Fatigue Life Assessment in Bainitic Steels Based on The Cumulative Strain Energy Density

Carbide-free bainitic steels are an example of high-strength steels that show an excellent combination of strength, ductility, toughness and rolling fatigue contact resistance and are progressively being introduced in the production of railways, crossings and automotive components. Although there are Mn-free approaches able to produce carbide-free bainitic steels, those based on the addition of Mn are less expensive. Therefore, it is important to fully understand the mechanical behavior of such materials to develop reliable engineering products. In this paper, three low-carbon bainitic steels, differing in Mn content, namely 0%, 2.3% and 3.2%, designated as steel A, B and C, respectively, were studied in a systematic manner. Low-cycle fatigue tests were conducted under symmetrical strain-controlled conditions for different strain amplitudes (0.6%, 0.7%, 0.8% and 1%). Independent of Mn content, a strong relationship between cumulative strain energy density and number of cycles to failure was found. Based on this relationship, a new predictive model, capable of estimating the fatigue lifetime, was developed. Predictions based on the new model were close to the experimental lives and were more accurate than those computed via the well-known Smith-Watson-Topper (SWT) and Liu criteria.


Introduction
Diminishing fuel consumptions and mandatory lower CO 2 emissions are both pushing the automotive industry to weight reduction in the design of cars and increasing the use of railway transportation of goods and people. Therefore, to achieve those goals, recently advanced high-strength steels are currently being used to reduce components' thicknesses in the automotive industry and to produce improved railway rails and crossings.
In fact, carbide-free bainitic steels are an example of these new materials which show an excellent combination of strength, ductility and toughness that satisfies demands of high-strength steels in various industrial applications [1,2]. Moreover, bainite covers some different microstructures, from the classical upper and lower bainite to the carbide-free structures found in lower carbon steels [3], and is produced from the transformation of austenite at temperatures below the pearlite range and above the martensite starting temperature [4]. In the production of bainitic steel, it is paramount that perlite and ferrite regions be pushed as far to the right as possible on the continuous cooling transition diagram to allow for bainitic formation, and alloying elements such as nickel, molybdenum, manganese and chromium are selected for this purpose [4]. Besides, the high silicon content of 1.3 wt% suppresses the Table 1. Chemical compositions of tested steels (wt%) (Data from [6]).  Monotonic mechanical properties of the tested steels (Data from [6,9] 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 (dε/dt = 6 × 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.

Property
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 (∆We) and plastic strain energy density per cycle (∆Wp) of the tested steels (see Figure 1). Total strain energy density per cycle (∆Wt) was defined as the sum of both components.
Appl. Sci. 2020, 10, x FOR PEER REVIEW 3 of 11   Monotonic mechanical properties of the tested steels (Data from [6,9] Table 3. Low-cycle fatigue properties of the tested steels (Data from [9]).  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.

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 were 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 ∆We and ∆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 ∆ /2 = 0.6%, where steel C exhibited higher plastic strain energy density values than steel B. 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 ∆We and ∆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 ∆ε/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, ∆We increased for lower contents of Mn, except for ∆ε/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 ∆We, ML , ∆Wp, ML and ∆Wt, ML . 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%.     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%. It is interesting to note that, at a fixed strain amplitude, steel A presented the greatest values of cumulative plastic strain energy density, similarly to plastic strain energy density per cycle; while steel C exhibited the highest cumulative elastic strain energy density, as with elastic strain energy density per cycle.  Concerning cumulative total strain energy density, Wt, it was possible to conclude that steel C with higher Mn content disclosed the highest values at strain amplitudes lower than 1.0%, while for ∆ɛ/2 = 1.0%, the highest value was exhibited by steel A. In addition, in a log-log scale, there was a    It is interesting to note that, at a fixed strain amplitude, steel A presented the greatest values of cumulative plastic strain energy density, similarly to plastic strain energy density per cycle; while steel C exhibited the highest cumulative elastic strain energy density, as with elastic strain energy density per cycle.
Concerning cumulative total strain energy density, Wt, it was possible to conclude that steel C with higher Mn content disclosed the highest values at strain amplitudes lower than 1.0%, while for ∆ε/2 = 1.0%, the highest value was exhibited by steel A. In addition, in a log-log scale, there was a direct proportionality between cumulative total strain energy density and the determined fatigue resistance of the tested steels (see Figure 5). As shown, the data could be successfully fitted by a single power function independent of Mn content and with a high correlation coefficient, which is a major and interesting outcome.
It is also interesting to note that elastic strain energy density of the mid-life cycle (∆We, ML ) could be correlated with cumulative strain energy density in log-log scales by means of a straight line, as shown in Figure 6. The values of ∆We, ML are summarized in Table 4. In this case, unlike cumulative total strain energy density versus the number of cycles to failure, the computed curves did not overlap. In the absence of Mn, the curve was shifted to the left. The increase of Mn content moved the curves to the right. Moreover, the slope tended to decrease as the content of Mn rose.
It is also interesting to note that elastic strain energy density of the mid-life cycle (∆We,ML) could be correlated with cumulative strain energy density in log-log scales by means of a straight line, as shown in Figure 6. The values of ∆We,ML are summarized in Table 4. In this case, unlike cumulative total strain energy density versus the number of cycles to failure, the computed curves did not overlap. In the absence of Mn, the curve was shifted to the left. The increase of Mn content moved the curves to the right. Moreover, the slope tended to decrease as the content of Mn rose.

Fatigue Life Prediction
Numerical fatigue lives were calculated for the three steels under study and the four strain amplitudes applied in strain-controlled fatigue tests using the well-known SWT model (Equation (1)) and the Liu model (Equation (2)). These calculations were performed with the stress and strain values of the mid-life cycle (see Table 5). After that, numerical results were compared with the obtained and interesting outcome.
It is also interesting to note that elastic strain energy density of the mid-life cycle (∆We,ML) could be correlated with cumulative strain energy density in log-log scales by means of a straight line, as shown in Figure 6. The values of ∆We,ML are summarized in Table 4. In this case, unlike cumulative total strain energy density versus the number of cycles to failure, the computed curves did not overlap. In the absence of Mn, the curve was shifted to the left. The increase of Mn content moved the curves to the right. Moreover, the slope tended to decrease as the content of Mn rose.

Fatigue Life Prediction
Numerical fatigue lives were calculated for the three steels under study and the four strain amplitudes applied in strain-controlled fatigue tests using the well-known SWT model (Equation (1)) and the Liu model (Equation (2)). These calculations were performed with the stress and strain values of the mid-life cycle (see Table 5). After that, numerical results were compared with the obtained

Fatigue Life Prediction
Numerical fatigue lives were calculated for the three steels under study and the four strain amplitudes applied in strain-controlled fatigue tests using the well-known SWT model (Equation (1)) and the Liu model (Equation (2)). These calculations were performed with the stress and strain values of the mid-life cycle (see Table 5). After that, numerical results were compared with the obtained experimental data (N f ) listed in Table 5. For the sake of clarity, a graphical comparison between the predicted and experimental lives is displayed in Figure 7. As can be seen, numerical predictions were almost exact for steel B and resulted in nonconservative values for steel A and steel C. Furthermore, we could conclude that the Liu criterion led to predictions tendentially safer than the SWT criterion. strain energy density and the number of cycles to failure (see Figure 5), and the power functions relating elastic strain energy density of the mid-life cycle and the associated cumulative total strain energy density (see Figure 6) found for the high-strength bainitic steels under study.  Regarding the determination of elastic strain energy density, defined here as the sum of both the area associated with the linear portion of the descending branch from maximum stress to zero and the area associated with the linear portion of the ascending branch from minimum stress to zero, the calculation could be carried out by means of the following equation: Taking into account some inaccuracies of the tested criteria, a new predictive model, based on the cumulative total strain energy density, was developed. The proposed model lies on energy relationships presented in Section 3.2, more precisely, the power function relating cumulative total strain energy density and the number of cycles to failure (see Figure 5), and the power functions relating elastic strain energy density of the mid-life cycle and the associated cumulative total strain energy density (see Figure 6) found for the high-strength bainitic steels under study.
Regarding the determination of elastic strain energy density, defined here as the sum of both the area associated with the linear portion of the descending branch from maximum stress to zero and the area associated with the linear portion of the ascending branch from minimum stress to zero, the calculation could be carried out by means of the following equation: where ∆σ is stress range, and E is Young's modulus. The stress range for a specific strain amplitude could be estimated from the cyclic curve, i.e., Appl. Sci. 2020, 10, 7774 where k is the cyclic hardening coefficient and n is the cyclic hardening exponent (see Table 3). Once the applied stress was known, the associated elastic strain energy density per cycle could be computed from Equation (3). Thus, based on the energy relationships of Figure 6, we could compute the cumulative total strain energy densities, which allowed the calculation of the corresponding fatigue life from the power function of Figure 5. The predicted values (N CTSED ) for different strain amplitudes of the three high-strength bainitic steels are presented in Table 5. Figure 8 compares the predicted lives with those determined in the experiments. As can be seen, the N CTSED values were more accurate than those computed through SWT and Liu criteria (see Figure 7).
where ∆σ is stress range, and E is Young's modulus. The stress range for a specific strain amplitude could be estimated from the cyclic curve, i.e., where k′ is the cyclic hardening coefficient and n′ is the cyclic hardening exponent (see Table 3). Once the applied stress was known, the associated elastic strain energy density per cycle could be computed from Equation (3). Thus, based on the energy relationships of Figure 6, we could compute the cumulative total strain energy densities, which allowed the calculation of the corresponding fatigue life from the power function of Figure 5. The predicted values (NCTSED) for different strain amplitudes of the three high-strength bainitic steels are presented in Table 5. Figure 8 compares the predicted lives with those determined in the experiments. As can be seen, the NCTSED values were more accurate than those computed through SWT and Liu criteria (see Figure 7). In order to analyze the predictive capabilities of the proposed model in a more systematic manner, probability density functions were used, defined from the following error parameter: where Np is the predicted lifetime, and Nf is the experimental life. In theory, more accurate models are those with lower standard deviations and mean errors close to zero. As shown in Figure 9, the proposed model combined these two features, since it had a mean error closer to zero and a lower standard deviation than the other two. Furthermore, it led to conservative predictions, which was another important outcome. Table 6 presents several statistical parameters computed from the Nf/Np ratios obtained for each model, namely minimum error, maximum error, mean, standard deviation and variance. In fact, the cumulative total strain energy density (CTSED) approach led to more accurate results, since its mean deviation was closer to zero and its standard deviation was smaller. In order to analyze the predictive capabilities of the proposed model in a more systematic manner, probability density functions were used, defined from the following error parameter: where N p is the predicted lifetime, and N f is the experimental life. In theory, more accurate models are those with lower standard deviations and mean errors close to zero. As shown in Figure 9, the proposed model combined these two features, since it had a mean error closer to zero and a lower standard deviation than the other two. Furthermore, it led to conservative predictions, which was another important outcome. Table 6 presents several statistical parameters computed from the N f /N p ratios obtained for each model, namely minimum error, maximum error, mean, standard deviation and variance. In fact, the cumulative total strain energy density (CTSED) approach led to more accurate results, since its mean deviation was closer to zero and its standard deviation was smaller.

Conclusions
The present paper studied cumulative strain energy density behavior in three bainitic rail steels with similar chemical compositions but different Mn contents under a symmetric strain-controlled mode. Based on the high correlation between cumulative total strain energy density and fatigue lifetime, a new predictive model, capable of estimating the durability of high-strength bainitic steels, was developed. Predictions were compared to those of the well-known SWT and Liu criteria. The following conclusions can be drawn: • The cyclic stress-strain response was affected remarkably by Mn content. For each steel, the stress ranges associated with a given strain amplitude were significantly different, as was the shape of the hysteresis loops.

•
In the absence of Mn, cumulative plastic strain energy density was higher than cumulative elastic strain energy density. Nevertheless, when Mn content was maximum, opposite behavior was observed.

•
Regarding cumulative total strain energy density at higher strain amplitudes, the maximum values were found for the steel with the highest Mn content. On the contrary, at lower strain amplitudes, the maximum Wt values were found for the Mn-free bainitic steel. • A very good correlation between cumulative total strain energy density and fatigue life was found. These two variables were successfully fitted by a single power function independent of Mn content.

•
In the form of a power function, a strong correlation between cumulative total strain energy density and elastic strain energy density at the mid-life was found. Nevertheless, in this case, these relationships depended on Mn content. • The new model founded on cumulative total strain energy density led to more accurate fatigue life predictions than those of the well-know SWT and Liu criteria, more specifically, lower standard deviations and mean errors closer to zero.

Conclusions
The present paper studied cumulative strain energy density behavior in three bainitic rail steels with similar chemical compositions but different Mn contents under a symmetric strain-controlled mode. Based on the high correlation between cumulative total strain energy density and fatigue lifetime, a new predictive model, capable of estimating the durability of high-strength bainitic steels, was developed. Predictions were compared to those of the well-known SWT and Liu criteria. The following conclusions can be drawn:

•
The cyclic stress-strain response was affected remarkably by Mn content. For each steel, the stress ranges associated with a given strain amplitude were significantly different, as was the shape of the hysteresis loops.

•
In the absence of Mn, cumulative plastic strain energy density was higher than cumulative elastic strain energy density. Nevertheless, when Mn content was maximum, opposite behavior was observed.

•
Regarding cumulative total strain energy density at higher strain amplitudes, the maximum values were found for the steel with the highest Mn content. On the contrary, at lower strain amplitudes, the maximum Wt values were found for the Mn-free bainitic steel. • A very good correlation between cumulative total strain energy density and fatigue life was found. These two variables were successfully fitted by a single power function independent of Mn content.

•
In the form of a power function, a strong correlation between cumulative total strain energy density and elastic strain energy density at the mid-life was found. Nevertheless, in this case, these relationships depended on Mn content.

•
The new model founded on cumulative total strain energy density led to more accurate fatigue life predictions than those of the well-know SWT and Liu criteria, more specifically, lower standard deviations and mean errors closer to zero.