Review Article Volume 5 Issue 2
Center for composite materials and structure, Harbin institute of technology, China
Correspondence: Xiangqiao Yan, Center for composite materials and structure, Harbin institute of technology, 150080, China
Received: July 15, 2020  Published: April 26, 2021
Citation: Yan X. A mean stress model of fatigue life of metal materials under multiaxial loading. Material Sci & Eng. 2021;5(2):6069. DOI: 10.15406/mseij.2021.05.00157
Due to the complexity of multiaxial fatigue damage of metal materials, up to date, it is still a challenging task to establish a multiaxial fatigue model with influence of different mean stress. In this paper, a linear mean stress model is presented on basis of the multiaxial model of fatigue life of metal materials by Liu and Yan. By using the experimental data of fatigue life of metal materials reported in the literature, the model is systematically validated
Keywords: multiaxial fatigue, mean stress, metallic materials
Many critical mechanical components experience multiaxial cyclic loading during their service life, such as railroad wheels, crankshafts, axles, and turbine blades, etc. Different from the uniaxial fatigue problem, the multiaxial fatigue problem is more complex due to the complex stress states. In recent decades, a significant amount of research has been devoted to acquire a better understanding of the failure mechanisms under multiaxial loading, including theoretical model study (e.g., a stress invariant method,^{1,5–10 }and a critical plane method,^{11–13} fatigue tests of metallic materials under multiaxial loading (e.g., fatigue test for engineering steels by Gough^{3} and fatigue of wrought hightensile alloy steel by Frith.^{4} Due to the complexity of multiaxial fatigue damage of metal materials, up to date, it is still a challenging task to establish a multiaxial fatigue model with influence of different mean stress. In this paper, a linear mean stress model is presented on basis of the multiaxial model of fatigue life of metal materials by Liu & Yan^{1,2} By using the experimental data of fatigue life of metal materials reported in the literature, the model is systematically validated.
In this section, the Marin’s mean stress model proposed by Liu and Yan^{1} is first described briefly. Then a linear mean stress model is given. After the reader reads this paper, the reason that the linear mean stress model is presented by the author is naturally understood.
A Marin’s mean stress model
A model for fatigue life prediction under multiaxial stress states can be expressed mathematically as follows:
$F({\sigma}_{ea},{\sigma}_{em},\rho ,N,{c}_{1},{c}_{2},\mathrm{...})=0$ （1）
where ${\sigma}_{e}$ is a mechanical parameter which is a measure of stress states in multiaxial loading, and here, the von Mises equivalent stress is adopted ; ${\sigma}_{ea}$ and ${\sigma}_{em}$ are the amplitude and the mean value of the equivalent stress, respectively. Multiaxial parameter $\rho $ is defined as follows:
$\rho =\frac{{\sigma}_{11,a}}{{\sigma}_{e,a}}$ （2）
where ${\sigma}_{11,a}$ is the amplitude of the first invariant of stress tensor. It is evident that, for the axial and shear fatigue loading, the value of the multiaxial parameter $\rho $ is equal to 1 and 0, respectively ${c}_{1},{c}_{2},\mathrm{...}$ . in the Eq. (1) are material coefficients, which are varied with the multiaxial parameter $\rho $ . Based on the previous wide investigations (e.g., Tao and Xia^{14} and Marin’s general equation^{15}) of engineering rules suitable for predicting the mean stress effect under the bending loading, it is assumed here that the model of multiaxial fatigue life has the following form:
$\mathrm{log}{\sigma}_{ea}{(1{\alpha}_{\rho}\frac{{\sigma}_{em}}{{\sigma}_{ea}})}^{{n}_{\rho}}={A}_{\rho}\mathrm{log}N+{C}_{\rho}$ （3）
where ${A}_{\rho}$ 、 ${C}_{\rho}$ 、 ${\alpha}_{\rho}$ 、 ${n}_{\rho}$ are material parameters dependent on the multiaxial parameter. For the axial and shear fatigue loading, Eq. (3) can be simplified as, respectively:
$\mathrm{log}{\sigma}_{a}{(1{\alpha}_{1}\frac{{\sigma}_{m}}{{\sigma}_{a}})}^{{n}_{1}}={A}_{1}\mathrm{log}N+{C}_{1}$ （4）
And
$\mathrm{log}\sqrt{3}{\tau}_{a}{(1{\alpha}_{0}\frac{{\tau}_{m}}{{\tau}_{a}})}^{{n}_{0}}={A}_{0}\mathrm{log}N+{C}_{0}$ （5）
In the absence of mean stress, the Eq. (3) is simplified as:
$\mathrm{log}{\sigma}_{ea}={A}_{\rho}\mathrm{log}N+{C}_{\rho}$ （6）
Thus under the axial and shear fatigue loading, the Eqs (4) and (5) can be written as respectively:
$\mathrm{log}{\sigma}_{a}={A}_{1}\mathrm{log}N+{C}_{1}$ （7）
And
$\mathrm{log}\sqrt{3}{\tau}_{a}={A}_{0}\mathrm{log}N+{C}_{0}$ （8）
Eq. (7) is the well known SN curve equation. Eq. (8) is the variant form of the SN curve equation under the shear fatigue condition, which can be written as:
$\mathrm{log}{\tau}_{a}={A}_{0}^{\text{'}}\mathrm{log}N+{C}_{0}^{\text{'}}$ （9）
in which ${A}_{0}^{\text{'}}$ and ${C}_{0}^{\text{'}}$ are related with ${A}_{0}$ and ${C}_{0}$ through the following relationships:
${A}_{0}^{\text{'}}={A}_{0},{C}_{0}^{\text{'}}={C}_{0}\sqrt{3}$ （10）
In view of the complexity of fatigue life analysis in the multiaxial stress states, and also taking into account that the existing literature has accumulated a large number of fatigue experimental data under the axial and shear loading, from the point of application, it is assumed that the material parameters in Eq. (3) can be obtained by interpolating the material parameters in Eqs (4) and (5), i.e.:
${A}_{\rho}={A}_{1}\cdot \rho +{A}_{0}\cdot (1\rho )$ （11）
${C}_{\rho}={C}_{1}\cdot \rho +{C}_{0}\cdot (1\rho )$ （12）
${\alpha}_{\rho}={\alpha}_{1}\cdot \rho +{\alpha}_{0}\cdot (1\rho )$ （13）
${n}_{\rho}={n}_{1}\cdot \rho +{n}_{0}\cdot (1\rho )$ （14）
In this way, multiaxial fatigue life can be directly estimated from the stress invariant parameter and the multiaxial S–N curve with the attention on mean stress effect.
A linear mean stress model
According to the Marin’s mean stress model of multiaxial loading, i.e., Eq.(3), a linear mean stress model presented in this paper is as follows:
$\mathrm{log}{\sigma}_{ea}(\beta {}_{\rho}{\gamma}_{\rho}\frac{{\sigma}_{em}}{{\sigma}_{e0}})={A}_{\rho}\mathrm{log}N+{C}_{\rho}$ （15）
where
${\sigma}_{eo}={\sigma}_{o}{}^{\rho}{(\sqrt{3}{\tau}_{o})}^{(1\rho )}$ (16)
in which ${\sigma}_{o}$ and ${\tau}_{o}$ are, respectively, a tensile fatigue limit and a shear fatigue limit. Under the axial and shear fatigue loading, Eq (15) can be written as
$\mathrm{log}{\sigma}_{a}(\beta {}_{1}{\gamma}_{1}\frac{{\sigma}_{m}}{{\sigma}_{0}})={A}_{1}\mathrm{log}N+{C}_{1}$ (17)
and
$\mathrm{log}\sqrt{3}{\tau}_{a}(\beta {}_{0}{\gamma}_{0}\frac{{\tau}_{m}}{{\tau}_{0}})={A}_{0}\mathrm{log}N+{C}_{0}$ (18)
in which $\beta {}_{1}$ , ${\gamma}_{1}$ , $\beta {}_{0}$ and ${\gamma}_{0}$ are material constants determined by fitting experimental data of fatigue life.
By the way, material parameters and in Eq (15) can be determined by using the interpolation formulas similar to formulas (13) to (14). Eq (16) is a multiaxial fatigue limit prediction equation recently proposed by Liu & Yan^{2}
Based on the experimental data of fatigue life of metallic materials from literature, in this section, fatigue life analysis will be carried out by using fatigue life prediction equations with the linear mean stress effect and the Marin’s mean stress effect. The comparison of two results will be given. In order to quantitatively evaluate the accuracy of the fatigue life prediction, the following error indexes are defined:
$ER1=\frac{({N}_{cal}{N}_{\mathrm{exp}})\times 100}{{N}_{\mathrm{exp}}}$ (19)
$ER2=\frac{({N}_{cal}^{\text{'}}{N}_{\mathrm{exp}})\times 100}{{N}_{\mathrm{exp}}}$ (20)
$MER1=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{(ER1)}_{i}}$ (21)
$MER2=\frac{1}{n}{\displaystyle \sum _{i=1}^{n}{(ER2)}_{i}}$ (22)
where n is the number of experimental cases, ${N}_{\mathrm{exp}}$ is experimental fatigue life; ${N}_{cal}$ and ${N}_{cal}^{\text{'}}$ are computed fatigue lives by the Marin’s mean stress model and the linear mean stress model, respectively. In this section, the fatigue life prediction of three metallic materials, Cast iron, 18G2A steel, and S355J0 alloy steel under uniaxial and multiaxial loading are carried out. For the sake of clear discussions, they are described in the forms of examples, respectively. The material constants determined by using the experimental data of fatigue life of these materials are listed in Table 1 & Table 2. Note here material constants, and, can be evaluated respectively by means of Eqs (7) and (8). For example, for 18G2A steel, let ${N}_{f}$ =2000000 (cycles), the following two values can be obtained.
${\sigma}_{0}$ =286 (MPa), ${\tau}_{0}$ =188 (MPa)
Materials 
${A}_{1}$  ${C}_{1}$ 
Rsquare 
${\alpha}_{1}$  ${n}_{1}$  ${\beta}_{1}$  ${\gamma}_{1}$ 
Cast iron 
0.140 
2.858 
0.840 
0.499 
1.210 
1.663 
0.039 
18G2A steel 
0.131 
3.286 
0.932 
12.13 
0.139 
1.161 
0.279 
S355J0 alloy steel 
0.177 
3.535 
0.893 
4.316 
0.215 
1.059 
0.343 
Table 1 Material constants and Rsquare of fitted SN equation under the axial fatigue loading
Materials 
${A}_{0}$  ${C}_{0}$ 
Rsquare 
${\alpha}_{0}$  ${n}_{1}$  ${\beta}_{0}$  ${\gamma}_{0}$ 
Cast iron 
0.074 
2.887 
0.965 
0.604 
0.700 
1.357 
0.044 
18G2A steel 
0.074 
2.984 
0.884 
27.233 
0.133 
1.223 
0.379 
S355J0 alloy steel 
0.076 
2.994 
0.670 
138.02 
0.079 
1.290 
0.251 
Table 2 Material constants and Rsquare of fitted SN equation under the shear fatigue loading
Example 1: Cast iron
For cast iron reported by Berto, Lazzarin & Tovo^{16} under zero mean stress, predicted results of fatigue life are given in Table 3. Error ranges of fatigue life are, respectively, [47, 80](%), [41, 66](%) and [43,28](%), for the axial, shear and multiaxial fatigue loading, which shows that the fatigue lives predicted by means of Eqs (6), (7) and (8) are very satisfactory. Note here that there is an early failure case with a relative error 746%, see Table 3, which should be deleted. Under the axial fatigue loading with mean stress , the error ranges of fatigue life, and，are,respectively, [28,44] (%) and [29,45] (%), see Table 4, which illustrates that fatigue lives predicted by means of the linear mean stress model，i.e., Eq (17)，almost are same as those by means of the Marin’s mean stress model, i.e., Eq (4). Under the shear fatigue loading with mean stress, the error ranges of fatigue life, and，are, respectively, [34,30] (%) and [34,23] (%), see Table 4, from which it can be seen that accuracy of fatigue lives predicted by using the two mean stress models is very high, and that the fatigue lives predicted by the linear mean stress model, i.e., Eq. (18), are a little better than those by the Marin’s mean stress model, i.e., Eq. (5). Under the multiaxial fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [72, 30] (%) and [62, 16] (%), see Table 4, which shows that that accuracy of fatigue lives predicted by using the two mean stress models is satisfactory (Table 3).
${\sigma}_{a}$ (MPa) 
${\sigma}_{m}$ (MPa) 
${\tau}_{a}$ (MPa) 
${\tau}_{m}$ (MPa) 
${N}_{\mathrm{exp}}$ (Cycles) 
ER(%) 


0 

0 


Table 3 Comparison of experimental data and predicted results of fatigue life of Cast iron with zero mean stress
${\sigma}_{a}$ (MPa) 
${\sigma}_{m}$ (MPa) 
${\tau}_{a}$ (MPa) 
${\tau}_{m}$ (MPa) 
${N}_{\mathrm{exp}}$ (Cycles) 
ER1 (%) 
ER2 (%) 








Table 4 Comparison of experimental data and predicted results of fatigue life of Cast iron with nonzero mean stress
Example 2: 18G2A steel
For 18G2A steel reported by Gasiak & Pawliczek^{17}, under zero mean stress, predicted results of fatigue life are given in Table 5. Error ranges of fatigue life are, respectively, [28, 105] (%), [34, 81] (%) and [54,13] (%), for the axial, shear and multiaxial fatigue loading, which shows that the fatigue lives predicted by means of Eqs (6), (7) and (8) are very satisfactory. Under the axial fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [45,92] (%) and [40,92] (%), see Table 6, which illustrates that fatigue lives predicted by the linear mean stress model, i.e., Eq (17) almost are same as those by means of the Marin’s mean stress model, i.e., Eq (4). Under the shear fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [46,53](%) and [54,46] (%), see Table 6, from which it can be seen that accuracy of fatigue lives predicted by using the Marin’s mean stress model, i.e., Eq (5) almost is same as that by means of the linear mean stress model, i.e., Eq (18). Under the multiaxial fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [49,172](%) and[42,170](%), see Table 7, which shows that the fatigue lives predicted by using the linear mean stress model, i.e., Eq (15), are a little better than those by the Marin’s mean stress model, i.e., Eq (3) (Table 4 Table 7).
${\sigma}_{a}$ (MPa) 
${\sigma}_{m}$ (MPa) 
${\tau}_{a}$ (MPa) 
${\tau}_{m}$ (MPa) 
${N}_{\mathrm{exp}}$ 
ER(%) 







Table 5 Comparison of experimental data and predicted results of fatigue life of 18G2A steel with zero mean stress
${\sigma}_{a}$ (MPa) 
${\sigma}_{m}$ (MPa) 
${\tau}_{a}$ (MPa) 
${\tau}_{m}$ (MPa) 
${N}_{\mathrm{exp}}$ 
ER1(%) 
ER2(%) 








Table 6 Comparison of experimental data and predicted results of fatigue life of 18G2A steel with nonzero mean stress (under uniaxial fatigue loading)
${\sigma}_{a}$ (MPa) 
${\sigma}_{m}$ (MPa) 
${\tau}_{a}$ (MPa) 
${\tau}_{m}$ (MPa) 
${N}_{\mathrm{exp}}$ 
ER1(%) 
ER2(%) 








Table 7 Comparison of experimental data and predicted results of fatigue life of 18G2A steel with nonzero mean stress (under multiaxial fatigue loading)
Example 3: S355J0 alloy steel
For S355J0 alloy steel reported by Gasiak & Pawliczek^{18} under zero mean stress, predicted results of fatigue life are given in Table 8. Error ranges of fatigue life are, respectively, [25,17] (%) and [33,81] (%), for the axial and shear fatigue loading, which shows that the fatigue lives predicted by means of Eqs (7) and (8) are very accurate. Under multiaxial fatigue loading, error range of fatigue life, [68,13](%), is satisfactory. Under the axial fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [57, 45] (%) and [51, 39] (%), see Table 9, which illustrates that accuracy of fatigue lives predicted by using the two mean stress models is very high, and that the fatigue lives predicted by the linear mean stress model, i.e., Eq (17), are a little better than those by the Marin’s mean stress model, i.e., Eq (4). Under the shear fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [42, 79] (%) and [48, 86] (%), see Table 9, from which it can be seen that accuracy of fatigue life predicted by using the two mean stress models is very high, and that the fatigue lives predicted by the Marin’s mean stress model, i.e., Eq (5), are a little better than those by the linear mean stress model, i.e., Eq (18). Under the multiaxial fatigue loading with mean stress, the error range and the mean error of fatigue lives predicted by using the linear mean stress model, i.e., Eq (15), are, respectively, [38,99] (%) and 8%, see Table 9, which shows that that accuracy of fatigue lives predicted by using the linear mean stress model is very high. However, the error range and the mean error of fatigue lives predicted by means of the Marin’s mean stress model, i.e., Eq (3), are, respectively, [89,68] (%) and 84%, see Table 9, which shows that the predicted by the Marin’s mean stress model is very poor. The comparison of the two results is shown In Figure 1. What is the reason that, under the multiaxial fatigue loading with mean stress, the accuracy of fatigue lives of S355J0 alloy steel predicted by the Marin’s mean stress model is very poor? For this, the author observes the material constants listed in Table 1 & Table 2 and finds that, for S355J0 alloy steel, is much less than, (=0.03127, see Table 10), which is the reason that the accuracy of fatigue lives predicted by using the Marin’s mean stress model, i.e., Eq. (3), is very poor. In order to further illustrate it, in the following form of Marin’s mean stress model:
$\mathrm{log}{\sigma}_{ea}{(1\alpha \frac{{\sigma}_{em}}{{\sigma}_{ea}})}^{n}={A}_{\rho}\mathrm{log}N+{C}_{\rho}$ (23)
${\sigma}_{a}$ (MPa) 
${\sigma}_{m}$ (MPa) 
${\tau}_{a}$ (MPa) 
${\tau}_{m}$ (MPa) 
${N}_{\mathrm{exp}}$ (Cycles) 
ER(%) 







Table 8 Comparison of experimental data and predicted results of fatigue life of S355J0 alloy steel with zero mean stress
${\sigma}_{a}$ (MPa) 
${\sigma}_{m}$ (MPa) 
${\tau}_{a}$ (MPa) 
${\tau}_{m}$ (MPa) 
${N}_{\mathrm{exp}}$ 
ER1(%) 
ER2(%) 








Table 9 Comparison of experimental data and predicted results of fatigue life of S355J0 alloy steel with nonzero mean stress

Marin’s mean stress model 
Linear mean stress model 

Materials 
${\alpha}_{1}/{\alpha}_{0}$  ${n}_{1}/{n}_{0}$  ${\beta}_{1}/{\beta}_{0}$  ${\gamma}_{1}/{\gamma}_{0}$ 
Cast iron 
0.826159 
1.726845 
1.225497 
0.88636 
18G2A steel 
0.445415 
1.045113 
0.949305 
0.736148 
S355J0 alloy steel 
0.031271 
2.697616 
0.820739 
1.366534 
Table 10 Comparison of material constants used to describe the effect of mean stress on fatigue life
Figure 1 Comparison of fatigue lives predicted by the linear mean stress model and the Marin’s mean stress model under multiaxial fatigue loading with nonzero mean stress.
by letting, and, , fatigue lives predicted are listed in 11, in which the error ranges are, respectively, [60,16](%) and [37,188] (%), and the mean errors are, respectively, 21(%) and 38(%) (Table 12). Comparison of detail errors are given in Table 12, from which it can be seen that the accuracy of fatigue lives predicted by using the linear mean stress model, i.e., Eq (15), is much better than that by using the Marin’s mean stress model, Eq (3).
${\sigma}_{a}$ (MPa) 
${\sigma}_{m}$ (MPa) 
${\tau}_{a}$ (MPa) 
${\tau}_{m}$ (MPa) 
${N}_{\mathrm{exp}}$ 
$\alpha ={\alpha}_{1}$ , $n={n}_{1}$ ER1(%) 
$\alpha ={\alpha}_{0}$ $n={n}_{0}$ 








Table 11 Comparison of experimental data and predicted results of fatigue life of S355J0 alloy steel under multiaxial fatigue loading with nonzero mean stress
Marin’s mean stress model 
Linear mean stress model 

$\alpha ={\alpha}_{1}$ , $n={n}_{1}$ 
$\alpha ={\alpha}_{0}$ , $n={n}_{0}$ 
$\alpha ={\alpha}_{\rho}$ , $n={n}_{\rho}$ 

Error range (%) 
Mean error (%) 
Error range (%) 
Mean error (%) 
Error range (%) 
Mean error (%) 
Error range (%) 
Mean error (%) 
[60,16] 
21 
[37,188] 
38 
[89,68] 
84 
[38,99] 
8 
Table 12 General summary of the analysis of variance (ANOVA), mean and coefficient of variation (CV), of the productive and nutritional characteristics of the different proportions of oats and vetch
F tests: ***; P<0.001, **; P<0.01, *; P<0.05, ns; Not significant
On basis of the present investigation, it can be seen that:
Why is the linear mean stress model, which, in the mathematical formula, has no an advantage over the Marin’s mean stress model, presented in this paper? The reason is on basis of the following two aspects:
${\sigma}_{ean}={\sigma}_{ea}(1+\beta F)$ ${\sigma}_{ean}/{\sigma}_{ea}=\lambda +\gamma F$ (24)
where $\beta $ is material sensitivity parameter to loadpath nonproportionality defined on $\sigma \u2014\sqrt{3}\tau $ stress plane; F is usually called a nonproportional loading factor which expresses the severity of non proportional loading. The author finds that the prediction results of fatigue life obtained by using Itoh’s formula sometimes are not satisfactory. Thus the following attempt is made:
${\sigma}_{ean}/{\sigma}_{ea}=\lambda +\gamma F$ (25)
where $\lambda $ , $\gamma $ are constants to be determined by linear fitting of ${\sigma}_{ean}$ and F. Obviously, the Itoh’s formula (24) has mechanical advantage over the linear formula (25). But verification by experimental data of fatigue life of metallic materials shows that the accuracy of fatigue lives predicted by means of the linear formula (25) is higher than that by the Itoh’s formula (24), which will be reported in another paper.
None.
The authors declare that there is no conflict of interest.
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