STP32572S.1207635-1.pdf

A. Saxena, ^ T. T. Shih, ^ andH. A. Emst^

Wide Range Creep Crack Growth Rate Behavior of A470 Class 8 (Cr-IVlo-V) Steel

REFERENCE: Saxena, A Shih, T. T., and Ernst, H. A., "Wide Range Creep Crack Growth Rate Behavior of A470 Class 8 (Cr-Mo-V) Steel," Fracture Mechanics: Fifteenth Symposium, ASTM STP 833, R. J. Sanford, Ed., American Society for Testing and Materials, Philadelphia, 1984, pp. 516-531,

ABSTRACT: Wide range steady-state creep crack growth rate behavior of an A470 Class 8 steel was characterized at 538C (1000F) using constant load and constant deflection rate methods of testing. Both methods yielded mutually consistent results. Crack growth rates in the range oflO^^tolO-^mm/h were successfully correlated with the energy rate line in-tegral, C*. Results are discussed in light of the recent analytical studies that have appeared in the literature for growing cracks in creeping materials. Various methods of determining C* for CT specimens are also discussed.

KEY WORDS; creep, fracture, Cr-Mo-V steel, C*-mtegral

Creep deformation and crack growth are important design considerations for several components that operate in high temperature environments. Re-cently, several experimental studies have shown that the energy rate line inte-gral, C*, fttst introduced by Landes and Begley and independently by Nikbin et al, is the leading candidate parameter for characterizmg steady-state creep crack growth behavior in structural alloys [1-3].

In this study, the steady-state creep crack growth rate behavior of an ASTM A470 Class 8 steel (Cr-Mo-V) is characterized over a wide range of crack growth rates and C*-values. Two test techniques, the constant displacement rate method and the constant load method, were utilized to obtain creep crack growth rate data over a wide range. The advantages and limitations of the two techniques are discussed. The relevance of C* for characterizing steady-state creep crack growth behavior is discussed in light of the several analytical

'Materials Engineering Department, Westinghouse R&D Center, Pittsburgh, Pa. 15235. Dr. Shih is now with the Steam Turbine-Generator Division.

516

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SAXENA ET AL ON A470 CLASS 8 STEEL 517

Studies that have recently appeared in the literature.^ Different methods of calculating C* for the CT test specimen are also compared and discussed.

Experimental Procedniie

Material and Specimens Standard 25.4-mm (l-in.)-thick compact type (CT) specimens were ma-

chined from a large steel forging manufactured in accordance with ASTM Specification for Vacuum-Treated Carbon and Alloy Steel Forgings for Tur-bine Rotors and Shafts (A470, Class 8). The notches in these specimens were oriented along the radial direction of the forging. The chemical composition and the tensile properties of the material are shown in Tables 1 and 2, respectively.

Limited creep deformation testing was also conducted at 538C (1000F). The secondary creep rate as a function of applied stress is shown in Fig. 1 along with some previous data on the same material [4],

Creep Crack Growth Rate Testing Two types of test techniques were used to obtain creep crack growth rate

behavior: (1) constant displacement rate method and (2) constant load method. These are briefly described below.

Constant Displacement Rate MethodThis technique was first developed by Landes and Begley [/] for obtaining creep crack growth behavior of diskal-loy and was subsequently used by Saxena [3] on 304 stamless steel. In this tech-nique, the tests are conducted under constant ram deflection rate (approxi-mately constant load-line deflection rate) using a servohydraulic test system. The specimens are heated in a three-zone resistance furnace with a tempera-ture control capability of 2C in the test section of the specimen. A d-c electrical potential system is used to measure crack length. The details of this system are described elsewhere [5]. Typically, three specimens are subjected to constant ram deflection rates between 0.025 to 0.15 mm/h (0.001 to 0.006 in./h). Dur-ing the test, specimen load, load-line deflection, and crack length are mon-itored continuously on a strip chart recorder.

In the constant displacement rate tests conducted in this study, a slight vari-ation from this procedure was followed. The deflection rates in the three speci-mens were changed in steps after crack extensions of approximately 5 mm (0.2 in.) according to a predetermined scheme shown in Fig. 2. This technique opti-mized the number of data points that were obtained from three tests when the

^Steady-state creep crack growth rate is defined as follows. Consider a coordinate system with the origin at the crack tip and moving with the growing crack. Ideal steady-state conditions exist when the stresses and strain rates at any distance ahead of the crack tip are independent of time. Quasi-steady-state is defined when the stress and strain rates vary slowly in comparison to the crack growth rate da/dt.


518 FRACTURE MECHANICS: FIFTEENTH SYMPOSIUM

TABLE 1Chemical composition (weight percent) of A470 Class 8 steel.

c

0.32

Mn

0.78

Si

0.28

Cr

1.20

Mo

1.18

V

0.23

P

0.012

S

0.011

Ni

0.13

Cu

0.05

Al

0.005

Sn

0.010

As

0.008

TABLE 2Tensile properties qfA470 Class 8 steel.

0.2% Test Temperature Yield Strength Ultimate Strength

C MPa ksi MPa ksi % Elongation Reduction

(5 cm gage length) of Area, %

24 427 538

(75) (800)

(1000)

623 515.7 464

(90.4) (74.8) (67.3)

775.6 624.6 522.6

(112.5) (90.6) (75.8)

14.2 14.2 17.5

39 53 75

ao 100

10

TS -A t 10

10

10 .-6

- I 1 1 A 470 Class Steel

538C (1000FI This Study o Leven and Marloff

oin ksi 0 In MPa

22 -^1 13x10 5.18x10 "

n=10.5

e = Ao

600

: 10

- 10

J I I u

,-3

,-4

10 20

Stress (ksi) 60 80 100

FIG. 1Secondary creep rate versus stress for A470 Class 8 steel at 538C (1000F).


.008

_ .006

f -""^ * .002

.008

.006

f .004

^ .002

.008

.006

f .004

> .002

-

-

Specimen fl

-

SAXENA ET AL ON A470 CLASS 8 STEEL 5 1 9

2.0

H . 1 5 -

. 1 0 | .05

.20

. 1 5 | "i

.10^

.05

.20

. 1 5 |

. l o e

.05

-

Specimen #2

-

-

.

Specimen #3

-

1 0.5 0.6 0.7

a/W 0.8

FIG. 2Deflection rate (V) as a function of crack length in various specimens.

data were reduced by the multiple-specimen graphical technique described later in this section.

Constant Load TestsThese tests were conducted using deadweight-type creep machines. The specimen was heated in a three-zone resistance furnace to the test temperature and then the load was applied by adding weights on the loading pan. The changes in load-line deflection were measured with the help of a dial gage attached to the setup. The deflection from the dial gage was peri-odically recorded and plotted with time. Typical deflection versus time records for some of the tests are shown in Fig. 3. The deflection versus time behavior ex-hibited a primary creep region and a steady-state creep region. When substan-tial amount of deflection had accumulated on the specimen, it was unloaded and subsequently fatigued to failure at ambient temperature. The amount of crack extension due to creep was measured directly on the fractured surface at five points across the thickness of the specimen. Thus a through-the-thickness average crack extension was obtained. See Table 3 for test results.

Data Reduction

The data were plotted as da/dt as a function of C*. In the deflection rate con-trolled tests, crack growth rate da/dt was simply obtained by calculating the



U. 36

0.30

J r

I Odd - 28984 N ( 6500 lb)

r I r

A470 Class SStsel 538C IIOOCFI

22295N(500Olbl

0.05

14

12

10

8

6

4

100 300 500 700 Time (hrsi

900 1100

FIG. 3Load-line displacement as a function of time for constant-load creep crack growth tests,

slope of the crack length versus time curve at various crack lengths. Crack growth rate data were obtained excluding crack extensions of approximately 0.5 mm following a step change in deflection rate. This was considered as the transient region during which the specimen response (the load) readjusts to the new deflection rate. In the constant load tests, da/dt was obtained by dividing the crack extension due to creep by the total test time. The transient time which immediately follows loading is also included in calculating the rate. From Fig. 3, it appears that the transition time (time up to the onset of constant deflection rate in the specimen) is negligible compared to the overall test duration; hence inclusion of that in calculating da/dt does not influence the value of da/dt significantly.

Three techniques were used for estimating C* for compact specimens tested in this study. They were (1) the multiple-specimen graphical method, (2) the fully plastic /-solutions, and (3) a method utilizing the load-deflection r^te behavior called the area technique. These methods are described below.

Multiple-Specimen Graphical MethodThis method of obtaining C* is based on the energy rate interpretation [/] of C* given by

C*= - dU* da (1)

where U* is power or the time rate of energy per unit thickness applied at the loading points. A detailed description of this method is given elsewhere [1,3].

Fully Plastic J-Solutions^/-solutions for compact specimens under fully plastic conditions of loading are available from the work of Kumar and Shih [6]. By analogy, these solutions can also be used to calculate C* when secondary



I

5

m

tS o

EE

jSaa

1^ 00 00 -H ^ ov r-i

(N O

^ n fo 1 1 t

00

TT

^

in T ri

*

X

f n 1 1 1 1 1 1

o o o o o o X X X X X X >-0 -^ (N 00 r


creep conditions dominate in the entire specimen. The relevant equation for calculating C* by this technique is

C*=A{W-a)hi{a/W, n) [P/{lA55a{W~ a))]" + ^ (2) where W is specimen width, P is load, a is crack length, and A and n are material constants in Eq 3 valid in the secondary creep regime.

e=.4(r" (3) where i is strain rate, a is stress, and h^ia/W, n) is a dimensionless function shown in Fig. 4 for selected values of . The term a{a/W) is given by

2a W-a + 2

2a W-a

+ 2 1/2 2a

W-a + 1 (4)

Area TechniqueShih [7] and Smith and Webster {8\ have independently suggested methods for using the measured loads and deflection rates at the loading points for estimating C* for CT specimens. This technique is analo-gous to the area technique commonly used to determine/-integral [9,10]. Here, the area under the load-deflection rate curve is taken instead of the area under the load-deflection plot used for calculating/. Shih's method, following that of Rice et al [9], models CT specimens as a pure bending case and applies only for deeply cracked specimens {a/W 1.0). Smith and Webster's method follows that of Merkle and Corten [10], which considers tension as well as bending and

2.0

1.8

1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

T 1 1 1 r CT Specimen

J I I I I 1 l_ 0 2 0.4 0.6 0.1

a/W

FIG. 4Dimensionless function hj versus a/vifor calculating C* in compact specimens.


SAXENA ET AL ON A470 CUSS 8 STEEL 523

thus can be used for a/Ty > 0.4. Since the latter is more general, it is adopted in this study. The relevant equations are given below.

PV n B{W a) n + \

where B is the thickness of the specimen, V is the deflection rate of the load point, n is a material constant defined in Eq 3, and r){a/W) is a simple function of a/W'given in Eqbiora/W > 0.4 [11].

i?(a/W0 = 2[1.261-0.261 (a/WO] (6)

The itrm PV{n/n + 1) in Eq 5 is the area under the load-deflection rate plot.

Results and Discussion In this section the creep crack growth results are presented and discussed.

The various methods of calculating C* are evaluated and the two techniques used for obtaining the data are compared with each other.

Creep Crack Growth Behavior The da/dt versus C* data for A470 Class 8 steel obtained from the constant

load and constant deflection rate tests are plotted in Fig. 5. The justification for the use of C* for plotting da/dt is provided subsequently. Equation 5 has been used to calculate C* for both types of tests. The C*-values for the constant deflection rate tests were also calculated using the multiple-specimen tech-nique for comparison. The data from the two data reduction techniques have been plotted using different symbols. All the data appear to follow the same general trend with some scatter.

Good correlation is obtained between da/dt and C* over a wide range of C*-values (five orders of magnitude) and crack growth rates (three orders of magnitude). Similar correlation between da/dt and C*-values was also re-ported by Shih for 304 stainless steel [7]. The creep crack growth rate behavior in the range of da/dt values of 2.5 X lO""* mm/h (10~^ in./h) reported in this study is of considerable practical significance. It is needed for accurately predicting life of structures designed for long life. The da/dt versus C* relation-ship for A470 Class 8 steel at 538C (1000F) is given by

da/dt = \0-HC*)'^-^'' (7)

where da/dt is in mm/h and C* is in J/m^/h. Or,



1 .

1

s f 1 1

8 6 4

2 , 3 10

8 6 4

2 -4 10 ^

8 6 4

2

10-5 8 6 4

2

in-

2 4 6 8 10 2 4 6 8 10^ 1 1 1 M ' ' ' ' 1

A 470 Class 8 Steel Air, 538C(1000f)

-

-

- Mean Trend from Smith and Webster, y,

ICr-Mo-V Steel ^ ^ 565''C \ y^

'^ , - ' ^ B(\N-a & ^ '

dt 1

: Lmm/hr J""

1 1 1 1 1 1 1 1 1

Joules/m /hr

2 4 6 8 10^ 1 1 1 1 1

^r

V ^

/^ ^fi' y A

d

iTTT '^^ /VV)

^/hr

1 I 1 l_J-

2 4 6 810* 2 4 6 8 10^ 1 1 1 1 1 1 1 1 1 1

. ^

y^ ^ ' ^ yii^^ /^ . - ' ^ 0 '

.

"

-

-

Constant Deflection Rate _ ^ Data

""^^ Constant Load Data,

o Constant Deflection Rate Data from Multiple- ~ Specimen C* Calculation -

1 1 1 1 1 1 1 I I I

2

10-1 8 6 4

2 10-2 8 * u 4 ^

2 ^ i r 3 8 6 4

2 10"* 8 6 4

10 ,-2 10 1.0 10 10' 10^ Energy Rate Line Integral, C* (In.lbs/in /hr )

FIG. 5Creep gmwth rate as a function of C*for A470 Class 8 steel.

d a / A = 1.3 X 10-4 (C*)0-67 (7a)

where da/dt is in in./h and C* is in in.-lb/in.^/h. The creep crack growth behavior at a higher temperature of 565C (1050F)

for Cr-Mo-V steel from the work of Smith and Webster [12] is also plotted in Fig. 5 for comparison. There appears to be a significant influence of tempera-ture on the creep crack growth behavior in this material. It should be pointed out, however, that there could be differences in the microstructure between the steels used in Ref 12 and this study and the difference in da/dt versus C* may be caused by that.

Comparison of Methods for Calculating C* Table 4 lists some representative values of C* obtained from the three proce-

dures described in the previous section of the paper. The discrepancies be-tween the multiple-specimen graphical method and the area technique range between factors of 1.04 to 2. In this range of discrepancies, no quantitative conclusions about the accuracy of the area technique are possible. The C*



1 S

I

!

H

S.^

till E l

a^

a

X X X (^ ^ lA 1/) 1/1 rr ^ ' W -H '^ ^ '^

^ m n O O O

X X X o r- r*^ vO (N -H

^H T H 11

X X X p S

o o o o o o X X X X X X

1 - CT> "^ a TT (N Tf 00 t-- QO ^ ^ - ^ -H (N

O O O

X X X

^ (N

X X X

^ ^ w r- irt r^

lO I/) lO o d d

s;-

o o o X X X Tf 3 S

X X X 3 \0 O

o o o o o o

o o o X X X X X X



calculations from the fully plastic /-solutions are different by up to two orders of magnitude compared to the C* values from the multiple-specimen graph-ical method and the area technique. The exact reason for this discrepancy is unclear at this point and will be the subject of future studies. One reason may be that in order for these solutions to be applicable/the entire specimen should be in secondary stage creep. This condition may not have been met by the tests conducted in this study. Because of these large discrepancies, the data from this method were not used in the plot shown in Fig. 5.

Both the multiple-specimen graphical method and the area technique utilize two measured quantities to calculate C*, namely the load and deflection rate. The fully plastic/-solutions require only the load and some material con-stants to calculate C*. Further, in the use of/-solutions, an assumption has to be made on whether the specimen is under plane strain or under plane stress. The C*-values calculated assuming plane stress will be 33.9 times larger than those calculated for plane strain. The values reported in Table 4 are for plane strain. The specimens are probably somewhere in between plane stress and plane strain; hence calculations based on either of the two states of stress will be in error.

A power law relationship between load and steady-state deflection rate of the type shown in Eq 8 is assumed in deriving the expression for calculating C* in the area technique [7,8].

V=CP" (8) where C is a material constant for a given crack length and n is the exponent in Eq 3. When steady-state deflection rate is plotted against the applied load for a series of specimens with nearly identical crack lengths (Fig. 6) the value of the

IKIIo-Newtons, KNI

r - ^

S 10" g 1 1 10-5

10 A470 - Ctaa 8 Stwl 538C(1000F) IT - C T Specimen, a/w = 0.52

2 4 6 Load, P IKilo-Pounils, KIP I

FIG. 6Load versus steady-state deflection rate {V)/or a IT-CT specimen at a/w = 0.52; temperature = 538C.



exponent n was found to be 11.58 as compared to 10.5 which was fitted through the stress versus strain rate data in Fig. 1. Both these values of n are the best estimates derived from linear regression of the respective data sets. No significance can be attached to the small difference between the two values. Hence the limited data available covering only one crack length appear to sup-port the validity of Eq 8.

The value of C* in the area technique is not sensitive to the value of K . In our calculations we have assumed = 11, which is the average of the two values obtained from the data shown in Figs. 1 and 6. Use of either 10.5 or 11.58 would not have yielded significantly different results.

From comparisons made between the three techniques, it appears that the area technique has the most promise in the future for determining the values of C*. More work is needed for its further verification.

Comparison of Test Techniques To obtain creep crack growth rate data over a wide range of growth rates of

C*-values, it is necessary to use both the constant deflection rate and the con-stant load techniques. The advantages and disadvantages of both techniques are discussed below.

The constant deflection rate method is useful at relatively higher crack grovrth rates of da/rfr > 5 X 10"3mm/h(2 X lO-^in./h). It allows data to be obtained over a crack extension of 10 to 15 mm (0.4 to 0.6 in.); thus several data points can be obtained from each test. This technique is not practical for obtaining data at lower crack growth rates, however, because it requires an expensive servohydraulic machine for a long time. Also, under nearly static conditions for long periods of time, the performance of servohydraulic ma-chines can be erratic.

The constant load test is conducted on a relatively inexpensive creep test ma-chine which also provides excellent load stability over a long period of time (the longest test duration in this study was seven months). This setup is therefore very suitable for obtaining data at cfa/rfr < 5 X 10~3mm/h(2 X 10"~''in./h). The major disadvantage with this technique is that only data over relatively short crack extensions can be obtained. This is because C* increases rapidly with crack extension in this configuration, causing the crack growth to become unstable. At higher loads or da/dt values, the amount of crack extension prior to instability can be so small that the technique may become impractical [/].

More work is needed in order to optimize test techniques for obtaining creep crack growth rate data. In the constant displacement rate tests, data over rela-tively large crack extensions can be obtained; however, the data are clustered around the same da/dt and C* values. Therefore, if a technique for gradually increasing deflection rate (instead of constant deflection rate) with crack ex-tension can be developed, it will significantly reduce test times and the number of specimens required for obtaining data.



Evaluation of C*for Characterizing Creep Crack Growth Rate For stationary cracks in creeping materials, it has been shown that a

C*-controlled HRR-type field (after Hutchinson and Rice and Rosengren [13,14]) develops at the crack tip [15,16]. This has previously been used as a basis for justifying the use of C* for characterizing creep crack growth rate [1,3]. This argument is not strong. We cannot be sure that for growing cracks, the stress and strain rate fields are controlled by C* and that the HRR singu-larity is valid. Recently, several analytical studies of growing cracks in creep-ing materials have appeared in the literature [17-19]. It will be of considerable interest to discuss our experimental results in light of the conclusions of these studies. First, the significant conclusions of the analytical studies are briefly reviewed.

Hui and Riedel [17] have shown that for growing cracks in creeping ma-terial, a new type of singular stress and strain rate field develops which is con-siderably different from the HRR field. This new singularity is independent of the load parameter and is dependent only on the current crack growth rate da/dt (or d) and material parameters (Eq 9). Its derivation is based on the con-tention that the elastic strain rate for growing cracks cannot be ignored in com-parison to the creep strain rate in a region approaching the crack tip {r -* 0). Consideration of these elastic strain rate terms results in the new singularity

ayoc d(t) AEr (9)

where a,-, is the stress tensor at the crack tip, E is the elastic modulus, TQ is the distance from the crack tip, and/I and n are material constants defined in Eq 3.

Far away from the crack tip, it is reasonable to assume that the creep strain rates dominate over the elastic strain rates for cracked bodies undergoing sec-ondary creep. Hence an expression derived by Riedel and Wagner [18] to char-acterize the stress distribution under steady-state conditions becomes appli-cable. The equation is

ay = (^-j ^{R,0} (10)

where R is the distance from the crack tip normalized with respect to the size (ro) of the region of the dominance of the new singularity, d is the angular loca-tion of the element under consideration, and E;, is a function of R and 9. Fur-ther, by equating Eqs 9 and 10 at rg, an approximate value of TQ has been ob-tained [18]:

1 / d \ ( + l)''2'/ 1 \(n-l)/2



If ro is small compared with the crack length and the other pertinent dimen-sions of the specimen such as the ligament width, C* can be expected to char-acterize the steady-state crack growth behavior because it characterizes the stress and strain rates in the bulk of the region ahead of the crack tip (Eq 10). An order of magnitude calculation of ro for one of the tests conducted in this study yielded a value of 10^2'' mm using a value of 1.69 X 10^ MPa (24.5 X 10^ ksi) for the E-value. This value is certainly negligible compared with the crack length. It thus appears that the small region in which the new singularity dominates is engulfed in a much larger zone in which C* characterizes the stress and strain rate behavior. Hence da/dt is expected to correlate with C*, which is in agreement with the experimental results of this study.

McMeeking and Leckie [19] have used an incremental crack growth model to justify the use of C* for characterizing creep crack growth rate. In their model they assume that the crack growth occurs in steps and that following each step of cr^ck growth there is a transient period during which C*-controlled stress and strain rate fields are re-established. Between two steps of crack growth, the crack is considered stationary. They also show that the transient time for re-establishing the C*-controlled field is negligible in comparison to the total time that elapses between two steps of crack growth. Hence the macroscopic crack growth rate is expected to correlate with C* because it completely characterizes the steady-state crack tip conditions when the crack is stationary.

From the analytical studies of growing cracks in creeping materials dis-cussed in the preceding paragraphs, it can be concluded that C* is a good can-didate parameter for characterizing steady-state crack growth behavior. Our experimental results over a wide range of C* and da/dt values provide some ex-perimental support to these conclusions. However, more work on different materials, specimen geometries, and temperatures should be performed in order to further substantiate these conclusions and establish the limitations on the use of C*. Some specific concerns are discussed below.

When large amounts of crack extensions occur, there are questions about the influence of prior crack growth history on the crack growth rate and on the load-deflection rate behavior of the specimen. There is some experimental evi-dence in the data obtained in the present study that these history effects may not be important. In our deflection rate controlled tests, the crack extensions were large (10 to 15 mm) and C* appeared to correlate well with da/dt over the entire crack extension range. Despite these data, a more in-depth experimen-tal study to further investigate the influence of prior growth history is needed. Another important issue is the extent of creep deformation in the specimen re-quired before the use of C* can be justified for characterizing da/dt. This is particularly important for creep-resistant superalloys or for steels such as the one used in this study at lower temperatures than used here. The transient time required for C*-controlled field to establish may be large, and significant crack extension may occur under transient conditions. Some quantitative


530 FRACTURE MECHANICS: FIF=TEENTH SYMPOSIUM

criteria to determine when C* can be used should be established. These issues will be addressed in detail in forthcoming papers [20,21].

Summaiy and Conclusions Wide range steady-state creep crack growth rate behavior of A470 Class 8

steels was characterized at 538C (1000F) using constant load and constant deflection rate methods of testing. The following conclusions can be derived from the results of these tests:

1. The constant load and the constant deflection rate methods for creep crack growth rate testing yield mutually consistent results. The constant load method is more suitable for testing at da/dt values less than 5 X 10~^ mm/h (2 X 10"'' in./h), while the constant deflection rate method is more suitable for testing at da/d^ values larger than 5 X 10~^mm/h(2 X 10"''in./h).

2. Wide range creep crack grovrth rate behavior in A470 Class 8 steels cor-relates well with C* and can be represented by a simple power law relationship (Eq 7).

3. There is good analytical justification for using C* for characterizing the steady-state creep crack growth rate behavior.

4. The C*-values calculated from the area technique were comparable to those calculated by the multiple-specimen graphical technique. The fully plastic /-solution for CT specimen was not found suitable for calculating C*-values in A470 Class 8 steels at 538C (1000F).

Acknowledgments The authors are indebted to P. J. Barsotti, R. B. Hewlett, and C. Fox for

conducting the creep crack growth rate tests. Financial support was provided by the Steam Turbine-Generator Division of Westinghouse Electric Company. The constant encouragement of V. P. Swaminathan and also fruitful discus-sions with him are gratefully acknowledged.

References [/] Landes, ] . D. and Begley, J. A. in Mechanics of Crack Growth, ASTM STP 590, American

Society for Testing and Materials, 1976, pp. 128-148. [2] Nikbin, K. M., Webster, G. A., and Turner, C. E. in Cracks and Fracture, ASTM STP 601,

American Society for Testing and Materials, 1976, pp. 47-62. [3] Saxena, A. in Fracture Mechanics: Twelfth Conference, ASTM STP 700, American Society

for Testing and Materials, 1980, pp. 131-151. [4\ Leven, M. M. and Marloff, R. H., unpublished data, Westinghouse R & D Center, Pitts-

burgh, March 1977. [5] Saxena, k.. Engineering Fracture Mechanics, Vol. 13, 1980, pp. 741-750. [6] Kumar, V. and Shih, C. F. in Fracture Mechanics: Twelfth Conference, ASTM STP 700,

American Society for Testing and Materials, 1980, pp. 406-438. [7\ Shih, T. T. in Fracture Mechanics: Fourteenth SymposiumVolume II: Testing and Ap-


SAXENA ET AL ON A470 c u s s 8 STEEL 531

plications, ASTMSTP 791, J. C. Lewis and B. Sines eds., American Society for Testing and Materials, 1983, pp. II-232-II-247.

[8] Smith, D. J. and Webster, G. A. in Elastic-Plastic Fracture: Second Symposium, Volume IInelastic Crack Analysis, ASTM STP 803, American Society for Testing and Materials, 1983, pp. I-654-I-674.

[9] Rice, J. R., Paris, P. C , and Merkle, J. G. in Flaw Growth and Fracture Toughness Testing, ASTMSTP 536, American Society for Testing and Materials, 1972, pp. 231-245.

[10] Merkle, J. G. and Corten, H. T., Journal of Pressure Vessel Technology, Transactions of ASMS. Vol. 96, 1974, pp. 286-292.

[//] Clarke, G. A. in Fracture Mechanics: Thirteenth Conference, ASTM STP 743, American Society for Testing and Materials, 1981, pp. 553-575.

[12] Smith, D. J. and Webster, G. A., Journal of Strain Analysis, Vol. 16, No. 2, 1981, pp 137-143.

[13] Hutchinson, J. yv.. Journal of Mechanics and Physics of Solids, Vol. 16, 1968, pp. 13-31. [14] Rice, J. R. and Rosengren, G. F.,Joumalof Mechanics and Physics of Solids, Vol. 16, 1968

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11, 1975, pp. 575-591. [16] Riedel, H. and Rice, J. R. in Fracture Mechanics: Twelfth Conference, ASTM STP 700,

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