Shift of S-N curves in some fatigue models due to loading cycle asymmetry

Heliyon 10 (2024) e26306 Available online 15 February 2024 2405-8440/© 2024 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Research article Shift of S–N curves in some fatigue models due to loading cycle asymmetry Jan Kohout a , * , Stanislav Vˇ echet b a Dept. of Mathematics and Physics, Military Technology Faculty, University of Defence, Kounicova 65, 662 10, Brno, Czech Republic b Institute of Materials Science and Engineering, Faculty of Mechanical Engineering, Brno University of Technology, Technick´ a 2896/2, 616 69, Brno, Czech Republic ARTICLE INFO Keywords: S-N curves Fatigue modelling Asymmetry of loading cycle Palmgren model Kohout-Vˇ echet model Logistic S–N model ABSTRACT Fatigue (W¨ ohler’s or S–N) curves are usually represented by upper stress of loading cycle in dependence on the logarithm of numbers of cycles to fracture. Increasing mean stress of loading cycle causes a shift of these curves towards higher values of fatigue strength. A successful quantitative description of the high cycle shift was published by Walker. The aim of the paper consists in deriving and verifying the relations describing the shift of fatigue curves in the whole cycle region from ultimate tensile stress to permanent fatigue limit, for the Palmgren, the KohoutVˇ echet and the logistic S–N models, using the high-cycle Walker approach. 1. Introduction Any reliable application of structural materials for dynamically loaded products should be based on a thorough study of their fatigue behaviour, usually at various loading cycle asymmetries. A suitable description of the position of S–N curves in dependence on loading cycle asymmetry allows reducing the extent of fatigue tests because not all asymmetries expected in service life of considered product is necessary to test. In high-cycle region the relation published by Walker [1] σ max(N,R) = σ max(N,0) (1−R)m(1) can be successfully used for describing the position of high-cycle fatigue curves in the dependence on the asymmetry of loading cycle where σ max means the upper stress of loading cycle, N is the number of cycles to fracture, R is the parameter of loading cycle asymmetry defined as R= σ min σ max (2) with σ min standing for the minimum stress of loading cycle, and an exponent m is a parameter describing the influence of various cycle asymmetries. The Walker equation (1) was successfully used for various materials [2–4] as well as for various loading modes [3,4], i.e. for various R values, see Eq. (2), and not only for the S–N curves but also for the fatigue crack growth curves. Many other functions have been tested for the description of the influence of mean stress e.g., very well-known the Goodman and the Gerber relations or the * Corresponding author. E-mail addresses: [email protected] (J. Kohout), [email protected] (S. Vˇ echet). Contents lists available at ScienceDirect Heliyon journal homepage: www.cell.com/heliyon https://doi.org/10.1016/j.heliyon.2024.e26306 Received 1 March 2023; Received in revised form 27 November 2023; Accepted 9 February 2024 Heliyon 10 (2024) e26306 2 Kwofie exponential function [5]. All of them are relatively simple but, according to the authors’ experience, none of them has as universal application as the Walker equation (1). Non-phenomenological approaches based on theoretical considerations (e.g. Ref. [6] based on fracture mechanics) unfortunately lose the application simplicity of the phenomenological Walker equation. Chandran [7,8] took an original approach. Based on the phenomenological description of fatigue crack growth, he derived a new model for the S–N curve σ a(N) = σ 1− σ ∞ exp(CNm)+ σ ∞(3) where σ a =( σ max − σ min )/2 is the stress amplitude of loading cycle, σ 1 =UTS (ultimate tensile stress), σ ∞ represents the fatigue limit for an infinite number of cycles to fracture i.e., permanent fatigue limit, and C and m are the parameters of the Chandran model. Then the maximum stress of loading cycle can be simply expressed σ max(N,R) = 2 1−R σ a(N,R), σ max(∞,R) ≡ σ ∞(R) = 2 1−R σ a(∞,R)(4) Considering that σ 1 =UTS is independent of the R parameter, Chandran obtained equations for σ a (N,R) as well as for σ max (N,R) and successfully applied them to experimental results of fatigue tests of polymers [7] and metals [8]. It should be noted, however, that while for most S–N models as well as for most experimental S–N curves with two bends (low-cycle bend towards σ 1 and high-cycle bend towards σ ∞ ) the curvature of the two bends is comparable, in the Chandran model the curvature of the low-cycle bend is significantly less than the curvature of the high-cycle bend. Also interesting is a certain similarity between the Chandran model and the Stüssi model [9,10] described by the relation σ max(N) = σ 1− σ ∞ 1+ α Nβ+ σ ∞(5) because for small x it is possible to write exp(x) ≈1 +x (see the denominator of Eq. (3)). And finally, both of these models have the same beauty defect: σ 1 ≡UTS = σ (0) ∕= σ (1) i.e., UTS is not reached in the first cycle but in the zeroth cycle. Three papers [11–13] tried to use the Kohout-Vˇ echet model [14] for describing S–N curve family for various values of the parameter of loading cycle asymmetry R. Correia et al. [11,12] used for each of the R values the regression function with specific values of regression parameters, while Barbosa et al. [13] introduced so-called normalized stress range Δ σ norm =Δ σ f(R)where f(R) = 1−R 1−a R (6) The normalized function f(R) in Eq. (6) is dependent on the positive a parameter less than 1. Its value is determined experimentally and is different not only for different materials, but also for positive and negative values of the R parameter. Fern´ andez-Canteli et al. [15] took a different approach; they divided fatigue models into three classes and for the Class III covering low-cycle, high-cycle and very high-cycle domains they solved the courses of the S–N curves for the values of the R parameter equal to 0, −0.5, and −1. The aim of the paper consists in the generalization of Walker’s access for the whole region of cycles from ultimate tensile stress to permanent fatigue limit. From the abundance of S–N models, the authors focus on only three models with which they work the most and therefore have the most experience with them: the Palmgren model, the Kohout-Vˇ echet model and the logistic S–N model. The Stromeyer model is also mentioned, the generalization of which leads to the Palmgren model. It follows from the form of Walker’s relation that it is simply applicable only to the σ =f(N) dependences, not to the N =f( σ ) dependences. 2. Generalization of the Walker approach 2.1. Stromeyer model High-cycle region of S–N curve is often described using the Stromeyer equation [16,17] (hereafter σ (N) or σ (N, R) instead of σ max (N) or σ max (N, R) will be simply written) σ (N) = a Nb+ σ ∞(7) where a, b <0 and σ ∞ are parameters determined using regression of the results of fatigue tests. This equation can be rewritten in a more suitable form σ (N) = ( σ C− σ ∞)(10−7N)b+ σ ∞(8) containing fatigue limit σ C for 10 7 cycles to fracture (equation can be simply changed for arbitrary but sufficiently high number of cycles). It is valid for arbitrary value of asymmetry parameter R including R =0. Using Eq. (1), it can be written for various R values σ (N,R) = [ σ C(0) − σ ∞(0)] (10−7N)b+ σ ∞(0) (1−R)m(9) because both the parameters σ C and σ ∞ relate to the high-cycle part of the S–N curve. It means that the shift of S–N curve can be J. Kohout and S. Vˇ echet Heliyon 10 (2024) e26306 3 described with the multiplication of fatigue limits σ C and σ ∞ by (1 −R) −m . 2.2. Palmgren model When considering the whole range of the number of cycles to fracture from the UTS to the permanent fatigue limit, the Palmgren function [17]. σ (N) = a(N+B)b+ σ ∞(10) can be used instead of Eq. (7) where B is the number of cycles to fracture in which the S–N curve bends to the UTS value. It can be rewritten similarly as Eq. (8) σ (N) = ( σ C− σ ∞) · [10−7(N+B)]b+ σ ∞(11) From this equation the parameter B can be simply calculated as B=107( σ 1− σ ∞ σ C− σ ∞)1/b =107( σ C− σ ∞ σ 1− σ ∞)−1/b (12) when σ 1 = σ (1) ≡UTS and B ≫ 1 is considered. In Eq. (11) the Walker equation (1) can be simply used as in Eq. (9) σ (N,R) = [ σ C(0) − σ ∞(0)] {10−7[N+B(R)]}b+ σ ∞(0) (1−R)m(13) but general dependence of parameter B on R value must be respected. As UTS ≡ σ 1 is not dependent on loading cycle asymmetry, Eq. (12) can be generalized as B(R) = 107[ σ C(0)(1−R)−m− σ ∞(0)(1−R)−m σ 1− σ ∞(0)(1−R)−m]−1/b ≡107[ σ C(0) − σ ∞(0) σ 1(1−R)m− σ ∞(0)]−1/b (14) Using Eqs (13) and (14), the family of empiric S–N curves determined for various loading cycle asymmetry can be fitted where σ 1 , σ C (0), σ ∞ (0), b, and m are the regression parameters. This derivation presented also in Ref. [18] has an alternative. Eq. (11) can be replaced by equation σ (N) = ( σ 1− σ ∞)(N+B 1+B)b + σ ∞(15) also following from Eq. (10). For B ≫ 1 Eq. (15) can be simplified as σ (N) = ( σ 1− σ ∞)(1+N/B)b+ σ ∞(16) and then σ (N,R) = [ σ 1− σ ∞(0)(1−R)−m] · [1+N/B(R)]b+ σ ∞(0)(1−R)−m(17) is obtained. So Eq. (17) as well as Eq. (13) (both supplemented by Eq. (14)) can be used for fitting the family of empiric S–N curves determined for various loading cycle asymmetry with the same set of regression parameters σ 1 , σ C (0), σ ∞ (0), b, and m. The regression results are of course the same in both the cases but using Eq. (17), the regression calculations are usually easier and faster. 2.3. Kohout-Vˇ echet model The simplest forms of the Kohout-Vˇ echet model [14] are σ (N) = σ ∞(N+B N+C)b (18) and σ (N) = σ 1(1+N/B 1+N/C)b (19) where the meaning of parameter B is the same as in the Palmgren function (10) and C means the number of cycles in which fatigue curve bends to permanent fatigue limit σ ∞ . Taking σ 1 , σ C , σ ∞ , and b as the basic parameters of the S–N curve, the B parameter can be expressed as J. Kohout and S. Vˇ echet Heliyon 10 (2024) e26306 4 B=107 σ C−1/b− σ ∞−1/b σ 1−1/b− σ C−1/b(20) (note the striking similarity to Eq. (12)) and the C parameter as C=107 σ C1/b− σ ∞1/b σ 11/b− σ C1/b(21) (note that the last two equations differ only in the sign of the exponent). Then in the case of different asymmetry of loading cycles it can be written σ (N,R) = σ ∞(0) (1−R)m[N+B(R) N+C(R)]b (22) or more simply σ (N,R) = σ 1[1+N/B(R) 1+N/C(R)]b (23) (see Eqs (18) and (19)) where [19] B(R) = 107 σ C(0)−1/b− σ ∞(0)−1/b [ σ 1(1−R)m]−1/b− σ C(0)−1/b(24) and C(R) = 107 σ C(0)1/b− σ ∞(0)1/b [ σ 1(1−R)m]1/b− σ C(0)1/b(25) are used instead of B and C from Eqs (20) and (21). So Eq. (22) or Eq. (23) together with Eqs (24) and (25) can be used for fitting the family of empiric S–N curves determined for various loading cycle asymmetry with the same set of regression parameters σ 1 , σ C (0), σ ∞ (0), b, and m as in the Palmgren model. 2.4. Logistic S–N model This model [20], which is practically identical to the Stüssi model [9,10], see Eq. (5), can be described by the equation for the S–N curve σ (N) = σ 1− σ ∞ 1+Eb(N−b−1)+ σ ∞(26) where E means the number of cycles to fracture determining the position of midpoint (identical with the inflection point) of the mentioned curve i.e., σ (E) =( σ 1 + σ ∞ )/2. Taking again σ 1 , σ C , σ ∞ , and b as the basic parameters of the S–N curve, the whole expression E b can be expressed as Eb= σ 1− σ C σ C− σ ∞ ·1 10−7b−1(27) When different asymmetry of loading cycles is considered, it can be written σ (N,R) = σ 1− σ ∞(0)(1−R)−m 1+Eb(R)(N−b−1)+ σ ∞(0)(1−R)−m(28) where E b (R) means [E(R)] b and Eb(R) = σ 1− σ C(0)(1−R)−m σ C(0)(1−R)−m− σ ∞(0)(1−R)−m·1 10−7b−1= σ 1(1−R)m− σ C(0) σ C(0) − σ ∞(0)·1 10−7b−1(29) is used instead of E b from Eq. (27). So Eq. (28) together with Eq. (29) can be used for fitting the family of empiric S–N curves determined for various loading cycle asymmetry with the same set of regression parameters σ 1 , σ C (0), σ ∞ (0), b, and m as in the Palmgren model and the Kohout-Vˇ echet model. 3. Empirical data for verification of derived equations For verification of derived equations in mentioned S–N curve models, empirical data obtained from fatigue tests of very different materials were used. J. Kohout and S. Vˇ echet Heliyon 10 (2024) e26306 5 3.1. Nodular cast iron Chemical composition of tested nodular cast iron was 3.51 wt% C, 0.15 % Mn, 2.40 % Si, 0.05 % P, 0.007 % S, 0.02 % Cu, and 0.050 % Mg [18]. The size of graphite nodules was 30–60 μ m, their shape was mostly perfectly granular. Test bars of 7 mm in diameter with threaded heads were made of nearly fully ferritic matrix, which was obtained already in cast state. As surface relief plays crucial role in fatigue behaviour, the bars were grinded to surface roughness Ra ≈0.4 μ m. For both tensile and fatigue tests the same type of test bars was used to avoid contingent differences caused by their different geometrical shape. UTS together with the other tensile properties were determined using the Zwick Tensile Tester 1478 at a strain rate of 5 ×10 −4 s −1 ; fatigue tests were performed using high-frequency resonance Amsler 10 HFP 1478 pulsator at frequency lower than 200 Hz, both at room temperature. Seven various asymmetries of loading cycle are tested in the range of asymmetry parameter R from −1 to 0.71. For determination of one S–N curve i.e., for one given value of the R parameter, 12 to 15 test bars were tested. 3.2. Aluminum alloy For testing the rolled bars made of 2024-T4 aluminum alloy with the diameter of 1.125 in (28.575 mm) and circumferential Vgroove (K t =2.4) were used by Lazan and Blatherwick [21] (the empirical results were published also in a much more accessible handbook [22]). Axial loading with the frequency of 30–60 Hz was applied at room temperature in air. Five various asymmetries of loading cycle are tested in the range of asymmetry parameter R from −1 to 0.74, with 5–9 test bars for each asymmetry. 3.3. Titanium alloy Test bars with diameter of 15 mm and length of 20 mm studied by de Krijger et al. [23] were manufactured from Ti–6Al–4V-ELI powder according to ASTM F3001. They were built in the chamber with inert Ar atmosphere with an oxygen level below 50 ppm on top of a solid titanium build plate from which they were subsequently removed using wire electric discharge machining. The fatigue tests were carried out at a stress ratio of R =0.1, 0.3, 0.5, 0.7, and 0.8 (compressive loading), on an MTS 100 kN hydraulic test machine, at a loading frequency of 15 Hz with a sinusoidal wave shape (for more details, see Ref. [23]). 4. Results of regression The fatigue test results of all the studied materials are fitted using the regression functions of the presented fatigue models, namely: •the Palmgren model with the regression function (17) supplemented by Eq. (14), •the Kohout-Vˇ echet model with the regression function (23) supplemented by Eqs (24) and (25), •the logistic S–N model with the regression function (28) supplemented by Eq. (29). Fig. 1. Fatigue test results of nodular cast iron [18] fitted using the regression function (17) supplemented by Eq. (14) (Palmgren model). J. Kohout and S. Vˇ echet Heliyon 10 (2024) e26306 6 4.1. Nodular cast iron The fatigue test results of nodular cast iron [18] fitted using the regression functions of various fatigue models are presented in Figs. 1–3, namely the Palmgren model in Fig. 1, the Kohout-Vˇ echet model in Fig. 2, and the logistic S–N model in Fig. 3. All three presented fits are essentially successful. However, at first glance it is obvious that: •the curves of the Palmgren model (Fig. 1) have a greater slope in the high-cycle region than the curves of the other models, •the curve of the logistic S–N model (Fig. 3) for R = − 1 lies too high (too close to the curve for R = − 0.67), especially for numbers of cycles to fracture lower than 10 5 . For data fitting the method of least squares was used which minimizes the sum of the squares of the residuals (the differences between observed values and the fitted values provided by a model). Obtained sum of least squares S characterizes the success of the data fitting but better criterion  σ = S/(n−p) √(where n is the number of experimental points and p is the number of regression parameters of considered model) is called standard deviation of regression model in the ISO 12107:2012 Standard. Its values for the considered models are shown in Table 1. Table 1 shows that the best fit is obtained for the Kohout-Vˇ echet model and the worst fit for the logistic S–N model. 4.2. Aluminum alloy The fatigue test results of 2024-T4 aluminum alloy [21] fitted using the regression functions of various fatigue models are presented in Figs. 4–6, namely the Palmgren model in Fig. 4, the Kohout-Vˇ echet model in Fig. 5, and the logistic S–N model in Fig. 6. Here, too, all three presented fits are essentially successful, but some characteristics of the individual fatigue models are reflected here as well: •the curves of the Kohout-Vˇ echet model (Fig. 5) have a smaller slope in the high-cycle region than the curves of the other models, •the curve of the logistic S–N model (Fig. 6) for R = − 1 fits the empirical results in the high-cycle region best, but in the mediumcycle region worst (see especially the empirical point for 260 MPa). The standard deviations of regression models  σ (square root of the quotient of the sum of least squares and the number of degrees of freedom) representing the success of the regression are shown in Table 2. Table 2 shows as well as Table 1 that the best fit is obtained for the Kohout-Vˇ echet model and the worst fit for the logistic S–N model also for fatigue test results of 2024-T4 aluminum alloy. Fig. 2. Fatigue test results of nodular cast iron [18] fitted using the regression function (23) supplemented by Eqs (24) and (25) (Kohout-Vˇ echet model). J. Kohout and S. Vˇ echet Heliyon 10 (2024) e26306 7 Fig. 3. Fatigue test results of nodular cast iron [18] fitted using the regression function (28) supplemented by Eq. (29) (logistic S–N model). Table 1 The  σ values for fits of fatigue test results of nodular cast iron [18] fitted using the regression functions of various fatigue models. model Palmgren Kohout-Vˇ echet logistic S–N Equations (17) with (14) (23) with (24) and (25) (28) with (29)  σ [MPa] 11.971 11.405 13.027 Fig. 4. Fatigue test results of 2024-T4 aluminum alloy [21] fitted using the regression function (17) supplemented by Eq. (14) (Palmgren model). J. Kohout and S. Vˇ echet Heliyon 10 (2024) e26306 8 Fig. 5. Fatigue test results of 2024-T4 aluminum alloy [21] fitted using the regression function (23) supplemented by Eqs (24) and (25) (Kohout-Vˇ echet model). Fig. 6. Fatigue test results of 2024-T4 aluminum alloy [21] fitted using the regression function (28) supplemented by Eq. (29) (logistic S–N model). Table 2 The  σ values for fits of fatigue test results of 2024-T4 aluminum alloy [21] fitted using the regression functions of various fatigue models. model Palmgren Kohout-Vˇ echet logistic S–N Equations (17) with (14) (23) with (24) and (25) (28) with (29)  σ [MPa] 18.090 15.351 19.858 J. Kohout and S. Vˇ echet Heliyon 10 (2024) e26306 9 4.3. Titanium alloy The fatigue test results of porous Ti–6Al–4V titanium alloy [23] fitted using the regression functions of various fatigue models are presented in Figs. 7–9, namely the Palmgren model in Fig. 7, the Kohout-Vˇ echet model in Fig. 8, and the logistic S–N model in Fig. 9. In this case, all three presented fits are really successful, only the curve of the logistic S–N model (Fig. 9) for R =0.1 lies systematically above the empirical points for the number to fracture lower than 10 5 . The standard deviations of regression models  σ are shown in Table 3. Table 3 shows that, for fatigue test results of porous Ti–6Al–4V titanium alloy, the best fit is obtained in this case for the Palmgren model and the worst fit again for the logistic S–N model. 4.4. Partial conclusion The equations describing the shift of S–N curves in the presented fatigue models due to the loading cycle asymmetry were derived using only two following assumptions: •fatigue limit σ C for N =10 7 cycles and permanent fatigue limit σ ∞ for N → ∞ depend on the parameter of loading cycle asymmetry R according to Eq. (1), •stress σ 1 = σ (1) ≡UTS is independent of the R parameter. Based on these two assumptions, the equations for the characteristic numbers of cycles to fracture were derived, which represent the parameters of the presented fatigue models, namely: for B(R) in the Palmgren model (see Eq. (14)), for B(R) and C(R) in the KohoutVˇ echet model (see Eqs (24) and (25)), and for E(R) in the logistic S–N model (see Eq. (29)). The B(R) parameter in the Palmgren model as well as in the Kohout-Vˇ echet model characterizes the low-cycle region and varies by several orders of magnitude depending on the R parameter. In contrast, the C(R) parameter in the Kohout-Vˇ echet model, which characterizes the high-cycle region, changes depending on the R parameter by a maximum of a few percent. Regression calculations are successful even in cases where the C parameter is considered independent of the R parameter; in a number of such cases a smaller standard deviation of regression model is even achieved, although this decrease is not significant. On the other hand, the E(R) parameter in the logistic S–N model, which characterizes the position of S–N curve midpoint, changes depending on the R parameter by fewer orders of magnitude than the parameter B(R) in the Palmgren model and in the Kohout-Vˇ echet model. While when regressing individual S–N curves, the logistic S–N model usually leads to the best fits [20], here when studying the family of S–N curves for various values of the R parameter, it gives the worst fits for all three materials studied. Therefore, the question arises whether the two assumptions mentioned above (dependence of limits σ C and σ ∞ on the R parameter according to Eq. (1) and independence of the σ 1 stress on the R parameter), which are sufficient for the Palmgren model as well as for the Kohout-Vˇ echet model, Fig. 7. Fatigue test results of porous Ti–6Al–4V titanium alloy [23] fitted using the regression function (17) supplemented by Eq. (14) (Palmgren model). J. Kohout and S. Vˇ echet Heliyon 10 (2024) e26306 16 Conceptualization, Formal analysis, Investigation, Methodology, Supervision, Validation, Writing – review & editing. Declaration of competing interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements Financial support from the Ministry of Defence of the Czech Republic within the Project for the Development of the Organization “DZRO Military autonomous and robotic systems” is gratefully acknowledged. In addition, the authors are grateful to all authors whose original experimental results were used in this paper. References [1] K. Walker, The effect of stress ratio during crack propagation and fatigue for 2024-T3 and 7075-T6 aluminum, in: M.S. 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