Fatigue crack propagation in steels

Engine&g Fracture hfechonics Vol. 18. No. 5. pp. %5-973, 1983 Printed in Great Britain.

&w-7944/83 nM)+ .w @ 1983 Petgamon Press Ltd.

FATIGUE CRACK PROPAGATION IN STEELS XIULINZHENGt Department of Material Engineering, North-Western Polytechnical University, Xian, Shaanxi, The People’s Republic of China

MANFRED A. HIRT Department of Civil Engineering, ICOM, Swiss Federal Institute of Technology, Lausanne, Switzerland Abstract-From previous investigations of the mechanisms of both fracture and fatigue crack propagation, the static fracture model proposed by Lal and Weiss may be thought as reasonable for describing fatigue crack propagation in metals at both low and interm~iate stress intensity factor ranges AK. Recent progress in fatigue crack propagation indicates that it is not only possible, but also necessary, to modify this static fracture model. Based on the modified static fracture model, the effective stress intensity factor range A&, which is defined as the difference between AK and the fatigue crack propagation threshold value A&, is taken as the governing parameter for fatigue crack propagation. Utilising the estimates of the theoretical strengths of metals employed in industry, a new expression for fatigue crack propagation, which may be predicted from the tensile properties of the metals, has been derived. The correlation between the fatigue crack propagation rate and the tensile properties is thus revealed. The new expression lits the test results of fatigue crack propagation of steels below lo-’ mm~cycleand indicates well the effect of stress ratio on the fatigue crack propagation rate.

1. INTRODUCTION

THESTUDY of fatigue crack propagation (FCP) examines how a fatigue crack grows under cyclic load. This topic is cu~ently the subject of considerable research, mainly dealing with the development of various models to best explain the crack propagation phenomenon. Most of these models, which are based on different theoretical assumptions, are summarized by Yokobori[l] and by Bathias and Bailon[2]. Fatigue crack propagation is, paradoxically, both simple and difficult. The problem is simple in the sense that, considering only region B of the fatigue crack propagation curve (Fig. l), it can often be successfully resolved by using the basic Paris’s equation[3]. It is, however, diticult in that it may require many parameters and bring together different physical mechanisms verging, in some unfavorable

g

[mm/cycle]

KC Final fracture

1o-2 1o-3

10-4

1r5

10-6

AK

[MPa $iij

Fig. 1. Schematic representation of fatigue crack propagation rate da/dPI versus stress intensity factor range AKon a log-log scale.

tProf. Zheng was a visiting professor at ICOM, at the Swiss Federal Institute of Technology, Lausanne, from January 1980to March 1982. 965 EFM

Vol. 18, No. S-u

966

XIULIN ZHENG and MANFRED A. HIRT

situations, toward impossibility. It is clear, for example, that the governing mechanisms of fatigue crack propagation in region C are quite different from those in regions A and B [4, S]. Hence, with present knowledge, any attempt to explain the fatigue crack propagation curve empirically and in its entirety has no physical basis. This also explains why Paris’s equation fails in regions A and C. The majority of the existing models predict values of the exponent M in Paris’s equation to be a constant of either 2 or 4 [2, 61. However, the value of m, determined by experiment, varies between 1.4 and 9.66 [7,8]. Fatigue crack propagation may be assumed to occur due to the plastic deformation and fracture of material elements located immediately ahead of a crack tip. Thus, fatigue crack propagation must be related to some other material property parameters. Several attempts have been made to find an appropriate correlation between the fatigue crack propagation and the said material property parameters although little progress has been made to date [5, 6, 9, 101. This indicates clearly that the existing fatigue crack propagation (FCP) models cannot perfectly account for the phenomenon in metals. From studies into the subject of FCP and fracture [ 1l-141, the static fracture model proposed by La1 and Weiss [lo] gives a good insight into the problem in regions A and B of fatigue crack growth. However, even here the model needs some modifications. Recent progress [15-171 has gone a considerable way in solving some of the physical constraints of fatigue crack propagation. This paper presents modifications made to La1 and Weiss’s model in order to develop a new FCP expression. The intention of ?he new expression is to incorporate the important feature of the correlation between fatigue crack propagation and the metal property parameters. Several tests results on various steels are then used to check the validity of the new expression. 2. FATIGUE CRACK PROPAGATION MODEL 2.1 Fracture mode Fatigue crack propagation may be assumed to occur due to the fracture of the so called “Fatigue Elements” located ahead of the crack tip as illustrated in Fig. 2. Depending on the microstructure, mechanical properties and the stress state, the fracture may occur either by shearing or in tension [13, 141.The tension fracture caused by normal stress is also known as the static fracture [lo]. The FCP in regions A and B, for most steels, is due to static fracture of fatigue elements [ll, 121. 2.2 Comments on La1 and Weiss’s model La1 and Weiss proposed a static fracture model where, during each loading cycle, the crack advances by the distance over which the maximum normal stress u exceeds the critical fracture stress uff of the metal, as shown in Fig. 3. The Lal and Weiss model has, however, some imperfections. First, it does not consider the blunting effect at the crack tip which has been observed under the scanning electron microscope [15]. Instead it employs the micro-support effect constant, proposed by Neuber, to prevent the normal stress at the crack tip tending to infinity. Second, the three material constants used have no evident physical significance. Hence, it is quite difficult to predict such material constants from the tensile properties of metals.

Fatigue

element

Fig. 2. Schematic illustration of the fatigue element along the potential crack path.

%7

Fatigue crack propagation in steels

1

Fig. 3. Schematic illustration of the assumption of the amount of incremental fatigue crack propagation [17].

2.3 Modified static fracture model Recently, Schwalbe[X] showed that not only a crack propagation threshold value A&, exists during the cyclic loading, but that a crack threshold value & also exists during the monoto~c loading. On loading, if K 5 & or AK 5 AK,,, the crack does not advance but the crack tip bluntens. Schwalbe also suggested that K,, has a value comparable with that of A&,. Primarily based on Lal and Weiss’s assumption and the recent research results mentioned above [ 15, 161,a new static fracture model for FCP has been developed, as shown in Fig. 4. In this section, the case of the general stress ratio of R = 0 will be considered. Then the general effect of stress ratio will be discussed later. When there is no loading (Fii. 4a) the crack remains closed and the radius of the crack tip tends to be extremely small. When the load is increased to point B the stress intensity factor K reaches the value of I&,, the crack opens elastically and the crack tip becomes blunt. This action relieves the stress at the crack tip such that the fictitious normal stress at the crack tip could be kept below or equal to the critical fracture stress uff of the metal. At this stage, the crack tip is stationary, i.e. no forward motion occurs (Fig. 4b). Increasing the load over point B, the crack tip advances and becomes more blunt (Fig. 4c). Finally, on loading to the maximum value, the crack tip reaches point 0” with au increased radius (Fig. 4d). On unloading, the crack tip does not displace but become resharpened and the crack closes (Figs. 4d-g). In all cases u 5 08. It can be seen from the model described above that the difference (K,, - &,) between the maximum stress intensity factor and the crack propagation threshold value is the governing parameter for FCP rather than I(,,. This hypothesis was first proposed by Schwalbe[ 161. This difference can be called the effective stress intensity factor I(,* [l]. In each loading cycle the crack advanced by an amount of xf and hence

which is the same as Lal and Weiss’s assumption. Thus, if the stress distribution ahead of the crack tip and the critical fracture stress are known, it is then easy to derive a mathematical expression for FCP. 2.4 Mathematical expression for FCP From linear elastic fracture mechanics [18], the distribution of the normal stress 0; in the y-direction, along the x-axis can be expressed by:

(2)

XIUCPJ ZHENG and MANFRED A. HIRT Y f Off

L

0

i

_--_-_-..-_--I,

0"

x

cl’) ~

Y Off

1

Y

I

+

1

uff

0

0

j

L!

I -,,=~J-------~

--_---_Lt X

0'

x

0" e)

Cl

~

~

Y

Y t

4 t

Off

Q k __-_-_--), 0"

X

f) ~

--_--_a

x

9) LOAD

0

P

a

9

Fig. 4. Schematic illustration of the static fracture model for fatigue crack

propagation.

Since the effective stress intensity factor is the governing parameter for FCP, it should be used in eqn (2) in place of &. So when x becomes xr = da/dN, eqn (2) can be rewritten as:

The criterion for the crack initiation at the notch tip is given by Ref. 1171as: K,S = qEc~~*

(4)

where K, is the theoretical stress conce~~ation factor, 5’ is the nominal stress applied to the notched

Fatigue crack propagation in steels

969

element, E is Young’s modulus and ur and ef are the material’s fracture strength and fracture ductility, respectively. Having accepted qEoref as the theoretical strength of the metallic materials [19], it is then possible to explain the physical si~ificance of eqn (4). The material at the notch root will fracture to initiate a crack if the fictitious elastic stress K,S exceeds the theoretical strength of metals. If the bluntened crack is taken as a sharp notch, the stress (r at the crack tip should then be equal to KS. When crack propagation at the crack tip is stopped, the stress u is considered to be not more than q/Eor~+ Hence, it can be deduced that the theoretical strength of metals may be taken as equal to the critical fracture stress in La1 and Weiss’s model. Then taking:

and, in addition, when R = 0: K ,,,* = AK,

K,,, = AK,,,,

Ice@= AK,e.

(6)

Substituting eqn (6) into eqn (3) yields: $=BAK&=

&

(AK - AK&,

(7)

and with eqn (5) follows:

+1

(8)

27rEure;

Equation (7) is the mathematics expression for FCP at the stress ratio of R = 0. When a crack propagates according to the static fracture model, the FCP coefficient B is a constant related to the tensile properties of metals. 3. EFFECT OF STRESS RATIO The stress ratio has an effect on the FCP rate according to the static fracture model [12]. In fact, AKth in eqn (7) is a material parameter sensitive to the stress ratio. Experimental resultsl2, 11, 201 indicate that the correlation with R can be expressed as: AKtr,= AK& I- R)Y,

(9)

where AKthOis the crack propagation threshold value for the stress ratio R = 0, and y is a constant which varies from zero to unity (111. It can be seen from eqns (7) and (9) that AKe, becomes smaller as R increases and hence the FCP rate increases. However, the variation of AKth has a greater effect on the FCP rate at lower values of AK (region A) and less effect at higher values of AK (region B). Tensile properties and the corresponding numerical values of AKti, at various stress ratios are given by Ritchie[21]. It is then easy to write the expression for FCP based on eqns (7) and (8): $

= 4.05 - lo-“(AK - 2.58)2

(for R = 0.05),

(10)

-$

= 4.05 s 10-‘“(AK - 1.80}2

(for R = 0.70).

(10

The curves for FCP predicted by eqns (10) and (11) are in good agreement with the test results shown in Fig. 5. However, it should be pointed out that, due to different definitions, the above values of AK,, are smaller than those determined experimentally [2]. In actual fact, the AKti, values obtained from tests are defined at about du/dN = 10m6to lo-’ mm/cycle [ll]. In eqn (7), the theoretical value is defined as the value of AK for du/dN = 0.

XIULIN ZHENG and MANFRED A. HIRT

970

A 300

M

alloy

steel

=

0.05 l

R

=

0.70

-6 _ 10

2

4

6

0

10

20

40

60

Fig. 5. Predicted curves and test results showing the influence of stress ratio on FCP for 300M alloy steel, austenitized at 87o”C,oil-quenched and tempered at 3WC 1211.

4. EXPERIMENTALRESULTS 4.1 Methods for the analysis of test data After log~~mi~ ~ansformation, eqn (7) becomes: da log do = log B + 2log(AK - AK,,,),

(12)

which represents a straight line with slope 2 when da/dN versus (AX - A&J is plotted on a logarithmic scale. Using a trial and error approach, a computer program for linear regression analysis can be written to obtain the values of B and A&, given the condition that the slope is within the range of 2 k 0.004. 4.2 Analysis of test results Test data for FCP in a new high strength low-alloy (HSLA) steel and a regular high strength steel are given in [22] and [23] respectively. The results obtained by the above mentioned regression analysis of the test data and the tensile properties of these two steels are listed in Table 1. Figure 6 shows the least squares fit to these test data. As may be seen, eqn (7) gives good agreement with the test data, particularly in region A (Fig. 6a). The predicted values of the coefficient B using eqn (8) are quite close to those determined by tests. The calculated values of A& may also be considered reasonable since they too are within the expe~mental ranges [20]. 4.3 Reanalysis of some existing test data Test data for the FCP of various steels which have been analysed based on Paris’s equation are given in Refs. [23-311. By reanalysing these results, one can easily obtain B and A&, in eqn (7). The tensile properties and the predicted values of B from eqn (8) are also listed in Table 2. The predicted values of B are very close to those obtained by reanalysis. However, with a factor of difference of two or three

Table 1. Tensile properties and test results of the FCP for two steels oU

MATERIAL

u

EMPa] HSLA

steel

30CrMnSiNi2A

Of

= ou

r is

the

RA

Y

R

[MPa+]

AKth

R

[%I

EXPERIMENT

PREDICTED

[MPa

r

s

w]

[22]

591

452

65.0

0.1

6.6?~.10-~~

7.55.1O-'O

10.10

0.979

0.08‘

[ZS]

1703

1357

45.7

0.2

6.51+10-'"

5.36*10-lo

1.61

0.985

0.051

(1 + RA)

and

correlation

of

= - In

coefficient

(I - RR) and

in

s the

eq.

(8)

standard

[17] deviation

971

Fatigue crack propagation in steels

10

-8 10

50

100

5

10

50

Fig. 6. Regression analysis of test data of fatigue crack propagation according to eqn (7). (a) High-strength, low-alloy steel [22]. (b) High-strength martensite steel [23]. to say the same thing for some pearlite-ferrite steels or for some quenched-tempered steels. Why this difference appears is still not clear and supplementary research is needed. However, it should be pointed out that the FCP is a highly localized phenomenon and the microstructure has thus a great influence on the FCP rate. In his work, Pelloux[32] suggested that the FCP in an aluminium alloy is controlled principally by the matrix properties when the plastic zone width is smaller than the interparticle spacing of the inclusions. From this, it may be suggested that the differing values of B for those steels will be more dependent upon the tensile properties of the matrix rather than those of the steels. If it is assumed that 0.1 E is the critical fracture stress, or theoretical strength, of the steel matrix phase [6], the value of B according to eqn (8) will become 3.97 * lo-“. This value is closer to that of B for those steels previously mentioned.

times in the value of B, it is not possible

5. DISCUSSION The modified static fracture model is still based on the tensile stress failure criterion. According to this hypothesis, the maximum stress at the crack tip in all cases should be either equal to, or below, the critical fracture stress of the metal employed. Consequently, the crack tip must be bluntened during loading in order to relieve the stress at the crack tip. This is actually required by the equilibrium between the applied stress and the resistance of the metal. Hence, the higher the applied stress, the bigger the crack tip radius. (b) Applicable domain of the new expression As mentioned before, the new expression can only be used to describe the crack propagation process in regions A and B. Hence, eqn (7) is only valid for regions A and B. The upper limit of region B may occur at a rn~~~de order of da/dN = lo-’ mm/cy~le. Thus again, eqn (7) can only be applied when da/dN I 10V3mm/cycle. (c) Efectiue stress intensity factor range The existence of the fatigue crack propagation threshold has been shown and the values of Ati, determined by experiment, have been summarized in [9, 20, 333. Since the fatigue crack will not propagate when AK 5 I&,, the difference of AK -AK,,, in order words A&, must be the goveming parameter in FCP instead of AK alone. Using the A& concept, the modified FCP model and the new expression derived from it, provide a better description of the crack propagation behaviour. In particular, it gives an explanation of the effect of stress ratio on the FCP rate.

972

XIULIN ZNENG and MANFRED A. HIRT Table 2. Tensile properties and reanalysed results of the FCP for various steels 0

I

RA

ci

i

I

B

fMPa-2]

T

i

lSNi(VOO)

[24]

1764 1 1679

31.0

0.05-0.2

8.67.10-l*

9.29.lcr'C

18Ni(1100)

[24]

1587

1493

35.6

0.05-0.2

6.37.10-lo

3.50.10-1c

4340(200)

[24]

2324

1325

17.2

0.05-0.2

1.97.10-8

1.55*10-"

19.30

4340(500)

[24]

1714

1521

26.3

0.05-0.2

1.37.10-9

1.36.10-g

14.05

H-11

[24]

195E

1424

3i.e

0.05-0.2

1.37.10-8

a.lo.io-1C

19.56

D6AC

[zs]

1618

1499

36.5

cl.i-0.3

4.53.10-'0

7.73*10-1c

6.78

12Ni-SCr-3Mo

[26]

129C

1269

64.0

o-o.7

2.72.10-'"

3.70.10-'C

2.73

H-130

[26]

1015

936

70.0

O-O.7

3.14.10-'0

3.70.10-'C

2.24

lONi-Cr-MD-Co

1261

1434

1310

72.0

o-o.7

2.11.10-'0

2.47*10-1c

8.53

3Ni-4Co-0,2X

[26]

1333

1257

61.0

-0.1-0.25

4.10.10-9

3.87.10-'0

27.88

10Ni-BCo-Mc

1271

951

917

69.3

3.06~10-~~

1.11.10-'0

4.65

30CrMnSiNi2

[23]

1676

1387

45.0

0.2

7.42.10-"

5.S9~10-lc

2.26

4357A

[2S]

569

405

73.0

o-o.7

5.38.10-10

5.99*10-10

10.17

43OZB

[2S3

604

384

67.0

o-o.7

4.27.10-'"

s.93.lo-10

11.31

936

[2S]

514

247

$8.0

o-o.7

3.59.10-'0

7.k?4*10-10

10.06

4B5-C

[2S]

432

268

66.0

O-G.7

3,21.10-10

9.9S.l!F"

9.62

jAEOO30

[29J

494

302

46.0

0

7.54.1r'C

1.74.10-"

17.60

-1

5.95.10-10

1.74.10-9

20.01

5AElO20

[293

412

261

58.0

0

l.aS.lo-9

1.37.10-g

24.71

-1

1.68.10-g

1.37.10-g

21.00

0

5.95.10-10

3.92*10-'0

20.01

-1

9.09~10-~0

3.92.10~'0

20.88

4572

[ZSJ

535

370

11.15 7.16

l340(1000)

[24]

1205

1120

38.5

0.05-0.2

9.03.10-10

3.06.10-'0

13.28

1340(1400)

[24]

665

431

53.3

0.05-0.2

1.64.10-Y

9.97.10-'C

21.00

4151403jHeat637)

[30]

769

641

56.5

2.61.10-10

7.65.10-'C

6.45

4151403(Heat933)

[30]

748

611

53.1

4.23.10-10

3.92.10-'0

8.31

4ISI403fHeat484)

[303

817

678

48.5

2.23.lo-'*

?.60.10-1c

7.43

SNi-Cr-Mo-V

[31]

1043

974

68.0

0.1

2.30.10-'0

3.87.10-l"

5.79

5AE4140(720~

[29]

1468

1372

55.0

0

1.25.10-10

1.25.10-1"

-4.48)

-1

1.43.10-'0

1.25.10-10

-1.20)

jAE4140(970)

[291

1146

1098

59.0

0

1.32.10-10

1.72.10-10

5.79

-1

2.95.10-'0

1.72.10-10

9.40

0

2.07.10-'0

5.63.10-10

13.23

-1

2.66.10-l"

;.63‘10-'0

11.59

jAEA140(1230)

[ZS]

720

617

63.0

(d) Constants in the new expression The new expression predicts a second power relationship between daldN and AK-A&,, hence a constant exponent value of 2 in eqn (7). This is the same as that predicted by other theoretical models [ 1,2,6]. The coeficient B is a material constant, Equation (8) gives the relationship between B and the tensile properties of steels. Experiments results for the values of I3 show good agreement with those predicted by eqn (8) (Tabk 1). In addition, most values of 3 have the same order of magnitude, i.e. IO-” [IMPa-‘]. The quantity AKtr.,in eqn (7) is also a material constant. The values of A&, in Tables 1 are obtained by regression analysis of the FCP data derived from eqn (12). These values are within the limits of AK,,, determined by tests and, therefore, may be considered to be adequate. Such a method for determining the value of AKtr, is very useful. In order to obtain an accurate value of A&,, da/dN must be measured at less than lop5 mm/cycle.

Fatigue crack propagation in steels

973

6. CONCLUSIONS 1. During loading, the bluntening phenomenon at the crack tip has been taken into account in the new static fracture model for the FCP developed in the present paper. It is clearly necessary to maintain mechanical equi~brium at the crack tip. 2. Considering the existence of the FCP threshold A&,, the effective stress intensity factor range A&, which is defined as the difference AK -A&,, is thought to be the governing parameter in FCP. 3. Based on the new FCP model, a new expression for FCP has been derived:

-&=~(AK - AK,,J2. where B is a material constant and can be predicted from the tensile properties. This expression for FCP can be applied to give a good description of the fatigue crack propagation at various stress ratios below da/dN rs W3 mm/cycle. 4. This study has introduced a very simple and useful relationship between the coefficient B and the tensile properties of metals leading, consequently, to their correlation with fatigue crack propagation. 5. ,I\&, is a very important parameter affecting the FCP rate, in particular at the lower values of AK, in region A. The required value of AK,,, may be determined from the test results of FCP by a trial and error approach using linear regression analysis and thus without any additional testing. 6. A final important conclusion is that the model includes the most essential crack propagation parameters such as AK, A&,, material properties and the stress ratio. Finally, although the existing experimental results justify the use of the new expression, more test results with different types of steels are necessary to show its range of applicability. Acknowledgements-We are grateful to Prof. J.-C. Badoux, director of the Institute for Steel Construction of the Federal Institute of Technology in Lausanne, Switzerland, for his support of our work. Appreciation is also expressed to Messrs. F. Celebi, B. Kerridge and M. Fiaux for their help in preparing this paper.

REFERENCES [I] T. Yokobori, ASTM STP 675,683-706 (1979). [2] C. Bathias and J.-C. Bailon, La Fatigue des Matlriaux des Structures. Les Presses de !‘Universit6 de Montrta!, MontrCa!(1980). [3] P. Paris and F. Erdogan, Trans. ASME, J. Bas. Engng 85528-534 (1983). [4] R. W. Hertzberg, ASTM STP 415.205-223 (1968). [S] R. J. Donahue, C. M. Clark, P, A~nmo, R. Kumble and A. J. McEvily, Inf. 1. Pratt. Meek. l&209-219(1972). 161M. P. Ashby, Micromechanjsms of Fracture in Static and Cyclic Failure in Fracfure Meckanics (Edited by R. A. Smith). Pergamon Press, Oxford, England (1979). [7] W. G. Clark Jr., Engng Fract. Meek. 2,287-299 (1971). [8] H. Suzuki and A. J. McEvily, Met. Trans. lOA, 475-481(1979). [9] P. E. Irving and L. N. McCartney, Metal Science, 351-361(1977). [lo] D. N. La! and V. Weiss, Met. Trans. 9A, 413-426(1978). [ll] C. J. Beevers, Metal Science, 362-367(1977). f12] C. E. Richards and T. C. Lindley, Engng &act. Meek. 4,951-978 (1972). [13] Xian Jiaotong University et a!., Mechanical Properties of Metals (!%l) (in Chinese). [ 141Zhang Xinggian et al., Mechanical Behaviour of Metals and A!!oys (1961)(in Chinese). [15] M. Kikukawa, M. Jono and M. Adachi, ASTM STP 675,23C247 (1979). [16] K. H. Schwalbe, Engng Fract. Meek. 9,547-556 (1977). [!7] Xiulin Zheng, Local strain range and fatigue crack initiation life. IABSE Proc. Fatigue Colloquium, pp. 169-178(1982). [!8] P. C. Paris and G. C. Sih, ASTM STP 381, 30-81 (1965). [19] Xiulin Zheng, Estimation of Notch Strength of Metals (unpublished~. 12010. Vosikovsky, Engng Fmcf. Meek. 11,592-204 (1977). [21] R. 0. Ritchie, J. Engng Muter. Tech., Trans. ASME, 195-204(1977). [22] M. A. Hirt, (unpublished data). [23] Quangli Hu, (unpublished data). 1241G. A. Miller, Trans. ASM 61,442*8 (1968). [25] P. C. Paris, R. J. Bucci and C. D. Little, ASTM, STP 513, 196-217(1972). [26] J. M. Barsom, E. J. Imhof and S. T. Rolfe, Engng Fract. Me&. 2,301~317(1971). [27] W. G. Clark Jr. and S. T. Hudak Jr., f. Testing and Eun!uafjon %6), 454-476 (1975). [28] J. M. Barsom, J. Engng Industry, Trans. ASME, 1190-11%(1971). [29] R. I. Stephens, P. H. Benner, C. Mauritzson and G. W. Tinda!!, J. Testing and Evaluation S(2),68-81 (1979). [30] W. A. Logsdon, Engng Fract. Meek. 7, 23-40 (1975). [31] A. M. Sullivan and T. W. J. Crooker, Testing and Evaluation s(2), 96-101 (1977). [32] R. Pelloux, Trans. ASM 57,511-518 (1964). [33] V. Weiss and D. N. La!, Met. Trans. 5A, 1946-1947(1974). (Received 25 May 1982;receiuedfor p~b!icat~on20 Jicly 1982)