# Fretting Fatigue Behavior under Tension-Bending Mixed-Mode Loading
## 3 Numerical Model and Method
### Fretting Damage Initiation Prediction Method  
High-gradient multiaxial and non-proportional stress produced by fretting fatigue will cause the fatigue life of the component to be lower than the fatigue limit of the material, which presents a great challenge to predict the fretting damage initiation. Several multiaxial fatigue criteria such as the critical plane approach, continuum damage mechanics approach, and stress invariance approach have been developed [36; 37; 38; 39] and successfully applied to the numerical prediction of fretting fatigue initiation. Among them, the critical plane approach, which is based on historical stress-strain parameters and agrees well with many experimental observations, has been extensively used to predict the crack initiation location,  
Figure 5: Dimensionless shear stress, \(q(x)/E_{1}\), at the contact surface for different mesh sizes.  
Figure 6: Comparison between FEM result and Hertz analytical solution of the contact pressure.  
angle, and life of fretting fatigue. The key to the critical plane approach is to determine the maximum fatigue damage plane of the material based on the selected fatigue parameters. The maximum value of the damage parameter in conjunction with some material constants is used to calculate the fatigue life and takes the peak value point as the crack initiation position. Different categories of damage parameters are suitable for different fatigue mechanisms (high-cycle fatigue or low-cycle fatigue) and different failure modes, and the classification of the categories is based on different combinations of mechanical parameters.  
Tension-bending mixed-mode fretting fatigue is numerically assessed by two critical plane parameters--McDiarmid (MD) [40] and Smith-Watson-Topper (SWT) [41]--to predict the fretting damage initiation location and to estimate the extent of the fretting damage. [PERSON] used two shear fatigue limits: one for the case where cracks grow parallel to the surface (Case A), and another for the case where cracks grow towards the surface (Case B). In our model, the torsional fatigue limit \(\tau_{f-1}\) was used, following the assumption of the original MD model in which cracks initiate parallel to the surface (Case A), which is consistent with the mechanism of fretting fatigue crack initiation. The stress-based criterion parameter MD is suitable for materials whose failure is dominated by tensile-mode cracking, while the energy-based criterion parameter SWT is suitable for materials whose failure is dominated by shear-mode cracking. The value of these two fatigue parameters can be calculated following Equations (5) and (6):  
\[MD=\frac{\Delta\tau_{\max}}{2}+\bigg{(}\frac{\tau_{f-1}}{2\sigma_{u}}\bigg{)} \sigma_{n,\max} \tag{5}\]  
\[SWT=\bigg{(}\sigma_{\max}\frac{\Delta\varepsilon}{2}\bigg{)}_{\max} \tag{6}\]  
where \(\frac{\Delta\tau_{\max}}{2}\) is the maximum shear stress amplitude, and \(\sigma_{n,\max}\) is the maximum normal stress perpendicular to the direction of maximum shear stress amplitude. Material constants \(\tau_{f-1}\) and \(\sigma_{u}\) are the torsional fatigue limit and ultimate tensile strength, respectively, and their values are derived from the literature [13] for Al2024-T351. The \(\sigma_{\max}\) is the maximum normal stress on a plane of the material point, and \(\Delta\varepsilon\) is the maximum strain range on the same plane. The predicted fretting damage initiation life can be calculated by the damage parameter combined with Basquin's equation, and the extent of the component under fretting fatigue is inversely related to the value of the fretting damage parameter [1, 12, 13].