Cracking instabilities in a low-carbon steel susceptible to dynamic strain aging

PII:

Acta mater. Vol. 46, No. 6, pp. 1933±1948, 1998 # 1998 Acta Metallurgica Inc. Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain 1359-6454/98 $19.00 + 0.00 S1359-6454(97)00423-0

CRACKING INSTABILITIES IN A LOW-CARBON STEEL SUSCEPTIBLE TO DYNAMIC STRAIN AGING R. MOHAN{ and C. MARSCHALL Battelle Memorial Institute, Columbus, OH 43201, U.S.A. (Received 23 May 1997; accepted 24 October 1997) AbstractÐCrack jumps occurring in concert with stable cracking in a low-carbon steel, used in nuclear power plants, are examined. These jumps occur in steels susceptible to dynamic strain aging at liquid water reactor temperatures. Through fractographic studies, a connection between the nature of cracking and fracture morphology is attempted. Numerical simulation of a crack growth experiment is carried out. The results of the study indicate that there may be certain conditions, such as crack-tip constraint and triaxiality and the dominance of elastic stored energy over plastic dissipation (in addition to the material's susceptibility to dynamic strain aging), which need to be met for crack jumps to occur. The numerical results also indicate that approximate methods, such as ASTM E1152 method, for estimating driving force for growing cracks may not be viable for materials susceptible to dynamic strain aging. # 1998 Acta Metallurgica Inc.

1. INTRODUCTION

It is well known that some low-carbon steels are susceptible to strain aging, involving a gradual change in certain properties with time (usually an increase in hardness and strength) following plastic straining. The property changes generally occur slowly near room temperature and more rapidly at higher temperatures, because di€usion of the elements responsible for aging is aided by raising the temperature. When property changes occur slowly, the phenomenon is commonly referred to as static strain aging; when they occur rapidly, speci®cally when the aging occurs simultaneously with the straining, the phenomenon is commonly called dynamic strain aging (DSA). Hence, the terms static and dynamic as used here refer to the rate of aging rather than to the rate of straining. If the rate of plastic straining is increased, the temperature at which dynamic strain aging is observed will be raised to enable the di€using atoms to keep pace with the moving dislocations, which produce the plastic strain. That the dynamic strain aging phenomenon in steels depends both on temperature and on strain-rate was demonstrated by Manjoine [1]. He showed that the tensile strength peak as well as the temperature range in which serrated stress±strain curves are observed, move to higher temperatures as strain rate is increased and vice versa. It is well established that both static and dynamic strain aging in steels are the result of interactions between dislocations and dissolved interstitial solute {E-mail: [email protected]

atoms, principally nitrogen and carbon [2±4]. Various features of the stress±strain curve, particularly in the DSA region, have provided a fertile ground for developing dislocation-based models [5± 8]. Atoms, such as nitrogen and carbon, that occupy the interstitial positions are slightly too large for the interstitial sites in the body-centered-cubic iron crystal lattice [9] and thus create a local mis®t stress when they are present. This mis®t energy can be mitigated when the interstitial atoms are relocated to the core region of dislocations, because in the dislocation core region the normal atom spacing is expanded slightly. If sucient time is available for di€usion (dependent on temperature), the nitrogen and carbon atoms will move to the dislocation core region, thereby lowering the mis®t energy. According to Baird [3], nitrogen and carbon have similar di€usion coecients in iron and distort the ferrite lattice nearly identically. Hence, it would be expected that the two elements will produce similar strain aging e€ects in steel. However, near room temperature, the solubility of carbon in ferrite is so smallÐa factor of perhaps 100 less than the solubility of nitrogen [10]Ðthat it is commonly assumed that nitrogen, rather than carbon, is chie¯y responsible for strain aging. While this may be the case for temperatures below approximately 1008C (2128F), at higher temperatures the increased solubility of carbon can cause strain aging even in the absence of nitrogen [10]. Thus, when both elements are present, both can contribute to strain aging at room temperatures above about 1008C (2128F), although the nitrogen may play a larger role because of its greater solubility at all temperatures. The amount of dissolved nitrogen or carbon needed

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to cause some evidence of strain aging is extremely small (within the range of 0.0004±0.002 wt%) [10, 11]. Some carbon steel pipes and pressure vessels used in the nuclear-power industry are susceptible to DSA. From a positive standpoint, carbon steels that are susceptible to DSA exhibit improved strength, creep resistance, and fatigue resistance within the DSA temperature range. However, they also exhibit several features that may be detrimental to their performance in reactor coolant pipes or pressure vessels. For example, when a steel susceptible to DSA is plastically deformed in the DSA temperature range, an unusually high dislocation density may result coupled with the immobilization of many of the dislocations by nitrogen and/or carbon atmospheres. Such microstructural changes might be expected to produce a shift in the ductileto-brittle fracture transition temperature. Li and Leslie [12] have shown that a tensile prestrain of only 3% at 2508C in a steel susceptible to DSA can cause the ductile-to-brittle fracture transition temperature to increase by as much as 458C. Formby and Charnock [13], report an increase in the ductile-to-brittle transition temperature of 76 to 1188C, depending on the number of fully reversed bend cycles (1±5 cycles) applied to a pre-cracked C±Mn steel specimen. Formby and Charnock [13] likened this situation to a pressure vessel containing a sharp defect that experiences plastic strain near the crack tip when subjected to service stresses in the DSA range. It is widely assumed that carbon steels tested above their ductile-to-brittle fracture transition temperature will show constant or slightly increasing fracture resistance as the temperature is raised. This assumption de®nitely is not true of steels that are susceptible to DSA, as was demonstrated over 70 years ago by Maurer and Mailander [14], who showed that the energy required to break the keyhole-notched, mild-steel bend specimen in quasi-static tests at 200±2508C (390±4808F) was about half that required at room temperature. Miglin et al. [15], in a study of strain aging e€ects in unloading-compliance J tests, reported that several steels susceptible to DSA displayed lowered Jlc values in the DSA temperature range. Miglin et al. [15] also reported that the tearing modulus, proportional to the slope of the J-resistance curve, diminished signi®cantly with increasing temperature in the DSA temperature range for a steel that was susceptible to DSA. Mukherjee [16] also has reported that Jlc and J-R curves for SA 106 Grade B pipes of several diameters were lower at 2008C (3928F) than at 208C (688F), both in base metal and in weld metal. In addition to having reduced fracture resistance at liquid water reactor (LWR) temperatures, carbon {Unpublished data from DTRC

steels susceptible to DSA frequently exhibit bursts of rapid, unstable crack growth (ductile in nature) at those temperatures, interspersed with periods of slow, stable crack growth. Tests conducted as part of the Nuclear Regulatory Commission's (NRC) Degraded Piping Program (Wilkowski et al., [17]) were among the ®rst in which crack jumps were reported to occur at service temperatures. Such instabilities in crack growth were observed both in the case of laboratory specimens and in full-scale tests of pipe used in the nuclear-power industry. Another example of a signi®cant ductile crack instability was reported by David Taylor Research Center{ (DTRC) for a through-wall circumferentially cracked pipe in a metal inert-gas weld, tested at 2888C (5508F). Such instabilities raise concern over their e€ects on leak-before-break and ¯aw evaluation criteria used by the nuclear industry. The objective of this study was to evaluate the e€ects of ductile crack instabilities, believed to be due to dynamic strain aging, on the fracture behavior and morphology of ferritic pipe steel at LWR temperatures. In addition, results are presented which evaluate the ability of current J-based analysis methodology to assess the e€ect of unstable crack jumps on the fracture behavior of ferritic steel pipe. Prior to discussing the results, a brief discussion addressing the causes of the observed cracking instabilities in DSA sensitive steels is presented. 2. CAUSES OF DUCTILE CRACKING INSTABILITIES

Although the presence of ductile cracking instabilities in a DSA sensitive ferritic steel was documented only recently by Wilkowski et al. [17] and Marschall et al. [18], it appears that this phenomenon was disregarded as experimental error in measurement in several tests conducted previously by several investigators. Possible explanations for the phenomenon of ductile cracking instability may include excessive machine compliance and the occurrence of dynamic strain aging. With respect to the ®rst candidate, it is well established that crack instabilities are favored in both laboratory specimens and pipes by steep unloading slopes in the load/displacement curve beyond the maximum load point, and by highly compliant loading systems. In both the pipe fracture experiments and the compact specimen tests reported by Wilkowski et al. [17] and Marschall et al. [18], the load-train compliance was substantially less than that necessary to cause crack instabilities, based on an analysis described by Wilkowski et al. [19]. In addition, the pipe test at DTRC, described previously, in which the crack jumped approximately one-quarter of the pipe circumference, was reported to be a low-compliance test. It was argued by Marschall et al. [18, 20] that the excessive machine compliance can cause a ductile-

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fracture instability only under falling load in displacement-controlled testing conditions; in their tests of cracked pipes and some laboratory C(T) specimens the ®rst crack jump occurred on the rising part of the load/displacement curve (prior to maximum load). They also found that in many C(T) tests relatively larger crack jumps were accompanied by lower negative slope of the load-displacement curve than that resulting from smaller crack jumps. Had the instabilities been caused by excessive machine compliance, an opposite result would have been expected. Based on the work of Marschall et al. [18] on steels with di€erent degrees of susceptibility to DSA, it appears that there is some connection between DSA and crack jumps. However, data exist which do not support a direct link between crack instabilities and DSA. For example, some of the pipe steels, which have shown a marked susceptibility to DSA in tensile tests, have not displayed unstable cracking in compact-specimen tests. It appears that in addition to the material's susceptibility to DSA certain other conditions need to be met to cause cracking instability. In addition to a lack of consistent results supporting a direct link between DSA and crack instabilities, no satisfactory explanation has been o€ered of how DSA can cause crack jumps. While

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an explanation of this connection is outside the scope of this work, it can be conjectured that crack jumps are analogous to serrated yielding which sometimes accompanies DSA and which results from a sudden tearing away of dislocations from their pinning atmospheres of nitrogen and carbon atoms. Crack jumps might also be likened to the occurrence of an upper yield point in steels susceptible to strain aging. At the point of yielding, the load falls sharply and localized straining occurs (Luders bands) with little or no strain hardening. At the tip of a growing crack, a similar lack of strain hardening could produce highly localized strain and promote crack extension with little energy absorption from the surroundings. Indirect evidence of this possibility is obtained from the fracture morphology study detailed in the next section. 3. EXPERIMENTAL RESULTS

In this section, results on the occurrence of crack jumps and on the fracture morphology C(T) specimens machined from A515 Grade 60 low carbon steel pipe are presented. This steel contained 0.13 wt% C, 0.13% Ni, 0.8% Mn, 0.25% Si, 0.13% Cr, 0.027% S, 0.13% Cu and much smaller amounts of

Fig. 1. Optical microstructure of the steel along the plane of crack (100). Notice that the ferrite grains are fairly equiaxed and that there is no evidence of deformation banding.

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P, Mo, Sn and Al. The microstructure of the steel along the crack propagation plane, shown in Fig. 1, indicates the presence of fairly equiaxed ferrite grains intermixed with pearlite regions. The ®gure reveals no evidence of deformation banding and of the presence of elongated inclusions along the fracture plane. Both of these features, if present, are likely to further complicate the process of cracking instability of the steel in the DSA range. Tensile and hardness tests indicated that this steel was susceptible to DSA within the temperature range of 150±3858C. More comprehensive experimental information on several steels having di€erent degrees of susceptibility to DSA obtained to develop correlation among material properties and DSA, is given by Marschall et al. [18]. 3.1. Crack growth tests Precracked and side-grooved 1 T compact specimens were machined from the A515 Grade 60 carbon steel pipe. The specimens were tested in crosshead control in a screw-driven Instron machine at several temperatures that ranged from 150 to 3858C (300 to 7258F). The crosshead speed was selected to cause crack initiation in approximately 5±10 min. Data obtained during each test included load, load-line displacement, and direct-current electric potential. The latter quantity was monitored to indicate the point of crack initiation and the amount of crack extension. Details of the experimental estimation of the point of crack initiation are found in Marschall et al. [18]. Figure 2 shows load-displacement curves obtained from the C(T) tests. The ®gure reveals that this steel exhibited crack jumps, as evidenced by sharp load drops, at

temperatures (232 and 2888C) within the DSA sensitive range as determined from tensile and hardness tests. Less pronounced crack jumps are evident at other temperatures. This ®nding lends additional support to the hypothesis that the occurrence of crack jumps is associated with a steel's susceptibility to DSA. Additional compact-specimen tests were conducted at 2888C (5508F) to determine the reproducibility of the crack-jump phenomenon. Each of the compact specimens was a 1 T planform size and had a thickness of 20.8 mm (0.82 in.); side grooves had a depth of 10% per side. A total of four tests were conducted at 2888C (5508F) on compact specimens. Each of the four specimens exhibited at least two signi®cant crack jumps, as indicated by the sharp load drops in the load-displacement curves shown in Fig. 3. It can be seen that the load-displacement curves and crack jumps obtained from the di€erent specimens show marked variability. The di€erence in the maximum load measured from the four tests indicates that there may have been some di€erences in the specimen thickness (in the side grooved region) and the initial crack length. These di€erences may in part have caused the observed variability in the nature of unstable cracking. On the other hand, the fracture morphologies of these specimens, as described in the following, do show similarities. In addition to DSA, other necessary conditions that trigger crack jumps are not conclusively known at this time. Thus, the phenomenon of ductile cracking instabilities may not necessarily be as random as indicated by the load-displacement curves. Analysis of the data obtained during the crack jumps indicates that the crack velocity during

Fig. 2. Load vs displacement curves for C(T) specimens of A515 Grade 60 steel.

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Fig. 3. Load±displacement curves for four C(T) specimens of A515 Grade 60 steel.

a crack jump was orders of magnitude slower than a cleavage crack, no greater than approximately 0.5 m/s. Nonetheless, that velocity is approximately 40,000 times greater than the value of 1.25  10ÿ5 m/s estimated for the slow, stable crack extension during the remainder of the compact-specimen test.

3.2. Fractographic and metallographic features associated with crack jumps Visual examination of the fracture surface of the four specimens tested at 2888C revealed both dark and light bands, indicative of di€erent degrees of oxidation on the fracture surfaces. The dark bands

Fig. 4. Photograph of a portion of the fracture surface of Compact Specimen F26-108 (A515 Grade 60 steel); specimen was tested at 2888C (5508F) and displayed two signi®cant crack jumps in Areas B and D.

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Fig. 5. SEM fractographs of compact Specimen F26-108 (A515 Grade 60 steel) from dark and light regions in Fig. 4: (a) Area A and (b) Area B.

MOHAN and MARSCHALL: CRACKING INSTABILITIES

appeared to be associated with slow, stable crack growth and the light bands with rapid crack jumps. The dark/light sequence seemed contrary to intuition which might have supposed that there would

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be a more gradual gradation in fracture-surface color from dark to light as the crack progressed from its origin to its ®nal position at the end of the test, as a result of oxidation time. However, it

Fig. 6. Magni®ed fractographs of regions shown in Fig. 5 and corresponding to stable and unstable crack propagation regions: (a) Area A and (b) Area B.

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appears that the dark/light sequence may possibly result from di€erences in re¯ectivity of the fracture surfaces due to di€erences in surface roughness, which in¯uence the degree of oxidation. The fractographic observations, described in the following, indicate that a rougher surface may result during stable crack extension. Figure 4 is an enlarged photograph of a portion of the fracture surface of one of the specimens which displayed two signi®cant crack jumps when tested at 2888C (5508F). The boundaries between the light regions (Regions B and D) and dark regions (Regions A, C and E) have been highlighted with a dark ink line. Also shown on the photograph is the crack length at the dark/light interface, relative to the fatigue-precrack tip, along the mid-thickness line. These dimensions were useful in positioning the specimen in subsequent examinations in a scanning electron microscope (SEM). It appears in Fig. 4 that the crack front extended fairly uniformly during the initial stable growth. However, the crack front at the end of the ®rst crack jump showed greater advance at mid-thickness than near the sides, indicative of tunneling. Some straightening of the crack front appears to have occurred during the second period of stable growth but, once again, additional tunneling is evident after the second crack jump. Specimen No. F26-108 was subjected to examination in a scanning electron microscope in an attempt to discern fractographic di€erences between the slow-crack-growth and the fast-fracture areas. Areas along the specimen mid-thickness line were examined, both within the dark regions and the light regions and at the boundaries between the two regions. Figure 5 shows SEM fractographs taken in Areas A and B (refer to Fig. 4). Each region consisted entirely of ductile dimpled rupture. However, some di€erences were apparent in the fracture appearance of the two regions; stable crack growth was accompanied by smaller void size (i.e. smaller mean void spacing) and greater void density as can be seen in Fig. 5(a), than the corresponding fracture appearance for unstable crack growth as shown in Fig. 5(b). The fracture morphologies of these regions at a higher magni®cation of 5000 shown in Fig. 6(a) and (d) further reinforce this conclusion. Fracture surfaces of regions C and E were similar to that shown in Fig. 5(a) corresponding to stable crack growth, while that of region D was similar to Fig. 5(b) (see Marschall et al. [18]). Similar observations made on a di€erent specimen (No. F26± 107) are reported in Marschall et al. [18]. The process of growth and coalescence of microvoids involving ductile tearing appears to be the predominant mode of failure during the stable crack growth region, as demonstrated by Fig. 5(a) and 6(a), corresponding to region A. On the other hand, to accommodate the crack jump, a few of the larger microvoids, which probably had nucleated prior to

the jump, grow rapidly at the expense of other microvoids, as shown in Fig. 5(b) and 6(b), corresponding to region B. Notice that these larger voids in region B show no evidence of being nucleated at a particle±matrix interface. These observations indicate that the fracture surface resulting from stable crack extension may possess a higher degree of roughness than the one resulting from a crack jump. This di€erence could have contributed to dark/light sequence shown in Fig. 4. Additional microscopic observations reported by Marschall et al. [18] reveal no discernible di€erences in the microstructure of regions associated with stable crack extension and crack jumps. Based on these observations, it appears that the absorbed energy associated with the separation of a unit length of crack face will be much smaller during the crack jumps than that during the stable crack extension. Since the load-displacement curves shown in Fig. 3 indicate load drops during crack jumps, the energy absorbed during crack jumps primarily comes from the release of stored elastic energy, which is triggered when the conditions are sucient for ferrite grain(s) immediately ahead of the crack tip to experience dynamic strain aging. 4. ANALYSIS OF FRACTURE RESISTANCE DURING STABLE AND UNSTABLE CRACK GROWTH

A numerical simulation of one of the experiments was attempted in order to gain some understanding of the e€ect of unstable cracking processes on the crack-tip deformation and its implication on the evolution of fracture parameters. As was seen earlier, DSA is a result of repeated pinning and unpinning of dislocations by interstitial atoms during straining. Such a behavior at the lattice level is re¯ected macroscopically by the formation of deformation bands and serrations in the stress±strain curve, both of which represent material instability. An additional concern involves the fact that only the ferrite phase experiences DSA. As seen in the previous section, the considered steel, though predominantly ferrite, does have about 10±15% pearlite. These aspects along with the available experimental data for this material make appropriate constitutive modeling of such a material quite dicult. In this analysis, the material is modeled within the framework of J2 ¯ow theory as having a smooth stress± strain response that corresponds to the quasi-static tensile test at 2888C (5508F). In addition, the basic input to the analysis is the experimentally measured relationship between crack growth and load-line displacement. Thus, it is impossible for the analysis to attempt an explanation for the cause of repeated stable and unstable cracking in these materials. However, it may provide some insight into the crack-tip behavior when crack jumps do occur. A numerical simulation of crack growth behavior in the 1 T C(T) test using Specimen No. F26-103,

MOHAN and MARSCHALL: CRACKING INSTABILITIES

performed at 2888C (5508F), was conducted. This specimen was chosen because of the excellent correlation obtained between the crack length measurements by d.c electric potential method and from fracture surface appearance, in addition to its exhibiting the ®rst crack jump very early in the decreasing-load part of the load displacement curve. 4.1. Modeling approach The analysis conducted relied on a small-displacement-gradient formulation of rate-independent J2 ¯ow theory of plasticity [21]. It is well known that under small-scale yielding conditions, the stationary crack-tip deformation and stress ®elds obtained from ®nite deformation and small strain formulations di€er only in a small region immediately ahead of the crack tip [22]. In addition, recent work on ductile crack growth [23] demonstrated that ®nite deformations have very little e€ect on the crack opening pro®les and the crack-tip stress ®eld. Although these observations are made only for small-scale yielding (SSY), it is not clear whether ®nite deformations play an important role under large scale yielding. From some preliminary analyses involving a very small amount of crack growth, it was observed that the results obtained using ®nite-deformation formulation di€ered insigni®cantly from the corresponding results obtained using small-strain formulation. The analysis assumed rate-independent behavior at 2888C (5508F), since the C(T) tests were conducted at very low displacement rates. Of more importance is the issue of inertial e€ects on the crack growth process during and after crack jumps. As stated earlier, the crack speed during the crack jumps is over 40,000 times the speed during stable crack growth. However, the crack speed during jumps is only a

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few percent of the Rayleigh wave speed in steels; thus, inertial e€ects were also neglected in this analysis. The analysis was carried out under both planestrain and plane-stress conditions. Since the C(T) specimen had side-grooves, the crack opening displacement and energy release rate tend to be fairly uniform across the thickness [24], thus making twodimensional analysis adequate. As mentioned earlier the experimentally observed relation between crack growth and applied load-line displacement was used to advance the crack in the analysis. The ABAQUS ®nite-element software was used in conducting the analysis [25]. The ®nite element mesh developed to simulate the C(T) test consisted of 887 eight-node iso-parametric (2D) elements using the commercially available ®nite-element software ABAQUS. The mesh is shown in Fig. 7. The crack-tip region was modeled with rectangular elements with an aspect ratio of 1.25. The smallest element at the crack-tip region was 0.1524 mm (0.006 in.). The extent of crack growth, occurring through 40 elements, including the crack jump examined, was about 6 mm (0.24 in.). Since the remaining ligament for this specimen was 22.5 mm (0.886 in.), the crack extended through 27% of the remaining ligament. The stress±strain curve used in the analysis is shown in Fig. 8. Quasi-static elastic±plastic analysis was performed to ensure adequacy of the mesh re®nement. The computed J-integral values along eight di€erent contours were almost path-independent (less than 3%). During stable crack growth the crack was advanced by one element at each step via a node release technique. When the crack extended by one element, the displacement boundary condition was removed for the current crack-tip element and the

Fig. 7. (a) Finite element mesh of the 1 TC(T) specimen.(b) Crack-tip region of the mesh shown in Fig. 7(a).

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Fig. 8. Stress±strain curve for the steel at 2888C (5508F).

reaction forces were reduced to zero. This reduction was achieved incrementally through 3±4 steps. This process is illustrated in Fig. 9. The procedure adopted in this analysis released the nodes to allow crack growth by one element (except during jumps) and increased the applied displacement simultaneously. The relation between the applied displacement and crack extension used in the analysis is shown in Fig. 10. During crack jumps, nodes pertaining to several elements were released at the same time with the applied displacement held ®xed. This process was achieved in 5±6 reduction steps. In order to understand the e€ect of crack jumps on the fracture process and parameters, the J-integral and the crack-tip opening angle (CTOA) were calculated from the analysis. J-integral values computed using near-tip contours are likely to exhibit strong path dependency due to the non-proportional straining that occurs in the wake of the growing crack, especially so when the amount of crack growth is substantial as in the present analysis. Thus, reliable estimates of J-integral values can be obtained only using far-®eld contours. Since ABAQUS does not permit the user to de®ne any

arbitrary far-®eld contours, far-®eld J-integral values were calculated using a post-processing code based on the work of Li et al. [26] on the domain integral method. The CTOA was calculated through the opening displacement one element behind the current crack-tip. 4.2. Numerical results The predicted variations of load versus applied load-line displacement for both the plane-strain and plane-stress cases are shown in Fig. 11. For comparison purposes the experimentally measured variation is also shown in the ®gure. The results for the plane-strain case agreed well with the experimental results, whereas the plane-stress case underpredicted the load signi®cantly. This result was not unexpected since the conditions at the center of the crack front were better represented by the planestrain assumption. The load-line displacements corresponding to crack initiation and crack jumps are also shown. Notwithstanding the overall agreement between the plane-strain analysis and experiment, the analytical results indicate that the rate of decrease in load prior to and after the ®rst crack

Fig. 9. Illustration of node release technique used to model crack extension.

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Fig. 10. Experimentally measured relation between crack extension and applied load-line displacement.

jump was much more than that revealed by the experiment. Such a signi®cant decrease in load during crack extension may o€er a plausible explanation for the onset of crack instabilities when the decrease in elastic stored energy overrides plastic dissipation. It is also worth noting that a plastic hinge formed even prior to crack initiation, as evidenced from the e€ective plastic strain contours shown in Fig. 12. Thus, large scale plasticity conditions prevailed even prior to crack extension. The variation of far-®eld J-integral values with crack growth for the plane-strain case is shown in Fig. 13. The experimental determination of J-integral based on the ASTM E 1152 [27] procedure, which used the load versus load-line displacement

and crack growth data, is also shown in the ®gure. It can be seen that the J-integral values calculated from far-®eld path are in agreement with the corresponding values calculated using the ASTM procedure only for small amounts of crack growth. Beyond crack growth of 1.2 mm (0.047 in.), corresponding to a crack growth of 5.5% of the original uncracked ligament, the ASTM procedure tended to overpredict the J-integral values. Similar disagreement between the J values calculated from the far-®eld path and the corresponding values calculated using the ASTM procedure was noted by Papaspyropoulos et al. [28] in their analyses of crack growth in C(T) 304 stainless steel specimens. Further discussion on this discrepancy will be pre-

Fig. 11. Comparison of measured and predicted load vs load-line displacement.

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Fig. 12. Contours of e€ective plastic strain, ePe€, just prior to the start of crack extension. Note the formation of plastic hinge.

sented later. It is also noted that the J-integral values calculated using the ASTM procedure were similar using the experimental and ®nite-element data for load versus load-line displacement. Despite the disagreement between the J values mentioned earlier, both the far-®eld path and the ASTM procedure predict similar magnitude of the drop in J values during the crack jumps. This ®gure also indicates that prior to the ®rst crack jump the planestress assumption underpredicted the J-integral values compared with the results using a planestrain assumption. It is interesting to note that

while the J-integral for the plane-stress case increased monotonically with crack growth until the onset of a crack jump, the corresponding values for the plane-strain case reached a maximum and decreased slightly immediately prior to the crack jump. This decrease in J-integral with crack extension during stable crack growth probably occurred when the contribution to the J-integral from the decrease in elastic strain energy overrode the contribution from the plastic dissipation. Owing to the higher triaxiality that existed at the crack-tip for the plane-strain case compared with the plane-stress

Fig. 13. Variation of J-integral with crack growth. The J-integral values are calculated using far-®eld contours using the domain integral method as well as the ASTM procedure utilizing the load vs loadline displacement data.

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Fig. 14. Crack opening displacement pro®les during crack extension.

case, this behavior was more likely to occur under plane-strain conditions than under plane-stress conditions as is evidenced from the present results. The decrease in J-integral values with crack extension may indicate the onset of cracking instabilities. In both the plane-stress and plane-strain cases, the Jintegral values dropped sharply immediately following the crack jump. This result was expected since the load dropped sharply while the load-line displacement was held ®xed during the crack jump. It is also worth noting that the J-integral de®ned by the far-®eld contour around the crack-tip may not o€er any energetic interpretation when cracking instabilities occur. The crack-opening-displacement pro®les at initiation, during extension, and following the crack jumps are shown in Fig. 14. At crack initiation the crack-tip was blunted and thereafter remained

sharp during crack extension. The ®rst and second crack jumps occurred after the crack extended by 2.8 mm (0.11 in.) and 5.49 mm (0.214 in.), respectively. It is interesting to note that the crack-tip blunted immediately after the crack jumps. The variation of CTOA with crack extension is shown in Fig. 15. This ®gure reveals that the crack-tip became sharper as the crack extended in a stable manner and blunted immediately following the crack jumps. During the stable crack extension following the crack jump the crack-tip tended to be sharp.

5. DISCUSSIONS

A connection between the nature of cracking and fracture morphology is attempted. To aid the correlation between the experiment and analysis, it is

Fig. 15. Variation of CTOA with crack extension. Note that the crack tip which remained sharp during the initial stable crack growth blunts immediately following crack arrest after the crack jump.

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worthwhile to summarize some of the salient features of the typical fracture morphology of a specimen tested at 2888C (5508F). As discussed in Section 3, the fracture surface of a typical C(T) specimen exhibits light and dark bands which generally indicate unstable and stable crack growth, respectively (see Fig. 4). Detailed SEM pictures of the fracture surface revealed that the fracture process was governed by ductile void nucleation and growth in both the stable and unstable cracking regions (Figs 5 and 6). However, the fracture morphologies produced during stable and unstable crack growth showed some di€erences. When the crack grew in a stable fashion, the void size and mean void spacing were smaller and the density of voids was higher [Fig. 5(a) and 6(a)] than the corresponding quantities that characterize the fracture surface during the crack jump [Fig. 5(b) and 6(b)]. It is apparent from Fig. 4 that while the crack front grew fairly uniformly during the initial stable extension, the crack front corresponding to the restart of the second stable extension following the ®rst jump was no longer straight, somewhat indicative of crack tunneling. By the end of the second stable extension, the crack front appears to have regained its original straight pro®le, which was lost again by the end of the second crack jump. From these crack pro®les, it appears that the onset of instability in crack extension may have been triggered around the center of the crack front where the crack-tip experienced higher constraint. This observation indicates that there are certain necessary conditions for unstable crack jumps to occur, including crack-tip constraint, stress triaxiality ahead of the crack tip and the dominance of elastic stored energy over plastic dissipation. The di€erence in fracture morphology in the stable and unstable crack regions as observed in the SEM indicates the dominant role of crack-tip plasticity during the crack growth process. The fact that there are no distinguishable di€erences in the microstructure of the material in the stable and unstable cracking regions lends further support to this argument. During stable crack growth after initial blunting, the active plastic zone is intense in the vicinity of the crack-tip. Thus a large number of voids nucleates and grows to coalesce when the crack is growing steadily. When the crack jumps, the crack faces have to separate rapidly and sucient time is not available to nucleate large numbers of voids. However, the few voids that had nucleated in the crack jump region prior to the crack jump may grow rapidly to accommodate the rapid separation of crack faces. Immediately following the crack jump, crack-tip blunting occurs as indicated by the analysis, accompanied by intense plastic deformation as the stable crack growth is resumed, thus leading to similar fracture morphology as in the initial stable growth region.

One of the objectives of the ®nite element simulation was to examine the e€ects of the nature of cracking on the near-tip behavior and compare the far-®eld J-integral values with the ASTM E1152 procedures used to assess the crack growth after a crack instability occurs. The experimentally observed relation between crack extension and applied load-line displacement served as the basic input to the model. In this section, an attempt is made to correlate the analytical results with the observed fracture morphology of the failed specimen. The reader is cautioned that the following correlation is by no means conclusive, since it relies only on one simulation. Clearly, there is a de®nite need to conduct additional analytical simulations and detailed characterization of fracture morphologies in order to understand the source, nature and e€ects of the unstable cracking in ferritic steels susceptible to DSA. Using the plane-strain assumption, the crack growth analysis predicted load versus load-line displacement in close agreement with the experimental measurements. However, the J-resistance curve obtained using the domain integral method on a far-®eld path deviated from the corresponding curve using the ASTM E1152 procedure beyond crack growth of 5.5% of the original ligament. It is well known that the validity of J as a crack-tip characterizing parameter ceases after some amount of crack growth. Hutchinson and Paris [29] de®ned a parameter o given by    b dJ oˆ …1† J da where `b' denotes the length of the remaining ligament. They argued that when `o' is much greater than unity, the conditions for J-controlled crack growth prevail. Based on detailed numerical analyses of crack growth in 4 T-C(T) A533B steel specimen, Shih et al. [30] determined that o = 40 for this geometry and J-controlled crack growth is valid up to 4% of the original ligament. Using the present analysis for the 1 T-C(T) specimen, o = 27. Thus, J-controlled crack growth may not be valid for crack growth amounts exceeding 4±5% of the original ligament, as evidenced by the results of the analysis. Since the ®rst crack jump occurred after a crack extension of 13% of the original ligament, it may not be possible to make meaningful comparison between far-®eld J values with the ASTM E1152 procedures used to assess crack growth after a crack instability occurs. In addition, based on the present analysis and the work of Hutchinson and Paris [29], it appears that any estimation procedure for determining J-R curve for large amounts of crack growth may not be appropriate. As seen in the comparison between the ASTM E1152 and domain integral methods of determining J, the ASTM method overpredicts the crack driving force

MOHAN and MARSCHALL: CRACKING INSTABILITIES

beyond crack growth of 5.5%. This implies that a ¯aw assessment methodology based on ASTM E1152 procedure may in fact be non-conservative when applied to cracked DSA-sensitive steel structures operating at LWR temperatures. The loads predicted by the plane-strain analysis agreed very well with the load measured experimentally up to a load-line displacement of 2.54 mm (see Fig. 11). Just prior to the onset of unstable cracking, the plane-strain analysis predicted a rate of decrease in the load that was not observed in the experiment. It was also found that there was a decrease in load just prior to unstable cracking even under plane-stress conditions. However, the far-®eld J-integral monotonically increased with stable crack extension until the ®rst crack jump under planestress conditions, whereas it dropped slightly just prior to the ®rst crack jump under plane-strain conditions. The analysis indicates that the onset of instability was triggered when release of elastic stored energy overrode plastic dissipation. The plane-strain simulation appears to have met this condition. Since plane-strain conditions are more likely to exist at the center of the crack front even in the side-grooved C(T) specimen, it might be anticipated that the onset of cracking instability would occur at the center of the crack front. The appearance of the fracture surface (Fig. 4) seems to support this conjecture. The correlation between the analysis and the experiment is by no means conclusive since it is based on only one experiment. Threedimensional analysis would provide more insight into the phenomenon of crack jumps due to DSA. It must be noted that the present work does not conclusively establish the connection between DSA and unstable cracking in low carbon steels. Crack growth experiments, involving unloading prior to and after the occurrence of crack jumps, will shed more light in understanding this phenomenon. Clearly, other critical experiments need to be performed, preferably in single phase low-carbon steels, not only to establish this connection but also to provide input for more suitable constitutive relations of the kind described by McCormick and co-workers [8, 31, 32]. AcknowledgementsÐThis work was supported in part by the U.S. Nuclear Regulatory Commission through the Materials Engineering Branch of the Oce of Nuclear Regulatory Research under Contract No. NRC-04-90-069 and in part by the Internal Research and Development Program by Battelle. Mr A. Hiser and Mr M. May®eld were the NRC Program Managers. The C[T] tests reported in this paper were performed by Mr P. Held and Dr C. Marschall. The authors are extremely grateful to Dr G. Wilkowski for his support as well as for the fruitful discussions with them. REFERENCES 1. Manjoine, M. J., J. appl. Mech., 1944, p. A211. 2. Baird, J. D., Iron and Steel, 1963, p. 450.

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3. Baird, J. D., Metall. Rev., 1971, 16, 1. 4. Ho, E. T. C. and Lepik, O. E., E€ects of strain aging and related phenomenon upon the toughness of carbon and carbon±manganese weld metals-a literature review, Ontario Hydro Research Division, Report 86323-K, 1987. 5. Keh, A. S., Nakada, Y. and Leslie, W. C., in Dislocation Dynamics, 381, ed. A. R. Rosen®eld et al.. McGraw-Hill, New York, 1968. 6. Van den Beukel, A., Acta metall., July 1980, 28(7), 965. 7. Van den Beukel, A. and Kocks, U. F., Acta metall., May 1982, 30(5), 1027. 8. McCormick, P. G., Acta metall. mater., 1988, 36, 3061. 9. Hume-Rothery, W. H. and Raynor, G. V., The Structure of Metals and Alloys, The Institute of Metals, London, 1954, p. 58. 10. Leslie, W. C. and Rickett, R. L., Trans. of AIME, 1983, p. 1021. 11. Wilson, D. V. and Russell, B., Acta metall., 1960, 8, 468. 12. Li, C. C. and Leslie, W. C., Metall. Trans. A, 1978, 9A, 1765. 13. Formby, C. L. and Charnock, W., in Practical Application of Fracture Mechanics to Pressure-Vessel Technology. Inst. of Mech. Engineers, London, 1971, pp. 1±7. 14. Maurer, E. and Mailander, R., Stahl u Eisen, 1925, 45, 895. 15. Miglin, M. T., Van Der, Sluys W. A., Futato, R. J. and Domian, H. A., in Elastic±Plastic Fracture Test Methods: The User's Experience, ASTM STP 856, ed. E. T. Wessel and F. J. Loss. American Society for Testing and Materials, Philadelphia, 1985, p. 150. 16. Mukherjee, B., Int. J. Pressure Vessels and Piping, 1988, 31, 363. 17. Wilkowski, G. M. et al., Degraded piping programÐ Phase II. Summary of technical results and their signi®cance to leak-before-break and in-service ¯aw acceptance criteria, NUREG/CR-4082, Vol. 8, Nuclear Regulatory Commission, Washington, DC, March 1989. 18. Marschall, C. W., Mohan, R., Krishnaswamy, P. and Wilkowski, G. M., E€ect of dynamic strain aging on the strength and toughness of nuclear ferritic piping at LWR temperatures, NUREG/CR-6226, BMI-2176, 1994. 19. Wilkowski, G. M. et al., Degraded piping program± Phase II. NUREG/CR-4082, Vol. 3, Appendix B, Nuclear Regulatory Commission, Washington, DC, March 1986. 20. Marschall, C. W., Landow, M. P. and Wilkowski, G. M., in Fracture Mechanics, 21st Symposium, ed. J. P. Gudas et al.. ASTM-STP 1074, 1990, p. 339. 21. Hill, R., The Mathematical Theory of Plasticity. Clarendon Press, Oxford, 1950. 22. McMeeking, R. M., J. Mech. Phys. Solids, 1977, 25, 357. 23. Dodds, R. H., Tang, M. and Anderson, T. L., Engng Fract. Mech., 1993, 46, 253. 24. deLorenzi, H. G. and Shih, C. F., J. Fract., 1983, 21, 195. 25. ``ABAQUS'' Version 5.4, Finite Element User Manual, HKS, Pawtucket, RI, 1995. 26. Li, F. Z., Shih, C. F. and Needleman, A., Engng Fract. Mech., 1985, 21, 405. 27. ASTM ``Standard Test Method for Determining J-R Curves'', ASTM E 1152-87, 1991 Annual Book of ASTM Standards, Section 3, 1991, p. 825. 28. Papaspyrpoulos, V., Marschall, C. and Landow, M., Predictions of J-R curves with large crack growth from small specimen data, NUREG/CR-4575, Nuclear

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Regulatory Commission, Washington, DC, September 1986. 29. Hutchinson, J. W. and Paris, P., Elastic Plastic Fracture, ASTM STP 668. American Society for Testing and Materials, 1979, p. 37. 30. Shih, C. F., deLorenzi, H. G. and Andrews, W. R., Elastic±Plastic Fracture, ASTM STP 668.

American Society for Testing and Materials, 1979, p. 65. 31. Ling, C. P. and McCormick, P. G., Acta metall. mater., 1990, 38, 2631. 32. Ling, C. P., McCormick, P. G. and Estrin, Y., Acta metall. mater., 1992, 41, 3323.