## 1 Introduction
### Origin of Residual Stresses during Quenching  
The residual stresses in the finished product are a consequence of the asynchronous thermal shrinkage and phase transformation of the core and the surface regions. During quenching all materials are exerted to shrinkage stresses since the surface always cools faster than the core. Additionally, if phase transformations occur during quenching, transformation stresses ensue as well and superimpose on the aforementioned shrinkage stresses. In order to understand the origin of the residual stresses during quenching and thereby the factors that affect it, a closer look at the generation of these two types of stress is necessary. The top part of Figure 2 shows the cooling curves of the surface and the core of a cylinder during quenching \((T^{\mathrm{s}},T^{\mathrm{c}})\) together with the yield stress, \(R_{\mathrm{y}}\) of these regions, which depends on the temperature and hence on process time and is assumed to be the same for compression and tension. For simplicity, only the longitudinal stress component is shown, although criteria for plastic flow also consider the tangential and radial components [14; 15; 16]. Concentrating only on the shrinkage stresses on the left side, the surface cools faster than the core at the initial stage of the quench. This results in longitudinal tensile stresses at the surface, whereas the core experiences longitudinal compressive stresses. As long as there is no plastic deformation, these stresses will be balanced at the end of the process with no residual stresses, as can be seen in the middle part of Figure 2. Usually, however, these shrinkage stresses exceed the yield stress of the material shown as a gray shaded area in the figure. This is promoted by the low yield stress at high temperatures where the temperature gradients are very high [10; 15]. Therefore, the material flows plastically leaving residual stresses at the end as shown in the bottom part of Figure 2. Upon further  
Figure 1: Schematic representation of the Tempcore(tm) process. cooling the core temperature falls rapidly while the surface gradually reaches the temperature of the cooling medium, thus reducing the magnitude of the shrinkage stresses and eventually reversing their signs. The plastic stretching of the surface causes compression at the surface and vice versa in the core. Once the temperature balance is established, the cylinder is left with compressive stresses at the surface and tensile stresses in the core due to the shrinkage stresses. If the phase transformation in the material is to be considered, the same line of reasoning could be laid out for pure transformation stresses ignoring the thermal shrinkage stresses and transformation plasticity. The top part of Figure 2b shows the cooling curves of the surface and the core of the cylinder, the martensite start and finish temperatures, \(M_{\rm s}\) and \(M_{\rm f}\), and the temperature-dependent yield stress of the material. Upon reaching \(M_{\rm s}\) it is assumed that the whole material transforms into martensite, which occurs at times \(t_{1}\) at the surface and \(t_{2}\) in the core. The volume expansion associated with the austenite to martensite transformation, which is ca. 3.4% for 0.19 % C steel [17], leads to compressive stresses arising near the surface compensated by tensile stresses in the core as shown in the middle part of Figure 2b. Once the core reaches \(M_{\rm s}\) and starts to transform and expand, the stresses are reduced in both regions. Due to the transformation into much harder  
Figure 2: Origin of longitudinal residual stress component during quenching of cylinders. Adapted from [14]. martensite, the yield stress curves undergo a much larger increase in magnitude than in the case of mere thermal shrinkage as shown in Figure 2a, left. The plastic flow occurring due to the transformation stresses exceeding the yield point (shown in gray) results in volume incompatibilities between the core and the surface at the end of the cycle.