## 4 Results
### Deep Rolling Analysis  
The 2D finite element model simulates (Figure 12) the residual stress distribution across the material's surface and depth, focusing primarily on surface interactions during the deep rolling process. In this simulation, the residual stress is extracted from node paths defined along points (P1 to P7), with the values averaged to estimate the compressive stress at specific depths. The 2D model, while computationally simpler, provides a generalized view of the stress distribution. However, it lacks the precision needed to fully capture how stresses propagate through the material's depth. The stress distribution in the 2D model is largely concentrated at the surface, and the simulation tends to overlook variations in stress intensity in deeper layers. This limitation can lead to higher error margins when comparing simulated values to experimental data, particularly when dealing with complex processes like deep rolling that involve dynamic interactions between the tool and the material. In the given deep rolling scenario (150 bar pressure, 1800 mm/min speed, 0.1 mm offset), the 2D simulation provided valuable insights into the overall stress behavior, but the error margin in comparison to experimental results was significant. While useful for preliminary analysis, the 2D model fails to capture the full extent of the stress distribution, especially in the material's deeper regions where stress plays a critical role in fatigue resistance.  
Figure 11: Boxplot of compressive residual stress with hypothesis testing and 95% confidence interval.  
Figure 12: Selected nodes in the 2D model.  
The 3D finite element mode (Figure 13), by contrast, provides a more detailed and accurate representation of stress distribution, particularly across the material's depth. This model accounts for the full range of interactions between the rolling tool and the material surface, offering a three-dimensional view of how compressive residual stresses are induced by the deep rolling process. In this simulation, node paths are defined within the midsection of the material, capturing the stress distribution at varying depths and angular orientations in the circumferential direction. The 3D model, under the same conditions (150 bar pressure, 1800 mm/min speed, 0.1 mm offset), predicted a compressive residual stress of 541 MPa, closely aligning with the experimental mean of 498.9 MPa. This proximity to the experimental results indicates that the 3D model provides a more reliable and accurate prediction of the residual stress profile. The smaller error margin in the 3D model makes it a superior tool for predicting the stress variations that occur not only at the surface but also in deeper layers of the material, which are crucial for improving mechanical properties such as fatigue resistance.  
The 3D model offers a higher level of accuracy compared to the 2D model. The FEA prediction of 541 MPa from the 3D model is closer to the experimental mean, with the results of the paired \(t\)-test confirming the model's reliability. The 2D model, while useful for surface-level analysis, exhibits higher discrepancies when compared to experimental data due to its inability to capture stress variations at different depths.  
The 3D simulation excels in capturing stress propagation through the material's depth, offering a more detailed view of how compressive stresses are distributed along both the longitudinal and circumferential directions. In contrast, the 2D simulation focuses primarily on surface-level stresses, which may lead to oversimplified conclusions about the material's behavior under deep rolling conditions.  
In industrial applications where precise stress analysis is crucial, the 3D FEA model is the preferred tool for optimizing deep rolling parameters, such as rolling pressure and speed. The 2D model, while faster and computationally less demanding, is better suited for preliminary studies or applications where in-depth stress analysis is not required.