# A Novel Low-Cost DIC-Based Residual Stress Measurement Device
## Appendix A  
In order to quantify the experimental residual stresses, the Integral Method with Tikhonov Regularisation was used, as suggested in ASTM E837-13a [73; 74], which calculates the non-uniform stresses at each step. Figure 6 shows the physical interpretation of a non-uniform stress. The key results of a residual stress measurement are the maximum principle stress \(\sigma_{\text{max}}\), minimum principle stress \(\sigma_{\text{min}}\), and the angle, \(\beta\) at each depth increment. These results are positioned relative to the strain gauge orientation and could be compared against the yield criterion of the material and the application requirements. The integral method involves calculating the combinational strains, shown in Equations (A1)-(A3), where \(\varepsilon_{1},\varepsilon_{2},\varepsilon_{3}\) are strains measured via the gauges, \(p_{j}\) is the isotropic combinational strain, \(q_{j}\) is the shear strain acting at 45 degrees, and \(t_{j}\) is the xy shear strain. \[p_{j}=(\varepsilon_{3}+\varepsilon_{1})_{j}/2\] (A1)\[q_{j}=\frac{\left(\varepsilon_{3}-\varepsilon_{1}\right)_{j}}{2} \tag{10}\]  
\[t_{j}=\left(\varepsilon_{3}+\varepsilon_{1}-2\varepsilon_{2}\right)_{j}/2 \tag{11}\]  
where \(j\) refers to the number of hole depth steps. The standard errors of these strains are calculated using Equations (10)-(11). \[p_{std}^{2}=\sum_{j=1}^{n-3}\frac{\left(p_{j}-3p_{j+1}+3p_{j+2}-p_{j+3}\right) ^{2}}{20(n-3)} \tag{12}\]  
\[q_{std}^{2}=\sum_{j=1}^{n-3}\frac{\left(q_{j}-3q_{j+1}+3q_{j+2}-q_{j+3}\right) ^{2}}{20(n-3)} \tag{13}\]  
\[t_{std}^{2}=\sum_{j=1}^{n-3}\frac{\left(t_{j}-3t_{j+1}+3t_{j+2}-t_{j+3}\right) ^{2}}{20(n-3)} \tag{14}\]  
where \(n\) is the number of steps and \(1\ \leq\ j\ \leq\ n-3\). Using the calibration factors provided in ASTM E837-13a, the calibration vectors can be formed at each hole depth increment, denoted as \(\underline{a}\) and \(\underline{b}\). The calibration constants are derived from FEA [75,76], however, they are nearly material independent, varying by less than 2% between materials. Note the calibration coefficients provided are for a 2 mm hole and therefore need to be adjusted with Equation (12), where \(d\) is the hole diameter. \[\underline{a}_{adjusted}=\left(\frac{d}{2}\right)^{2}*\underline{a} \tag{15}\]  
The residual stress at each hole depth is estimated using Equations (13)-(14). \[\underline{a}P=\frac{E}{1+v}p \tag{16}\]  
\[\underline{b}Q=Eq \tag{17}\]  
\[\underline{b}T=Et \tag{18}\]  
For a few milling steps, Equations (16)-(18) can be effective. However, with more hole steps, matrices \(\underline{a}\) and \(\underline{b}\) can become ill-conditioned, leading to significant errors. Therefore, Tikhonov regularisation needs to be applied. This involves forming a \"second derivative\" matrix \(c\), Equation (19):  
\[q_{j}=\frac{\left(\varepsilon_{3}-\varepsilon_{1}\right)_{j}}{2} \tag{10}\]0  
where the number of rows equals the number of hole depth steps used. The first and last rows contain zeros; all other rows have [-1 2 -1] centred along the diagonal. By implementing Tikhonov second-derivative regularisation on Equations (20)-(21), \(P\), \(Q\) and \(T\) are calculated:  
\[q_{j}=\frac{\left(\varepsilon_{3}-\varepsilon_{1}\right)_{j}}{2} \tag{10}\]1\[q_{j}=\frac{\left(\varepsilon_{3}-\varepsilon_{1}\right)_{j}}{2} \tag{10}\]2 \[q_{j}=\frac{\left(\varepsilon_{3}-\varepsilon_{1}\right)_{j}}{2} \tag{10}\]3  
The factors \(\alpha_{p}\), \(\alpha_{Q}\) and \(\alpha_{T}\) control the amount of regularisation used which smooth results. Small numbers ranging from \(10^{4}\) to \(10^{6}\) are suitable for the initial guess.