## 2 Materials and Methods
### Digital Volume Correlation  
DVC is a measuring tool for detecting the internal displacement field of material based on volume images. Its basic principle is to take the volume image before deformation as the reference image and the volume image after deformation as the target image. The 3D displacement vector is calculated by tracking the position of each point in the reference image after deformation. In order to accurately track the displacement of a certain point P (x, y, z), it is necessary to take a small cube (sub-volume) centered on this point. The displacement components u, v, and w of the sub-volume center point can be determined by tracking the selected position P (x\({}^{\prime}\), y\({}^{\prime}\), z\({}^{\prime}\)) of the deformed sub-volume in the target image.  
The subset-based local DVC method selected in this text is the spatial domain algorithm. In order to obtain the displacement of sub-voxel precision, the non-linear iterative algorithm is usually adopted. The non-linear problem gradually converges from the initial deformation vector to the local optimal extremum (the deformation vector that makes the reference and the deformation sub-volume match best) through this algorithm. An iterative optimization algorithm (IC-GN algorithm) is used to optimize the optimization function (ZNSSD) to obtain sub-pixel displacement.  
The most important parameters that affect the calculation accuracy of displacement are the subset size and shape function [24]. The subset size, also known as the sub-volume size, has a direct impact on the precision of the DVC measurements [25]. The larger the subset size is, the greater the consumption of calculation resources and time; thus, the deformation at the smaller scale can be ignored. When a small subset size is selected, the calculation time decreases and errors occur; therefore, the subset size cannot be too large or too small. The subsets of 8 \(\times\) 8 \(\times\) 8 voxel, 16 \(\times\) 16 \(\times\) 16 voxel, and 32 \(\times\) 32 \(\times\) 32 voxel were selected for calculation in [26]. The results showed that as the subset size increased, the mean error and standard deviation dropped, and the DVC computation results were more precise and accurate. The calculation inaccuracy was considerable when the subset size was 8 \(\times\) 8 \(\times\) 8 voxel. The increased size of the subset will have an impact on the measurement outcomes. The subset's size should be chosen to be as minimal as possible while maintaining accuracy. The subset size selected in this paper is 31 \(\times\) 31 \(\times\) 31 voxel. The shape function should be able to accurately describe the local deformation of the target subset, including the zero-order function, the first-order function, and the second-order function [27]. The more commonly used first-order function is selected in this text, as shown in Equation (1):  
\[\begin{bmatrix}U_{1}(\Delta x,\Delta y,\Delta z)\\ V_{1}(\Delta x,\Delta y,\Delta z)\\ W_{1}(\Delta x,\Delta y,\Delta z)\end{bmatrix}=\begin{bmatrix}u\\ v\\ w\end{bmatrix}+\begin{bmatrix}u_{x}&u_{y}&u_{z}\\ v_{x}&v_{y}&v_{z}\\ w_{x}&w_{y}&w_{z}\end{bmatrix}.\begin{bmatrix}\Delta x\\ \Delta y\\ \Delta z\end{bmatrix} \tag{1}\]  
where \(\Delta x\), \(\Delta y\), and \(\Delta z\), respectively, represent the distance from point (x, y, z) to the center point (x\({}_{0}\), y\({}_{0}\), z\({}_{0}\)) of the reference image subset in X, Y, and Z directions; \(u\), \(v\), and \(w\) are the displacement of the subset center in X, Y, and Z directions; and \(u_{x}\), \(u_{y}\), \(u_{z}\), \(v_{x}\), \(v_{y}\), \(v_{z}\), \(w_{x}\), \(w_{y}\), and \(w_{z}\) are the displacement gradient of the subset, which is also called sub-voxel precision displacement. The accuracy of the sub-voxel displacement measurement is better than 0.01 voxel.