Mean stress effect correction using constant stress ratio S–N curves

Abstract

This paper presents a stress based approach to take into account the influence of the mean stress value on fatigue strength of constructional materials. Elaborated model uses two S–N curves, i.e. for alternating stress (R = −1) and another one obtained under stress ratio R ≠ −1, for calibrating the equations of boundary condition. Two particular equations for the coefficient of intensification in stress transformations were proposed. The main advantage of the proposed solution is that the mean stress effect correction depends on the number of cycles to failure, what corresponds to the observed changes in experimental results presented in the literature. Proposed relations were compared with popular models for mean stress correction. The verification was made using selected series of experimental results taken from the literature. It was shown that the proposed solution is well correlated with experimental results.

Introduction

It is well known that structures and machine elements subjected to time-varying loads on the appropriate level are subject to the effect of material fatigue. Often such kind of loading is accompanied by a significant mean value, for example as a result of self-weight of the construction, which considerably influences the damaging process [1], [2]. Generally in fatigue life assessment algorithms stress and strain are used [3], [4], [5], rather than force or energy parameters, for description of the materials effort. In particular, stress is the commonly used quantity in middle- and high-cycle fatigue, where plastic deformation does not play any major role. In such a case S–N curves, i.e. curves where the stress amplitude is presented versus number of cycles to failure, provide the basic information about fatigue properties of material or machine components. An engineer or designer can conclude about the number of cycles to failure based on such curves, knowing the stress amplitude resulted from loading and vice versa. Due to the simplicity of this approach the S–N curves are still widely used in design applications. Although this method is simple, if the mean stress effect should be also taken into account it requires preparation of a number of curves for different mean stress values. Usually such curves are prepared for constant values of stress ratio

$$R=\frac{{\sigma }_{\mathit{min}}}{{\sigma }_{\mathit{max}}}\text{,}$$

where σ_min and σ_max are minimum and maximum stress under constant amplitude loading condition. Typical values of the stress ratios used for S–N curves with significant mean stress are R = 0 and R = −0.5. Direct use of S–N curves for R ≠ −1 leads to the need of performing a series of expensive fatigue tests [6], [7], [8]. For this reason scientists based on the results of experimental and theoretical foundations are proposing a number of models to take into account mean stress [9], [10], [11], [12], using only selected material parameters and a basic S–N curve prepared for R = −1, i.e. for zero mean stress. Most of them define a boundary condition of stress state as a function of stress amplitude σ_a, the mean stress σ_m, and parameters for a chosen material beyond which the material will be destroyed. Some commonly used models were selected and are presented in Fig. 1 using a line resulting from the boundary condition given by a specific model. If any point drawn in Fig. 1 for the material and stress state under consideration is found below the selected boundary line, this means that fatigue failure will not appear according to that model. Gerber ([13], Germany) proposed a parabola to model the boundary line. To calibrate the model the ultimate strength of the material σ_uts was used [13]

$$\frac{{\sigma }_a}{{\sigma }_{\mathit{aT}}}=1-{\left(\frac{{\sigma }_m}{{\sigma }_{\mathit{uts}}}\right)}^2\text{,}$$

where σ_aT is the transformed alternating stress amplitude, σ_a and σ_m are the stress cycle amplitude and mean value, respectively. Using the stress amplitude σ_aT and S–N curve for R = −1 the expected number of cycles to fracture can be calculated

$$N_{\mathit{cal}}=N_k{\left(\frac{{\sigma }_{\mathit{aT}}}{{\sigma }_{\mathit{af}}}\right)}^k\text{,}$$

where k is the slope of the fatigue curve and N_k the number of cycles to failure corresponding to the stress amplitude σ_af. The posted Eq. (3) allows to determine the expected number of cycles until failure on the basis of the S–N fatigue characteristic. In this equation there is a constant S–N curve slope value for R = −1 load ratio. For different values of load ratios R the S–N curve may manifest different slopes. This phenomena should be taken into account to properly estimate the fatigue life in a broader range of the number of cycles. This may be obtained by taking into account the change of sensitivity of the material due to mean stress, depending on the number of cycles and, consequently, designate the transformed stress amplitude σ_aT.

The main drawback of the model proposed by Gerber (2) is that it does not distinguish between compression and tension but in most cases of constructional materials compressive stress increases the fatigue life in comparison to the tensile stress. This disadvantage is not present in Goodman’s model ([14], England). The relation proposed by Gerber has been modified and takes the following form [14]

$$\frac{{\sigma }_a}{{\sigma }_{\mathit{aT}}}=1-\frac{{\sigma }_m}{{\sigma }_{\mathit{uts}}}\text{,}$$

which leads to a straight line presented in Fig. 1. Similar concept was also used by Susmel et al. [10] (1930, USA), who proposed the use of yield strength σ_y instead of ultimate strength; however, it also leads to a straight line presented in Fig. 1, but with a different slope in comparison to the Goodman’s model

$$\frac{{\sigma }_a}{{\sigma }_{\mathit{aT}}}=1-\frac{{\sigma }_m}{{\sigma }_y}\text{.}$$

A well-known modification of the ‘straight line’ concepts is the model elaborated by Dowling [15] (1960, USA) where in the place of static material properties the fatigue strength coefficient is used

$$\frac{{\sigma }_a}{{\sigma }_{\mathit{aT}}}=1-\frac{{\sigma }_m}{{\sigma }_f^{\prime }}\text{.}$$

Another interesting and popular proposal to this issue was presented by Morrow [16], [17] in terms of a strain-based approach

$${\epsilon }_a=\frac{{\sigma }_f^{\prime }-{\sigma }_m}{E}(2N_f{)}^b+{\epsilon }_f^{\prime }(2N_f^{\prime }{)}^c\text{.}$$

This approach was later expanded by Manson and Halford [18]

$${\epsilon }_a=\frac{{\sigma }_f^{\prime }-{\sigma }_m}{E}(2N_f{)}^b+{\epsilon }_f^{\prime }{\left(1-\frac{{\sigma }_m}{{\sigma }_f^{\prime }}\right)}^{\frac{c}{b}}(2N_f{)}^c$$

where E is the Young’s modulus, ε′_f is the fatigue coefficient of the plastic strain, b and c are the fatigue strength and ductility exponents, respectively, and N_f is the number of cycles until failure. Another common model has been presented by Smith et al. [19] (USA) in their common paper [19], [20]. This model is usually used in strain based fatigue analysis by combining the SWT parameter with strain-life equation proposed by Manson [21] multiplied by Basquin relation [22]

$${\sigma }_{\max }{\epsilon }_a=\left(\frac{{\sigma }_f^{\prime }}{E}(2N_f{)}^b+{\epsilon }_f^{\prime }(2N_f{)}^c\right)({\sigma }_f^{\prime }(2N_f{)}^b)=\frac{({\sigma }_f^{\prime }{)}^2}{E}(2N_f{)}^{2b}+{\sigma }_f^{\prime }{\epsilon }_f^{\prime }(2N_f{)}^{b+c}\text{,}$$

where σ_max = σ_m + σ_a is the maximum stress level and ε_a is the strain amplitude. With the assumption that the plastic part of the strain amplitude is small and can be neglected during fatigue life assessment, which is a typical practice in high-cycle fatigue, the relationship for transformed stress amplitude can be derived [15], [20] from the left hand side of Eq. (9) as

$${\sigma }_{\mathit{aT}}=\sqrt{({\sigma }_a+{\sigma }_m){\epsilon }_{a}E}=\sqrt{({\sigma }_a+{\sigma }_m){\sigma }_a}\text{.}$$

Eq. (10) can be also expressed by stress ratios coefficient R [15]

$${\sigma }_{\mathit{aT}}={\sigma }_{\mathit{max}}\sqrt{\frac{1-R}{2}}={\sigma }_a\sqrt{\frac{2}{1-R}}\text{.}$$

Walker [23] (1979, USA) has developed a model for mean stress effects based on his experimental tests of aluminum alloys in the following form

$${\sigma }_{\mathit{aT}}=({\sigma }_{\max }{)}^{1-\gamma }{\sigma }_a^{\gamma }\text{,}$$

which can be also expressed by a stress ratios coefficient R

$${\sigma }_{\mathit{aT}}={\sigma }_{\mathit{max}}{\left(\frac{1-R}{2}\right)}^{\gamma }={\sigma }_a{\left(\frac{2}{1-R}\right)}^{1-\gamma }\text{.}$$

The main advantage of this model is the material-dependent γ parameter which allows to calibrate the model for various groups of materials. By setting the parameter γ = 0.5 the model proposed by Walker simplifies to those proposed by Smith–Watson–Topper. Kwofie [24], [25] (2001, Ghana) proposed an exponential function which uses a mean stress sensitivity factor α for calibration of the model and the factor depends on the type of material

$$\frac{{\sigma }_a}{{\sigma }_{\mathit{aT}}}=\exp \left(-\alpha ·\frac{{\sigma }_m}{{\sigma }_{\mathit{uts}}}\right)\text{.}$$

A careful analysis of presented mean stress effect models leads to the following conclusions:

• Some of the models, e.g. the models proposed by Gerber, Goodman and Soderberg, are using material constants determined on the basis of monotonic tensile tests. This offers a simple implementation of these models, since these properties are commonly available for most materials. Please note also that these models are developed for the high cycle fatigue or so called fatigue limit assessment what partly justifies the use of static properties of the material. However, this is the main reason why it is not possible to describe the fatigue behavior of the material in the right way, since these parameters do not involve information on how the material behaves under time-variable loading, especially when cyclic hardening or softening of the material occurs in the middle-cycle fatigue. Only the model by Morrow uses fatigue strength coefficient but the coefficient is assessed on the basis of experiential data without mean stress. This also shows that in the case of the mean stress taken into account, fatigue life assessment will be realized without real information of the mean stress sensitivity of the materials.

• Only a few propositions use material-dependent parameters which enable the model to be calibrated and follow the real behavior of the material, e.g. Walker’s and Kwofie’s models have an γ and α factor, respectively. Such kind of coefficients increases the accuracy of fatigue life estimation and make the models more flexible. Although the need to determine these factors for each material makes these models difficult to apply and, consequently, they are uncommonly referred to.

• Most of the models allow the calculation of the transformed amplitude according to the following general formula [4].

$${\sigma }_{\mathit{aT}}={\sigma }_{a}K({\sigma }_m\text{,}P)\text{,}$$

where intensifications coefficient K can be expressed as a function of mean stress σ_m and some material parameters generally represented by P. The exception are the SWT and Walker models which take into account the mean stress using also the stress amplitude, what leads to

$${\sigma }_{\mathit{aT}}={\sigma }_{a}K({\sigma }_a\text{,}{\sigma }_m\text{,}P)={\sigma }_{a}K(R\text{,}P)\text{.}$$

• All the presented models do not take directly into account the relationship between the number of cycles to failure and the mean stress sensitivity of the material. This can be regarded as a big deficiency of the common models because the mean stress sensitivity depends considerably on the cycle range in which the fatigue failure is expected or was observed. To explain the effect on an example experimental results presented by Kim et al. [26] were used. The material used for research in that paper was a 5% chrome steel. The microstructure of the material was primarily tempered martensite matrix with carbides (FeCr)₇C₃ dispersed in the matrix and on the prior austenite grain boundary. Specimens used in that study were solid specimens with a round cross section. The fatigue tests were carried out under load control. The results were taken from this study only for uniaxial axial tests under constant amplitude loading. Failure was defined as a complete separation of specimen. Two figures were prepared using data presented in [26]. Fig. 2 presents experimental fatigue results of smooth samples under uniaxial stress state for four mean stress values σ_m = −300, −100, 0, and 300 MPa. The series of experimental results for σ_m = 0 (R = −1) were also approximated with S–N curve by means of the following equation of regression σ_a,_R=−1 = σ_af(N/N_k)^(1/−k). As a result three material constants were obtained: σ_af = 507.6, N_k = 2 × 10⁶, and k = 13.78. Such kind of data presentation is usual for stress based fatigue tests and it can be noticed that on the basis of it, that obtained sets of data manifested a small scatter. Fig. 3 has been prepared according to the graph presented in Fig. 1, but the experimental results and the S–N curve from Fig. 2, has been used for this purpose. The scatter of points in Fig. 3 depends on individual slopes of curves prepared for constant R because we divide σ_a,_R/σ_a,_R=−1. At different slopes we obtain a large scatter and the direction of changes corresponds to the gap between the slopes, as indicated by the arrows. This is most clear for the mean stress value of 300 MPa. Only for the series of experimental results with the mean stress value equal to zero, experimental points are grouped around the value 1/K = 1 and their scatter is small. Series of experimental results for significant mean stress values are characterized with larger scatters. Moreover, for particular series the points are in sequence corresponding to the number of cycles to failure, which is marked in Fig. 3 with arrows. Such huge differences in scatter and the sequence of each series of experimental points indicate that the K coefficient does not depend only on the mean stress value, but also on the number of cycles to failure. Gasiak and Pawliczek [27] made some deliberations about this effect and propose to correct them by means of simple functions estimated from experimental data without additional theoretical background. Please note that this effect is often considerable and cannot be neglected during fatigue life assessment for most of constructional materials.

It is often found that for the material from which the machine components are made, S–N curves for repeated (R = −1) and alternating (R = 0) stress are present. For this reason it was decided that a simple model could be developed for mean stress correction which is using the stress amplitudes read directly from such S–N curves for fixed number of cycle. In this case the general equation for transformed stress amplitude takes the following form

$${\sigma }_{\mathit{aT}}={\sigma }_{a}K({\sigma }_m\text{,}{N}_f)\text{or}{\sigma }_{\mathit{aT}}={\sigma }_{a}K({\sigma }_m\text{,}{\sigma }_{\mathit{aN}\text{,}R=-1}\text{,}{\sigma }_{\mathit{aN}\text{,}R})\text{.}$$

It is expected that the model will take into account the different sensitivity of the material on mean stress due to the number of cycles to failure. This will result in a better description of the influence of mean stress on material fatigue.