Influence of the Ductility Exponent on the Fatigue of Structural Steels

Abstract

Fatigue models using the strain-life method do not show exact conformity with the empirical results. Therefore, the use of the mean-stress correction approach is to be evaluated, with a particular focus on mild and higher-strength steel. The influence of the ductility parameters will be studied. A potential favorable development of structural steels with regard to ductility will be checked. The paper will focus on two types of structural steel: S355 and S700. Initially, the mechanical properties of the steel test specimens were measured via a tensile testing rig. In addition, a fatigue test was carried out by applying various mean-stresses. Surface roughness was measured at the notch and introduced into the initial model. The strain amplitudes were determined using the Ramberg-Osgood and Masing material models. Subsequently, a curve fitting was applied to the strain-life data for the fatigue ductility exponent. The multiparameter model was fitted with only one parameter. The resulting model showed a good fit between the strain-life curve and the test results. During the course of the optimization, the error and the scatter were calculated separately for steel types S355 and S700. Based on the ductility exponent, a favorable behavior of the materials was determined.

1. Introduction

The research focuses on two structural steels, S355 and S700. The strength characteristics and the elongation at break are determined in the tensile testing, and the tolerable cycles are tested in fatigue. The types of steel, hereby, differ in terms of strength characteristics and ductility.

The strain-life approach is a concept that is appropriate for the calculation of fatigue life for details, as shown in [1]. The strain-life curve helps to understand how fatigue life and strain amplitude are interconnected.

The strain-life curve is described by: $\sigma {{}^{\prime }}_{\mathrm{f}}$ the fatigue strength coefficient, $b$ the fatigue strength exponent, $\epsilon {{}^{\prime }}_{\mathrm{f}}$ the fatigue ductility coefficient, and c the fatigue ductility exponent [2]. Equation (1) of Manson-Coffin-Morrow from [3] also recourses to these same parameters. These cyclic parameters for the strain-life curve are shown in Figure 1.

(1)${\epsilon }_{\mathrm{a}}={\epsilon }_{\mathrm{a},\mathrm{el}}+{\epsilon }_{\mathrm{a},\mathrm{pl}}=\frac{\sigma {{}^{\prime }}_{\mathrm{f}}}{E}{\left(2N_{\mathrm{f}}\right)}^b+\epsilon {{}^{\prime }}_{\mathrm{f}}{\left(2N_{\mathrm{f}}\right)}^c$

The literature provides values for the cyclic parameters; this is the case, for instance, in [3,4,5,6]. The underlying values of Figure 1 are based on the Uniform Material Law (UML) from [7] cited in [4].

Ductility is a required property of steel, especially in civil engineering. The strain-life curve offers the possibility to give a statement about the ductility value. Here, the plastic part ${\epsilon }_{a,pl}$ is taken into consideration. The strain-life equation from Coffin-Manson-Morrow (1) provides an indication of the amplitude of the plastic strains on the number of cycles. The two parameters, which are part of the equation, evaluate ductility.

The plastic part of the strain-life curve is represented as a line in the double-logarithmic coordinate system, with the load reversals and the plastic strain amplitudes plotted [3]. Therefore, the reversals are drawn on the horizontal axes as described in [5]. Hereby, the strain amplitude is given in mm/mm, and fatigue life is measured in cycles or reversals. The plastic part is characterized by the fatigue ductility coefficient $\epsilon {{}^{\prime }}_{\mathrm{f}}$ and the fatigue ductility exponent $c$.

(2)${\epsilon }_{\mathrm{a},\mathrm{pl}}=\epsilon {{}^{\prime }}_{\mathrm{f}}·{\left(2N_{\mathrm{f}}\right)}^c$

As discussed in [8], several factors have an influence on fatigue life, and mean-stress is one of them. In the approach of Morrow, mean stress has a decisive influence on fatigue life. In this approach, the shown curve is changed as it takes into consideration a value for mean-stress. This adaptation by Morrow has influences exclusively on the elastic part of the equation, which means that the influence on the low cycle fatigue is lower [9]. Smith-Watson-Topper established their own approach to calculating fatigue life under the influence of mean-stress. In their method, fatigue life is expressed in relation to the damage parameter $P_{\mathrm{SWT}}$ [10]. In [11], available approaches for the procedure of correcting mean-stress are further developed. For this purpose, additional parameters are used, and a distinction is made between tensile and compressive mean stresses.

(3)${\epsilon }_{\mathrm{a}}={\epsilon }_{\mathrm{a},\mathrm{el}}+{\epsilon }_{\mathrm{a},\mathrm{pl}}=\frac{\sigma {{}^{\prime }}_{\mathrm{f}}-{\sigma }_{\mathrm{m}}}{E}{\left(2N_{\mathrm{f}}\right)}^b+\epsilon {{}^{\prime }}_{\mathrm{f}}{\left(2N_{\mathrm{f}}\right)}^c$

Surface roughness is one of several parameters that influence fatigue strength [8]. According to the Forschungskuratorium Maschinenbau (FKM) guideline, the influence of roughness can be calculated using specific material properties and roughness parameters. The influence of roughness is considered in the strain-life curve at parameter $b$, as shown in [12]. In [13], the influence of surface roughness is also applied by means of parameter $b$. According to those guidelines, the impact of surface roughness increases with higher cycle numbers.

In [14], similar materials are tested in fatigue testing. A comparison is made between mild steel and high-strength steel to make a statement for the application of components. In [15], similar materials are also tested. In the tensile test, stress-strain curves are determined. These curves correspond to the results of the tensile tests implemented in this case. A fitting of the strain-life curve is carried out based on the cyclic material parameters.

A fitting of the determined test results is performed in [16] to compare the base material of higher-strength steels with welded structures. The ductility parameters are adjusted. Flat specimens with a drilled hole are used to investigate the basic material of steel (type S355 and S700). The strain amplitudes, corresponding to the Manson-Coffin-Morrow equation, are determined for the tests. In this work, the multiparameter model from Equation (1) is fitted by optimizing a single parameter. For this purpose, the ductility exponent $c$ is taken. This intends to provide a more accurate prediction of fatigue life.

The fatigue crack is induced at the round notch. The notch is produced by drilling a hole in the specimen. Neuber’s rule is used to determine the stress and strain amplitudes. In addition, the notch is investigated using an FE analysis. The material models are used to determine the stress and strain amplitudes that occur.

2. Materials and Methods

Cyclic material parameters. In order to determine the cyclic material parameters for both structural types of steel, two methods are selected from a whole series of possible material laws [17]. Ductility is not directly included in both methods of determining the cyclic material parameters. Firstly, the cyclic material parameters of the Uniform Material Law are defined [7] as cited in [4]. Furthermore, the FKM method is used [6].

The FKM method used to determine the cyclic material parameters specifies set values for $b$ and $c$. The values $\sigma {{}^{\prime }}_{\mathrm{f}}$ and $\epsilon {{}^{\prime }}_{\mathrm{f}}$ are calculated using the formulas shown, being dependent upon $R_{\mathrm{m}}$. The tensile strengths result from calculating the average of the values measured in the tensile testing. The calculated material parameters are used in the figures of the following chapters for steel types S355 and S700. The FKM Method’s validity range is given by [6] and is from $R_{\mathrm{m}}=121$ to $2296 \mathrm{MPa}$.

(4)$\sigma {{}^{\prime }}_{\mathrm{f}}=3.1148 \mathrm{MPa} · {\left(\frac{R_{\mathrm{m}}}{\mathrm{MPa}}\right)}^{0.897}$

(5)$b=-0.097$

(6)$\epsilon {{}^{\prime }}_{\mathrm{f}}=\mathit{min}\left(\begin{array}{c}0.338\\ 1033·{\left(\frac{R_{\mathrm{m}}}{\mathrm{MPa}}\right)}^{-1.235}\end{array}\right)$

(7)$c=-0.52$

(8)$K^{\prime }=\frac{\sigma {{}^{\prime }}_f}{{\left(\epsilon {{}^{\prime }}_{\mathrm{f}}\right)}^{n^{\prime }}}$

The cyclic material parameters can also be calculated using the Uniform Material Law. The values $\sigma {{}^{\prime }}_{\mathrm{f}}$ and $\epsilon {{}^{\prime }}_{\mathrm{f}}$ are determined as a function of the tensile strength. Young’s modulus is also used to assess $\epsilon {{}^{\prime }}_{\mathrm{f}}$. The parameter $n^{\prime }$ is set to the constant value. The value $K^{\prime }$ is determined in the UML only as a function of the tensile strength, in contrast to the previous formulation. In [4], the estimation formulas are given:

(9)${{\sigma }^{\prime }}_{\mathrm{f}}=1.5 R_{\mathrm{m}}$

(10)$b=-0.087$

(11)$\epsilon {{}^{\prime }}_{\mathrm{f}}=0.59 \mathsf{\Psi }$

(12)$c=-0.58$

(13)$K^{\prime }=1.65·R_{\mathrm{m}}$

(14)$n^{\prime }=0.15$

(15)$\mathsf{\Psi }=1.0 \mathrm{for} R_{\mathrm{m}}\le 630 \mathrm{MPa}; \mathsf{\Psi }=1.375-125 · \frac{R_{\mathrm{m}}}{E} \mathrm{for} R_{\mathrm{m}}>630 \mathrm{MPa};$

3. Experiment

Tensile Testing. Tensile testing is performed on specimens of the listed structural steels. From these, the stress-strain curve is determined. Two curves are plotted together in a graph for comparison (Figure 2). Steel type S355 achieves a favorable elongation at break. Type S700 is characterized by a higher tensile strength. The initial cross-section before the start of the tensile test is the same for both types of steel, of 20 mm width and 5 mm thickness. The tensile testing is performed according to EN 6892. After the tensile testing, the remaining cross-sections are measured on the specimens. Due to the cross-section, the shape of the necking is different on the materials tested. The measurement of the fracture zone and the initial cross-section are evaluated, resulting in a value for the necking. In the tensile testing, the elongation at fracture, of the steel type S355 and S700, is determined. Consequently, the fracture area is measured. This is used to determine the degree of necking. For the steel type S355, an elongation at break of 30.97% and necking of 55.40% is reached. The steel type S700 achieves an elongation at fracture of 20.05% with a necking of 50.74%.

Material parameters. The two structural steels are tested in tensile testing. The characteristic values that are determined are listed in Table 1. The structural steels differ in terms of yield strength and tensile strength. They also differ in the elongation at fracture. Both the elongation at fracture and the necking are expected to be related to ductility. The result of the tensile test is considered when determining the cyclic material parameters. It is assumed that a small value of necking leaves a larger cross-section at fracture. This is assumed to be favorable for ductility. A large necking with an associated smaller cross-section at fracture is less favorable.

The influence of the previous factors on fatigue strength is to be investigated through a test and measurements on the specimen. Sheets are used as test specimens. In the middle of the sheet, there is a hole. The sheet is clamped at its ends in the testing machine. The test material is described in Figure 3. During the test, the force is applied to the specimen. The experiments were performed on a hydropulse machine (Schenk Instron, Darmstadt, Germany) with a maximum force of 1600 kN. The force becomes introduced at the sheet and alternates between the maximum value $F_{\max }$ and the minimum value $F_{\min }$. The reversals are repeated cyclically until the macrocrack appears [18]. The macrocrack is detected at an opening length of roughly 0.5 mm.

As Table 2 shows, the number of cycles to failure is known from the fatigue test. The number of cycles can be assigned to the loads applied. In addition, they are converted to reversals when inserted in the strain-life curve. Hereby, it is considered that: $1·\mathrm{Cycl}\triangleq 2·\mathrm{Reversals}$ [9]. The cyclic material parameters and Young’s modulus are used to generate the strain-life curve. Based on the corresponding strains from the model according to Masing, the strain-life curve is used to calculate fatigue life. The strain-life curve is generated according to the equation of Manson-Coffin-Morrow. The values corresponding to the FKM method [6] are used as cyclic material parameters.

Surface roughness. The roughness of the surface has an influence on the test results, which is why surface roughness is measured to determine the effect on the outcome. A profile of the roughness is determined, as well as the roughness values $R_{\mathrm{a}}$ and $R_{\mathrm{z}}$. Furthermore, thanks to the results of all measurements, a mean value is determined. A distinction is made between steel of type S355 and S700. The values determined are, hence, used to consider the influence of roughness on fatigue. According to [12], the achieved number of cycles is reduced by high roughness. The influence of roughness is more significant in the area of high-cycle fatigue than in the area of low-cycle fatigue [8].

Roughness measurements are carried out using the Handysurf + roughness measuring device from Accretech (Tokyo Seimitsu, Tokyo, Japan), allowing the identification of the needed sizes by focusing on the measuring point at the notch. For this purpose, the device passes through the drill hole, taking individual measurements. The measurement is performed on the specimens 760 times. The mean value is then calculated. The single measurement is performed with the surface measuring device on the component contour. Taking into consideration different locations and calculating the mean value allows for the determination of the roughness values. For series steel type S355, these are determined as $R_{\mathrm{a}}=1.85 \mathsf{\mu }\mathrm{m}$ $\left(s=0.66\right)$ and $R_{\mathrm{z}}=10.42 \mathsf{\mu }\mathrm{m}$ $\left(s=3.26\right)$. For steel type S700, the roughness values are identified as follows: $R_{\mathrm{a}}=0.90 \mathsf{\mu }\mathrm{m}$ $\left(s=0.32\right)$ and $R_{\mathrm{z}}=5.48 \mathsf{\mu }\mathrm{m}$ $\left(s=1.65\right)$.

4. Calculation Model

FKM Method and UML. The equations from Section 2 for determining the cyclic material parameters are calculated separately, according to the FKM method and the Uniform Material Law. This procedure is applied to both steel type S355 and steel type S700. The resulting individual values are correlated; these concerns ${{\sigma }^{\prime }}_{\mathrm{f}}$ and b, such as $\epsilon {{}^{\prime }}_{\mathrm{f}}$ and the value $c$. The $\epsilon {{}^{\prime }}_{\mathrm{f}}$ and ${{\sigma }^{\prime }}_{\mathrm{f}}/E$ result in the value of strain at $0.5·\mathrm{N}\triangleq 1 Reversal$ [4]. The parameter $n^{\prime }$ indicates the position of the elastic and plastic life lines. This value results from the ratio of the exponents $b$ and $c$. Considering the two methods, there is a difference in ${{\sigma }^{\prime }}_{\mathrm{f}}$ and $b$. However, there is also a relatively larger difference in the values of $\epsilon {{}^{\prime }}_{\mathrm{f}}$ and $c$. This difference has a significant effect on the low cycle fatigue (LCF). The calculated parameters are enumerated in Table 3.

Stress concentration. The maximum tension at the notch is calculated according to the formula from [20], a formula specifically meant for the calculation of round notches, just like the drilling holes at the center of the sample sheets from this experiment. The calculated factor is multiplied by the elastic calculated tension from the load case. Subsequently, the cyclic material behavior is taken into consideration. With the help of Neuber’s rule and the hyperbola, plastic deformation is considered, as described in [18] and mentioned in [5]. Hereby, the Ramberg-Osgood relation, as mentioned in Equation (16), is used; this relation has been taken from [21] as cited in [19].

(16)${\epsilon }_{\mathrm{a}}={\epsilon }_{\mathrm{el}}+{\epsilon }_{\mathrm{pl}}=\frac{{\sigma }_{\mathrm{a}}}{E}+{\left(\frac{{\sigma }_{\mathrm{a}}}{K^{\prime }}\right)}^{\frac{1}{n^{\prime }}}$

In order to generate the hysteresis loop under mean stress influence also, the model of Masing as cited in [3], is used. In the model of Masing, the plastic strains are multiplied with factor 2, as shown in Equation (17). The material model in the just-mentioned equation is activated after reaching the force’s turning point, as in [3].

(17)$\Delta \epsilon =\frac{{\sigma }_{\mathrm{a}}}{E}+2{\left(\frac{\Delta \sigma }{2K^{\prime }}\right)}^{\frac{1}{n^{\prime }}}$

Parameter $n^{\prime }$. The values of the parameter $n^{\prime }$, the cyclic strain hardening exponent, and K′, the cyclic strain hardening coefficient, are used for determining the plastic strain in Equations (16) and (17). While $K^{\prime }$ is determined according to the Uniform Material Law, the value of $n^{\prime }$ depends on the parameters b and c [9]. The value of $n^{\prime }$ has a significant influence on the calculation of the plastic strain. Thanks to the value n′, a statement about ductility can be made.

(18)$n^{\prime }=\frac{b}{c}$

Hysteresis loop of stress. In Figure 4, the used material model UML is applied to the specific sample for $R=-1/3$. As Figure 3 illustrates, the applied force changes from the maximum value 322 kN to the minimum value −107.5 kN. The beginning point in the figure is located at point zero. Step one hereby follows the Equation (16) until the tension reaches the maximum value ${\sigma }_{\max }$ and finds itself at the turning point. As a result, the course of tension follows Equation (17) and the second turning point takes place when the minimum value of tension has been reached. In the further course of the cyclic hysteresis loop, steps two and three are then repeated continuously. The material properties are ‘bimodular’ as described in [3].

Mean-stress. As noted in [22], hysteresis is influenced by mean-stress. There is no mean-stress influence for a load ratio $R=-1$ on the hysteresis loop. In the case of mean-stress of ${\sigma }_{\mathrm{m}}=0$, the center of the hysteresis loop is at point zero. Under the influence of mean-stress, the center is displaced from point zero. The load ratio $R\ne -1$ shows that mean-stress is nonzero:${\sigma }_{\mathrm{m}}\ne 0$. The same can be observed in some of the tested specimens. An overview of the applied maximum and minimum forces and load ratios is given in Table 2.

The mean-stress value of the specimens from the present experiment is zero or positive; in this regard, there is no influence of mean-stress, according to the equation by Manson-Coffin-Morrow, or the value of mean-stress is a value of tension expressed by ${\sigma }_{\mathrm{m}}$. A positive value of mean-stress has a negative influence on fatigue life as mentioned in [8]. The present experiment aims to provide fatigue life results under the influence of mean stress.

Plastic strain simulation. The finite element method is used to compare the results from the calculation according to [20]. An example is taken from the test series with the load ratio $R=-1$. The upper and lower loads are 215 kN and −215 kN. The first step from Figure 5 represents the assumed material behavior. A multilinear kinematic hardening model in ANSYS is used for this purpose. The applied material model follows Equation (16). The strain is composed of an elastic and plastic part: ${\epsilon }_{\mathrm{el}}+{\epsilon }_{\mathrm{pl}}$.

Due to the concentration of stress in the notch, local plasticizing is expected. The plastic strain component appears in the zone of high stress. The stress peaks even out at the edges. The location of maximum strain is expected to take place where the fatigue crack occurs. The fatigue crack appearing in the notch has been confirmed in the test in the form of a macrocrack as described by [23]. The predicted plastic strains are determined by material law and thus depend on the parameter $n^{\prime }$. This is compatible with the equation of Manson-Coffin-Morrow from [4], as it relates to the elastic and plastic part of the strain amplitude. The influence of ductility on the plastic part ought to be shown. The parameter $c$ is supposed to allow a statement about ductility.

The strain amplitude is derived from the hysteresis loop [5] and is equal to one-half of ∆ε. It is composed of elastic and plastic components, whereby the influence of the plastic part is much smaller in the region of very high cycle fatigue. The number of achievable load reversals according to the Manson-Coffin-Morrow equation and the measured load reversals are evaluated. The differences between the calculated load reversals and the number of cycles to failure can be seen in Figure 6 and Figure 7 for S355 and S700. The differences become clear when comparing the points from the experiments with the calculated corresponding number of cycles or reversals on the existing strain-life curve. They are plotted logarithmically. In the boundary regions, the total strain amplitude corresponds to the separate elastic and plastic life lines.

The fatigue cycles can be determined according to the Manson-Coffin-Morrow equation and differ from the measured load cycles. The deviation is influenced by several possible factors [8]. A significant influence is on the material of the experiment. However, it is to be expected that influencing factors also interfere with one another during the modification. With the optimization of the parameter $c$, however, the material-specific ductility value is to be shown. The variation of $c$ is associated with the favorable influence of the ductile behavior of the structural steels. This will be used to conclude the material behavior in the tensile testing. Therefore, ductility becomes evident on the basis of necking and elongation.

Modification of parameter $b$. The surface roughness measurements carried out in the laboratory are performed at the relevant position in the drill hole, since stress concentration is present in this position, as shown in Figure 5. Surface treatment and the resulting roughness have an influence on parameter $b$ [12]. In this case, parameter $b$ is an exponent in the Manson-Coffin-Morrow equation. Therefore, the parameter should be set in advance to match the associated roughness. Based on the value for $b$ from the FKM method specification, a modified value $b_{\mathrm{sur}}$ is calculated. The calculation of the roughness factor $K_{\mathrm{S}}$ is derived from the FKM guideline [24]. The influence of roughness is considered according to Equation (20). Here, $K_{\mathrm{S}}$ is used at a specific number of cycles, which is $N_{\mathrm{f},\mathrm{limit}}={10}^6 \mathrm{cycles}$. The influence increases continuously in the high cycle fatigue [12].

(19)$K_{\mathrm{S}}=1-a_{\mathrm{R}}·\log \left(R_{\mathrm{z}}\right)·\log \left(\frac{2·R_{\mathrm{m}}}{R_{\mathrm{m},\mathrm{N},\min }}\right)$

Calculation of the parameter $b_{\mathrm{sur}}.$ The value $K_{\mathrm{S}}$ given in the equation is determined according to the FKM guideline [24]. The parameters $a_{\mathrm{R}}$ and $R_{\mathrm{m},\mathrm{N},\min }$ depend on the material group. The value $R_{\mathrm{m}}$ corresponds to the tensile strength and it is determined in the tensile test. The value $R_{\mathrm{z}}$ corresponds to the average roughness of the surface and it is determined by the measured roughness profile. The tensile strength is much higher for steel type S700, as shown in the stress-strain curve. However, the roughness is finer than in the case of steel type S355. According to the categories in [12], the range of values goes between polished and ground. The determined values for $b_{\mathrm{sur}}$ are given in Figure 8 and Figure 9. The influence of roughness on the parameter $b$ is also specified in the design standard [13] for cranes. There, a method for modifying $b$ is shown, which considers the average roughness as the roughness’s parameter $R_{\mathrm{a}}$. Besides roughness, residual stress can also have an impact on the outcomes.

(20)$b_{\mathrm{sur}}=b+\frac{log\left(K_{\mathrm{S}}\right)}{log\left(N_{\mathrm{f}, \mathrm{limit}}\right)}$

Mean-stress influence, according to Morrow. Mean-stress also influences the strain-life curve. Morrow [25], cited in [3], carries out a modification of the strain-life curve. The value of mean stress is considered in Equation (21). However, mean stress influences the elastic life line only, i.e., the first part of the Manson-Coffin-Morrow equation. Since in the low cycle fatigue, the plastic life line has an even more significant influence, the effect of mean-stress in high cycle fatigue becomes decisive. The mean-stress influence is characterized by a change in the elastic life line. In the case of tensile mean-stress, the elastic life line is shifted downwards, and the cycles that can be sustained are reduced.

(21)${\epsilon }_{\mathrm{a},\mathrm{el}}=\frac{\sigma {{}^{\prime }}_{\mathrm{f}}-{\sigma }_{\mathrm{m}}}{E}{\left(2N_{\mathrm{f}}\right)}^b$

The test results in Figure 8 are sorted according to the load ratios. This provides evidence about which test results have no mean-stress influence $\left(R=-1\right)$ and the results with mean-stress influence $\left(R\ne -1\right)$. Since only tensile mean-stress is present, the cycles that can be supported are reduced. The tensile mean-stress reduces the cycles that can be sustained on the strain-life curve. The strain amplitude ranges within the studied area at the specified levels. This is defined prior by means of the strain amplitudes. In this way, the results are grouped with regard to the load ratio and the strain amplitudes in the figure. A similar approach is taken for Figure 9. The strain-life curve is calculated from the plastic and modified elastic life lines.

According to Morrow’s approach, the influence of mean-stress on the results should be considered. Based on the results, a fitting of the Manson-Coffin-Morrow curve to the present experimental results needs to be done. For this purpose, the optimization will include mean-stress correction so that the associated mean-stress can be considered.

Optimization. The fitting of the strain-life curve is done by changing parameter $c$. The abbreviation OPT stands for optimized. An optimization function from MATLAB is used for this purpose. In the error function, the equation of Manson-Coffin-Morrow is accessed. The method is applied to the mentioned mean-stress correction and the modified value $b_{\mathrm{sur}}$. The experimental results are considered in the error function. The least squares method is used for this task in the double logarithmic coordinate system. The associated values are specified according to the steel type and are determined using the FKM method.

(22)$error\left(c_{\mathrm{opt}}\right)=min{\displaystyle \sum } {\left(\log \left(N\right)-\log \left(N_{\mathrm{f}}\right)\right)}^2$

5. Results

The function of error. The error function has been found in the selected region in Figure 10. The region is determined by parameter $c$. The minimum of the found error function is located in the region of interest. The minimum fulfills the above condition. Parameter $c_{\mathrm{opt}}$ is associated with the smallest error size in the specified region. Thus, the optimum value of parameter $c$ is specified. This is done for steel type S355 and steel type S700. Parameter $c$ and the error functions are different when referring to the steel types. The error size is related to the number of tests and is plotted correspondingly.

Size of error. The error size is calculated from the equation described above. A smaller error size obtained indicates a more favorable fit of the strain-life curve. The error size for steel type S700 is larger than the one for steel type S355. This indicates a higher scattering of the test results of steel type S700. With the curve shown in Figure 10, the searched value of parameter $c$ is shown. Subsequently, the scattering is illustrated. Table 4 presents the results of fitting the strain-life curve for the parameter c and the size of the minimum error.

In examining the influence of the parameters $b$ and $c$, a fitting is to be carried out. The change of $b$ has, among other effects, a higher effect on the range of very high cycle fatigue. Parameter $c$ is in the part of plastic strains in the equation of Manson-Coffin-Morrow. Thereby, the influence of parameter $c$ is essential in the low and high cycle fatigue. The parameter $c$ is considered to be a measure of the ductility of the two structural steels.

The fitting of parameter $c$ for steel type S355 is conducted. The result is summarized in Figure 11. The other cyclic parameters correspond to the FKM method. Except for the optimized parameter $c$, the parameter $b$ is modified. The calculated values for parameter $b$ are under the influence of roughness. Parameter $c$ from the FKM method is listed in the figure based on the result for parameter $c$ from the optimization. The experimental results are plotted in the figures and are listed in Table 2 for steel type S355. The plastic line, with the value $c$ from the FKM method and with the optimized parameter $c$, is shown. The strain-life curve is composed of the modified and optimized elastic and plastic lines. There is an intersection point between the modified elastic and optimized plastic life lines. This intersection point is offset from the original intersection point. The value for $n^{\prime }$ in the figure is calculated from the ratio of the modified parameter $b$ and the optimized value of parameter $c$.

Parameter $c$ and $n^{\prime }$. The parameter $c$ has a significant influence on the region in which the test results are located. The ductility exponent determines the slope of the plastic part. Up to the point of the intersection of the plastic line with the elastic line, the part of plastic strain in the total strain is larger. The intersection point is in the region of the experimental results. By optimizing parameter $c$, as a result of ductility, the intersection of the elastic and plastic lines is moved in the direction of the fatigue strength. Thus, ductility is assigned an influence on the plastic life line. Parameter $n^{\prime }$ depends on $c$. By modifying parameter $c$, it should become possible to draw conclusions about ductility. For this purpose, the strain-life curves of the two steels are considered. In addition, for the steel type S700, the strain-life curve is shown in Figure 12.

Apart from parameter $c$, the value $\epsilon {{}^{\prime }}_{\mathrm{f}}$ has an influence on the plastic life line. Since the experimental data are in the mentioned region, a fitting of $\epsilon {{}^{\prime }}_{\mathrm{f}}$ turns out to be difficult. This is because an extrapolation from the mentioned region would be needed, as mentioned in [3]. This leads to uncertainty in the output. Therefore, modification of both ductility parameters is avoided, and only parameter $c$ is optimized. The adjustment of both parameters leads to small error sizes but to an unsatisfying value of $\epsilon {{}^{\prime }}_{\mathrm{f}}$. The fitting is shown in Figure 13 and Figure 14. The previous Figure 11 and Figure 12 differ with respect to the material model.

As the experimental values are under the influence of mean-stress, mean-stress is considered for all individual values. To plot the strain-life curve, the highest appearing mean-stress is assigned to the elastic component. The curve is shown with only one mean-stress for the purpose of visual comparison. However, to fit the parameter $c$, the corresponding assigned value of mean-stress is considered. Furthermore, the fitting is carried out for steel type S700. For this purpose, the corresponding test results and cyclic material parameters are used. The test results are taken from Table 2. The fitting of the strain-life curve by optimizing parameter $c$ is shown for steel type S700. Again, the elastic part is considered under the influence of mean-stress and roughness. The plastic part is determined by the optimization of parameter $c$.

When fitting the parameter $c$, the value of parameter $n^{\prime }$ is determined over again. In this process, the result of the calculation due to the compatibility condition is included in the calculation of the strain amplitudes. For this purpose, the parameters are inserted and changed step by step. The strain amplitudes are then determined again and used for the fitting. The determined results of the cycle numbers are taken from Table 2 and afterward inserted again. Subsequent to the fitting of the parameters, the strains and stresses are recalculated. The fitted parameter $c$, and consequently, parameter $n^{\prime }$, influence the calculated strain amplitudes according to the Ramberg-Osgood relation and the Masing model. Thereby, the course of the hysteresis curve from Figure 4 changes after the fitting of parameters $c$ and $n^{\prime }$. The calculated strain amplitudes are assigned to the test results.

6. Discussion

The parameters $b$ and $c$ have a decisive influence on the strain-life curve, as discussed in [26]. This is one of the reasons the two parameters stand in the exponent. Parameter $c$ is used as an indicator for the level of ductility of a material. The ductility is also related to fracture elongation and fracture necking. Modern structural steels achieve a lower fracture necking than expected by comparable elongation at fracture. This results in a larger residual cross-section.

Parameter $c$ presents here at a lower value than expected. The larger residual cross-section at a lower necking fracture is considered an advantage. Parameter $c$ is used as a value to measure ductility. Ductility thus influences fatigue behavior. This is the case in low cycle fatigue in particular. Since only the parameter $c$ is fitted and the remaining parameters stay the same, a deviation from the estimation formula is assumed. The increase of parameter $c$ is observed for both types of structural steel. In the data collection [27], values are given for comparison. When fitting the cyclic parameters, the values for parameter c are related to the other values.

Necking and elongation at break. It is assumed that the smallest possible necking is favorable, especially in the case of large strain amplitudes. The necking and elongation at break are not included in the calculation of the cyclic material parameters of the FKM Method or in the method of the Uniform Material Law. It is assumed that the influence of ductility is implied in the specification of the parameters $c$ and $\epsilon {{}^{\prime }}_{\mathrm{f}}$. When the fatigue crack occurs in the specimens, the necking at the edges occurs. Based on the measured necking and fracture strain, a different range of ductility is assigned to the diverse steel types. Therefore, different values are assumed for parameter $c$.

The strain amplitudes are determined based on the optimized parameter $c$, the modified parameter $b$, and the remaining calculated cyclic parameters. Subsequently, the newly calculated strain amplitudes are compared. For this comparison, the parameters from the FKM Method are used to determine the strain amplitudes. The result of the comparison for steel type S355 is shown in Figure 15. The scatter bands have been drawn in the figure. A normal distribution is assumed. In both very high cycle fatigue and low cycle fatigue, due to the size of the results, no fitting of all parameters is applied. In fact, the relevance of the fitting is limited to the range in which the test results are located. The variation of the results from the center line shows the difference in the strain amplitudes before and after the optimization with parameter $c$ has taken place. Since the values given by the FKM method are estimated formulas, deviations from them are to be assumed.

For steel type S700, the strain amplitudes are also plotted according to the strain-life curve, and the scatter bands are drawn. For the determination of the multipliers, ${10}^{\pm 1.28·s}$ from [4] is applied. Thereby the standard deviation (s) is determined with a normal distribution. As can be seen from the strain-life curve, the values calculated after optimization are higher or lower than the strain amplitude determined on the basis of cyclic parameters from the FKM method. The steel type S700 shows a considerably larger scatter of results as the fitting has not been done as conveniently as for steel type S355. This is also evident when looking at the error function in Figure 10. Here, a larger error is found. The resulting values for steel type 355 identified a scatter comparable to that of [7] cited in [4], which refers to unalloyed and low-alloy steels from the data collection [27].

The cycles determined in the experiment are compared to the calculated cycles in Figure 15 and Figure 16. For this purpose, the calculated cycles are determined according to the Manson-Coffin-Morrow equation under the influence of mean stress. The cyclic parameters are calculated based on the FKM method. In addition, the obtained optimized parameter $c$ and the modified parameter $b$ are applied. The deviation of the results indicates the fit of the strain-life curve to the existing test results. A more significant deviation is found for steel type S700. Therefore, the fitting could not be performed as precisely as was the case for steel type S355. This is related to the determined curve of the error functions. A normal distribution was assumed for the calculation of the scatter bands. The normal distribution was tested according to Kolmogorov-Smirnov.

As described, influencing variables such as mean-stress and roughness determine the result for parameters $b$ and $c$. Differences in the test results are already found when determining the influence of mean-stress. Part of the higher scatter for steel type S700 is assigned to the mean-stress correction. Due to the ductility in steels with higher strength, the correction of mean-stress also has a deviation. In the case of steel type S355, the mean-stress correction applied is more appropriate. The influence of roughness becomes apparent in the high cycle fatigue. An optimization without roughness correction does not lead to a satisfactory result with the available test results.

Range of experimental results and extrapolation. The test results are listed in Table 2. The tests are presented in a specific range. This range is limited by the position of the experiments. The lower limit is the smallest measured value in the Low Cycle Fatigue with a number of 2.525 cycles for steel type S355. Steel type S700 reaches a number of 3.240 cycles. The higher limit is described by the largest measured value. For steel type S355, this is 243.762 cycles, and for steel type S700, it is 335.211 cycles. Thus, the test results are in the range of both low cycle fatigue and high cycle fatigue. In the described range, fatigue behavior is to be studied.

7. Conclusions

The results lead to the following arguments:

Statement on mean-stress. Morrow’s mean stress correction is related to mean-stress magnitude and is considered in the Manson-Coffin-Morrow equation for the elastic strain part. This correction is used here both for steel type S355 and for the higher-strength steel type S700. While for the steel type S355, a good fit is obtained, for steel type S700, a larger scatter of results can be noted. The selected mean-stress correction gives a more favorable outcome for steel type S355 than for steel type S700.

Statement on ductility. There has been a positive trend in the ductility of structural steels in the past decades. This is evident from the favorable fracture necking of the structural steels used. Another criterion is parameter $c$. By changing this parameter, an evident good fitting is achieved. A positive influence of the parameter $c$ can be seen for both types of steel chosen. The second ductility parameter in the Manson-Coffin-Morrow equation $\epsilon {{}^{\prime }}_{\mathrm{f}}$ is not fitted. This is due to the associated uncertainty in extrapolation and the available experimental results. Therefore, the fitting is not valid, especially for very small numbers of cycles. In high cycle fatigue, the size of the plastic part grows. However, the fitting is not applicable at very high cycle numbers because the missing fitting of $\epsilon {{}^{\prime }}_{\mathrm{f}}$ leads to a lack of precision.

Fitting the strain-life curve for the two steel types, S355 and S700, results in a change in the fatigue ductility exponent $c$. The change in the ductility exponent $c$ is greater for steel type S355 than for S700. A correlation is established between the change of parameter $c$ and the static ductility parameters necking and elongation at fracture. Consequently, these parameters mainly affect fatigue at low cycles. The result is a more favorable fatigue life due to higher ductility.

Statements about parameter $c$. The multiparameter model of Manson-Coffin-Morrow is well adaptable by one parameter. The ductility exponent $c$ is used for this purpose. The parameter $c$ differs from the estimation formulas applied. The difference indicates the more favorable ductility of the material. The statement generated by the parameter fitting is limited to the range of evaluation in this work and does not suit the range of very high cycle fatigue or under low cycle fatigue. The fitting allows a statement on the ductility of the material and is related to fracture elongation and necking. Ductile behavior is particularly evident in the tensile test. An alternative is to fit the strain-life curve using the results from the fatigue tests.

Author Contributions

Conceptualization, M.K.; methodology, M.K.; software, M.K.; validation, A.N.; formal analysis, M.K.; investigation, A.N. and M.K.; resources, A.N. and M.K.; data curation, M.K.; writing—original draft preparation, M.K.; writing—review and editing, A.N., R.L. and M.K.; visualization, M.K.; supervision, R.L.; project administration, R.L. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Not applicable.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

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Figures and Tables

Figure 1. The strain-life curve according to Equation (1) with the cyclic parameters from [1] in [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.]. The graphs were created with MATLAB.

Figure 2. Examples of two stress-strain curves of the tested steels. Young’s modulus is determined. The strength and the strain differ. These two stress-strain curves are shown as examples. The stress is plotted in (MPa) and the strain in (%). Next to the stress-strain curves, the cross sections after break are illustrated. The upper cross-section relates to the S700 type steel, and the lower part to the S355 type steel.

Figure 3. Setup and the procedure of the experiment: (a) The specimen in the load situation. On the one hand, under a load of pressure, and on the other hand, under a load of tensile [19]. The dimensions given are in mm. (b) The cyclic course of the applied stress in the specimen over time, as in [3]. (c) The hydropulse testing machine in the TVFA laboratory of the University of Innsbruck.

Figure 4. A hysteresis loop from a laboratory-tested experiment for steel type S355. After the first load path, the loop is repeated multiple times under the influence of mean stress. In the figure, the step-by-step procedure is marked.

Figure 5. The state of maximum and minimum stress, and also the strain when the maximum load states are reached without the influence of mean-stress. In Figure (a), the maximum stress concentrated in the area of the notch is given. Figure (b), gives the equivalent situation under a compressive force. The Figure (c) illustrates the local concentration of strain. The states are performed using the finite element method with ANSYS.

Figure 6. The strain-life curve using the cyclic material parameters from the FKM method. Furthermore, the measured reversals are shown together with the corresponding strains calculated by Equation (16). The figure shows the results for steel type S355. The plastic and elastic life lines are plotted, and the total strain is calculated as amplitude. The exponents [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] used for the calculation are listed in the figure. The FKM index at the two exponents refers to the FKM method.

Figure 7. For steel type S700, the cyclic material parameters according to the FKM method have been determined. Once more, the load reversals are plotted. Furthermore, the elastic and plastic parts are indicated.

Figure 8. The strain-life curve is under the influence of mean-stress according to Morrow’s approach concerning surface roughness for steel type S355. The applied mean-stress corresponds to the largest associated value from the tests. The modified elastic life line is drawn below the original elastic life line. In the figure, all measured test values for steel type S355 are listed. They are sorted according to the corresponding load ratio.

Figure 9. The strain life line and the results from the test for steel type S700. The corresponding mean-stress influences the test results. The approach of Morrow takes this into consideration. The elastic component is changed by the effect of mean-stress. The modified elastic life line is shown beside the original elastic life line.

Figure 10. The error is a function of parameter [Forumla omitted. See PDF.] for both tested steel types. Both curves find a minimum error and the corresponding parameter value. Starting from the minimum, a change of parameter [Forumla omitted. See PDF.] causes an increase in error.

Figure 11. The strain-life curve fitted according to the test results for steel type S355. The cyclic material parameters are calculated from the FKM method. The strain-life curve crosses the experimental results in the figure. Mean-stress influence can change the curve.

Figure 12. The strain-life curve for steel type S700 after the fitting of parameter [Forumla omitted. See PDF.]. The cyclic material parameters are calculated from the FKM method. The original plastic line, before the fitting of the parameter [Forumla omitted. See PDF.], is presented according to the specification of the FKM method. The change of parameter [Forumla omitted. See PDF.] is shown after fitting the plastic line. The curve of the plastic line before and after the optimization is plotted.

Figure 13. The strain-life curve fitted to the test results for the steel type S355. The cyclic material parameters are derived from the Uniform Material Law. The modified parameter [Forumla omitted. See PDF.] and the optimized parameter [Forumla omitted. See PDF.] are used to create the strain-life curve.

Figure 14. The strain-life curve fitted according to the test results for the steel type S700. The cyclic material parameters are derived from the Uniform Material Law. The modified parameter [Forumla omitted. See PDF.] and the optimized parameter [Forumla omitted. See PDF.] are used to create the strain-life curve.

Figure 15. The strain amplitudes are plotted according to the equation of Manson-Coffin-Morrow and the equation of Masing for steel type S355 in Figure (a). For the scatter bands, multipliers are calculated to be ([Forumla omitted. See PDF.]) and ([Forumla omitted. See PDF.]. On the right, in Figure (b), the cycle numbers are calculated and plotted. For comparison, the cycles experimentally measured are inserted. The scatter bands are determined with multipliers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] at survival probability of 10 and, respectively, 90 percent. The indicated strain amplitudes and calculated cycles are generated from the cyclic material parameters of the FKM method. The optimized parameter [Forumla omitted. See PDF.] and the modified parameter [Forumla omitted. See PDF.] are considered.

Figure 16. The strain amplitudes are plotted according to Manson-Coffin-Morrow’s equation and Masing’s equation for steel type S700 in Figure (a). For the scatter bands, multipliers are calculated to be ([Forumla omitted. See PDF.]) and ([Forumla omitted. See PDF.]. On the right, in Figure (b), the cycle numbers are calculated and plotted. For comparison, the experimentally measured cycles are inserted. The scatter bands are determined with multipliers [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] at survival probability of 10 and, respectively, 90 percent. The indicated strain amplitudes and calculated cycles are generated from the cyclic material parameters of the FKM method. The optimized parameter [Forumla omitted. See PDF.] and the modified parameter [Forumla omitted. See PDF.] are considered.

Table 1

The values given in the table were determined in tensile testing at the TVFA laboratory of the University of Innsbruck. The values are mean values from six single tests.

Steel Type	Yield Strength	Tensile Strength	Elongation	Necking	Young’s Modulus
S355	$R_{\mathrm{e}}=410$ MPa	$R_{\mathrm{m}}=497$ MPa	$A=30.97\%$	$Z=55.40\%$	$E=\mathrm{208,935}$ MPa
S700	$R_{\mathrm{e}}=725$ MPa	$R_{\mathrm{m}}=814$ MPa	$A=20.05\%$	$Z=50.74\%$	$E=\mathrm{202,471}$ MPa

Table 2

Calculated stress and strain. Results of the fatigue tests. The note (1) or (2) refers to the steel type S355 or S700.

$\mathit{R}$	${\mathit{F}}_{\mathbf{max}}$ $\left(\mathbf{kN}\right)$	${\mathit{F}}_{\mathbf{min}}$ $\left(\mathbf{kN}\right)$	${\mathit{S}}_{\mathit{m}\mathit{a}\mathit{x}}$ $\left(\mathbf{MPa}\right)$	${\mathit{S}}_{\mathit{m}\mathit{i}\mathit{n}}$ $\left(\mathbf{MPa}\right)$	${\mathit{e}}_{\mathit{m}\mathit{a}\mathit{x}}$ $\left(\mathbf{mm}/\mathbf{mm}\right)$	${\mathit{e}}_{\mathit{m}\mathit{i}\mathit{n}}$ $\left(\mathbf{mm}/\mathbf{mm}\right)$	$\mathbf{Steel} \mathbf{Type} \mathbf{S}355$ $\left(\mathbf{Cycles}\right)$	$\mathbf{Steel} \mathbf{Type} \mathbf{S}700$ $\left(\mathbf{Cycles}\right)$
$R=-1$	$285$	$-285$	894	−894	$0.0044{ }^{\left(2\right)}$	$-0.0044{ }^{\left(2\right)}$	-	$4140$ $3600$ $3240$ $3780$
	$215$	$-215$	674	−674	$0.0032{ }^{\left(1\right)}$	$-0.0032{ }^{\left(1\right)}$	$5500$ $4317$ $5035$ $3960$	$\mathrm{22,147}$ $\mathrm{19,961}$ $\mathrm{17,287}$ $\mathrm{18,000}$
	$145$	$-145$	455	−455	${0.0022}^{ \left(1\right)}$	$-0.0022{ }^{\left(1\right)}$	$\mathrm{40,285}$ $\mathrm{34,425}$ $\mathrm{38,087}$ $\mathrm{38,579}$	$\mathrm{119,880}$ $\mathrm{132,120}$ $\mathrm{162,180}$ $\mathrm{83,700}$
	$110$	$-110$	345	−345	${0.0017}^{ \left(1\right)}$	$-0.0017{ }^{\left(1\right)}$	$\mathrm{206,591}$ $\mathrm{210,674}$ $\mathrm{231,610}$ $\mathrm{243,763}$	$\mathrm{332,192}$$\mathrm{335,211}$$\mathrm{285,842}$
$R=-\frac{1}{2}$	$287$	$-143$	899	−449	$0.0043{ }^{\left(1\right)}$	$-0.0022{ }^{\left(1\right)}$	$2520$ $4317$ $4090$ $4389$	-
$R=-\frac{1}{3}$	$323$	$-108$	1011	−337	$0.0048{ }^{\left(1\right)}$	$-0.0016{ }^{\left(1\right)}$	$3955$ $4062$ $4023$ $3801$	$\mathrm{15,217}$ $\mathrm{17,325}$ $\mathrm{13,976}$ $\mathrm{13,939}$
	$218$	$-73$	682	−227	$0.0032{ }^{\left(1\right)}$	$-0.0011{ }^{\left(1\right)}$	$\mathrm{35,508}$ $\mathrm{33,916}$ $\mathrm{36,627}$ $\mathrm{24,840}$	$\mathrm{70,560}$ $\mathrm{57,240}$ $\mathrm{50,580}$ $\mathrm{48,420}$
	$165$	$-55$	517	−172	$0.0025{ }^{\left(1\right)}$	$0.0008{ }^{\left(1\right)}$	$\mathrm{132,501}$ $\mathrm{128,523}$ $\mathrm{116,879}$ $\mathrm{154,504}$	$\mathrm{190,475}$ $\mathrm{162,547}$ $\mathrm{170,144}$ $\mathrm{132,120}$
$R=0$	$430$	$0$	1348	0	$0.0067{ }^{\left(2\right)}$	0	-	$\mathrm{10,389}$ $\mathrm{10,152}$ $\mathrm{10,562}$ $\mathrm{11,940}$
	$290$	$0$	909	0	$0.0044{ }^{\left(1\right)}$	0	$\mathrm{20,700}$ $\mathrm{21,780}$ $\mathrm{26,280}$ $\mathrm{30,000}$	$\mathrm{31,320}$ $\mathrm{31,860}$ $\mathrm{36,900}$ $\mathrm{28,440}$
	$220$	$0$	790	0	$0.0033{ }^{\left(1\right)}$	0	$\mathrm{82,800}$ $\mathrm{84,901}$ $\mathrm{83,568}$ $\mathrm{89,385}$	$\mathrm{96,603}$$\mathrm{123,684}$$\mathrm{99,900}$

Table 3

According to the FKM method and the UML, the cyclic material parameters are inserted in this table. The exponents $b$ and $c$ are given uniformly in both methods. The values are presented next to each other in the table.

Cyclic MaterialProperties	FKM		UML
	S355	S700	S355	S700
${{\sigma }^{\prime }}_{\mathrm{f}}$ $\left(\mathrm{MPa}\right)$	817	1271	746	1221
$b$	−0.097		−0.087
$\epsilon {{}^{\prime }}_{\mathrm{f}}$	0.338	0.263	0.59	0.515
$c$	−0.52		−0.58
$K^{\prime }$ $\left(\mathrm{MPa}\right)$	1001	1631	820	1343
$n^{\prime }$	0.187		0.15

Table 4

Summary of the results from the optimization process. The results are listed separately for steel types S355 and S700. Furthermore, they are separated by the FKM Method and Uniform Material Law. The result of the parameter $c$, the total corresponding error size, and the error size referred to the number of corresponding tests is given.

S355 FKM	$c_{\mathrm{opt}}=-0.444$	$Error=3.71$	$\raisebox{1ex}{$Error$}\!\left/ \!\raisebox{-1ex}{$n$}\right.=0.103$
S700 FKM	$c_{\mathrm{opt}}=-0.461$	$Error=12.66$	$\raisebox{1ex}{$Error$}\!\left/ \!\raisebox{-1ex}{$n$}\right.=0.333$
S355 UML	$c_{\mathrm{opt}}=-0.498$	$Error=4.65$	$\raisebox{1ex}{$Error$}\!\left/ \!\raisebox{-1ex}{$n$}\right.=0.129$
S700 UML	$c_{\mathrm{opt}}=-0.545$	$Error=15.4$	$\raisebox{1ex}{$Error$}\!\left/ \!\raisebox{-1ex}{$n$}\right.=0.405$

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