Probabilistic failure assessment of ferritic steels using the master curve approach including constraint effects

Engineering Fracture Mechanics 74 (2007) 1274–1292 www.elsevier.com/locate/engfracmech

Probabilistic failure assessment of ferritic steels using the master curve approach including constraint effects Jo¨rg Hohe *, Jochen Hebel 1, Vale´rie Friedmann, Dieter Siegele Fraunhofer Institut fu¨r Werkstoffmechanik, Wo¨hlerstr. 11, 79108 Freiburg/Brsg., Germany Received 9 March 2006; received in revised form 13 July 2006; accepted 18 July 2006 Available online 15 September 2006

Abstract The present study is concerned with an enhancement of the master curve approach for the probabilistic failure assessment of ferritic steel structures considering constraint effects. In an experimental program based on a variety of different specimens, distinct effects of the specimen geometry on the reference temperature T0 are observed. The experimental data base is examined in terms of the K–Tstress-, J–A2-, J–Q- and J–h-concepts. Based on the results, a constraint enhancement for the master curve concept is suggested consisting of a constraint dependent temperature shift or an alternative constraint dependent scaling of the stress intensity factor.  2006 Elsevier Ltd. All rights reserved. Keywords: Constraint effect; Transferability; Two parameter concepts; Master curve; Probabilistic failure assessment

1. Introduction The brittle failure of laboratory specimens and structural components by transgranular cleavage is a stochastic process. The cleavage failure process is usually initiated by the brittle failure of carbides or other hard inclusion due to local overloading caused by a plastic deformation of the surrounding metallic matrix and subsequent local stress redistribution. Since the carbide inclusions – or other weak spots – are randomly distributed on the microstructural level of the material, the macroscopic failure of the material is a stochastic process, even if a deterministic failure criterion is postulated on the microscopic level of structural hierarchy. For the enhanced failure assessment of ferritic steels and other engineering materials exhibiting strong effects of randomness, different probabilistic failure concepts have been proposed in the literature. These concepts can be divided into two groups depending on the type of loading parameter employed. Local approach concepts are based on the local mechanical stress and strain fields in the cleavage fracture process zone ahead of the crack front of precracked specimens or structural components (or in the vicinity of other stress

*

1

Corresponding author. Tel.: +49 761 5142 340; fax: +49 761 5142 401. E-mail address: [email protected] (J. Hohe). Present address: Technische Universita¨t Darmstadt, Institut fu¨r Mechanik, Hochschulstr. 1, 64289 Darmstadt, Germany.

0013-7944/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.engfracmech.2006.07.007

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Nomenclature a Ai B B0 c d E ðkÞ fij h J Jc J2 K KJ K ref J K_ J KJc K 1T Jc K med Jc Kmin K0 l m n Pf Q r rcl rpl si T T0 T ci0 T deep 0 Tstress w Y dij g u m rij rref ij re rI r0

crack depth intensities of the first three terms in the HRR field specimen thickness or crack front length reference thickness for master curve analysis slope of the linear regression curve for the constraint correction shift parameter of the linear regression curve for the constraint correction Young’s modulus circumferential stress distribution stress triaxiality coefficient J-integral fracture toughness second invariant of the Cauchy stress tensor stress intensity factor stress intensity factor determined from J-integral constraint corrected stress intensity factor loading rate fracture toughness fracture toughness related to reference thickness B0 median fracture toughness master curve threshold fracture toughness reference fracture toughness reference length scale parameter master curve Weibull exponent hardening exponent in the Ramberg-Osgood constitutive equation accumulated failure probability Q-parameter, additional hydrostatic stress superimposed to the HRR field radial distance from the initial fatigue crack front (radius of crack front polar reference system) distance of cleavage origin from the initial fatigue crack front width of strongly deformed region ahead of the initial crack (blunting and ductile crack growth) exponents in the power-series expansion of the HRR field actual testing or service temperature master curve reference temperature constraint indexing master curve reference temperature master curve reference temperature determined from deep crack specimens T-stress, intensity of second term in the asymptotic elastic crack front field specimen nominal width general constraint parameter components of the unit tensor plastic correction factor angle of crack front polar reference system Poisson’s ratio components of the Cauchy stress tensor reference crack front stress field v. Mises equivalent stress maximum principal Cauchy stress yield stress

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concentrations). Macroscopic or global concepts – on the other hand – are based on a macroscopic loading parameter such as the stress intensity factor K or the J-Integral. The advantage of the local concepts is their direct foundation on the local mechanical situation at the cleavage origin. Therefore, these concepts are able to re-capture the physics of the failure process and thus account for all effects of the local stress and strain fields as well as all possible interaction or gradient effects in a natural manner. The most important model in this group is the well-known Beremin model [2] which is based on the assumption of a power-law probability density distribution for the size of the brittle inclusions and thus the size of freshly nucleated microcracks. According to the weakest-link concept [12], overall failure of the entire structure is assumed in the event of the brittle failure of any control volume. In order to overcome some of the shortcomings of the simple formulation of the original Beremin model, different enhanced formulations have been proposed (see [5,18]). An alternative model based on an exponential probability distribution of the size of the hard inclusions has recently been presented by Kroon and Faleskog [4,9]. Disadvantage of the local approach models is the extreme numerical effort required due to extremely high mesh resolution necessary for an adequate stress analysis in the cleavage fracture process zone. Especially the Beremin model is rather sensitive to inaccuracies in the stress results due to its power-law formulations with powers in the range of 20 and higher. Hence, macroscopic concepts based on global or integral loading parameters such as K or J are used in most structural integrity analyses performed in engineering applications. The main advantages of the macroscopic concepts compared to the local approach concepts are their high numerical efficiency and their relative robustness. The most important macroscopic probabilistic fracture concept is the master curve concept according to Wallin [16] which employs the stress intensity factor KJ computed from the J-integral as a loading parameter. For the corresponding material property K J c , a three parameter Weibull distribution is assumed in order to account for probabilistic effects. Temperature effects are described in terms of an exponential dependence of the median fracture toughness of the material on the actual temperature which is assumed to be unique (‘‘master curve’’) for all types of ferritic steels with a reference temperature T0 for adjustment of the curve as the only remaining material dependent parameter. The motivation of using the parameters K or J as fracture parameters derives from the fact, that these quantities govern the first term of the Williams expansion of the elastic crack front stress field and the elastic–plastic HRR field, respectively, in a unique manner. Hence, the respective stress fields directly ahead of the crack front are governed by these quantities alone. On the other hand, fractographic observations in the present study as well as in previous studies by other authors show that the cleavage fracture originates from a point in some distance ahead of the crack front, which may be located outside of the dominance area of the respective asymptotic field. Hence, the stress and strain state at the cleavage origin is usually not controlled by the parameters K and J in a unique manner so that the fracture toughness determined in fracture mechanics experiments depends on the constraint situation of the respective test specimen rather than being a unique material parameter. To include this effect into the master curve concept, Wallin [17] has suggested the use of linear shift of the master curve reference temperature T0 depending on the intensity Tstress of the second term of the elastic Williams expansion. In the present study, the constraint effect in the probabilistic failure assessment is analyzed experimentally and numerically. In the experimental investigation, a variety of fracture mechanics specimens with different overall geometries, sizes and crack depths are tested, including SE(B), C(T) and CC(T) specimens with thicknesses in the range from 10 mm to 50 mm and crack depths ranging from a/w = 0.13 to a/w = 0.51. As expected, a strong geometry dependence of the master curve reference temperature is observed. Fractographic analyses of the tested specimens show that the cleavage origin is located in a range from 20 lm up to 200 lm depending on the test temperature and the final fracture load level. Finite element analyses of the test reveal that this location is in most cases located outside the area of dominance for the asymptotic K- or J-fields. To assess this effect, different two-parameter concepts are applied, including the mathematically rigorous K–Tstress- and J–A2-concepts to increase the area of dominance of the asymptotic fields as well as the phenomenological J–Q- and J–h-concepts using the Q-parameter and the stress triaxiality coefficient h as secondary fracture parameters. In all cases, the different constraint situations can be described in a unique manner in terms of the respective secondary fracture parameter. Based on these results, a constraint correction for the master curve concept is proposed, consisting of a scaling of the measured fracture toughness in dependence

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on the secondary fracture parameter similar to the specimen thickness correction. As a result, a constraint indexing reference temperature T ci0 is obtained. Alternatively, a the result can be interpreted as a failure assessment procedure using the standard reference temperature in conjunction with a two-dimensional probabilistic fracture toughness locus instead of the standard one-dimensional failure criterion. 2. Experimental investigation 2.1. Material and methods In a first approach, the effect of the stress triaxiality on the fracture toughness is investigated experimentally. The investigation is performed on a nuclear grade 22 NiMoCr 3–7 pressure vessel steel (similar to A 508, class 2). The material is taken from the cylindrical shell of a reactor pressure vessel originally intended for a power plant that finally has not been constructed. The material was forged followed by a heat treatment consisting of an austenization at T = 900 C. Subsequently, the shell was quenched in water, tempered at T = 650 C for 7.5 h followed by cooling in air. Afterwards, the vessel was cladded on its inner surface followed by a heat treatment at about T = 600 C. The chemical composition of the material is given in Table 1. For the basic mechanical characterization of the material, round tensile specimens with 5 mm diameter and 12 mm gauge length were tested in the temperature range from T = 150 C up to ambient temperature. All specimens were located between 1/4 and 3/4 of the shell thickness. The temperature dependent Young’s modulus E(T) was determined directly from the experimental output. The true stress vs. logarithmic strain curves in the plastic range were determined by a finite element simulation of the tests with an adaption of the material data such that the numerically determined load vs. elongation and load vs. necking diameter curves fit with their experimental counterparts. For investigation of the effect of the crack front constraint on the fracture parameters, an extensive experimental investigation was performed involving specimen geometries in accordance with the restrictions of ASTM Standard E 1921 [1] as well as specimen geometries violating these restrictions. In accordance to the standard, 33 SE(B) 10 · 10 specimens, 18 C(T) 25 specimens and five C(T) 50 specimens with a relative crack depth of a/w  0.5 have been tested. The tests in excess of the limits of ASTM E 1921 [1] involved 20 shallow crack SE(B) 10 · 10 specimens with relative crack depths of a/w  0.18 and a/w  0.13 (ten specimens for each crack depth) as well as seven CC(T) 100 specimens with a width of 100 mm, a thickness of 20 mm and a relative crack depth of a/w  0.5. All specimen were machined from the shell such that the crack front was located at either 1/4 or 3/4 of the shell thickness (one of the crack fronts for the CC(T) specimens). All specimens were orientated in the T–S direction with the crack growth direction parallel to the inward transverse (S) direction for the SE(B) and C(T) specimens. For the larger CC(T) specimens, the crack growth direction for the crack front at the location 3/4 was the outward transverse direction whereas the second crack front was located approximately at the center of the pressure vessel shell. The fatigue crack starter notch of the specimens was fabricated using electrical discharge machining. Fatigue precracking was performed in four stages with a maximum final load level of KJ = 17 MPa m1/2. In this context, the shallow crack SE(B) specimens were initially oversized in width in order to ensure a sufficient growth of the precrack and thus a sufficiently straight crack front. After precracking, the specimens were carefully machined Table 1 Chemical decomposition of the material (22 NiMoCr 3–7) C

Si 0.20 0.17 0.19

Mn 0.95 0.84 0.89

P

S

Ni

Cr

Cu

0.008 0.006 0.007

0.009 0.005 0.007

0.091 0.083 0.087

0.42 0.39 0.40

0.04 0.03 0.04

Maximum Minimum Average

0.25 0.18 0.22 Mo

V

Ta

Co

Al

Sn

As

Sb

Maximum Minimum Average

0.59 0.51 0.55

<0.01 <0.01 <0.01

<0.005 <0.005 <0.005

0.011 0.010 0.011

0.027 0.017 0.019

0.010 0.005 0.008

0.011 0.007 0.009

0.002 0.001 0.001

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to their final size. None of the specimens in the experimental investigation was side grooved. The fracture mechanics tests were performed in the lower transition range at T = 120 to 30 C. The C(T) and CC(T) specimens were tested in an insulated cooling chamber with temperature control by liquid nitrogen vapour. The SE(B) specimens were tested in a cooled ethanol bath. All tests were performed at a constant crosshead speed which was chosen such that an initial loading rate of K_ J ¼ 1–2:5 MPa m1/2 s1 was achieved for the C(T) and SE(B) specimens. The CC(T) specimens were tested at a lower initial loading rate of K_ J  0:5 MPa m1/2 s1 to avoid strain rate effects due to the larger plastic deformation in the vicinity of the crack front of these specimen geometry. Full details on the experimental procedures are presented in a previous report [6]. 2.2. Master curve analysis The fracture mechanics experiments are evaluated in terms of the probabilistic master curve concept according to ASTM Standard E 1921 [1] which is intended to characterize the fracture toughness under smale scale yielding conditions along the entire crack front. Using this concept, the fracture toughness K J c is determined from the J-integral at final failure  12 E KJc ¼ J c ð1Þ 1  m2 where Jc is decomposed into an elastic and a plastic part. For deep crack C(T) and SE(B) specimens, the elastic part is determined from the resulting external load F at failure using the approximate formulae given in ASTM E 1921 [1]. For the shallow crack SE(B) specimen and for the CC(T) specimen which are not permitted according to the standard, similar formulae are used as given by Tada et al. [15]. The plastic part of the J-integral Jc is evaluated from the plastic work in terms of the applied force vs. crack opening displacement record as described in ASTM E 1921 [1]. The plastic correction factors g for the non-permitted shallow crack bend bars and the center cracked panels are determined by means of finite element analyses of the tests (see Section 3.1), resulting in g = 1.61 and g = 1.26 for the SE(B) specimens with a/w = 0.18 and a/w = 0.13, respectively, as well as g = 1 for the CC(T) specimens. All fracture toughness results are thickness corrected according to  14 B 1T K J c ¼ K min þ ðK J c  K min Þ ð2Þ B0 from their actual thickness (or crack front length) B to the reference thickness B0 = 25 mm in order to account for the fact that the probability of finding a weak spot in a critical distance ahead of the crack front increases with increasing crack front length. According to the master curve concept [1,16], it is assumed that the accumulated failure probability Pf can be described by means of the three parameter Weibull distribution  m Pf ¼ 1  e



K J K min K 0 K min

ð3Þ

in terms of the stress intensity factor KJ. For ferritic steels, the threshold fracture toughness and the Weibull modulus are assumed to be fixed at Kmin = 20 MPa m1/2 and m = 4, respectively. The reference fracture toughness K0 is related to the median fracture toughness K med according to: Jc 1

4 K med J c ¼ K min þ ðK 0  K min Þðln 2Þ

ð4Þ

For the median fracture toughness, a temperature distribution (master curve) according to 1=2 K med þ 70 MPa m1=2 e0:019 J c ðT Þ ¼ 30 MPa m

 C1 ðT T

0Þ

ð5Þ

is assumed, where the reference temperature T0 is the only remaining material parameter. The master curve reference temperature is determined by adaption to the experimental data using a maximum likelihood optimization procedure (see ASTM E 1921 [1]). If only the specimen types permitted in ASTM E 1921 [1] (deep crack SE(B) and C(T) specimens) are included into the master curve analysis, a reference temperature of T0 = 63.8 C is obtained. A master curve

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Fig. 1. Experimental results.

analysis based solely on the deep crack SE(B) specimens results in a reference temperature of T0 = 70.2 C whereas a reference temperature of T0 = 51.9 C is obtained, if only the results obtained on C(T) 25 and C(T) 50 specimens are considered. The difference of DT0 = 18.3 C coincides with the results presented earlier by Joyce and Tregoning [7] for A 533 B steel. The experimental results for the fracture toughness K 1T J c are presented in Fig. 1 together with the fractiles for 5%, 50% and 95% failure probability under the assumption of T0 = 63.8 C as determined according to the current version of ASTM E 1921 [1]. The individual experimental results are also listed in Tables 3 and 4. It is observed that the 5% and 95% fractiles bound most of the experimental results obtained on specimen types permitted according to the standard. On the other hand, most of the shallow crack SE(B) specimens as well as most of the CC(T) specimens exhibit a fracture toughness significantly beyond the 95% fractile of the master curve concept, indicating a much lower reference temperature T0 if these specimen were included in the master curve analysis. Determining a master curve reference temperature for these specimen types alone results in T0 = 76.0 C for the SE(B) specimens with a/w  0.18, T0 = 92.7 C for the SE(B) specimens with a/w  0.13 and T0 = 105.3 C for the CC(T) specimens. Thus, a distinct effect of the specimen geometry on the master curve reference temperature and thus on the failure probability at a specific load level is observed. Indicated by the different reference temperatures obtained solely on deep cracked SE(B) specimens or solely on C(T) specimens, this effect occurs even for specimen types permitted by the current version of the standard. 2.3. Fractographic investigation In order to investigate the effects leading to the different failure load levels and thus to the different master curve reference temperatures T0 for the different specimen types, the fracture surfaces of most specimens tested have been analyzed fractographically. Objective of these investigations was the determination of the fracture mechanism as well as the determination of the location of the cleavage triggering sites. All fractographic investigations were performed using a scanning electron microscope. Table 2 Linear regression parameters Y

c [C]

d [C]

Yref [–]

T ci0 ½ C

Tstress/r0 Q A2 h

51.14 47.23 203.4 40.59

68.6 59.8 0.16 157.0

0.403 0.29 0.22 2.723

56.9 53.9 55.8 53.7

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Table 3 Experimental results a/w [–]

T [C]

rcl [mm]

K J c [MPa m1/2]

1/2 K 1T J c [MPa m ]

Q [–]

A2 [–]

h [–]

10 · 10 10 · 10 10 · 10 10 · 10 10 · 10

0.526 0.5321 0.535 0.531 0.5265

86 85 85 85 85

0.055 0.131 0.153 0.047 0.13

103.5 100.6 110.0 86.2 138.6

86.4 84.1 91.6 72.6 114.3

0.218 0.217 0.22 0.211 0.232

0.043 0.044 0.01 0.088 0.033

0.292 0.293 0.294 0.293 0.287

2.404 2.404 2.367 2.46 2.316

SE(B) SE(B) SE(B) SE(B) SE(B) SE(B)

10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10

0.5275 0.5263 0.5232 0.5227 0.5257 0.5259

75 75 75 75 75 75

0.046 0.101 0.09 0.094 0.076 0.063

94.1 117.0 131.7 126.0 138.6 103.2

78.9 97.1 108.8 104.3 114.3 86.2

0.216 0.224 0.231 0.229 0.235 0.219

0.062 0.0 0.021 0.009 0.038 0.034

0.299 0.296 0.292 0.293 0.29 0.298

2.443 2.369 2.347 2.358 2.329 2.41

SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B)

10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10

0.5203 0.5218 0.519 0.5214 0.5257 0.5257 0.5228 0.5269 0.5286 0.527 0.5246 0.5312 0.5274 0.5233 0.5222

67 67 70 67 70 70 70 70 70 70 70 70 70 70 70

0.089 0.159 0.116 0.054 0.059 0.01 0.037 0.113 0.078 0.035 0.063 0.146 0.163 0.237 0.119

123.4 143.5 159.6 105.9 98.5 105.7 79.9 131.4 101.1 83.7 120.5 133.5 132.7 141.0 120.4

102.2 118.2 131.0 88.3 82.4 88.2 67.6 108.6 84.5 70.7 99.9 110.3 109.6 116.2 99.8

0.224 0.234 0.24 0.217 0.233 0.236 0.223 0.251 0.234 0.224 0.245 0.229 0.229 0.233 0.223

0.016 0.047 0.075 0.021 0.031 0.012 0.094 0.039 0.021 0.075 0.013 0.055 0.044 0.049 0.016

0.295 0.29 0.286 0.3 0.308 0.306 0.312 0.297 0.308 0.313 0.3 0.295 0.294 0.29 0.296

2.362 2.325 2.3 2.402 2.506 2.489 2.573 2.441 2.495 2.551 2.463 2.321 2.332 2.328 2.364

KJ 1-5-16 KJ 1-5-17

SE(B) 10 · 10 SE(B) 10 · 10

0.5276 0.5268

65 65

0.047 0.172

89.5 139.2

75.3 114.8

0.23 0.259

0.05 0.055

0.314 0.296

2.541 2.443

KJ KJ KJ KJ KJ

SE(B) SE(B) SE(B) SE(B) SE(B)

10 · 10 10 · 10 10 · 10 10 · 10 10 · 10

0.5204 0.5245 0.5262 0.5205 0.5236

50 50 50 50 50

0.172 0.109 0.023 0.196 0.15

147.0 155.4 76.7 187.5 188.8

121.0 127.7 65.1 153.2 154.2

0.242 0.245 0.21 0.262 0.262

0.053 0.08 0.089 0.144 0.157

0.287 0.286 0.305 0.285 0.286

2.358 2.334 2.522 2.271 2.259

0.185

90

0.033

97.8

81.9

0.253

0.34

0.407

2.025

0.1832 0.1756 0.1862 0.1891 0.1819 0.1823 0.1825 0.1899

85 85 85 85 85 85 85 85

0.078 0.061 0.061 0.167 0.134 0.052 0.109 0.052

111.9 110.5 97.4 148.0 122.8 82.8 131.0 75.9

93.1 92.0 81.6 121.8 101.8 69.9 108.3 64.5

0.258 0.258 0.253 0.279 0.263 0.246 0.27 0.24

0.321 0.303 0.34 0.328 0.311 0.347 0.306 0.358

0.39 0.389 0.407 0.358 0.377 0.425 0.37 0.438

2.036 2.055 2.022 2.016 2.045 2.027 2.045 2.023

0.191

80

0.087

110.5

92.0

0.258

0.341

0.39

2.016

0.1325 0.1287 0.1309 0.126 0.1302 0.1319 0.1245 0.1348 0.1283 0.126

85 85 85 85 85 85 85 85 85 85

0.15 0.121 0.135 0.179 0.101 0.161 0.069 0.202 0.072 0.09

178.4 138.8 130.1 144.6 109.9 165.8 127.0 168.7 106.2 119.6

146.0 114.5 107.6 119.1 91.5 136.0 105.1 138.3 88.6 99.2

0.418 0.387 0.385 0.388 0.373 0.405 0.375 0.411 0.366 0.375

0.898 0.724 0.696 0.751 0.655 0.839 0.681 0.856 0.651 0.672

0.427 0.434 0.437 0.433 0.453 0.429 0.438 0.429 0.456 0.444

1.337 1.57 1.601 1.523 1.656 1.454 1.591 1.386 1.639 1.603

No.

Type

KJ KJ KJ KJ KJ

1-5-11 CL3 B23 CL3 B24 CL3 B25 CL3 B26

SE(B) SE(B) SE(B) SE(B) SE(B)

KJ KJ KJ KJ KJ KJ

1-5-3 1-5-5 1-5-6 1-5-7 1-5-8 1-5-9

KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ

1-5-1 1-5-2 1-5-4 1-5-10 1-5-18 1-5-19 1-5-20 1-5-21 1-5-22 1-5-23 1-5-24 CL3 B27 CL3 B28 CL3 B29 CL3 B30

1-5-12 1-5-13 1-5-14 1-5-15 1-6-2

KJ CL3 B51

SE(B) 10 · 10

KJ KJ KJ KJ KJ KJ KJ KJ

SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B)

CL3 CL3 CL3 CL3 CL3 CL3 CL3 CL3

B52 B55 B56 B57 B58 B59 B60 B61

10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10

KJ CL3 B54

SE(B) 10 · 10

KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ

SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B) SE(B)

CL5 CL5 CL5 CL5 CL5 CL5 CL5 CL5 CL5 CL5

B1 B2 B3 B4 B5 B6 B7 B8 B9 B10

10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10 10 · 10

Tstress/r0 [–]

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Table 4 Experimental results (continued) K J c [MPa m1/2]

1/2 K 1T J c [MPa m ]

No.

Type

a/w [–]

T [C]

rcl [mm]

A2 [–]

h [–]

KJ 1-7-3 KJ 1-8-4

C(T) 25 C(T) 25

0.5149 0.5151

90 90

0.036 0.026

42.5 65.3

42.5 65.3

0.187 0.273

0.122 0.239

0.357 0.266

2.588 2.663

KJ KJ KJ KJ KJ KJ KJ KJ KJ KJ

1-7-1 1-7-2 1-7-4 1-8-1 1-8-2 1-8-3 1-9-1 1-9-2 1-10-3 1-10-4

C(T) C(T) C(T) C(T) C(T) C(T) C(T) C(T) C(T) C(T)

25 25 25 25 25 25 25 25 25 25

0.5142 0.5141 0.5155 0.5167 0.5169 0.5152 0.5167 0.5186 0.5184 0.5153

60 60 60 60 60 60 60 60 60 60

0.032 0.109 0.03 0.075 0.095 0.036 0.04 0.042 0.042 0.027

93.9 105.0 75.2 102.0 94.8 68.5 93.1 84.2 99.7 74.4

93.9 105.0 75.2 102.0 94.8 68.5 93.1 84.2 99.7 74.4

0.382 0.425 0.353 0.411 0.39 0.324 0.382 0.35 0.444 0.353

0.294 0.304 0.301 0.29 0.291 0.295 0.286 0.276 0.303 0.302

0.22 0.208 0.236 0.213 0.22 0.247 0.222 0.234 0.21 0.236

2.707 2.706 2.784 2.695 2.697 2.794 2.694 2.695 2.752 2.785

KJ KJ KJ KJ KJ KJ

1-9-3 1-9-4 1-10-1 1-10-2 CL3 CT6 CL3 CT7

C(T) C(T) C(T) C(T) C(T) C(T)

25 25 25 25 25 25

0.5155 0.5175 0.5184 0.5172 0.5146 0.512

30 30 30 30 30 30

0.104 0.082 0.075 0.257 0.142 0.159

133.2 115.0 97.8 155.6 128.5 113.5

133.2 115.0 97.8 155.6 128.5 113.5

0.543 0.509 0.42 0.542 0.503 0.465

0.343 0.31 0.299 0.322 0.319 0.32

0.179 0.189 0.198 0.168 0.177 0.183

2.73 2.752 2.744 2.724 2.742 2.759

KJ KJ KJ KJ KJ

CL4 CL4 CL4 CL4 CL4

C(T) C(T) C(T) C(T) C(T)

50 50 50 50 50

0.5107 0.5107 0.5096 0.5107 0.5105

60 60 60 60 60

0.093 0.148 0.076 0.077 0.079

105.8 101.6 109.3 91.2 81.6

122.0 117.0 126.2 104.7 93.3

0.316 0.305 0.328 0.272 0.25

0.267 0.266 0.266 0.251 0.249

0.218 0.221 0.215 0.235 0.246

2.665 2.668 2.666 2.664 2.669

CT1 CT2 CT3 CT4 CT5

Tstress/r0 [–]

Q [–]

KJ CL2 MT3 KJ CL3 MT1

CC(T) 100 CC(T) 100

0.5153 0.5099

120 120

0.0 0.078

58.4 59.0

63.2 63.9

0.242 0.251

0.302 0.303

0.425 0.426

1.612 1.643

KJ KJ KJ KJ KJ

CC(T) CC(T) CC(T) CC(T) CC(T)

0.5144 0.5123 0.5156 0.5118 0.5137

90 90 90 90 90

0.113 0.047 0.083 0.099 0.092

143.4 125.3 121.5 122.3 130.8

154.4 138.4 134.2 135.1 144.6

0.522 0.506 0.518 0.512 0.524

0.657 0.611 0.601 0.608 0.619

0.427 0.436 0.438 0.438 0.433

1.613 1.685 1.729 1.729 1.644

CL2 CL2 CL3 CL3 CL5

MT1 MT2 MT2 MT3 MT1

100 100 100 100 100

For all specimens considered in the fractographic analysis, the cleavage triggering site were determined by tracing back the macroscopic and microscopic fracture lines towards the origin of the cleavage process. Both, the distance rcl of the cleavage triggering site to the front of the fatigue precrack and the distance of the cleavage origin from the specimen surface in the thickness direction were measured. In addition, the width rpl of the strongly deformed region ahead of the fatigue crack front covering both, blunting and limited ductile crack growth was determined. In the fractographic analysis it was observed that all specimens investigated failed by transgranular cleavage, irrespectively of the specimen size and the specimen geometry. For the specimens tested at higher temperatures and thus at higher final load levels, some ductile tearing might have preceeded the final failure event by unstable cleavage. Nevertheless, even in these cases, the length of the strongly deformed region ahead of the fatigue precrack was found to be smaller than rpl  60lm for the deep crack compact and bending specimens. Slightly larger sizes were observed for a few shallow crack bend bars, indicating a generally larger amount of plastification for this specimen type. In most cases, the cleavage triggering site was found on a grain boundary, misorientated grain boundary or grain boundary triple point. Whether the grain boundary itself or inclusions in the vicinity of the grain boundary caused the failure could not be determined. Only in few cases, isolated carbides were detected at the cleavage origin. The type of the fracture origin was found to be independent of the fracture toughness in the considered range. Nevertheless, for some specimens tested at higher temperatures and thus at higher fracture load levels, isolated accumulations of ductile voids were observed in the vicinity of carbide inclusions.

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Fig. 2. Position of the cleavage triggering points.

Manganese sulfides were observed only on few fracture surfaces. In all cases, the manganese sulfides were detected at distances of more than 300 lm from the cleavage origin and thus outside the cleavage process zone. The distance of the cleavage origin from the initial fatigue crack front increases with increasing failure load level as it can be observed from Fig. 2, where the fracture toughness of the specimens is plotted as a function of the total distance rcl of the cleavage triggering site from the fatigue crack front (including the blunting and ductile ranges). In this context, no effect of the specimen geometry is observed. All data points are found within the same scatter band. For the specimens reaching a failure load level of K J c  100 MPa m1/2 (corresponding to tests at T  T0), the cleavage origin is found in the range rcl  40–120 lm from the initial crack front. Regarding the position of the cleavage triggering site in the specimen thickness direction, no systematic effect of the failure load level or the specimen geometry was observed. All cleavage triggering sites were detected in the interior of the specimens, far from the specimen surfaces. 3. Enhanced fracture concepts 3.1. Finite element analysis To enable the assessment of the cleavage failure in terms of enhanced fracture concepts, all experiments were simulated numerically using the finite element method. The specimens were meshed by three-dimensional, pure displacement based eight-node elements using a tri-linear interpolation scheme. Full integration was used in order to circumvent the occurrence of spurious modes of deformation. The crack front was modeled as a slightly blunted crack front with a radius of 5 lm in order to avoid numerical instabilities at higher load levels. In the vicinity of the crack front, an extremely fine mesh was used to ensure a high resolution of the mechanical fields in the cleavage fracture process zone whereas a coarse mesh was sufficient far from the crack front. As an example, the mesh for the simulation of the C(T) 25 specimens is presented in Fig. 3. The bolts of the experimental loading devices were modeled as rigid bodies interacting with the specimens via a unilateral contact formulation. The analyses were performed within the geometrically nonlinear framework in order to re-capture realistic conditions for the highly deformed cleavage process zones. The finite element models were loaded by prescribed displacements of the experimental loading devices. The material was described in terms of standard incremental J2 plasticity with isotropic hardening using the yield curves for the respective temperature up to 100% plastic strain determined as described in Section 2.1. No ductile tearing was considered, since no significant ductile crack growth was observed in the experimental study. From the numerical simulation, the stress field in the ligament was determined in terms of the components rij of the Cauchy (‘‘true’’) stress tensor related to the actual configuration. The J-integral and the related stress intensity factor KJ during the loading process were determined from the resulting forces at the rigid bodies

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Fig. 3. Finite element mesh for the C(T) 25 specimens.

modeling the loading device and the crack opening displacement in a similar manner as in the experimental investigation. For comparison and for determination of the plastic correction factor g for specimen types exceeding the limits of ASTM E 1921 [1], the J-integral was determined directly from the stress and displacement results using its definition. Both approaches were found in a good agreement. In addition to the elastic– plastic analyses, purely elastic analyses with sharp cracks were performed in order to determine the elastic Tstress (see Section 3.2) at the respective failure load level. In order to investigate the reason for the different fracture toughness levels K J c obtained from experiments using different specimen geometries, the stress fields in the ligament of all six specimen types employed in the experimental investigation are computed for a unique load level of KJ = 100 MPa m1/2 at a temperature of T = 75 C. As the reference stress component, the maximum principal Cauchy stress rI is considered, since this stress component is assumed to control the possible instability of freshly nucleated microcracks in the ligament ahead of the original crack front. The results are presented in Fig. 4 as a function of the distance r to the fatigue crack front. As expected, an equivalent stress distribution for all six specimen geometries is observed directly ahead of the fatigue crack front up to r  35 lm. In this area, the stress field is dominated by the intensity J or KJ of the

Fig. 4. Stress distribution in the ligament of different specimen geometries.

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first (singular) term in the asymptotic power-series expansion. In the representation of Fig. 4, the singular nature of the stress distribution is not clearly visible since the stress is considered in terms of the Cauchy stress components related to the actual (deformed) configuration rather than to the reference configuration as considered in the original derivation of the singular fields. For increasing distances r from the fatigue crack front, increasing differences between the stress distributions of the different specimen types are observed. In this range, the stress field is no longer dominated by the first term in the power-series expansion alone. Instead, the higher order terms become increasingly important. A comparison of Fig. 2 showing the location of the cleavage triggering sites and Fig. 4 showing the corresponding stress fields reveals that the majority of the specimens which failed approximately at the considered load level of K J c = 100 MPa m1/2 had a cleavage origin outside the KJ-dominated stress field. Hence, all fracture concepts, whether deterministic or probabilistic, based on these loading parameters feature the problem of restricted transferability of the experimentally determined fracture parameters. 3.2. Two parameter approaches A possibility of considering the cleavage initiation site being outside the K- or J-dominated field but still retaining the advantages of the macroscopic approach of fracture, is the extension of the area of dominance by inclusion of additional higher order terms as stress intensities in the respective power-series expansion. Different approaches of this type are available in literature. The historically oldest two parameter concept is the K–Tstress-concept considering both the first (singular) and the second (nonsingular) term in the Williams expansion rij ðr; uÞ ¼

KI ð2pÞ

1

1 2

ð1Þ

ð2Þ

r2 fij ðuÞ þ T stress r0 fij ðuÞ þ    ð1Þ

ð6Þ ð2Þ

of the plane strain crack front stress field where fij ðuÞ and fij ðuÞ are dimensionless circumferential stress distributions. The intensity Tstress of the second term in Eq. (6) forms a secondary independent fracture parameter in addition to the primary fracture parameter KI [3]. Both parameters together define a failure envelope in the corresponding generalized plane. In the present study, Tstress is determined from an elastic simulation of the respective test loaded up to the same elastic stress intensity factor using the interaction integral [8,10]. A similar two parameter concept can be based on the power-law expansion of the elastic–plastic HRR field if higher-order singular or non-singular terms are considered [19]. In this case, the crack front stress field is represented by    rs2 r2s2 s1 r s1 ð1Þ ð2Þ ð3Þ rij ðr; uÞ ¼ A1 fij ðuÞ þ A2 fij ðuÞ þ A22 fij ðuÞ þ    ð7Þ l l l with s = 1/(n + 1) and A1  J1/(n+1) where n is the hardening exponent according to the Ramberg-Osgood constitutive equation, s2 is the exponent of the second singular or non-singular term and l is a reference length. Within this concept, the intensity A2 forms an independent secondary fracture parameter in addition to the primary fracture parameter J. The secondary fracture parameter A2 is determined as presented by Nikishkov [11] by comparison of the actual radial and circumferential stresses in the specimen and the stresses according to the asymptotic field (7) at two points located in the ligament and at u = p/4 both at a distance of r = 2J/r0 to the crack front. Both, the K–Tstress- and the J–A2-concepts are mathematically rigorous concepts to enhance the area of dominance of the asymptotic crack front fields which are characterized by the respective parameters in a unique manner. Hence, both concepts are able to provide an improved approximation of the stress fields in the cleavage process zone (see Section 2.3) compared to their single parameter counterparts. On the other hand, the fact that the cleavage origins might be located outside the area of dominance of the asymptotic K- or J-fields is not the only effect resulting in the restricted transferability of the fracture parameters obtained from experiments on different specimen types. Another important effect is the transition from a plane stress state on the surfaces of the specimen towards an approximate plane strain state in the center of the body. In contrast to the ‘‘in-plane constraint’’ effect, this ‘‘thickness constraint’’ effect cannot be re-captured by

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the K–Tstress- and the J–A2-concepts since even in their extended version, both concepts still assume a pure plane strain state. As an alternative accounting for both, the ‘‘in-plane constraint’’ effect and the ‘‘thickness constraint’’ effect, a number of phenomenological two parameter concepts have been developed. In the context of the present study, the J–Q-concept and the J–h-concept are considered. The J–Q-concept according to O’Dowd and Shih [13,14] is based on a decomposition rij ðr; uÞ ¼ rref ij ðr; uÞ þ r0 Qdij

ð8Þ rref ij

of the crack front field in the ligament into the stress field belonging to a reference case and an additional hydrostatic stress field characterized by the dimensionless intensity Q where dij is the unit tensor. The reference stress field rref ij can either be the asymptotic single parameter K- or J-field or a plane strain reference field determined numerically in a boundary layer analysis if the respective material constitutive behavior cannot adequately be approximated by Hooke’s law (K-field) or the Ramberg-Osgood constitutive equation (J-field). In the present study, the secondary fracture parameter Q is determined as the difference in the circumferential stress obtained in the simulation of the respective fracture test and the circumferential stress obtained in a plane strain elastic–plastic boundary layer analysis under prescribed displacements applied to a circular contour far from the crack front according to the load level K J c reached in the experiment. The boundary layer analysis is performed by means of the finite element method using the material parameters and yield curves as determined in the basic characterization of the material (see Section 2.1). The stress difference is evaluated at a single location in the ligament at a distance of r = 2J/r0 from the crack front. Due to the pointwise evaluation in the center of the specimen, the ‘‘thickness constraint’’ effect is included only in a simplified manner, since the variations in the mechanical field close to the specimen surfaces do not affect its determination. Nevertheless, since the main focus of the present study is the ‘‘in-plane constraint’’ effect, this simplification might cause no difficulties. As an alternative to the J–Q-concept, the J–h-concept is considered. This concept employs the ratio rkk h¼ ð9Þ 3re of the hydrostatic stress rkk/3 and the v. Mises equivalent stress re as a secondary fracture parameter. Both stresses are determined from the results of the simulation of the respective test. In contrast to the previous approach, no reference stress field is considered. The stress triaxiality ratio h is evaluated similar to the J–Q-concept at a single point in the ligament located at a distance of r = 2J/r0 to the fatigue crack front. 3.3. Results The results of the determination for the secondary fracture parameters for all four two parameter concepts considered are presented in Fig. 5a (Tstress/r0), b (Q), c (A2) and d (h) as a function of the test temperature T. The individual results are listed in Tables 3 and 4. In all four cases, the experiments performed on a specific specimen geometry yield similar levels of the respective secondary fracture parameter. For all four concepts, the highest levels of the secondary fracture parameter are obtained for the tests of C(T) specimens followed by the deep crack SE(B) specimens. In the tests of the shallow crack SE(B) specimens, significantly lower levels of the respective secondary fracture parameter are obtained, especially for the specimens with a crack depth ratio of a/w  0.13. The secondary fracture parameters of the CC(T) specimens are found approximately in the same range as the parameters obtained for the shallow crack bend bars with the smaller crack depth. As it has to be expected, no distinct difference between the the C(T) 25 specimens and the C(T) 50 specimens are observed, indicating that there is no specimen size or thickness effect on the secondary fracture parameters (although the considered thickness range is limited. Notice that the sequence of the levels of the secondary fracture parameters for the different specimen types reflects the sequence of the maximum principal stress distributions obtained in the cleavage process zone for the individual specimen types at a common external load level as presented in Fig. 4. Hence, all four secondary fracture parameters considered can be employed for a characterization of the external geometry, at this stage at least in a qualitative manner. Regarding the temperature dependence of the results, a strong effect of the test temperature is observed for the elastic Tstress levels of both the C(T) and the CC(T) specimens. This effect is caused by the different failure

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Fig. 5. Secondary fracture parameters.

load levels K J c obtained at different temperatures (see Fig. 1). In the present study, the parameter Tstress/r0 is determined in a purely elastic analysis up to the same external load level as in the (elastic–plastic) experiment in a similar manner as suggested in an earlier study by Wallin [17] by the use of approximate formulae based on linear elastic solutions. Hence, the parameter Tstress/r0 is directly proportional to the applied load level and thus combines the influence of the local crack front constraint situation and the external load level. No such temperature effect on Tstress/r0 is observed in case of the SE(B) specimens with a/w = 0.51. The majority of these specimens has failed in a fully plastic state, where the external load level and thus the elastic stress intensity factor increases only slightly with increasing prescribed displacement of the loading device. Hence, although the elastic–plastic fracture toughness K J c of these specimens increases with the test temperature, the resulting external force at failure and thus the elastic T-stress does not exhibit a significant increase with temperature. A similar behavior is observed for the Q parameter. The other two secondary fracture parameters A2 and h are normalized quantities where A2 is normalized directly with the J-integral whereas h is the ratio of two stress quantities. Both parameters are determined direcly from elastic–plastic solutions. Hence, none of these parameters exhibits a distinct explicit load or temperature dependence when determined at the instant of specimen failure. The temperature effect observed in case of the stress intensity A2 for the C(T) specimens in Fig. 5c might be explained by the slow convergence of the determination procedure requireing a sufficient load level. The most stable results are obtained in case of the stress triaxiality coefficient h determined as the ratio of two stress components determined from the same

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elastic–plastic analysis. In this case, except some experimental scatter, constant values for the secondary fracture parameter are obtained throughout the considered temperature range for all six groups of specimens. 4. Constraint correction within the master curve concept 4.1. Temperature shift In order to account for constraint effects in the master curve analysis, Wallin [17] has proposed a simple approach assuming a constraint dependent master curve reference temperature T0. The constraint dependent reference temperature is defined in terms of Tstress according to ( deep for : T stress > 0 T0 T 0 ðT stress Þ ¼ ð10Þ deep T stress  C T 0 þ 10 MPa for : T stress 6 0 where T deep denotes the reference temperature determined on deep crack SE(B) and C(T) specimens according 0 to ASTM E 1921 [1]. Since the elastic Tstress used in Eq. (10) is load dependent, the average value for each group of specimen types is used in most cases. As a result, a component related master curve for the actual constraint situation can be established. In order to assess the validity of Eq. (10) for the present material concerning the scatter in both the fracture toughness K J c and the respective secondary fracture parameter Tstress/r0, Q, A2 or h, an individual master curve reference temperature T0 is computed for each individual specimen in the experimental data base. The result is presented in Fig. 6a–d as a function of the respective secondary fracture parameter. In addition, a linear regression analysis of the experimental data is performed, where the results are approximated by a function T 0 ðY Þ ¼ cY þ d

ð11Þ

in terms of the constraint parameter Y (Tstress/r0, Q, A2 or h, respectively). The parameters c and d are determined by means of a least square minimization. The results are given in Table 2. As expected, a distinct scatter is observed in the individual reference temperatures T0 for all six specimen types considered in the experimental investigation. This scatter is directly caused by the scatter in the fracture toughness values K J c measured in the individual tests. The scatter in the secondary fracture parameters in most cases is found to be much less distinct. This result again underlines the capability of the secondary fracture parameters to characterize the constraint situation of the individual specimen types. The constraint correction to the master curve reference temperature according to Eq. (10) as proposed by Wallin [17] is found to be in good agreement with the present experimental results (see Fig. 6a). The linear regression analysis of the present data in terms of Tstress/r0 (see Fig. 6a) results in a regression line with almost identical slope, shifted slightly towards lower reference temperatures. By means of this regression line, Wallin’s [17] constraint correction, which was originally restricted to negative Tstress values can easily be extended into the positive range. As it can be observed in Fig. 6b to d, a similar linear constraint correction of the master curve reference temperature can be performed with equally good results in terms of the alternative secondary fracture parameters Q, A2 and h. As pointed out before, the advantage of these alternative secondary fracture parameters is that – in contrast to the elastic Tstress – these parameters are load independent due to their definition. Furthermore, especially the stress triaxiality ratio h has the advantage to account for thickness constraint effects which cannot be captured by the plane KJ–Tstress- and J–A2-concepts. 4.2. Fracture toughness scaling The definition of a constraint dependent master curve reference temperature in terms of the secondary fracture parameter Tstress/r0, Q, A2 or h provides a simple engineering approach for a constraint dependent probabilistic assessment of cleavage fracture. Through this procedure, a specimen or component specific master curve is established. Using this constraint specific master curve requires that the local constraint situation

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Fig. 6. Effect of the secondary fracture parameters on the master curve reference temperature T0.

at the crack front remains constant during the loading history of the structural component to be assessed, at least in an approximate manner. On the other hand, complex loading histories in structural application might in many cases involve transient local stress situations with constraint parameters Tstress/r0, Q, A2 or h varying with time. In these cases, the use of a constraint dependent master curve can be difficult, since a different constraint and material dependent limit curve has to used for the assessment of each individual instant of time. Furthermore, the definition of a constraint dependent master curve reference temperature T0(Y) results in the introduction of a material parameter T0 which additionally depends on the actual loading situation. Depending on the philosophy in the assessment of structural integrity, the combination of material and (local) loading effects in a single parameter might be considered as unsatisfactory from the theoretical point although there is no ‘‘correct’’ or ‘‘incorrect’’ treatment of this effect. Nevertheless, an alternative concept based on a constraint related reference temperature and an alternative constraint correction strategy might be advantageous. A concept of this type can be defined based on the linear shift of the reference temperature as defined in Section 4.1. Since the temperature distribution (5) assumed in the master curve concept depends on both, the actual temperature T and the reference temperature T0 solely in terms of the difference T  T0, the constraint dependent linear shift dT0 can be applied with the identical result either to the reference temperature T0, as proposed before, or to the actual temperature T. In the first strategy, the material curve would be shifted in dependence on the actual constraint situation as in Fig. 7a. In the latter

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Fig. 7. Strategies for constraint corrected master curve analysis.

Fig. 8. Constraint corrected master curve analysis.

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strategy, the material curve would remain independent from the loading situation Y, whereas the data points KJ(T) would be shifted in temperature direction, resulting in the identical failure probability (see Fig. 7b). Although the alternative concept based on a constraint dependent shift of the actual testing or service temperature T implies a proper separation of material and loading parameters, the strategy is still unsatisfactory from the theoretical point of view, since it is based on the definition of a constraint dependent temperature. This problem can easily be avoided since the master curve (5) is defined as an exponential function. One of the basic features of this type of function is that a shift in the direction of the argument coincides with a scaling of the function itself. Hence, the temperature shift of the data points as indicated in Fig. 7b can be replaced by a constraint dependent scaling of the stress intensity factors according to 1=2 K ref þ ðK J ðT Þ  30 MPa m1=2 Þe0:019cðY Y J ðT ; Y Þ ¼ 30 MPa m

ref Þ

ð12Þ

where c is the slope of the linear regression curves in Fig. 6 or Eq. (11), respectively, whereas Yref denotes the constraint parameter for a reference case. By means of Eq. (12), the current loading situation in terms of the actual stress intensity factor is corrected to a reference case, e.g. the case of the C(T) 25 specimens. Through this procedure, a constraint indexing master curve reference temperature T ci0 is established, defining a material curve independent from the actual loading situation. Advantage of this constraint correction is the proper separation of the material properties in terms of T ci0 , the actual loading situation in terms of K ref J ðT ; Y Þ and the

Fig. 9. Two parameter fracture toughness envelopes at T ¼ T ci0 .

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actual temperature T. Furthermore, the constraint correction (12) of the stress intensity factors agrees with the crack front length correction (2). The constraint correction defined by Eq. (12) is applied to the present experimental data base using all four two parameter concepts considered in the present study. The results are presented in Fig. 8a for the K–Tstressconcept, in Fig. 8b for the J–Q-concept, in Fig. 8c for the J–A2-concept and in Fig. 8d for the J–h-concept. In all four cases, the case of the C(T) 25 specimens was adopted as the reference case, resulting in the reference constraint parameters Yref and the constraint indexing master curve reference temperatures presented in Table 2. It is observed that in all cases, all experimental data obtained in the experiments using different specimen geometries can be described in a unique scatter band where the 5%- and the 95% fractile bound more than 90% of the data. Hence, the present constraint correction procedure provides a transferable probabilistic fracture concept based on the global loading parameter KJ and a secondary fracture parameter Tstress/r0, Q, A2 or h, respectively. An alternative interpretation of Eq. (12) is that together with the fracture toughness curve (5) and the corresponding constraint indexing reference temperature, this equation defines a two parameter toughness locus in the plane defined by the primary loading parameter KJ and the respective secondary fracture parameter Tstress/r0, Q, A2 or h. The corresponding 5%-, 50%- and 95% toughness loci at T ¼ T ci0 are presented in Fig. 9a–d, respectively. For a more distinct representation, the experimental data points obtained at different test temperatures T are projected to T ci0 retaining their individual failure probabilities from Fig. 8. It is clearly observed that the fracture toughness K 1T J c increases exponentially with decreasing constraint parameter Tstress/r0, Q, A2 or h, respectively. This exponential decrease results in strong effects especially for low constraint situations, whereas only less distinct effects have to be expected for constraint situations beyond the case of the C(T) specimens. A comparison of the different two parameter concepts shows that all four concepts yield almost equivalent results, except some outliers far below the 5% toughness locus of the J–A2-concept. 5. Conclusion The present study is concerned with the probabilistic failure assessment of specimens and structures consisting of ferritic steels considering constraint effects. In an experimental investigation, a variety of different fracture mechanics specimens with different overall and local geometries including C(T), SE(B) and CC(T) specimens with different crack depths has been tested. In a fractographic analysis, the cleavage triggering sites have been determined, which were found outside the range where the mechanical stress and strain fields can be described uniquely by the stress intensity factor or the J-integral, characterizing the first (singular) terms of the Williams expansion of the elastic crack tip field and the elastic–plastic HRR field, respectively. In order to enhance the are of dominance of the asymptotic fields, different two parameter approaches are applied, including the K–Tstress-, J–Q-, J–A2- and J–h-concepts. All concepts are found to be able to characterize the local constraint situation of the different specimen geometry types considered. In a master curve analysis, it is shown that the master curve reference temperature depends approximately linearly on the respective secondary fracture parameters of all four concepts. By the definition of a constraint dependent reference temperature, a constraint dependent and thus component specific material curve can be established. This material curve can be used in a failure assessment of the respective component, provided that the constraint situation remains at least approximately constant during the loading history. On the other hand, difficulties may arise, if the distinct variations of the constraint situations occur during a complex transient loading history. As an extension, an alternative constraint correction for the master curve concept is proposed. This approach consists in a scaling of the stress intensity factor as the primary loading parameter and thus implies a clear separation between the material properties, the loading parameters and the temperature. Furthermore, the constraint dependent scaling of the fracture toughness is a similar procedure as the crack front length correction and therefore fits with the general procedure of the master curve concept in a natural manner. In an application of the proposed procedure to the experimental data base it is shown that all experimental data, irrespectively of the local stress state in the ligament can be described in a unique scatter band defined by a single constraint indexing reference temperature which does not depend on the specimen geometry. Hence, the proposed procedure defines a transferable scheme for the probabilistic failure assessment of ferritic steels.

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It is shown that distinct effects of the secondary fracture parameter have to be expected especially for low constraint situations whereas the effect at high constraint situations is much less distinct. Advantage of the concept for the engineering failure analysis is that it accounts for the local stress situation in the cleavage fracture process zone but retains the numerical efficiency associated with the global approach in terms of a macroscopic loading parameter. Hence, the concept can be adopted for an efficient reliable failure assessment with enhanced precision and reduced uncertainties compared to the standard single parameter concepts. It enables the reliable reduction of conservativities in structural application and therefore can be employed towards an economically more efficient failure tolerant design. Acknowledgement This work has been financially supported by the German Federal Ministry of Economics and Labor (BMWA) under contract no. 150 1239 (reactor safety research program). References [1] ASTM Standard E 1921-02. Standard test method for determination of reference temperature T0 for ferritic steels in the transition range. Annual Book of ASTM Standards. West Conshohocken, PA: American Society for Testing and Materials; 2002. [2] Beremin FM. A local criterion for cleavage fracture of a nuclear pressure vessel steel. Metallurg Transact 1983;14:2277–87. [3] Du ZZ, Hancock JW. The effect of non-singular stresses on crack-tip constraint. J Mech Phys Solids 1991;39:555–67. ¨ berg H. A probabilistic model for cleavage fracture with a length scale – parameter estimation and [4] Faleskog J, Kroon M, O predictions of stationary crack experiments. Engng Frac Mech 2004;71:57–79. [5] Gao X, Ruggieri C, Dodds RH. Calibration of Weibull stress parameters using fracture toughness data. Int J Frac 1998;92:175–200. ¨ berpru¨fung des Mastercurve-Ansatzes im [6] Hohe J, Tanguy B, Friedmann V, Sto¨ckl H, Bo¨hme W, Varfolomeyeva V, etal. Kritische U Hinblick auf die Anwendung bei deutschen Kernkraftwerken. Report No. S8/2004. Freiburg: Fraunhofer Institut fu¨r Werkstoffmechanik; 2004. [7] Joyce J, Tregoning R. Investigation of specimen geometry effect and material inhomogeneity effects in A 533 B steel. ECF 14. In: Neimitz A, Rokach I, Kocanda K, Golos K, editors. Fracture mechanics beyond 2000. Sheffield: EMAS Publishing; 2002. [8] Kfouri AP. Some evaluation of the elastic T-term using Eshelby’s method. Int J Frac 1986;30:301–15. [9] Kroon M, Faleskog J. A probabilistic model for cleavage fracture with a length scale-influence of material parameters and constraint. Int J Frac 2002;118:99–118. [10] Nakamura T, Parks DM. Determination of elastic T-stress along three-dimensional crack fronts using an interaction integral. Int J Solids Struct 1992;29:1596–611. [11] Nikishkov GP. An algorithm and computer program for the three-term asymptotic expansion of elastic–plastic crack tip stress and displacement field. Engng Frac Mech 1995;50:65–83. [12] Mudry M. A local approach to cleavage fracture. Nucl Engng Design 1987;105:65–76. [13] O’Dowd NP. Applications of two-parameter approaches in elastic–plastic fracture mechanics. Engng Frac Mech 1995;52:445–65. [14] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter I. Structure of fields. J Mech Phys Solids 1991;39:989–1015. [15] Tada H, Paris P, Irwin G. The stress analysis of cracks handbook. St Louis, MO: DEL Research Corp; 1985. [16] Wallin K. Statistical modeling of fracture in the ductile-to-brittle transition range. In: Blauel JG, Schwalbe KH, editors. Defect assessment in components. Fundamentals and applications. London: Mechanical Engineering Publications; 1991. p. 415–45. [17] Wallin K. Quantifying T-stress controlled constraint by the master curve transition temperature T0. Engng Frac Mech 2001;68: 303–28. [18] Xia L, Shih CF. Ductile crack growth – III. Transition to cleavage fracture incorporating statistics. J Mech Phys Solids 1996;1996: 603–39. [19] Yang S, Chao YJ, Sutton MA. Higher order asymptotic crack tip fields in a power-law hardening material. Engng Frac Mech 1993; 45:1–20.