method of Andrieu (2012). Also, a proposal is made for another method which is based only on the Master Curve temperature T0. BACKGROUND AND INTRODUCTION
Protection against cleavage fracture (brittle fracture of steel at low temperatures) remains a key factor in many material requirements in the maritime and offshore industry. For example, the requirement of a Charpy impact energy of 27 J for Grade A steel specified by IACS (2017) and other classification body requirements is designed to prevent cleavage fracture. Other grades of steel or parts of the structure may have different Charpy requirements to account for yield strengths, application temperature, or safety relevance of the structure. Charpy require- *Corresponding author:
Carey.Walters@tno.nl
ements are typically sufficient to prevent cleavage in most scenarios. However, as the offshore industry moves to colder climates and thicker and stronger steels are applied in new structural configurations, there is a need to reduce conservatism and increase knowledge on the detailed testing and modelling of cleavage. Sometimes, engineering critical assessments, like the one given in the standard BS 7910 “Guide to methods for assessing the acceptability of flaws in metallic structures”, are a sufficient next step. For some combinations of materials and environment, yet more detailed analysis can still be necessary, and even portions of BS 7910 require advanced modelling parameters that are typically not measured for structural-grade steel. These detailed analyses need sophisticated failure criteria that are compatible with finite element calculations.
Beremin (1983) proposed a model of cleavage which relates the probability of fracture to the so-called Weibull stress, which is given in Eq. (1). 𝜎𝑤= √∑(𝜎1 𝑗 )𝑚𝑉𝑗 𝑉0 𝑗 𝑚 (1) where σw is the Weibull stress, m is the Weibull shape parameter,
σ1 is the first principal stress, Vj is the incremental volume of summation, and V0 is an arbitrary volume used for normalization. The summation is typically taken over the volume that is plastically strained, but other criteria have been proposed. Because Eq. (1) represents a summation over elements (or an integral over volume), it can be convenient to substitute the stress field ahead of a crack into this formula. The stress field in a plastic material was developed by Hutchinson (1968) and Rice, Rosengren (1968) and is known as the HRR field. The magnitude of the HRR scales with the J-integral. When the HRR field is substituted into Eq. (1) and the assumption is made that the J-integral is approximately equal to K2(1-ν2)/E, the Weibull stress is given by Eq. (2).
𝜎𝑤𝑚=
𝐵𝜎0𝑚−4𝐾4𝐶𝑚,𝑛
𝑉0
(2) where B is the specimen thickness, K is the applied stress intensity factor, Cm,n is a constant that depends only on m and n,
σ0 is a parameter that scales with yield stress, and n is the strain hardening exponent. Further details on the derivation of Eq. (2) can be found in Andrieu et al. (2016) or Beremin (1983). The probability of failure is given in terms of the Weibull stress according to Eq. (3). 𝑃𝑓 = 1 − 𝑒𝑥𝑝 [− ( 𝜎𝑤 𝜎𝑢 ) 𝑚 ] (3)
where Pf is the probability of fracture, and σu is the Weibull scale parameter. Other forms of the Weibull probability function have also been proposed, especially one that accounts for a threshold below which there is zero probability of fracture (Gao et al, 1998), but those developments are not considered here.
The currently-accepted method to find the parameters for the Beremin model was proposed by Gao et al. (1998). This method is referred to in internationally recognized standards, such as BS 7910:2013 and ISO 27306. The method of Gao et al. (1998) requires ten repetitions of two different test geometries, for a total of twenty tests that are similar to CTOD or J-integral tests. It is interesting to note that because fracture tests are typically done in sets of three, twenty tests is almost seven sets of standard CTOD tests. It is therefore probably less expensive to test several sets of CTOD specimens at various conditions (e.g. at different temperatures, stain rates, constraints, etc.) than to find the Beremin parameters and then use formulas based on the Beremin parameters to predict different conditions. For example, BS 7910:2013 Annex N allows for fracture toughness values to be modified for constraint, but the parameter m could be required as input for these analyses. Therefore, several CTOD tests at different constraint levels could have been performed for a lower cost than applying the analytics of BS 7910:2013 Annex N, even neglecting the analytical effort involved. Gao et al. (1998) gave no concrete justification of the number of specimens, so Walters and Dragt (2016) performed an analysis to see if fewer specimens would be acceptable. Their analysis supported the same number as suggested by Gao et al. (1998). A commercial lab will charge on the order of one thousand euros per CTOD specimen, so in addition to being more expensive than alternatives, finding the Beremin parameters can be expensive in its own right. While Charpy testing is offered as an alternative for one of the test configurations, it is expected to be static and instrumented, thus leading to a cost that is more similar to CTOD testing than to standard Charpy testing. Therefore, there is a strong interest in finding less expensive methods of identifying Beremin parameters. A low cost method of identifying the Beremin parameters is not straightforward because of the statistical nature of the parameters and because determination of these parameters from fracture tests can be ambiguous. Specifically, Gao et al. (1998) showed that, due to the
self-scaling nature of the stress field, the parameters m and σu are interrelated, and tests that rely exclusively on small-scale yielding fracture testing are ambiguous. They used this to explain the large amount of scatter in Beremin parameters that had been previously shown in the literature. Their analysis is seen to invalidate most of the procedures for finding Beremin parameters that had been presented prior to 1998, such as the one by Minami (1992).
Some methods have been proposed in the years since Gao et al. (1998). In general, these methods are less expensive than the method of Gao et al. (1998), but they have not been directly compared with the method of Gao et al. (1998), which remains the method recognized by international standards. In this paper, some of the less expensive methods will be considered, and an alternative approach of estimating the Beremin parameters will be proposed.
MATERIALS AND TESTS
Twenty five millimeter thick S355J2+N steel from SSAB was obtained from a commercial source. The mechanical properties of this steel are reported in Table 1, and the chemical properties are reported in Table 2. Both Table 1 and Table 2 were taken from the material certificate. This is the same material as was reported in Walters et al. (2017).
Table 1: Mechanical properties according to the material certificate, with minimum values indicated in parentheses
ReH [MPa] Rm [MPa] A % Charpy V @ -20 °C [J]
375 (345) 532 (470) 30 (22) 212, 226, 248 (27)
Table 2: Chemical composition of the plate according to the material certificate, in weight %
Cev C Si Mn P S Al Nb
0.42 0.160 0.44 1.44 0.010 0.006 0.030 0.039
V Ti Cu Cr Ni Mo N B
0.011 0.003 0.017 0.05 0.04 0.006 0.005 0.0003 Tension tests with a 12.5 mm diameter round bar specimen were performed on this material at room temperature and -60°C. Data from eighteen CTOD tests with the Single Edge Notched (SENB) geometry were available from Walters et al. (2017). Triplicate CTOD tests were conducted at each of temperatures: -130 °C, -120 °C, -110 °C, -100 °C, -90 °C, and -70 °C. In addition to the data from Walters et al. (2017), further tests were carried out at -110 °C. Seven of the new tests had the standard geometry (a/W=0.5), and ten of the new tests had a shallow pre-crack a/W=0.1. With the exception of the shallow cracks, all of the CTOD tests were on Bx2B full thickness specimens according to BS 7448-1. All tests were carried out in a commercial lab. The CTOD results are shown in Figure 1.
The tension tests at room temperature and -60°C were insufficient to provide plasticity data over the range of temperatures that CTOD tests were performed at. Therefore, the stress-strain curves were inverse engineered from previously
available small punch tests, which were available from Walters et al. (2017). The stress-strain data was first found at room temperature (for which both small punch test and tension test data was available) by the inverse method. This showed that the inverse engineering method produced the same results for small punch testing and tension testing. From there, inverse engineering was used to find the stress-strain curves from small punch tests that were performed at -186 °C, -175 °C, -170 °C, -131 °C, -112 °C, -70 °C, and room temperature. The following stress-strain relation was fitted to the observations:
𝜎𝑦(𝑇ℎ, 𝜀𝑝) = 𝑚𝑎𝑥 { 1 𝐴 + 𝐶𝑇ℎ 1 𝐷 + 𝐹𝑇ℎ (1 + 𝐺(𝜀 − 𝜀0)𝑛) (4)
where Th is the homologous temperature (absolute temperature divided by the absolute melt temperature, taken as 1500 °C), and
A, C, D, F, G, ε0, and n are fitting constants. The values taken for the fitting constants are given in Table 3. The material parameters were chosen so that the curve fits worked best for the test conditions at -131 °C and -112 °C, in the range approximately corresponding to the T0 temperature.
Figure 1: CTOD test data compared to a Master Curve fit. Black dots represent standard tests, and red x’s represent shallow-crack (a/W=0.1) results. Note that overlapping points at -130 °C and -110 °C have been offset by a few degrees Celsius so that they can be seen separately.
Table 3: Fitting constants for Eq. (4)
A [MPa-1] C [MPa-1] D [MPa-1] F [MPa-1] G [-]
0.000459 0.0155 156. 2097. 402000.
ε0 [-] n [-]
0.0223 0.162 FEA MODEL
A relationship between the stress field in the specimen and either the CTOD or J-integral was necessary for the calibrations performed below. An FEA model for each geometry (namely,
a/W=0.5 and a/W=0.1) was created and used for each of the
calibrations. The mesh for the a/W=0.5 condition is shown in Figure 2; the other model was created with similar principles and parameters. The crack tip was modeled as a finite notch with a crack tip that is 0.01 mm in diameter. This diameter is mostly in line with the guidelines given by Andrieu (2012) for the CTOD values experienced in this work. The smallest elements near the crack tip were 0.00125 mm on one edge. The rollers were modeled as analytical rigid bodies. A mesh refinement study was performed to assure that the results were element size independent. The temperature-dependent stress-strain relationship from Eq. (4) was implemented as a piecewise linear plasticity model that was created separately for each temperature. The models were solved with Abaqus 6.14-1. Quadratic hexahedral elements with reduced integration (C3D20R) were used throughout. The Weibull stress was calculated through a Python-based postprocessing script. The V0 was chosen to be 0.000125 mm3 as suggested by Beremin
(1983), and the summation was performed over the plastically strained elements.
Figure 2: FEA model of CTOD tests, used for calculating the Weibull stress. For scale, the diameter of the crack tip is 0.01mm in diameter.
METHOD OF GAO ET AL. (1998)
This method relies heavily on the observation that the Weibull distribution of a set of Small-Scale Yielding (SSY) J-integral results has a known theoretical slope (namely, 2). This helps to characterize the scatter in J-integral tests before calculating the Weibull slope (m) of the results when expressed in terms of Weibull stresses (σw). The individual steps are given below and summarized in Figure 3.
1. Perform at least ten J-integral tests for two different specimen geometries: one with small-scale yielding, corresponding approximately to standard testing practice with deep notches; and the other with low constraint conditions, likely from a shallow crack. The low-constraint case is also called the Large-Scale Yielding (LSY) case. All of the specimens should feature cleavage with no prior ductile tearing and be tested at the same temperature. 2. Perform detailed Finite Element Analysis for both specimen
Figure 3: Schematic of the procedure by Gao et al. (1998)
3. Assume an m value and use it to compute the σw vs. J relationships for both configurations. Use the σw vs. J maps to calculate the SSY J-integral that corresponds to the same
σw value as the LSY J-integral measured in the LSY tests. This “constraint corrected” value is referred to as Jc. Determine the β value of Eq. (5)
𝑃𝑓= 1 − 𝑒𝑥𝑝 [− (
𝐽 𝛽)
2
] (5)
by applying the maximum likelihood method according to Eq. (6). 𝛽 = [1 𝑁(∑ 𝐽 2 𝑁 𝑖=1 )] 1/2 (6) The β values should be calculated for JSSY and Jc as two separate populations, resulting in βSSY and βc. The error shall be determined as R(m) = (βc-βssy)/βssy.
4. Iteratively adjust the assumed m value and re-do Step 3 until
R is zero.
5. Calculate σu as the σw when J=βSSY.
The reader is referred to the original paper of Gao et al. (1998) for a full description of their method. In total, this procedure requires the testing of twenty fracture specimens at a single temperature and presumably a tension test at the same temperature. Analytical solutions for the relationship between σw and J-integral can be used for the SSY case, but they likely cannot be avoided for the LSY condition. Therefore, detailed FEA is required if this calibration process is to be used.
Figure 4: The σu and m parameters calculated based on the method of Gao et al. (1998). The 5th and 95th percentile (dashed line) are superimposed with
the mean (solid line). Note that each experiment set contains two experiments: one SSY and one LSY.
1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 4250 2 4 6 8 10 σu [M P a ]
This method was applied to the twenty CTOD tests performed at -110 °C in the aforementioned data set. In order to assess whether or not fewer experiments could be used to find a sufficient calibration, the same calibration procedure was performed with all permutations of the data grouped into sets of two experiments. Each set of two experiments contained one LSY and one SSY tests. The results of this study are shown in Figure 4. As can be seen, the method of Gao et al. (1998) using the recommended twenty specimens produced m = 21.3 and σu = 2370. MPa. Performing permutations on the various combinations of test results produced high scatter and indeed a bimodal distribution until at least 16-18 experiments (thus, 8-9 sets) were performed. Therefore, it can be seen from this test that the procedure of Gao et al. (1998) cannot effectively be used for much fewer than twenty total experiments. This is supported by the recommendations of Gao et al. (1998) and the numerical exercise performed by Walters and Dragt (2016).
METHOD OF ANDRIEU ET AL. (2012)
The method of Andrieu et al. (2012) offers an alternative to the approach of Gao et al. (1998). It appears to address the ambiguity imposed by the self-similarity of the stress field around a crack tip with an observation that the relationship between fracture toughness and yield strength is governed by the parameter m. Specifically, Eq. (2) can be expressed in the form of Eq. (7): ln(𝐾) =4 − 𝑚 4 𝑙𝑛 ( 𝐶𝑜𝑛𝑠𝑡 𝜎0 ) (7)
when the parameters σw, B, V0, and Cm,n are taken as constant, which is reasonable for a single material over a limited range of temperatures. Because yield strength and fracture toughness both depend on the temperature, the relationship between the two can be found by testing at multiple temperatures, then finding the slope of the relationship. Once m is found, determination of σu is straightforward. Andrieu et al. (2012) further streamline the process of finding σu by performing detailed finite element calculations and giving a way of finding Cm,n through a series of curve-fitted equations and a table. Walters and Dragt (2016) have also provided a curve fit for the table provided by Andrieu et al. (2012) to make the process easier. Specifically, the steps of Andrireu et al. (2012) are:
1. Perform SSY J-integral tests at different temperatures. All of the specimens should feature cleavage with no prior ductile tearing.
2. Determine m by plotting the KJc results versus yield strength in log-log scale and taking the slope. The value of m is related to slope by a simple formula, as shown in Eq. (7). 3. Determine Cm,n by the formulas and tables given by Andrieu
et al. (2012), which are not repeated here. These formulas and tables are straightforward to automate in software. 4. From here, Andrieu et al. (2012) offer two different
procedures:
a. For small data sets: Calculate σu from Eq. (8)
𝜎𝑢= [ 3 4 𝐵𝐶𝑚,𝑛𝜎0𝑚−4𝐾̅4 𝑉0 ] 1/𝑚 (8) where 𝐾̅ is the mean toughness of the fracture tests corresponding to the yield strength used in the formula.
b. For large data sets: Rank-order the fracture results for a single temperature and determine the failure probability for each of the specimens. Find the toughness of the value with the probability of failure closest to 0.528. This value will be called
KPf. The σu value is then taken from Eq. (9).
𝜎𝑢= [ 𝐵𝐶𝑚,𝑛𝜎0𝑚−4𝐾̅𝑃𝑓4 𝑉0𝑙𝑛 ( 1 1 − 𝑃𝑓) ] 1/𝑚 (9)
This procedure is summarized in Figure 5.
Figure 5: Schematic of the procedure by Andrieu et al. (2012)
This method was applied to all of the CTOD tests in the aforementioned data set except for 17 of those performed at -110 °C. Therefore, sets of three experiments performed at six different temperatures were used. In order to assess whether or
not fewer experiments could be used to find a sufficient calibration, the same calibration procedure was performed with all permutations of the data grouped into sets of three experiments, representing one temperature condition. The results of this study are shown in Figure 6. As can be seen, the method of Andrieu et al. (2012) using the six sets (18 experiments) produced m = 19.2 and σu = 2521. MPa. Performing permutations on the various combinations of test results produced high scatter and until at least 9-12 experiments (thus, 3-4 sets) were performed.
Figure 6: The σu and m parameters calculated based on the method of Andrieu et al. (2012). The 5th and 95th percentile (dashed line) are
superimposed with the mean (solid line). A temperature set is a set of three CTOD tests performed at the same temperature.
The very high scatter in the results is notable, especially at conditions in which very few (e.g. only two) data sets were taken. Non-physical negative values of m can also be seen. The high scatter and negative values of m can be seen as a product of the confluence of various factors. Specifically, there is a high level of scatter in cleavage fracture toughness, and a weak dependence
of yield strength (which correlates with σ0) on temperature. Therefore, when only two data sets that are only 10 °C apart are taken, then there is a high scatter in the slope of the ln(K) versus ln(σ0) relationship. A particularly stark example can be seen when comparing the results of the CTOD tests at -100 °C to those at -90 °C. If only these two results were taken, then it would appear that the toughness increases with lower temperature, which would give the wrong sign of the slope of the ln(K) versus ln(σ0) relationship, which would give a negative value for m. From this observation, it may appear that higher differences between the maximum and minimum test temperatures could mitigate the effect of scatter and allow for a more accurate measurement of m and thus also σu. To test this hypothesis, the same set of permutations was considered, but this time, they were sorted by the difference between the maximum temperature (Tmax) and the minimum temperature in the data set (Tmin). The results are shown in Figure 7. As can be seen, the value of m appears to be converged when the difference between Tmax and
Tmin is at least 40 °C. Furthermore, for those large temperature differences, the answer appears to be invariant of the total number of sets that is used for the calibration. Therefore, the conclusion of this section is that the method of Andrieu et al. (2012) is able to produce a converged value of m and σu with as few as two sets (six experiments) as long as the difference in testing temperature is at least 40 °C.
METHOD OF CAO ET AL. (2011)
The Master Curve concept was first published in the open literature by Wallin (1989). This concept makes use of the observation that in the lower ductile to brittle transition, the fracture toughness of ferritic steels has an approximately constant scatter and relationship with temperature. The result is Eq. (10), which relates fracture toughness to the temperature (T), thickness (B), and probability of fracture (Pf).
𝐾𝐽𝑐 = 20 + (11 + 77𝑒𝑥𝑝(0.019(𝑇 − 𝑇0))) (25 𝐵) 1 4 (𝑙𝑛 1 1 − 𝑃𝑓 ) 1 4 (10)
The concept and the methodology for finding T0 have been standardized in ASTM E1921. After they calibrate the Master Curve, Cao et al. (2011) make use of the known scatter profile of fracture toughness testing by using the Master Curve to randomly generate imaginary fracture toughness results (which they denote as Kgen) that might be expected if thousands of tests are performed on the same material. They then uses these thousands of results to create an estimate of the Beremin parameters. Their procedure is outlined in Figure 8, and the specific steps are:
1. Perform SSY J-integral tests and find T0 in accordance with ASTM E1921.
2. Using the Master Curve, randomly generate thousands of imaginary test results at the same temperature. Cao et al. (2011) recommend 10,000. -45 -30 -15 0 15 30 45 60 75 90 2 3 4 5 6 m [ -]
Number of temperature sets
1500 1750 2000 2250 2500 2750 3000 3250 3500 3750 4000 4250 2 3 4 5 6 σu [M P a ]
3. Perform FE analysis to find the stress fields corresponding to the generated Kgen results.
4. Make an initial guess for m, then use the stress field from (3) to evaluate σw for each of the randomly generated results from (2).
5. Perform linear regression analysis on the relationship between ln(ln(1/(1-Pf))) vs. ln(σw) to find m. This relationship (and the associated slope) can be anticipated based on Eq. (3).
6. Iterate steps (4) and (5) until the m calculated by linear regression analysis is equal to the initial guess.
7. Once m is converged, use the σu and m from the last iteration.
Figure 7: The σu and m parameters calculated based on the method of Andrieu et al. (2012). The convergence of the results as a function of the difference in temperature between the highest and lowest data set is shown. (Colors are available in the electronic version)
A similar calibration method was produced by Qian et al. (2015). This paper acknowledged the non-uniqueness issue identified by Gao et al. (1998), and answered it by indicating that the Monte Carlo technique compensates for the non-uniqueness identified by Gao et al. (1998). It may seem counterintuitive that random generation of data could compensate for the fact that the SSY test is in fact non-unique. However, by a similar technique as described in Walters and Dragt (2017), Dragt et al. (2017) showed that even if only SSY conditions are considered, increasing the number of specimens ultimately produces an answer which converges to the correct Beremin parameters. Walters and Dragt (2017) concluded that the number of specimens required to find the correct Beremin parameters was far too large and abandoned this approach. However, by randomly generating data based on the known statistical distribution of the Master Curve, Cao et al. (2011) and Qian et al. (2015) appear to have been able to “generate” ten thousand samples based on only performing a limited number of tests.
Figure 8: Schematic of the procedure by Cao et al. (2011)
-45 -30 -15 0 15 30 45 60 75 90 0 20 40 60 m [ -] Tmax-Tmin [ C] 2 sets 3 Sets 4 Sets 5 Sets 6 Sets
While the method of Cao et al. (2011) specifies that FE analysis should be run, the procedure itself makes use of SSY specimens, so it should be possible to use analytical methods like those of Andrieu (2012) to avoid the need of FE modelling.
Before the procedure was applied, the effect of the number of generated data was considered. Figure 9 shows that randomly generating 10,000 samples is indeed adequate, though up to a million points can reduce uncertainty. The authors of this paper believe that because of the very low cost of generating additional data points, it makes more sense to generate one million data points.
Figure 9: Effect of the number of generated samples on the determination of σu and m. The points represents individual calibrations, and the 5th, 50th,
and 95th percentile of the distributions based on a Gaussian copula are
over-plotted.
Performing the method of Cao et al. (2011) gives m = 24.5 and σu = 2364 MPa. A similar permutation study was performed for this procedure as was performed with the method of Andrieu (2012), above. The results of that study are shown in Figure 10. It is notable that this procedure produces considerably less scatter than that of Andrieu (2012) for a low number of tests, but
the amount of scatter is still significant. This method depends very highly on a correct determination of the T0. Therefore, the same permutation study was performed on the value of T0, and the results of that study are shown in Figure 11. What this figure shows is that with the given experimental data, using fewer sets of three experiments results in a high scatter in the measured value of T0. The scatter in the measured T0 is then naturally inherited into the calculation of m and σu. It is known that the calibration of T0 becomes less certain as the test temperature deviates further from the T0 itself. Therefore, the relationship between the T0 and the average temperature of the data set for each of the permutations was calculated. It was found that the coefficient of correlation between T0 and the average temperature of the data set used to calculate T0 was -0.16, meaning that there was no apparent relationship between these two parameters. What this means is that the scatter found in Figure 11 (and consequently Figure 10) cannot be explained by the difference between test temperature and T0.
Figure 10: The σu and m parameters calculated based on the method of Cao et al. (2011). The 5th and 95th percentile (dashed line) are superimposed with
the mean (solid line). A temperature set is a set of three CTOD tests performed at the same temperature.
All of the above studies were also performed for the method by Qian et al. (2015). However, the method of Qian et al. (2015) produced almost identical results as Cao et al. (2011), so they were omitted.
ALTERNATIVE APPROACH OF FINDING σu AND m
In the prior sections, it was shown that the methods proposed by Cao et al. (2011) and Qian et al. (2015) are able to find approximately the same Beremin parameters as Gao et al.’s (1998) method. This is important because the only input that Cao and Qian use is the T0 temperature, though there is an implicit temperature-dependent stress-strain relationship behind their assumption. Based on the observation that a unique relationship can be made between the Beremin parameters and the T0, it is possible to create relationships based on some standard assumptions. These relationships for the material under investigation are shown in Figure 12 and the fits to these are given in Eq. (11) and (12).
𝑚 = −0.043𝑇0+ 12.63 (11)
𝜎𝑢= −2.68𝑇0+ 2528 (12)
where T0 is given in °C and σu is in MPa.
In the future, it may be possible to create a series of these relationships that are a function of yield strength and/or yield strength to ultimate tensile strength ratio. If such a table existed, then it would serve as a practical tool for finding the m value needed in standards like BS 7910 Annex N or ISO 27306.
Figure 11: The dependence of T0 on the number of temperature sets. A temperature set is a set of three CTOD tests performed at the same temperature.
DISCUSSION
The Beremin parameters found by the various methods given above are summarized in Table 4. Because these parameters are statistical in nature, it is not surprising that there
are different answers given the vastly different approaches. Nevertheless, the values found between the various methods are uncannily close when the maximum number of experiments for each method is used. Gao et al. (1998) has shown mathematically that any method that only uses SSY testing is ambiguous, and all of the other methods do just that. Andrieu et al. (2012) give a satisfactory explanation of why they can determine an unambiguous value of m despite this shortcoming; namely, they show that m governs the relationship between the fracture toughness and the yield strength, and they use the temperature dependence of yield strength as the way to vary the yield strength. The reason that the methods of Cao et al. (2011) or Qian et al. (2015) work is less satisfying. It appears that the strength of this method comes from two main sources: (1) the known statistical distribution of the Master Curve provides additional information beyond what is known directly from testing, and (2) the ability to create a very large number of imaginary tests enables for these statistical values to be determined with a higher level of confidence. Dragt et al. (2017) has assessed the number of tests required to find the Beremin parameters using only SSY test results. In that study, the number was unacceptably high (in the hundreds or thousands), but it ultimately did converge to a realistic value of the known Beremin parameters. The method of Cao et al. (2011) and Qian et al. (2015) evidently relies on that property. Therefore, it can be seen that a known T0, which can be measured with as few as six tests, together with the assumption of the Master Curve can generate thousands of imaginary tests, which can accurately give the Beremin parameters.
The goal of this paper is ultimately to assess whether or not the parameters m and σu can be found less expensively than by the method of Gao et al. (1998). Therefore, it is interesting to determine how few experiments can be used. Toward that end, the Coefficient of Variation (COV, standard deviation normalized by the mean) for m and σu was calculated as a function of the number of total experiments in Figure 13. What this shows is that Andrieu et al. (2012) produce extremely high scatter when only a few experiments are randomly chosen. However, it was shown in the section dedicated to Andrieu et al. (2012) that much better results can be obtained when a high range of test temperatures is intentionally chosen. Other than that, to achieve a COV less than 10%, one would need 15 experiments for Cao et al. (2011), Qian et al. (2015), and Andrieu et al. (2012) and at least 18 experiments for Gao et al. (1998). This does not represent a large cost savings, so the final conclusion is that the method of Andrieu et al. (2012) produces the best cost savings, with as few as two sets, when a difference between the test temperature of the sets is at least 40 °C. This would require 70% fewer fracture tests than required by Gao et al. (1998) and 67% fewer fracture tests for the same procedure with only 18 experiments.
Figure 12: Dependence of m and σu on T0, based on the method of Cao et al. (2011)
Throughout this paper, considerable attention has been given to the contribution of the cost of CTOD testing to the total cost of finding σu and m. However, the cost of finding the stress-strain data has not been given as much attention. Because the temperature dependence of the stress-strain curve is integral to the development of cleavage fracture, this is a very important factor to consider. Furthermore, the cost of finding the true stress-strain curves is far greater than the cost of the test itself, which is typically quite inexpensive in a commercial lab. The reason for the added cost is that the true stress-strain curve is required far past the onset of necking in typical materials. Therefore, it is necessary to either inverse engineer the true stress-strain data or use a Bridgman correction. The inverse engineering method requires a full force-deflection curve to be collected, and that a true stress-strain curve is iteratively modified and substituted into a detailed FEA model of the test. In the authors’ experience, this process costs a couple of days of effort for a junior-level engineer for 5 different temperatures and is quite difficult to automate. The Bridgman correction (from
Bridgman, 1952) requires a little bit more testing effort and considerably less engineering effort. However, since unusual measurements are required during the testing (namely, the specimen diameter in the center of the neck and the radius of curvature of the neck), many commercial testing houses are unable to accommodate this and may add extra costs for the measurement of these parameters. Once those parameters are measured, the engineering effort to convert the data into a true stress strain to very high strains is minimal. Therefore, the number of tension conditions is also included in Table 4 as a cost parameter. (Note that this is just the number of conditions; the number of repetitions at each condition is left to the experience of the practitioner). It is notable that this is an important area where the procedure of Gao et al. (1998) is less expensive than the other methods. Correlations between yield strength and test temperature exist in the open literature, so it is anticipated that testing at only one temperature may be possible in the future.
Figure 13: Coefficient of Variation (COV) of m and σu depending on the total number of experiments performed for the four different methods.
Table 4: Beremin parameters found by the various different methods, together with parameters affecting the cost
Gao Andrieu Cao Qian
m [-] 21.3 19.2 18.7 18.7 σu [MPa] 2370. 2521. 2772. 2775. Min. num. fracture tests 20 6 6* 6* Min num. tension conditions 1 2+ 2+ 2+ FEA required Yes No No** No**
* = Number dictated by ASTM E1921, which is at least 6,
depending on the results
** = FEA is indicated in the procedure, but could be avoided if
the formulas and tables of Andrieu et al. (2012) are used CONCLUSIONS
The parameters necessary for calculating the probability of fracture according to Beremin (1983) were determined by three different methods. It was found that all three methods produce adequate estimates of these parameters when enough experimental data is used. It was observed that the lowest cost approach to finding the Beremin parameters was to apply the method of Andrieu (2012) with fracture tests that are at least 40 °C apart. This allowed for the Beremin parameters to be estimated with only two sets (six specimens) of data.
The method of Cao et al. (2011) depends only on the ability of the researchers to find an accurate T0. Based on this observation, an alternative approach was proposed that allows for the determination of Beremin parameters based on tables of
T0, yield strength, and yield to ultimate strength ratio. However, this approach was not fully developed.
NOMENCLATURE
a Total notch and fatigue pre-crack depth.
A, C, D, F, G, ε0 Curve-fitting constant for stress-strain relation
A% Permanent elongation of the gage length after fracture
B CTOD test specimen thickness
Cm,n Constant that depends on m and n CTOD Crack Tip Opening Displacement
E Young’s modulus FEA Finite Element Analysis
j Summation index
J J-integral
Jc Constraint-corrected J-integral of LSY test
JLSY J-integral measured in LSY test
JSSY J-integral measured in SSY test
K Applied stress intensity factor LSY Large-Scale Yielding
m Weibull exponent
n Strain hardening exponent for stress-strain relation
N Number of specimens used for determination of β
Pf Probability of failure
R Error between βc and βSSY
ReH Upper yield strength
Rm ultimate tensile stress SSY Small-Scale Yielding
T Temperature
T0 Master curve transition temperature
Vj Integral volume over which summation is taken
V0 Arbitrary volume used for normalization
W Total height of the specimen
β Weibull scale factor for J-integral results
βc βLSY constraint-corrected to the equivalent σw as SSY
βLSY β for LSY condition
βSSY β for SSY condition
ν Poisson’s ratio
σu Weibull stress scale parameter
σw Weibull stress
σ0 Curve fitting parameter that scales with yield strength σ1 First principal stress
ACKNOWLEDGMENTS
The funding of the Early Research Program (ERP) of TNO through the Structural Integrity project is gratefully acknowledged.
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