# Fatigue Life Assessment in Bainitic Steels Based on The Cumulative Strain Energy Density
## 3 Results  
### Cumulative Strain Energy Density  
Low-cycle fatigue behavior of the tested steels under symmetrical strain-controlled conditions for a strain amplitude of 1.0% is presented in Figure 2. As shown, the cyclic stress-strain responses were  
Figure 1: Graphical definition of elastic strain energy density (darker grey areas) and plastic strain energy density (light grey area) considered in the calculations.  
significantly different for the three materials. Overall, it was clear that the stress range associated with this strain level increased with higher Mn contents.  
Moreover, irrespective of the tested steel, the shapes of the hysteresis loop changed significantly throughout the lifetime. These changes were particularly evident by comparing the second and the mid-life loops. Since the uncontrolled stress range reduced over time, the three materials exhibited a cyclic strain-softening behavior for this strain level. Literature suggests that at lower strain amplitudes Steel C maintains a similar trend, but Steel A and Steel B tend to strain-harden [9].  
Figure 3 plots elastic and plastic strain energy densities against the number of cycles for different strain amplitudes. These variables, as can be seen, were relatively constant along time. In most cases, the curves present three main stages: a short initial period with rapid changes; a dominant saturated stage with stable values; and a final stage, close to total failure, characterized by sudden changes. As expected, elastic and plastic strain energy densities per cycle increased with strain amplitude. On the other hand, at a given strain amplitude, the values of \(\Delta\)_We_ and \(\Delta\)Wp depended on the tested steel, which agreed with the hysteresis loop shapes displayed in Figure 1.  
At a fixed value of strain amplitude, a close analysis of the results showed that plastic strain energy density per cycle was higher for steel A and tended to diminish as Mn content increased, except for \(\Delta\varepsilon\)/2 = 0.6%, where steel C exhibited higher plastic strain energy density values than steel B.  
As far as elastic positive strain energy density is concerned, opposite behavior was observed, i.e., the maximum values at a given strain amplitude were found for steel C. Furthermore, \(\Delta\)_We_ increased for lower contents of Mn, except for \(\Delta\varepsilon\)/2 = 0.6%, where steel A had higher plastic strain energy density than steel B.  
Figure 4 plots cumulative elastic (_We_), plastic (Wp) and total (Wt) strain energy densities as a function of strain amplitude. The computed values of the aforementioned variables are listed in Table 4, as well as the three components of strain energy density of the mid-life cycle, identified as \(\Delta\)_We_,ML, \(\Delta\)Wp_ML and \(\Delta\)WtML. At first sight, it could be concluded that steel A without Mn revealed higher plastic strain energy densities than elastic strain energy densities for all the strain amplitudes applied, showing a high fatigue toughness capability. On the contrary, the other two steels revealed higher elastic than plastic strain densities, except for steel B at a strain amplitude of 1.0%.  
Figure 2: Cyclic stress–strain response of: (**a**) steel A; (**b**) steel B; and (**c**) steel C (Data from [9]).  
Figure 3: Plastic (\(\Delta\)Wp) and elastic (\(\Delta\)_We_) strain energy densities per cycle as a function of strain amplitude for: (**a**) steel A; (**b**) steel B; and (**c**) steel C.