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Prediction of non propagating cracks
PREDICTION OF NON PROPAGATING CRACKS M. H. EL HADDAD. T. H. TOPPER and K. N. Shill-H Departmentof Civil Engineering,University of Waterloo,Waterloo,Ontario,CanadaN2L 3Gt Ah&act-An explanationfor non propagatingfatigue cracks is presentedbased on the criterionthat once the value of a particularstrain intensity factor reduces lo the threshold value for the materialthe crack should stop. Predicted lengths of these cracks based on solutions for the intensity factor are in good agreementwith the experimentaldata Intensityfactor trends for cracks in notches pn shown to vary from an initial decrease to a minimumvalue followed by an increaseand eventualconvergencewith the trendfor the equivalent long crack for sharp notches to the blunt notch curves that continuouslyincreased during their approachto the long crack trend.The type of trendexhibited by a given notch depends both on notch geometry and notch size. In blunt notches the maximumvalue of the thresholdstress for crack propagation is at initiation. However, for sharp notches the peak value of the thresholdstress vs crack length curves shifts to a finite length. Stresses above the initiationlevel but below this peak stress kvel result in fatigue crackswhich start but do not propagateto failure. Predictedvalues of the fatiguelimit stresses for a variety of sizes in a circularand an elliptical notch are in good agreementwith experimentalresults.
NOTATION modulusof elasticity i theoreticalstress concentrationfactor, a function of crack length k stress concentrationfactor of uncrpckednotches
kb stress concentrationfactor, a function of crack length k: strain concentrationfactor, a function of crack length N numberof cycles R ratio of minimumto maximumstress da/dN : : c A[ AL AS AC A0 AK A&I AU,, A.% AS,
crack growth rate crack length measuredfrom the edge of the notch materialconstant total crack length includingthe depth of the notch half the plate width depth of the notch notch root radius fatigue limit nominalstrain range nominal stress range local strainrange aroundcrack tip in notched specimens local stress rangearoundcrack tip in notched specimens stress intensity factor thresholdstress intensity factor thresholdstress in a smooth specimen thresholdstress in notchedspecimen minimumpropagation stressin notched specimen
INTRODUCI’ION FATIGUECRACKSin
structural elements usually initiate at a geometric stress raisers such as holes, changes of member cross section or weld toes. Cracks thus formed advance fist through the highly stressed region close to the notch and then continue through the body of the structural element until final fracture occurs. Cracks form quickly at the roots of sharp notches, even at low stress levels, but if the notch is very sharp and the nominal stress range sufficiently small, they do not continue to grow across the specimen cross section. Thus, the fact that a notched specimen is unbroken after testing does not necessarily imply that the material at the notch root is untracked. When the stress necessary to initiate a crack will not cause it to grow to failure the notched fatigue limit which is generally equated to this initiation stress is instead equal to the minimum stress level necessary to continue crack propagation to fracture. For blunt notches these two fatigue limits are identical, but for some sharply notched specimens they are different, resulting in a stress regime in which nonpropagating cracks are found[14]. In this regime a crack is formed at the notch root when the nominal stress is equal to the fatigue limit divided by the stress concentration factor but the crack will not continue ‘growing unless a certain higher threshold stress level is applied[lA]. Several investigators have attempted to predict nonpropagating cracks and the notched fatigue limit using empirical formulations [ IA],
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a volumetric strength effect[5], crack opening and closure[6] and the elastic fracture mechanics approach[7]. The failure of existing elastic solutions for stress intensity factors to explain nonpropagating cracks is believed to be due to local plasticity around notch roots and the inabiiity of these solutions to correlate the behaviour of short cracks@-111. Previous work[ll121has shown that the growth rates for very short cracks considerably exceed those given by conventions stress intensity crack growth laws derived from larger cracks, and cracks in a specimen with a sharply decaying notch strain field exhibit a crack growth rate that initially decreases, reaches a minimum, and thereafter increases[8, 131. This early decrease in crack growth rates was predicted based on the argument that the initiation and early growth of a crack is controlled by the phstic strain field of the notch which diminishes rapidly f9, lo]. Also, a rapid early growth stage was predicted using a strain based intensity factor[8] and the f-integral method[l4]. The latter investigations suggested that the effect of plasticity is to raise the effective stress intensity factor and J-integral values above the elastic solution in the vicinity of the notch. In the fatter prediction[8. 141an effective crack length which is equal to the actual length increased by a length constant which accounts for non-continuum behaviour of‘very small cracks is employed. This paper presents an analytical explanation for nonpropagating cracks and notch sensitivities based on solutions for a strain based intensity factor and in addition explains the factors governing the propagation of fatigue cracks and correlates the behaviour of cracks in notches of various severities. INTRNSITY FACTORS FOR SHORT CRACRS The present authors have proposed[l l] the folIowing strain based intensity factor for a crack of length 1. AK = EAem where he and E are the applied strain range and the modulus of elasticity of material respectively and f,, is a constant for a given material and material condition. For elastic levels of the stress AS, eqn (1) becomes AK = AS-)
(2)
Taking the usual defmition of the threshold stress intensity A&& as the ~nimum stress intensity that wih give rise to crack propagation, the relation between A&, and AQ,, the threshold stress range for crack propagation at crack length f may be obtained from eqn 2 as:
A&, = has,, w
(3a)
or
(3b) Since the threshold stress at a very short crack length has been shown [I 11 to approach the fatigue limit of the material AGOf. may be obtained from eqn (3b) as: AK,I, = Aa,-
Ma)
or
Mb) As the crack length decreases, the length 1, constitutes an increasing fraction of the effective length until at very short lengths it represents the crack length at which the fatigue limit stress can propagate a crack into the interior of the specimen. It should be noted that intensity factors defined by eqns (1) and (2) predict higher crack propagation rates for short cracks than the more usual representation of stress intensity which deletes the term 4. A series of experiments on CSA CM.1 1 steel were performed to examine the accuracy of
Predictionof non propagatingcracks
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the term lo in correlating experimental threshold and short crack results. In these experiments short cracks were initiated at the end of a slit cut in a sheet specimen by applying a very small cyclic load. The slit was then machined away and the threshold stress levels for various crack lengths determined using a series of specimens. Figure I presents these experimental results together with the theoretical curve given by eqn (3b). Agreement of the predictions of eqn (3b) and the test data is excellent indicating that the equation describes the effect of proximity of the crack tip to the specimen surface on threshold stress intensity. Figure 2 which reproduces previous experimental data on the basis of AK,k v crack length, shows that while the threshold stress intensity factor AK,, calculated via (3a) is dependent on crack length if the term lo is dele@d from the equation, A& remains constant if lo is included in the equation. The threshold stress intensity A&, was estimated using eqn (4a) as shown in Fig. 2. This observation is also supported by the experimental results given in Refs.[2] and [7]. Frost[7] argued that the dependency of AK,,, on crack length may be explained by a variation of crack closure with crack length. The constant lo has been defined empirically in terms of AK,,, and Aa, through eqn (4b) without advancing a model for its significance in the crack propagation process, but it can be considered at the surface as a measure of the reduced flow resistance of surface grains due to their lack of constraint[ 111. To examine the ability of the term lo to predict crack growth rates for short cracks, measured crack growth rates in CSA G40.11 steel plates as a function. of the elastic stress intensity factor which deletes the term 1, are given in Fig. 3. Results for crack lengths less than 1.0 mm have higher crack growth rates than predicted by the long crack trend. A similar trend in experimental results was noticed in Refs. [l&16]. However, as shown in Fig. 4, if eqn (2) is used to correlate short and long crack results the discrepancy shown in Fig. 3 is eliminated. f., loor
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Effect of crack length on thresholdstress amplitudefor fatigue crack growth in G40.11 steel. L.mm 0 IO
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20-
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Fig. 2. Estimationof thresholdstress intensity factor.
M. H. EL HADDAD et 01.
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. l f
6’
8
> 100
AK KSI fi Fig.3. Comparison of short crack with long crack results for G40.11 steel without b included in the AK
solution. AK, MPa fi
.
Fig. 4. Fatigue crack propagation rates as a function of AK given by eqn (2).
Prediction of non propagating cracks
5n
CONDITIONS FOR NON PROPAGATING CRACRS Intensitysolutionsfor cracks in notches In using eqn (1) to calculate intensity factors for cracks emanating from notches the nominal strain term Ae should be replaced by the local strain in the vicinity of the crack tip, Ar. This gives AK = EAcm). Estimates of local strain Ar can be obtained using finite element plastic solutions161 or may be derived from a relationship between concentration factors proposed by Neuber[17] which has been shown to give reasonably accurate results[8]. If eqn (5) is recast in terms of the strain concentration factor k: we have AK = Ek:Aem.
(6)
When applied stress levels are low enough that notch strains remain elastic, solutions for the elastic stress concentration factor k’ for a crack in a notch [ 181 may be used to determine the local stress or strain in terms of the nominal stress or strain and eqn (6) becomes AK = Ek' Aem)
= k' ASm).
Here the crack length 1 is measured from the notch root and lo is again the material constant defined by eqn (4b). The stress concentration factor k’[18] &creases from an initial maximum value approximately equal to 1.12 kt, where k, is the theoretical stress concentration factor for an tutcracked notch, to a value of d(/ + c + &J/(1+ lo), where c is the notch depth, as the crack passes outside the field of influence of the notch. Thereafter, the crack may be analyzed as a simple crack with a length equal to the actual crack length plus the notch depth. Intensity factor solutions given by eqns (S)-(7) successfully correlated short crack growth data for .notches of varying severity with elastic long crack data[8]. Although constant stress amplitude tests of these notches gave crack growth rate vs crack length curves which varied from monotonically increasing for blunt notches to an initial decrease followed by an increase for sharp notches, all the data fell within the long crackdata when correlated by the intensity factors given by eqns (5)-o. Conversely solutions for the intensity factors were successfully used predict elastic and inelastic short crack growth curves for notches of various severities. Crack arrest condition Previous observations[8, 131have shown that cracks in a sharply notched specimen exhibit a crack growth rate that initially decreases, reaches a minimum, and thereafter increases. Stress intensity vs crack length curves for a constant load level in these specimens also initially decreased and then increased. When the applied load level was such that the stress intensity at the specimen surface was above the threshold but the minimum value of the stress intensity was below the threshold, crack growth started and then ceased when the stress intensity decreased to the threshold resulting in dormant cracks[8-101. To verify this criterion for predicting non propagating cracks, crack growth data were obtained from a specimen with a 0.40 mm dieter circular notch in a G40.11 steel, and plotted in IQ. 5 on scales of AK vs crack length. In this figure, experimental values of AK are obtained by equating the unknown intensity factor at a given crack propagation rate in the notched specimen to the known stress intensity factor in a centrally cracked specimen having the same crack propagation rate[8]. Predicted values of AK based on eqn (5) are in agreement with experimental results, but on the other hand elastic solutions given by eqn (7) (also plotted in the figure) resulted in an underestimate of AK for short cracks. The figure illustrates that for stress levels of 483 MP,, 414 MP,, and 339MP,, at which specimens were tested, AK values initially decreased with crack length but reached a minimum above the threshold value of the material AKlh estimated from eqn (4a). At stress levels of 310 IMP, and 241 MP, where AK values based on eqn (5) decrease to a minimum below the threshold value, tests were run for ten million cycles without failure and no cracks were seen on the surface of specimen. When the stresses in each of these specimens were increased to 339 MP,, they failed and a subsequent examination of the fracture
hi. H. EL HADDAD et al. I
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1. IfI Pii 5. Predictionof non propagatingcracks in a circulat notch for G40.1t steel.
surface of the specimens indicated that cracks had initiated at the mid-plane of the specimens (presumably due to the biaxial stress state[l91) and crack fronts corresponding to the hrst applied stress levels were noted on the fracture surface. Non propagating crack lengths aJong the mid-plane were measured and arc in good agreement with predicted lengths based on AK solutions given by eqns (5) as shown in Fii. 5. Elastic solutions for AK based on eqn (7) also predict the same non propagating crack lengths as those predicted based on eqn (5) and both solutions are expected to accurately predict the minimum stress level corresponding to failure. This coincidence between the two sotutions could have been anticipated for this metal because its local stress-strain behaviour at the endurance limit strain amplitude is essentially eiastic. Since eqn (3b) shows that the local threshold stress levei will decrease with increasing crack length from this already essentially elastic level at the surface, the local material response will become elastic as the threshold stress intensity is approached making eqn (5) equivalent to eqn 17). Experimental results for non propagating cracks in SAE lOi steel specimens containing an elliptical notch are given in Fig. 6. In two specimens tested at a stress level of 69 iUP,, cracks started and were measured along the surface of the specimen using a travelling microscope until they ceased to propagate. After ten million cycles, the load was increased to yield a stress levef of 76 Mp, and both specimens then failed as shown in Fig. 6. Predicted non prop~~g crack lengths based on the value of AK given by eqn (5) are in good agreement with the experiment. Also, the solution for AK at a stress level equal to 76 MP, predicts that the specimen will fail since the minimum of the AK vs crack length curve is above the threshold of the material. Experimental test data found in Ref. [3] for non propagating cracks in a V-edge notched mild steel plate are given in Fig. 7.3oth qns (5) and (7) give accurate predictions of non propagating crack lengths and the minimum stress level correspondiig to failure. The following section examines in more detail the factors governing the propagation of fatigue cracks in notches of varying geometry and size, and the conditions for non propagating cracks. TERRSEOLD !3TRE!BE!3FOR NOTCHES Notch size effect
Further insight into the dependence of AK vs crack length trends on the notch size and the amount of local plasticity may be obtained if AK is plotted vs crack length in a non~mensional form by rearrangingeqn (6) as follows: AK = k:,/(y). AS%$&j
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Predictionof non propagating cracks
44’ 002
0 01
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Fig, 6. Prediction of non propagatingcracks in an elliptical notch for SAE1015steel.
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. Fii. 7. Predictionof non propagatingcracks in a V-notch for Mild steel.
Curves plotted on this basis in Fig. 8 compare the behaviour of an elliptical notch, a circular notch and a long crack for various notch diameters assuming elastic nominal strains, i.e. EAe = AS. Curves in the bottom half of the figure representing elastic notch root material response are independent of load level since k: in eqn (6) will be equal to the elastic stress concentration factor k’ given in eqn (7). Here intensity factor values for small diameter circular and elliptical notches both initially decrease with crack length and then increase to converge with long crack values. As the notch size is increased this trend changes to one of continuously increasing intensity with crack length for both geometries. This change occurs at a smaller size for the circular than for the elliptical notch with, for example, the minimum diameter for which subsequent intensities do not fall below the initial value being 2.54 mm for the former and 10.00mm for the latter. This is the minimum diameter for which a non propagating crack is possible at a constant cyclic load level. At smaller diameters, if the load level applied results in an initial intensity above the threshold intensity for crack propagation but a minimum intensity below the threshold, crack growth will start and then cease when the applied intensity decreases to the threshold. The upper part of Fig. 8 illustrates changes in curves of intensity vs length for a 5.08 mm EFlfVd.II.No.3-G
M. H. EL HADDAD et al. a,mm
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CIRCULAR NOTCH WITH C EQUAL TO 254mm I ELASTIC NOTCH ROOT RESPONSE
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Fig 8. Behaviour of short cmclrs at various notch sizes.
diameter circular notch as the load is increased to levels causing notch plasticity. Changes in k:, the strain concentration factor in eqn (6) at the two inelastic levels, increase intensities at the short crack lengths within the plastic zone close to the notch and raise the initial parts of the curves. The curves, however, rejoin the elastic curve as the crack tip propagates beyond this inelastic region and k: becomes equal to k’. Previous observations of an effect of notch size on the behaviour of short cracks at notches can also be explained in terms of the applied nominal threshold stress A&,. This stress may be obtained by substituting the value of AK,,, into eqn (7) which has already been shown to predict non propagating cracks and the loads required for specimen failure. Rearranging eqn (7), we obtain;
Substituting for the value of AK,,, using eqn (4a), the last equation reduces to:
(10) Solutions for AS,,, given by eqn (10) are shown in Fig. 9 for cracks initiating from circular notches with dilTerent diameters. For the largest notch the threshold stress corresponding to failure is at initiation and once cracks initiate they will continue growing until final fracture. This initiation stress level is equal to AaJb However, as the notch size decreases the peak value of the threshold stress vs length curve shifts to the right as shown in Fig. 9. This peak value gives the minimum stress level required to fail the specimen. At stress levels below this value but above the value of Audkr which corresponds to initiation, cracks will start but will not propagate. Therefore, for a given notch geometry, decreasing the size of the notch results in an increase in the stress level required for failure and a decrease in notch sensitivity. Experimental results for non propagating cracks and the stress levels corresponding to failure given in Fig. 9 for a G40.11 steel lie close to the curves predicted by eqn (10).
Prediction of non propagating cracks
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Fii. 9. Notch size effect on the threshold stresses for circular notches in CM.1
1steelplates.
The e#ect of notch geometry Curves plotted on the basis of eqn (8) compare the behaviour of a V-notch having a 5.08 mm depth and a long crack for various notch root radii as shown in Fig. 10. These curves representing only elastic notch root material response are independent of load level. Here intensity factor values for small notch radii are initially greater than those for long cracks but decrease rapidly to approximate the intensity values of the latter. As the notch mot radius increases this trend changes to one of continuously increasing intensity values which start below the long crack curve but approach and then converge with it. At notches with small root mdii, applied load levels resulting in an initial intensity above the threshold intensity for crack propagation but a minimum intensity below the threshold result in cracks that start and then stop when the applied intensity decreases to the threshold. At larger notch root radii cracks once initiated continue propagating until final failure. Also, shown in Fii. 10 is the intensity vs crack length curve for the critical notch root radius (1.27 mm) above which cracks once initiated will propagate to specimen failure--only below this radius are non propagating cracks possible. Figure 11 gives experimental and predicted results for three radii values of this V notch in terms of A&, given by eqn (10) and crack length. Results agree well with predictions with crack arrest close to the curve of eqn (10) in the sub critical radius notch and continuing crack growth at stress levels close to AuJk, in the notches with larger radii. Fatiguelimitfor notches Equation (10) was used to calculate the minimum stress necessary to propagate a crack to specimen failure-that is, the peak value of threshold stress illustrated in Figs. 9 and 11. The curve given by these calculations for circular notched plate specimens at G40.11 steel together
0
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10. Behaviourof short cracks at a V-notch.
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M. H. EL HADDAD et ol. L,mm 1.0 I
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II. Notch root radiuseffect on the thresholdstresses for V-notch in Mild steel plates.
4 LO- loo IO -
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C. NOTCH DEPTH, in Fig. 12. Predict& of fatigue limit stresses for circularnotches in WI. 1I steelplates.
with experimental results is given in Fig. 12. Agreement between theory and experiment are excellent. The calculations suggested that the fatigue limit stresses increased with decreasing notch diameters closely approximating the fatigue limit of the material when the notch diameter decreased below I,,. Hence, increases in the fatigue limit of the notch and reductions in notch sensitivity are predicted for both reductions in the size of a notch and increases in the value of lofor a material. Similar calculations for the fatigue limit of the V-notched specimens of Ref.[21 were performed and the results are plotted together with experimental data in Fig. 13. Again, the agreement between the predicted curve and the data is excellent. The calculations predicted the trends observed in Fig. 10. Fatigue limits for notches with radii above the critical radius for the existence of a peak threshold stress (1.27 mm) were governed by the initiation stress while fatigue limits for notches with smaller radii were governed by a peak threshold stress occurring at a finite crack length. In this latter regime the notch fatigue limit remains virtually constant. As shown in Fig. 10 the length at which the minimum intensity which corresponds to the critical length occurs is for all notches in this regime, small compared to c. Hence, we may approximate
Predictionof non propagatingcracks
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28 i?
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EDGE NDTCHED PLATE
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Fig. 13. Predictionof fatigue limit stresses for a V-notch in Mild steel specimens.
the constant minimum fatigue limit stress by substituting the long crack value of k’ in eqn (10) to give
(11) hence, since 1 and lo are much smaller than c. AS, = Au,
(12)
AS, = 3.
(13)
This approximation which has previously been suggested by Smith and Miller[9] provides predictions for the fatigue limit of the V-notch virtually identical with those given by eqn (10). It should however, as they suggest, be restricted to sharp notches since it provides a much less satisfactory approximation for the blunter circular notch.
CONCLUSIONS (1) Non propagating fatigue cracks are predicted based on the criterion that once the value of the intensity factor reduces to the threshold value for the material the crack should become dormant. Predicted lengths for these cracks are in good agreement with the experimental data. (2) Intensity factor trends for cracks in notches are shown to vary from an initial decrease to a minimum value followed by an increase and eventual convergence with the trend for the equivalent long crack for sharp notches to the blunt notch curves that continuously increase during their approach to the long crack trend. The transition from sharp notch to blunt notch trends depends both on notch geometry and notch size. (3) In blunt notches the maximum value of the threshold stress for crack propagation is at initiation. However, for sharp notches the peak value of the threshold stress shifts to a finite length on threshold stress crack length curves. Stresses above the initiation level but below this peak stress level result in fatigue cracks which start but do not propagate to failure. (4) Predicted values of the fatigue limit stresses for a variety of sizes in two notch geometries are in good agreement with experimental results. Also, increases in the fatigue limit stress of a notch and reductions in notch sensitivity are correctly predicted for both reductions in the size of the notch and increases in the value of lo for a material of which the specimen is made.
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M. H. EL HADDAD et al. REFEIENCES
[I] N. E. Frost, T/tr Arto~uticrJ Quutrerfy 8. I-20 (Feb. 1957). [2] N. E. Frost, Pmt. Institt~tionof Mechanical Engineers.Vol. 173.No. 35, pp. 81l-835, London (1959). (31 N. E. Frost and D. S. I)ugdale. l Mechanic end Physics of Solids 5, 182-192,London (1937). [41 N. E. Frost, 1. Mechanical Engng ScL 2(2), lop-1 19 (I%@. [SI D. N. Lai and V. Weiss. Metalfwgicai Tmnsactfons LA 1623-1629(1975). 161K. Ohji. K. 08ura. H. T&ii. H. Ohkita and Y. Ohkuba. An analytical approach to the non propagating crack problem lwing the finite eiement method. Pmt. ISth Japan Congresson Matetials Research. pp. 91-94. The Society of Materials Science, Japan, Kyoto (1972). [A N. E. Frost. K. 1. Marsh and L. P. Pook. Metei Fatfgu. Clarendon Press, Oxford (1974). 181k41H. El Haddad. K. N. Smith and T. H. Topper, A strain based intensity factor solution for short fatigue cracks initiating from notches. II th Nat. Symp. Fmctum M&t., ASTM. American Society for Testing and Materials, June 12-14, Bfacksburg, V.A. R. A. Smith and K. J. Miller, ht. 1. Mcch. Engng and Sci., submitted June 1977. H. hf. Hammouda and K. J. Miller. Symp. Efastic Pfastic Pmctun. American Society for Testing and Materials, Atlanta Nov. 16-18 (1977). M. H. El Haddad. K. N. Smith and T. H. Topper, Fatigue Crack Pmpagation of Short CM&, (1978)ASME/CSME Joint Conference on Pressure Vessets and Pipin Nuclear Energy and Materials, Montreal (June Z-29 1978).Paper No. 78Mat-7 if21 A. Talug and K. Reiisnider, Analysis and Inoestigation of Smaff Flows. Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Cmck Gmwth, pp. 81-96. STP 637, ASTM, American Society for Testing and Materials. Phiiadelphia (1977). [I31 D. Broek, The Propagation of Fatigue Cmtks Emanating fmm Holes. NLR TR 72134 U, National Aerospace L&rauuy, The NetherIands (1972). T. H. Topper and K. N. Smith. /. Integmt Applicar~nr for Short Fatigue Cm&r at 1141M. H. El Haddad, N. E. DOW& Notches, ht. /. Fmctum. submitted for publication. (May 1978).
[ISI S. Pearson, Engng Fracture MecA 1,23S-247 (1975). 1161N. E. Dowlii, Cmck Growth Bving Low Cyclk Fatigue of Smooth Axial Specimens, ASTM STP (37, pp. 97-121. Arm&an Society for Testing and MWxiala, Phifadelphia (lmr). 1.54c550 (1961). [f7l H. Heuber, 1. Applied Mech., ASME, [I81 H. Tada. P. C. Paris and G. R. Irwin, 7’heStress Analysis of Cracks Handbook. Del Research Corp., Hellertown. PA [19] !9?*Leis and T. H. Topper, 1. Engng Mat. and Tech.. ASME Vol. 99, Series H., No. 3, pp. 215-221(July 1977). (Receioed5 June 1978: receivedfor publication 5 September1978)
NOTATION modulusof elasticity i theoreticalstress concentrationfactor, a function of crack length k stress concentrationfactor of uncrpckednotches
kb stress concentrationfactor, a function of crack length k: strain concentrationfactor, a function of crack length N numberof cycles R ratio of minimumto maximumstress da/dN : : c A[ AL AS AC A0 AK A&I AU,, A.% AS,
crack growth rate crack length measuredfrom the edge of the notch materialconstant total crack length includingthe depth of the notch half the plate width depth of the notch notch root radius fatigue limit nominalstrain range nominal stress range local strainrange aroundcrack tip in notched specimens local stress rangearoundcrack tip in notched specimens stress intensity factor thresholdstress intensity factor thresholdstress in a smooth specimen thresholdstress in notchedspecimen minimumpropagation stressin notched specimen
INTRODUCI’ION FATIGUECRACKSin
structural elements usually initiate at a geometric stress raisers such as holes, changes of member cross section or weld toes. Cracks thus formed advance fist through the highly stressed region close to the notch and then continue through the body of the structural element until final fracture occurs. Cracks form quickly at the roots of sharp notches, even at low stress levels, but if the notch is very sharp and the nominal stress range sufficiently small, they do not continue to grow across the specimen cross section. Thus, the fact that a notched specimen is unbroken after testing does not necessarily imply that the material at the notch root is untracked. When the stress necessary to initiate a crack will not cause it to grow to failure the notched fatigue limit which is generally equated to this initiation stress is instead equal to the minimum stress level necessary to continue crack propagation to fracture. For blunt notches these two fatigue limits are identical, but for some sharply notched specimens they are different, resulting in a stress regime in which nonpropagating cracks are found[14]. In this regime a crack is formed at the notch root when the nominal stress is equal to the fatigue limit divided by the stress concentration factor but the crack will not continue ‘growing unless a certain higher threshold stress level is applied[lA]. Several investigators have attempted to predict nonpropagating cracks and the notched fatigue limit using empirical formulations [ IA],
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M.H.ELHADDAD etal.
a volumetric strength effect[5], crack opening and closure[6] and the elastic fracture mechanics approach[7]. The failure of existing elastic solutions for stress intensity factors to explain nonpropagating cracks is believed to be due to local plasticity around notch roots and the inabiiity of these solutions to correlate the behaviour of short cracks@-111. Previous work[ll121has shown that the growth rates for very short cracks considerably exceed those given by conventions stress intensity crack growth laws derived from larger cracks, and cracks in a specimen with a sharply decaying notch strain field exhibit a crack growth rate that initially decreases, reaches a minimum, and thereafter increases[8, 131. This early decrease in crack growth rates was predicted based on the argument that the initiation and early growth of a crack is controlled by the phstic strain field of the notch which diminishes rapidly f9, lo]. Also, a rapid early growth stage was predicted using a strain based intensity factor[8] and the f-integral method[l4]. The latter investigations suggested that the effect of plasticity is to raise the effective stress intensity factor and J-integral values above the elastic solution in the vicinity of the notch. In the fatter prediction[8. 141an effective crack length which is equal to the actual length increased by a length constant which accounts for non-continuum behaviour of‘very small cracks is employed. This paper presents an analytical explanation for nonpropagating cracks and notch sensitivities based on solutions for a strain based intensity factor and in addition explains the factors governing the propagation of fatigue cracks and correlates the behaviour of cracks in notches of various severities. INTRNSITY FACTORS FOR SHORT CRACRS The present authors have proposed[l l] the folIowing strain based intensity factor for a crack of length 1. AK = EAem where he and E are the applied strain range and the modulus of elasticity of material respectively and f,, is a constant for a given material and material condition. For elastic levels of the stress AS, eqn (1) becomes AK = AS-)
(2)
Taking the usual defmition of the threshold stress intensity A&& as the ~nimum stress intensity that wih give rise to crack propagation, the relation between A&, and AQ,, the threshold stress range for crack propagation at crack length f may be obtained from eqn 2 as:
A&, = has,, w
(3a)
or
(3b) Since the threshold stress at a very short crack length has been shown [I 11 to approach the fatigue limit of the material AGOf. may be obtained from eqn (3b) as: AK,I, = Aa,-
Ma)
or
Mb) As the crack length decreases, the length 1, constitutes an increasing fraction of the effective length until at very short lengths it represents the crack length at which the fatigue limit stress can propagate a crack into the interior of the specimen. It should be noted that intensity factors defined by eqns (1) and (2) predict higher crack propagation rates for short cracks than the more usual representation of stress intensity which deletes the term 4. A series of experiments on CSA CM.1 1 steel were performed to examine the accuracy of
Predictionof non propagatingcracks
575
the term lo in correlating experimental threshold and short crack results. In these experiments short cracks were initiated at the end of a slit cut in a sheet specimen by applying a very small cyclic load. The slit was then machined away and the threshold stress levels for various crack lengths determined using a series of specimens. Figure I presents these experimental results together with the theoretical curve given by eqn (3b). Agreement of the predictions of eqn (3b) and the test data is excellent indicating that the equation describes the effect of proximity of the crack tip to the specimen surface on threshold stress intensity. Figure 2 which reproduces previous experimental data on the basis of AK,k v crack length, shows that while the threshold stress intensity factor AK,, calculated via (3a) is dependent on crack length if the term lo is dele@d from the equation, A& remains constant if lo is included in the equation. The threshold stress intensity A&, was estimated using eqn (4a) as shown in Fig. 2. This observation is also supported by the experimental results given in Refs.[2] and [7]. Frost[7] argued that the dependency of AK,,, on crack length may be explained by a variation of crack closure with crack length. The constant lo has been defined empirically in terms of AK,,, and Aa, through eqn (4b) without advancing a model for its significance in the crack propagation process, but it can be considered at the surface as a measure of the reduced flow resistance of surface grains due to their lack of constraint[ 111. To examine the ability of the term lo to predict crack growth rates for short cracks, measured crack growth rates in CSA G40.11 steel plates as a function. of the elastic stress intensity factor which deletes the term 1, are given in Fig. 3. Results for crack lengths less than 1.0 mm have higher crack growth rates than predicted by the long crack trend. A similar trend in experimental results was noticed in Refs. [l&16]. However, as shown in Fig. 4, if eqn (2) is used to correlate short and long crack results the discrepancy shown in Fig. 3 is eliminated. f., loor
‘.
.
IO
100
loo
I
I
I
I
EXPERIMENTAL
RESULTS
G40 II
R--I
-
mm
010
STEEL
.
.lo=0.24mm
PREDICTION
I-
I .Ol
OS0 001
/ 010
I ID
420 loa
L,m Fig.
1.
Effect of crack length on thresholdstress amplitudefor fatigue crack growth in G40.11 steel. L.mm 0 IO
1.0
I
10.0
I
100
20-
f Y a -lO
0.00001
ORI
o
AK,=Au+,fi
l
AK,,= Aqhm
0.10 I,
1.0
In
Fig. 2. Estimationof thresholdstress intensity factor.
M. H. EL HADDAD et 01.
.
. l f
6’
8
> 100
AK KSI fi Fig.3. Comparison of short crack with long crack results for G40.11 steel without b included in the AK
solution. AK, MPa fi
.
Fig. 4. Fatigue crack propagation rates as a function of AK given by eqn (2).
Prediction of non propagating cracks
5n
CONDITIONS FOR NON PROPAGATING CRACRS Intensitysolutionsfor cracks in notches In using eqn (1) to calculate intensity factors for cracks emanating from notches the nominal strain term Ae should be replaced by the local strain in the vicinity of the crack tip, Ar. This gives AK = EAcm). Estimates of local strain Ar can be obtained using finite element plastic solutions161 or may be derived from a relationship between concentration factors proposed by Neuber[17] which has been shown to give reasonably accurate results[8]. If eqn (5) is recast in terms of the strain concentration factor k: we have AK = Ek:Aem.
(6)
When applied stress levels are low enough that notch strains remain elastic, solutions for the elastic stress concentration factor k’ for a crack in a notch [ 181 may be used to determine the local stress or strain in terms of the nominal stress or strain and eqn (6) becomes AK = Ek' Aem)
= k' ASm).
Here the crack length 1 is measured from the notch root and lo is again the material constant defined by eqn (4b). The stress concentration factor k’[18] &creases from an initial maximum value approximately equal to 1.12 kt, where k, is the theoretical stress concentration factor for an tutcracked notch, to a value of d(/ + c + &J/(1+ lo), where c is the notch depth, as the crack passes outside the field of influence of the notch. Thereafter, the crack may be analyzed as a simple crack with a length equal to the actual crack length plus the notch depth. Intensity factor solutions given by eqns (S)-(7) successfully correlated short crack growth data for .notches of varying severity with elastic long crack data[8]. Although constant stress amplitude tests of these notches gave crack growth rate vs crack length curves which varied from monotonically increasing for blunt notches to an initial decrease followed by an increase for sharp notches, all the data fell within the long crackdata when correlated by the intensity factors given by eqns (5)-o. Conversely solutions for the intensity factors were successfully used predict elastic and inelastic short crack growth curves for notches of various severities. Crack arrest condition Previous observations[8, 131have shown that cracks in a sharply notched specimen exhibit a crack growth rate that initially decreases, reaches a minimum, and thereafter increases. Stress intensity vs crack length curves for a constant load level in these specimens also initially decreased and then increased. When the applied load level was such that the stress intensity at the specimen surface was above the threshold but the minimum value of the stress intensity was below the threshold, crack growth started and then ceased when the stress intensity decreased to the threshold resulting in dormant cracks[8-101. To verify this criterion for predicting non propagating cracks, crack growth data were obtained from a specimen with a 0.40 mm dieter circular notch in a G40.11 steel, and plotted in IQ. 5 on scales of AK vs crack length. In this figure, experimental values of AK are obtained by equating the unknown intensity factor at a given crack propagation rate in the notched specimen to the known stress intensity factor in a centrally cracked specimen having the same crack propagation rate[8]. Predicted values of AK based on eqn (5) are in agreement with experimental results, but on the other hand elastic solutions given by eqn (7) (also plotted in the figure) resulted in an underestimate of AK for short cracks. The figure illustrates that for stress levels of 483 MP,, 414 MP,, and 339MP,, at which specimens were tested, AK values initially decreased with crack length but reached a minimum above the threshold value of the material AKlh estimated from eqn (4a). At stress levels of 310 IMP, and 241 MP, where AK values based on eqn (5) decrease to a minimum below the threshold value, tests were run for ten million cycles without failure and no cracks were seen on the surface of specimen. When the stresses in each of these specimens were increased to 339 MP,, they failed and a subsequent examination of the fracture
hi. H. EL HADDAD et al. I
,mm
010 i
30c
\
IO I
,+‘”//
\\
I
001
IO
001
-
F+LASllC SOWTtON
----
ELASnc SOWlION
I
0.10
I
1. IfI Pii 5. Predictionof non propagatingcracks in a circulat notch for G40.1t steel.
surface of the specimens indicated that cracks had initiated at the mid-plane of the specimens (presumably due to the biaxial stress state[l91) and crack fronts corresponding to the hrst applied stress levels were noted on the fracture surface. Non propagating crack lengths aJong the mid-plane were measured and arc in good agreement with predicted lengths based on AK solutions given by eqns (5) as shown in Fii. 5. Elastic solutions for AK based on eqn (7) also predict the same non propagating crack lengths as those predicted based on eqn (5) and both solutions are expected to accurately predict the minimum stress level corresponding to failure. This coincidence between the two sotutions could have been anticipated for this metal because its local stress-strain behaviour at the endurance limit strain amplitude is essentially eiastic. Since eqn (3b) shows that the local threshold stress levei will decrease with increasing crack length from this already essentially elastic level at the surface, the local material response will become elastic as the threshold stress intensity is approached making eqn (5) equivalent to eqn 17). Experimental results for non propagating cracks in SAE lOi steel specimens containing an elliptical notch are given in Fig. 6. In two specimens tested at a stress level of 69 iUP,, cracks started and were measured along the surface of the specimen using a travelling microscope until they ceased to propagate. After ten million cycles, the load was increased to yield a stress levef of 76 Mp, and both specimens then failed as shown in Fig. 6. Predicted non prop~~g crack lengths based on the value of AK given by eqn (5) are in good agreement with the experiment. Also, the solution for AK at a stress level equal to 76 MP, predicts that the specimen will fail since the minimum of the AK vs crack length curve is above the threshold of the material. Experimental test data found in Ref. [3] for non propagating cracks in a V-edge notched mild steel plate are given in Fig. 7.3oth qns (5) and (7) give accurate predictions of non propagating crack lengths and the minimum stress level correspondiig to failure. The following section examines in more detail the factors governing the propagation of fatigue cracks in notches of varying geometry and size, and the conditions for non propagating cracks. TERRSEOLD !3TRE!BE!3FOR NOTCHES Notch size effect
Further insight into the dependence of AK vs crack length trends on the notch size and the amount of local plasticity may be obtained if AK is plotted vs crack length in a non~mensional form by rearrangingeqn (6) as follows: AK = k:,/(y). AS%$&j
579
Predictionof non propagating cracks
44’ 002
0 01
DOS
05
CRACK LENGTH,
0 IO
020
1, In
Fig, 6. Prediction of non propagatingcracks in an elliptical notch for SAE1015steel.
I,
mm IO !
010 I PRESENT
PLASTIC
ELASTIC
SOLUTIONS
EXPERIMENTAL EDGE
SOLUTIONS FOR
RESULTS
NOTCHED
PLATES
SD
AK
FOR
1
AK ( FROST f ’ I] MILD
STEEL
SI
0001
0.003
I
I
/
001
0.05
0.10
CRACK
LENGTH,
0.30
1, in
. Fii. 7. Predictionof non propagatingcracks in a V-notch for Mild steel.
Curves plotted on this basis in Fig. 8 compare the behaviour of an elliptical notch, a circular notch and a long crack for various notch diameters assuming elastic nominal strains, i.e. EAe = AS. Curves in the bottom half of the figure representing elastic notch root material response are independent of load level since k: in eqn (6) will be equal to the elastic stress concentration factor k’ given in eqn (7). Here intensity factor values for small diameter circular and elliptical notches both initially decrease with crack length and then increase to converge with long crack values. As the notch size is increased this trend changes to one of continuously increasing intensity with crack length for both geometries. This change occurs at a smaller size for the circular than for the elliptical notch with, for example, the minimum diameter for which subsequent intensities do not fall below the initial value being 2.54 mm for the former and 10.00mm for the latter. This is the minimum diameter for which a non propagating crack is possible at a constant cyclic load level. At smaller diameters, if the load level applied results in an initial intensity above the threshold intensity for crack propagation but a minimum intensity below the threshold, crack growth will start and then cease when the applied intensity decreases to the threshold. The upper part of Fig. 8 illustrates changes in curves of intensity vs length for a 5.08 mm EFlfVd.II.No.3-G
M. H. EL HADDAD et al. a,mm
1’~l
1.
CIRCULAR NOTCH WITH C EQUAL TO 254mm I ELASTIC NOTCH ROOT RESPONSE
CRACK
LENGTH,
a,
In
Fig 8. Behaviour of short cmclrs at various notch sizes.
diameter circular notch as the load is increased to levels causing notch plasticity. Changes in k:, the strain concentration factor in eqn (6) at the two inelastic levels, increase intensities at the short crack lengths within the plastic zone close to the notch and raise the initial parts of the curves. The curves, however, rejoin the elastic curve as the crack tip propagates beyond this inelastic region and k: becomes equal to k’. Previous observations of an effect of notch size on the behaviour of short cracks at notches can also be explained in terms of the applied nominal threshold stress A&,. This stress may be obtained by substituting the value of AK,,, into eqn (7) which has already been shown to predict non propagating cracks and the loads required for specimen failure. Rearranging eqn (7), we obtain;
Substituting for the value of AK,,, using eqn (4a), the last equation reduces to:
(10) Solutions for AS,,, given by eqn (10) are shown in Fig. 9 for cracks initiating from circular notches with dilTerent diameters. For the largest notch the threshold stress corresponding to failure is at initiation and once cracks initiate they will continue growing until final fracture. This initiation stress level is equal to AaJb However, as the notch size decreases the peak value of the threshold stress vs length curve shifts to the right as shown in Fig. 9. This peak value gives the minimum stress level required to fail the specimen. At stress levels below this value but above the value of Audkr which corresponds to initiation, cracks will start but will not propagate. Therefore, for a given notch geometry, decreasing the size of the notch results in an increase in the stress level required for failure and a decrease in notch sensitivity. Experimental results for non propagating cracks and the stress levels corresponding to failure given in Fig. 9 for a G40.11 steel lie close to the curves predicted by eqn (10).
Prediction of non propagating cracks
581
Fii. 9. Notch size effect on the threshold stresses for circular notches in CM.1
1steelplates.
The e#ect of notch geometry Curves plotted on the basis of eqn (8) compare the behaviour of a V-notch having a 5.08 mm depth and a long crack for various notch root radii as shown in Fig. 10. These curves representing only elastic notch root material response are independent of load level. Here intensity factor values for small notch radii are initially greater than those for long cracks but decrease rapidly to approximate the intensity values of the latter. As the notch mot radius increases this trend changes to one of continuously increasing intensity values which start below the long crack curve but approach and then converge with it. At notches with small root mdii, applied load levels resulting in an initial intensity above the threshold intensity for crack propagation but a minimum intensity below the threshold result in cracks that start and then stop when the applied intensity decreases to the threshold. At larger notch root radii cracks once initiated continue propagating until final failure. Also, shown in Fii. 10 is the intensity vs crack length curve for the critical notch root radius (1.27 mm) above which cracks once initiated will propagate to specimen failure--only below this radius are non propagating cracks possible. Figure 11 gives experimental and predicted results for three radii values of this V notch in terms of A&, given by eqn (10) and crack length. Results agree well with predictions with crack arrest close to the curve of eqn (10) in the sub critical radius notch and continuing crack growth at stress levels close to AuJk, in the notches with larger radii. Fatiguelimitfor notches Equation (10) was used to calculate the minimum stress necessary to propagate a crack to specimen failure-that is, the peak value of threshold stress illustrated in Figs. 9 and 11. The curve given by these calculations for circular notched plate specimens at G40.11 steel together
0
.n
Fii.
I .LO
.30 CRACK
40 .50 .*0 LENGTH, a, an
TO
10. Behaviourof short cracks at a V-notch.
a0
582
M. H. EL HADDAD et ol. L,mm 1.0 I
IO I
00 I
V-NOTCH -
C -5mm
- 60
PREDICTION
EXPORIMENTAL
RESULTS REFF(3 1
FAILURE
-60
MO.26 mm 0-c 1.27 mm p Wt.6 mm I
NON
3i
a
PRopuiATION
c
CRACKS -40
a
+I
a %
z +I
-20
1 .a
.ool
I 0.10
I 1.0
1, in Fig.
II. Notch root radiuseffect on the thresholdstresses for V-notch in Mild steel plates.
4 LO- loo IO -
I
ZOOI
.a
0.10
1.0
C. NOTCH DEPTH, in Fig. 12. Predict& of fatigue limit stresses for circularnotches in WI. 1I steelplates.
with experimental results is given in Fig. 12. Agreement between theory and experiment are excellent. The calculations suggested that the fatigue limit stresses increased with decreasing notch diameters closely approximating the fatigue limit of the material when the notch diameter decreased below I,,. Hence, increases in the fatigue limit of the notch and reductions in notch sensitivity are predicted for both reductions in the size of a notch and increases in the value of lofor a material. Similar calculations for the fatigue limit of the V-notched specimens of Ref.[21 were performed and the results are plotted together with experimental data in Fig. 13. Again, the agreement between the predicted curve and the data is excellent. The calculations predicted the trends observed in Fig. 10. Fatigue limits for notches with radii above the critical radius for the existence of a peak threshold stress (1.27 mm) were governed by the initiation stress while fatigue limits for notches with smaller radii were governed by a peak threshold stress occurring at a finite crack length. In this latter regime the notch fatigue limit remains virtually constant. As shown in Fig. 10 the length at which the minimum intensity which corresponds to the critical length occurs is for all notches in this regime, small compared to c. Hence, we may approximate
Predictionof non propagatingcracks
583
28 i?
PREGENT STUDY EXPERIMENrAL
RESULTS
0
EDGE NDTCHED PLATE
.
CYLINDRICAL
DIRECT
NDTCIIED
&AR
STRESS
Fig. 13. Predictionof fatigue limit stresses for a V-notch in Mild steel specimens.
the constant minimum fatigue limit stress by substituting the long crack value of k’ in eqn (10) to give
(11) hence, since 1 and lo are much smaller than c. AS, = Au,
(12)
AS, = 3.
(13)
This approximation which has previously been suggested by Smith and Miller[9] provides predictions for the fatigue limit of the V-notch virtually identical with those given by eqn (10). It should however, as they suggest, be restricted to sharp notches since it provides a much less satisfactory approximation for the blunter circular notch.
CONCLUSIONS (1) Non propagating fatigue cracks are predicted based on the criterion that once the value of the intensity factor reduces to the threshold value for the material the crack should become dormant. Predicted lengths for these cracks are in good agreement with the experimental data. (2) Intensity factor trends for cracks in notches are shown to vary from an initial decrease to a minimum value followed by an increase and eventual convergence with the trend for the equivalent long crack for sharp notches to the blunt notch curves that continuously increase during their approach to the long crack trend. The transition from sharp notch to blunt notch trends depends both on notch geometry and notch size. (3) In blunt notches the maximum value of the threshold stress for crack propagation is at initiation. However, for sharp notches the peak value of the threshold stress shifts to a finite length on threshold stress crack length curves. Stresses above the initiation level but below this peak stress level result in fatigue cracks which start but do not propagate to failure. (4) Predicted values of the fatigue limit stresses for a variety of sizes in two notch geometries are in good agreement with experimental results. Also, increases in the fatigue limit stress of a notch and reductions in notch sensitivity are correctly predicted for both reductions in the size of the notch and increases in the value of lo for a material of which the specimen is made.
s84
M. H. EL HADDAD et al. REFEIENCES
[I] N. E. Frost, T/tr Arto~uticrJ Quutrerfy 8. I-20 (Feb. 1957). [2] N. E. Frost, Pmt. Institt~tionof Mechanical Engineers.Vol. 173.No. 35, pp. 81l-835, London (1959). (31 N. E. Frost and D. S. I)ugdale. l Mechanic end Physics of Solids 5, 182-192,London (1937). [41 N. E. Frost, 1. Mechanical Engng ScL 2(2), lop-1 19 (I%@. [SI D. N. Lai and V. Weiss. Metalfwgicai Tmnsactfons LA 1623-1629(1975). 161K. Ohji. K. 08ura. H. T&ii. H. Ohkita and Y. Ohkuba. An analytical approach to the non propagating crack problem lwing the finite eiement method. Pmt. ISth Japan Congresson Matetials Research. pp. 91-94. The Society of Materials Science, Japan, Kyoto (1972). [A N. E. Frost. K. 1. Marsh and L. P. Pook. Metei Fatfgu. Clarendon Press, Oxford (1974). 181k41H. El Haddad. K. N. Smith and T. H. Topper, A strain based intensity factor solution for short fatigue cracks initiating from notches. II th Nat. Symp. Fmctum M&t., ASTM. American Society for Testing and Materials, June 12-14, Bfacksburg, V.A. R. A. Smith and K. J. Miller, ht. 1. Mcch. Engng and Sci., submitted June 1977. H. hf. Hammouda and K. J. Miller. Symp. Efastic Pfastic Pmctun. American Society for Testing and Materials, Atlanta Nov. 16-18 (1977). M. H. El Haddad. K. N. Smith and T. H. Topper, Fatigue Crack Pmpagation of Short CM&, (1978)ASME/CSME Joint Conference on Pressure Vessets and Pipin Nuclear Energy and Materials, Montreal (June Z-29 1978).Paper No. 78Mat-7 if21 A. Talug and K. Reiisnider, Analysis and Inoestigation of Smaff Flows. Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Cmck Gmwth, pp. 81-96. STP 637, ASTM, American Society for Testing and Materials. Phiiadelphia (1977). [I31 D. Broek, The Propagation of Fatigue Cmtks Emanating fmm Holes. NLR TR 72134 U, National Aerospace L&rauuy, The NetherIands (1972). T. H. Topper and K. N. Smith. /. Integmt Applicar~nr for Short Fatigue Cm&r at 1141M. H. El Haddad, N. E. DOW& Notches, ht. /. Fmctum. submitted for publication. (May 1978).
[ISI S. Pearson, Engng Fracture MecA 1,23S-247 (1975). 1161N. E. Dowlii, Cmck Growth Bving Low Cyclk Fatigue of Smooth Axial Specimens, ASTM STP (37, pp. 97-121. Arm&an Society for Testing and MWxiala, Phifadelphia (lmr). 1.54c550 (1961). [f7l H. Heuber, 1. Applied Mech., ASME, [I81 H. Tada. P. C. Paris and G. R. Irwin, 7’heStress Analysis of Cracks Handbook. Del Research Corp., Hellertown. PA [19] !9?*Leis and T. H. Topper, 1. Engng Mat. and Tech.. ASME Vol. 99, Series H., No. 3, pp. 215-221(July 1977). (Receioed5 June 1978: receivedfor publication 5 September1978)