The effect of notch geometry on critical distance high cycle fatigue predictions

International Journal of Fatigue 27 (2005) 1623–1627 www.elsevier.com/locate/ijfatigue

The effect of notch geometry on critical distance high cycle fatigue predictions* David B. Lanninga,*, Theodore Nicholasb, Anthony Palazottob a

College of Engineering, Embry-Riddle Aeronautical University, 3700 Willow Creek Road, Prescott, AZ 86301-3720, USA b Air Force Institute of Technology, 2950 Hobson Way, Building 640, Wright-Patterson AFB, OH 45433-7765, USA Available online 21 July 2005

Abstract A critical distance method for predicting the fatigue limit stresses of notched specimens was implemented for notched specimens with a wide range of notch dimensions. Circumferentially notched cylindrical specimens (ktZ1.97–4.07) taken from Ti–6Al–4V forged plate were cycled to failure (RZ0.1 and 0.5) using a step loading method for estimating the 106 cycle fatigue limit stresses. These experimental data were used in combination with finite element solutions for all specimen geometries to determine a ‘critical distance’, a quantity or parameter determined from the stress distribution surrounding the notch in combination with fatigue limit stress data from unnotched specimens. A unique parameter was not found for all of the specimen geometries. However, predictions for the fatigue limit stresses of the larger notch geometries may be made with some amount of accuracy using a single value of the critical distance parameter, while reasonable predictions for the specimens with the smallest notch dimensions may be made upon the recognition of an apparent size effect. q 2005 Elsevier Ltd. All rights reserved. Keywords: High cycle fatigue; Notches; Critical distance; Ti–6Al–4V

1. Introduction Critical distance methods for predicting the fatigue behavior of laboratory specimens and components with stress raisers are an outgrowth of well-known design methods that have been in use for many decades. It has been long understood that the use of the ‘hot spot’ stress at the root of the notch or stress raiser is often overly conservative when used for the prediction of fatigue strength. Design rules, such as those from Neuber [1] and Peterson [2], make use of the stress gradient at the notch root in recognition of both the conservatism of ‘hot spot’ stress predictions, as well as the observed increase of the fatigue strength with increasing severity of the stress gradient for notches of similar kt, known as the notch size effect. The critical distance approach, investigated by several workers during the last dozen or so years [3–6], is a search * The views expressed in this article are those of the authors and do not reflect the official policy or position of the United States Air Force, Department of Defense, or the US Government. * Corresponding author. Tel.: C1 928 777 3930; fax: C1 928 777 6952. E-mail address: [email protected] (D.B. Lanning).

0142-1123/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijfatigue.2005.06.017

for a distance over which a stress, strain, or energy-based quantity from the notched geometry may be related to a similar quantity for an unnotched specimen in a predictive scheme. The aforementioned traditional design rules [1,2] for stress concentrations are essentially critical distance methods where the stress distribution in the vicinity of the notch is linearized to that at the notch tip, although a ‘critical distance’ is not explicitly calculated. Rather, a material parameter in the form of a length is calculated, often assumed to be related to a characteristic microstructural feature. More recent studies make use of the non-linear stress distribution away from the notch root. However, most studies appear to have been of limited scope in terms of the range of notch geometries and loading conditions investigated [3–5]. Recently, the authors undertook a broad study [6] on the use of critical distance methods for notched Ti– 6Al–4V, including a range of notch geometries and loading conditions (RZ0.1, 0.5, 0.65 and 0.8). The methods exhibited some promise as predictive tools. A possible size effect was uncovered with the smallest geometries, and techniques for handling the notch tip plasticity occurring at high stress ratios (RZ0.65 and 0.8) were introduced. The current study builds upon the effort to determine if a critical distance technique can lead to a robust predictive

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D.B. Lanning et al. / International Journal of Fatigue 27 (2005) 1623–1627

Nomenclature d kf kt Nf R

r smax smin snet Ds

critical distance fatigue notch factor elastic stress concentration factor number of cycles to failure stress ratio, smin/smax

methodology by introducing additional geometries having similar notch root radii but dissimilar kt, focusing on the lower stress ratios (RZ0.1 and 0.5) where the strains at the notch root remain elastic, and by further investigating the possible critical distance size effect.

2. Experimental method and numerical simulation Smooth and notched cylindrical fatigue specimens were taken from Ti–6Al–4V forged plate, the heat treatment details of which are found elsewhere [6,7]. Table 1 provides tensile properties of the Ti–6Al–4V. Circumferential V-notches were machined using a low-stress grinding technique and subsequently stress-relieved. All specimens were 130 mm in length. The smooth specimens had a gage section diameter of 5.08 mm while the nominal gage section diameter of the notched specimens was 5.72 mm. Fig. 1 shows a diagram of the notched gage section and Table 2 provides corresponding dimensions for the seven geometries. The elastic stress concentration factors for the seven notch geometries were estimated from both finite element analysis and solutions in the literature [8]. The fatigue limit stresses at NfZ106 cycles, with RZ0.1 and 0.5 and testing frequencies from 50 to 70 Hz, were estimated using a step-loading technique. This step-loading technique has previously been shown to produce accurate results for the Ti–6Al–4V product form used in the current investigation for both notched and smooth specimens [9–11]. The details of this procedure may be found elsewhere [6,9–11]. Finite element solutions for the seven specimen geometries at all loading conditions were determined using four-node quadrilateral elements in an axisymmetric mesh of the gage section. The material behavior was assumed to be elastic-perfectly plastic, which is a good assumption with the Ti–6Al–4V under investigation for modest plastic straining, together with an effective stress

notch root radius maximum stress minimum stress net section stress at reduced cross-section stress range

yielding criterion. Mechanical properties at an elevated strain rate may be more representative of conditions at the notch root than quasi-static mechanical properties. Table 1 gives both quasi-static mechanical properties and those at an elevated strain rate on the order of magnitude as the average strain rate calculated at the notch tip during cycling at 50 Hz. The elevated strain rate mechanical properties were used in the finite element simulations. All finite element solutions for specimens cycled under conditions of RZ0.1 and 0.5 predicted solely elastic straining. Notch geometries with similar notch root radii (rZ0.127 and 0.330 mm) were selected in light of an appreciable amount of scatter in the fatigue limit stresses for the Ti–6Al–4V found in previous work by the authors [11,12]. Notches with similar notch root radii have features that are alike, regardless of kt. Fig. 2 shows the normalized stress distributions for the four specimens with notch root radii of rZ0.330 mm. The stress distributions are quite similar near the notch root, with the exception of the hoop stress, which decreases with decreasing kt. Therefore, if the stress state is controlling the high cycle fatigue lives and there is significant difference in the results as a function of kt, the difference should be due to the hoop stress. Normalized stress distributions for the notch geometries with rZ0.127 mm, not provided here, show similar trends.

Table 1 Mechanical properties of the forged Ti–6Al–4V plate Strain rate (sK1)

0.0005

0.05

Yield stress (MPa) Ultimate stress (MPa) Elastic modulus (GPa)

930 978 119

1003 1014 127

Fig. 1. Gage section of a cylindrical circumferentially-notched fatigue specimen.

D.B. Lanning et al. / International Journal of Fatigue 27 (2005) 1623–1627 Table 2 Elastic stress concentration factors and notch dimensions for seven specimen geometries r (mm)

h (mm)

D (mm)

d (mm)

2.85 3.51 4.07 1.97 2.30 2.58 2.72

0.127 0.127 0.127 0.330 0.330 0.330 0.330

0.127 0.279 0.635 0.100 0.203 0.381 0.730

5.72 5.72 5.72 5.72 5.72 5.72 5.72

5.47 5.16 4.45 5.52 5.31 4.96 4.26

200 106 cycles, Ti-6Al-4V ρ = 0.330 mm Alternating Stress (MPa)

Notch, kt

1625

150

R = 0.1 R = 0.5

100 k t = 1.97 k t = 2.30

50

k t = 2.58 k t = 2.72

3. Haigh diagrams 0

Figs. 3 and 4 provide all of the experimental data on two constant-life Haigh (Goodman) diagrams, one plot for each notch root radius. While significant scatter is apparent, the reduction in the net section stress with increasing kt is apparent in both figures. 1.2 Elastic stress distributions normalized with respect to the axial stress at the notch root

1

k = 1.97, 2.30, 2.58, 2.72 Normalized Stress

t

0.8

σaxial

0.6 0.4

σ hoop

Increasing diameter, decreasing k t

0.2

σradial

0 0

0.1 0.2 0.3 0.4 Distance from Notch Root (mm)

0

200 300 Mean Stress (MPa)

500

Fig. 4. Constant-life Haigh diagram for notched Ti–6Al–4V in terms of net section stresses, rZ0.330 mm.

If the stress state indeed controls the fatigue behavior, plotting the experimental data in terms of estimated notch tip stresses may shed light on the similarities between the specimen geometries. Figs. 5 and 6 show the same data, now in terms of notch tip stresses (axial direction) which are merely ktsnet. The data are somewhat collapsed, and while significant scatter in the data remains, an ordering of the data in terms of higher kt leading to slightly higher fatigue limit stresses can be observed. If an equivalent stress or some similar parameter describing the three-dimensional stress state is expected to influence high cycle fatigue behavior, then this ordering is expected upon consideration of the stress distributions from Fig. 2. The use of equivalent stresses in such a constant-life Haigh diagram, not shown here, further collapses the data only slightly, however. 4. Critical distance analysis The critical distance concept hypothesizes a single distance or depth inward from the ‘hot spot’ stress at a stress 500

150 6

106 cycles, Ti-6Al-4V ρ = 0.127 mm

10 cycles, Ti-6Al-4V

ρ = 0.127 mm Alternating Stress (MPa)

400 R = 0.1

100 R = 0.5 50

kt = 2.85 kt = 3.51

R = 0.1

Notch tip stresses

R = 0.5

300

200

kt = 2.85 2.85 100

kt = 3.51 3.51

kt = 4.07 0

400

0.5

Fig. 2. Normalized stresses inward from the notch root for specimens with rZ0.330 mm.

Alternating Stress (MPa)

100

0

50

100

150 200 250 Mean Stress (MPa)

300

kt = 4.07 4.07 350

Fig. 3. Constant-life Haigh diagram for notched Ti–6Al–4V in terms of net section stresses, rZ0.127 mm.

0

0

200

400 600 Mean Stress (MPa)

800

1000

Fig. 5. Constant-life Haigh diagram for notched Ti–6Al–4V in terms of notch tip stresses, rZ0.127 mm.

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D.B. Lanning et al. / International Journal of Fatigue 27 (2005) 1623–1627 0.08

500

Alternating Stress (MPa)

400

Notch tip stresses

300

k t = 1.97 Stress range criterion 6 k = 2.30 Ti-6Al-4V, 10 cycles

0.06

k t= 2.58

0.05

k t= 2.72

t

R = 0.5

200

kt = 1.97 kt = 2.30

100

R = 0.1

k t= 2.85 0.04

k t= 3.51

0.03

k t= 4.07

0.02

kt = 2.58 0.01

kt = 2.72

0

0.07 R = 0.1 Critical Distance (mm)

106 cycles, Ti-6Al-4V ρ = 0.330 mm

0

200

400 600 Mean Stress (MPa)

800

0

1000

0

0.05

0.1 0.15 0.2 0.25 Notch Root Radius (mm)

0.3

0.35

Fig. 6. Constant-life Haigh diagram for notched Ti–6Al–4V in terms of notch tip stresses, rZ0.330 mm.

Fig. 7. Critical distances based on stress range versus notch root radius, RZ0.1.

concentration, over which a stress or strain-related quantity controls fatigue crack initiation and therefore the majority of high cycle fatigue life. This is analogous to a volume effect, where a volume of material near the notch surface controls crack initiation and early propagation. Beyond this distance, the stress state has little influence on initiation and therefore does not affect the majority of high cycle fatigue life. Previous work by the authors identified one of several critical distance criteria that best fit the data in that study [6]. The most successful critical distance criterion was one of the simplest and was based on stress range:

rZ0.330 mm appears significant, but no ordering of kt’s appears when examining the data in both figures together. The last objective was to investigate the potential of a critical distance method for making predictions on fatigue behavior. Such a technique might be to calculate the best critical distance or set of critical distances possible from a given amount of experimental data, and then apply it to subsequent testing. There is not much difference between the average d’s in both Figs. 7 and 8 for rZ0.127 mm. The average of all of these values is dZ0.028 mm. This average value was used for predictions of the fatigue limit stresses for ktZ2.85, 3.51, and 4.07, the results of which are found in Table 3. While one must exercise caution when using experimental data to make predictions about that particular data set, these results do give some measure of the range of expected predictions from the analytical technique. The results in terms of percent difference with respect to the experimental fatigue limit stress appear reasonable. Table 4 provides similar predictions for the specimens with rZ0.330 mm. The critical distance used was

where

Dsnotch ðxÞ Z Dssmooth

(1)

where d is the critical distance to be evaluated, x is the distance inward from the notch root in the direction normal to the loading, and Ds is the stress range. The theoretical condition of kfZkt leads to a critical distance of zero, where kf is the fatigue notch factor (unnotched fatigue limit stress/ notched fatigue limit stress) and DsnotchZDssmooth at the notch root surface (i.e. xZ0). Figs. 7 and 8 provide the calculated critical distances in this study as a function of notch root radius for loading conditions of RZ0.1 and 0.5, respectively. The figures do not show the data of individual tests, but instead show results from the average fatigue limit stresses of several identical specimens tested at the same stress ratio. The hope here is that all of the critical distances would be approximately the same value, but there is some amount of scatter clearly visible. Comparing Fig. 7 to Fig. 8, the scatter is about the same at both values of R. However, a distinct difference is obvious between the data from the two notch root radii. Specimen data for the smaller r exhibit far less scatter at both R’s than data for the larger r. At RZ0.1, a size effect is apparent, whereas the geometries with rZ0.127 mm lead to a smaller critical distance d than at rZ0.330 mm. This scatter for specimens with

0.06 0.05 Critical Distance (mm)

d Zx

kt = 1.97

Stress range criterion

kt = 2.30

Ti-6Al-4V, 10 cycles

kt = 2.58 0.04

6

R = 0.5

kt = 2.72 kt = 2.85

0.03 0.02

kt = 3.51 kt = 4.07

0.01 0

0

0.05

0.1 0.15 0.2 0.25 Notch Root Radius (mm)

0.3

0.35

Fig. 8. Critical distances based on stress range versus notch root radius, RZ0.5.

D.B. Lanning et al. / International Journal of Fatigue 27 (2005) 1623–1627 Table 3 Fatigue limit stress predictions based on critical distances of dZ0.028 mm, for rZ0.127 mm, net section maximum stresses Notch kt

R

Experimental stress (MPa)

Predicted stress (MPa)

Difference (%)

2.85 2.85 3.51 3.51 4.07 4.07

0.1 0.5 0.1 0.5 0.1 0.5

294 387 225 320 213 273

307 380 248 307 214 264

4.7 K1.7 10 K4.0 0.1 K3.0

Table 4 Fatigue limit stress predictions based on critical distances of dZ0.048 mm, for rZ0.330 mm, net section maximum stresses Notch, kt

R

Experimental stress (MPa)

Predicted stress (MPa)

Difference (%)

1.97 1.97 2.30 2.30 2.58 2.58 2.72 2.72

0.1 0.5 0.1 0.5 0.1 0.5 0.1 0.5

405 501 360 421 341 352 274 310

400 495 342 423 303 375 284 352

K1.2 K1.3 K5.1 0.5 K11 6.5 3.7 13

dZ0.048 mm, the average of specimen results at the larger notch root radius. The range of difference in the predictions is about the same, which at first may appear erroneous since the scatter in the critical distance results was clearly larger. However, the fact that the stress gradient is lessened for the notches with the larger notch root radii leads to reduced sensitivity in the predictions using critical distances in the face of such scatter. Earlier work by the authors included a notched specimen geometry with an intermediate notch root radius, rZ 0.203 mm [6]. At the time of this previous study, the critical distances calculated using the same stress range criterion resulted in average values of dZ0.040 mm for both limited rZ0.330 notch root radii specimen results as well as for this intermediate geometry. While more data are warranted, in light of the additional results of the current study where the average at rZ0.330 mm is dZ0.048 mm, a continuous decrease in the critical distance parameter is a possibility within the range of notch sizes tested.

5. Conclusions Constant-life 106 cycle Haigh diagrams for Ti–6Al–4V notched specimens loaded at stress ratios of RZ0.1 and 0.5 have been presented. A critical distance criterion in terms of stress range has been investigated for a wide range of

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geometries of circumferentially-notched fatigue specimens. Stress-life fatigue data typically exhibit scatter, which is particularly evident as more and more data are accumulated, as in the present study. However, the critical distance method used here for notched Ti–6Al–4V has shown promise as a predictive tool. Testing at RZ0.1 and 0.5 led to similar values of critical distances for a given notch root radius. However, the appearance of a possible size effect with the smallest notches, where the average critical distance was almost half that of the specimens with the larger notch root radius, suggests that the method be implemented with caution.

Acknowledgements This work was supported by the US Air Force National Turbine Engine High Cycle Fatigue (HCF) Program.

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