HCF notch predictions based on weakest-link failure models

International Journal of Fatigue 25 (2003) 835–841 www.elsevier.com/locate/ijfatigue

HCF notch predictions based on weakest-link failure models夽 David B. Lanning a,∗, Theodore Nicholas b, Anthony Palazotto b a b

Embry-Riddle Aeronautical University, College of Engineering, 3700 Willow Creek Road, Prescott, AZ 86301-3720, USA Air Force Institute of Technology, AFIT/ENY, Hobson Way, Building 640, Wright-Patterson AFB, OH 45433-7765, USA

Abstract Weakest-link models for the prediction of the high cycle fatigue (HCF) limit stress of notched components were investigated. The models employed surface area elements in the high-stress region surrounding the root of a notch. Statistical descriptions of unnotched specimen experimental data were developed using Weibull distributions and median ranking, and incorporated into some of the weakest-link notch formulations. The predictions from each of the failure models were compared to the experimental 106 cycle fatigue limit stresses. The fatigue limit stresses were estimated using a step-loading technique at stress ratios from R = ⫺1 to 0.8, for five geometries (elastic stress concentration factors of Kt = 2.0, 2.8 and 4.1) of circumferentially notched Ti–6Al–4V specimens. The methods were shown to be modestly successful, although data obtained at R = ⫺1 and R ⬎ 0.65 could not be correlated well with data at intermediate values of R.  2003 Elsevier Ltd. All rights reserved. Keywords: High cycle fatigue; Notches; Ti–6Al–4V; Weakest link; Weibull distribution

1. Introduction Detrimental effects of foreign object damage (FOD) to critical aircraft engine components have sustained interest in the high cycle fatigue (HCF) behavior at notches and stress concentrations. While fatigue predictions for components with FOD must be made with consideration for the presence of residual stresses and cracks, notch analysis attempts to assess the singular influence of the FOD geometry. The notch size effect, observed when notches with larger overall dimensions lead to lower fatigue limits than those with smaller dimensions yet similar Kt, is partial evidence that the stress field surrounding the notch root in addition to the “hot spot” stress contributes to fatigue behavior. The stresses and stress gradient inward from the notch root are typically used in notch analyses, such as with the design rules formulated by Neuber [1] and Peterson [2]. The argument is that for the same nom∗ Corresponding author. Tel.: +1-928-777-3930; fax: +1-928-7776945. E-mail address: [email protected] (D.B. Lanning). 夽 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.

0142-1123/$ - see front matter  2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0142-1123(03)00156-7

inal applied stress and Kt, notches with sharper stress gradients have a lower average stress near the notch root, and therefore lead to a higher fatigue limit. However, a similar argument might be made with consideration of the stresses along the notch surface near the notch root. A surface region with a higher average stress should lead to a greater probability of failure (i.e. lower fatigue limit) than a region with a lower average stress. The use of weakest-link theory is well-established for describing the failure strength of brittle materials such as ceramics and high-modulus fibers used in composite materials [3,4]. A unit length, area, or volume of material is assigned a probability of failure associated with a set of loading conditions and material properties, and these elements are combined by means of probability theory to model the failure probability of the entire component. The application of weakest-link theory to metallic materials is less established. Hudak et al. [5] developed a weakest-link model to predict the failure of notched Ti–6Al–4V using the surface area in the notch region, and calculated the Weibull modulus for their model from a best fit of experimental notched specimen data. Surface area was used since most cracks initiate at or very near the surface of Ti–6Al–4V fatigue test specimens. Therefore, the justification for the weakest-link approach applied to notches was that it is statistically likely that

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Nomenclature A d D f F h Kt n Pf R a r s s0

element surface area reduced cross-section diameter nominal gage section diameter frequency cumulative distribution function (CDF) notch depth elastic stress concentration factor number of elements probability of failure stress ratio = s min / s max Weibull modulus or slope notch root radius stress Weibull characteristic stress

fewer weak grains (i.e. crack initiation sites) will exist along the smaller highly-stressed surface of a small notch than on the larger surface of a large notch that is similarly stressed. Similar reasoning was used in the current investigation, although the model was formulated and implemented in a different way. Weakest-link theory was used to predict the Ti–6Al–4V fatigue limit stress for a wide range of notch geometries and applied stress ratios. The range of geometries tested (K t = 2.0 to 4.1) was representative of a significant portion of the range of Kt’s observed in typical FOD [5]. Fatigue models were constructed from surface area elements in the high-stress notch root region. Smooth specimen statistical properties were developed using the Weibull distribution and used with the weakest-link model to predict the notched specimen fatigue limit stresses. The predictions were compared to experimental results obtained from a step-loading technique on Ti–6Al–4V specimens corresponding to an HCF life of 106 cycles.

2. Experimental All cylindrical fatigue specimens were machined from forged Ti–6Al–4V plate which was heat treated to obtain a product form similar to that used in typical aircraft engine airfoil components. Mechanical properties are provided in Table 1, and details of the heat treatment can be found elsewhere [6]. Cylindrical axial fatigue specimens were machined with circumferential Vnotches of five sizes as well as without notches. Notched specimens were 130 mm in length, with a gage section diameter of 5.72 mm and gage section length of 10 mm. A diagram of the notched gage section is shown in Fig. 1 and corresponding dimensions and Kt’s (2.0, 4.1, and three sizes of 2.8) for the five notch sizes are provided

Table 1 Mechanical properties for the Ti–6Al–4V forged plate Mechanical property

Strain rate (s⫺1) 0.0005

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

930 978 119

0.05 1003 1014 127

in Table 2. The three notches having the same elastic stress concentration factor (K t = 2.8 ± 0.07, with the variability due to both machining tolerances and method for calculating Kt) are additionally referred to as small, medium, and large, corresponding to their relative dimensions. The Kt’s were calculated from finite element (FE) solutions and compared to solutions from the literature [7]. Smooth unnotched specimens had dimensions of specimen length of 130 mm, gage section length of 10 mm, and gage section diameter of 5.08 mm. Most of the specimens were subsequently stress relieved at 704 °C for 1 h in vacuum to remove any residual stresses after machining. The fatigue limit stresses at 106 cycles were estimated using a step-loading method [8–11], with most testing performed at a frequency of f = 50 Hz. Coaxing, also referred to as “under-stressing”, was a possible concern with a step-loading method, but it has been addressed elsewhere [9] and has been shown to not be a concern with this Ti–6Al–4V product form [10,11]. The Haigh (Goodman) diagram with all notched and smooth specimen data is shown in Fig. 2. Stress distributions were taken from ABAQUS FE solutions for each of the experimental loading conditions leading to 106 cycle fatigue failure. The Ti–6Al–4V was modeled as an elastic-perfectly plastic material, which

D.B. Lanning et al. / International Journal of Fatigue 25 (2003) 835–841

837

60o

d

h

Fig. 2. Haigh diagram for notched and smooth Ti–6Al–4V specimens for a constant life of 106 cycles, with net reduced-section stresses plotted (∗Specimens not stress relieved).

Table 3 Von Mises equivalent and axial 106 cycle fatigue limit stresses at the notch root Kt

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

Table 2 Elastic stress concentration factors and dimensions for five circumferential V-notched specimens Kt

r (mm)

h (mm)

D (mm)

d (mm)

2.0 2.8 (large) 2.8 (medium) 2.8 (small) 4.1

0.330 0.330 0.203 0.127 0.127

0.100 0.730 0.254 0.127 0.635

5.72 5.72 5.72 5.72 5.72

5.52 4.26 5.21 5.47 4.45

2.0 2.8 (large) 2.8 (medium) 2.8 (small) 4.1 a

R = 0.1

R = 0.5

svm (saxial) (MPa)

svm (saxial) (MPa)

731 660 954 704 744

906 747 997 993 913

3. Weibull distribution Fatigue limit stress data generated by the step-loading method exhibit a significant amount of scatter, typical

(991) (842) (1140)a (1130)a (1040)

Specimens were not stress relieved.

of all fatigue data. Therefore, stress will be treated as the random variable in the following analysis. The Weibull distribution is often used to describe fatigue data. The cumulative distribution function (CDF) for the two-parameter Weibull distribution is [12]:

冋 冉 冊册

F(s) ⫽ 1⫺exp ⫺ is reasonable for modest amounts of plastic straining. Mechanical properties at a higher strain rate (Table 1) may be more representative of conditions at the notch root during experimental testing, and were therefore used in the FE yielding criterion. Predicted axial stresses and von Mises equivalent stresses at the notch root for each geometry and R are provided in Table 3.

(800) (744) (1070)a (782) (846)

s s0

a

(1)

Weibull CDFs for smooth bar data at different values of stress ratio, R, are presented in Fig. 3. The interpolated fatigue limit stress (maximum stress) from each step-loading test was plotted using median ranking [13]. Also plotted in Fig. 3 is the best fit of a two-parameter Weibull distribution found by least-squares regression of data at each R. Fitting statistical distributions to limited data, such as presented in Fig. 3, for use in design may be quite risky. Five successful tests, each were performed at R = 0.1 and 0.5, which are rather low sample numbers with which to calculate accurate CDF parameters. Testing at the other stress ratios comprised of only two successful

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the constant-diameter gage section of the smooth specimens is A smooth = 160 mm2. Inserting Eq. (1) into Eq. (2) gives the CDF for the strength of a smooth specimen:

冋 冉 冊 册

Fsmooth(s) ⫽ 1⫺exp ⫺n

s

aelem

s0,elem

(4)

where the characteristic stress is that of the elements. This CDF remains in Weibull form. The Weibull modulus for the chain is equal to the Weibull modulus for the individual elements and the subscript will henceforth be left off. The characteristic stress for the chain is now: s0,smooth ⫽ n⫺1/as0,elem Fig. 3. CDFs for the 106 cycle fatigue limit stress data for smooth specimens estimated using the step-loading method and median ranking.

results each. Although one may speculate that two data points can provide a very rough estimate for the mean value of the specimen population, it is not reasonable to believe that two data points can provide a reliable estimate for the Weibull exponent. In this study, therefore, data for R = 0.1 and 0.5 was predominantly used. The weakest-link model assumes that a structure or a “chain” consists of many structural elements or “links”, each having a strength described by a statistical distribution. The element probability of failure, the result of the CDF evaluated at a particular value of the random variable, is dependent on the loading imparted to the structure and the strength of the element, but is otherwise unrelated to the failure of any other element in the chain. That is, strengths of the elements are assumed to be statistically independent. If the CDF for each element is identical, the CDF for the strength of the entire chain of n elements is: Fchain(s) ⫽ 1⫺(1⫺Felem(s))n

(2)

where the CDFs for the strength of the structure and elements are denoted Fchain(s) and Felem(s), respectively. Eq. (2) assumes the applied stress is the same for each element. A useful approach may be to develop a model that uses smooth specimen results to make notch fatigue predictions. First, the elements composing a structure may be characterized in several ways. If the model parameter under consideration is area, the number of elements is defined: n⫽

A0 Aelem

(3)

where Aelem is the area of each element and A0 is the total area of the model. Dividing the entire smooth specimen surface into elements of equal area, Weibull distribution parameters can be developed. The total surface area of

(5)

By choosing the number of links dividing the smooth specimen surface, the characteristic stress of the individual link may be found. However, the number of links chosen does not affect the end solution for the chain.

4. Weakest-link notch model The development of a notch fatigue model applicable to a wide range of notch sizes, Kt’s, and loading conditions is desired. Weakest-link theory is applied with the intent of predicting notch fatigue behavior in Ti– 6Al–4V and the notch size effect. A larger notch is typically associated with a lesser fatigue life or a lower fatigue limit stress than a smaller notch of similar Kt. An analogy could be made between this observation and weakest-link theory. Namely, a notch with a greater highly-stressed surface area (i.e. larger notch consisting of a greater number of elements) has a greater probability of failure, all else being equal. The CDF for the notched specimens must be written in a generalized form to allow for the varying stress field across the notch surface. Eq. (2) is rewritten to reflect groups of elements that have both the same loading and statistical distribution parameters, so that m groups of elements with different applied stresses exist: Fchain(s) ⫽ 1⫺(1⫺F1(s1))n1(1⫺F2(s2))n2%

(6)

(1⫺Fm(sm))nm where ni is the number of elements in the chain loaded at the ith stress level. The elements are now characterized using surface areas: Fchain(s) ⫽ 1⫺(1⫺F1(s1))A1/Aelem

(7)

(1⫺F2(s2))A2/Aelem%(1⫺Fm(sm))Am/Aelem where Ai is the total surface area of the ith group of elements loaded at the ith stress level. The two-parameter Weibull CDF is used for each group of elements:

D.B. Lanning et al. / International Journal of Fatigue 25 (2003) 835–841

冋 冉 冊册 冋 冉 冊册 冋 冉 冊册 a

A1 s1 Fchain(s) ⫽ 1⫺exp ⫺ Aelem s0,elem exp ⫺

a

A2 s2 Aelem s0,elem

%exp ⫺

Am sm Aelem s0,elem

(8)

冋

冉 冘 冊册 冉 冘 冊册

冋

⫺exp ⫺

m

1 Aelemsa0,elem

sai Ai

⫽1

(9)

i⫽1

m

1

a 0,smooth i ⫽ 1

A0s

sai Ai

It can be seen that the magnitude of n does not affect the solution. FE solutions and component geometry, along with the smooth specimen distribution parameters, can be used to calculate a probability of failure at any stress level. For convenience, Eq. (9) will be modified using the normalized stress: qi ⫽

si s1

(10)

where s1 is the notch tip stress. Eq. (9) provides a probability of failure, but cannot directly predict a fatigue limit stress. As long as the strains in the notch region remain elastic, the stress field may be normalized and s1 becomes the only unknown on the right-hand side of the following equation:

冋 冉

冊 冉 冘 冊册

1 s1 Fchain(s) ⫽ 1⫺exp ⫺ A0 s0,smooth

a

m

qai Ai

(11)

i⫽1

If a number of notched specimens have been experimentally tested, the average of the estimated fatigue limit stresses should roughly correspond to the mean value of the Weibull distribution. The mean value for the two-parameter Weibull CDF (Eq. (1)) is well known:

冉 冊 1 a

m ⫽ s0⌫ 1 ⫹

(12)

where ⌫ is the gamma function. Eq. (11) is in two-parameter Weibull form, and the Weibull characteristic stress is:

冘

⫺1/a

冢 冣 m

s0,chain ⫽

qai Ai

i⫽1

A0

s0,smooth

specimens will have failed. The mean stress was less than the stress calculated at P f = 0.5 by a few percent for these Weibull distributions.

a

Combining terms and further simplifying with Eq. (5): Fchain(s) ⫽ 1⫺exp ⫺

839

(13)

and therefore the mean value of the chain can be computed easily. It is worth the reminder that the mean stress value is not the same value as the stress at P f = 0.5, which is the predicted value at which one half of the

5. Results and discussion The predictions for this model are the mean stresses, and are tabulated for each notch geometry at R = 0.1 and 0.5 in Table 4. The results for the local stress state along the notch surface are taken from the FE solutions. The stresses are von Mises equivalent stresses, and the element size corresponds to that in the FE model. Comparison of Tables 3 and 4 shows that accuracy of the results is mixed, and several trends are apparent. Predictions for K t = 2.8 are generally conservative. They also show a small increase in the fatigue limit stress with smaller notches at R = 0.1, which is consistent with a notch size effect. For higher mean stress (R = 0.5), small amounts of plasticity were predicted at the notch root, and the notch size effect is not portrayed accurately by the model. Of importance is the fact that the notch size effect is not readily apparent in Ti–6Al–4V for the specimen geometries tested (Table 3), although it has been observed in flat dogbone specimens for the same Ti– 6Al–4V product form [6]. Predictions for the other notches (K t = 2.0 and 4.1) are not conservative, especially at R = 0.1. A concern with the model is that predictions for dissimilar notches having the same magnitude notch root radii are almost identical, because that is not what the experimental data show. Specimens with notches of K t = 2.0 and 2.8 (large) have notch root radii of r = 0.33 mm, and the fatigue limit stress predictions are almost equal. The same is observed with notches of r = 0.127 mm. These predictions may be expected by a comparison of stress distributions along the surface of the notch root. The normalized shapes of the stress distributions for different notches with the same notch root radii are nearly identical in close proximity to the notch root. If the surface stress distributions are the same, then the current predictions should be the same. A final concern is not visible from the data presented Table 4 Von Mises equivalent 106 cycle fatigue limit stresses at the notch root predicted by Eq. (12) Kt

2.0 2.8 (large) 2.8 (medium) 2.8 (small) 4.1

R = 0.1

R = 0.5

svm (MPa)

svm (MPa)

837 834 851 875 877

904 902 884 891 932

840

D.B. Lanning et al. / International Journal of Fatigue 25 (2003) 835–841

here, but has been noted through various manipulations of the solution. It was found that the fatigue limit stress predictions are sensitive to the Weibull distribution parameters. The amount of data used in calculating the Weibull parameters was limited in this investigation, as noted earlier. An extra data point from a suspect step-loading test was temporarily included in the analysis to observe the sensitivity of the solutions to changes in Weibull parameters. This suspect result (R = 0.1) was of greater magnitude than any of the other data at the same condition, and raised the Weibull parameters by a small amount. However, the model predictions were increased by almost 10%, making the results non-conservative. It is noted here that sufficient smooth specimen testing is necessary for this model to succeed. A different approach to this weakest-link model was presented in the study by Hudak et al. [5]. They opted to use a model similar in form to Eq. (11) and compare either data from two notched specimen geometries or one notched specimen to smooth specimen data. The Weibull slope a that best fit the experimental and FE data was calculated, and not evaluated as a parameter of a Weibull distribution found from smooth specimen testing data. Their study found values for the exponent to be around a = 35. The same thing can be done with Eq. (11), where two specimen cases are assumed:

冋 冉 冋 冉

冊 冉 冘 冊册 冊 冉 冘 冊册

1 s1,1 Fchain,1(s) ⫽ 1⫺exp ⫺ A0 s0,smooth

a

1 s1,2 Fchain,2(s) ⫽ 1⫺exp ⫺ A0 s0,smooth

a

m1

qai,1Ai,1

(14)

qaj,2Aj,2

(15)

i⫽1 m2

j⫽1

Equating these probabilities of failure: qai,1Ai,1 ⫽ sa1,2

i⫽1

冉冘 冊 m2

qaj,2Aj,2

a2.0 a2.8 large a2.8 medium a2.8 small a4.1

R = 0.1

R = 0.5

35.7 14.9 11.2 14.6 35.5

23.8 13.3 13.2 14.3 24.3

higher, but are more in line with those calculated by Hudak et al. [5]. The fatigue limit stresses for R = 0.65 and 0.8 loading were not calculated in a similar fashion. The difficulty with Eq. (11) is that FE solutions predict a significant amount of plasticity at high R, and therefore normalized elastic stresses cannot be used with this technique. Fig. 4 shows an example stress distribution of the axial stress inward from the notch root of the large K t = 2.8 at R = 0.8. As plastic straining occurs, the loading conditions in terms of stress ratio change from point to point throughout the stress field. Plastic straining has the effect of reducing the local stress ratio. In the case of applied R = 0.8, the local stress ratio is reduced to approximately R = 0.65 at the notch tip. This stress ratio increases towards the value of the applied stress ratio away from the notch root as plastic straining lessens. Predictions were also not made for R = ⫺1. Experimental results using the step-loading technique show the full theoretical reduction in the fatigue limit stress by the magnitude Kt (the hot spot stress is roughly equal to the smooth specimen stress required for fatigue failure at the same loading conditions, see Fig. 2). Recently, it has been speculated that specimens tested at negative values of R had already developed cracks in earlier cycle blocks during step-loading [9]. If these cracks did not

(16)

j⫽1

which is a comparison between any two specimen geometries labeled 1 and 2. The Weibull slope that best fits the combination of experimental data can be found by iteration. If this is done when the geometries are both notched specimens as was done by Hudak et al. [5], it turns out that for several of the notch combinations, a unique value of a cannot be found with the current data. Where a unique value can be found, the magnitude varies from a = 3 to 34. However, if one of the specimens is the smooth specimen, a unique value of a can always be found. These values are given in Table 5. There is some consistency between the values for the K t = 2.8 notches, and these values are fairly close to the Weibull slope from the experimental smooth specimen data, which is approximately a = 14 at R = 0.1. This may explain some of the reasonably close predictions found in Table 4. The a values for K t = 2.0 and 4.1 are much

1500

σ max (notch)

Stress (MPa)

冉冘 冊 m1

sa1,1

Table 5 Best fit Weibull exponents based on Eq. (16)

1000

∆σ (notch)

x

500

σ min (notch) 6

Ti-6Al-4V, 10 cycles Kt = 2.8, R = 0.8 0

0

0.1

0.2

0.3

0.4

0.5

Distance from Notch Root (mm) Fig. 4. Axial stress distributions for smax and smin for large notch (K t = 2.8) at R = 0.8.

D.B. Lanning et al. / International Journal of Fatigue 25 (2003) 835–841

propagate until some higher stress level was reached in the final block, then the values of stress obtained correspond to a crack growth criterion, whereas tests at positive R correspond to crack initiation. The reason for the difference is based on the speculation that initiation is based on total stress (or strain) range whereas propagation is based on only the positive stress range. Thus the data obtained at R = ⫺1 may be inconsistent with data at the positive values of R because of the difference in mechanism (initiation or onset of propagation), and the true initiation criterion may correspond to a lower stress. Further, the reduction of the values of initiation versus propagation stress levels may differ between a notched and a smooth specimen. For these reasons, it is difficult to project any criteria developed from positive R data to R = ⫺1. 6. Conclusions A weakest-link model based upon surface area stresses has been developed and used to predict the fatigue limit stresses for several notched specimen geometries and loading conditions. All the predictions are within about 20% of experimental estimates, but predictions are affected by several shortcomings. Several conclusions can be made from this investigation: 1. Predictions for the K t = 2.8 notches tend to be more conservative than those for other notches. 2. A small notch size effect is predicted in the weakestlink models. 3. Sufficient smooth specimen data are necessary for confidence in Weibull distribution parameters. 4. Differing notch geometries with similar notch root radii tend to produce similar predictions with the variable stress surface area model. 5. Localized plastic straining (R ⬎ 0.65) and negative stress ratios (R = ⫺1) lead to difficulties with the weakest-link method that are not adequately addressed in this investigation.

841

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

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