# Fretting Fatigue Experiment and Finite Element Analysis for Dovetail Specim at High Temperature
## 3 Numerical Simulation Method  
### Modeling of Crystal Plastic  
In previous research, the majority of the constitutive model was linear elastic, and thus the material's plastic behavior was ignored [27]. Plastic characteristics, on the other hand, have been thought to have a significant impact on fretting wear and fracture [28]. The theory of crystallographic plastic was added to the ANSYS program as a supplement to the analysis of finite element contact strength in the present research. The rate-dependent constitutive formulation [33; 34] is taken up with continuum frame theory to describe the dislocation slip mechanism quantitatively. The multiplicative decomposition of the deformation gradient in Figure 4 shows a free stress configuration during the deformation process of the deformable body from the reference configuration to the current configuration. This free stress configuration contains only the deformation caused by the plastic slip, which is the intermediate configuration. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Temperature & 630 \({}^{\circ}\)C \\ Load amplitude & 13,500 N,14,956 N,16,451.6 N,17,947.2 N \\ Load ratio & 0.05 \\ Frequency & 1 Hz \\ Load spectrum type & Sine wave \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameters for the fretting fatigue test. The intermediate configuration only contains plastic slip deformation, and the dislocation plastic slip does not change the orientation of the crystal grains, which can be described as:  
\[\overline{g}^{\alpha}=s_{0}^{\alpha},\ \overline{m}^{\alpha}=m_{0}^{\alpha} \tag{1}\]  
where \(\overline{g}^{\alpha}\) and \(\overline{m}^{\alpha}\) are the slip direction and the normal direction of the slip surface in the intermediate configuration, respectively. The subscripts of \(s_{0}^{\alpha}\) and \(m_{0}^{\alpha}\) correspond to those vectors in the original configuration. The superscript \(\alpha\) represents the number of the slip system. In the current configuration, the orientation of the slip system evolves as follows:  
\[\mathbf{s}^{\alpha}=F^{*}.\mathbf{s}_{0}^{\alpha},\ \mathbf{m}^{\alpha}=\mathbf{m}_{0}^{\alpha}.F^{*}{}^{-1} \tag{2}\]  
where \(s^{\alpha}\) and \(\mathbf{m}^{\alpha}\) are the slip direction and the normal direction of the slip surface in the current configuration. \(F^{*}\) represents the elastic part of the deformation gradient. It can be seen that the crystal plasticity model based on the current configuration needs to update the slip system orientation for each constitutive iteration. In contrast, the model based on the intermediate configuration does not need to perform this calculation, which significantly reduces the scale of constitutive calculations. Most of the crystal plastic models established in the current configuration are the linear elastic relationship between the stress rate and the deformation rate tensor, as shown in the following formula [14]:  
\[\tau^{(\alpha)}=P^{(\alpha)}:T \tag{3}\]  
where \(P^{(\alpha)}\) is the Schmid factor of slip system \(\alpha\) and \(T\) represents the second Piola-Kirchhoff stress tensor relative to the intermediate configuration. The strain rate was expressed using a power-law relationship:  
\[\dot{\gamma}^{(\alpha)}=\dot{\gamma}_{0}^{(\alpha)}\left[\frac{\tau^{(\alpha) }}{g^{(\alpha)}}\right]\left[\left|\frac{\tau^{(\alpha)}}{g^{(\alpha)}}\right| \right]^{(1/m)-1} \tag{4}\]  
where \(g^{(\alpha)}\) is the slip system strength or shear resistance, \(m\) is the strain rate sensitivity exponent, and \(\dot{\gamma}_{0}^{(\alpha)}\) represents the reference shear strain rate. The current crystal state of strain hardened is characterized by functions \(g^{(\alpha)}\). Simply, each \(g^{(\alpha)}\) is taken to depend on the sum of the slip magnitudes, i.e.,  
\[g^{(\alpha)}=g^{(\alpha)}(\gamma) \tag{5}\]  
Figure 4: Schematic diagram of the multiplicative decomposition of the deformation gradient. where \(\gamma=\sum\limits_{a}|\gamma^{(a)}|\).