## 2 A Quantitative Description of the Peening Coverage  
In this section, the peening coverage is described quantitatively by analyzing the effect of the distance between shots, dent size, and the number of shots on the peening coverage. Generally speaking, peening coverage is defined as the percentage of a surface area indented. For shot peening, the peening coverage is determined by three factors, (a) dent diameter, (b) the number of dents in each unit area of the peening surface, and (c) the time of peening. In this study, 'cycle' is used to describe the time of peening for FE simulation.  
Before the peening coverage is described quantitatively, some assumptions should be put forward about locations and sequence of shots impact. Firstly, the locations and sequence of the multiple shots impacting is not generated randomly. Secondly, peening coverage depends on both dent size and the number of dents generated on Almen strip surface in the unit area, rather than the impact sequence. Thirdly, the locations of the multiple shots are defined in the same position and the shots impact one by one.  
With assumptions above, the process of shots impacting is analyzed. Figure 1 shows the Schematic of dent size on Almen strip surface in 1-cycle, the second, fourth and fifth row of shots is omitted in Figure 1 for convenience. In Figure 1a, the shot balls are adjacent to each other in the row and impact Almen strip surface at the same time. The width of the shadow is defined as the size of each dent (\(D_{\text{s}}\)), and the distance between the center of the shots is defined as \(C_{\text{d}}\). Figure 1b-d shows the process of dent expansion. In Figure 1d, peening coverage reaches 100%. This impact pattern, from the first to the sixth row of shots, is called one 'impact cycle'. The overlap of the dents is inevitable in the peening process, however, with the reasonable distance between shots, there is little overlap when full peening coverage is reached. Therefore, higher peening coverage could be got by the repetition of this cycle.  
Figure 1: Schematic of dent size on the surface of Almen strip in 1-cycle (100% peening coverage). (**a**) the first row impacts the Almen strip surface, (**b**) the dent position of the first-row impact, (**c**) the dent position of the third-row impact, (**d**) the dent position of the 1-cycle.  
With the analysis above, a quantitative description of peening coverage is proposed, and the peening coverage is computed by Equation (1).  
\[R_{o}=\frac{D_{s}\cdot N_{l}-C_{d}}{N_{l}}\times\frac{1}{D_{s}}\times 100\%, \tag{1}\]  
\[N_{l}=a\cdot\frac{C_{d}}{D_{s}}, \tag{2}\]  
where \(R_{o}\) is the peening coverage, \(D_{s}\) is dent size, \(C_{d}\) is the distance between the centers of shots; and \(N_{l}\) defined in Equation (2) characterizes the number of dents between the centers of the shot. As shown in Equation (2), the number of dents \(N_{l}\) is determined by \(C_{d}\), \(D_{s}\), and the coefficient of proportionality (\(a\)). The coefficient of proportionality can be determined by the overlapping dents of the shot impacts [28]. The dent size is related to the size of the ball and the impact velocity. The horizontal distance between the centers of shots (\(C_{d}\)) is twice the radius of the shots. The effectiveness of the quantitative description of peening coverage will be proved in Section 3.