# Multiscale Damage Evolution Analysis of Aluminum Alloy Based on Defect Visualization
## 4 Damage Model  
### Establishment of an Equivalent Damage Model  
It was difficult to conduct a damage evolution analysis and reduce the calculation time due to the shape and position of the plenty of mesoscopic defects inside the specimen. Therefore, a simplified equivalent multiscale damage model, which could reflect both mesoscopic damage and macroscopic fractures based on the grading of an internal defect, was established. The equivalent mesoscopic void defects volume \(V_{n}\) and position information \(P_{n}\) were calculated based on original damage voids feature information. The volume of voids appeared in three orders of magnitude (\(10^{-6}\) mm\({}^{3}\), \(10^{-7}\) mm\({}^{3}\), and \(10^{-8}\) mm\({}^{3}\)), which were, respectively, recorded as level 1, level 2, and level 3. The volume and position information of \(n^{th}\) void for level 1 defects were recorded as \(V_{1n}\), \(P_{1n}\), and the remaining level 2 and level 3 were, respectively, recorded as \(V_{2m}\), \(P_{2m}\) for \(m^{th}\) void and \(V_{3q}\), \(P_{3q}\) for \(q^{th}\) void. The volume of the level 3 voids (\(10^{-8}\) mm\({}^{3}\)) was averaged according to Equation (1) to obtain the average volume \(\overline{V_{3}}\) since the volume of the level 3 voids was small. The level 1 voids were selected as base points, which were the main points with volume and location information; the level 2 and level 3 voids were selected as the reference points; the smallest volume in the level 3 voids was eliminated based on the effect of size on defect. Three levels (level 1, level 2, and level 3) of voids position information were placed in matrices \(P_{1}\), \(P_{2}\), and \(P_{3}\), respectively. Matrix \(P_{1}\) was a position matrix for the base points, and matrices \(P_{2}\) and \(P_{3}\) were recorded as the position matrices for the reference points, respectively, as shown in Equation (2). \[\overline{V_{3}}=\frac{\sum\limits_{i=1}^{q}V_{3i}}{q} \tag{1}\]  
\[P_{1n}=\left[ \begin{array}{ccc}x_{11}&y_{11}&z_{11}\\ x_{12}&y_{12}&z_{12}\\ \vdots&\vdots&\vdots\\ x_{1n}&y_{1n}&z_{1n}\end{array} \right]=\left[ \begin{array}{c}p_{11}\\ p_{12}\\ \vdots\\ p_{1n}\end{array} \right],P_{2}=\left[ \begin{array}{ccc}x_{21}&y_{21}&z_{21} \\ x_{22}&y_{22}&z_{22}\\ \vdots&\vdots&\vdots\\ x_{2m}&y_{2m}&z_{2m}\end{array} \right]=\left[ \begin{array}{c}p_{21}\\ p_{22}\\ \vdots\\ p_{2m}\end{array} \right],P_{3}=\left[ \begin{array}{c}x_{31}&y_{31}&z_{31} \\ x_{32}&y_{32}&z_{32}\\ \vdots&\vdots&\vdots\\ x_{3q}&y_{3q}&z_{3q}\end{array} \right]=\left[ \begin{array}{c}p_{31}\\ p_{32}\\ \vdots\\ p_{3q}\end{array} \right], \tag{2}\]  
\(p_{1n}=\left[ \begin{array}{cc}x_{1n}&y_{1n}&z_{1n}\end{array} \right]\) is a coordinate vector of the \(n^{th}\) void of level 1, and \(p_{2m}=\left[ \begin{array}{cc}x_{2m}&y_{2m}&z_{2m}\end{array} \right]\) and \(p_{3q}=\left[ \begin{array}{cc}x_{3q}&y_{3q}&z_{3q}\end{array} \right]\) are the coordinate vector of the \(m^{th}\) void of the level 2 and the \(q^{th}\) void of the level 3, respectively. The MATLAB was applied to solve the aggregation  
The schematic diagram of the distance between the base point voids and the reference point voids is shown in Figure 10.  
In addition, the voids defect was equivalent to a sphere model based on the simulation analysis by Digmat software. A lossless solid model was created by ABAQUS, and the equivalent position information \(P_{n}\) was assigned to the corresponding void defect model after the equivalent void model was created by the ABAQUS subroutine based on the equivalent void volume \(V_{n}\). Then, the voids defect model assigned to the position information was moved to the corresponding position of the solid model by the ABAQUS subroutine and subjected to an ablation processing. Finally, the damage model with the equivalent void information was obtained.