A new model of multiaxial fatigue life prediction with the influence of different mean stresses

Abstract

In this paper, based on the thought of Modified Wöhler Curve Method (MWCM), a new general model for predicting multiaxial fatigue life with influence of mean stress is presented. Different from the MWCM, the expressions of multiaxiality effect and mean stress effect are located separately in the proposed fatigue equation, so that the new model can consider the impact of both axial and torsional mean stresses, and the equation form possesses excellent extendibility and variability. The wildly used von Mises equivalent stress is adopted as the fatigue parameter to improve computational efficiency. Finally, in conjunction with the Itoh criterion, the model can be trivially extended to perform non-proportional fatigue prediction with different mean stresses. Some representative fatigue tests published in the previous literature are used to verify this study.

Keywords

Multiaxial fatigue mean stress metallic materials proportional loading non-proportional loading

Introduction

As one of the main failure modes in real components, multiaxial fatigue has attracted much attention. The problem of performing fatigue assessment is further complicated when mean stress is included. This makes it evident that the sound engineering tools capable of accurately and efficiently estimating multiaxial fatigue damage with mean stress inference is needed.

Plenty of fatigue criteria have been derived, which can be subdivided into three major categories: stress-, strain- and energy-based approaches (Cristofori et al., 2008; Crossland, 1956; Fatemi and Socie, 1988; Findley, 1957; Gasiak and Pawliczek, 2003; Kluger et al., 2014; Liang and Chen, 2016; Liu and Yan, 2018ab; Marin, 1956; Mcdiarmid, 1994; Mishra et al., 2016; Papadopoulos, 2001; Shang and Wang, 1998; Shen and Akanda, 2016; Sines, 1961; Susmel and Lazzarin, 2002; Zghal et al., 2016; Zhan et al., 2016; Zhang et al., 2018; Zhu et al., 2016, 2017). From a different perspective, they can also be classified as the critical plane approaches and the stress invariant-based approaches. Susmel and Lazzarin (2002) ever considered that the most interesting and promising methodologies are those based on the critical plane concept. Thus, over the last decade, Susmel and his collaborators made a great effort to devise a novel approach that is suitable for estimating fatigue damage not only in plain specimens, but also in notched components, the well-known Modified Wöhler Curve Method (MWCM). The critical plane approaches seem to have more physical significance, especially when the fatigue parameters adopt the shear mechanical components, which are associated with the occurrence of slip bands on the surfaces. Unfortunately, the calculation of the range or mean value of the stress parameter on a critical plane is sometimes extremely time-consuming, limiting the application of these criteria to address practical problems.

As demonstrated by Cristofori et al. (2008): advanced design methodologies, such as computer-aided design, require the formalization of criteria which have to be efficient in calculation. In this scenario, stress invariant-based criteria are generally deemed to be the most promising ones to meet the above requirement. Meanwhile, Papadopoulos put forward a deviatoric stress space theory (Papadopoulos et al., 1997), so that the non-proportional loading paths can be properly described by the invariant quantities. This is why increased stress invariant-based criteria have been proposed and validated by researchers all along (Cristofori et al., 2008; Crossland, 1956; Marin, 1956; Sines, 1961).

The mean stress has always been an important fatigue influence. Recently, a comparison study of existing multiaxial fatigue criteria related to metallic materials with different mean stresses was performed by Kluger and Lagoda (2014). It showed that whether the empirical models (e.g., Goodman, 1899; Morrow, 1968) or the critical plane approaches (Findley, 1957; McDiarmid, 1994; Papadopoulos, 2001) could produce satisfactory predictions in the absence of mean stresses. However, when subjected to the loads including non-zero mean stresses, the calculated results are seriously different from the measured values, and the representative instances are the S355J0 alloy steel and 30NCD16 steel.

For the multiaxial fatigue assessment combined with mean stresses, a focus of dispute is whether the influence of shear mean stresses on multiaxial fatigue damage deserves to be taken into account. On the one hand, through a series of experiments, Sines (1961) achieved a conclusion that the presence of superimposed static torsional stresses can be neglected as long as maximum shear stress $τ_{max}$ to shear yield stress $τ_{y}$ lower than unity. This standpoint has a far-reaching impact on the follow-up researches, and even once became the criterion for the correctness of the subsequent fatigue models. On the other hand, Gough (1949) and Wang and Miller (1991) presented an opposite viewpoint by investigating the materials S65A and 1.99% NiCrMo steel, respectively. Furthermore, Papadopoulos demonstrated that the shear mean stress influence is more and more obvious with the decrease of life region (Papadopoulos et al., 1997), indicating the shear mean stresses should not be ignored in finite fatigue life prediction.

In view of the above comparison of the multiaxial fatigue criteria and the ambiguous cognition on the contribution of shear mean stress, it is still a challenging issue to establish a general multiaxial mean stress fatigue life prediction model. Based on the thought of MWCM, a new general model aimed at addressing multiaxial mean stress effect is proposed in this paper. The wildly used von Mises equivalent stress is adopted as the fatigue parameter to improve computational efficiency. Different from the MWCM, the stress multiaxiality and mean stress effect are expressed separately in the fatigue equation, thus it has the capability to reflect the contribution of both axial and torsional mean stresses to multiaxial fatigue damage. In the final part, the authors make an attempt to extend the proposed model to evaluate non-proportional fatigue lives by combining with the Itoh criterion. The present study will be systematically validated with the experimental data reported in the previous literature.

A new model of multiaxial fatigue life prediction

The fatigue equation forms of the conventional mean stress methods can be reduced to the following expression

F_{c} (σ_{ea}, σ_{em}, α, n, \dots) = S_{1 or 0} (N, A, C, \dots)

(1)

where F is the function of stress quantities and material parameters. In detail,

σ_{e}

is the mechanical parameter measuring multiaxial stress states;

σ_{ea}

and

σ_{em}

are the amplitude and mean value of

σ_{e}

, respectively; α, n... are the material parameters describing mean stress influence; function

S_{1 or 0}

represents the uniaxial basic fatigue reference curve, and the superscripts 1 and 0 express axial and torsional loading cases respectively; N is the number of cycles of fatigue failure; A, C... are the material parameters depicting the fatigue curve. It can be noted that these methods can be explicitly formulated by changing the term of function F_c on the left end of equation, and they depend on only one type of uniaxial fatigue curve.

Different from the above traditional methods, a significant innovation point of MWCM is to comprehensively utilize both axial and torsional fatigue curves when estimating multiaxial fatigue life, as written

F_{MWCM} (τ_{c, a}) = S_{ρ_{eff}} (N, ρ_{eff}, A_{ρ_{eff}}, C_{ρ_{eff}}, \dots)

(2)

where

τ_{c, a}

is maximum shear stress amplitude;

ρ_{eff}

is the effective multiaxial parameter, which controls the material parameters

A_{ρ_{eff}}

C_{ρ_{eff}}, \dots

; and MWCM can also consider the mean stress effect by introducing a mean stress sensitivity index m with the range from 0 to 1, as defined

\begin{matrix} ρ_{eff} = \frac{σ_{n, a} + m σ_{n, m}}{τ_{c, a}} \\ ρ_{0} = \frac{σ_{n, a}}{τ_{c, a}} \end{matrix}

(3)

where

ρ_{0}

is the so-called multiaxial parameter, which quantifies the degree of multiaxiality of stress field;

σ_{n, a}

and

σ_{n, m}

are the amplitude and mean value of the normal stress perpendicular to the critical plane, respectively. It is worthwhile pointing out that based on the definition of

ρ_{eff}

, the detrimental contribution of shear mean stress is neglected by MWCM.

Inspired by the formulations of the conventional methods and MWCM, the new model adopts the following mathematical expression

F_{3} (σ_{ea}, σ_{em}, ρ, α_{ρ}, n_{ρ}, \dots) = S_{ρ} (N, ρ, A_{ρ}, C_{ρ}, \dots)

(4)

where von Mises equivalent stress is selected as the fatigue parameter

σ_{e}

. The amplitude and mean value of von Mises stress can be calculated according to the deviatoric stress space theory proposed by Papadopoulos et al. (1997). In detail, under a proportional loading, the amplitude is numerically equal to the distance between the two terminals of loading path, and the mean stress is the corresponding value of loading midpoint. Multiaxial parameter ρ in the present paper is defined similar to the triaxial stress ratio as follows

ρ = \frac{σ_{11, a}}{σ_{e, a}}

(5)

where

σ_{11, a}

is the range of the first invariant of stress tensor.

It is trivial to validate that, in the axial and torsional cases, the values of multiaxial parameter ρ are equal to 1 and 0, respectively. Moreover, the material parameters $α_{ρ}, n_{ρ}, A_{ρ}, C_{ρ} \dots$ are also varied with multiaxial parameter ρ.

Based on the previous investigations on mean stress influence (e.g., Tao and Xia model (2007) and Marin general equation (1956)), and the fact that a constant (such as material tensile strength) does not affect the fitting accuracy mathematically, the proposed model selects the equivalent stress amplitude instead of the ultimate tensile strength in Marin equation as the second calibration information. The specific equation form of the new model can be expressed as

log σ_{ea} (1 - α_{ρ} \frac{σ_{em}}{σ_{ea}})^{n_{ρ}} = A_{ρ} log N + C_{ρ}

(6)

When subjected to pure tension and torsion, equation (6) can be respectively simplified as

log σ_{a} (1 - α_{1} \frac{σ_{m}}{σ_{a}})^{n_{1}} = A_{1} log N + C_{1}

(7)

log \sqrt{3} τ_{a} (1 - α_{0} \frac{τ_{m}}{τ_{a}})^{n_{0}} = A_{0} log N + C_{0}

(8)

Equation (7) is consistent with the axial mean stress model reported by Tao and Xia (2007). From the brief review of the previous discussion on the shear mean stress influence in introduction, it is reasonable to believe that the sensitivity to shear mean stresses is an individual material property. Therefore, the present model introduces a set of material parameters ( $α_{0}$ and n₀) in equation (8) to address this property.

In the absence of mean stresses, equation (6) can be rewritten as

log σ_{ea} = A_{ρ} log N + C_{ρ}

(9)

Then equations (7) and (8) under the axial and torsional loadings can be respectively simplified as

log σ_{a} = A_{1} log N + C_{1}

(10)

log \sqrt{3} τ_{a} = A_{0} log N + C_{0}

(11)

Equations (10) and (11) are the well-known axial and torsional S–N curve equation forms. In view of the complexity of multiaxial fatigue life prediction, and based on the interpolation law suggested by MWCM for characterizing the transition of multiaxial fatigue curves from axial to torsional cases, from the application point, it is assumed that the material parameters in equation (6) can be obtained by interpolating the material parameters in equations (7) and (8), i.e.

A_{ρ} = A_{1} \cdot ρ + A_{0} \cdot (1 - ρ)

(12)

C_{ρ} = C_{1} \cdot ρ + C_{0} \cdot (1 - ρ)

(13)

α_{ρ} = α_{1} \cdot ρ + α_{0} \cdot (1 - ρ)

(14)

n_{ρ} = n_{1} \cdot ρ + n_{0} \cdot (1 - ρ)

(15)

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. In addition, from the perspective of equation structure, MWCM concentrates the descriptions of both mean stress and multiaxiality influences in one parameter $ρ_{eff}$ , and further reflected in the parameters $A_{ρ_{eff}}$ , $C_{ρ_{eff}}$ . However, the present model separately represents the two influences on two sides of the equation. This enables the equation form to have well extendibility and variability, as generally expressed

log (M (σ_{ea}, σ_{em}, α_{ρ}, n_{ρ}, \dots)) = A_{ρ} log N + C_{ρ}

(16)

where

M (σ_{ea}, σ_{em}, α_{ρ}, n_{ρ}, \dots)

is an equivalent form transformed from the traditional axial mean stress criteria according to the diverse application conditions.

It is worth emphasizing that the fundamental purpose of this study is to propose a general multiaxial mean stress model with an invariance equation form as equations (4) and (6). Therefore, when the fatigue experiment amount is limited, the formulations with fewer material parameters can be adopted in the left part of equation (6) or (16). For instance, the modified verification equations for the 30NCD16 steel and 7075-T651 aluminum alloy in the subsequent sections.

Experimental verification and discussions

This section takes advantage of the representative fatigue tests of materials 18G2A steel and 30NCD16 steel to examine the proposed model. The experiments were carried out under the proportional loadings accompanied by non-zero mean stresses, and detailed experimental information can be found in Refs (Froustey and Lasserre, 1989; Gasiak and Pawliczek, 2001). Simultaneously, the famous MWCM is used as a comparison object in this validation process.

The round smooth specimen made of 18G2A steel (Gasiak and Pawliczek, 2001) is referenced as the first example. Initially, on the basis of the experimental data of fully reversed (R=−1) bending and torsional loadings, the parameters A₁, A₀, C₁ and C₀ in equations (10) and (11) can be determined by means of numerically fitting technique, and the corresponding results are listed in Table 1. Subsequently, combined with the parameters calculated in the above procedure and the measured lives under two uniaxial loadings with non-zero mean stresses (R ≠ −1), the material parameters

α_{1}

, n₁,

α_{0}

and n₀ in equations (7) and (8) can be obtained by fitting as well.

Table 1.

Calculated material coefficients and corresponding R-square values of the predicted materials.

Materials	Axial			Torsion
Materials	A ₁	C ₁	R-square	A ₀	C ₀	R-square
18G2A	−0.13	3.28	0.93	−0.07	2.98	0.88
30NCD16	−0.09	3.37	0.99	−0.12	3.53	0.99
6082-T6	−0.14	3.02	0.99	−0.12	2.87	0.94
7075-T651	−0.10	2.64	0.96	−0.12	2.83	0.96

The axial fitting accuracy can be illustrated in terms of the relationship between the normalized equivalent stresses $σ_{a, e} / σ_{a}$ and the normalized mean stresses $σ_{m} / σ_{a}$ in Figure 1, where $σ_{a, e} = σ_{a} (1 - α_{1} \frac{σ_{m}}{σ_{a}})^{n_{1}}$ is calculated by the R ≠ −1 axial experimental life according to the R = −1 fatigue curve. Figure 2 shows a similar comparison for the torsional cases. Meanwhile, the mean stress sensitivity index (m = 2.79) in the MWCM is determined by the both R = −1 and R ≠ −1 axial data as listed in Table 2.

Figure 1.

A comparison of experimental and fitting results of $σ_{a, e} / σ_{a}$ vs. $σ_{m} / σ_{a}$ under bending loadings with non-zero mean tensile stresses.

Figure 2.

A comparison of experimental and fitting results of $τ_{a, e} / τ_{a}$ vs. $τ_{m} / τ_{a}$ under torsional loadings with non-zero mean shear stresses.

Table 2.

The calculated material parameters describing mean stress effect for 18G2A steel, 30NCD16 steel and 7075-T651 aluminum alloy.

Materials	Axial			Torsion			MWCM
Materials	$α_{1}$	n ₁	R-square	$α_{0}$	n ₀	R-square	m	R-square
18G2A	−17.36	0.12	0.63	−30.30	0.117	0.88	2.79	−0.40
30NCD16	−0.30	1.05	0.66	0	1.05	–	1.20	0.77
7075-T651	0.33	1	0.68	0.17	1	0.68	0.81	0.68

MWCM: Modified Wöhler Curve Method.

In order to quantify the prediction accuracy, the following two error indexes are introduced

E_{N} = \frac{N_{f, cal} - N_{f, exp}}{N_{f, exp}} 100 %

(17)

ER = \frac{1}{n} \sum_{i = 1}^{n} | (E_{N})_{i} |

(18)

where E_N is the relative error for one experimental sample, while error index

ER

represents the general precision of a set of data. In detail,

N_{f, cal}

and

N_{f, exp}

are the calculated and experimental fatigue life, respectively; n is the number of experimental samples in the corresponding comparison set.

The comparisons of experimental and calculated lives by the new model and MWCM under bending loadings are shown in Figure 3, and the detailed values are listed in Table 3. For R = −1 cases, it can be observed that the results of the present model are same as those of the MWCM, with the relative error range [−27, 104] (%) and the error index 23%. Meanwhile, the vast majority of results with few exceptions are included in the scatter band of factor 2. However, turning to the R ≠ −1 cases, the present model provides intuitively better predictions compared to the MWCM, included in the scatter band of coefficient 2, with the relative error range [−45, 91] (%) and error index 29%. Whereas several results of MWCM fall outside the lines of factor 2 and even 3, within the relative error interval of [−75, 204] (%) and the error index 56%.

Figure 3.

Comparisons of experimental and calculated lives by the new model and MWCM under bending loadings for 18G2A steel.

Table 3.

Comparisons of experimental and calculated lives by the new model and MWCM under bending loadings for 18G2A steel.

$σ_{x, a}$ (MPa)	$σ_{x, m}$ (MPa)	$N_{f, exp}$ (Cycles)	$N_{f, cal}$ (Cycles)	$E_{N} (%)$	$τ_{c, a}$ (MPa)	$σ_{n, a}$ (MPa)	$σ_{n, m}$ (MPa)	$N_{f, cal}$ (Cycles)	$E_{N} (%)$
399.6	0	128,630	157,205	22.21	199.8	199.80	0	157,205	22.21
399.8	0	153,881	156,610	1.77	199.9	199.90	0	156,610	1.77
399.8	0	168,309	156,610	−6.95	199.9	199.90	0	156,610	−6.95
364.1	0	327,016	318,441	−2.62	182.0	182.05	0	318,441	−2.62
365.3	0	375,852	310,590	−17.34	182.7	182.65	0	310,590	−17.36
367.0	0	415,067	299,839	−27.76	183.5	183.50	0	299,839	−27.76
275.0	0	1,306,495	2,678,087	104.98	137.50	137.50	0	2,678,087	104.98
275.4	0	2,869,140	2,648,716	−7.68	137.65	137.70	0	2,648,716	−7.68
275.5	0	3,364,694	2,641,430	−21.49	137.75	137.75	0	2,641,430	−21.49
369.5	123.2	57,172	49,692	−13.08	184.75	184.75	61.60	125,527	119.56
369.6	123.2	65,725	49,601	−24.53	184.80	184.80	61.60	125,363	90.73
369.1	123.0	72,614	50,123	−30.97	184.55	184.55	61.50	126,305	73.94
298.3	99.4	131,431	252,242	91.92	149.15	149.15	49.70	399,938	204.30
298.5	99.5	168,583	250,897	48.82	149.25	149.25	49.75	398,368	136.30
298.5	99.5	178,962	250,897	40.19	149.25	149.25	49.75	398,368	122.60
268.8	89.6	578,316	555,687	−3.91	134.40	134.40	44.80	702,403	21.45
268.9	89.6	671,481	554,281	−17.45	134.45	134.45	44.80	701,272	4.43
268.9	89.6	712,823	554,281	−22.24	134.45	134.45	44.80	701,272	−1.62
247.3	82.4	976,665	1,046,320	7.13	123.65	123.65	41.20	1,103,368	12.97
247.1	82.4	1,356,769	1,052,101	−22.45	123.55	123.55	41.20	1,107,127	−18.40
290.2	290.2	95,759	125,446	31.00	145.10	145.10	145.10	137,137	43.21
288.6	288.6	133,071	130,820	−1.69	144.30	144.30	144.30	139,769	5.03
265.7	265.7	157,011	244,959	56.01	132.85	132.85	132.85	185,733	18.29
274.3	274.3	173,047	192,366	11.16	137.15	137.15	137.15	166,461	−3.81
255.0	255.0	241,679	334,608	38.45	127.50	127.50	127.50	213,934	−11.48
254.5	254.5	422,196	339,628	−19.55	127.25	127.25	127.25	215,383	−48.99
235.3	235.3	887,342	615,843	−30.59	117.65	117.65	117.65	282,075	−68.21
235.4	235.4	1,126,890	613,861	−45.52	117.70	117.70	117.70	281,663	−75.00

Note: The left table part corresponds to the new model; the right is for MWCM. Tables 4 to 6 employ the same table layout.

An identical comparison with the axial situations is carried out on torsional loadings. As illustrated in Table 4 and Figure 4, the predictions of the two compared approaches are still consistent in the R = −1 torsional cases, with the relative error range [−33, 80] (%) and the error index 32%. When subjected to the R ≠ −1 torsional loadings, the new model is obviously superior to the MWCM, and produces the predicted points located near the diagonal with the relative error range [−46, 53] (%) and an error index 24%. In contrast, the results calculated by the MWCM are unacceptable accompanied by an order of magnitude error.

Figure 4.

Comparisons of experimental and calculated lives by the new model and MWCM under torsional loadings for 18G2A steel.

Table 4.

Comparisons of experimental and calculated lives by the new model and MWCM under torsional loadings for 18G2A steel.

$τ_{xy, a}$ (MPa)	$τ_{xy, m}$ (MPa)	$N_{f, exp}$ (Cycles)	$N_{f, cal}$ (Cycles)	$E_{N} (%)$	$τ_{c, a}$ (MPa)	$N_{f, cal}$ (Cycles)	$E_{N} (%)$
223.0	0	224,794	201,785	−10.24	223.0	201,785	−10.24
222.6	0	278,939	206,676	−25.91	222.6	206,676	−25.91
222.7	0	310,708	205,442	−33.88	222.7	205,441	−33.88
195.7	0	637,980	1,152,073	80.58	195.7	1,152,073	80.581
195.4	0	785,106	1,175,895	49.78	195.4	1,175,895	49.775
196.2	0	874,370	1,113,516	27.35	196.2	1,113,516	27.351
183.9	0	2,296,066	2,641,153	15.03	183.9	2,641,153	15.029
184.0	0	3,095,310	2,622,068	−15.29	184.0	2,622,068	−15.29
183.0	0	4,173,870	2,819,793	−32.44	183.0	2,819,793	−32.44
175.0	58.3	157,758	118,650	−24.79	175.0	5,119,127	3144.9
174.5	58.2	229,202	123,066	−46.31	174.5	5,318,302	2220.4
151.6	50.5	551,271	805,290	46.08	151.6	34,739,540	6201.7
151.3	50.4	640,123	826,855	29.17	151.3	35,669,785	5472.3
151.3	50.4	689,753	826,855	19.88	151.3	35,669,785	5071.4
138.4	46.1	2,176,479	2,715,068	24.75	138.4	117,115,316	5281
138.4	46.1	3,187,921	2,715,068	−14.83	138.4	117,115,316	3573.7
138.2	138.2	356,608	546,615	53.28	138.2	119,396,645	33,381
137.8	137.8	526,773	568,165	7.86	137.8	124,103,897	23,459
135.4	135.4	710,704	718,240	1.06	135.4	156,884,638	21,975
135.2	135.2	860,208	732,544	−14.84	135.2	160,009,079	18,501
131.4	131.4	1,123,272	1,071,535	−4.61	131.4	234,054,539	20,737
131.2	131.2	1,501,895	1,093,532	−27.19	131.2	238,859,312	15,804

From the above comparisons, it can be found that, since the fatigue equations of the two models can be reduced to the same form under two basic R = −1 uniaxial loadings, there is no deviation in prediction accuracy between them. Perhaps due to the fact that there is only one material parameter m in the MWCM, the axial mean stress effect on 18G2A may not adequately addressed, so that the R ≠ −1 axial predictions of MWCM are not as good as those of the new model. Nevertheless, the new model can obtain the R ≠ −1 uniaxial results with the similar accuracy to those without shear mean stresses. Thus it can be concluded that equations (7) and (8) are capable of properly revealing the impact of the mean stress on both axial and torsional loadings in terms of an effective stress parameter $σ_{a, e}$ (i.e. $σ_{e, a} (1 - α \frac{σ_{m, e}}{σ_{a, e}})^{n}$ ).

Furthermore, Figure 5 and Table 5 summarize the comparison results with regard to the multiaxial loadings. The R = −1 results calculated by the present model and MWCM are well correlated with the experiment, leading to the relative error ranges [−54,−12] (%), [−65, −37] (%) and error indexes 33%, 50%, respectively. This indicates that the definition of multiaxial parameter ρ in equation (5) has the ability to appropriately quantify the stress multiaxiality under proportional loadings.

Figure 5.

Comparisons of experimental and calculated lives by the new model and MWCM under multiaxial loadings of R=−1 for 18G2A steel.

Table 5.

Comparisons of experimental and calculated lives by the new model and MWCM under multiaxial loadings (18G2A).

$σ_{x, a}$ (MPa)	$σ_{x, m}$ (MPa)	$τ_{xy, a}$ (MPa)	$τ_{xy, m}$ (MPa)	$N_{f, exp}$ (Cycles)	$N_{f, cal}$ (Cycles)	$E_{N} (%)$	$τ_{c, a}$ (MPa)	$σ_{n, a}$ (MPa)	$σ_{n, m}$ (MPa)	$N_{f, cal}$ (Cycles)	$E_{N} (%)$
199.7	0	199.7	0	168,497	146,721	−12.92	223.3	99.9	0	105,537	−37.37
199.7	0	199.7	0	192,588	146,721	−23.82	223.3	99.9	0	105,537	−45.20
199.5	0	199.5	0	226,543	148,150	−34.60	223.1	99.8	0	106,596	−52.95
180.2	0	180.2	0	609,639	396,401	−34.98	201.5	90.1	0	293,732	−51.82
180.2	0	180.2	0	615,487	396,401	−35.60	201.5	90.1	0	293,732	−52.28
180.0	0	180.0	0	737,954	400,682	−45.70	201.3	90.0	0	297,000	−59.75
164.5	0	164.5	0	1,303,741	957,372	−26.57	183.9	82.3	0	728,356	−44.13
164.6	0	164.6	0	1,407,201	951,760	−32.37	184.0	82.3	0	723,959	−48.55
164.4	0	164.4	0	2,101,434	963,019	−54.17	183.8	82.2	0	732,781	−65.13
146.5	48.8	146.5	48.8	87,304	237,451	171.98	163.8	73.3	24.4	898,190	928.81
146.7	48.9	146.7	48.9	93,327	234,175	150.92	164.0	73.4	24.5	887,905	851.39
146.5	48.8	146.5	48.8	121,935	237,451	94.74	163.8	73.3	24.4	898,190	636.61
129.8	43.3	129.8	43.3	382,788	764,496	99.72	145.1	64.9	21.7	2,379,290	521.57
129.8	43.3	129.8	43.3	417,127	764,496	83.28	145.1	64.9	21.7	2,379,290	470.40
121.4	40.5	121.4	40.5	1,408,351	1,460,198	3.68	135.7	60.7	20.3	4,079,074	189.63
121.4	40.5	121.4	40.5	1,505,671	1,460,198	−3.02	135.7	60.7	20.3	4,079,074	170.91
118.5	39.5	118.5	39.5	2,317,350	1,846,489	−20.32	132.5	59.3	19.8	4,960,948	114.08
118.3	39.4	118.3	39.4	3,056,739	1,878,526	−38.54	132.3	59.2	19.7	5,033,601	64.67
139.4	139.4	139.4	139.4	115,525	118,814	2.85	155.9	69.7	69.7	396,522	243.23
139.4	139.4	139.4	139.4	138,497	118,814	−14.21	155.9	69.7	69.7	396,522	186.30
129.2	129.2	129.2	129.2	211,846	247,787	16.97	144.5	64.6	64.6	617,608	191.54
129.5	129.5	129.5	129.5	233,041	242,290	3.97	144.8	64.8	64.8	609,311	161.46
123.2	123.2	123.2	123.2	257,221	392,505	52.59	137.7	61.6	61.6	814,980	216.84
123.2	123.2	123.2	123.2	366,179	392,505	7.19	137.7	61.6	61.6	814,980	122.56
120.9	120.9	120.9	120.9	681,921	470,992	−30.93	135.2	60.5	60.5	909,653	33.40
120.9	120.9	120.9	120.9	794,444	470,992	−40.71	135.2	60.5	60.5	909,653	14.50
118.0	118.0	118.0	118.0	1,165,727	595,674	−48.90	131.9	59.0	59.0	1,048,012	−10.10

Turning to the R ≠ −1 multiaxial conditions shown in Figure 6, the predictions of the new model are satisfactory, falling within the interval of factor 2 lines, accompanied by a relative error range [−48, 171] (%) and an error index 49%. Meanwhile, it is simple to catch the phenomenon that the forecast precision of the new model under R ≠ −1 multiaxial loadings are close to that under R = −1 multiaxial loadings. The similar phenomenon can also be observed in the axial and torsional cases. This is benefited from that the proposed fatigue equation possesses invariable mathematical form independent of loading cases as given in equation (6); the simplified uniaxial equations (equation (7) and (8)) and multiaxial parameter ρ have been proved to be effective in representing the influences of mean stress and stress multiaxiality, respectively.

Figure 6.

Comparisons of experimental and calculated lives by the new model and MWCM under multiaxial loadings of R ≠ −1 for 18G2A steel.

On the other hand, almost all the results of MWCM are located out of the factor 2 lines, with several points even out of the five or more scatter bandwidth on the non-conservative side in Figure 7. As to the reason for the poor accuracy of MWCM under the R ≠ −1 torsional and multiaxial loadings, an explanation was suggested by Cristofori et al. (2008) that fatigue behavior of 18G2A steel is highly sensitive to the presence of non-zero mean torsional stresses. In spite of this, the MWCM has no ability to handle this situation, due to the definition of effective multiaxial parameter $ρ_{eff}$ . As discussed above, it is logical to propose a set of new parameters characterizing the shear mean stress effect, which is also expected to be true from the standpoint of Papadopoulos et al. (1997).

Figure 7.

Comparisons of experimental and calculated lives by the new model and MWCM under fully reversed loadings for 30NCD16 steel.

In addition, by comparing the R ≠ −1 torsional and multiaxial predictions of MWCM (Figures 4 and 6), it can be seen that the multiaxial accuracy is no longer as bad as the torsional one. This indicates that the influence of shear mean stress on multiaxial loadings is indeed slighter than that on torsional loadings, and it can be intuitively embodied by the proposed model. In detail, as listed in Table 2 that $α_{1} < α_{2}$ and $n_{1} \approx n_{2}$ , it shows that the weight of mean stress influence on axial loadings is less than that on torsional loadings for 18G2A steel. Moreover, under the combined tension–torsion situations, the multiaxial parameter ρ increases with the load from torsion to tension, and such variation is further reflected in the parameters $α_{ρ}$ and $n_{ρ}$ , so that the degree of mean stress influence on multiaxial cases can be directly reduced. This also verifies the linear relationship in equations (14) and (15) is applicable.

To conclude, the present model has a capacity to properly consider the contribution of not only the axial mean stress but also the torsional mean stress to multiaxial fatigue life assessment, so that it can be applied to more diverse metallic materials, especially those have non-ignorable sensitivity to the shear mean stress effect.

As the second part, abundant conditions can be encountered in the fatigue researches. Such as the standpoint that the presence of mean torsional stresses can be neglected as long as it is within a certain range (Davoli et al., 2003; Sines, 1961), and some materials are indeed less sensitive to the impact of shear mean stresses. Furthermore, sometimes the R ≠ −1 torsional loading data is absence due to the limit of experiment amount. The effectiveness of the present model to the above conditions will be verified with the tests of 30NCD16 steel (Froustey and Lasserre, 1989). The torsional and multiaxial loadings of the tests do not include shear mean stresses, so that the shear mean stress effect is neglected for this material. Therefore, a new assignment operation of the parameters $α_{0}$ and n₀ that describe the shear mean stress effect is needed instead of the above fitting approach.

In detail, it can be found from Figures 1 and 2 that the relationships between $σ_{a, e} / σ_{a}$ vs. $σ_{m} / σ_{a}$ and $τ_{a, e} / τ_{a}$ vs. $τ_{m} / τ_{a}$ are approximately linear, and the calculated parameter n₀ is nearly consistent with n₁ for 18G2A steel. Therefore, for the materials that do not account for the shear mean stress influence, it is logical to postulate that the parameter $α_{0}$ in equation (8) is taken as 0, and n₀ in equation (8) is equal to n₁ in equation (7). In this manner, if the two parameters ( $α_{1}$ and n₁) describing axial mean stress effect are considered as a whole, the number of fitting parameters in the MWCM and new model can be regarded as equivalent. The remaining material parameters of 30NCD16 steel can be calculated by the procedure similar to that of 18G2A steel, as summarized in Tables 1 and 2.

Figure 7 presents an overall comparison for the fully reversed uniaxial and multiaxial loadings. The perfect results can be observed in the two basic uniaxial cases. Simultaneously, the R = −1 multiaxial results calculated by the present model are also slightly superior to those of MWCM, with the minor error indexes 29%. Furthermore, the predictions of two compared approaches applied to the loadings with different mean stresses are shown in Figures 8 and 9, and detailed numerical values are listed in Table 6. A set of close results can be achieved related to the R ≠ −1 axial loadings, with the error indexes 41% and 37%.

Figure 8.

Comparisons of experimental and calculated lives by the new model and MWCM under bending loadings with different mean stresses for 30NCD16 steel.

Figure 9.

Comparisons of experimental and calculated lives by the new model and MWCM under multiaxial loadings with different mean stresses for 30NCD16 steel.

Table 6.

Experimental and computational results related to 30NCD16 steel.

$σ_{x, a}$ (MPa)	$σ_{x, m}$ (MPa)	$τ_{xy, a}$ (MPa)	$N_{f, exp}$ (Cycles)	$N_{f, cal}$ (Cycles)	$E_{N} (%)$	$τ_{c, a}$ (MPa)	$σ_{n, a}$ (MPa)	$σ_{n, m}$ (MPa)	$N_{f, cal}$ (Cycles)	$E_{N} (%)$
765	0	0	120,000	94,146	−21.54	382.5	0	0	94,146	−21.54
790	0	0	90,000	67,761	−24.71	395	0	0	67,761	−24.71
795	0	0	80,000	63,526	−20.59	397.5	0	0	63,526	−20.59
780	0	0	100,000	77,189	−22.81	390	0	0	77,189	−22.81
725	0	0	200,000	163,055	−18.47	362.5	0	0	163,055	−18.47
708	0	0	250,000	207,836	−16.87	354	0	0	207,836	−16.87
720	0	0	210,000	175,013	−16.66	360	0	0	175,013	−16.66
753	0	0	140,000	110,669	−20.95	376.5	0	0	110,669	−20.95
820	0	0	51,000	46,284	−9.25	410	0	0	46,284	−9.25
785	0	0	95,000	72,306	−23.89	392.5	0	0	72,306	−23.89
715	0	0	230,000	187,942	−18.29	357.5	0	0	187,942	−18.29
0	0	482	120,000	117,499	−2.08	482	0	0	117,499	−2.08
0	0	500	90,000	86,718	−3.65	500	0	0	86,718	−3.65
0	0	505	80,000	79,856	−0.18	505	0	0	79,856	−0.18
0	0	495	100,000	94,248	−5.75	495	0	0	94,248	−5.75
0	0	450	200,000	207,593	3.80	450	0	0	207,593	3.80
0	0	440	250,000	250,076	0.03	440	0	0	250,076	0.03
0	0	446	210,000	223,531	6.44	446	0	0	223,531	6.44
0	0	470	140,000	144,793	3.42	470	0	0	144,793	3.42
0	0	527	51,000	56,089	9.98	527	0	0	56,089	9.98
0	0	497	950,000	91,152	−4.05	497	0	0	91,152	−4.05
0	0	445	230,000	227,727	−0.99	445	0	0	227,727	−0.99
660	290	0	250,000	140,792	−43.68	330	330	145	158,010	−36.80
695	290	0	120,000	88,375	−26.35	348	348	145	93,030	−22.47
620	450	0	140,000	122,404	−12.57	310	310	225	131,065	−6.38
640	450	0	51,000	93,900	84.12	320	320	225	94,655	85.60
0	290	460	120,000	172,142	43.45	375	161	234	93,121	−22.40
0	450	430	250,000	300,988	20.40	357	129	341	66,757	−73.30
0	450	460	120,000	172,142	43.45	414	202	225	35,972	−70.02
500	290	290	120,000	135,226	12.69	382	228	249	44,401	−63.00
490	450	285	95,000	103,244	8.68	375	249	397	8,963	−90.56
600	0	335	100,000	103,788	3.79	335	600	0	62,457	−37.54
562	0	315	200,000	202,651	1.33	315	562	0	139,158	−30.42
500	290	290	210,000	119,623	−43.04	290	500	290	5,127	−97.55
490	450	285	230,000	60,366	−73.75	285	490	450	7,659	−96.67

Note: The last four rows are out-phase loadings with 90° shift.

Turning to the R ≠ −1 multiaxial cases, the evaluated results by the present model are significantly superior to those of MWCM, resulting in the relative error range [8, 43] (%) and an error index 25%, and the points are included in the two scatter bandwidth in the chart. Whereas the MWCM leads to the results accompanied by the error range [−90, −22] (%) and the error index 63%, and most of them fall out of the three scatter bandwidth, with the error of 4 of the total 5 points being over 60%.

The scattered results produced by MWCM in Figure 9 can be explained by that, the axial mean stress effect on multiaxial loadings is slighter than that on axial loadings; in other words, following the same mean stress sensitivity index (m) determined by the axial cases may lead to large deviations in multiaxial situations. This further illustrates the rationality of the transition process expressed by equations (14) and (15). Additionally, it is worth mentioning that the calculated sensitivity index, m, of both validation materials are out of the range between 0 and 1, which is suggested by Susmel (2008), and thus it also implies that the range of MWCM performing mean stress fatigue assessment may be limited.

Different from the MWCM, which concentrates the descriptions of mean stress effect and multiaxial influence on one effective multiaxial parameter $ρ_{eff}$ , the new model has the capability of separately expressing them on two sides of the fatigue equation (equation (3)). Thus, the detrimental contribution of both axial and shear mean stresses can be considered, and it can be felicitously applied to the circumstance without the attention on mean shear stress influence as well. In addition, the separated expression makes the fatigue equation structure more reasonable and extensible; that is, for each influence factor (mean stress or multiaxiality), the corresponding formulation can vary with applicable conditions. For example, the description of axial mean stress effect with few fitting parameters can be used to simplify the model by replacing the formula on the left side of the equation.

The extended non-proportional fatigue life prediction equation

As discussed in the last section, the effectiveness of the proposed model for proportional aspect has been verified, and the fatigue equation structure is well extensible. Thus, in this section, the authors make a trial extension to make the proposed model applicable to non-proportional cases, which is a common loading form in engineering.

It has to admit that the non-proportional fatigue assessment is still an open problem. Recently, a widely developed viewpoint suggested that cyclic stress hardening response is closely contact with non-proportional fatigue damage. Therefore, in some criteria, typically Fatemi and Socie criterion (1988), both stress and strain components are adopted to form fatigue parameters. In conjunction with the cyclic constitutive relation of the evaluated material, the cyclic stress hardening or softening response can be introduced to account for non-proportional additional detriment.

However, there are always two sides to everything. The aforementioned cyclic constitutive relation is normally not readily accessible. Moreover, as reviewed on some non-proportional tests (Pejkowski, 2017), a conclusion can be drawn that the variation tendency of cyclic stress hardening and non-proportional hardening is inconsistent for some metallic materials. Consequently, the authors believe that a general engineering fatigue approach has to acknowledge that the sensitivity of fatigue life to non-proportional loadings is a material property.

For the purpose of extending the present stress controlled model to implement life forecast under non-proportional loadings, the widely approved non-proportional fatigue criterion proposed by Itoh et al. (1995) is used for reference, and it can be converted to the form of stress components

σ_{NP, a} = σ_{ep, a} (1 + {aF}_{np})

(19)

where

σ_{NP, a}

σ_{ep, a}

are the amplitudes of equivalent non-proportional stress and corresponding effective proportional stress, respectively; a is a material parameter that reveals the material sensitivity to non-proportional induced additional hardening;

F_{np}

is the non-proportionality factor illustrating the non-proportional severity of loading paths.

The purpose of Itoh criterion is to establish the relationship between the amplitudes of equivalent non-proportional stress and corresponding effective proportional stress, and the proposed mean stress model is precisely to describe the behavior of corresponding effective proportional stress with mean stresses. Thus, the extended non-proportional fatigue life prediction equation can be obtained by substituting $σ_{NP, a}$ of equation (19) into equation (6) instead of $σ_{ea}$

log σ_{ep, a} (1 + {aF}_{np}) (1 - α_{ρ} \frac{σ_{ep, m}}{σ_{ep, a}})^{n_{ρ}} = A_{ρ} log N + C_{ρ}

(20)

where

σ_{ep, m}

is equal to the value of von Mises stress on the midpoint of effective proportional loading path. In this article, the effective proportional loading definition employs the path-dependent maximum-range cycle counting procedure (PDMR) suggested recently by Dong et al. (2010).

The extended model will be validated with three sets of fatigue experimental data from Refs (Froustey and Lasserre, 1989; Susmel and Petrone, 2003; Zhang et al., 2016). The first instance is the out-phase part of 30NCD16 steel experiments, whose proportional part has been applied in the verification in the section Experimental verification and discussions. The non-proportionality effect is introduced by using the combined sinusoidal bending and torsion with 90° phase shift, and the out-phase loadings with non-zero mean axial stresses are also included. Under such out-phase loadings, the non-proportionality factor can be concisely decided as the ratio of the two principal axes of the elliptical loading path, and the material parameter a can be obtained by fitting the proportional and 90° out-phase experimental data, a=0.29.

The out-phase comparison results of the both models for 30NCD16 steel are shown in Figure 10, and the detailed values are listed in Table 6. As can be seen that, removing the fitting data of R = −1 out-phase loadings, i.e., $σ_{ep, m} = 0$ , the results calculated by the new model under the out-phase loadings with non-zero mean stresses have obviously higher accuracy than those by MWCM. This further verifies the standpoint that non-proportionality effect is a material property and the success of the extended model equation expressed in equation (20).

Figure 10.

Comparisons of experimental and calculated lives by the new model and MWCM under 90° out-phase loadings for 30NCD16 steel.

The second example consists of the tests on 6082-T6 solid cylindrical specimens conducted by Susmel and Petrone (2003). The combined sinusoidal axial and torsional loadings comprise two phase shifts (90° and 126°), and the material parameter a can be obtained by the identical measure as above. For comparison fairness, it have to declare that in this instance the parameter a is fitted by using the proportional loading data and only one point of 90° out-phase loading (Specimen Code: P46BT20), a=0.23.

Figure 11 shows the comparisons of experimental and calculated lives by the proposed model and MWCM for 6082-T6. It illustrates that each prediction of the new model under multiaxial in-phase loadings is better than the corresponding one of MWCM with an error reduced by approximately 10%. Meanwhile, for the non-proportional loadings with 90° and 126° phase shifts, it can be intuitively found that the precision of the present model is generally higher than that of MWCM accompanied by a less maximum error. In addition, as to the non-proportional fatigue assessment without mean stresses, a specialized study has been published by the present authors in the strain components form (Liu and Yan, 2018b), and readers are advised to acquire more information from it.

Figure 11.

Comparisons of experimental and calculated lives by the new model and MWCM for 6082-T6.

The final examination is on the 7075-T651 aluminum alloy (Zhang et al., 2016), which includes four phase shifts (30°, 45°, 90° and 180°). To reduce the number of required material parameters, the parameters n₁ and n₀ are taken as 1 for this example, and the calculated material parameters are listed in Tables 1 and 2, and a is equal to 0.23. Figure 12 presents an overall comparison of measured and calculated lives.

Figure 12.

Comparisons of experimental and calculated lives by the new model and MWCM for 7075-T651 aluminum alloy.

In detail, both models can obtain the axial satisfactory results, whether or not the mean stresses are contained. However, MWCM is once again invalid in the R ≠ −1 torsional cases, thus further affecting its multiaxial prediction accuracy in both in-phase and out-phase conditions. Contrarily, the extended model can achieve multiaxial results consistent with the accuracy of uniaxial results, regardless of whether the loads include the mean stresses. This indicates that the extended model has the ability to take full account of the influencing factors of each concern (multiaxial, mean stress and non-proportional effects).

To conclude, under the non-proportional loadings, although the proposed model needs to fit a material parameter a, it possesses the characteristics of high prediction accuracy and computational convenience, which is cost-efficient to make up for this shortcoming. In addition, for the energy-based methods, the process of calculating cyclic constitutive law will increase the computational complexity and time consuming. Meanwhile, as mentioned in above, the conclusion that the trend of cycle stress hardening and non-proportional hardening is uncertain for some materials may also affect their reliability of application.

Conclusions

In this paper, a new general model for predicting multiaxial fatigue life with influence of different mean stresses is presented. The following major conclusions can be drawn:

Based on the famous MWCM and conventional mean stress criteria, the new model adopts a mathematical equation with invariable form to take into account the multiaxial mean stress effect. A typical difference from MWCM is that the proposed model can consider the impact of shear mean stresses on finite fatigue life, which is in sympathy with the opinion of Papadopoulos on this aspect.

The contribution of both axial and torsional mean stresses to multiaxial fatigue damage can be taken into account by the new model, which has been proved to be reasonable through experiments of 18G2A steel. Moreover, the model can be reduced to the form without considering the contribution of shear mean stresses as validated with material 30NCD16 steel.

Benefited from the proposed fatigue equation structure, which places the part representing the multiaxial S–N curves (right part of equation (6)) and that indicating the mean stress influence (left part of equation (6)) on the two sides of equation individually, the model can separately address the two aspects for a given loading. In other words, these influence factors are separately located in the equation form, thus the present model possesses well extendibility and variability, and it can be trivially optimized by using the expressions of criteria that are more reasonable for the corresponding factor to replace the original one. Furthermore, an extended model applied to non-proportional loads is proposed by combining with the Itoh criterion, whose effectiveness has been proved by satisfactory prediction accuracy.

A wildly used stress invariant (von Mises stress) is taken as the fatigue parameter, so that the model is convenient for application in the design phase of mechanical components, especially when the fatigue life is included in the objective function of optimal design.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

References

Cristofori

Susmel

Tovo

(2008) A stress invariant based criterion to estimate fatigue damage under multiaxial loading. International Journal of Fatigue 30: 1646–1658.

Crossland B (1956) Effects of large hydrostatic pressure on torsional fatigue strength of an alloy steel. In: Proceedings of international conference on fatigue of metals, institution of mechanical engineers, London, pp. 138–149.

Davoli

Bernasconi

Filippini

, et al.(2003) Independence of the torsional fatigue limit upon a mean shear stress. International Journal of Fatigue 25(6): 471–480.

Dong

Wei

Hong

(2010) A path-dependent cycle counting method for variable amplitude multi-axial loading. International Journal of Fatigue 32: 720–734.

Fatemi

Socie

(1988) A critical plane approach to multiaxial fatigue damage including out of phase loading. Fatigue & Fracture of Engineering Materials & Structures 11(3): 149–165.

Findley

(1957) Fatigue of metals under combinations of stresses. Transactions ASME 79: 1337–1348.

Froustey

Lasserre

(1989) Multiaxial fatigue endurance of 30NCD16 steel. International Journal of Fatigue 11(3): 169–175.

Gasiak G and Pawliczek R (2001) The mean loading effect under cyclic bending and torsion of 18G2A steel. In: Proceedings of 6th international conference on biaxial/multiaxial fatigue and fracture, pp. 213–222.

Gasiak

Pawliczek

(2003) Application of an energy model for fatigue life prediction of construction steels under bending, torsion and synchronous bending and torsion. International Journal of Fatigue 25(12): 1339–1346.

10.

Goodman

(1899) Mechanics Applied to Engineering, New York: Longmans Green and Co.

11.

Gough

(1949) Engineering steels under combined cyclic and static stresses. Journal of Applied Mechanics 160: 417–440.

12.

Itoh

Sakane

Ohnami

, et al.(1995) Nonproportional low cycle fatigue criterion for type 304 stainless steel. Journal of Engineering Materials and Technology-ASME 117(3): 285–292.

13.

Kluger

Łagoda

(2014) New energy model for fatigue life determination under multiaxial loading with different mean values. International Journal of Fatigue 66: 229–245.

14.

Liang

Chen

(2016) A regularized Miner’s rule for fatigue reliability analysis with Mittag-Leffler statistics. International Journal of Damage Mechanics 25(5): 691–704.

15.

Liu

Yan

(2018a) An extension research on the theory of critical distances for multiaxial notch fatigue finite life prediction. International Journal of Fatigue 117: 217–229.

16.

Liu

Yan

(2018b) A new method for studying the effect of multiracial strain states on low cycle non-proportional fatigue prediction. International Journal of Fatigue 117: 420–431.

17.

Marin J (1956) Interpretation of fatigue strength for combined stresses. In: Proceedings of international conference on fatigue of metals, institution of mechanical engineers, London, pp. 184–194.

18.

Mcdiarmid

(1994) A shear stress based critical-plane criterion of multiaxial fatigue failure for design and life prediction. Fatigue & Fracture of Engineering Materials & Structures 17(12): 1475–1484.

19.

Mishra

Dutta

Ray

(2016) Fatigue life estimation in presence of ratcheting phenomenon for AISI 304LN stainless steel tested under uniaxial cyclic loading. International Journal of Damage Mechanics 25(3): 431–444.

20.

Morrow

(1968) Fatigue Design Handbook. Advances in Engineering (4), Warrendale: Society of Automotive Engineers.

21.

Papadopoulos

(2001) Long life fatigue under multiaxial loading. International Journal of Fatigue 23: 831–849.

22.

Papadopoulos

Davoli

Gorla

, et al.(1997) A comparative study of multiaxial high-cycle fatigue criteria for metals. International Journal of Fatigue 19: 219–235.

23.

Pejkowski

(2017) On the material's sensitivity to non-proportionality of fatigue loading. Archives of Civil & Mechanical Engineering 17(3): 711–727.

24.

Shang

Wang

(1998) A new multiaxial fatigue damage model based on the critical plane approach. International Journal of Fatigue 20(3): 241–245.

25.

Shen

Akanda

(2016) A modified closed form energy-based framework for fatigue life assessment for aluminum 6061-T6: Strain range approach. International Journal of Damage Mechanics 25(5): 661–671.

26.

Sines

(1961) The prediction of fatigue fracture under combined stresses at stress concentrations. Transactions of the Japan Society of Mechanical Engineers 4(15): 443–453.

27.

Susmel

(2008) Multiaxial fatigue limits and material sensitivity to non-zero mean stresses normal to the critical planes. Fatigue & Fracture of Engineering Materials & Structures 31(3–4): 295–309.

28.

Susmel

Lazzarin

(2002) A bi-parametric modified Wöhler curve for high cycle multiaxial fatigue assessment. Fatigue and Fracture of Engineering Materials and Structures 25(1): 63–78.

29.

Susmel

Petrone

(2003) Multiaxial fatigue life estimations for 6082-T6 cylindrical specimens under in-phase and out-of-phase biaxial loadings. European Structural Integrity Society 31(3): 83–104.

30.

Tao

Xia

(2007) Mean stress/strain effect on fatigue behavior of an epoxy resin. International Journal of Fatigue 29(12): 2180–2190.

31.

Wang

Miller

(1991) The effect of mean shear stress on torsional fatigue behaviour. Fatigue and Fracture of Engineering Materials and Structures 14: 293–307.

32.

Zghal

Gmati

Mareau

, et al.(2016) A crystal plasticity based approach for the modelling of high cycle fatigue damage in metallic materials. International Journal of Damage Mechanics 25(5): 611–628.

33.

Zhang

Shang

Sun

, et al.(2016) Multiaxial high-cycle fatigue life prediction model based on the critical plane approach considering mean stress effects. International Journal of Damage Mechanics 27(1): 32–46.

34.

Zhang

Shang

Sun

, et al.(2018) Multiaxial high-cycle fatigue life prediction model based on the critical plane approach considering mean stress effects. International Journal of Damage Mechanics 27(1): 32–46.

35.

Zhan

Meng

, et al.(2016) Continuum damage mechanics-based approach to the fatigue life prediction for 7050-T7451 aluminum alloy with impact pit. International Journal of Damage Mechanics 25(7): 943–966.

36.

Zhu

Lei

Huang

, et al.(2016) Mean stress effect correction in strain energy-based fatigue life prediction of metals. International Journal of Damage Mechanics 26(8): 1219–1241.

37.

Zhu

Lei

Huang

, et al.(2017) Mean stress effect correction in strain energy-based fatigue life prediction of metals. International Journal of Damage Mechanics 26(8): 1219–1241.