Cyclic Tests of Smooth and Notched Specimens Subjected to Bending and Torsion Taking into Account the Effect of Mean Stress

The paper contains the results of fatigue tests of smooth and notched specimens made of 10HNAP (S355J2G1W) subjected to proportional cyclic loading with use of mean values stress. The results obtained for specimens under bending, torsion and one combination of bending with torsion for four mean values have been compared. The experimental data have been collected in the tables and shown in the figures with use of various σa(τa)-N fatigue characteristics for which parameters of the regression equations have been determined. The influence of average values on the allowable stress amplitudes and amplitude of moments at the level close to the fatigue limit depending on the angle α determining loading combination and the average stress is also shown. The greatest effect of the notch on fatigue life compared to smooth specimens is observed at symmetrical loads. At unsymmetrical loads with non-zero mean stress, this effect clearly weakens or disappears.


Introduction
Some structures are subjected to bending, torsion or a combination of both loads, which cause fatigue processes in materials.
In addition, mean stresses occur in the structure load, resulting e.g., from the self-weight of the structure or the initial loads associated with the functions fulfilled by the structure. It is then necessary to adopt certain calculation models for fatigue life, taking into account the multiaxial load status and the effect of mean load values, and to perform their experimental verification [1][2][3][4][5]. In real structures, stress concentrations often occur due to the notch in the form of changes in cross-sectional dimensions, which significantly affect the fatigue strength of materials. Therefore, it becomes important to examine the impact of the notch on the fatigue life of the structure [6,7]. Modern methods of assessing fatigue life of machine elements can be divided into two groups. The first is characterized by an analytical approach based on spectral analysis of stochastic processes, while the second is characterized by an algorithmic approach based on numerical methods. These methods are constantly modified and improved so that you can meet the requirements currently set by designers.
Smaga et al. [8] present description on fatigue and transformation behavior at surrounding temperature and 300 • C of austenitic stainless steel AISI 347 in the whole fatigue range from Low Cycle Fatigue (LCF) to Very High Cycle Fatigue (VHCF). The test results show a significant effect of the chemical composition and temperature of induced martensite creation and the behavior of cyclic strain.
The local approach to predict the fatigue strength of sharply notched components can be applied to welded joints in a simplified form oriented to practical applications [9]. In this paper, satisfactory calculation results, based on the analytic expressions of the local stress field have been shown. McDiarmid introduced some modifications of criterion determining the shear stress amplitude to take into account the stress mean value and the impact of the stress concentrator on the final result [10].
Rozumek et al. [11] gave a comparison between the experimental and calculated fatigue lives of specimens made of 10HNAP steel under non-proportional random bending with torsion including changes of strain energy density in the critical plane.
Literature sources present various review papers describing the characteristics of materials [12], which allow estimating the durability of structures and devices [13]. The ASTM [14] standards, which define and describe the method of determining stress (σ-N) and strain (ε-N) characteristics, are one of the most commonly used in the description of tests.
The purpose of the research described in the work is to investigate the behavior of smooth and notched specimens made of 10HNAP (S355J2G1W) steel under conditions of cyclic loading with the presence of various mean values. A wide database of material parameters describing its response for fatigue loads is necessary for the formulation and verification of strength calculation models. This data requires research and improvement for notched structures exposed to multiaxial fatigue loads.

Materials and Fatigue Test Stand
The subject of the study is 10HNAP (S355J2G1W) constructional steel included in the European Standard (EN 10155) "Low-alloy rust-resistant steel". It is a general purpose weldable steel, with increased resistance to atmospheric corrosion. The content of elements in the tested steel is included in Table 1, and strength properties in Table 2. The specimens were cut from 16 mm thick sheet in accordance with the rolling direction. Figure 1 shows the shape and dimensions of the specimens. For a notched specimen the theoretical stress concentration factor, determined in accordance with Neuber's model [15], is for bending K tB = 2.46 and for torsion K tT = 1.76. McDiarmid introduced some modifications of criterion determining the shear stress amplitude to take into account the stress mean value and the impact of the stress concentrator on the final result [10].
Rozumek et all [11] gave a comparison between the experimental and calculated fatigue lives of specimens made of 10HNAP steel under non-proportional random bending with torsion including changes of strain energy density in the critical plane.
Literature sources present various review papers describing the characteristics of materials [12], which allow estimating the durability of structures and devices [13]. The ASTM [14] standards, which define and describe the method of determining stress (σ-N) and strain (ε-N) characteristics, are one of the most commonly used in the description of tests.
The purpose of the research described in the work is to investigate the behavior of smooth and notched specimens made of 10HNAP (S355J2G1W) steel under conditions of cyclic loading with the presence of various mean values. A wide database of material parameters describing its response for fatigue loads is necessary for the formulation and verification of strength calculation models. This data requires research and improvement for notched structures exposed to multiaxial fatigue loads.

Materials and Fatigue Test Stand
The subject of the study is 10HNAP (S355J2G1W) constructional steel included in the European Standard (EN 10155) "Low-alloy rust-resistant steel". It is a general purpose weldable steel, with increased resistance to atmospheric corrosion. The content of elements in the tested steel is included in Table 1, and strength properties in Table 2. The specimens were cut from 16 mm thick sheet in accordance with the rolling direction. Figure  1 shows the shape and dimensions of the specimens. For a notched specimen the theoretical stress concentration factor, determined in accordance with Neuber's model [15], is for bending KtB = 2.46 and for torsion KtT = 1.76. The tests were carried out on the fatigue test stand MZGS-100 [16][17][18]. The stand used in the experiment allows testing for fatigue bending, torsion and proportional bending with torsion. The The tests were carried out on the fatigue test stand MZGS-100 [16][17][18]. The stand used in the experiment allows testing for fatigue bending, torsion and proportional bending with torsion. The tests Materials 2020, 13, 2141 3 of 13 of cyclic bending and torsion were performed in low cycle fatigue (where number of cycles to failure is 10 4 -10 5 ) and high cycle fatigue (where number of cycles to failure is 10 5 -10 7 ) regime [19] with controlled load (in the considered case, the moment amplitude was controlled). The loads were of a sinusoidal nature under a frequency between 25-29 Hz. The amplitudes and the mean value were changed in accordance with the applied load. The nominal stress amplitude and nominal mean stress value (stress ratio) were used in calculations.
The load state of the specimen depends on the angular position of the rotary head of the stand, which is defined by the angle α determining the combination of bending and torsion ( Figure 2). Materials 2020, 13, x FOR PEER REVIEW 3 of 13 tests of cyclic bending and torsion were performed in low cycle fatigue (where number of cycles to failure is 10 4 -10 5 ) and high cycle fatigue (where number of cycles to failure is 10 5 -10 7 ) regime [19] with controlled load (in the considered case, the moment amplitude was controlled). The loads were of a sinusoidal nature under a frequency between 25-29 Hz. The amplitudes and the mean value were changed in accordance with the applied load. The nominal stress amplitude and nominal mean stress value (stress ratio) were used in calculations. The load state of the specimen depends on the angular position of the rotary head of the stand, which is defined by the angle α determining the combination of bending and torsion ( Figure 2).
where ω is the frequency of load changes. The vectors of the torsional moment M Tα (t) and bending moment M Bα (t) (Figure 2) are related to the following relationships [20] ( ) In the extreme position α=0 rad the specimen is bent with a variable moment and in the position α=π/2 rad, the specimen is twisted with a torque The bending and torsional moments are related to the angle of rotation of the head with clamp ( Figure 2 The result of the operation of both synchronized moments is a state of stress in which nominal normal stresses σ α (t) and shear stresses τ α (t) change their value according to phase and with the same angular frequency (ω = 29 Hz): The specimen load consists of a cyclically variable moment with amplitude M a and its mean value M m : where ω is the frequency of load changes. The vectors of the torsional moment M Tα (t) and bending moment M Bα (t) ( Figure 2) are related to the following relationships [20] M In the extreme position α = 0 rad the specimen is bent with a variable moment and in the position α = π/2 rad, the specimen is twisted with a torque The bending and torsional moments are related to the angle of rotation of the head with clamp ( Figure 2) with proportion: In intermediate positions 0 < α < π/2 rad, both moments occur simultaneously and their values are determined from: Materials 2020, 13, 2141

of 13
The result of the operation of both synchronized moments is a state of stress in which nominal normal stresses σ α (t) and shear stresses τ α (t) change their value according to phase and with the same angular frequency (ω = 29 Hz): where: W x and W 0 are section modulus for bending and torsion, respectively. A stress ratio for bending R B and torsion R T are equal [21], since: and Main stresses in the plane stress state considered here on the surface of the specimen (σ 1 0, σ 3 0 and σ 2 = 0) can be saved for the condition R B = R T and compatible phases as where: Directions of main stresses in relation to stress σ α (t) are rotated by angles

The Terms of Fatigue Tests
Smooth and notched specimens made of 10HNAP (S355J2G1W) steel were subjected to fatigue testing. Fatigue tests included synchronous, sinusoidal variable loads with an additional mean loads.
The tests covered three states of specimens loading. The following mean stresses were used for individual values of the angle α of rotation of the machine head (a) α = 0 rad (bending), The tests were performed for 4-5 levels of stress amplitude, 2-3 specimens at each load level.

Fatigue Characteristics of the Material
The results of fatigue tests were approximated by the following regression Equations [14]: where: σ a -bending stress amplitude, τ a -torsional stress amplitude, N-number of cycles for failure, A and m-parameters of the regression equation. Tables 3 and 4 contain the numerical values of the parameters of the regression Equations (14) and (15), as well as the values of the correlation coefficient r for variables logN and logσ a and logarithm variance values from the number of cycles N, i.e., µ logN for smooth and notched specimens at various load combinations. The parameter m, which in the double-logarithmic coordinate system is the slope coefficient σ a -N, takes negative values and shows fluctuations. The largest changes are observed in the case of bending, where m is clearly smaller for smooth specimens compared to notched specimens. For torsion there is no such clear tendency and differences between m coefficient are not so strong as in bending case. The opposite trend to bending occurs for combining bending with torsion where mean stress value is greater than 0. Negative values of the correlation coefficient r in both tables indicate a strong relationship between the logarithm of the number of cycles to failure and the logarithm of the stress amplitude for smooth and notched specimens at all types of loads. Table 3. Parameters of regression equations describing fatigue life of smooth specimens.

Bending
Bending with Torsion Torsion

Bending
Bending with Torsion Torsion     Figure 3 shows that with pure bending, the presence of a notch in the specimens significantly reduces the allowable stress amplitudes in relation to smooth specimens for the same fatigue life. The increase in mean stresses σm when bending smooth samples (Figure 3a) in the range of 0-150 MPa reduces (as expected) the allowable stress amplitudes σa, while at σm = 225 MPa it increases unexpectedly σa. The growth trend σa, with the increase σm, for a constant number of cycles to damage, is more clearly manifested in notched specimens (Figure 3b). These observations lead to the conclusion that from certain, sufficiently high levels of mean stress in smooth specimens and from  Figure 3 shows that with pure bending, the presence of a notch in the specimens significantly reduces the allowable stress amplitudes in relation to smooth specimens for the same fatigue life.
The increase in mean stresses σ m when bending smooth samples (Figure 3a) in the range of 0-150 MPa reduces (as expected) the allowable stress amplitudes σ a , while at σ m = 225 MPa it increases unexpectedly σ a . The growth trend σ a , with the increase σ m , for a constant number of cycles to damage, is more clearly manifested in notched specimens (Figure 3b). These observations lead to the conclusion that from certain, sufficiently high levels of mean stress in smooth specimens and from much lower levels σ m notched specimens have large plastic strains and cyclic strengthening of the material.
Additional factors that should be taken into account when explaining these observations are the presence of biaxial stress states in the notch bottom and stress gradients in the cross sections of the specimens. With pure torsion, the presence of a notch in the samples weakens them compared to smooth specimens (Figure 4), but to a lesser extent than is observed when bending (Figure 3).    Table 5 shows values of the fatigue limits for the material. In the case of bending with torsion, providing the fatigue limit value requires the adoption of an appropriate multiaxial fatigue criterion, which will be the subject of analysis in a separate work. Based on the parameters of the regression equations in Tables 3 and 4, charts were drawn for the impact of mean values σm = τm = 0, 36, 75 and 114 MPa on the allowable stress amplitudes for the three selected number of cycles for damage N = 10 4 , 10 5 and 10 6 smooth and notched specimens ( Figure 6). Lines approximating the results of Figure 6 obtained by interpolation. Thanks to these graphs, forecasting of fatigue life or allowable stress amplitudes for other load combinations than α = 1.107 rad becomes possible without conducting additional tests. Figure 6a shows that with symmetrical Additional factors that should be taken into account when explaining these observations are the presence of biaxial stress states in the notch bottom and stress gradients in the cross sections of the specimens. With pure torsion, the presence of a notch in the samples weakens them compared to smooth specimens (Figure 4), but to a lesser extent than is observed when bending (Figure 3).
The influence of mean shear stresses τ m on the permissible amplitudes of shear stresses τ a for smooth specimens (Figure 4a) and notched specimens (Figure 4b) with constant life is similar here as for bending. When comparing the fatigue diagrams of smooth specimens (Figures 3a and 4a) and notched specimens (Figures 3b and 4b) under bending and torsion, it can be seen that the mean stresses σ m and τ m strongly affect the allowable stress amplitudes σ a and τ a in smooth specimens than in notched specimens, where in turn the effects that can be the result of cyclic strengthening of the material are more pronounced. Manifested in an increase in allowable stress amplitudes with an increase in mean stresses for the same number of cycles for damage. With the combination of bending and torsion (Figure 5), the strongly weakening effect of the specimens notch is observed only with symmetrical cycles (σ m = τ m = 0), while with asymmetrical cycles (σ m = τ m > 0) this effect is blurred and can be assume that it disappears. Increasing mean stresses σ m = τ m for smooth specimens (Figure 5a) basically reduces the allowable stress amplitudes σ a = τ a , while for notched specimens (Figure 5b) it unexpectedly increases the allowable stress amplitudes. To determine the equivalent stress for a complex load, the Huber-Mises hypothesis was used. The effect of the mean load value was taken into account with the Goodman correction and with the use of the Neuber's model notch effect was included (K tB = 2.46, K tT = 1.76). For smooth specimens the maximum stress varies from 260 MPa for HCF up to 500 MPa for LCF regime. In the case of notched specimens (Figure 5b Table 5 shows values of the fatigue limits for the material. In the case of bending with torsion, providing the fatigue limit value requires the adoption of an appropriate multiaxial fatigue criterion, which will be the subject of analysis in a separate work. Based on the parameters of the regression equations in Tables 3 and 4, charts were drawn for the impact of mean values σ m = τ m = 0, 36, 75 and 114 MPa on the allowable stress amplitudes for the three selected number of cycles for damage N = 10 4 , 10 5 and 10 6 smooth and notched specimens ( Figure 6). Lines approximating the results of Figure 6 obtained by interpolation. Thanks to these graphs, forecasting of fatigue life or allowable stress amplitudes for other load combinations than α = 1.107 rad becomes possible without conducting additional tests. Figure 6a shows that with symmetrical load combinations, the presence of a notch on the specimens reduces the allowable stress amplitudes by about 50% compared to the stress amplitudes for smooth specimens at the same lifetime. However, in Figure 6b-d it can be seen that the impact of the notch on the permissible stress amplitudes decreases when non-zero mean stresses appear. In Figure 6d an unexpectedly favorable effect of the notch is observed at more cycles for damage (10 5 ; 10 6 cycles) for the combination of bending and torsion with the mean stress σ m = τ m = 114 MPa.
The parameters of the regression equations from Tables 3 and 4 were also used to illustrate in Figure 7 the impact of mean stresses σ m and τ m on the permissible stress amplitudes σ a and τ a for three different combinations of loads and three fatigue life N = 10 4 , 10 5 and 10 6 cycles.
In the graphs (Figure 7) there is a lack of a clear decrease in the stress amplitudes σ a and τ a with an increase in mean stresses, respectively, σ m and τ m as would be expected according to many well-known graphs, e.g., Haigh, Soderberg [21], for smooth specimens. In the case of notched specimens is surprising increase in the stress amplitudes σ a and τ a with the increase in mean stresses σ m and τ m , which with pure bending and pure torsion is insignificant and explicit with the combination of bending and torsion. The parameters of the regression equations from Tables 3 and 4 were also used to illustrate in In the graphs (Figure 7) there is a lack of a clear decrease in the stress amplitudes σa and τa with an increase in mean stresses, respectively, σm and τm as would be expected according to many wellknown graphs, e.g., Haigh, Soderberg [21], for smooth specimens. In the case of notched specimens is surprising increase in the stress amplitudes σa and τa with the increase in mean stresses σm and τm, which with pure bending and pure torsion is insignificant and explicit with the combination of bending and torsion. [MPa] [MPa] □ -smooth specimens ■ -notched specimens Permissible values of amplitudes of the total moment of the loading force at a level close to the fatigue limit (N = 10 6 cycles) depending on the type of nominal stresses (α = 0, 63 • and 90 • ), as well as the values of mean stress (σ m , τ m ) and notch shown in Figure 8. Figure 8 shows that the lowest values of moment amplitudes are required to damage notched specimens under bending and they are on average twice as large as the moment amplitudes for damage of these specimens when torsion. Such significant differences in the values of amplitudes for pure bending and pure torsion are not observed in the case of smooth specimens. At asymmetrical loads of notched specimens, the levels of torque amplitudes under pure torsion and the tested combination of bending and torsion are similar (Figure 8b-d). Figure 8d shows again that with the combination of bending and torsion larger moments are needed to damage notched specimens than smooth specimens. This anomaly deserves a separate, elastic-plastic analysis of the phenomena observed in specimens from the tested material. Permissible values of amplitudes of the total moment of the loading force at a level close to the fatigue limit (N = 10 6 cycles) depending on the type of nominal stresses (α = 0, 63° and 90°), as well as the values of mean stress (σm, τm) and notch shown in Figure 8. Figure 8 shows that the lowest values of moment amplitudes are required to damage notched specimens under bending and they are on average twice as large as the moment amplitudes for damage of these specimens when torsion. Such significant differences in the values of amplitudes for pure bending and pure torsion are not observed in the case of smooth specimens. At asymmetrical loads of notched specimens, the levels of torque amplitudes under pure torsion and the tested combination of bending and torsion are similar ( Figure  8b-d). Figure 8d shows again that with the combination of bending and torsion larger moments are needed to damage notched specimens than smooth specimens. This anomaly deserves a separate, elastic-plastic analysis of the phenomena observed in specimens from the tested material.
The effect of negligible small influence of mean shear stress on fatigue life, especially for high cycle fatigue regime, is well known [21,22]. The mean stress value can involve the ratcheting effect in material and phenomena of ratcheting strains can influence on damage accumulation process.  The effect of negligible small influence of mean shear stress on fatigue life, especially for high cycle fatigue regime, is well known [21,22]. The mean stress value can involve the ratcheting effect in material and phenomena of ratcheting strains can influence on damage accumulation process.
Additionally, investigation of 10HNAP (S355J2G1W) steel shows that for fatigue limit the coefficient of the cycles asymmetry sensitivity for torsion is about two times smaller in comparison to bending loads [23]. Assuming a negligible effect of the shear means stress the allowable amplitude of the loads at the fatigue limit.
Additionally, investigation of 10HNAP (S355J2G1W) steel shows that for fatigue limit the coefficient of the cycles asymmetry sensitivity for torsion is about two times smaller in comparison to bending loads [23]. Assuming a negligible effect of the shear means stress the allowable amplitude of the loads at the fatigue limit.

Conclusions and Finding
The following conclusions can be drawn from the tests of smooth and notched specimens made of 10HNAP steel at cyclic loads, taking into account the effect of mean stress values 1 Notch significantly reduces the allowable nominal stress amplitudes in the case of pure bending and pure torsion, while the effect is smaller in the combination of bending and torsion. These observations confirm the typical behavior of structural steels in machine elements where stress concentrations occur. 2 The greatest effect of the notch on fatigue strength compared to smooth specimens is observed at symmetrical loads. At unsymmetrical loads with non-zero values of mean stress, this effect [MPa] [MPa] □ -smooth specimens ■ -notched specimens [MPa] [MPa] □ -smooth specimens ■ -notched specimens

Conclusions and Finding
The following conclusions can be drawn from the tests of smooth and notched specimens made of 10HNAP steel at cyclic loads, taking into account the effect of mean stress values 1 Notch significantly reduces the allowable nominal stress amplitudes in the case of pure bending and pure torsion, while the effect is smaller in the combination of bending and torsion. These observations confirm the typical behavior of structural steels in machine elements where stress concentrations occur. 2 The greatest effect of the notch on fatigue strength compared to smooth specimens is observed at symmetrical loads. At unsymmetrical loads with non-zero values of mean stress, this effect clearly weakens or disappears and incidentally turned out to be unexpectedly opposite to the general trend in a large number of cycles. 3 In smooth specimens with sufficiently high mean stresses (σ m = τ m = 225 MPa) and for significantly lower mean stress levels (σ m = τ m = 36 MPa) in notched specimens there appears the effect of strengthening the material, which indicates the presence of significant plastic deformations. 4 The calculation models proposed for assessing the fatigue life of 10HNAP steel under bending, torsion and combination of bending with torsion taking into account the value of mean stresses and the presence of notches should take into account the appearance of significant local plastic strains, cyclic strengthening of the material and the occurrence of stress gradients in the cross-sections of the rods. Proposals for such models will be the subject of a separate publication by the authors.
Author Contributions: Conceptualization, R.P. and D.R.; methodology, R.P. and D.R.; formal analysis, R.P. and D.R.; investigation, R.P. and D.R.; writing-original draft preparation, R.P. and D.R. All authors have read and agreed to the published version of the manuscript.