## 2 Experimental Methods and Simulations
### Simulation Methods
#### 2.2.1 Crystal Plasticity Models
\begin{split} g=\left[\begin{array}{cccc}\cos\varphi_{2}&\sin \varphi_{2}&0\\ -\sin\varphi_{2}&\cos\varphi_{2}&0\\ 0&0&1\\ \end{array}\right]\left[\begin{array}{cccc}1&0&0\\ 0&\cos\phi&\sin\phi\\ 0&-\sin\phi&\cos\phi\\ \end{array}\right]\left[\begin{array}{cccc}\cos\varphi_{1}&\sin\varphi_{1}&0 \\ -\sin\varphi_{1}&\cos\varphi_{1}&0\\ 0&0&1\\ \end{array}\right]\\ =\left[\begin{array}{cccc}\cos\varphi_{1}\cos\varphi_{2}-\sin\varphi_{1} \sin\varphi_{2}\cos\phi&\sin\varphi_{1}\cos\varphi_{2}+\cos\varphi_{1}\sin \varphi_{2}\cos\phi&\sin\varphi_{2}\sin\phi\\ -\cos\varphi_{1}\sin\varphi_{2}-\sin\varphi_{1}\cos\varphi_{1}\cos\phi&-\sin \varphi_{1}\sin\varphi_{2}+\cos\varphi_{1}\cos\varphi_{2}\cos\phi&\cos \varphi_{2}\sin\phi\\ \sin\varphi_{1}\sin\phi&-\cos\varphi_{1}\sin\phi&\cos\phi\\ \end{array}\right]\end{split}  \tag{1}\]  
(3) Start the simulation under the current loading step and the mechanical response behavior of the model is obtained. Based on the projection relationship of the strain increment of the grain, the change of the Euler angle is calculated, which is added to the original Euler angle to represent the rotation of the grain. (4) In the crystal coordinate system, the total stress \(\Sigma\sigma_{c,ij}\) is obtained by accumulating the stress increments calculated from the previous steps. The \(\tau_{RSS}\) is defined as:  
\[\tau_{RSS}=\sum_{i}\sigma_{c,ij}\times\cos\varphi_{1}\times\cos\varphi_{2} \tag{2}\]  
Figure 2: Process of developing the crystal plasticity finite element simulation method (CPFEM): (**a**) the Electron Backscatter Diffraction (EBSD) inverse pole figure (IPF) mapping, (**b**) 2.5D finite element model, (**c**) boundary condition, (**d**) displacement-time curve. where \(\varphi_{1}\) and \(\varphi_{2}\) are respectively the angles between the normal direction and the tangential direction of the slip surface and the \(\Sigma\sigma_{c,ij}\) direction, \(c\) is the crystal coordinate system. The activation of slip systems is determined by comparing with the CRSS, \(\tau_{0}\). If \(\tau_{RSS}>\tau_{0}\), the slip system is activated and the grain enters the plastic deformation stage. With the activation of slip systems, the CRSS changed because of the strain hardening:  
\[\tau^{i}=\frac{d\tau^{i}(\Gamma)}{d\Gamma}\sum\limits_{j}h^{ij}\gamma^{j} \tag{3}\]  
where \(\tau^{i}\) is the increment of CRSS, \(\gamma^{j}\) is the shear strain rate of the slip system \(j\), \(\Gamma\) is the accumulating increment of shear strain of the slip system, \(h^{ij}\) is the hardening coefficient. (5) In this simulation, it is considered that plastic deformation is caused by slip and the plastic strain \(\varepsilon_{c}^{p}\) is the cumulative value of resolved shear strain of all slip systems. According to the state of the grain, it is determined that the grain is in the elastic stage or plastic stage. After that, the instantaneous stiffness matrix is obtained, and the total stiffness coefficient matrix can be calculated:  
\[\sigma_{c}=\left\{ \begin{array}{c}L_{c}^{e}\times\varepsilon_{c}^{e}\\ L_{c}^{e}\times\varepsilon_{c}^{e}+L_{c}^{p}\times\varepsilon_{c}^{p}\end{array} \right. \tag{4}\]  
where \(L\) is the stiffness coefficient matrix, \(p\) and \(e\) represent respectively in the plastic stage and elastic stage. (6) Use the updated constitutive parameters to continue the next step of the finite element calculation until the last step.