a new method for predicting fatigue life in notched geometries

Fatigue & Fracture of Engineering Materials & Structures 1998; 21: 1–15

A NEW METHOD FOR PREDICTING FATIGUE LIFE IN

NOTCHED GEOMETRIES

C.-H. C* and L. F. CRensselaer Polytechnic Institute, Troy, NY 12180, USA

Received in final form August 1997

Abstract—The objective of this paper is to develop a notch crack closure model, called NCCM, basedon plasticity-induced effects and short fatigue crack growth in the vicinity of the notch, and to predictthe fatigue failure life of notched geometries. By using this model the regime for non-propagating cracks(n.p.c.) and the relationship between the fatigue strength reduction factor, Kf , and the elastic stressconcentration factor, Kt , under mean stress conditions, can be determined quantitatively. A crack closuremodel is assumed to apply in the notch regime based on an approach developed to explain the crackgrowth retardation behavior observed in smooth specimen geometries after an overload. Notch plasticityeffects are also applied in the NCCM model. Fatigue failure life is calculated from both short fatiguecrack growth in the notch region where elastic–plastic fracture mechanics (EPFM) is applied and fromlong fatigue crack growth remote from the notch where linear elastic fracture mechanics (LEFM) occurs.This prediction is obtained using a quantity called the effective plasticity-corrected pseudo-stress. TheNCCM can be used to account quantitatively for various observed notch phenomena, including boththe relationship between Kf and Kt and n.p.c. The effects of the tensile mean stress on the Kf versus Ktrelationship is investigated and leads to the little recognized but technologically important observationthat mean stress conditions exist where Kf can be greater than Kt . The role of notch radius and tensilemean stress on n.p.c. behavior is also explored. The model is verified using experimental data for notchgeometries of aluminum alloy 2024-T3, alloy steel SAE 4130 and mild steel specimens tested at zero andtensile mean stress.

Keywords—Fatigue life; Notch crack closure model; Non-propagating cracks; Plasticity-correctedpseudo-stress.

NOMENCLATURE

a=crack lengtha∞=material constant

A, m=crack growth parametersc=crack closure constantd=notch depthe=remote strain amplitudeE=Young’s modulusF=notch geometry correction factor

Kf= fatigue strength reduction factorKt=elastic stress concentration factorNf= fatigue failure lifera=notch elastic zonerp=notch plastic zoneR=the R-ratio, Smin/SmaxS=remote stress amplitude

Sfl= fatigue limit of smooth specimensSm=mean stress

Smax=remote maximum stressSop=crack opening stressSult=ultimate tensile strengthSyo=cyclic yield strengthU=ratio of crack opening strain, Deeff/De

*Now Senior Engineer, Central Research Institute, Tatung Co., Taipei, Taiwan.

© 1998 Blackwell Science Ltd 1

2 C.-H. C and L. F. C

x=distance ahead of notch rootDSeff=effective stress range

De, Deeff=strain range and effective strain rangeDeeffp=the effective plasticity-corrected pseudo-stress range

en=peak strain amplitude at notch rooteop=crack opening strain

ey(x)= local strain distribution ahead of a notch rooteyo=cyclic yield strain

r=notch radiussn=peak stress amplitude at notch root

snm= fatigue limit of notched geometriessy(x)= local stress distribution ahead of a notch root

INTRODUCTION

The growth of short cracks under high stress levels (such as growth in notches), invalidates theuse of linear elastic fracture mechanics (LEFM) procedures because the local plastic yielding zoneis large compared to the crack length. Here non-linear elastic or elastic–plastic fracture concepts,together with a variety of crack closure models, have been used to predict the observed shortcrack effects [1–6]. In the case of notches, the work of Ting and Lawrence [1], Tanaka andAkiniwa [2], and Abdel-Raouf et al. [3] can be cited. Ting and Lawrence combine the two crackclosure models developed by Tanaka et al. [4] and Sun and Sehitoglu [5] to describe the crackgrowth in notched geometries as well as the failure life and the Kf under fully reversed stress.Tanaka and Akiniwa [2] modified the crack closure model of Tanaka et al. [4] to describe thethreshold stress intensity behavior of a short crack growing in notch geometries. Abdel-Raoufet al. [3] used a crack opening stress equation developed by McEvily et al. [6], together with Ktto predict notch fatigue failure.

This paper is a further treatment of the notch fatigue problem. The approach uses a notch crackclosure model called NCCM in which the effect of notch plasticity on crack closure behavior isintroduced to describe crack growth and to predict fatigue failure life in notches.

DEVELOPMENT OF NCCM

Equation for the local strain distribution equation—uniaxial model

Some years ago Blatherwick and Olson conducted fatigue tests on an edge-notched low carbonsteel plate with Kt=1.8 [7]. The fatigue tests were controlled by the strain amplitude measuredat the notch tip. They showed that, beyond the notch root, the local strain distribution followedthe same form as the fully elastic stress distribution. This local strain distribution varies both withthe distance from the notch root and with the magnitude of the notch radius. An equation for thelocal strain distribution can thus be estimated.

An equation suggested by Lukas and Klesnil [8] is used to describe the distribution of the fullyelastic stress, sy . Here

sy(a)=KtS

S1+4.5x

r

(1)

where Kt is the elastic stress concentration factor, S is the remote (nominal) stress, x is the distancefrom the notch root and r is the notch root radius. It is then assumed that the elastic–plasticstrain distribution, ey(x), in the notch region follows the same form as the local elastic stress

Fatigue & Fracture of Engineering Materials & Structures, 21, 1–15 © 1998 Blackwell Science Ltd

Predicting fatigue life in notched geometries 3

distribution [9,10], Hence, the local strain distribution ahead of the notch root is

ey(x)=en

S1+4.5x

r

(2)

where x is the distance from the notch root. The peak strain amplitude, en , can be determinedusing Galinka’s equivalent strain energy method [10]. Here the strain energy density at the notchroot can be calculated on the basis of the elastic stress distribution. Detailed development of thisapproach is shown elsewhere [11].

Notch plastic zone, rp, and notch elastic zone, r

aWhen the peak strain amplitude exceeds the cyclic yield strain, a plastic zone develops around

the notch root. The notch plastic zone size in the x-direction can be derived from Eq. (2).Substituting the depth of x=rp into Eq. (2), where rp is the plastic zone size and ey (rp )=eyo , rpbecomes

rp=CA eneyoB2−1D A r

4.5B (3)

The value of rp is thus related to the notch root radius, the material strength and the magnitudeof the peak strain.

In fully reversed loading, the magnitude of the notch plastic strain in the reverse loading directionequals the notch plastic strain in the forward direction. In the presence of a tensile mean stress,the notch plastic strain in the reversed direction is less in magnitude than the notch plastic strainin the forward direction. Crack reopening occurs at some forward or tensile load, followed bycrack growth. It is assumed here that fatigue crack growth is related to the tensile notch plasticstrain only, and is unaffected by the magnitude of the notch plastic strain in the reverse direction.

When the value of the local strain amplitude ahead of the notch reaches the value of the remotestrain amplitude, e, that depth is defined as the notch elastic zone size, ra . Substituting the valueof ey (rn )=e and x=ra into Eq. (2) gives

ra=CAeneB2−1D A r

4.5B (4)

Crack retardation due to overloading in an unnotched cracked specimen and its extension to notchbehavior

Crack closure is attributed here to the elastic unloading of the plastically deformed materialahead on the crack tip as the crack propagates, causing residual displacements in the wake of thefatigue crack. In unnotched specimens, this plasticity-induced closure can result in a crack thatremains closed for a significant portion of a fatigue cycle. This behavior has been widely used toexplain a variety of fatigue crack growth phenomena [12]. In an unnotched, cracked specimen,plasticity-induced closure can be used to explain the retardation in crack growth following atransient overload. The behavior is illustrated in Fig. 1(a). The upper part shows the distributionof the maximum applied stress, Smax versus time when a transient overload has occurred. A crack

<iiFF56362.2.1,l(p5,p5)i(0,0)>Fig. 1. Illustrations of the effect of (a) a single tensile overload in a smooth specimen and (b) effective

stress range of a notch.

Fatigue & Fracture of Engineering Materials & Structures, 21, 1–15© 1998 Blackwell Science Ltd


is also shown where a is the crack depth. In the lower part of Fig. 1(a), the crack opening stress,Sop , is included and its behavior with crack position during and after the overload is described.Note that DSeff , the difference between Smax and Sop , decreases and then slowly increases as thecrack responds to the overload transient. This behavior in DSeff provides the explanation for crackretardation in unnotched specimens subject to overload.

Extending this behavior to geometries containing notches, the upper part of Fig. 1(b) shows thedistribution of the local stress, sy , ahead of the notch which results when the applied stress is atits maximum value. This stress equals the flow stress in the notch region until rp is reached, afterwhich it decreases with increasing distance from the notch root. A crack is now assumed to growfrom the notch root and, as shown in the lower part of Fig. 1(b), produces the crack openingstress, Sop , which increases in the yield region, eventually reaching a constant value beyond thefield of the notch. The magnitude of DSeff for the two regions is given in the figure. Note thatthe minimum value of DSeff occurs in the elastic field of the notch. This assumed behavior for thecrack opening stress of a notched specimen when the crack grows into an elastic field is similarto the behavior of an unnotched specimen with overload and becomes the basis for the developmentof NCCM. No attempt is made to obtain DSeff quantitatively. Rather, its shape is approximatedand is applied as described below.

T he notch crack closure model (NCCM)—assumptions

In the analysis that follows, the following assumptions are made:

(1) For unnotched geometries, there is no large plastic field around the fatigue crack and Sopcan be used to describe the crack closure behavior of the growing crack. For notchedgeometries, the presence of a notch plastic zone makes the use of strains attractive, so thatan analysis based on the crack opening strain, eop , rather than the crack opening stress, Sop ,seems appropriate.

(2) The local stress and strain distribution within the notch region is assumed to be the appliedstress and strain for a fatigue crack growing in the notch root.

(3) The effect of the crack tip plastic zone is assumed to be negligible when the fatigue crackgrows in a notch plastic-influenced field. Crack growth is determined by the notch plasticstrain only.

(4) The magnitude of the crack opening strain, eop , is related to the local strain distribution ofa notch, applying Assumption 3.

T he crack opening strain of NCCM

The form of the crack opening strain in notch geometries can now be determined using twoassumed functions to describe its behavior within and beyond the plastic zone of the notch. Bothfunctions can be represented in terms of the variable x, measured from the notch root. Since thefatigue crack of length a is also measured from the notch root, and crack opening is identifiedwith crack length, x and a are identical.

The first function describes the behavior within the notch plastic zone, 0<x<rp . Starting atthe notch root, the crack opening strain increases. This is because opening or closure depends onthe length of the crack (i.e. on the compliance of the system) as well as the extent and gradientof the plastic deformation in the region of the crack. The opening strain continues to increase upto the point where the fatigue crack equals the notch plastic zone size. After the elastic–plasticboundary is reached, i.e. at a=rp , the second function takes over. Here rp<x<ra and themagnitude of the crack opening strain decreases as crack growth continues until the notch elastic



zone size, a=ra , is reached. This behavior is a result of the change from plastic to elastic actionahead of the crack. Beyond a=ra , the magnitude of the crack opening strain remains at a constantvalue, defined as the steady-state crack opening strain for a long fatigue crack in a uniformstress field.

Reference is made to Fig. 2(a), where the two curves are shown, together with the notch plasticzone, defined by rp , the notch elastic zone, ra , the local strain ey , and the local strain at the notchelastic–plastic boundary, eyo . Note that the combined curve bears a resemblance to Sop in Fig. 1(a).The specific shape is approximate and no attempt is made to determine a quantitative solution here.

The crack opening strain in the notch plastic zone is defined as eop1 and its assumed form is

eop1−emineopp−emin

= (1−e−ka ) (5)

Here eopp is the opening strain at a=rp , emin is the minimum applied strain at a=0, and k=10/rp .This definition for k reduces the value of the exponential term to zero when a=rp .

Defining R as emin/en , emin is eliminated in Eq. (5) and

eop1= (eopp−Ren ) (1−e−ka )+Ren (6)

Here eopp , the maximum crack opening strain, occurs at a=rp .In the same manner the form of the second approximate function, eop2 , can be determined.

Referring again to Fig. 2(a),

eopp−eop2eopp−eops

=1−e−10Ca−rpra−rpDc (7)

Here c is a constant, called the crack closure constant, and the strain eops is the steady-state crackopening strain when e is the remote applied strain.

Specific applications of these equations require knowledge of the opening strain relative to themaximum strain as the crack grows. The quantity U is introduced [13,14] as a specific measureof crack opening, defined as U=Deop/De. U is a function of the material, the R value and thestrain. Thus

U=DeopDe

=emax−eopemax−emin

=1−

eopemax

1−R(8)

eop=emax−Uemax(1−R) (9)

In the case where x=rp , emax=eyo , the yield strain, and

eopp=eyo−Ueyo(1−R) (10)

where eopp is the maximum opening strain. When the crack reaches x=ra , eop=eops and Eq. (9)can be expressed as

eops=De A 1

1−R−UB (11)

<iiFF56362.2.2,l(p5,p5)i(0,0)>Fig. 2. The relation between (a) opening strain and crack length and (b) non-propagating behavior

illustrated in the NCCM approach.



where De is the remote strain range. In Eqs (10) and (11) U is assumed to be U=0.55+0.33R+0.12R2 for aluminum alloys after Schijve [13].

Non-propagating crack behavior in NCCM

A fatigue crack becomes non-propagating when the crack opening strain exceeds the maximumapplied strain at a specific crack depth. When this occurs the fatigue crack cannot open at thecrack tip and crack arrest occurs. This will happen only when the crack has grown beyond thenotch plastic zone. The situation is shown in Fig. 2(b) where the shape of the decreasing appliedstrain amplitude curve and that of the crack opening curve are such that contact occurs. It happenswhen r>rp . Thus, the NCCM approach explains in a comprehensible way the experimentalobservation [15] that the non-propagation behavior always occurs at rp<a<ra .

LIFE PREDICTION

T he eVective plasticity-corrected pseudo-stress intensity range

In linear elastic fatigue life prediction, the elastically determined stress intensity factor, togetherwith the Paris law relationship, is used to obtain the crack growth rate. Various modifications canbe made in this approach. For example, Elber’s effective stress intensity factor range [12] can beintroduced to account for crack closure effects. In the case of short fatigue crack growth, Newman[16] modified Elber’s stress intensity range by adding a portion of the Dugdale cyclic plastic zonelength. Here

DKeff=FDseffEp(a+wc) (12)

where w=0.25 and c is determined from the Dugdale plastic zone size as c= (1−Syo/Smax)2 (rc/4),where Syo is cyclic yield strength and rc is the Dugdale plastic zone size of a crack tip undermonotonic load.

In this paper, the concept of the pseudo-stress intensity factor is introduced to provide a betterdescription of the crack growth behavior. Here E×the strain replaces the stress. This change isconsistent with the existence of plastic strains in the notch zone. As a consequence, Newman’seffective plasticity-corrected stress intensity factor in Eq. (12) is converted to a strain intensityfactor, called the effective plasticity-corrected pseudo-stress intensity range, DKeffp , or

DKeffp=FEDeeffpEp(a+d+wc) (13)

The particular strain used is called the total effective strain range, Deeffp , where

Deeffp=emax−eop (14)

obtained at the specific crack depth a, as discussed above. Equation (13) includes the notch depthd, such that a threshold pseudo-stress intensity factor exists when a=0. The geometric factor Fappearing in Eqs (12) and (13) is a function of (a+d+wc) [17].

A Paris law relationship of the form da/dN=A(DKeffp )m can now be introduced to representboth short and long crack growth, where the necessary adjustments are made as appropriate (i.e.a<rp or a>rp ) in Eq. (14). Integrating to determine the fatigue life,

Nf= P aoai

da

A(DKeffp )m(15)

where ai is the initial crack size, considered to be of the order of the grain size, ao is the criticalcrack size, determined from the critical stress intensity factor, KIc . Also m is the Paris exponentand A is a constant.



The calculation procedure for determining the fatigue failure life was to (1) obtain the localstrain distribution at the crack root, (2) obtain the opening strain using the NCCM approach,(3) obtain the effective plasticity-corrected pseudo-stress intensity factor using Eq. (13) and(4) obtain fatigue life, using Eq. (15). A variety of results can be obtained following these procedures.The detailed calculations are described elsewhere [11].

S–N curves

Using the experimental data of Illg [18], Fig. 3 shows S–N curves of aluminum alloy 2024-T3and SAE 4130 under tensile fatigue loading for a double-edge-notched plate whose width was57.15 mm. Unnotched and notched specimens were tested, the latter had a notch depth of 9.525 mmand two distinct radii (8.06 and 1.45 mm) resulting in values of Kt of 2.0 and 4.0. Test resultsincluded two mean stresses, Sm=0 and Sm=138 MPa. The symbols such as ‘+’, ‘1’ and ‘#’, inall figures refer to the actual experimental data of Illg [18]. These experimental data cover fatiguelives ranging from 103 to 108 cycles to failure. The solid line curves are the predicted fatigue failurecurves obtained from NCCM. The procedure in obtaining these predicted curves is as follows:

(1) The calculated fatigue life curve at Kt=4 was adjusted to fit the experimental data, afterintroducing the known dimensions and material properties. The parameters used to obtainthe fit are A, m and c, and are given in Table 1 for the three alloys used in the investigation.

(2) The same constants are then used to calculate the fatigue failure lives when Kt=1 and 2.(3) The data so calculated for Kt=1 and 2 are plotted in Fig. 3 for comparison with the actual

experimental data.

The predicted failure lives shown in Fig. 3 are in good agreement with the observed experimentaldata when Sm=0 and Sm=139 MPa. Under conditions of identical remote stress amplitude andnotched geometry, it is noted that the fatigue failure lives of specimens tested with tensile meanstress are less than the fatigue failure lives of specimens tested under fully reversed loadingconditions (Sm=0).

a versus N and da/dN versus a curve

Figures 4 and 5 show the calculated fatigue crack growth behavior under four different appliedloading conditions for aluminum alloy 2024-T3. Similar results were obtained for SAE 4130[11,18]. The curve for the lowest applied stress level shows the arrest or cessation of the fatigue

<iiFF56362.2.3,l(p5,p5)i(0,0)>Fig. 3. Endurance data and predicted curves: (a) Al 2024-T2 at Sm=0 MPa; (b) Al 2024-T2 atSm=138 MPa; (c) SAE 4130 at Sm=0 MPa; (d) SAE 4130 at Sm=138 MPa. Note that in (b) only the

predicted curve for Kt=1 is presented.

Table 1. The values of constants forAl 2024-T3, SAE 4130 and mild steel

Al 2024-T3 SAE 4130 Mild steel

A 0.8×10−10 0.6×10−12 3.6×10−10m 3.1 3.25 3.5c 0.86 0.92 1.14

<iiFF56362.2.4,l(p5,p5)i(0,0)>Fig. 4. Crack growth characteristics of Al 2024-T3 at Sm=0 MPa and Kt=4: (a) crack growth rateversus crack length, Sa at A=42.7 MPa, B=51.7 MPa, C=65.1 MPa, D=75.8 MPa; (b) crack length

versus number of cycles, Sa at A=75.8 MPa, B=65.1 MPa, C=51.7 MPa, D=42.7 MPa.



<iiFF56362.2.5,l(p5,p5)i(0,0)>Fig. 5. Crack growth characteristics of Al 2024-T3 at Sm=138 MPa and Kt=4: (a) crack growth rateversus crack length, Sa at A=10.3 MPa, B=20.7 MPa, C=41.4 MPa, D=69 MPa; (b) crack length

versus number of cycles, Sa at A=69 MPa, B=41.4 MPa, C=20.7 MPa, D=10.3 MPa.

crack growth (n.p.c.). Continuous growth to failure occurs at higher applied stresses and showssome retardation in growth rate.

It can be noted that the fatigue crack length is greater when a tensile mean stress, Sm , is present(Fig. 5), than when the stress is fully reversed, Sm=0 (Fig. 4), at the same value of N. This isbecause the crack growth rate, da/dN, can be described in terms of Smax , that is, Sa+Sm . Notealso in these two figures that the value of a for crack growth cessation is greater when a meanstress is present than when the stress is fully reversed.

Sop

versus a

The crack opening strain distribution was developed in Eqs (5) and (7), following the NCCMmodel. Accordingly, the crack opening stress can be determined, using the cyclic stress–strainpower law relationship, for aluminum alloy 2024-T3 and SAE 4130 at four stress amplitudes whereKt=4. The results for 2024-T3 aluminum are shown in Figs 6 and 7. Results for SAE 4130 aregiven elsewhere [11]. The dashed line in each sketch provides the distribution of the crack openingstress versus a for the conditions indicated. The solid lines show the maximum and minimumapplied stresses. To cause a non-propagating crack, the crack opening stress must exceed themaximum applied stress. Under such conditions, the crack cannot reopen and growth ceases. Thissituation is seen to occur in each case after the fatigue crack has grown into the notch elastic zone(as evidenced by the decreasing values of the maximum and minimum stresses).

If the crack opening stress is less than the minimum stress, crack closure will not influencefatigue crack growth. The crack opening stress is then assumed to be equal to the minimum stress.This situation is shown in Fig. 6 and particularly in Fig. 7 for low values of crack growth.

DISCUSSION

The NCCM model allows insight both into the relationship between Kf and Kt , particularlywhen mean stresses are applied, and into the regime of non-propagating cracks. We first considerthe Kf versus Kt relationship.

Using the methodology described above, predicted Kf values were derived for aluminum alloy2024-T3 and SAE 4130 for various values of Kt and the mean stress. Existing analytical methodsfor determining such relationships are introduced for the purpose of comparison. For example,when the mean stress is zero, Neuber’s plastic stress concentration factor, Kp , can be used [19,20].

<iiFF56362.2.6,l(p5,p5)i(0,0)>Fig. 6. Crack opening stress and applied stress distribution versus crack length in Al 2024-T3 at

Sm=0 MPa, Kt=4: (a) Sa=42.7 MPa; (b) Sa=52.7 MPa; (c) Sa=65.1 MPa; (d) Sa=75.8 MPa.

<iiFF56362.2.7,l(p5,p5)i(0,0)>Fig. 7. Crack opening stress and applied stress distribution versus crack length in Al 2024-T3 at

Sm=138 MPa, Kt=4: (a) Sa=10.3 MPa; (b) Sa=20.7 MPa; (c) Sa=41.4 MPa; (d) Sa=69 MPa.



Here

Kp=1+Kt−1

1+Sa∞r

=Kf (16)

where a∞ is a material constant. When mean stresses exist, an approximate procedure based on themodified Goodman relationship, can be considered. Here

snm=ssmKt

=SflKtC1− Sm

sultD (17)

where ssm is the fatigue strength of a smooth specimen with a tensile mean stress, Sm , while snmis the fatigue strength of a notched specimen under the same mean stress. Also sult is the tensilestrength of the material. From this, Kf can be calculated, where Kf is defined as the ratio of thesmooth specimen fatigue strength, Sfl , to the notched specimen fatigue strength, snm , each subjectedto the mean stress, Sm . Thus from Eq. (17)

Kf=Sflsnm

=Kt

C1− SmsultD (18)

Comparisons of Kf , determined both from NCCM prediction and those derived from Eqs (16)and (18), are included in Table 2 and Fig. 8. With reference to Fig. 8, the solid lines represent thepredicted values by NCCM. The points represented by the ‘*’ symbol give the calculated valuesof Kf from Neuber’s equation, while those points represented by ‘#’ reveal the calculated values

Table 2. Predicted values of Kf from NCCM, Neuber and themodified Goodman relationship for Al 2024-T3 and SAE 4130

Kt1 2 4 6 8 10

Sm=0 MPa

Al 2024-T3 (Kf or Kp)NCCM 1 1.65 3.00 3.74 3.99 4.17Neuber (Kp) 1 1.81 2.93 3.65 4.17 4.56

SAE 4130 (Kf or Kp)NCCM 1 1.61 3.77 4.57 4.96 5.15Neuber (Kp) 1 1.86 3.18 4.15 4.88 5.45

Sm>0 MPa

Al 2024-T3 (Kf)NCCM 1 2.14 6.92 9.65 11.75 14.40Modified Goodman

relationship 1 2.8 5.6 8.4 11.2 14.0SAE 4130 (Kf)

NCCM 1 2.86 8.16 11.57 15.25 19.9Modified Goodman

relationship 1 3.2 6.5 9.7 13.0 16.2

<iiFF56362.2.8,l(p5,p5)i(0,0)>Fig. 8. Curves of Kf versus Kt relationships for (a) Al 2014-T3 and (b) SAE 4130. 1 : Data from Neuber

Eq. # : Data from modified Goodman relationship. Solid line: NCCM.



of Kf using the modified Goodman approach. Additionally, a solid line representing the situationwhere Kf=Kt is constructed.

According to Fig. 8, when the stress is fully reversed, predicted values of Kf are less than Kt forboth materials. This is consistent with experience and with the results obtained from Neuber’sequation. Note that the predicted and empirically derived results are in close agreement. For thecase of a high tensile mean stress, values of Kf predicted by NCCM for each material are foundto exceed those determined by Eq. (16). Of particular interest in Fig. 8 (and Table 2) is theobservation that Kf>Kt in all those cases where tensile mean stresses exist in notched geometries.Fuchs and Stephens [20] reported earlier that tensile mean stresses cause the notch factor Kf toexceed Kt , based on Grover’s results [21], even adding the comment that ‘tensile mean stress canbe fatal in fatigue loading’. This statement presumably referred to the common practice of equatingKf to Kt in fatigue design applications. From this statement and, more quantitatively, from Fig. 8,predicting the notch fatigue strength by using Kt can be an unsafe procedure.

Non-propagating crack behavior is an important consideration when fatigue cracks propagatefrom notched geometries, especially when the notches are sharp. The crack opening strain criterionfor non-propagating cracks was described above for the NCCM, see Figs 5 and 6. To illustratethe applicability of the NCCM to such situations, Fig. 9 was constructed, comparing experimentaldata from Frost [15,22] (as indicated by the various symbols) for mild steel with the predictionsobtained from the NCCM model (the solid lines). The dashed line comes from the work of Smithand Miller [23] and applies only when the mean stress was zero. The crack initiation curve, shownalso in Fig. 9, is constructed using Sfl/Kt .

The curves describing non-propagating crack behavior as predicted by NCCM agree well withthe experimental results both under zero mean stress and when tensile mean stresses are present.For a given notch, the stress amplitude regime for non-propagating cracks is lowered for increasingtensile mean stress. The stress amplitude of a non-propagating crack with zero mean stress predictedby Smith and Miller is conservative relative to the NCCM prediction.

CONCLUSIONS

(1) The crack opening strain and its variation with notch plasticity and crack length for afatigue crack emanating from a notch can be modeled by NCCM. The NCCM approachcan be applied to describe the crack closure behavior during fatigue crack growth in boththe non-linear and linear stress–strain behavior field of the notch.

(2) Fatigue life predictions were performed using NCCM together with the effective plasticity-corrected pseudo-stress intensity range. They agreed with experimental data for the fullyreversed stress and the tensile mean stress test conditions.

(3) The predicted relationship between Kf and Kt in this research was in good agreement withthe Kf versus Kt relationship obtained from Neuber’s equation for fully reversed stress. Fornotched bars with high tensile mean stresses, the predicted Kf values are found to be greaterthan the Kt values.

(4) Non-propagating cracks are predicted to occur by the NCCM model when the crack openingstrain exceeds the applied maximum strain. The nominal stress level for this condition canthen be determined for notched bars, both for a fully reversed stress and for a tensile mean

<iiFF56362.2.9,l(p5,p5)i(0,0)>Fig. 9. Stress amplitude versus Kt graph of mild steel.



stress. The predicted nominal stress level for non-propagating cracks is in good agreementwith reported experimental data.

REFERENCES

1. J. C. Ting and F. V. Lawrence Jr (1993) A crack closure model for predicting the threshold stresses ofnotches. Fatigue Fract. Engng Mater. Struct. 16, 93–114.

2. K. Tanaka and Y. Akiniwa (1988) The Propagation of Short Fatigue Cracks at Notches. AST M ST P924, 281–298.

3. H. Abdel-Raouf, T. H. Topper and A. Plumtree (1992) A model for the fatigue limit and short crackbehaviour related to surface strain redistribution. Fatigue Fract. Engng Mater. Struct. 15, 895–909.

4. K. Tanaka, Y. Nakai and M. Yamashita (1981) Fatigue growth threshold of small cracks. Int. J. Fracture17, 519–532.

5. W. Sun and H. Sehitoglu (1991) Modeling of plane strain fatigue crack closure. J. Engng Mater. T echnol.113, 31–40.

6. A. J. McEvily, K. Minakawa and H. Nakamura (1984) Fracture mechanics, microstructure and thegrowth of short and long fatigue cracks. Fract.: Interact. Microstructure, Mechanisms, Mechanics 215–233.

7. A. A. Blatherwick and B. K. Olson (1968) Stress redistribution in notched specimens during fatiguecycling. Exp. Mech. 8, 356–361.

8 P. Lukas and M. Klesnil (1978) Fatigue limit and notched bodies. Mater. Sci. Engng 34, 61–66.9. H. Y. Ahmad, E. R. de Los Rios and J. R. Yates (1994) The influence of notch plasticity on short fatigue

crack behaviour. Fatigue Fract. Engng Struct. 17, 235–242.10. G. Glinka (1985) Energy density approach to calculation of inelastic strain–stress near notches and

cracks. Engng Fract. Mech. 22, 485–508.11. C. H. Chien (1996) Modeling for fatigue life prediction of notched geometries, Ph.D thesis, Rensselaer

Polytechnic Institute, Troy, NY.12. W. Elber (1971) The significance of fatigue crack closure. AST M STP 486, 230–242.13. J. Schijve (1988) Fatigue crack closure: observations and technical significance. ASTM ST P 924, 281–298.14. S. Suresh and R. O. Ritchie (1984) Propagation of short fatigue cracks. Int. Metals Res. 29, 445–476.15. N. E. Frost and D. S. Dugdale (1975) Fatigue tests on notched mild steel plates with measurement of

fatigue cracks. J. Mech. Phys. Solids 5, 182–192.16. J. C. Newman, Jr (1994) A review of modeling small-crack behavior and fatigue-life prediction for

aluminum alloy. Fatigue Fract. Engng Struct. 17, 429–439.17. H. Tada, P. C. Paris and G. R. Irwin (1985) T he Stress Analysis of Cracks Handbook. Paris Productions

Inc., St.Louis, MO, U.S.A.18. W. Illg (1956) Fatigue tests on notched and unnotched sheet specimens of 2024-T3 and 7075-T6 aluminum

alloys and of SAE 4130 steel with special consideration of the life range from 2 to 10,000 cycles. NACATN 3866.

19. H. Neuber (1945) Kerbspannungslehre. Springer, Berlin; (1961) Trans. T heory of Notch Stress. U.S. Officeof Technical Services.

20. H. O. Fuchs and R. I. Stephens (1980) Metal Fatigue in Engineering. A Wiley-Interscience Publication,New York.

21. H. J. Grover (1966) Fatigue of aircraft structure. NAVAIR 01-1A-13, Department of the Navy.22. N. E. Frost (1959) A relation between the critical alternating propagation stress and crack length for

mild steel. Proc. Inst. Mech. Engrs 173, 811–836.23. R. A. Smith and K. J. Miller (1978) Prediction of fatigue regimes in notched components. Int. J. Mech.

Sci. 20, 201–206.


a new method for predicting fatigue life in notched geometries

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