ABSTRACT
In order to validate an existing energybased fatigue life prediction understanding, the strain energy accumulation for interrupted loading cycles was analyzed. The life prediction method being validated was developed based on the understanding that strain energy density accumulated during monotonic fracture is a physical damage quantity that is equal to total cumulative hysteresis strain energies in a fatigue process. If this understanding is true, it is possible to suspend cyclic loading for long periods during a fatigue procedure, and then resume the procedure to failure, resulting in the same fatigue life as if the fatigue test was conducted continuously to failure. This assumption, along with critical analyses such as surface roughness and loading frequency, is tested empirically on Titanium 6Al4V (Ti64) axial tensioncompression specimens in the lifetime regime of 5×103 and 6×104. The failure results are compared, with encouraging results, to the aforementioned energybased prediction method, thus validating the theory and the prediction capability.
INTRODUCTION
Fatigue life behavior is one of the most critical material properties required for gas turbine engine component design. The widely used design tools for characterizing fatigue life are the modified Goodman diagram and a Stress versus Fatigue Life (SN) curve [1, 2]. For an accurate characterization of fatigue life, tens to hundreds of experimental results are required to construct Goodman diagrams and SN curves. Depending on the desired lifing limit, significantly long time periods could be necessary to gather these experimental results. Therefore, a fatigue life prediction method that would require considerably less data and time than conventional empirical fatigue data would be an improvement to the aforementioned design tools.
In order to reduce the amount of empirical data necessary to construct a fatigue life design tool, the discovery of a physical fatigue damage quantity was required. The simplest way to attain a damage quantity that is cumulative with fatigue cycles is by exploring the correlation between fatigue life and energy. This correlation was studied as early as 1923 by Jasper [3]; however extensive/successful research on energy/failure correlation was not conducted until the second half of the 20th century. In 1955, Enomoto validated that under cyclic loading there exists a critical energy value where failure occurs, thus validating the existence of a physical damage quantity for fatigue [4]. This critical energy value was defined in later research efforts as the following: the accumulation of hysteresis plastic energy during fatigue, and the strain energy accumulated during monotonic fracture [5, 6, 7]. Both of these physical damage quantities have been used to predict fatigue life of materials such as steel, aluminum and 60Sn/40Pb solder [8, 9, 10, 11]. Also, modifications to the method proposed by Stowell has provided an energybased fatigue life prediction method capable of determining fatigue life of uniaxial bending and tensioncompression at various stress ratios [12, 13]
Based on the definition of the fatigue life damage quantity, it is fair to say that cumulative hysteresis damage is irreversible; thus, if cyclic loading is suspended for a significant time frame during fatigue testing, fatigue life will not be affect. This theory was tested for uniaxial tensioncompression loading of Ti64 at two distinct fullyreversed stress amplitude values. Also analyzed was the acceptability of the variation in the empirical fatigue life results associated with each of the two stress amplitudes. The results of the finding are presented in the proceeding sections.
EXPERIMENTAL PROCEDURE
Cyclic loading experiments on Ti64 were conducted on two different specimens: (1) a continuous radius specimen and (2) a uniform gagesection specimen, both shown in Figure 1 [14]. The continuous radius specimen is used for acquiring fatigue results because it is less susceptible than the uniform gagesection specimen to buckling at higher compressive stress magnitudes. The uniform gagesection, however, is more compatible with a MTS (Material Testing Systems) 634.12E24 model extensometer because they both have the same gage length (25.4mm); thus, it is used for low frequency, cyclic testing to acquire precise hysteresis stressstrain results.
Both the continuous radius and the uniform gagesection specimens were machined via waterjet on a 3.175mm thick Ti64 plate, where the geometry is based on recommendations from ASTM E466 standard for load controlled fatigue testing [14]. The chosen grain direction of each specimen was designated to be perpendicular to the eventual loading axis. No delicate postmachine polishing or stress relieving procedure was conducted on any specimen after the waterjet cut. The effect of this decision is analyzed later in the manuscript.
The fatigue tests were conducted on an axial MTS servohydraulic load frame with a 100KN load capacity. The load frame was controlled using a MTS TestStarIIs model controller, which stores load, displacement, and also strain via a MTS 634.12E24 model extensometer mounted on the specimen; however, the only data required by the TestStarIIs during fatigue testing was load amplitude and cycles to failure. The tensioncompression loads applied during the cyclic tests were fullyreversed at an operating frequency between the range of 515Hz, where the frequency is chosen based on the allowable stroke distance and load amplitude of the 100KN load frame performance chart [15].
Low frequency, cyclic tests were conducted on the same 100KN MTS load frame as the fatigue tests. The tensioncompression loads applied during the cyclic tests were fullyreversed at an operating frequency of 0.1Hz. This frequency was chosen because it provides the optimal rate for accurate hysteresis strain energy calculation without the effects of anelasticity [16]. The TestStarIIs controller was used to acquire time, load, strain and displacement data during testing at a rate of 2 points per second, which is 20 points per cycle.
Monotonic fracture results are a critical part of the energybased life prediction method. These results were attained from specimens that were waterjet cut, with no postmachine polishing or stress relieving procedure, from a 3.175mm thick Ti64 plate according to the recommendations by ASTM E8 standard for tension testing of metals [17]. Tests were conducted on the 100KN MTS load frame. Experiments were carried out at a displacement control rate of 0.0254 mm/sec. Data was acquired at a rate of 0.5 data points per second.
Figure 1. ASTM fatigue specimen dimensions (mm): (a.) Continuous radius, (b.) uniform gagesection.
FATIGUE LIFE ANALYSIS
Unlike Aluminum 6061T6 results from previous research [12, 13], Ti64 has a wide fatigue life scatter. Therefore, the effects of several material behaviors and trends, due to test setup, on fatigue life were analyzed. Though most of these trends were previously observed for Al 6061T6, it was important to show that the material behavior of Ti64 showed a consistent scatter and validated irreversible plastic energy accumulation during cyclic loading.
Frequency Effect: The first testing effect observed was the MTS load frame frequency versus fatigue life. This effect was observed for the fullyreversed stressamplitudes of 534MPa and 724MPa. These stress levels were chosen for two reasons: 1.) the values were high enough to avoid the region where the endurance limit phenomenon occurs, and 2.) time constraints prevented testing the theory of irreversible cumulative hysteresis damage at higher cycle counts. The energybased life prediction calculation determined that the expected life for stressamplitudes of 534MPa and 724MPa was approximately 4600 and 39000 cycles, respectively, and it was assumed that fatigue failure of each stress level would occur around those values regardless of the loading frequency. The analyzed frequencies were 15Hz, 10Hz and 5Hz for 534MPa, and 10Hz and 5Hz for 724MPa. These frequencies were chosen based on the allowable stroke distance of the 100KN load frame performance chart [15]. The result for each stress level is shown in the plots of Figure 2. Also, statistical results are shown in Table 1. Though the results of this analysis shows a wide scatter, primarily the 18% relative standard deviation of the 534MPa results, there is no noticeable trend based on loading frequency. Therefore, fatigue results at stress amplitudes above 700MPa, which are run at 10Hz or less, can be plotted with fatigue results at stress levels below 500MPa, which are conducted at 15Hz.
Figure 2. Frequency versus failure cycles: a.) 534MPa and, b.) 724MPa.
Table 1. Statistical results of frequency effect.

534MPa 
724MPa 
Lifetime STD = 
7859 
724 
Lifetime Mean = 
43545 
6643 
Rel. STD (%) = 
18.05 
10.91 
Load Effect: The applied stress amplitude for fatigue experiments are determined by measuring the crosssectional area of the specimen being tested. Based on variation with the waterjet machine tolerance (+/0.127mm), the +/ 0.2032mm manufacturer tolerance of the Ti64 plate thickness and slight variance in hand measurements, the possibility of a fatigue life trend with respect to the applied load amplitude was likely. This trend was observed for 534MPa and 724MPa. The graphical observation is shown on the plots of Figure 3, and the statistics are in Table 2. Like the frequency effect analysis, these results cannot verify that there is a fatigue life trend based on loading values.
Figure 3. Load amplitude versus failure cycles: a.) 534MPa and, b.) 724MPa.
Table 2. Statistical results of load effect.

534MPa 
724MPa 
Load STD (KN) = 
0.039 
0.231 
Load Mean (KN) = 
10.77 
14.44 
Rel. STD (%) = 
0.358 
1.599 
Surface Roughness Effect: Ti64 is a more sensitive material than Al 6061T6, which is the material the energybased life prediction method was developed upon. Though previous research shows there is no correlation between fatigue performance and surface roughness, this analysis was conducted by comparison with two postmachining surface polishing techniques and does not regard a nonpolished surface roughness [18]. Since there is no postwaterjet polish or LSG (Low Stress Grind) process for the fatigue specimens in this study, the surface roughness of each specimen have a higher probability of lacking consistency. Furthermore, handling the specimens (delivery, storage, etc.) could also induce unwanted residual stresses. Therefore, fatigue results of the three different batches of Ti64 specimens were compared; note, the batches are from the same Ti64 plate stock and are designated as specimens machined at three different times. The results, which are compared on the SN plot of Figure 4, show a consistent trend for fatigue behavior.
Figure 4. Fatigue results for Ti64.
Loading Delay Analysis: Previously stated, the energybased fatigue life prediction method was developed based on the understanding that a physical damage quantity exists for fatigue failure. The damage quantity is accumulated plastic strain energy from hysteresis loading, which is irreversible. In order to validate this understanding, fatigue analysis was conducted at the same stress amplitude values used in the previous analysis (534MPa and 724MPa). The scope of the analysis was to see if suspending cyclic loading during fatigue testing would affect fatigue life. The analysis was conducted to answer two key questions: 1.) will suspending cyclic loading present a noticeable trend between fatigue life and loading delay time, and 2.) will the fatigue life results from the delayed tests fall within the scatter of continuous cyclic loaded fatigue results? To answer the first question, empirical Ti64 results were viewed on the plots of Figure 5 for 534MPa and 724MPa. The results show no trend for both stress levels and minimal variation at 724MPa.
In order to see if fatigue life results fell within acceptable Ti64 scatter, all fatigue data was plotted on the normal distribution constructed by fatigue data of continuous cyclic loading results [19]. Since fatigue life shows no noticeable trend with respect to loading frequency, fatigue data at different frequencies were used in the construction of the normal distribution. The results for both stress levels are shown on the plots of Figure 6 and 7. Both stress levels show that all but one of the time delayed fatigue life data falls within onesigma of the distribution. Therefore, this validates that suspending cyclic loading for significant time frames during fatigue testing will not affect fatigue life and the notion of a physical damage quantity for fatigue still holds true.
Figure 5. Ti64 fatigue life comparison with load delays: a.) 534MPa and, b.) 724MPa.
Figure 6. Normal distribution of fatigue life at 534MPa.
Figure 7. Normal distribution of fatigue life at 724MPa.
ENERGYBASED LIFE PREDICTION
Data from the fatigue analysis section is used here to validate the capability of the energybased fatigue life prediction method. This method was developed from the stressstrain representations of Equation (1) – (3) [7, 12]. Equation (1) represents the monotonic stressstrain relationship, Equation (2) represents the expression for the parameter ao in Equation (1), and Equation (3) is the expression for the cyclic strain. Equation (3) was created based on a simplified coordinate system, where the horizontal versus vertical axes represents peaktopeak strain versus peaktopeak stress, respectively. On this coordinate system, shown in Figure 8, the origin is defined as the minimum fullyreversed point of a hysteresis loop; in other words, both the stress and the strain values are read from zero to peaktopeak magnitudes.
The parameters for Equation (1) – (3) are defined as follows: cris the nominal applied monotonic stress value, s is the strain corresponding to the applied monotonic stress, app is the generalized/peaktopeak stress value corresponding to the generalized/peaktopeak cyclic strain scycie (2aa replaces app in Equation (3) after all necessary derivations), af is the fracture stress, sf is the ductility, ay is the yield stress, E is the modulus of elasticity, and the variables ctc, ct0, so, and C are curve fit parameters [12]. The curve fit parameters for the cyclic and monotonic representations are statistically acquired by comparison between the equations and the respective experimental results [20, 21].
The energybased prediction method calculates fatigue life by dividing the total monotonic strain energy density by the strain energy density for one cycle. The total strain energy density accumulated during a monotonic process is determined as the area underneath the curve constructed by Equation (1), and the strain energy density for one cycle is represented by the area within the hysteresis loop formed by Equation (3). Calculating the monotonic strain energy density from experimental results is a straightforward task, whereas the strain energy density in one cycle is determined by making an assumption that the tensile stressstrain behavior of the hysteresis loop is the same as the compressive behavior. This assumption is a simplification for the strain energy density per cycle calculation because Bauschinger effect shows that the tensile and compressive behaviors in a hysteresis loop are not identical [22]. The resulting hysteresis strain energy regarding the simplification, which is calculated by Equation (4), shows a minor deviation with Al 6061T6 data when compared with the correlating strain energy that incorporates Bauschinger Effect on Figure 9.
The energybased fatigue life calculation is represented by Equation (5) [23], where aa is the applied stress amplitude. Using the material parameters in Table 3, Equation (5) results compare well to the empirical Ti64 fatigue results in Figure 10.
Figure 8. Energybased hysteresis loop schematic.
Figure 9. Hysteresis energy comparison.
Figure 10. Energybased life prediction comparison for Ti64.
Table 3. Material parameters for Ti64.
1.80E+05 

113.6 

6.94E10 

4.53E01 

345.7 

1161 

66.24 

1029 
CONCLUSION
It has been proven that hysteresis strain energy density is irreversible due to the analysis showing that suspending cyclic loading during fatigue testing of Ti64 does not affect the expected fatigue life in the lifetime regime of 5×103 – 6×104. Other effects that do not show a noticeable trend on fatigue life are surface roughness and loading frequency. Although these assumptions, along with the omission of Bauschinger effect, have been used in energybased theory prior to this manuscript, validating the assumption and constructing a wellcompared energybased life prediction for Ti64 fatigue has been encouraging for the future direction of energy/failure correlation studies.