# Fatigue Damage Evaluation of Aviation Aluminum Alloy Based on Strain Monitoring
## 2 Crystal Plasticity Modeling and Numerical Implementation  
### Theory of Crystal Plasticity  
The kinematic theory of the classical crystal plasticity model follows the pioneering work of [PERSON], whose exact mathematical theoretical model was successively developed by [PERSON] and [PERSON] (1972) [35], [PERSON] and [PERSON] (1977) [36], [PERSON] and [PERSON] (1983) [37], [PERSON] and [PERSON] (1985) [38], and others. The total deformation gradient tensor of a crystal, \(F\), can be decomposed into two parts, the elastic deformation gradient, \(F^{*}\), and the plastic deformation gradient, \(F^{p}\), viz:  
\[F=F^{*}F^{p} \tag{1}\]  
We define the plastic part \(L^{p}\) of the velocity tensor deformation tensor as follows:  
\[L^{p}=F^{p}{(F^{p})}^{-1}=\sum_{\alpha=1}^{n}\dot{\gamma}^{\alpha}s^{\alpha} \otimes m^{\alpha} \tag{2}\]  
where \(s^{\alpha}\) and \(m^{\alpha}\) refer to the direction of shear slip and normal to the slip plane in the reference configuration, respectively. \(n\) is the number of slip systems of the corresponding grain. \(\dot{\gamma}^{\alpha}\) denotes the slip shear rate on the \(\alpha\)th slip system and is summed over all activated slip systems. The elastic strain is defined as the Green-Lagrange strain tensor \(E^{e}\):  
\[E^{e}=\frac{1}{2}\Big{[}{F^{*}}^{T}F^{*}-I\Big{]} \tag{3}\]  
where \(I\) is a second-order constant tensor. The second type of Piola-Kirchhoff stress \(S\) is related to \(E^{e}\) by the following:  
\[S=C^{e}E^{e} \tag{4}\]  
where \(C^{e}\) is the fourth-order elastic constant tensor. The decomposed shear stress on the slip surface \(\alpha\) is then derived from the following:  
\[\tau^{\alpha}=S:(s^{\alpha}\otimes m^{\alpha}) \tag{5}\]  
Cauchy stress is derived as follows:  
\[\sigma=\det(F^{*})^{-1}F^{*}SF^{*T} \tag{6}\]  
The plastic shear rate of each slip regime obeys a power-law function of the resolved shear stress \(\tau^{\alpha}\)[39]:  
\[\dot{\gamma}^{\alpha}=\dot{\gamma}_{0}sign(\tau^{\alpha}-\chi^{\alpha})\Big{(} \frac{\tau^{\alpha}-\chi^{\alpha}}{g^{\alpha}}\Big{)}^{n} \tag{7}\]where \(\dot{\gamma}^{(\alpha)}\) is the reference shear strain rate, \(n\) is the rate sensitivity index related to the material properties, \(g^{\alpha}\) is the critical shear stress in the \(\alpha\)th slip regime for isotropic hardening, and \(\chi^{\alpha}\) is the back stress for kinematic hardening. When the material parameter \(n\) tends to \(\infty\), it is rate-independent [40]. The initial value of \(g^{\alpha}\) is denoted as the initial slip resistance \(g^{0}\), assuming uniformity across all slip systems. It is defined as follows:  
\[\dot{g}^{(\alpha)}=\sum_{\beta}h_{\alpha\beta}\gamma^{(\beta)} \tag{8}\]  
where \(h_{\alpha\beta}\) is the sliding hardening modulus due to self-hardening and potential hardening, which is defined as follows:  
\[h_{\alpha\beta}=q_{\alpha\beta}h(\gamma) \tag{9}\]  
where \(q_{\alpha\beta}\) is the self- and latent hardening ratio coefficient. When \(\alpha\
eq\beta\), \(h_{\alpha\beta}\) represents the latent hardening constant, and when \(\alpha=\beta\), \(h_{\alpha\alpha}\) represents the self-hardening constant. Here, \(h(\gamma)\) is defined as follows:  
\[h(\gamma)=h_{0}\;\text{sech}^{2}\bigg{|}\frac{h_{0}\gamma}{\tau_{s}-\tau_{0}} \bigg{|} \tag{10}\]  
where \(h_{0}\) represents the initial hardening modulus, \(\tau_{0}\) represents the yield stress which equals the initial critical resolved shear stress, \(\tau_{s}\) represents the stress at the limit of \(g^{\alpha}\), and \(\gamma\) represents the cumulative shear strain on all slip systems, which is calculated as follows:  
\[L^{p}=F^{p}{(F^{p})}^{-1}=\sum_{\alpha=1}^{n}\dot{\gamma}^{\alpha}s^{\alpha} \otimes m^{\alpha} \tag{2}\]0  
To describe the ratcheting behavior of single crystals, a hardening criterion is introduced on each slip system. This criterion incorporates a nonlinear dynamic recovery term, building upon the classical Armstrong-Frederick [41] hardening with a modification of the dynamic recovery term.