1. INTRODUCTION AND LITERATURE REVIEW
1.6. Important Parameters in Fretting Fatigue Evaluation
1.6.5. Stress-Strain Averaging Methods
Figure 32: Running Condition Fretting Map (RCFM) (A) and Material Response Fretting Map (MRFM) (B) damage (Reproduced based on [117⎼118])
Figure 33: Flowchart towards capturing effect of slip magnitude on fretting fatigue results
To capture the resultant stress gradient, there are different stress averaging methods considered in the literature [59⎼60, 75⎼77, 100⎼101, 112]. Using these methods, the stress averaging can be carried out around the contact surface nodes, over the certain critical dimension. By considering such averaged stress-strain values in the fretting fatigue initiation criteria, more realistic damage value can be obtained to get better experimental correlation. Figure 34 shows the schematic representation of different averaging methods.
Figure 34: Schematic representation of different stress-strain averaging methods damage (Reproduced based on [59])
Based on the corresponding domain considered for averaging, these averaging methods can be further classified into four methods, as given below.
▪ Surface method: This method considers the contact nodal stress values only. Often the results predicted with this method are highly conservative.
▪ Point method: In this method, stress-strain state at a point located at a certain distance inside the material along the critical plane is considered in the fretting fatigue life evaluation. Schematic representation of this method is similar to that of line method, as shown in Figure 35. Compared to the line method, here the averaging along the critical plane is not carried out.
▪ Line method: As shown in Figure 35, line method averages the damage parameter data (DP) for the nodes along the critical plane till critical dimension. The averaged damage parameter (DPavg) data is then considered to predict crack initiation life. This method can be considered as the simplest and most intuitive averaging method among the mentioned stress averaging methods [52].
▪ Volume/Area method: Volume/Area method is similar to the line method except here instead of line, the averaging of damage parameter is done over the selected volume/area.
The radius of the averaging volume/area can be evaluated in the same way as the critical dimension for line method is evaluated. Figure 36 shows the schematic representation of this method.
The critical dimension, 𝑟𝑐 [101], needed for stress-strain averaging is derived as
𝑟𝑐 = (1 𝜋⁄ )[∆𝐾𝑡ℎ⁄𝜎𝑓−1]2 (33) where 𝑟𝑐 is the critical dimension for stress-strain averaging, ∆𝐾𝑡ℎ is the crack threshold stress intensity factor range, 𝜎𝑓−1 is fatigue limit in tension.
Often it is observed that this critical dimension is generally of the similar order of the magnitude as the material’s grain size [112].
Figure 35: Line averaging method damage (Reproduced based on [52])
Figure 36: Volume/area averaging method damage (Reproduced based on [52])
Table 1: Experimentally observed relationship between critical dimension and failure cycles ( 𝑟𝑐 is the critical dimension for stress averaging, 𝑁𝑖 is the cycle to crack initiation)
[121]
Sr. No. Material Relationship
1 Al/4%Cu 𝑟𝑐 = 15.2𝑁𝑖−0.312
2 Al2024-T351 𝑟𝑐 = 12.6𝑁𝑖−0.231
3 Al7075-T6 𝑟𝑐 = 21.7𝑁𝑖−0.347
4 Ti-6Al-4 V, 𝑟𝑐 = 0.42𝑁𝑖−0.135
Table 2: Statistic results considering the ratio of estimated cycles and actual cycles [59]
𝑟𝑐 Surface
method
Point method
Line method
Area/
Volume method
Variable initiation
length method Mean
1 mm 0.027 1.102 1.133 1.067 0.913
0.5 mm 0.153 1.088 1.120 1.066
0.2 mm 0.428 1.053 1.080 1.038
Standard Deviation
1 mm 0.0087 0.395 0.371 0.364 0.286
0.5 mm 0.054 0.363 0.349 0.341
0.2 mm 0.151 0.326 0.323 0.309
Resultant value of the critical dimension 𝑟𝑐, from Equation 33, can be used in the point method (i.e., considering the stress at a point located at distance of 𝑟𝑐⁄2 from critical stress point), line method (i.e., averaging stress evaluated over a distance 2𝑟𝑐) or
volume/area method (i.e., averaging stress semi-circular area / half sphere of radius 𝑟𝑐) [120].
In another method for stress-strain averaging i.e., the variable length method, the critical dimension is not constant but is observed to be function of number of failure cycles. The resultant experimental relationship between the critical dimension for point method and the number of failure cycles is given in the Table 1, for various materials [121]. From Table 1, it is observed that the critical dimension decreases as the number of fatigue cycles increases and vice-versa. It is based on the theory of critical distance [122] considered in in the form of point method [123]. The observed relationship is thus a fatigue property of individual material and is based on the assumption that linear elastic constitutive law can be used to predict mechanical behavior for most of the engineering materials. Hence, the change in size of plastic zone near the notch tip with change in cyclic loadings depends upon the magnitude of cyclic loading. With the corresponding critical dimension, stress values are obtained using critical plane method and considered to evaluate resultant fatigue life. The predicted fatigue results found to be within the scatter band of + 3N.
Navarro et al. [59] have mentioned that with the above averaging methods, estimated fatigue life correlates better with the experimental life as observed through the corresponding mean and standard deviation values of the ratio of estimated fatigue life and experimental fatigue life. As per his study, volume method results are close to line method results. Also compared to the results obtained using variable initiation length method, they are on conservative sign, which is good from design perspective. Also, comparatively lesser dependence on averaging dimension is observed for the volume/area method. Thus, it indicates the averaging stress/strains over a critical dimension can contribute towards the better prediction of crack initiation life, especially in cases where high stress-gradients are present. Table 2 shows the comparative assessment of results obtained using different stress averaging methods.