# Fatigue Life Assessment in Bainitic Steels Based on The Cumulative Strain Energy Density
## 2 Materials and Methods  
The chemical composition of each high-strength low-carbon bainitic steel under study, referred to as steels A, B and C, is presented in Table 1. As clearly seen, Mn content varied from 0% to 3.2%, while the other chemical elements were maintained constant. Static and cyclic mechanical properties, which depend on microstructure [9], were obtained for the three materials through monotonic axial tensile and low-cycle fatigue tests, respectively (see Tables 2 and 3).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Steel** & **C** & **Si** & **Mn** & **Cr** & **Ni** & **Mo** & **Al** & **Fe** \\ \hline A & 0.26 & 1.6 & 0 & 1.9 & 0.4 & 0.3 & 0.6 & Bal. \\ B & 0.27 & 1.7 & 2.3 & 1.9 & 0.4 & 0.4 & 0.7 & Bal. \\ C & 0.27 & 1.7 & 3.2 & 1.9 & 0.4 & 0.4 & 0.7 & Bal. \\ \hline \hline \end{tabular}
\end{table}
Table 1: Chemical compositions of tested steels (wt%) (Data from [6]).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Property** & **Steel A** & **Steel B** & **Steel C** \\ \hline Fatigue strength coefficient, \(\prime\) [MPa] & 1585 & 2029 & 2501 \\ Fatigue strength exponent, b & \(-\)0.0721 & \(-\)0.06885 & \(-\)0.074 \\ Fatigue ductility coefficient, \(\prime\) & 0.7432 & 1.9401 & 0.201 \\ Fatigue ductility exponent, c & \(-\)0.608 & \(-\)0.8167 & \(-\)0.5535 \\ Cyclic hardening coefficient, \(\prime\) (MPa) & 1684 & 1931 & 3085 \\ Cyclic hardening exponent, n\(\prime\) & 0.1172 & 0.0722 & 0.1330 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Low-cycle fatigue properties of the tested steels (Data from [9]).  
The fatigue tests were performed under a strain-controlled mode with strain amplitudes, De/2, of 0.6%, 0.7%, 0.8% and 1%, using sinusoidal waveforms (strain ratio, R = \(-\)1), with the aid of a mechanical extensometer, model MTS 634.31F-24, directly clamped to the gauge section in an MTS (MTS Systems Corporation, Eden Prairie, MN, USA) servo-hydraulic testing machine equipped with a 10 kN capacity load cell for a constant strain rate (\(de/\)dt = \(6\times 10^{-3}\) s\({}^{-1}\)). The specimens were machined and polished according to the specifications outlined in ASTM E606 and had a gauge section with a length of 10 mm and a diameter of 5 mm. The tests were initiated in compression and stopped when total failure occurred. During the test, the hysteresis loops were stored when the stress range of a given cycle altered more than 5% with respect to the previous recorded cycle, resulting typically between 50 and 65 hysteresis loops for each strain amplitude applied. Moreover, at least 200 points were saved for each hysteresis loop.  
Spline-based techniques were used for data interpolation and smoothing of all hysteresis loops recorded during low-cycle fatigue tests, namely cubic and thin plate splines. Then, numerical integration was carried out to determine the values of elastic strain energy density per cycle (\(\Delta\)\(W\)e) and plastic strain energy density per cycle (\(\Delta\)Wp) of the tested steels (see Figure 1). Total strain energy density per cycle (\(\Delta\)Wt) was defined as the sum of both components.  
Concerning fatigue life predictions, apart from the new proposed predictive model, the SWT model [11], as well as the Liu model [12], were used. In the case of the former, the damage parameter is based on the maximum tensile stress applied on a critical plane and the strain amplitude acting on the same plane (Equation (1)); regarding the Liu model (Equation (2)), which is based on the virtual strain energy applied, DW, it takes into account both elastic and plastic strain energy densities of the hysteresis loops collected in fatigue experiments.  
\[\sigma_{\text{max}}\ \frac{\Delta\varepsilon}{2}=\frac{\sigma_{f}^{\prime 2}}{E} \ (2N_{f})^{2b}+\sigma_{f}^{\prime}\ \varepsilon_{f}^{\prime}\ (2N_{f})^{b+c} \tag{1}\]  
\[\Delta W=\Delta\sigma\ \Delta\varepsilon\ =\frac{4\ \sigma_{f}^{\prime 2}}{E} \ (2N_{f})^{2b}+4\ \sigma_{f}^{\prime}\ \varepsilon_{f}^{\prime}\ (2N_{f})^{b+c} \tag{2}\]