## 2 Material and Methods  
The material in focus is the high-Mn austenitic TWIP steel X40 MnCrAl19-2 (precidur(r) H-Mn LY) provided by thyssenkrupp Hohenlimburg GmbH (Hagen, Germany). As reported in [39], the initial state is characterized by an average hardness of 250 HV as well as a homogenous microstructure with an average grain size of about 20 \(\upmu\)m and a high fraction of recrystallization twins. Furthermore, tensile tests disclosed a YS and UTS of about 400 and 880 MPa respectively, demonstrating the high work-hardening potential of the material, as well as an elongation at fracture of approximately 52%.  
The present work reports a follow-up study. For detailed information on the experimental setup, i.e., specimen preparation, machines and parameters used, the reader is referred to [39] for the sake of brevity. In addition to RT deep rolling, deep rolling at an elevated temperature of 550 \({}^{\circ}\)C was performed for the first time in the present study. According to the previous results, a deep rolling force of 855 N was chosen. Conditions deep rolled at RT and 550 \({}^{\circ}\)C are referred to as dr_855_RT and dr_855_550 \({}^{\circ}\)C in the remainder of the manuscript. In order to expand the fatigue database shown for RT deep rolling in [39] and in order to conduct a minimum of at least four tests for all stress amplitudes considered, several additional fatigue tests were also performed for the dr_855_RT condition.  
The residual stress stability at half-life was investigated for 5 different stress amplitudes, i.e., 425, 460, 480, 500 and 540 MPa. Based on data obtained, the arithmetic mean of the number of cycles to failure was calculated from all tests being conducted on a stress level considered. Prior to the fatigue test, residual stress values were determined at the direct surface in the center of the gauge length of the specimen. The spot of the measurement was marked by a color marker. Subsequently, the specimens were fatigued to the averaged half-life before surface residual stresses were measured again at the marked spot. Additionally, post-fatigue residual stress depth profiles were determined for the lowest and highest stress amplitudes of 425 and 540 MPa, respectively. All tests that have either been conducted within the present work or recompiled from [39] are summarized in Table 1.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & **EBSD** & **Hardness** & \begin{tabular}{c} **Residual** \\ **Stress Profiles** \\ \end{tabular} & \begin{tabular}{c} **Tensile** \\ **Tests** \\ \end{tabular} & \begin{tabular}{c} **Fatigue Tests** \\ \end{tabular} & \begin{tabular}{c} **Residual** \\ **Stress** \\ \end{tabular} & \begin{tabular}{c} **Fracture** \\ **Surface** \\ **Stability** \\ \end{tabular} & \begin{tabular}{c} **Fracture** \\ **Analysis** \\ \end{tabular} \\ \hline dr\_855\_RT & - & \begin{tabular}{c} Recompiled \\ from [39] \\ \end{tabular} & \begin{tabular}{c} Recompiled \\ from [39] \\ \end{tabular} & \
\begin{tabular}{c} Partly recompiled \\ from [39] \\ \end{tabular} &
\begin{tabular}{c} X \\ \end{tabular} & - \\ \hline dr\_855\_550 \({}^{\circ}\)C & X & X & X & X & X & X \\ \hline \hline \end{tabular}
\end{table}
Table 1: Summary of the conducted and recompiled tests for the different conditions considered.