Use of Strain-life Models with Wavelet Bump Extraction ( WBE ) for Predicting Fatigue Damage ABSTRACT

Shah rum Abdullah" Choi Jae-ChiI2,John R. Yates and Joseph A. Giacomin lJabatan Kejuruteraan Mekanik dan Bahan, Fakulti Kejuruteraan & Alam Sina, University Kebangsaan Malaysia 43600 UKM Bangi, Selangor, Malaysia 2Hyundai Motor Company, South Korea 3Department of Mechanical Engineering, University of Sheffield, United Kingdom Received Date: 28th August 2006 Accepted Date: 7th March 2007 This paper presents the use of strain-life fatigue damage models to observe the cycle sequence effects in the wavelet-based fatigue data editing algorithm. This algorithm is called Wavelet Bump Extraction (WBE), which was developed to produce a shortened signal by extracting fatigue damaging events from the original signal with the retention of the original cycle sequences. Current industrial practice uses the Palmgren-Miner linear damage rule to predictthefatigue life orfatigue damage under variable amplitude (VA) loadings. Using VA loadings, however, this rule does not have load interaction accountability in the analysis.Thus,a more suitable approach has been identified for predicting fatigue damage of VA loadings, i.e. the Effective Strain Damage (ESD) model. In this study, the cycle sequence effect observation was implemented in both analytical and experimental works using the WBE extracted VA loadings. The study includes the comparison between the experimental and the analytical (using four strain-life fatigue damage models: Coffin-Manson, Morrow, Smith-Watson-Topper and ESD) fatigue damage. The smallest average in the fatigue damage difference was found when using the ESD strain-life model, suggesting the suitability of the this model for analysing VA fatigue loadings.


INTRODUCTION
Fatigue life prediction is important in the design process of vehicle structural components, with the essential input variable for fatigue is the load history.Practically, automobile manufacturers go to great lengths to instrument vehicles and subject them to a variety of driving conditions.By necessity, vehicle development requires accelerated fatigue testing and this is often accomplished by correlating test tracks with public road data.Both roads and test tracks generate variable amplitude (VA) load time histories.Loads that are predicted to do little or no damage can be eliminated and large amplitude cycles are retained.The process can be performed using a wavelet-based fatigue data editing,known as Wavelet Bump Extraction (WBE) algorithm which was developed by Abdullah et al. (2003;2004) with the retention of the loading cycle sequences.
The situation where the order of loading affects the fatigue life is called a sequence effect.Sequence effects exist both in the crack initiation stage and crack propagation stage (Fuchs and Stephens 1980).Sequence effects are also related to overload and underload condition in VA loadings (DuQuesnay et al. 1993;Topper and Lam 1997).When overloads were inserted in the small cycle or below the material fatigue limit, the small cycles following the overloads contributed to the fatigue damage accumulation.Considering the importance of sequence effects in the fatigue damage prediction under VA loadings (Dowling 1999), therefore, a suitable approach was introduced (DuQuesnay et al. 1993) in order to have closer fatigue life predictions when compared to the respective experimental findings.
In this paper the cycle sequence effects are analysed using the VA loadings extracted by the WBE algorithm.The analytical fatigue lives using four strain-life models were compared to experimental findings in order to observe the load interaction effects in fatigue damage prediction.To date, there are no current studies that are related to the analysis of cycle sequence effects using the loadings extracted from fatigue data editing algorithms.This paper, finally, discusses one of the stages for validating the WBE effectiveness, i.e. fatigue cycle sequence effects observation.

BACKGROUND OF THE WAVElET BUMP EXTRACTION (WBE) ALGORITHM
The Wavelet Bump Extraction (WBE) algorithm was designed to identify and extract fatigue damaging events in orderto produce a shortened loading.This shortened loading should have equivalent fatigue damage and global signal statistical values to the original input loading.Using WBE, large amplitude segments were extracted with the retention of the original load cycle sequences.WBE has three main stages (Figure 1): the wavelet decomposition process, the identification and extraction of the fatigue damaging events, and the production of a mission signal.
WBE uses the orthogonal wavelet transform by means of the 12th order Daubechies wavelets which were chosen as the basis functions.Each wavelet level describes the time behaviour of the signal in a frequency band.Using the WBE algorithm,fatigue damaging events are identified in wavelet groups.A wavelet grouping stage permits the user to cluster wavelet levels into a single region of significant signal vibrational energy.A bump, which is an oscillatory transient with a monotonic decay envelope either side of a peak value (Figure 2a), is identified in each wavelet group by means of an automatic trigger level (Figure 2b).Bump identification is performed by means of a search which identifies the points at which the signal envelope inverts from decay behaviour.After all the bumps ~re identified in all the wavelet groups, b~mp segments are extracted (Figure 2c) by removing the original time history of the complete section between the start and the end of the bump.The extracted bump segments are combined to produce a mission time history which has equivalent signal statistics and fatigue damage potential of the original loading.

FATIGUE LIFE PREDICTION
Current industrial practice for fatigue life prediction is to use the Palmgren-Miner (PM) linear damage rule (Palmgren 1924;Miner 1945).This rule is normally applied with strain-life fatigue damage models for analyzing shorter fatigue life problems.The strain .. life fatigue life ' behaviour considers plastic deformation that occurs at the localised region where fatigue cracks begin with the influence of a mean stress.
The first strain-life model is the Coffin-Manson relationship (Coffin 1954;Manson 1965) and is defined as where E is the strain amplitude, a is the stress where am is the mean stress.The SWT strain-life model is mathematically defined as (Smith et al. 1970) where a max is the maximum stress.
Several limitations were found in the implementation of the PM linear damage rule.The fatigue damage is accurately calculated for constant amplitude (CA) loadings, but it may lead to the erroneous prediction for VA loadings (Dowling 1999).It is because the PM linear damage rule assumes no load sequence effect and lacks load interaction accountability.Considering the importance of sequence effects in VA loadings, therefore,a suitable approach was identified.Several studies related to the fatigue life prediction on metal components under VA loadings have been carried out by Fatemi and Yang (1998), such as a model derived from random vibration theory, the fracture mechanics approach, etc.In spite of better improvement in fatigue life prediction contributed by these models, however, they were difficult to associate for general use.
The use of overload events with a simple linear damage model (DuQuesnay et al. 1993) has been found to be suitable for this purpose.This strainli,fe fatigue damage model, or known as Effective Strain Damage (ESD), is based on crack growth and crack closure that works well for a wide variety of materials, load spectra, component geometries, strain magnitudes and mean-strain effects.This model was developed for safe-life or i life to crack detection, so that fatigue damage can be analysed based on the short crack growth.
The ESD model is mathematically defined as follows: where E is the elastic modulus of the material, AE* is a net effective strain range for a closed hysteresis loop that is related to fatigue crack growth.A and B are the material constants, and Nf is the number of cycles to failure.The magnitude of EAE* for a given cycle is a function of crackopening stress, Sop, level and it is dependant on the prior stress and strain magnitudes in the loading history.The expression of E,1.E* can be expanded to where Em •• and Eop are the maximum strain and the crack-opening strain of the particular cycle, respectively.E; is the intrinsic fatigue limit strain range under the VA loading condition.
In order to consider the cycle sequence effects in the fatigue life calculation, a decay parameter, m, is used to define th,f change in a crack-opening stress between two adjacent cycles.AS was first op defined as where 5 is the current opening stress and S is   The damage caused by each cycle of the repeated loading is calculated by reference to the material strain-life curve.The N f value can be obtained from Equation ( 1) -( 4) for all listed strain-life models.Finally, the fatigue damage (D)  potential for one cycle is defined as

APPLICATION OFTHE WBE ALGORITHM USING VARIABLE AMPLITUDE LOADINGS
The effectiveness of the WBE algorithm was evaluated using a VA strain loading which was measured on a van suspension arm while driving over a pave test track.The type of road surface is shown in Figure 3.The suspension arm pave test data set was chosen because it contained many small amplitude, high frequency and transient events in the signal background.The signal was sampled for 23,000 discrete points at a sampling rate of 500 Hz. Figure 4a shows a plot of the acquired 46-second time history.Using the WBE algorithm the bump segments that produced the majority of fatigue damage were identified and extracted, so as to achieve the mission signal which matched the original data within ± 10% in terms of root-mean-square and kurtosis value (Figure 4b).The segments were combined to produce the 18.8-second mission signal (Figure 4c).

LABORATORY FATIGUE TESTING AND RESULTS
The material chosen for the test sample was BS 080A42 steel, which is often used in the suspension components of passenger cars.Smooth speCimens were used forthe fatigue tests and the specimen geometry is shown in Figure 5.The speCimens were hand-polished with 60-1000 grades of silicone carbide abrasive papers and finished with 6-~m diamond compound in order to produce anhourglass profile.A servo-hydraulic test machine was used in displacement control mode for all tests.
A strain-life curve, as shown in Figure 6a, was obtained from CA fatigue tests at seven different strain amplitudes.Using the CA fatigue test data and a tensile test data, the mechanical properties is presented in Table 1.The cyclic mechanical properties were obtained using CA fatigue test data with Equation ( 1) and (4).Uniaxial VA fatigue

DISCUSSIONS
Four strain-life models were used for the analysis of fatigue damage using variable amplitude loadings, which is the subject of this paper.The models are the Coffin-Manson relationship, the Morrow mean stress correction effect, the SWT mean stress correction effect and the ESD model.The first three are suitable for the conventional .analysiswith the CA loading.These models are not suitable for analysing VA loadings,as the load interaction effects are not accounted for in the fatigue life prediction.Despite this, they are often used for VA problems in practice.To properly treat VA loading histories, the ESD strain-life model was developed in order to perform for life-to-crack detection.In the ESD model the damage parameter that is treated is the effective strain range.The fatigue damage IS analysed based on short crack growth concepts to incorporate retardation by changing crack closure levels.In order to calculate fatigue life using the ESD model for a VA loading history, the fatigue cycles should be reconstructed based on the original position in the time history.This reconstruction procedure is required in order to retain the original load cycle sequences present in the original VA loading.Cycle reconstruction is not required for CA loadings, since the cycle sequences are not affected in the fatigue life prediction using this type of 10ading.The fatigue cycle reconstruction consists of converting a time history into a series of peak-valley reversals.These reversals were then rainflow counted (Matsuishi  The cycles were then sorted based on the peakvalley history in order to produce a similar pattern to the original load sequences.For example, a reconstructed cycle history is shown in Figure 7 for the first bump segment (81) of the signal.
Table 2 shows the fatigue damage values of ten VA loadings calculated using four strainlife models.In addition, the findings obtained from the laboratory fatigue testing were also presented in this table.From the information in Table 2, the calculated fatigue damage for each bump segment and mission signal was compared to the respective experimental value.The total average fatigue damage difference (AD) between the Morrow strain-life model and experiment gave the highest value of 566%.Using the ESD model, the smallest difference was found to be at 21 %.These results show a better accuracy of the ESD model to predict the fatigue damage of VA loadings compared to the other models.
', is the fatigue strength coefficient, E is the modulus of elasticity, N, is the number of cycles to failure, b is the fatigue strength exponent, E', is the fatigue ductility coefficient and c is the fatigue ductility component.Some of the realistic service situations involve non-zero mean stresses.Two mean stress effect models are used in the strain-life fatigue damage analysis, i.e.Morrow and Smith-Watson-Topper (SWT) strain-life models.Mathematically, the Morrow's model is defined by (Morrow 1968) a ' ( a) Ea =-t 1-a~ (2N,t +E ' ,(2N,f(2) ru D the steady-state opening stress.S is defined as cu the 5 value of the previous cycle.5 is defined op 55 as S = as (1_(smax)2J+ RS .
a and ~ are the material constants, 5 is max the maximum stress of the previous largest cycle in the time history,5 . is the minimum stress mm of the previous largest cycle and 5 is the cyclic • yYield stress. 37

FIGURE 3 .
FIGURE 3. The pave test track used to measure fatigue road loading

FIGURE 4 .
FIGURE 4. Plot of time histories: (a) Experimetally measured VA loading, (b) Extracted bump segments with the bump segment label numbers, (c) The mission road loading

FIGURE 6 .
FIGURE 6.(a) Experimental strain-life curve, (b) Experimental fatigue lives Cycle Sequence of the PV History

FIGURE 7 .
FIGURE 7. Time history reconstruction for the first bump segment (B 1) of the signal FIGURE 8. Fatigue damage correlation between the calculation and the experiment using the WBE extracted loadings

CONCLUSION 1 .
The Effective Strain Damage (ESD) strain-life model was identified as a suitable strain-life model accounting for the cycle sequence effects.The average fatigue life difference 3. The combination of WBE and ESD provide a novel wavelet-based fatigue data editing technique which accurately extracts the most .fatigue damaging events (bump segments) and preserves of the original load cycle sequence within the fatigue loadings.Professor D.l.DuQuesnay (Royal Military College of Canada) for all their supports.A Material constant for the ESD strain-life model [

TABLE 1 .
Mechanical properties of BS 080A42 steel

Mechanical Properties Cyclic Mechanical Properties
Ultimate tensile strength, Su[MPa]