# Investigation on Residual Stress Loss during Laser
## 3 Simulation Procedure
### Derivation of Laser Shock Pressure  
The classic one-dimensional solution model proposed by [PERSON] et al. [32] was used to calculate the laser shock pressure. According to this model, during laser shock processing, the laser intensity absorbed by the opaque coating is:  
\[I_{p}(t)=AP\cdot I(t) \tag{3}\]  
where \(t\), \(AP\), and \(I(t)\) represent the time, the energy absorption coefficient of the opaque coating, and the incident laser intensity, respectively. Relationships between plasma thickness \(L_{p}(t)\) and plasma pressure \(p(t)\) are:  
\[\frac{dL_{p}(t)}{dt}=\frac{2}{Z}p(t) \tag{4}\]  
\[\frac{2}{Z}=\frac{1}{Z_{1}}+\frac{1}{Z_{2}} \tag{5}\] \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{5}{c}{**Mesh Size (\(\mu\)m)**} & \multirow{2}{*}{**Dimple**} & \multirow{2}{*}{**Calculation**} \\ \cline{2-2} \cline{4-7}  & & & & & & \\ \cline{2-2} \cline{4-7}  & \multirow{2}{*}{**Domain**} & \multirow{2}{*}{**Domain**} & \multirow{2}{*}{**Domain**} & \multirow{2}{*}{**Domain**} & \multirow{2}{*}{**Depth (\(\mu\)m)**} & \multirow{2}{*}{**Time(s)**} \\  & & & & & & \\ \hline
1 & 25 & 50 & 100 & 200 & 22.23 & 56 \\
2 & 20 & 40 & 80 & 160 & 22.15 & 98 \\
3 & 15 & 30 & 60 & 120 & 21.98 & 153 \\
4 & 10 & 20 & 40 & 80 & 21.85 & 314 \\
5 & 8 & 16 & 32 & 64 & 21.849 & 587 \\
6 & 5 & 10 & 20 & 40 & 21.847 & 1072 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Different meshing strategies. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Young’s Modulus**} & **Poison’s** & **Density \(\rho\)** & \multicolumn{4}{c}{**Parameters Used in Johnson-Cook Model**} \\ \cline{3-7}  & **Ratio \(\mu\)** & **(T/mm\({}^{3}\))** & **\(A\) (MPa)** & **\(B\) (MPa)** & **\(n\)** & **\(C\)** \\ \hline
210,300 & 0.30 & \(7.98\times 10^{-9}\) & 305 & 441 & 0.10 & 0.057 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The properties of 316 L stainless steel [31]. \[I_{p}(t)=p(t)\frac{dL_{p}(t)}{dt}+\frac{3}{2\alpha}\frac{d}{dt}\big{[}p(t)\,L_{p}( t)\big{]} \tag{6}\]  
where \(Z_{1}\) is the shock impedance of the target material, \(Z_{2}\) is the shock impedance of the transparent overlay, and \(\alpha\) is the ratio of plasma thermal energy to internal energy. According to whether there is laser radiation or not, the interaction process of laser and material mainly include two stages: heating stage and adiabatic cooling stage. Suppose that the initial thickness of the plasma is 0, that is \(L_{p}(0)=0\), and laser intensity is a constant value \(I_{0}\), then at the laser switched-off time \(T_{\text{off}}\) (\(T_{\text{off}}\) is equal to the laser pulse duration \(\tau\)), the laser shock pressure and the thickness of the plasma can be obtained through Equations (4) and (6):  
\[p\Big{(}T_{off}\Big{)}=0.10\bigg{(}\frac{\alpha}{2\alpha+3}\bigg{)}^{1/2}Z^{1 /2}I_{0}^{1/2} \tag{7}\]  
\[L_{p}(T_{\text{off}})=2\times 10^{4}p(T_{\text{off}})\cdot T_{\text{off}}\cdot Z ^{-1} \tag{8}\]  
where \(p(T_{\text{off}})\) is the laser shock peak pressure. Equation (7) describes the relationship between laser shock peak pressure and the laser intensity. In the first stage (\(t<T_{off}\)), the thickness of the plasma and laser shock pressure can be expressed as:  
\[L_{p}(t)=\Bigg{(}L_{p_{0}}^{2}+\bigg{(}\frac{2p(T_{off})t}{Z}\bigg{)}^{2} \Bigg{)}^{\frac{1}{2}} \tag{9}\]  
\[p(t)=p(T_{off})\Bigg{(}1-\frac{L_{p_{0}}^{2}}{L_{p}(t)^{2}}\Bigg{)}^{\frac{1}{ 2}} \tag{10}\]  
where \(L_{p0}\) is the initial plasma thickness. In the second stage (\(t>T_{off}\)), the thickness of the plasma and laser shock pressure could be obtained by:  
\[L_{p}(t)=L_{p}\Big{(}T_{off}\Big{)}\Bigg{(}1+\frac{\gamma+1}{T_{off}}\big{(}t -T_{off}\big{)}\Bigg{)}^{\frac{1}{\gamma+1}} \tag{11}\]  
\[p(t)=p\Big{(}T_{off}\Big{)}\Bigg{(}\frac{L_{p}\Big{(}T_{off}\Big{)}}{L_{p}(t)} \Bigg{)}^{\gamma} \tag{12}\]  
where \(\gamma\) represents the ratio of specific heat. The shock pressure of laser can be obtained through solving the equations from Equations (4) to (10) in MATLAB software.