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Procedía Engineering 101 (2015) 159-168
Procedía Engineering
www.elsevier.com/locate/procedia
3rd International Conference on Material and Component Performance under Variable Amplitude Loading, VAL2015
Fatigue life of 2024-T3 aluminum under variable amplitude multiaxial loadings: Experimental results and predictions
Nicholas R. Gatesa, Ali Fatemia*
aMechanical, Industrial, and Manufacturing Engineering Department, The University of Toledo, 2801 West Bancroft Street, Toledo, OH 43606,
Abstract
This paper investigates fatigue behavior under variable amplitude multiaxial service loadings using 2024-T3 aluminum alloy. Experimental results for smooth and notched specimen fatigue tests under axial, torsion, and combined axial-torsion loadings are compared to predictions based on von Mises equivalent stress-based approaches. Predictions using the SWT mean stress correction were in better agreement with experimental results than those based on Modified Goodman. The stress-based approaches predicted fatigue life reasonably well for notched specimens where a local uniaxial stress state always exists for the geometry considered, but not so well for smooth specimens where multiaxial loading effects come into play.
© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-reviewunderresponsibilityoftheCzech Society forMechanics
Keywords:Multiaxial Fatigue; Variable Amplitude; Service Loading; Non-proportional; 2024-T3; Notch
1. Introduction
Most engineering components and structures are subjected to variable amplitude cyclic loadings throughout their lifetime. These loadings often result in stresses along more than one axis within the component (i.e. multiaxial stress states). Additionally, notches, which serve as stress concentrations, are often unavoidable in practice and due to their geometry, may produce local multiaxial stress states regardless of the nominal loading. However, despite the significance of such conditions, the synergistic complexity of fatigue crack initiation with combined stresses and stress
* Corresponding author. Tel./fax: +1 419 530 8213. E-mail address: afatemi@eng.utoledo.edu
1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/).
Peer-review under responsibility of the Czech Society for Mechanics
doi:10.1016/j.proeng.2015.02.021
concentrations under variable amplitude loadings has only been evaluated in a limited number of studies (e.g. [1-3]). Available experimental evidence suggests that some commonly used fatigue damage analysis techniques may not be capable of producing accurate predictions for such complex and yet highly practical conditions.
Even in the absence of a notch, fatigue life analyses under multiaxial variable amplitude loadings can be quite complex. To simplify the analysis, equivalent stress methods such as von Mises stress, maximum shear stress, maximum principal stress, or Sine's Method are often used to compute the equivalent fatigue life of a cycle counted proportional loading history so long as the loading remains primarily elastic [4]. If mean stresses are present, an equivalent mean stress, to be used in a uniaxial mean stress correction model, may be computed using criteria such as von Mises effective mean stress or hydrostatic stress. For non-proportional loadings, however, these equivalent stress approaches fail to account for the increase in damage and/or non-proportional hardening resulting from the rotation of principal stress directions. As a result, alternative equivalent stress-based approaches have been proposed by Sonsino [5] and Lee et al. [6] which attempt to account for the effects of non-proportional loading through an appropriate correction factor.
Because load components remain in a constant ratio to one other under linear elastic proportional loading, cycle counting may be performed on any loading component, or an equivalent stress history, and the range and mean stresses of each component can subsequently be determined through appropriate scaling of the counting variable. For nonproportional loading, however, loading components may be applied out-of-phase or at different frequencies with respect to one another. In the case of asynchronous loadings, which are often encountered in service loading histories, there may be no clear definition of a cycle and the amplitudes and mean stresses of the different loading components, required as input for the equivalent stress approaches, can change depending on the load component used for cycle counting. This is true even for the equivalent stress criteria modified specifically for application to non-proportional loadings, thus making the equivalent stress methods vague under such conditions.
Notched specimen fatigue life prediction adds even further complexity. In a notch analysis, depending on the notch geometry, the magnitude and location of maximum stresses at the notch may change with a change in the nominal stress ratio. For example, the shift in location and magnitude of maximum stresses for the specimen geometry used in this study is shown in Ref. [7]. This makes fatigue life prediction for non-proportional nominal loadings particularly complex. Under these conditions, nominal stress approaches based on a modified stress-life fatigue curve for notched specimens (at a specific stress concentration value) lose their relevance. Instead, a local approach must be used in order to properly account for the variation of stresses at the notch root. However, if significant plasticity is present, multiaxial notch root stress-strain models must be employed to estimate local stresses and strains, many of which are complex and/or not suited for application to non-proportional loadings.
In this paper, fatigue behavior under variable amplitude multiaxial service loadings is investigated using 2024-T3 aluminum alloy. Experimental results for both smooth and notched specimen fatigue tests are compared to predictions based on simple equivalent stress-based fatigue life analysis techniques and some discussion is provided.
Nomenclature
Kf Fatigue Notch Factor
K Elastic Stress Concentration Factor
Nf Cycles to Failure
SNf Effective Fully-Reversed Stress Amplitude
Sqa Equivalent Stress Amplitude
Sqm Equivalent Mean Stress
Sq, max Maximum Equivalent Stress per Cycle
Su Material Ultimate Stress
Yield Stress in Monotonic Tension
v'y Cyclic Yield Stress
2. Experimental Procedures
The material chosen for all fatigue tests in this study was aluminum alloy 2024-T3, a common aerospace alloy since the 1930s. Material properties used in subsequent analyses were generated experimentally through a number of monotonic and cyclic deformation tests and can be found in Ref. [7], which contains stress analyses and crack initiation predictions for the same specimens investigated in this paper, but with emphasis on constant amplitude multiaxial loadings. The current paper extends the work of this reference to variable amplitude multiaxial loadings.
All tests were performed using specimens of a thin-walled tubular geometry. The specimens feature a 30 mm long gage section with an outside diameter of 29 mm and an inside diameter of 25.4 mm. Two variations of the specimen were produced: one with a smooth gage section and one with a 3.2 mm diameter circular transverse hole drilled through the thickness on one side of the specimen. Specimens were machined from drawn tubing with nominal dimensions of 34.9 mm outside diameter and 4.75 mm wall thickness. They were then fully polished, inside and out, to eliminate any influence of machining marks. For notched specimens, holes were produced by a drilling and reaming operation. Any burrs were subsequently removed by light polishing to avoid adverse effects on test results. More information on the test specimens, including technical drawings, can be found in Ref. [7].
All variable amplitude fatigue tests performed on both the smooth and notched specimens were based on a single stress-based simulated service loading history representing the nominal axial and shear loading conditions on the lower wing skin of a long-range military patrol aircraft as derived from flight test data. Tests were performed by repeating the entire load history until failure using the axial loading channel only, the torsion channel only loading, or the full combined axial-torsion loading. The entire loading history for each test performed was scaled by an appropriate factor to obtain stress levels that would produce fatigue lives ranging from less than one block to around 10 blocks. Further information on the loading history, including rainflow range-mean matrices and histograms, load-time histories, and axial-torsion stress path, can be found in Appendix A. All cycle counting was performed using the simplified uniaxial rainflow counting technique outlined in ASTM Standard E1049 [8] for repeating load histories. Nominal stresses and stress concentration factors were computed based on gross cross sectional area.
Testing was carried out in nominal load control in a closed loop servo-hydraulic axial-torsion load frame with a dynamic rating of 100 kN axial load and 1 kNm torsional load. Load train alignment was carefully inspected and maintained throughout testing. Crack initiation and growth was monitored via cellulose acetate replication for smooth specimen tests, and by using a 2.0 megapixel digital microscope, capable of 10-230x optical zoom levels, for notched specimens.
Although crack growth was monitored for both smooth and notched specimens, crack initiation on the inner surface of some smooth specimens made judging crack initiation difficult. Therefore, for consistency, the definition of crack initiation for all smooth specimens was considered a 3% change in displacement or rotation amplitude when compared to a stable reference cycle. This generally corresponded to final crack lengths of approximately 10-15 mm, with growth from 1 mm to final length occurring very rapidly. For notched specimens, crack initiation length was based on the crack transition length found from a Katagawa-Takahashi diagram for the material. This transitional crack length represents the length at which fatigue damage/failure switches from being controlled by the fatigue limit to being controlled by the threshold stress intensity factor for the material. A crack length of 0.2 mm was calculated and used as the crack initiation definition.
3. Results and Predictions
3.1. Smooth Specimen Fatigue Life Predictions
In this study, fatigue life predictions for smooth specimens were performed using an equivalent stress-based approach based on uniaxial rainflow counting of a signed von Mises stress history. The sign of the equivalent stress computed at each point in the loading history is based on the sign of the principal stress with the largest absolute value at that particular point in time [9]. For the case of axial-torsion loading, this is always the sign of the axial component of stress. This approach is similar to that described in Section 1 for proportional loading, but varies is a few key aspects.
The first is the addition of the sign to the equivalent stress value. This allows for proper identification of minimum and maximum equivalent stresses during cycle counting as equivalent stress is computed before cycle counting in this approach rather than after. The second difference is in the consideration of mean stresses.
In traditional equivalent stress-based approaches, an equivalent mean stress is computed from the mean values of various loading components over a given cycle. This equivalent mean stress is then used, along with the equivalent stress amplitude, in traditional uniaxial mean stress correction models to compute a new equivalent fully-reversed stress amplitude for calculation of fatigue damage. However, for many non-proportional loading histories, as mentioned in the introduction, the amplitudes and means of the stress components required for these calculations can be difficult to determine since the definition of a cycle is often not clear. Therefore, in subsequent fatigue life calculations, mean stress corrections are performed directly with the mean value of equivalent stress identified through cycle counting. Mean stress corrections were performed using two different models: Modified Goodman, given by Eq. (1), and Smith-Watson-Topper (SWT), given by Eq. (2).
Sf = (1)
The Modified Goodman mean stress correction is given above where SNf is the effective fully-reversed stress amplitude for use in fatigue damage computation, Sqa and Sqm are the equivalent stress amplitude and mean stress, respectively, identified from cycle counting of the equivalent stress history, and Su is the material ultimate strength.
SNf SqaSq,max (2)
In the SWT mean stress correction above, Sq>max is the maximum equivalent stress value in a particular cycle.
Once the value of SNf was determined for each counted cycle, fatigue damage was computed from the following fully-reversed uniaxial stress-life equation (obtained from fatigue tests of the same 2024-T3 tubular specimens):
SNf = 1089(Nf ) -0133 (3)
where Nf is the number of cycles to failure and SNf is in MPa. Damage from each cycle was summed using the Palmgren-Miner linear damage rule to obtain the number of loading blocks to failure. The results of the smooth specimen fatigue life predictions are shown as solid symbols in Fig. 1 for both Modified Goodman and SWT mean stress corrections.
3.2. Notched Specimen Fatigue Life Predictions
Notched specimen fatigue lives were predicted by using a local pseudo stress-based approach. This approach was chosen due to its simplicity and its ability to account for changes in local stress behavior as a result of changing ratios of nominal stresses. In order to apply such an approach, a suitable fatigue life curve must first be obtained for the notched component. This pseudo stress-life curve was constructed by following the procedure outlined in Refs. [10, 11]. First, the smooth specimen endurance limit (at 5 108 cycles in this case) is raised by a factor of Kt /Kf, where Kt is the uniaxial elastic stress concentration factor and Kf is the fatigue notch factor. Doing so allows for a qualitative consideration of stress gradient effects in the vicinity of the notch root. Kf can be determined experimentally, as the ratio between the smooth specimen and notched specimen endurance limits, or through a suitable approximation formula. For this study, Kf was determined using Neuber's formulation [12]. Details of the calculation can be found in Ref. [7]. Once the modified endurance limit has been determined, the remainder of the pseudo stress-life curve is constructed by retaining the same S-N slope factor as the smooth specimen fatigue curve. With a ratio of Kt /Kf equal
to 1.31, the resulting pseudo stress-life curve takes the same form as Eq. (3), but with the coefficient changing from 1089 MPa to 1426 MPa.
Fatigue life predictions were performed at the locations around the hole where cracks were experimentally observed to initiate (±90° to specimen axis for axial and combined loadings and ±45° to specimen axis for torsion). If the location was not known prior to performing fatigue life predictions, multiple locations around the hole would need to be analyzed to determine where the minimum fatigue life occurs. The pseudo-stress history for each location was computed by first obtaining elastic stress concentration factors, defined as local von Mises stress divided by nominal stress, from linear finite element analysis for each respective loading channel. The nominal loading history was then scaled by the appropriate concentration factor at the desired location. For combined loading situations, the principle of superposition was used to account for the combined stress concentration effect of both loading channels. A comparison of the predicted and experimental fatigue lives for all notched specimens is presented, using open symbols, in Fig. 1 for both Modified Goodman and SWT mean stress corrections.
"S 10"1 ts
• Smooth Axial ♦ Smooth Torsion la Smooth Combined / y
o Notched Axial O Notched Torsion □ Notched Combinée o ✓ / / ™ / o +/ , ' ßg A S 1 /
/ / > / a 0/ S / ' / / /
/ / // / ✓ - có' / / f / 1 11 Conservative
S io°
_o m ■ö
o Smooth ♦ Smooth □ Smooth Axial Torsion Combined * □ ca t '
wuiuieu rtXIdl O Notched Torsion □ Notched Combined my A -tn / 0/ y A /1 / o /6 ''' ' /
/ ✓ / / /_/ /' S / f / s f
/ ' ✓ / / Conservative
10"3 10"- 10"1 10" 101 Experimental Blocks to Failure
Experimental Blocks to Failure
Fig. 1. Predicted vs. experimental fatigue life using (a) Modified Goodman and (b) SWT mean stress corrections
4. Discussion and Conclusions
Overall, the equivalent stress-based approaches used in this study produced mixed results. Notched predictions based on Modified Goodman mean stress correction agree with the experimental data for the longer life tests, but become increasingly more conservative as life decreases. Notched specimen life predictions based on SWT, however, agree well with experimental data at all lives (9 of 11 predictions within a factor of ±3) while having the tendency to be only slightly non-conservative. Although it may appear surprising at first, that a simple equivalent stress-based approach produces such accurate predictions for these complex loading conditions, the notch geometry and loading spectrum considered in this study produce favorable conditions for this life prediction procedure.
Because of the free surface boundary conditions of the hole and a state of plane stress in the specimen thickness direction, a uniaxial stress state always exists at any location around the perimeter of the notch regardless of the degree of multiaxiality of the nominally applied loading [7]. Therefore, no complex effects due to multiaxial and/or nonproportional local stresses need be accounted for in this case. Additionally, because of the nature of the loading spectrum, a large percentage of the stress amplitudes, even at the notch, are elastic despite the large values of maximum stress (corresponding to occasional overloads). To help illustrate this point, box and whisker plots of the SWT mean
stress corrected local elastic equivalent stress amplitude for each cycle in the loading history are shown in Fig. 2a. In the figure, the y-axis labels give the maximum nominal stress, in MPa, and the type of loading (A: Axial, T: Torsion, AT: Axial-Torsion) while the blue highlighted region shown over the box and whisker plots, represents the range of stress amplitudes for 99% of the applied cycles. With a monotonic yield stress, oy, of 330 MPa and a cyclic yield stress, o'y, of 415 MPa, it is clear that most of the cycles, even for the highest loadings, remain completely elastic. For this special case of uniaxial and primarily elastic local stresses, it is expected that the equivalent local stress approach would produce reasonable life predictions. However, it should be emphasized that these conditions are not always present in multiaxial service loading histories (with or without a notch) and equivalent stress approaches would be expected to break down in the more complex cases.
Evidence of the shortcomings of equivalent stress approaches can be seen in the smooth specimen fatigue life predictions performed in this study. Although these predictions should be simplified in absence of a notch, in this case, they become more complex as a result of the multiaxial state of stress (excluding axial only loading) not present in the notched life predictions. Despite the fact that 99% of all mean stress corrected stress amplitudes remain completely elastic for all smooth specimen tests performed (as seen in Fig. 2b), predictions are still considerably worse than in the case of the notched specimens. Predictions based on Modified Goodman and SWT mean stress corrections were similar for smooth specimens with 2 of 6 (33%) and 4 of 6 (67%) predictions falling within factors of ±3 and ±10 of the experimental results, respectively, for each model.
In general, there are a number of additional drawbacks to equivalent stress approaches that should be considered when performing life predictions. These include the inability to account for changes in material constitutive behavior during cyclic plastic deformation and non-proportional loadings as well as the inability to predict failure/crack initiation orientation. Crack orientation is important information if subsequent crack growth analyses are to be performed. As another point worth mentioning, performing mean stress corrections using the mean value of equivalent stress obtained from cycle counting corrects for the presence of mean shear stresses. However, mean shear stresses have been shown to have little influence on fatigue damage when stresses are below yield [13].
Fig. 2. Box and Whisker plots of all mean stress corrected (SWT) equivalent von Mises stress amplitudes for (a) notched specimen tests and (b) smooth specimen tests. The blue highlighted region encompasses 99% of all cycles in the loading history.
In conclusion, stress-based fatigue life analyses of smooth and notched components under multiaxial variable amplitude service loading conditions using a signed von Mises stress history and SWT mean stress corrections may work well in some specific cases, but were unable to bring all predictions in this study within even a factor of ±10 of the experimental data. Although these approaches are simple to implement, the tradeoff is often a decrease in accuracy resulting from a number of shortcomings.
In order to overcome these limitations, more advanced fatigue life prediction methodologies should be employed. For example, one option would be to use a multiaxial cycle counting approach, such as that proposed by Wang and
Brown [14], on an equivalent elastic-plastic stress or strain history. Alternatively, Bannantine and Socie [15] proposed a method of variable amplitude fatigue life analysis based on a modification of uniaxial rainflow counting for use with the critical plane concept. These approaches, combined with advanced cyclic plasticity models, notch stress-strain estimation models, and fatigue damage parameters provide the ability to accurately reflect and account for the more complex aspects of material response and fatigue damage. As a result, it is reasonable to expect improved fatigue life predictions from such methods, similar to what was observed in Ref. [7] for constant amplitude loading. Performing variable amplitude fatigue life analyses using such techniques is a major area of focus for this research and is the subject of current ongoing investigation.
Acknowledgements
Special thanks are given to the United States Naval Air Systems Command (NAVAIR) for financial support of this study and to Dr. Nam Phan for serving as a technical point of contact.
Appendix A. Load History Information
The following table and figures give detailed information regarding the load ranges, sequence, and phasing of the service loading history used in this study. The maximum and minimum axial stresses in the unscaled load history are 144.8 MPa and -51.3 MPa, respectively, while the maximum and minimum shear stresses are 67.0 MPa and -15.9 MPa, respectively. The total length of the history is 914094 data points with each point approximately corresponding to one load reversal.
Simplified Rainflow Range-Mean Matrix for Axial Cycles
Mean Axial Stress (MPa)
MIN -40 -34 -23 -22 -16 -10 -4 2 3 14 20 26 32 33 44 50 56 62 63 74
MIN MAX -34 -23 -22 -16 -10 -4 2 3 14 20 26 32 33 44 50 56 62 63 74 30
0 10 32 2523 3459 47 ¿2 9 13149 3 512 320 221 43 233 170 140 169 639 531 66
10 20 57 13 13 9 3 3027 13 1359 309 250 55 166 715 1330 5963 14213 3377 ¿57
20 29 : 13 6 1166 233 ¿2 2 253 415 3 317 2950 67336 94914 12336 1435 203 54
29 39 : 27 11 73 91 1 35 13 190 1433 37690 53251 10902 392 70 22 4
39 49 1 7 43 2 25¿ 303 255 4955 12144 3047 353 20 6 3
49 59 24 1 557 1193 234 1366 3057 1356 151 33
59 69 4 266 5 ¿2 219 391 799 522 63 11 3
69 79 3 1141 129 2 97 139 176 233 37 4 1
79 33 3 256 1035 ¿3 10 33 46 44 23 4 1
33 93 94 322 444 231 1 15 6 5 11 3
93 103 33 935 251 35 11 4 4 4
103 113 ¿5 193 360 10 6 3
113 123 1 251 230 271
123 137 9 ¿53 65 22
137 147 31 219 4 2
147 157 65 55 1 1
157 167 1 59 9
167 177 5 13 1
177 136 10 3
136 196 3
Simplified Rainflow Range-Mean Matrix for Shear Cycles _Mean Shear Stress (MPa}_
MIN -9 -6 -3 -1 2 5 3 11 17 20 23 25 23 31 3¿ 37 40 ¿3 46
MIN MAX -6 -3 -1 2 5 3 11 14 17 20 23 25 23 31 3¿ 37 40 ¿3 46 49
0 4 25 1020 2355 5795 7995 4630 4351 1257 396 ¿53 316 296 2397 6311 335 162
4 3 3 196 11240 11041 53441 9923 1090¿ 1129 6635 1039 1399 775 373 6393 33223 4777 334 11
3 12 2 22 16311 7169 115970 11 ¿¿3 6326 296 7555 1236 7¿1 337 202 29¿ 2150 6459 273 46 11
12 17 1113 47¿5 6146 1331 309 143 1617 217 111 31¿ 131 12¿ 72 3¿3 560 311
17 21 50 1526 3032 970 561 2 ¿9 507 7 23 39 295 236 69 31 5¿ 23
21 25 1 76 700 463 149 537 63¿ 31 2 20 537 130 23 20 16 2
25 29 7 120 39 407 377 291 133 6 16 166 153 2 1 1 1
29 33 3 33 363 159 333 279 132 3 ¿2 2
33 37 21 30 157 37 3153 945 223 12 1 1
37 ¿2 2 12 29 1¿3 1236 1355 1029 3
¿2 46 1 1 267 1067 1211 253 63
46 50 359 Iü0¿ 313 ¿03 30 1
50 54 1 171 500 423 321 20 2
54 53 5 319 364 65 19 5
53 62 1 64 353 49 2 11
62 66 5 10¿ 56 22 2
66 71 20 112 21
71 75 3 95 3
75 79 25
79 33 1 1
Fig. 3. Range mean matrices for simplified rainflow counting of axial and shear channels of loading history. Headings show min and max values for each cycle counting "bin."
Fig. 4. Range-mean histograms of rainflow counted (a) axial cycles and (b) shear cycles.
-50 0 50 100
Time Axial Stress (MPa)
Fig. 5. (a) Representative load vs. time history for a sample of 1000 data points and (b) shear stress vs. axial stress plot showing load phasing.
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