## 4 Uniform In-Depth Residual Stress Field
### Linear-Elastic Material Properties  
First, the hole-drilling process is numerically calibrated with a test piece where a uniform in-depth residual stress field of 50 MPa is applied on and the calibration coefficients are determined. The uniform residual stress field is divided into equi-biaxial (\(\sigma_{\mathrm{x}}=\sigma_{\mathrm{y}}=\) 50 MPa) and deviatoric (\(\sigma_{\mathrm{x}}=-\sigma_{\mathrm{y}}=\) 50 MPa) residual stress fields and a linear-elastic material law is considered. Two 3D models with linear-elastic material behavior are evaluated. A known equi-biaxial stress field is applied on the first model while the second model is evaluated by applying a deviatoric stress field. The model with equi-biaxial residual stress field is used to determine calibration coefficient a. A deviatoric residual stress field gives calibration coefficient b. Ten depth-step increments of 0.1 mm are evaluated for both models until a final hole depth of 1 mm is reached. The calibration coefficients are calculated by evaluating the relaxed strains for each depth-step [10]. Uniform in-depth residual stress fields of 100 MPa and 150 MPa were also considered which resulted in the same calibration coefficients. In Table 3 the calculated calibration coefficients are shown. The coefficients for elastic-plastic material behavior are also already mentioned.  
The calibration coefficients from the finite element modeling with linear-elastic material behavior should equal the coefficients specified in ASTM. However, the coefficients calculated with the 3D model are on average 4% smaller compared with the ones from ASTM. By increasing the hole depth, the difference also increases because the distance to the strain gauge rosette on the surface increases. There is less contribution of the relaxed strain at the deeper hole depth increments because the distance to the surface of the strain gauge rosette increases. Geometric variables such as the hole diameter, plate thickness, and drilling depth have an influence on the determination of the calibration coefficients.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**Hole Depth** & \multicolumn{2}{c}{**ASTM E837-13a (Linear-Elastic**} & \multicolumn{2}{c}{**FEM Linear-Elastic Material**} & \multicolumn{2}{c}{**FEM Elastic-Plastic Material**} \\ **(mm)** & \multicolumn{2}{c}{**Material Behavior**} & \multicolumn{2}{c}{**Behavior**} & \multicolumn{2}{c}{**Behavior**} \\ \hline
**Hole Depth** & **Calibration** & **Calibration** & **Calibration** & **Calibration** & **Calibration** & **Calibration** \\
**(mm)** & **Coefficient a** & **Coefficient b** & **Coefficient a** & **Coefficient b** & **Coefficient a** & **Coefficient b** \\ \hline
0.1 & 0.015 & 0.028 & 0.016 & 0.028 & 0.013 & 0.034 \\
0.2 & 0.036 & 0.066 & 0.037 & 0.065 & 0.029 & 0.078 \\
0.3 & 0.059 & 0.110 & 0.058 & 0.108 & 0.043 & 0.119 \\
0.4 & 0.083 & 0.155 & 0.079 & 0.153 & 0.056 Measurement errors of strain, hole depth, and geometry cause a difference between the calibration coefficients determined with finite element modeling and specified in ASTM. Research conducted by [PERSON] and [PERSON] [16] shows that calibration coefficient a becomes ill conditioned faster than calibration coefficient b and this can also can be noticed in Table 3.  
There is also a comparison made with coefficients from other studies [16] and similar error rates and trends are observed in the experimental calibration. Therefore it is concluded that the calibration coefficients from the three-dimensional finite element modeling are in agreement with the conventional coefficients of the incremental hole-drilling method specified in ASTM. This finite element model is used to model the incremental hole drilling for the determination of residual stresses in an orthotropic bridge deck [10].