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Procedía Engineering 114 (2015) 574 - 582

Procedía Engineering

www.elsevier.com/locate/procedia

1st International Conference on Structural Integrity

Variable amplitude fatigue life in VHCF and probabilistic life

predictions

Attilio Arcaria *, Nicole Apetrea, Norman Dowlingb, Martin Meischelc, Stefanie Stanzl-Tscheggc, Nagaraja Iyyera, Nam Phand

aTechnical Data Analysis, Inc., 3190 Fairview Park Drive, Suite 650, Falls Church, VA 22042, USA bMaterials Science and Engineering Department, and Engineering Science and Mechanics Department (Jointly Appointed), Virginia Polytechnic

Institute and State University, Blacksburg, VA 24061, USA c University of Natural Resources and Life Sciences, BOKU, Vienna, Austria dUS Naval Air System Command, Patuxent River, MD 20670, USA

Abstract

Fatigue life in the very high cycle fatigue (VHCF) regime for aluminum alloy 7075-T6 in plate form is characterized in constant and variable amplitude loading using unique testing equipment that allows superposition of small amplitude vibrations on top of duty cycles [1]. Constant amplitude loading data from the current experimental effort and from literature sources are used to construct a strain-life input using a Walker mean stress correction method. Variable amplitude loading data are analyzed using the constructed strain-life input. A novel probabilistic approach based on the probabilistic framework of Castillo [2] and modified by using the proposed mean stress correction method is applied. Results are compared with experimentally obtained fatigue lives. Insights into modes of failure in very high cycle fatigue for constant and variable amplitude loading, the role of experimental scatter and interaction effects are presented.

©2015Published byElsevierLtd.Thisisanopen access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering Keywords: Very High Cycle Fatigue; Variable Amplitude Loading; Mean Stress Effects; Walker Equation; Weibull Regression;

1. Introduction

Mechanical components are often subjected to vibratory environments given by their elastic response to applied varying loads. Working loads and vibrations in aircraft structural applications may come from engines and rotating components, or dynamic loads on the airframe, such as gust and buffet loads [3]. The component locally experiences stresses of different amplitudes and applied at different frequencies and phases; areas of stress concentration within the component are of particular interest. Often components are designed such that the majority of these vibrations causes stresses of medium to small amplitude, near or below the conventional fatigue endurance stress for the material. These vibrations are however superimposed on events of larger amplitude, such as maneuver loads or on-off conditions, and

* Corresponding author. Tel.: +1-703-226-4075 ; E-mail address: aarcari@tda-i.com

1877-7058 © 2015 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 INEGI - Institute of Science and Innovation in Mechanical and Industrial Engineering doi:10.1016/j.proeng.2015.08.107

contribute to the development of a more complex stress spectrum at critical locations. The service spectrum therefore combines vibratory events and larger duty cycles and their superposition defines the stress ranges and mean stresses experienced at a critical location.

In fatigue this service stress spectrum is analyzed to determine whether at any time during the lifetime of the component a critical event, such as crack initiation or propagation, occurs. Within the philosophy of the safe-life approach to fatigue, the first critical event is the initiation of a fatigue crack at a stress raiser or critical area. Methodologies for the use of the stress spectrum for the calculation of fatigue life to crack initiation, such as the stress- or strain-life approaches, rely on one fundamental piece of information: the mechanical behavior of the material under static and fatigue loading. The fatigue behavior of the material is usually obtained in the form of a stress- or strain- life curve; in addition elastic and elastic-plastic material behavior in spectrum loading needs to be characterized.

The identification of specific events or cycles whose fatigue damage needs to be accounted for within a stress spectrum through the use of an appropriate methodology, such as the rainflow counting method, requires the characterization of stress- or strain-life behavior of the material in different fatigue regimes [4]. The total damage is obtained by linear summation using Miner's rule and the critical value of this summation is 1, number associated with a crack initiation event. Within the strain-life approach to fatigue different regions in the strain-life curve are identified, as material behavior in fatigue becomes more or less dependent on applied plastic strain amplitude, low cycle fatigue (LCF) and high cycle fatigue (HCF). More recently, driven by the need of reliable design for structural components expected to experience a very large number of fatigue cycles [5,6], fatigue characterization of material behavior beyond the HCF region is being pursued. Fatigue regimes beyond HCF are usually referred to as very high cycle fatigue (VHCF) and ultra-high cycles fatigue (UHCF), and conventionally indicate regions in the strain- or stress-life curve corresponding to cycles to failure ranging from 107 to 1010 cycles. Material characterization is inevitably subjected to experimental scatter; particularly in HCF and VHCF scatter represents a significant challenge when performing fatigue calculations [7]. Typically a least square regression is used to define the stress- or strain-life log-log linear relation in fatigue, however there are specific assumptions that need to be considered when using linear regression and least square approximations. When analyzing test data across several fatigue regimes these assumptions may not hold and new methods need to be used [8].

In this work the study of fatigue life for aluminum alloy 7075-T651 in plate form for variable amplitude loading in VHCF is presented, along with fatigue life predictions using a novel probabilistic approach for the characterization of material behavior in low-, high-, and very-high-cycle fatigue that includes a mean stress correction method. Experimental data in constant and variable amplitude loading in VHCF are presented in the first part of this work, with particular emphasis on the observed failures for critical superimposed stress spectrum, followed by test-analysis correlation results, conclusions and recommendations.

2. Materials and Test Methods

The study of the mechanical behavior of materials in fatigue typically involves the testing of material samples in

constant amplitude loading, often at different R-ratios, for several stress or strain levels. Traditionally, multiple fatigue tests are performed to obtain a statistically sound material characterization in LCF, starting from 102 - 103 number of cycles to failure, Nf, going to HCF up to about 106 cycles, limit conventionally referred to as: endurance limit.

While fatigue behavior in LCF and HCF was shown to follow similar trends across very different classes of materials, material behavior in VHCF and UHCF has been shown to vary significantly depending on the type of material, composition, microstructure, heat treatments, and consequent mechanical behavior. General consensus exists that, by using specific experimental techniques to allow testing at very high frequencies, several materials show fatigue failures in VHCF and UHCF [5,6]. The conventional endurance limit may therefore differ from the theoretical strain or stress level that is expected to cause infinite life, if such level exists [5,6]. However the effects of the sequence of stress levels in VHCF and UHCF for a complex spectrum need to be investigated, specifically in relation to stresses in the LCF and HCF regime [1,9].

Our experimental work combines two experimental techniques to produce novel and unique stress sequences representative of the highly vibratory environments that components experience during their useful life superimposed to larger duty cycles. The low frequency amplitude is obtained using a servo-hydraulic load-device with the signal generated in Force control mode [1]. The ultrasonic device is attached to the servo-hydraulic testing machine to allow

Fig. 1: Constant amplitude fatigue data for 7075-T6 plate from ultrasonic testing and compared to [10].

the application of fixed mean stress levels or varying mean stresses according to predefined sequences (sinusoidal wave, square wave, ramps, etc..). The resulting stress spectrum is shown in Figure 2 (a).

The material tested is aircraft aluminum alloy 7075-T651 in plate form, the dimensions and characteristics of the material are given in [1]. The specimens are cut along the rolling direction of the plate. A comprehensive testing program conceived and developed by the authors [1] included several spectrum types and sequences of stress amplitudes and mean stresses. One of these spectrum categories, named sine-on-ramp, and experimental results are presented in this work, along with constant amplitude results for the material.

Two different R-ratios have been tested in constant amplitude loading for this experimental program: R=-1 and R=0.5. For constant amplitude loading the strain levels tested and fatigue life results are shown in Figure 1. Results clearly show that fatigue failures occur even beyond 106 cycles, with fractured specimens beyond 109 cycles; the strain-life behavior shows a continuous decrease in stress amplitude for increasing number of cycles to failure, consistent with fatigue behavior previously reported for this class of aluminum alloys [5,10]. Fatigue test data for ultrasonic testing compare well with traditional testing methodologies.

(a) Test spectrum sine-on-ramp for 7075-T6 (b) Test results for sine-on-ramp spectrum for different <aj values. aluminum.

Fig. 2: Variable amplitude loading testing.

Variable amplitude loading tests have been performed by implementing the sine-on-ramp spectrum as shown in Figure 2 (a). One of the goals of the tests was to observe the influence of VHCF stresses on the total fatigue life. In order to maintain the same maximum and minimum stresses for each spectrum block, imax =400 MPa and imin=20MPa, the mean values of the ramp signal are developed as shown in the Figure 3. Three different sine-on-ramp spectrum are obtained each corresponding to one superimposed stress amplitude in the VHCF range, ia1=50MPa, ia2=65MPa, and ia3=80MPa.

Fatigue life is significantly shorter than the corresponding constant amplitude loading life for imax=400 MPa and imin=20MPa, estimated to be around 120,000 cycles; this is clearly due to the presence of a large number of small amplitude vibrations whose contribution to damage is significant in the total fatigue life. The current spectrum is composed in major part of VHCF cycles: 200,000 cycles per block are in fact near or below the endurance stress for the material. Calculations of equivalent stress amplitude, iar, based for example on Smith-Watson-Topper method shows that even for ia3=80MPa the equivalent stress amplitude corresponding to the highest mean stress (Spectrum 3, level-6, 320MPa) is around 178MPa. This is essentially the "traditional endurance stress" amplitude for the material.

Spectrum 1 - Sine-on-Ramp with Gal= 50MPa

Gmean-level-l (MPa) cycles at level-i (MPa) Gmax-level-l (MPa) Gmm-level-l (MPa)

level -1 70 20,000 50 120 20

level - 2 126 20,000 50 176 76

level - 3 182 20,000 50 232 132

level - 4 238 20,000 50 288 188

level - 5 294 20,000 50 344 244

level - 6 350 20,000 50 400 300

Spectrum 2 - Sine-on-Ramp with aa2= 65MPa

Gmean-level-l (MPa) cycles at level-i a,2 (MPa) Gmax-level-l (MPa) Gmm-level-l (MPa)

level -1 85 20,000 65 150 20

level - 2 135 20,000 65 200 70

level - 3 185 20,000 65 250 120

level - 4 235 20,000 65 300 170

level - 5 285 20,000 65 350 220

level - 6 335 20,000 65 400 270

Spectrum 3 - Sine-on-Ramp with Ga3= SOMPa

Gmean_|eve|_, (MPa) cycles at level-i (MPa) (MPa) ^min-level-l (MPa)

level -1 100 20,000 80 180 20

level - 2 144 20,000 80 224 64

level - 3 188 20,000 80 268 108

level - 4 232 20,000 80 312 152

level - 5 276 20,000 80 356 196

level - 6 320 20,000 80 400 240

Fig. 3: Table showing sine-on-ramp spectrum levels.

Note that in Figure 2 (b) the total number of cycles is reported; this value should be divided by the total number of cycles per block, approximately 200,000 cycles, to obtain the number of spectrum blocks to failure. The average number of blocks to failure is 3,770 blocks for ia1 =50MPa, 1,979 blocks for ia2=65MPa, and 185 blocks for ia3=80MPa. There is a strong correlation between fatigue life and the applied superimposed vibration amplitude. Fatigue life is significantly reduced by the application of ia3=80 MPa superimposed amplitude with respect to 50 and 65MPa. Some difference between the two latter amplitudes is also evident from Figure 2 (b).

Experimental scatter is significant in this fatigue regime, and particularly evident for the case of ia3=80MPa. One data point is more than one order of magnitude apart from the majority of the remaining experimental data. An unusually large particle observed in the crack initiation area may be responsible for the significantly shorter fatigue life, as it will be shown in the next section.

Fractographic images have been collected for all broken samples, the area observed to be the origin of the fatigue crack which progressively leads to failure has been identified for all tests. Internal fatigue crack initiations have been reported for the majority of the fatigue tests in constant amplitude loading, one example is reported in Figure 4 (a).

Cracks were observed to initiate in all cases from constituent particles or inclusions. Back scatter (BSED) frac-tographic analysis revealed the different composition of these particles with respect to the surrounding matrix. They

appear lighter than the surrounding matrix owing to the higher atomic number of their constituents compared to aluminum. Analysis of these particles using Energy-Dispersive X-ray spectroscopy (EDX) revealed the presence of Fe, Zn, Mg, and Cu, indicating that possibly the composition of the particle may have been Al7Cu2Fe or MgZn2 as reported in other studies on this structural aluminum alloy [11].

Analysis of size and distribution of cracked particles observed in the SEMs BSED images has been performed to study the fracture surface and investigate the role (position and size) of these particles as crack initiation points in constant and variable amplitude loading. The particles are expected to be elongated in the rolling direction [11], therefore the observations on the fracture surface should be interpreted as the description of the size and characteristics of constituent particles for a random section of the material volume tested along the rolling direction.

Note that the distribution of particles within this or part of this cross section may not be statistically representative of the distribution within the entire material volume tested; however the collection of cross sections images is considered a representative sample of the population of critical cross sections within the volume of the material tested in HCF.

The analysis was performed by using image analysis software that identified the cracked particles based on reproduced grey scale on the SEM and measured some of their fundamental shape characteristics: area, greatest Ferret diameter (FD), and Aspect Ratio (AR). The aspect ratio is calculated as the ratio of the major and minor axis of a circumscribed ellipse. As shown in Figure 4 (b), once the particles are identified, they are numbered starting with the largest particle within the identified crack initiation area. The results for all constant amplitude and variable amplitude loading tests are then compiled as a single data set, for a cumulative description of the characteristics of the identified particles.

Figure 5 (a), (b), and (c) show the histograms describing the distribution of Area, Feret diameter, and aspect ratio. The large majority of particles on the fracture surfaces observed shows an area between 10 and 200 ¡m2 and a greatest Feret diameter between 10 and 20 ¡m, note that the histogram bins are equally spaced on a logarithmic scale. The aspect ratio is about 1.5 to 2.5 for most of the particles analyzed. The dimensions show that a significant part of the inclusions/particles have large dimensions, possibly due to the thickness of the plate tested (20 mm).

Data concerning the size and characteristics of only the largest particle in the area of crack initiation for each test from constant amplitude loading are compared to the data obtained from variable amplitude loading. It is interesting to note that a difference exists between the average area of the observed critical particles in constant amplitude loading with respect to critical particles in variable amplitude loading. The average value for constant amplitude loading is 1,800 ¡m2, while in variable amplitude loading is 400 ¡m2 (not shown in Figure 5). However from the results in constant amplitude loading, a significantly higher dispersion was observed. No significant difference in greatest Feret diameter or aspect ratio for these particles is observed.

Systematic measurements of the distance of the internal particles in the area of crack initiation are also obtained from the SEM images. Note that in variable amplitude loading only one clearly identifiable internal initiation was observed, while in constant amplitude loading the majority of initiations occurred at the interior of the specimen. The

(a) Internal Initiation

(b) Particle Analysis Example

Fig. 4: SEM images for fractures 7075-T6 aluminum specimens.

5 16 ',!) 158 500 i,5SÖ 5,000 10,000 2 A 3 16 .t.' 64 128 More

Area, füll' Feret Diameter, |im

(a) Area Distribution (b) Feret Diameter Distribution

1 1.5 2 3 r.'l 4 4.5 5 More

Aspect Ratio

(c) Aspect Ratio Distribution

Fig. 5: Histograms of particle characteristics on fracture surface of 7075-T6.

average distance of the particle from the surface of the specimen was 0.684 mm (diameter of the specimen is 4mm). The observed trends seem to indicate a definite change in critical defect location and average size.

One fatigue test (Test 96) in variable amplitude loading showed an internal initiation particle of unusually large area, 1,980 ¡m2. The fatigue life for this variable amplitude loading test is significantly lower than all others tested with the same spectrum.

2.1. Comments on Fractographic Results

It was shown that for this class of materials the incubation process is majorly impacted by the brittle failure of Al7Cu2Fe or fractured Fe-bearing constituent particles located on the specimen surface [11]. In HCF up to 90% of the total fatigue life can be spent in nucleating a defect or propagating a small crack that eventually becomes a dominant fatigue crack [11,15]. Although the analysis performed in this work seems to indicate that mostly larger particles within the volume tested are critical in VHCF, but also that size is not the only element that determines the criticality of the particle or the area surrounding it.

Additional factors, such as the relative orientation of the grains around the particle or the presence of multiple particles in the same critical area, can contribute to the criticality of the site [11,15]. If size were the only parameter, initiation in variable amplitude loading or constant amplitude HCF loading would also be equally likely at the subsurface, however this is not the case. Even if the size of a critical particle is on average higher in VHCF, the variance for the data collected is also higher, therefore large particles may represent one of the possible critical sites or drastically increase the probability of the site to be critical given the occurrence of concomitant factors.

Bozek et al. [11] hypothesized that both particle aspect ratio and size, along with grain orientation and strain level are the key parameters to explain the stochastic nature of particle cracking in 7075-T6 aluminum. Barter et al. [13] showed crack initiation in 7050-T7451 aluminum is affected by inclusion shape and size, and that often initiation may result from coalescence of cracks from multiple cracked inclusions within a critical area. Salajegheh [14] also hypothesized that multiple factors influence the number and type of fatigue "hot spots" for Inconel alloy in HCF and VHCF. The identification of the largest particles or defects as the location of crack initiation within the material volume tested in VHCF was also shown by Kazymyrovych for a tool steel [15]. He also argues that VHCF testing is a

useful tool for quality control of materials, given the inherent ability to find information on the weakest microstructural link within the volume tested.

It is interesting to note that in variable amplitude loading, with alternating stresses of different amplitude, some in HCF and some in VHCF regime, the size of the critical particle becomes a secondary factor and most initiations happen on the surface of the sample. One of our test cases (Test 96) seems to indicate that a particularly unfavorable combination of particle size and additional determining factors may yield very short lives in variable amplitude.

The contribution of VHCF cycles to fatigue damage exists as described in this work, and also shown by [1,9,12], however the critical area or defects impacted by this contribution is different than the naturally critical area in constant amplitude. This may have implications on the quantitative contribution of VHCF cycles to total fatigue damage, given the different mechanical constraints dictated by particle location and environment experienced (subsurface vacuum vs surface environment). This also in part demonstrates that stress levels of different amplitudes indeed interact at the microstructural level. The mechanisms of different modes of damage accumulation, however, still need to be investigated. Additional work is also needed to better rationalize the criticality of the area of fatigue crack initiation.

3. Analysis Methods

Common methods rely on least square approaches for the determination of material parameters that describe the strain-life curve of the material. However, inherent assumptions are required when performing least-square linear regression; an assumption of normality and constant variance at each strain level is typically required to correctly justify the regression process. This assumption is reasonable in LCF and can be extended in part to HCF, however it may not hold near or below the endurance limit of the material. The increase in experimental scatter for decreasing strain amplitude requires proper mathematical formulations.

In this work a probabilistic strain-life Weibull regression model, based on the work by Castillo [2] and modified by the authors [8] using a Walker mean stress-like equation is used. The model deals with total strain amplitude corrected for mean stress, ea (jT^) 7, and gives explicitly the probabilistic P-e-N field. Emphasis is placed on the Walker method, as previous work [16] in a stress-life context has shown that it is superior to other common methods of handling mean stress effects.

The resulting model is advantageous with respect to other means stress correction methods as it gives the possibility of including and regressing fatigue data for several different R-ratios, while calculating and optimizing the mean stress sensitivity parameter y. The current form differs from the Walker mean stress correction method as it applies the correction factor to strain amplitude. Note that this is a simplification with respect to the method of [16], made to facilitate the regression analysis.

3.1. 7075-T6 Data Collection

Fatigue data for aluminum alloy 7075-T651 were collected by the authors from several sources and compiled into a single data set. Results of the regression for the strain-life fatigue data collected are shown in Figure 6.

The strain-life input curves obtained from regression of the fatigue data in constant amplitude loading for 7075-T6 are used within a strain-life approach to fatigue life calculation. The methodology uses the rainflow counting method to identify fatigue cycles in the spectrum and Miner's rule to sum the damage of each cycle in the spectrum block. The number of blocks to failure is obtained and the total number of cycles to failures is calculated from the total number of cycles per block.

3.2. Observations

The developed model shows good correlation with the constant amplitude loading data set and it allows calculating the mean stress sensitivity parameter y, whose value was estimated to be 0.522. Figure 7 shows a comparison with a conventional strain-life fit by using linear regression and the estimated value of y (plastic and elastic strain vs life linear segments are also shown as dashed lines). The developed models show reasonable proximity, however, at very long life (> 108 cycles) it is evident that the two formulations start to differ more significantly. The conventional fit shows a much steeper slope at long life, dictated by the influence of fatigue data in low and high cycle fatigue.

Fig. 6: Constant amplitude fatigue data in LCF-HCF-VHCF for 7075-T6 from several sources.

Fig. 7: Probabilistic fatigue life predictions with developed model.

The conventional fit progressively diverges from the VHCF data, while the current model follows more closely the observed trends at long life. Additionally, the developed model formulation inherently allows obtaining a strain-life input that corresponds to different probabilities of failure and that can be used for fatigue life calculations. Results for variable amplitude loading show fair agreement with experimental data for the sine-on-ramp spectrum developed in this experimental work and that most of the data points fall on the left side of the 50% Pof curve and all data but one data point is within the developed bounds. This data point could perhaps be considered an outlier, given the observations on the crack surface mentioned earlier. The observed trend in fatigue life predictions seems to indicate that small cycles in variable amplitude loading produce more damage than in constant amplitude loading.

It has been shown previously how interaction effects may play a significant role in the accumulation of fatigue damage in LCF and HCF [1,9]. More recently the authors consistently showed the role of interaction effects within HCF and VHCF regimes for several other spectrum types [1]. These interaction effects should be included in the material strain-life input or in the damage calculation based on postulated mechanisms responsible for increased damage of small amplitude vibrations in variable amplitude loading. The use of a more conservative curve, such as the one corresponding to a lower percentile, may also yield safe predictions for cases of interaction effects. Different

spectrum sequences may however show different interaction levels, therefore detailed investigations, based on sound experimental and modeling work starting at the microstructural level are needed.

Overall, the developed model shows good promises and advantages with respect to traditional methods for an appropriate statistical characterization of material strain-life behavior. Fatigue life predictions can highly benefit from the use of this model as they become rooted in sound mathematical grounds to account for the variance of fatigue data in different fatigue regimes.

4. Conclusions

Material characterization is needed in VHCF if real-life applications need to be analyzed; of particular interest should be the material behavior in variable amplitude loading, considering that mechanical components in many applications are subjected to both vibratory cycles and duty cycles. Observations from the fatigue tests in this work indicate that all failure started from particles within the 7075-T6 aluminum matrix. A model able to satisfactorily describe the fatigue behavior in LCF-HCF-VHCF was developed and used for fatigue-life predictions. Results show significant interaction effects. Future work will study interaction effects starting from microstructural observations to develop a material input or a damage accumulation model able to capture this interaction.

Acknowledgements

Special thanks are given to the United States Naval Air Systems Command (NAVAIR) for financial support of this study and to Nam Phan for serving as a technical point of contact.

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