## 1 Introduction
### Origin of Residual Stresses during Quenching
This leads to residual stresses, which are shown at the bottom of Figure 2b by arrows, which are reversed in sign compared to the left image. Combining these two types of stress by simple superposition is not accurate, since the material has deformed plastically. The fundamental concept from this discussion is, however, helpful in understanding the factors involved in solving such problems and the material parameters required for modeling such a quenching process successfully. It is clear that the yield stress of the material at high temperatures determines plastic flow and has to be quantified. The relative position of the time in which the transformation initiates and ends at the surface and in the core significantly influences the magnitude as well as the sign of the final residual stresses [14]. To the best of the authors' knowledge these parameters are not available in literature for the B500B alloy, which constitutes a large fraction of commercially available rebars and, hence, have to be determined experimentally. Some attempts have been made to predict the strength and transformation temperatures with models and theoretical calculations [5]. The thermal history or the cooling curve across the cross section of the bar during quenching and tempering is, however, impractical and prohibitively expensive to obtain directly from the manufacturing facility, if for this purpose the production of large tonnages has to be stopped. Therefore, several studies have been conducted to predict the transient temperature profile across the cross-section of the rebar with computational fluid dynamics simulations of the coolant coupled to the heat transfer problem in the rebar [5; 18; 19]. In summary, the following facts are important to consider. The plastic deformation determines the magnitude of the residual stresses. The temperature difference between the surface and the core is dependent on the following factors: the starting temperature of the quenching process, the diameter, the heat transfer coefficient, and the thermal properties of the material. The residual stresses are larger in magnitude if the yield strength at high temperature is very low, due to larger plastic deformation. Finally, the residual stresses greatly depend on the relative initiation time of the transformation in the surface and core regions. The phenomena that were neglected in the above discussion for simplicity need to be considered for solving real problems. The material as a whole will not transform into martensite, but rather a mixture of all the phases including ferrite, pearlite, bainite, and martensite is present in the rebar as will be seen in Section 3.1. Only longitudinal stresses were considered, but the criteria for plastic flow usually consider all the components of stress [16]. Furthermore, transformation plasticity effects that were neglected may have a significant influence on the residual stress distribution and should be examined [20]. In addition to the material parameters, the geometry of the ribs on the rebar and its surface finish also significantly influence the fatigue behavior [8]. Experimental determination of residual stresses, e.g., using X-ray diffraction, will reach their limitations for such complex surface geometry. Furthermore, the resolution is not high enough to measure the residual stresses at the foot of the ribs, which usually act as the origin of cracks in these bars. Hence, it is only through simulating the quenching process with the correct material parameters, material constitutive model, and rebar geometry that a better understanding of the process can be gained.