A New Approach to Estimate the Fatigue Limit of Steels Based on Conventional and Cyclic Indentation Testing

Abstract

For a reliable design of structural components, valid information about the fatigue strength of the material used is a prerequisite. As the determination of the fatigue properties, and especially the fatigue limit σ_w, requires a high experimental effort, efficient approaches to estimate the fatigue strength are of great interest. Available estimation approaches using monotonic properties, e.g., Vickers hardness (HV), and in some cases the cyclic yield strength, only allow a rough estimation of σ_w. The approaches solely based on monotonic properties lead to substantial deviations of the estimated σ_w in relation to the experimentally determined fatigue limit as they do not consider the cyclic deformation behavior. In this work, an estimation approach was developed, which is based on a correlation analysis of the fatigue limit σ_w, HV, and the cyclic hardening potential obtained in instrumented cyclic indentation tests (CIT). For this, eleven conditions from five different low-alloy steels were investigated. The CIT enable an efficient and quantitative determination of the cyclic hardening potential, i.e., the cyclic hardening exponent_CHT e_II, and thus, the consideration of the cyclic deformation behavior in an estimation approach. In this work, a strong correlation of σ_w with the product of HV and |e_II| was observed. In relation to an existing estimation approach based solely on HV, considering the combination of HV and |e_II| enables the estimation of σ_w with an enormously increased precision.

1. Introduction

Materials used in engineering applications are oftentimes subjected to cyclic loadings. Therefore, the knowledge of cyclic properties, and especially the fatigue limit, is indispensable for a safe and reliable design. To determine the fatigue properties, a high experimental effort is required, i.e., a relatively large number of fatigue tests and specimens. Consequently, methods to reduce this effort are of great industrial and scientific interest. For this purpose, short-time procedures [1,2,3,4,5] and correlations between the fatigue limit and mechanical parameters that can be determined with a low experimental effort (e.g., hardness and ultimate tensile stress (UTS)) [6,7,8] were elaborated.

Hardness tests are one of the simplest methods to characterize the monotonic strength of a material. By investigating six different steels, Garwood et al. [6] showed that Rockwell hardness (HRC) roughly correlates with the fatigue limit σ_w. Note that the fatigue limit σ_w discussed in the presented work is defined as the stress amplitude σ_a that leads to an ultimate number of cycles of 10⁶–10⁷, depending on the material investigated and the test frequency used. According to Murakami [9,10], the linear relationship found by Garwood et al. [6] can be expressed based on the Vickers hardness (HV) using Equation (1) (compare Figure 1), with σ_w in MPa. This relationship is valid up to a material-specific hardness value, ranging between 400 and 500 HV [6,9,10]. Note that the fatigue limits given in Figure 1 were determined in rotation bending tests and the deviation observed for materials with a higher hardness is partially caused by residual stresses at the surface [11]. As this loading condition results in a stress gradient from the surface to the inner material, an increased influence of residual stresses in relation to uniaxial fatigue testing occurs. In addition to the residual stresses and the loading condition, it must be considered that a higher surface roughness significantly decreases the fatigue limit [9,12].

σ_w = 1.6 HV ± 0.1 HV

(1)

In addition to Equation (1), the ASTM Handbook [13] provides a linear correlation of the fatigue limit and Brinell hardness (HB) for some steels, which is valid for a hardness up to 500 HB. The resulting correlation factor is 1.72, which is relatively close to Equation (1). However, this is only a rough estimation, as it is based on the assumption that the fatigue limit is half of the UTS [13]. Furthermore, the hardness can also be used to estimate the fatigue limit in case of defect-based failure, as shown by the investigations of Murakami [9,10] and Casagrande et al. [14].

In addition to hardness, the tensile properties, i.e., the yield strength σ_y or the UTS, were also used in different scientific works to assess the fatigue limit [7,8,15,16,17].

Although the abovementioned models provide sufficient estimations for some steels, a reliable prediction of the fatigue limit exclusively based on hardness or tensile properties is not achieved [15,18,19]. This is illustrated by the big scatter band shown in Figure 1, while the deviation between the estimated and experimentally determined fatigue limit increases with increasing hardness [10]. Furthermore, Fleck et al. [15] and Rennert et al. [17] show that the relation between the fatigue limit and yield strength, i.e., the ratio σ_w/σ_y, depends on the type and the strength of the respective steel. In general, in these works, an increase in σ_y leads to a less pronounced growth of σ_w, and thus, with an increase in σ_y, the ratio σ_w/σ_y decreases. This is explained by Fleck et al. [15] with the cyclic deformation behavior: While steels with a lower strength in the initial state show cyclic hardening, most high-strength steels exhibit cyclic softening. This is also underlined by the work of Lopez and Fatemi [19], who observed for two steels with the same hardness a different fatigue limit caused by a different cyclic deformation behavior. Consequently, for a reliable estimation of the fatigue limit σ_w, the cyclic deformation behavior must be considered.

Beyond the possibilities of conventional hardness measurement, instrumented indentation testing enables the analysis of the deformation behavior. Thus, other material properties, e.g., the Young’s modulus E [20,21], can be determined, too. Moreover, the cyclic deformation behavior of a material can be characterized using instrumented cyclic indentation tests (CIT) [22,23]. The short-time procedure PhyBaL_CHT, which is based on CIT and enables determining the cyclic hardening potential, was used in [23] to describe the cyclic deformation behavior of differently heat-treated conditions of the low-alloy steel 42CrMo4. The cyclic hardening potential refers to the capacity of a metallic material to increase its strength during cyclic loading, especially via dislocation activities, which counteracts local stress concentrations, e.g., at microstructural inhomogeneities. As demonstrated in [23], the cyclic deformation behavior obtained with this testing approach corresponds to the cyclic deformation curves determined in uniaxial compressive fatigue tests. Furthermore, Kramer et al. [24] showed for the steel 18CrNiMo7-6 that the cyclic hardening potential determined in CIT correlates with the amount of cyclic hardening obtained in uniaxial push-pull fatigue tests.

In our various investigations, it was demonstrated that the cyclic hardening observed in CIT highly depends on different microstructural phenomena, e.g., the size and distribution of precipitates [25,26,27], the dislocation density [28], and the grain size [24]. Moreover, in [26,27], the results obtained in CIT showed a higher sensitivity to microstructural changes than conventional hardness measurements. Accordingly, in another of our own studies [29] on two differently heat-treated Cu-alloyed steels, X0.5CuNi2-2 and X21CuNi2-2, the ranking of the fatigue strength could only be explained by a combined consideration of the hardness and cyclic hardening potential, as conditions with nearly identical hardness showed a significant difference in σ_w, depending on their cyclic hardening potential.

Based on the results outlined above, PhyBaL_CHT is a promising method to improve the estimation of the fatigue limit of steels as it enables consideration of the cyclic deformation behavior, i.e., the cyclic hardening potential. The high potential of this method is especially underlined by the results reported in [29]. Based on that, in the present work, the relationship between the hardness, the cyclic hardening potential obtained in CIT, and the fatigue limit σ_w was analyzed. For this purpose, different types of low-alloy steels with a bcc lattice structure were investigated. In addition to existing data from [29] for Cu-alloyed steels and [30,31] for different batches of C50E, hardness measurements, CIT, and fatigue tests were performed on differently heat-treated 42CrMo4 and a batch of bainitic 100CrMnSi6-4.

2. Materials and Methods

2.1. Materials

For the correlation analysis presented in this work, which is the basis for an improved estimation of the fatigue limit σ_w, three differently heat-treated batches of 42CrMo4, one batch of 100CrMnSi6-4, five batches of differently heat-treated Cu-alloyed steels with two different Cu contents, and two batches of C50E (railway steel R7) were used. In summary, the presented analyses are based on eleven conditions of five different low-alloy steels.

As a condition with a relatively high hardness, a bainitic 100CrMnSi6-4 with the chemical composition given in

Table 1. Chemical compositions of 42CrMo4 [23] and 100CrMnSi6-4.

Element in wt.-%	C	Si	Mn	P	S	Cr	Mo	Ni	Cu	Al
100CrMnSi6-4	0.93	0.59	1.09	0.01	0.01	1.46	-	0.03	0.01	<0.001
42CrMo4	0.37	0.20	0.76	0.002	0.027	1.02	0.16	0.20	0.19	0.02

was investigated. To generate a bainitic microstructure, the material was austenitized at 880 °C for 30 min and subsequently bainitized at 220 °C for 8 h, resulting in a retained austenite fraction of 20 vol.-%.

The 42CrMo4 conditions investigated comprise one normalized as well as two quenched and differently tempered conditions. As these variants are from the identical batch investigated in [23], a more detailed description of the microstructure and heat treatment parameters can be found in this preliminary publication. However, the chemical composition is given in Table 1 while the annealing temperatures T_a and times t_a are listed in Table 3.

For the Cu-alloyed steels, a detailed description of the chemical compositions, heat treatment parameters, and the resulting microstructure is given in [29,32]. The variants considered in this work differ in their C contents, i.e., 0.005 wt.-% (X0.5CuNi2-2) and 0.21 wt.-% (X21CuNi2-2), and in their annealing times t_a, respectively. Moreover, a quenched condition of X0.5CuNi2-2 was analyzed. Please note that these laboratory melts were named with our own labeling based on DIN EN 10027-1 [33] and constitute two low-alloy steels.

Moreover, two batches of C50E, which were extracted from a railway wheel and differ in extraction positions and thus, cooling conditions, were used. As these batches were extracted from a component, no defined heat treatment parameters can be provided. However, the resulting microstructures are characterized in detail in [27,28].

2.2. Mechanical Characterization

To analyze the correlation of the fatigue limit σ_w and the cyclic deformation behavior obtained in CIT, the respective data is required. Thus, for all material conditions, the cyclic hardening potential was determined using the short-time procedure PhyBaL_CHT, which is described in detail in [23,24]. For this, instrumented cyclic indentation tests (CIT) were performed using a Fischerscope H100C device and a Fischerscope HM2000 device (both from Helmut Fischer GmbH, Sindelfingen, Germany), which are both equipped with a Vickers indenter and enable a continuous measurement of the indentation force F and the indentation depth h. For the specimens of the high-strength steel 100CrMnSi6-4, a maximum indentation force of F_max = 2000 mN was used while the other steels were tested with F_max = 1000 mN. In CIT, a sinusoidal load function with a frequency of 1/12 Hz and, in total, 10 cycles were used. For each variant, 20 indentations were performed at 2 different polished sections, respectively. Thus, the results of each variant were determined based on 40 CIT, resulting in a high statistical reliability. To exclude any interference between the indentation points, they were placed at a distance of at least five times the indent diagonal, which is in accordance with [34].

From the second cycle on, the continuously measured signals result in an F-h hysteresis (compare Figure 2a). In analogy to the plastic strain amplitude obtained from a stress–strain hysteresis, the half width of the F-h hysteresis at mean loading is defined as the plastic indentation depth amplitude h_a,p. Similar to the cyclic deformation curve, the development of h_a,p versus the number of cycles N is used to describe the cyclic deformation behavior (compare Figure 2b). The resulting h_a,p–N curve shows a stabilized slope from the fifth cycle on, which indicates saturation of the macro plastic deformation and a domination of microplasticity. This regime of the h_a,p–N curve can further be expressed by the power function h_{a,p II} given in Equation (2) [24]:

5 ≤ N ≤ 10: h_{a,p II} = a_II·N^eII

(2)

The exponent e_II describes the slope of h_{a,p II} and, thus, of the h_a,p–N curve in the stabilized regime. As a steeper slope of the h_a,p–N curve indicates a more pronounced cyclic hardening, e_II is used to quantify the cyclic hardening potential of the investigated material and, thus, is called the cyclic hardening exponent_CHT. A steeper slope of the curve, which is described by a higher absolute value of the exponent |e_II|, indicates a higher cyclic hardening potential, i.e., a higher capacity to counteract local stress concentrations [23,24,26,29]. As already discussed in the introduction, the fatigue strength relates to the cyclic hardening potential and, thus, |e_II| is considered for the correlation analyses. Note that the cyclic hardening exponent_CHT is not equivalent to the cyclic strain hardening exponent n’ in the Ramberg–Osgood curve as e_II is, in contrast to n’, determined in a multi-axial compressive stress state.

In addition to e_II, the Vickers hardness was measured for all material variants, as Equation (1) is the starting point of the correlation analyses presented in this work. The Vickers hardness (HV) was measured with a ZwickRoell ZHU 250top universal hardness testing device (ZwickRoell GmbH & Co. KG, Ulm, Germany). In the case of the conditions of the steels X0.5CuNi2-2, X21CuNi2-2, and C50E, a test force of 98.07 N was used (HV 10) while the variants of steel 42CrMo4 and 100CrMnSi6-4 were tested with 294.12 N (HV 30). However, as the Vickers hardness is considered to be independent of the test force, the hardness values can be used adequately. Each HV value was determined by calculating the arithmetic mean of eight measurements. Please note that the values of HV of the steels X0.5CuNi2-2 and X21CuNi2-2 previously published in [32] differ slightly from the ones used in this work because additional hardness measurements were taken into account.

To correlate the values of hardness and cyclic hardening exponent_CHT with the fatigue limits, valid data of σ_w is required for each condition. The fatigue limits of the different variants of X0.5CuNi2-2 and X21CuNi2-2 were obtained by extrapolating the power functions of the Woehler curves given in [29] to the ultimate number of cycles N_L = 2 × 10⁶, which is commonly used to determine the fatigue limit of these types of steels. Because of the small lifetime scatter observed for this material, this approach provides a sufficient reliability. For the steel C50E, σ_w was determined by VHCF experiments with N_L = 2 × 10⁸, as presented in [30]. However, because no failure occurred beyond ~10⁶ cycles, the fatigue limit can be defined at N_L = 2 × 10⁶.

As for the 100CrMnSi6-4 variant investigated in this work, no fatigue data were available, a Woehler curve was determined for this condition. To this end, fatigue specimens with a polished gauge length and the geometry given in Figure 3a were used. Before polishing, the specimens were hard-turned, which could lead to pronounced residual stresses in the surface. Thus, the process-induced residual stresses were measured at a 40 kV tube voltage, 40 mA tube current and a scanning speed of 0.004°/s using an X-ray diffractometer of the type PANalytical Empyrean (Malvern Panalytical B.V., Almelo, the Netherlands) equipped with a Cu Kα₁ tube. The fatigue tests were stress-controlled with a stress ratio R = −1 and a frequency of 10 Hz. For this, a servo-hydraulic testing system of the type Schenck PSA 40 was utilized. The tests were performed at ambient temperature up to N_L = 2 × 10⁶.

Similarly, for the variants of 42CrMo4, fatigue experiments were required, as in [23] only single fatigue tests were performed, which did not enable the estimation of σ_w. The fatigue specimens of 42CrMo4 had a geometry as illustrated in Figure 3b with a polished gauge length. For the quenched and tempered conditions, the fatigue tests were conducted in stress control with the resonant testing device RUMUL TESTRONIC 200 kN (RUSSENBERGER PRÜFMASCHINEN AG, Neuhausen am Rheinfall, Switzerland) and a frequency of approximately 86 Hz while for the normalized variant, the resonant testing device RUMUL TESTRONIC 100 kN (RUSSENBERGER PRÜFMASCHINEN AG, Neuhausen am Rheinfall, Switzerland) and a frequency of approximately 72 Hz were used. All tests were conducted at ambient temperature up to N_L = 10⁷ cycles.

To determine the fatigue limits σ_w of the 42CrMo4 variants, the staircase method, which is described in detail in [35], was used. For each condition, the first experiment of this approach was performed at a stress amplitude σ_a,start, being in the range of σ_w. To roughly estimate σ_a,start, preliminary fatigue tests were performed for each condition. Each following step depended on the results of the prior experiment. If the specimen had reached the maximum number of cycles N_L (run-out), the following experiment was conducted at a higher stress amplitude, which was obtained by multiplying the former stress amplitude with the staircase factor d_log. If an experiment led to failure, the stress amplitude was reduced by dividing the previous stress amplitude by d_log. Therefore, all stress amplitudes σ_n were calculated as described in Equation (3) [35].

The staircase factor d_log depends on the scatter of the tested material and is calculated using the expected or known standard deviation of the whole data set (compare Equation (4)). For the tested conditions, the exponent s_log was estimated based on the preliminary fatigue experiments. For the normalized condition and the variant annealed at T_a = 650 °C, s_log was set to 0.01 while for the variant annealed at T_a = 550 °C, an s_log of 0.03 was chosen. After finishing the test series, a fictitious data point can be set, depending on the result of the last experiment. The fictitious point was set to the next smaller stress amplitude in case of failure or at the next higher stress amplitude if the experiment resulted in a run-out [35]. An exemplary fictitious test series of the staircase method is illustrated in Figure 4.

σ_n = σ_a,start × (d_log)ⁿ

(3)

d_log = 10^slog

(4)

The fatigue limit σ_w was calculated based on the number of stress levels i and the number of events f_i on each stress level. The lowest stress level was assigned to i = 0 and each higher stress level had the next larger number i + 1. Using Equations (5) and (6), the factors F and A were calculated, which were used to determine σ_w with Equation (7). Based on this, the fatigue limit σ_w was determined with a failure probability of 50% [35]:

F = Σf_i

(5)

A = Σ(i × f_i)

(6)

σ_w = σ₀ × d_log^(A/F)

(7)

3. Results

3.1. Determination of the Fatigue Limit of Differently Heat-Treated 42CrMo4 and 100CrMnSi6-4

To obtain statically reliable fatigue limits σ_w of the different conditions of 42CrMo4, the staircase method was used, leading to the results summarized in

Table 2. Parameters of the staircase method F, A, and fatigue limit σ_w of the 42CrMo4 conditions.

Material Condition	F	A	σw in MPa
normalized	10	11	261
Ta = 650 °C, ta = 2 h	8	10	402
Ta = 550 °C, ta = 2 h	10	19	479

and Figure 5a. As expected, the normalized condition showed the lowest σ_w while the condition annealed at T_a = 550 °C yielded the highest fatigue limit.

Considering the failure mechanisms that led to these σ_w, differences can be observed between the conditions. While for the normalized condition, crack initiation was only observed at surface slip bands, the quenched and tempered conditions also exhibited crack initiation at nonmetallic inclusions. In addition to crack initiation at defects, the condition annealed at T_a = 650 °C also showed crack initiation at slip bands at the surface. Note that for this condition, only one defect with √area = 40 µm was observed, which did not lead to a decrease in the number of cycles to failure N_f. Thus, no relevant influence of microstructural defects on the σ_w determined is expected for this condition. In contrast to the other conditions, the variant annealed at T_a = 550 °C only showed fatigue crack initiation at nonmetallic inclusions. However, apart from one specimen fractured at a stress amplitude of σ_a = 450 MPa, which was caused by a defect with √area = 190 µm, only small defect sizes (√area < 90 µm) were observed. As for the small defects, no correlation between the defect size and N_f was obtained, and as the one specimen with a bigger defect has only an insignificant influence on the value σ_w for T_a = 550 °C, the influence of microstructural defects on σ_w could also be neglected.

Since for the 100CrMnSi6-4 a smaller number of specimens was available, for this condition, a conventional Woehler curve was determined, which is shown in Figure 5b. The results obtained show a, in comparison with typical high-strength steels, relatively low scatter, which is caused by the small crack-initiating defect sizes, i.e., √area < 25 µm. Consequently, it is expected that also for this variant, the influence of crack-initiating defects on the fatigue limit is weak. Because of the small lifetime scatter, the Woehler curve shown in Figure 5b enables a valid determination of the fatigue limit of this material. For this purpose, the regression line was extended to N_f = 2 × 10⁶ and based on this, the fatigue limit σ_w is assessed to be approximately 930 MPa (compare Figure 5b).

The fatigue limits of the material conditions of Cu-alloyed steels and the C50E were determined in analogy to 100CrMnSi6-4. For these conditions, only fatigue crack initiation at the surface and no crack-initiating defects were observed and thus, for these variants, the influence of microstructural defects can be neglected, too.

3.2. Relationship between the Fatigue Limit, Hardness, and Cyclic Hardening Potential

To elaborate an improved approach for an estimation of the fatigue limit based on cyclic indentation testing, for all material conditions considered, the fatigue limit σ_w, the macro hardness (HV), and the cyclic hardening exponent_CHT e_II were required. Thus, all conditions were characterized by hardness measurements and CIT. The results obtained in indentation testing and the experimentally determined fatigue limits are listed in

Table 3. Fatigue limit σ_w, Vickers hardness (HV), and cyclic hardening exponent_CHT |e_II| of the investigated steels.

Material	Condition	σw in MPa	HV	|eII|	References
X0.5CuNi2-2	quenched	400	182 ± 3	0.453 ± 0.027	[29,32]
	Ta = 550 °C, ta = 120 s	460	205 ± 2	0.463 ± 0.022
	Ta = 550 °C, ta = 2400 s	510	207 ± 3	0.537 ± 0.023
X21CuNi2-2	Ta = 550 °C, ta = 120 s	480	257 ± 3	0.383 ± 0.024	[29,32]
	Ta = 550 °C, ta = 2400 s	530	263 ± 3	0.429 ± 0.023
C50E	position 1	370	275 ± 3	0.290 ± 0.007	[30]
	position 3	340	260 ± 4	0.289 ± 0.008
42CrMo4	normalized	261	192 ± 6	0.292 ± 0.010	[23]
	Ta = 650 °C, ta = 2 h	402	286 ± 2	0.292 ± 0.009
	Ta = 550 °C, ta = 2 h	479	364 ± 5	0.270 ± 0.007
100CrMnSi6-4	-	930	709 ± 5	0.274 ± 0.007

, which is the data basis for the correlation analyses shown in the following. Table 3 also contains the standard deviation of HV and the 90% confidence interval of e_II, illustrating that the scatter of HV and |e_II| is low for all materials.

The σ_w values shown in Table 3 were determined using fatigue specimens with polished surfaces, leading to an elimination of the influence of the surface roughness. As the specimens were manufactured from bars with diameters significantly bigger than the gauge diameters using low feed rates and manual polishing, thermally and finishing-induced residual stresses are assumed to be relatively low. Additionally, only push-pull fatigue testing (R = −1) was applied and, thus, the whole material volume in the gauge length was loaded homogenously, leading to a less pronounced impact of the residual stresses and surface roughness in relation to other loading conditions, e.g., rotating bending. Consequently, the influence of the residual stresses was neglected.

Note that the specimens of 100CrMnSi6-4 were hard-turned and subsequently polished. Due to the hard-turning, for this condition, pronounced compressive residual stresses in the loading direction (−733 MPa) were measured in the surface-near area, which corresponds to [36]. However, the crack initiation started at defects located at a distance to the surface bigger than 800 µm, which, hence, significantly exceeds the depth of the residual stresses induced by hard-turning (compare [36]). Consequently, the influence of the residual stresses induced by hard-turning on σ_w can be neglected in this case, too.

In summary, the values of σ_w given in Table 3 are mainly related to the mechanical properties of the material volume. In this context, the properties of the material volume refer to the integral, global material properties in contrast to local ones, e.g., at the surface. More details about the microstructure and the properties of the material volume are given in the reference mentioned in Table 3. As the results determined in indentation testing are also dominated by the properties of the material volume, the material parameters listed in Table 3 constitute an excellent basis for the approach proposed.

Initially, for all material conditions considered in this work, the relation between the fatigue limit and the Vickers hardness was analyzed. For this purpose, the σ_w are plotted against the respective HV in Figure 6a. Moreover, in Figure 6a, the commonly used Equation (1) is represented by a straight line, showing a rough correlation with the experimental data. However, some conditions exhibit a rather large deviation from the relation given in Equation (1), especially in the case of 100CrMnSi6-4 and the Cu-alloyed steels. In accordance with [6,9], the relatively hard 100CrMnSi6-4 (706 HV) yields lower σ_w than expected from Equation (1). However, for the softer Cu-alloyed steels, a significantly higher σ_w can be observed in relation to Equation (1). It should, thus, be noted that the deviation from this estimation approach is not limited to high-strength conditions.

Comparing e_II of these materials, the overestimated 100CrMnSi6-4 exhibits low |e_II| while the underestimated Cu-alloyed steels all show strong cyclic hardening and, thus, a high |e_II| (see Table 3). Consequently, the relative deviation between the experimentally determined σ_w and the estimation based on Equation (1) was quantified by σ_w/(1.6 × HV). These relative deviations were related to |e_II|, which is illustrated in Figure 6b. Applying a linear regression of σ_w/(1.6 × HV) as a function of |e_II| results in an impressively good correlation, which can be expressed by Equation (8). Note that Equation (8) contains all three material parameters considered in this work. By rearranging Equation (8), the fatigue limit σ_w can be expressed as the dependency of HV and |e_II| (see Equation (9)):

σ_w/(1.6 × HV) = 2.946 × |e_II| + 0.006

(8)

σ_w = 4.7136 × |e_II| × HV + 0.0096 × HV

(9)

The summand 0.0096×HV of Equation (9) results from the mathematical regression analysis and represents the section point with the σ_w-axis. This summand is considered as a mathematical artifact. Note that even for a very high hardness of 1000 HV, this summand would shift the estimated fatigue limit by only 9.6 MPa and consequently, it can be neglected.

Considering Equation (9), the fatigue limit correlates with the product of HV and |e_II|. However, the factor (4.7136) given in Equation (9) was calculated based on the estimation from Equation (1), which yields pronounced deviations. Thus, a regression analysis of σ_w and HV × |e_II| was examined only based on the data given in Table 3, which is illustrated in Figure 7.

The diagram shown in Figure 7 demonstrates an excellent correlation between the fatigue limit σ_w and the product HV × |e_II|, which can be expressed mathematically by Equation (10). Thus, this equation can be used to estimate σ_w based on the Vickers hardness (HV) and the cyclic hardening exponent_CHT |e_II|. Note that this relation covers data with a high range of HV × e_II| (56 to 194) and σ_w (261 MPa to 930 MPa):

σ_w = 4.82 × HV × |e_II| − 7.18

(10)

However, the factor (4.82) and the subtrahend (7.18) of this equation result from regression analysis and, hence, highly depend on the data basis used. Although the data used in the presented work is sufficient to verify the fundamental applicability of the relation, the explicit quantification requires a substantially bigger data basis, especially in the range between σ_w = 500–900 MPa. Consequently, from the available data, only the general form of this relation can be derived, which is given in Equation (11):

σ_w = C₁ × HV × |e_II| + C₂

(11)

4. Discussion

The regression analysis conducted in Section 3.2 results in a new estimation approach of the fatigue limit σ_w based on the hardness and the cyclic hardening potential determined in CIT (see Equations (10) and (11)). To analyze the improvement of the estimation achievable with this approach, an additional regression analysis was performed on a reduced data set, without the conditions X0.5CuNi2-2 (t_a = 2400 s), X21CuNi2-2 (t_a = 2400 s), C50E (position 3), and 42CrMo4 (T_a = 650 °C). As illustrated in Figure 8a, the regression based on Equation (11) and the reduced data set leads to C₁ = 4.82 and C₂ = −2.34. These coefficients were used to estimate the fatigue limit of the omitted material conditions. Figure 8b demonstrates that the experimentally determined fatigue limits σ_{w, exp.} are very close to the estimated fatigue limit σ_{w, est.}. To quantitatively compare the estimation based on HV × |e_II| with the estimation based solely on HV, the relative differences (Δσ_{w, est.}) between σ_{w, exp.} and σ_{w, est.} estimated with Equations (1) and (11), respectively, were determined for each condition by Equation (12). The resulting deviations are illustrated in Figure 8c:

Δσ_{w, est.} = (σ_{w, exp.} − σ_{w, est.})/σ_{w, est.}

(12)

As demonstrated by the results shown in Figure 8c, a strong improvement of the fatigue limit estimation is realized by considering the cyclic hardening potential in addition to the hardness. In accordance with the results shown by Fleck et al. [15] and Fatemi and Lopez [19], this improvement is caused by the integration of the cyclic deformation behavior, i.e., the cyclic hardening exponent_CHT e_II.

Although e_II is determined based on a relatively small number of cycles (N = 10), it was shown in preliminary work that this material parameter strongly corresponds to the cyclic deformation curves [23,24] as well as the fatigue strength and lifetime [29,32,37] determined in uniaxial fatigue tests. Moreover, the limited number of cycles enables a fast (about 2 h) analysis of the cyclic deformation characteristics. Consequently, the cyclic hardening exponent_CHT obtained with PhyBaL_CHT allows the consideration of the cyclic deformation behavior in an efficient approach to estimate the fatigue limit σ_w.

Note that the estimation approach presented here was only validated for low-alloy steels with several carbon contents and a bcc lattice structure. Moreover, only push-pull fatigue experiments with R = −1 were considered and, thus, this approach needs to be extended to other loading conditions, e.g., rotational bending testing. In addition to this, the data basis is limited to polished surfaces and materials whose fatigue limit is not affected by microstructural defects and residual stresses.

Despite the abovementioned constraints, this approach yields promising results for a wide range of construction materials and, thus, has high potential for industrial and scientific applications. However, a bigger data basis is required, which is an objective of future research.

5. Conclusions and Prospects

In this work, a new approach to estimate the fatigue limit σ_w of low-alloy steels is presented. For this purpose, the correlation of the fatigue limit σ_w with the Vickers hardness (HV) and the cyclic hardening potential was analyzed based on eleven material conditions, which comprise five different low-alloy steels. To determine the cyclic hardening potential, instrumented cyclic indentation tests (CIT) were performed under each material condition. By analyzing the cyclic deformation behavior in CIT, the cyclic hardening exponent_CHT |e_II|, which represents the cyclic hardening potential of the material, was determined.

The results obtained demonstrate that an estimation of σ_w simply based on hardness measurement is insufficient. However, by considering the combination of HV and the cyclic deformation behavior, i.e., the cyclic hardening potential, the estimation of the fatigue limit σ_w can be improved enormously. Consequently, the cyclic hardening potential, which represents the cyclic deformation behavior, and can be easily determined by cyclic indentation testing, was used for the first time to obtain a valid estimation of σ_w, requiring only a small volume of material and a short testing time. Note that for this testing approach, the microstructure and residual stress state of the samples should be equivalent to the specimens or components considered. Thus, this approach has high potential for industrial and scientific applications.

Although the data considered in the presented work is sufficient for verification of the fundamental viability, an extended data base is required for a valid quantification. Moreover, the transferability of this approach to other materials, e.g., aluminum alloys, austenitic steels, or titanium alloys, needs to be investigated [38,39]. Furthermore, the cyclic hardening potential might also be used to improve the estimation of the fatigue limit in the case of defect-based failure. In addition to this, the scatter of the hardness, cyclic hardening potential, and resulting fatigue limit must be integrated [40]. Finally, an analysis of the influence of the loading condition, surface roughness, residual stress state, and microstructural defects, which were excluded in this work by the experimental design, is required, too. These aspects will be objectives of future work.