## 2 Theoretical Research  
The indentation model represents the contact between a deformable semi-infinite matrix and a rigid positive pyramid indenter, as shown in Figure 2. The matrix has equal biaxial residual stress \(\sigma_{R}\). The matrix is assumed to be a homogeneous elastoplastic material with yield stress \(Y\). The plastic yield adopts the [PERSON] criterion. The indentation modulus is \(E^{*}=E/(1-\
u^{2})\). \(E\) is defined as the elastic modulus and \(\
u\) is defined as the [PERSON]'s ratio. Assuming an indenter load of \(P\), which is pressed perpendicular to the surface of the specimen substrate, its orthorhombic indentation depth is \(\lambda\). The actual contact diagonal length \(a\) is extracted considering accumulation or subsidence. Therefore, the average pressure \(P_{m}\) of the entire substrate-indenter contact area is defined as Equation (1),  
\[P_{m}=\frac{2P}{a^{2}} \tag{1}\]  
\(P_{m}\) reaches its maximum value at fully plastic deformation, where the indentation depth is \(\lambda^{\text{max}}\). The maximum pressure is material hardness \(H\)[25; 26]. When there are no residual stresses in the matrix, the hardness of the matrix is \(H_{0}\) and the corresponding indentation depth of the indenter is \(\lambda_{0}^{\text{max}}\). The contact response differences between the materials with residual stress and those without residual stress are defined as \(\Delta H=H-H_{0}\) and \(\Delta\lambda^{\text{max}}=\lambda^{\text{max}}-\lambda_{0}^{\text{max}}\). \(\Delta H\) and \(\Delta\lambda^{\text{max}}\) are the functions of indentation modulus \(E^{*}\), [PERSON]'s ratio \(\
u\), yield strength \(Y\), residual stress \(\sigma_{R}\) and indenter diagonal length \(a\). \[\Delta H=H-H_{0}=f(\
u,E^{*},R,Y,\sigma_{R}) \tag{2}\]  
\[\Delta\lambda^{\text{max}}=\lambda^{\text{max}}-\lambda_{0}^{\text{max}}=g( \
u,E^{*},R,Y,\sigma_{R}) \tag{3}\]  
Figure 2: Positive rhomboidal indentation with equal biaxial residual stress: (**a**) 3D model and (**b**) 2D indentation. Because Poisson's ratio \(v\) has little effect on static indentation, the effect of Poisson's ratio is not considered in order to simplify the calculations [27,28]. Based on the \(\prod\) theorem of dimensional analysis, \(\Delta H\) and \(\Delta\lambda^{\text{max}}\) are represented as  
\[\frac{\Delta H}{Y}=\frac{H-H_{0}}{Y}=f(\frac{E^{*}}{Y},\frac{\sigma_{R}}{Y}) \tag{4}\]  
\[\frac{\Delta\lambda^{\text{max}}}{a}=\frac{\lambda^{\text{max}}-\lambda_{0}^{ \text{max}}}{a}=g(\frac{E^{*}}{Y},\frac{\sigma_{R}}{Y}) \tag{5}\]  
\(H_{0}\) and \(\lambda_{0}^{\text{max}}\) can be further simplified and integrated as follows [25]:  
\[\frac{H_{0}}{Y}=b\left[1-\left(\frac{E^{*}}{Y}\right)^{-0.05}\right] \tag{6}\]  
\[\frac{\lambda_{0}^{\text{max}}}{a^{\prime}}=\frac{1}{c\frac{E^{*}}{Y}+d} \tag{7}\]  
where \(b\), \(c\) and \(d\) are undetermined constant coefficients, which need to be obtained by fitting simulation data. Based on the geometric relationship, the truncated contact diagonal length \(a^{\prime}\) can be calculated from the indentation depth and the angle of indenter:  
\[a^{\prime}=\tan\frac{1}{2}\theta\times\lambda\]  
Organizing the above equations, the equation for the relationship between hardness deviation and residual stress can be derived as follows:  
\[\frac{H}{Y}-\frac{H_{0}}{Y}=\left(\frac{E^{*}}{Y}\right)^{0.5}\times\left\{ \begin{array}{l}-B\left(\frac{\sigma_{R}}{Y}\right)^{2}-C\frac{\sigma_{R}}{ Y},\frac{\sigma_{R}}{Y}\leq 0\\ D\frac{\sigma_{R}}{Y},\frac{\sigma_{R}}{Y}>0\end{array} \right. \tag{8}\]  
where \(H\) is hardness, \(Y\) is yield strength, \(E^{*}\) is indentation modulus and \(\sigma_{R}\) is residual stress. Coefficients \(B\), \(C\) and \(D\) are obtained by fitting the simulation and experimental data according to the determination of specific materials. Equation (8) is further arranged as a relation between residual stress and hardness. \[\sigma_{R}=\left\{