Multiaxial Fatigue Lifetime Estimation Based on New Equivalent Strain Energy Damage Model under Variable Amplitude Loading

Abstract

The largest normal stress excursion during contiguous turn time instants of the maximum torsional stress is presented as an innovative path-independent fatigue damage quantity upon the critical plane, which is further employed for characterizing fatigue damage under multiaxial loading. Via using the von Mises equivalent stress formula, an axial stress amplitude with equivalent value is proposed, incorporating the largest torsional stress range and largest normal stress excursion upon the critical plane. The influence of non-proportional cyclic hardening is considered within the presented axial equivalent stress range. Moreover, according to proposed axial equivalent stress amplitude, an energy-based damage model is presented to estimate multiaxial fatigue lifetime upon the critical plane. In order to verify the availability of the proposed approach, the empirical results of a 7050-T7451 aluminum alloy and En15R steel are used, and the predictions indicated that estimated fatigue lives correlate with the experimentally observed fatigue results well for variable amplitude multiaxial loadings.

1. Introduction

A lot of industrial structures and mechanical components, for example, chemical plants, airframes, turbines, landing gears, pressure vessels, and power generation plants, generally experience multiaxial loadings [1,2]. Moreover, these structural components typically experience random or out-of-phase variable amplitude loads [3,4]. Thus, the multiaxial fatigue lifetime evaluation of these mechanical parts is still an intractable problem [5]. The critical plane approach is universally deemed to be more applicable for multiaxial fatigue lifetime prediction, which can account for the fatigue damage mechanism and have a definite physical meaning. Three types of models including strain–stress-based, strain-based, and stress-based models, are concluded in the critical plane approach. For instance, Findley [6] and McDiarmid [7] models, which are regarded as representative stress-based critical plane models, include merely stress items and are able to be applicative for relatively long fatigue life regimes. The Brown–Miller model [8], which is deemed as a typical strain-based critical plane model, includes merely strain items and is able to be applicable for both long and short fatigue lifetime regions. But, the constitutive relationship of materials, for example, out-of-phase cyclic hardening, cannot be considered by these types of strain-based critical plane models. As a representative strain–stress-based critical plane model, the Socie–Fatemi model [9] includes both stress and strain items and is able to be suitable for both long and short fatigue lifetime regions. In particular, one category of especial strain–stress-based models, which are expressed as the energy-based models, are broadly applicable to multiaxial fatigue lifetime estimation. For example, the Smith–Watson–Topper (SWT) fatigue damage model [10], which was developed by Socie et al. [11], is utilized for estimating fatigue life for those materials showing a tensile crack failure pattern. Moreover, Yu et al. [12], Chen et al. [13], Li et al. [14], Jiang [15], and Ince-Glinka [16] proposed some modificatory versions of SWT models to perform the multiaxial fatigue life estimation of those materials showing a tensile failure pattern. Additionally, for materials showing a torsional crack failure pattern, Varvani-Farahani [17], Glinka et al. [18], Pan-Huang-Chen [19], Varvani-Farahani et al. [20], Chu [21], and Liu [22] developed some multiaxial fatigue damage models during the past few decades; many investigations focusing upon multiaxial fatigue damage parameters have been performed for variable amplitude loads [23,24,25]. But, it is still challenging work for multiaxial fatigue damage estimation under random or multiaxial variable amplitude loading because of a lack of some accurate and universal fatigue lifetime estimation approaches [26,27,28,29,30].

During the past few years, several new multiaxial fatigue life prediction models have been proposed. Luo et al. [31] proposed path-dependent damage parameters of pure fatigue damage and additional ratchetting, one in a strain form, and they constructed a multiaxial life prediction model through addressing the concurrent efforts of additional ratchetting damage and pure fatigue, one to predict the multiaxial fatigue life of a welded joint. Zhao et al. [32] presented a nonlocal multiaxial fatigue model based on artificial neural networks to estimate the fretting fatigue life of dovetail joints. Wang et al. [33] proposed a novel multiaxial fatigue life prediction model based on octahedral shear strain energy. In the proposed model, the maximum octahedral shear strain energy served as the damage parameter without requiring any additional material parameters. Zhao et al. [34] defined a new characteristic plane (subcritical plane) to describe the particularity of additional cyclic hardening under a non-proportional loading condition. On the new defined subcritical plane, a corresponding damage parameter containing the effect of additional hardening was also built, by which the dynamic path of the stress spindle, combining the material property and loading environment, was fully analyzed. Nourian-Avval and Khonsari [35] proposed a model to evaluate the fatigue damage based on heat dissipated under different multiaxial loadings. In the proposed model, the fatigue damage was evaluated by considering the different rates of dissipated energy at a given shear and tensile strain. Choi et al. [36] proposed a semi-empirical ε-N fatigue model, which accounted for both material anisotropy and complex stress states. Furthermore, six machine learning models have been employed for the fatigue life prediction. Amjadi and Fatemi [37] modeled the multiaxial fatigue behavior of short glass fiber-reinforced thermoplastics under different environmental and loading conditions using a critical plane-based damage model. The effects of fiber orientation, stress state, mean stress, and stress concentration on multiaxial fatigue behavior were considered in the model. Temperature and frequency effects on multiaxial fatigue behavior were also included by applying the proposed damage model in a general fatigue model. Zhang et al. [38] introduced the Symbolic Regression–Neural Network (SR-NN) framework, a novel integration of symbolic regression-derived expressions with neural networks aimed at enhancing predictive accuracy in this field. Almamoori and Alizadeh [39] proposed a simple and effective approach for measuring fatigue life in the presence of multiaxial loading. The proposed damage parameter included terms such as the critical plane and phase difference angles. Ferreira et al. [40] employed an optimized version of the maximum variance method (MVM) to determine maximum shear stress amplitudes in a high-cycle multiaxial fatigue analysis using the critical plane approach, increasing accuracy and reducing computational burden. This refined approach posits that the MVM assumes that the critical plane is the one subjected to the maximum variance of the shear stress history.

In this study, the objective of the current work is to develop a new multiaxial fatigue lifetime estimation method for engineering parts experiencing variable amplitude multiaxial loads. Within this presented approach, an innovative multiaxial fatigue damage quantity ${\sigma }_n^{*}$, which can consider the multiaxial out-of-phase cyclic hardening and has an explicit physical meaning, is developed for building a multiaxial fatigue damage quantity on the critical plane. A new equivalent stress expression is introduced based on the proposed multiaxial fatigue damage quantity ${\sigma }_n^{*}$. Furthermore, combined with the Manson–Coffin equation, a novel damage model for multiaxial fatigue is established upon the basis of equivalent strain energy. The presented multiaxial fatigue damage model is established upon the basis of the critical plane method and does not include any additional weight factors or material parameters. Moreover, the influence of out-of-phase cyclic hardening can be considered by presented fatigue damage parameters. The effectiveness of the proposed fatigue lifetime evaluation approach can be validated using the collected experimental data for circular tube components of a 7050-T7451 aluminum alloy and En15R steel, which are subjected to multiaxial variable amplitude loads.

2. Proposed Equivalent Strain Energy Method

For the purpose of considering the influences of out-of-phase cyclic hardening and mean stress upon fatigue strength, an equivalent energy damaging parameter (expressed as EBDP) upon the critical plane is presented below:

(1)$\mathrm{EBDP}={\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}\cdot {\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}$

where $\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}/2$ is the amplitude with equivalent axial strain; $\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}/2$ is the equivalent axial stress amplitude calculated using Equation (2) below:

(2)${\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}=\sqrt{{\left({\displaystyle \frac{{\sigma }_n^{*}}{2}}\right)}^2+3\cdot {\left({\displaystyle \frac{\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)}^2}$

where $\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the torsional stress range, and ${\sigma }_n^{*}/2$ is described as the amplitude of the maximum tensile stress excursion upon the critical plane during contiguous turn time instants of the maximum torsional stress.

Along the persistent slip areas or shear zones of the fatigue cracking tip, fatigue cracking growth can be a de-cohesion course from the perspective of the microscopic scale. The de-cohesion behavior can be assisted via the normal stress upon the cracking plane [41]. Within the damage and de-cohesion course at the area of the fatigue cracking tip, the reverse of the torsional deformation process can be associated with the turning point of torsional stress. Therefore, the fatigue cracking propagation is able to be enhanced via the tensile stress range during contiguous turn time instants of the maximum torsional stress. When the largest torsional stress changes direction, the influence of the normal stress loading history is eliminated. During a count reversal, the normal stress contribution can be considered to be the time interval of contiguous turn time instants of the maximum torsional stress, which is beneficial to promote fatigue cracking propagation.

In this current study, a path-dependent multiaxial fatigue damage quantity is proposed on the critical plane, which is denoted as the normal stress excursion with maximum value ${\sigma }_n^{*}$ during contiguous turn time instants of the maximum torsional stress. The tensile stress excursion ${\sigma }_n^{*}$ can be denoted below:

(3)${\sigma }_n^{*}=\underset{t_{\mathrm{a}}<t<t_{\mathrm{b}}}{\mathrm{m}\mathrm{a}\mathrm{x}}\left[{\sigma }_n\left(t\right)\right]-\underset{t_{\mathrm{a}}<t<t_{\mathrm{b}}}{\mathrm{m}\mathrm{i}\mathrm{n}}\left[{\sigma }_n\left(t\right)\right]={\sigma }_n^{\mathrm{m}\mathrm{a}\mathrm{x}}-{\sigma }_n^{\mathrm{m}\mathrm{i}\mathrm{n}}$

where $t_{\mathrm{a}}$ and $t_{\mathrm{b}}$ are, respectively, used for denoting the time points of the two contiguous turn time instants of the maximum torsional stress ${\sigma }_n^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and ${\sigma }_n^{\mathrm{m}\mathrm{i}\mathrm{n}}$ are the largest and least normal stresses within the time range $\left[t_{\mathrm{a}},t_{\mathrm{b}}\right]$.

For the purpose of clarifying the applicability of the presented multiaxial fatigue damage quantity ${\sigma }_n^{*}$ to estimate loading path non-proportionality, three 7050-T7451 aluminum alloy specimens are, respectively, used for the axial–torsional cyclic loading experiments under 90° out-of-phase, 45° out-of-phase, and in-phase strain sequences [42], as depicted in Figure 1. These experiments are full-reversal cyclic loading tests with the same equivalent strain value, $\Delta {\epsilon }_{\mathrm{eq}}/2=0.7\%$. Moreover, the adopted strain ratio with regard to shear and tensile strain ranges equals 1.7, i.e., $\lambda =1.7$, as depicted within Table 1.

For the three phase angles of concern ($\phi =0^{°}$, $\phi ={45}^{°}$, and $\phi ={90}^{°}$), the variation histories of tensile and torsional strain and stress are analyzed upon the critical plane, as depicted within Figure 2. For in-phase loading, the tensile stress excursion ${\sigma }_n^{*}$ during contiguous turn time instants (A and B) of the maximum torsional stress is quite little upon the critical plane, as shown within Figure 2a. In such a situation, the tensile stress excursion ${\sigma }_n^{*}$ upon the critical plane equals the tensile stress excursion $\Delta {\sigma }_n$. With comparison with proportional loading, the normal stress excursion ${\sigma }_n^{*}$ increases obviously for multiaxial 45° out-of-phase load, as depicted within Figure 2b. For multiaxial 90° out-of-phase load, the normal stress excursion ${\sigma }_n^{*}$ reaches its largest value, as shown in Figure 2c. Moreover, with the increasing in phase angle φ, the excursion of normal stress ${\sigma }_n^{*}$ increases as depicted within Figure 2d. Hence, the influence of out-of-phase cyclic hardening can be considered using the proposed multiaxial fatigue damage quantity ${\sigma }_n^{*}$.

With regard to Equation (1), the equivalent axial strain $\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}/2$ on the critical plane is able to be determined using the multiaxial fatigue damage model proposed by Shang and Wang [41], as shown below:

(4)${\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}=\sqrt{{\left({\epsilon }_n^{*}\right)}^2+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)}^2}={\displaystyle \frac{{{\sigma }^{\prime }}_{\mathrm{f}}}{E}}\cdot {\left(2N_{\mathrm{f}}\right)}^b+{{\epsilon }^{\prime }}_{\mathrm{f}}\cdot {\left(2N_{\mathrm{f}}\right)}^c$

where ${\epsilon }_n^{*}$ is the excursion of normal strain upon the critical plane, which is measured during adjacent turning time instants of the largest torsional strain, and $\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ is the maximum torsional strain range.

Furthermore, the equivalent axial strain parameter $\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}/2$ from Equation (4) and equivalent stress parameter $\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}/2$ from Equation (2) are plugged into Equation (1). Thus, the presented damage model with equivalent strain energy EBDP is able to be denoted upon the critical plane as shown below:

(5)${\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}\cdot {\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}=\sqrt{{\left({\displaystyle \frac{{\sigma }_n^{*}}{2}}\right)}^2+3\cdot {\left({\displaystyle \frac{\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)}^2}\cdot \sqrt{{\left({\epsilon }_n^{*}\right)}^2+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)}^2}={\displaystyle \frac{{\left({{\sigma }^{\prime }}_{\mathrm{f}}\right)}^2}{E}}\cdot {\left(2N_{\mathrm{f}}\right)}^{2b}+{{\sigma }^{\prime }}_{\mathrm{f}}{{\epsilon }^{\prime }}_{\mathrm{f}}\cdot {\left(2N_{\mathrm{f}}\right)}^{b+c}$

As depicted within Figure 3a, the maximum amplitude of torsional stress $\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$, maximum amplitude of torsional strain $\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$, and excursion of normal strain ${\epsilon }_n^{*}$ upon the critical plane have been shown for the three phase angles of concern ($\phi =0^{°}$, $\phi =0^{°}$, and $\phi =0^{°}$ as depicted within Figure 1). It is found that, with the increasing in multiaxial load history non-proportionality, all of the largest amplitudes of torsional stress $\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$, normal strain excursion ${\epsilon }_n^{*}$, and maximum amplitude of torsional strain $\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$ upon the critical plane are rising. In this investigation, the non-proportionalities of multiaxial load histories are able to be considered by the four damage parameters ${\sigma }_n^{*}$, $\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$, ${\epsilon }_n^{*}$, and $\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$. Additionally, we considered the situation that with the increasing in the phase angle, multiaxial fatigue life gets shorter. Therefore, the influence of out-phase cyclic hardening upon fatigue damage is able to be reflected using the multiaxial fatigue damage parameters ${\sigma }_n^{*}$, $\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$, ${\epsilon }_n^{*}$, and $\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$.

Moreover, the presented equivalent axial stress $\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}/2$, the Shang–Wang damage parameter $\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}/2$, and the $\mathrm{EBDP}$ damage parameter are drawn within Figure 3b. It should be highlighted that with the increasing in the phase angle, these three fatigue damage parameters all increase gradually. Thus, these three parameters can account for the out-of-phase degree of loading history. Furthermore, within comparison, the out-of-phase degree of multiaxial load history can be better considered by the constructed $\mathrm{EBDP}$ damage parameter.

Within Table 1, the predictive fatigue lives using the presented EBDP model are compared with the experimentally measured data. It can be found that, with the growth in the out-of-phase degree of the loading history, the experimentally obtained results of the 7050-T7451 aluminum alloy exhibit worse fatigue behavior. On the other hand, with the increase in the load path out-of-phase degree, the predictive fatigue lives using the EBDP model can become shorter. Hence, the proposed EBDP modeling is able to account for the influence of the load path out-of-phase degree on fatigue lifetime.

In particular, for multiaxial in-phase loading [41], the equivalent strain amplitude $\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}/2$ can reduce to the equivalent strain amplitude$\Delta {\epsilon }_{\mathrm{eq}}/2$ correctly as follows:

(6)${\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}=\sqrt{{\left({\epsilon }_n^{*}\right)}^2+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)}^2}={\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}}{2}}$

And, the equivalent stress amplitude $\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}/2$ can reduce to the equivalent stress amplitude $\Delta {\sigma }_{\mathrm{eq}}/2$ as follows:

(7)${\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}=\sqrt{{\left({\displaystyle \frac{{\sigma }_n^{*}}{2}}\right)}^2+3\cdot {\left({\displaystyle \frac{\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)}^2}={\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}}{2}}$

Therefore, the presented equivalent strain energy damage model EBDP can be denoted as Equation (8):

(8)${\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}\cdot {\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}={\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}}{2}}\cdot {\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}}{2}}={\displaystyle \frac{{\left({{\sigma }^{\prime }}_{\mathrm{f}}\right)}^2}{E}}\cdot {\left(2N_{\mathrm{f}}\right)}^{2b}+{{\sigma }^{\prime }}_{\mathrm{f}}{{\epsilon }^{\prime }}_{\mathrm{f}}\cdot {\left(2N_{\mathrm{f}}\right)}^{b+c}$

In addition, within the condition of axial compression–tension load, the amplitude of equivalent strain $\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}/2$ can degenerate into the amplitude of tensile strain $\Delta {\epsilon }_x/2$ correctly as shown below:

(9)${\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}=\sqrt{{\left({\epsilon }_n^{*}\right)}^2+{\displaystyle \frac{1}{3}}{\left({\displaystyle \frac{\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)}^2}={\displaystyle \frac{\Delta {\epsilon }_x}{2}}$

Furthermore, $\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$ and ${\sigma }_n^{*}/2$ are able to be denoted as Equations (10) and (11) within the case of uniaxial symmetric cyclic loadings:

(10)${\displaystyle \frac{{\sigma }_n^{*}}{2}}={\displaystyle \frac{\Delta {\sigma }_x}{4}}$

(11)${\displaystyle \frac{\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}={\displaystyle \frac{\Delta {\sigma }_x}{4}}$

Next, the presented axial equivalent stress amplitude $\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}/2$ is able to be denoted below:

(12)${\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}=\sqrt{{\left({\displaystyle \frac{{\sigma }_n^{*}}{2}}\right)}^2+3\cdot {\left({\displaystyle \frac{\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)}^2}={\displaystyle \frac{\Delta {\sigma }_x}{2}}$

Hence, based on the critical plane approach, the presented equivalent strain energy damage modeling EBDP is able to be expressed below:

(13)${\displaystyle \frac{\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}\cdot {\displaystyle \frac{\Delta {\epsilon }_{\mathrm{eq}}^{\mathrm{cr}}}{2}}={\displaystyle \frac{\Delta {\sigma }_x}{2}}\cdot {\displaystyle \frac{\Delta {\epsilon }_x}{2}}={\displaystyle \frac{{\left({{\sigma }^{\prime }}_{\mathrm{f}}\right)}^2}{E}}\cdot {\left(2N_{\mathrm{f}}\right)}^{2b}+{{\sigma }^{\prime }}_{\mathrm{f}}{{\epsilon }^{\prime }}_{\mathrm{f}}\cdot {\left(2N_{\mathrm{f}}\right)}^{b+c}$

It should be mentioned that the formalization of the equivalent axial strain energy (EBDP) model (Equation (13)) is the same as that of the universal Smith–Watson–Topper (SWT) model [10]. In particular, in the condition of uniaxial tension–compression load, the presented damage modeling (Equation (5)) can degenerate into the formalization of the Smith–Watson–Topper (SWT) model. Additionally, it can be found that the presented equivalent critical plane damage parameter is a damage quantity with equivalent strain energy on the basis of the critical plane approach within the essence.

3. Multiaxial Fatigue Lifetime Calculation

Based on the presented fatigue damage model, multiaxial fatigue lifetimes are estimated for variable amplitude loading. The flowchart of presented multiaxial fatigue lifetime estimation is depicted schematically within Figure 4.

3.1. Stress–Strain Analysis for Multiaxial Load

3.1.1. Three-Dimensional Stress Condition

For a general material point $O$ as shown within Figure 5, the actual elastic–plastic strains and stresses can be denoted as follows:

(14)${\sigma }_{ij}=\left[\begin{array}{ccc}{\sigma }_x& {\tau }_{xy}& {\tau }_{xz}\\ {\tau }_{xy}& {\sigma }_y& {\tau }_{yz}\\ {\tau }_{xz}& {\tau }_{yz}& {\sigma }_z\end{array}\right]$

(15)${\epsilon }_{ij}=\left[\begin{array}{ccc}{\epsilon }_x& {\epsilon }_{xy}& {\epsilon }_{xz}\\ {\epsilon }_{xy}& {\epsilon }_y& {\epsilon }_{yz}\\ {\epsilon }_{xz}& {\epsilon }_{yz}& {\epsilon }_z\end{array}\right]=\left[\begin{array}{ccc}{\epsilon }_x& 1/2{\gamma }_{xy}& 1/2{\gamma }_{xz}\\ 1/2{\gamma }_{xy}& {\epsilon }_y& 1/2{\gamma }_{yz}\\ 1/2{\gamma }_{xz}& 1/2{\gamma }_{yz}& {\epsilon }_z\end{array}\right]$

• (1). By employing the normal vector $\overrightarrow{X^{\prime }}$, the angle of a general plane $\Delta$ is calculated as shown within Figure 5. Via orientation angles $\varphi$ and $\theta$, the normal vector $\overrightarrow{X^{\prime }}$ can be denoted. $\varphi$ is the orientation angle between the $\overrightarrow{X^{\prime }}$ and normal vector Z-axis, and $\theta$ is the angle between the projection of normal vector $\overrightarrow{X^{\prime }}$ and X-axis on the X-Y plane. Additionally, an innovative coordinate $O{X}^{\prime }{Y}^{\prime }{Z}^{\prime }$ can be obtained as depicted within Figure 5, in which the $Y^{\prime }$-axis is situated at the intersection line related to the X-Y plane and general material plane $\Delta$, and the $Z^{\prime }$-axis lies upon the plane $\Delta$.

• (2). Upon the $i$th analytical plane $\Delta$, the calculated strain and stress tensors are able to be expressed using angles $\theta$ and $\varphi$ below:

  (16)${{\sigma }^{\prime }}_{ij}=M{\sigma }_{ij}{M}^T$

  (17)${{\epsilon }^{\prime }}_{ij}=M{\epsilon }_{ij}{M}^T$

  where $M^T$ is the transpose of the matrix, which is able to be calculated utilizing the transformational matrix $M$ as follows:

  (18)$M=\left[\begin{array}{ccc}\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi & \mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{s}\mathrm{i}\mathrm{n}\varphi & \mathrm{c}\mathrm{o}\mathrm{s}\varphi \\ -\mathrm{s}\mathrm{i}\mathrm{n}\theta & \mathrm{c}\mathrm{o}\mathrm{s}\theta & 0\\ -\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi & -\mathrm{s}\mathrm{i}\mathrm{n}\theta \mathrm{c}\mathrm{o}\mathrm{s}\varphi & \mathrm{s}\mathrm{i}\mathrm{n}\varphi \end{array}\right]$

3.1.2. Plane Stress State

For a thin-walled tubular component experiencing complicated axial–torsional loadings as depicted within Figure 6a, the investigative material planes are all orthogonal to the specimen surface, as shown within Figure 6b. In such a case, the applied strains and stresses are able to be denoted as Equations (19) and (20):

(19)${\sigma }_{ij}=\left[\begin{array}{ccc}{\sigma }_x& {\tau }_{xy}& 0\\ {\tau }_{xy}& 0& 0\\ 0& 0& 0\end{array}\right]$

(20)${\epsilon }_{ij}=\left[\begin{array}{ccc}{\epsilon }_x& {\epsilon }_{xy}& 0\\ {\epsilon }_{xy}& -{\nu }_{\mathrm{eff}}{\epsilon }_x& 0\\ 0& 0& -{\nu }_{\mathrm{eff}}{\epsilon }_x\end{array}\right]=\left[\begin{array}{ccc}{\epsilon }_x& {\gamma }_{xy}/2& 0\\ {\gamma }_{xy}/2& -{\nu }_{\mathrm{eff}}{\epsilon }_x& 0\\ 0& 0& -{\nu }_{\mathrm{eff}}{\epsilon }_x\end{array}\right]$

where the effective Poisson’s ratio ${\nu }_{\mathrm{e}\mathrm{ff}}$ is able to be calculated by utilizing Equation (21):

(21)${\nu }_{\mathrm{eff}}=0.5-{\displaystyle \frac{\left(0.5-{\nu }_{\mathrm{e}}\right)\Delta {\sigma }_{\mathrm{eq}}}{E\Delta {\epsilon }_{\mathrm{eq}}}}$

where $\Delta {\epsilon }_{\mathrm{eq}}$ and $\Delta {\sigma }_{\mathrm{eq}}$ are the equivalent strain and stress ranges using the von Mises criterion, respectively; E is the modulus of elasticity; and ${\nu }_{\mathrm{e}}$ is the Poisson’s ratio of elasticity, which can be calculated utilizing the computational procedure shown in Ref. [43].

3.2. Multiaxial Cycle Count Method

With regard to multiaxial variable amplitude load, an appropriate cycle count method is needed to enumerate numerous separate cycles within a load sequence. According to the equivalent von Mises strain rule and rain-flow count method, Wang and Brown suggested a multiaxial reversal count method [42,43,44]. The equivalent von Mises strain can be computed utilizing the algorithm as follows:

(22)${\epsilon }_{\mathrm{eq}}={\displaystyle \frac{1}{\sqrt{2}\left(1+{\nu }_{\mathrm{eff}}\right)}}\sqrt{{\left({\epsilon }_x-{\epsilon }_y\right)}^2+{\left({\epsilon }_y-{\epsilon }_z\right)}^2+{\left({\epsilon }_z-{\epsilon }_x\right)}^2+{\displaystyle \frac{3}{2}}\left({\gamma }_{xy}^2+{\gamma }_{yz}^2+{\gamma }_{xz}^2\right)}$

As depicted within Figure 7a, a case of multiaxial variable amplitude load history has been adopted to explain the thinking of the multiaxial reversal count method presented by Wang and Brown; the corresponding strain loading paths in $\gamma /\sqrt{3}-\epsilon$ strain space are depicted in Figure 7b. Based on the Wang–Brown’s multiaxial count method, the total number of nine reversals have been enumerated, i.e., $A\to A^{\prime }$, $B\to B^{\prime }$, $C\to C^{\prime }$, $D\to D^{\prime }$,$E\to E^{\prime }$, $F\to F^{\prime }$, $G\to G^{\prime }$, $H\to H^{\prime }$, $I\to I^{\prime }$, and $J\to J^{\prime }$, as shown within Figure 7c. The normal stress excursion ${\sigma }_n^{*}$ with regard to the J-J′ reversal is illustrated in Figure 7d.

3.3. Calculation of the Critical Plane with Regard to Every Count Reversal

The procedure of the critical plane calculation can be summed up for the three-dimensional stress state as shown below:

• (1). The shear strain range can be computed using angles $\theta$ and $\varphi$on the ith candidate plane.

  (23)$\Delta {\gamma }_i=\underset{t_1<t_2\le t_{\mathrm{finish}}}{\underset{t_{\mathrm{begin}}\le t_1\le t_{\mathrm{finish}}}{\mathrm{m}\mathrm{a}\mathrm{x}}}\left\{2\sqrt{{\left[{\epsilon }_{x^{\prime }{y}^{\prime }}\left(t_1\right)-{\epsilon }_{x^{\prime }{y}^{\prime }}\left(t_2\right)\right]}^2+{\left[{\epsilon }_{x^{\prime }{z}^{\prime }}\left(t_1\right)-{\epsilon }_{x^{\prime }{z}^{\prime }}\left(t_2\right)\right]}^2}\right\}$

  where ${\mathrm{t}}_{begin}$ and ${\mathrm{t}}_{finish}$ are the beginning time point and finishing time point during one count reversal, respectively.

• (2). Via making angles $\theta$ and $\varphi$ change between $0^{°}$ and $360^{°}$ and $0^{°}$ and $180^{°}$, respectively, the torsional strain amplitude can be computed. Subsequently, the direction angle of the plane with maximum torsional strain amplitude can be obtained.

• (3). By utilizing Equation (24), the ranges of normal strain can be calculated as follows:

  (24)${\epsilon }_n=\underset{t_p<t_q\le t_{\mathrm{finish}}}{\underset{t_{\mathrm{begin}}\le t_p\le t_{\mathrm{finish}}}{\mathrm{m}\mathrm{a}\mathrm{x}}}\left\{\left|{\epsilon }_{x^{\prime }}\left(t_p\right)-{\epsilon }_{x^{\prime }}\left(t_q\right)\right|\right\}$

• (4). For those planes with maximum range of torsional stress $\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}$, the normal strain amplitudes on the $\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}$ planes are made into a comparison. According to the largest torsional stress range plane with the maximum amplitude of normal strain, the orientation of the critical plane can be calculated. Furthermore, the angles ${\theta }_{\mathrm{cr}}$ and ${\varphi }_{\mathrm{cr}}$ are utilized to denote the pair of direction angles with regard to the critical plane.

Particularly, three stresses are equal to zero on the component surface with regard to the plane stress state, as shown within Figure 6b. In the current study, all of the analytical material planes are perpendicular to the surface of the investigated component. That means that the direction angle $\varphi$ equals $90^{°}$, i.e., $\varphi ={90}^{°}$. Hence, the plane Δ of concern can be determined by a direction angle $\theta$, and $\theta$ changes from ${-90}^{°}$ to ${90}^{°}$, or from $0^{°}$ to ${180}^{°}$. By making a comparison of the normal strain amplitudes upon every largest torsional strain amplitude plane, the critical plane can be determined as the plane of the largest torsional strain amplitude with the largest variable range of normal strain.

3.4. Computation of Multiaxial Fatigue Damage Quantities upon the Critical Plane

The angles ${\varphi }_{\mathrm{cr}}$ and ${\theta }_{\mathrm{cr}}$ are utilized for denoting the direction of the critical plane with regard to a count reversal. Upon the calculated critical plane, the computation of multiaxial fatigue damage parameters is summed up below:

• (1). By utilizing Equation (25), the largest torsional strain range is calculated:

  (25)$\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}=\underset{t_{\mathrm{p}}<t_{\mathrm{q}}\le t_{\mathrm{end}}}{\underset{t_{\mathrm{start}}\le t_{\mathrm{p}}\le t_{\mathrm{end}}}{\mathrm{m}\mathrm{a}\mathrm{x}}}\left\{2\sqrt{{\left[{\epsilon }_{x^{\prime }{y}^{\prime }}\left(t_{\mathrm{p}}\right)-{\epsilon }_{x^{\prime }{y}^{\prime }}\left(t_{\mathrm{q}}\right)\right]}^2{\left[{\epsilon }_{x^{\prime }{z}^{\prime }}\left(t_{\mathrm{p}}\right)-{\epsilon }_{x^{\prime }{z}^{\prime }}\left(t_{\mathrm{q}}\right)\right]}^2}\right\}$

  where $t_p$ and $t_q$ can be employed to represent two time instants within a count reversal, during which the maximum torsional strain range is obtained.

• (2). Between adjacent turning time instants $\left[t_1,t_2\right]$ of the largest torsional strain range, the maximum excursion of normal strain ${\epsilon }_n^{*}$ can be computed utilizing Equation (26):

  (26)${\epsilon }_n^{*}=\underset{t_1\le t\le t_2}{\mathrm{m}\mathrm{a}\mathrm{x}}{\epsilon }_{x^{\prime }}\left(t\right)-\underset{t_1\le t\le t_2}{\mathrm{m}\mathrm{i}\mathrm{n}}{\epsilon }_{x^{\prime }}\left(t\right)$

• (3). The largest torsional stress range ($\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}$) can be calculated adapting Equation (27):

  (27)$\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}=\underset{t_3<t_4\le t_{end}}{\underset{t_{start}\le t_3\le t_{end}}{\mathrm{m}\mathrm{a}\mathrm{x}}}\left\{2\sqrt{{\left[{\tau }_{x^{\prime }{y}^{\prime }}\left(t_3\right)-{\tau }_{x^{\prime }{y}^{\prime }}\left(t_4\right)\right]}^2+{\left[{\tau }_{x^{\prime }{z}^{\prime }}\left(t_3\right)-{\tau }_{x^{\prime }{z}^{\prime }}\left(t_4\right)\right]}^2}\right\}$

  where $t_3$ and $t_4$ are used for representing the correspondent pair of time instants; the largest torsional stress amplitude can be obtained in the time range [$t_3$, $t_4$].

• (4). By utilizing Equation (28), the largest tensile stress ${\sigma }_{n,\mathrm{m}\mathrm{a}\mathrm{x}}$ is calculated as shown below:

  (28)${\sigma }_{n,\mathrm{m}\mathrm{a}\mathrm{x}}=\underset{t_{\mathrm{start}}\le t\le t_{\mathrm{end}}}{\mathrm{m}\mathrm{a}\mathrm{x}}{\sigma }_{x^{\prime }}\left(t\right)$

  where ${\sigma }_{x^{\prime }}\left(t\right)$ is the normal stress in one reversal for all time points. In addition, the excursion ${\sigma }_n^{*}$ of maximum normal stress between adjacent turning time instants $\left[t_3,t_4\right]$ of the largest shear stress is obtained employing the algebra expression below:

  (29)${\sigma }_n^{*}=\underset{t_3\le t\le t_4}{\mathrm{m}\mathrm{a}\mathrm{x}}{\sigma }_{x^{\prime }}\left(t\right)-\underset{t_3\le t\le t_4}{\mathrm{m}\mathrm{i}\mathrm{n}}{\sigma }_{x^{\prime }}\left(t\right)$

3.5. Multiaxial Fatigue Damage Estimation

According to the presented multiaxial fatigue damage model (Equation (5)), the resulting multiaxial fatigue damage $D_k$ with regard to the k count reversal can be estimated as follows:

(30)$D_k={\displaystyle \frac{1}{2N_{\mathrm{f},k}}}$

where $N_{\mathrm{f},k}$ is used for denoting the fatigue lifetime with regard to the $k$th count reversal.

By adopting the Miner’s linear cumulative damage law [45], accumulative fatigue damage $D$ for the whole load history is calculated after the computation of fatigue damage for each count reversal:

(31)$D=\sum _{k=1}^{\mathrm{m}}{\displaystyle \frac{1}{2N_{\mathrm{f},k}}}$

where $m$ is the total counted number corresponding to all multiaxial reversals; $N_{\mathrm{f},k}$ is the fatigue lifetime for the $k$th count reversal.

At last, by using Equation (32), the fatigue lifetime ($N_{\mathrm{pre}}$) is able to be estimated as shown below:

(32)$N_{\mathrm{pre}}={\displaystyle \frac{1}{D}}$

4. Experiment Validation Results

For the purpose of validating the evaluation capability of the presented multiaxial fatigue lifetime estimation methodology, the experimentally collected results of the 7050-T7451 aluminum alloy and En15R steel, which are obtained via Refs. [44,46], are employed within this work. Both of the selected two kinds of specimens are thin-walled circular tubes. These two sorts of materials show an obvious out-of-phase cyclic hardening phenomenon and exhibit the shear fatigue failure mode. The fatigue and mechanics parameters of the investigated two sorts of materials are shown within Table 2.

A total number of twenty experimental data values for En15R steel, which are used within strain control under variable amplitude multiaxial block load, have been utilized for validating the presented multiaxial fatigue damage model (EBDP model). The experimentally acquired data were generated at the University of Sheffield by M.W. Brown et al. [34]. It can be found from Figure 8 that the experimental data include eight sorts of variable amplitude multiaxial load paths.

With regard to the 7050-T7451 aluminum alloy [46], the selected thin-walled circular components are processed by some recrystallized bars, which have outline dimensions with a gage distance of 25 mm, external radius of 6.25 mm, and internal radius of 5.25 mm. The fatigue experiments are performed under strain controlling, and the axial and shear loading histories are collected by means of four channels in accordance with the test output and command signals. Additionally, the axial and shear stress responses at the component gauge part are computed from the measured loading strain histories, which can be utilized as the entering quantity to the multiaxial fatigue life estimation model. Within this study, the presented multiaxial fatigue damage models are validated using seven categories of variable amplitude multiaxial load histories, as shown within Figure 9.

In the current investigation, predictive fatigue lifetimes with the presented damage models are compared with experimentally acquired data for En15R steel, as shown within Figure 10a. It should be mentioned that the estimated fatigue lifetimes relate well with experimentally measured data; the estimated fatigue lives are basically within two error factors. Moreover, for the 7050-T7451 aluminum alloy, satisfactory fatigue lifetime predictions are obtained by the presented multiaxial fatigue damage model, and the prediction lifetimes are basically within two error factors, as depicted within Figure 10b.

The presented multiaxial fatigue damage model is compared with other energy-based damage models for the purpose of verifying estimation accuracy and applicability. Another three multiaxial fatigue damage models, which were presented by Varvani-Farahani (VF) [17], Pan et al. (PHC) [19], and Varvani-Farahani, Kodric, and Ghahramani (VKG) [20], are selected to predict fatigue lifetime for experimentally obtained results of the 7050-T7451 aluminum alloy and En15R steel. Varvani-Farahani (VF) summed up the torsional and tensile strain energies upon the critical plane, and presented a multiaxial fatigue damage modeling, which is shown below [17]:

(33)$\left(1+{\displaystyle \frac{{\sigma }_n^{\mathrm{m}}}{{{\sigma }^{\prime }}_{\mathrm{f}}}}\right)\left(\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \frac{\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{ 2}}\right)+\left({\displaystyle \frac{{{\tau }^{\prime }}_{\mathrm{f}}{{\gamma }^{\prime }}_{\mathrm{f}}}{{{\sigma }^{\prime }}_{\mathrm{f}}{{\epsilon }^{\prime }}_{\mathrm{f}}}}\right)\Delta {\sigma }_n\Delta {\epsilon }_n=2{\displaystyle \frac{{{{\tau }^{\prime }}_{\mathrm{f}}}^2}{G}}{\left(2N_{\mathrm{f}}\right)}^{2b^{\prime }}+2{{\gamma }^{\prime }}_{\mathrm{f}}{{\tau }^{\prime }}_{\mathrm{f}}{\left(2N_{\mathrm{f}}\right)}^{b^{\prime }+c^{\prime }}$

where $\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$ and $\Delta {\sigma }_n/2$ are, respectively, the torsional and tensile stress amplitudes; ${\sigma }_n^m$ is the normal mean stress; and $\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}/2$ and $\Delta {\epsilon }_n/2$ are the torsional and tensile strain amplitudes on the critical plane, respectively.

Pan et al. (PHC) presented a multiaxial fatigue damage model constituted by shear and axial fatigue properties, in which the influence of mean stress is taken into consideration. The weighted fatigue damage model is shown below [19]:

(34)$\left(1+{\displaystyle \frac{{\sigma }_n^{\mathrm{m}}}{{{\sigma }^{\prime }}_{\mathrm{f}}}}\right){\displaystyle \frac{\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{ 2}}{\displaystyle \frac{\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}+\left({\displaystyle \frac{{{\gamma }^{\prime }}_{\mathrm{f}}{{\sigma }^{\prime }}_{\mathrm{f}}}{{{\epsilon }^{\prime }}_{\mathrm{f}}{{\tau }^{\prime }}_{\mathrm{f}}}}\right){\displaystyle \frac{\Delta {\epsilon }_n}{2}}{\displaystyle \frac{\Delta {\sigma }_n}{2}}={\displaystyle \frac{{{{\tau }^{\prime }}_{\mathrm{f}}}^2}{G}}{\left(2N_{\mathrm{f}}\right)}^{2b^{\prime }}+{{\gamma }^{\prime }}_{\mathrm{f}}{{\tau }^{\prime }}_{\mathrm{f}}{\left(2N_{\mathrm{f}}\right)}^{b^{\prime }+c^{\prime }}$

Varvani-Farahani, Kodric, and Ghahramani (VKG) proposed a multiaxial fatigue damage quantity, which combines the shear and tensile strain energies upon the critical plane, as depicted within Equation (35). It should be noted that, in terms of both torsional and tensile fatigue properties, the right side of Equation (35) is implemented as follows [20]:

(35)${\displaystyle \frac{1}{\left({{\sigma }^{\prime }}_{\mathrm{f}}{{\epsilon }^{\prime }}_{\mathrm{f}}\right)}}\left(\Delta {\sigma }_n\Delta {\epsilon }_n\right)+{\displaystyle \frac{1}{\left({{\tau }^{\prime }}_{\mathrm{f}}{{\gamma }^{\prime }}_{\mathrm{f}}\right)}}\left(\Delta {\tau }_{\mathrm{m}\mathrm{a}\mathrm{x}}{\displaystyle \frac{\Delta {\gamma }_{\mathrm{m}\mathrm{a}\mathrm{x}}}{2}}\right)=\left[{\displaystyle \frac{{{\sigma }^{\prime }}_{\mathrm{f}}}{E}}{\left(2N_{\mathrm{f}}\right)}^b+{{\epsilon }^{\prime }}_{\mathrm{f}}{\left(2N_{\mathrm{f}}\right)}^c\right]+\left[{\displaystyle \frac{{{\tau }^{\prime }}_{\mathrm{f}}}{G}}{\left(2N_{\mathrm{f}}\right)}^{b^{\prime }}+{{\gamma }^{\prime }}_{\mathrm{f}}{\left(2N_{\mathrm{f}}\right)}^{c^{\prime }}\right]$

As depicted within Figure 11a–d, the fatigue lifetime estimation results are supplied by the EBDP model, VF model, PHC model, and VKG model. It can be found that, for the investigated 7050-T7451 aluminum alloy and En15R steel, the proposed EBDP damage models can supply more satisfactory fatigue lifetime estimation results than VF, PHC, and VKG damage models. For the investigated two sorts of materials, the multiaxial fatigue lifetime estimation according to the presented EBDP damage model is able to supply satisfactory fatigue lifetime predictions. Furthermore, for the multiaxial fatigue lifetime evaluation of mechanical parts, the presented method can be robust and effective under variable amplitude multiaxial loadings. For the purpose of estimating the effectiveness of presented multiaxial fatigue damage modeling, the multiaxial fatigue lifetime estimation errors are estimated by using a probabilistic analytical method. Equation (36) is employed to estimate the prediction error E_N as shown below [47]:

(36)$E_{\mathrm{N}}={\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\left({\displaystyle \frac{N_{\mathrm{e}\mathrm{x}\mathrm{p}}}{N_{\mathrm{pre}}}}\right)$

where $N_{\mathrm{e}\mathrm{x}\mathrm{p}}$ is utilized for denoting the experimentally observed fatigue lifetime, and $N_{\mathrm{pre}}$ is the predictive fatigue lifetime.

With regard to the EBDP, VF, PHC, and VKG models, the frequent counting histograms of the logarithmical relative deviation exponent EN have been depicted within Figure 12 for the investigated two kinds of components constituted by the 7050-T7451 aluminum alloy and En15R steel, respectively. For the examined four multiaxial fatigue lifetime estimation models, the normal relative error curves have been depicted within Figure 12. Compared with the presented EBDP model, the prediction errors are negative and the prediction results are unconservative or unsafe for the VF, PHC, and VKG damage models. Additionally, it can be found that the presented EBDP model has higher estimation accuracy than the VF, PHC, and VKG models for investigated multiaxial variable amplitude loading histories. Therefore, the proposed damage model provides a robust and efficient tool to the multiaxial fatigue life prediction of engineering parts under variable amplitude multiaxial loadings.

5. Discussion

Within the current study, the presented multiaxial fatigue damage parameter is formed within a formalization of equivalent tensile strain energy, as shown within the presented multiaxial fatigue damage modeling (Equation (5)). Within the presented multiaxial fatigue damage modeling, the torsional and tensile strain and stress damage quantities upon the critical plane can be taken into consideration. Moreover, based upon the presented multiaxial fatigue damage quantities, the fatigue lifetime is able to be predicted efficiently and reasonably for variable amplitude multiaxial loads, as depicted within Figure 10a,b.

For the thin-walled tubular components with the 7050-T7451 aluminum alloy, both the proposed EBDP and the VF damage models can supply satisfactory fatigue lifetime estimations for axial–torsional constant amplitude cyclic loads, as shown in Table 1. In addition, upon the basis of the presented EBDP damage model, the multiaxial fatigue lifetime estimations are superior to those based on the VF, EBDP, and PHC damage models for En15R steel and the 7050-T7451 aluminum alloy experiencing multiaxial variable amplitude loads, as depicted in Figure 12.

For the investigated En15R steel and 7050-T7451 aluminum alloy, two representative multiaxial variable amplitude load paths are employed for the purpose of clarifying the deviations between the presented EBDP damage parameter and VF damage parameter, as depicted within Figure 13a,b. As shown within Figure 13a, a multiaxial variable amplitude load sequence with an experimental fatigue lifetime of 358 load blocks has been selected for the 7050-T7451 aluminum alloy. It can be found that twenty reversals are enumerated in total, and both the calculated fatigue damage within the proposed EBDP model and that in the VF model for all count reversals are shown in Figure 13a. It is found that, for most of the count reversals, the computed EBDP and VF fatigue damage are diverse, which contributes to the deviations in the fatigue lifetime estimations via the VF model (727 load blocks) and EBDP model (319 load blocks). Moreover, the proposed EBDP fatigue damage model, which is constructed with axial equivalent stress amplitude $\Delta {\sigma }_{\mathrm{eq}}^{\mathrm{cr}}/2$, is able to supply more accuracy and satisfactory fatigue lifetime estimations within the comparison with the VF model. As depicted within Figure 13b, multiaxial fatigue damage in the proposed EBDP model is compared with that in the VF model for all counted reversals with regard to En15 steel. It can be found that relatively large deviations exist between the multiaxial fatigue damage given by the proposed damage model and VF damage model for the first, second, and fifth count reversals, which can contribute to the difference of fatigue lifetime predictions provided using the EBDP model (2766 load blocks) and VF model (4796 load blocks). Additionally, the estimation fatigue lives by the EBDP model are more accurate than those by the VF model for variable amplitude multiaxial load histories.

Within the practical engineering, the stresses at the concerning position cannot be determined directly and the strains can be measured using strain measurement equipment [48,49,50,51,52,53,54,55,56,57,58,59,60,61,62]. For the purpose of evaluating the fatigue lifetime using the presented multiaxial fatigue damage parameters, a multiaxial constitutive relationship is able to be adopted to compute the stress quantities by means of the measured strain data [63,64,65,66,67,68,69,70,71,72]. Furthermore, integrating the multiaxial notch modification method with the multiaxial constitutive relation, the presented multiaxial fatigue damage modeling can be applied to perform fatigue lifetime estimation for notched specimens [73,74,75,76,77,78,79,80,81,82,83,84,85]. The applicability and accuracy of the presented approach within handling practical mechanical parts subjected to more complex load cases [86,87,88,89,90,91,92,93], containing a complicated triaxial stress condition, need to be further validated. Additionally, En15R steel and the 7050-T7451 aluminum alloy are selected to validate the proposed approach in this investigation; further works for checking the accuracy of the presented approach in dealing with other material and structural conditions are needed [94,95,96,97,98,99,100,101,102]. In addition, the effects of normal/shear mean stress on fatigue life should be further investigated in the proposed multiaxial fatigue damage model. Furthermore, in order to make the proposed approach more practicable in ordinary engineering design applications, the further simplification of the proposed model should be performed in the future.

6. Conclusions

For the purpose of estimating the fatigue lifetime of mechanical parts experiencing variable amplitude multiaxial loadings, an innovative computation approach is presented within the current investigation. The experimentally measured results of two thin-walled tube components, which are constituted by a 7050-T7451 aluminum alloy and En15R steel, have been employed to verify the presented approach. Some key conclusions are able to be made below based on the experimental validation results:

• (1). An important multiaxial damage quantity upon the critical plane, which is described as the tensile stress excursion ${\sigma }_n^{*}$ during contiguous turn time instants of the largest torsional stress, is proposed in this study. The proposed damage quantity ${\sigma }_n^{*}$ has been validated to be sensitive for the out-of-phase cyclic hardening of multiaxial loading paths.

• (2). By integrating the critical plane method with the thinking of strain energy, an axial equivalent strain energy damage model (EBDP model) is proposed upon the critical plane, which does not contain any supernumerary material parameters. The influences of mean stress and out-of-phase cyclic hardening are able to be taken into consideration within the presented multiaxial fatigue damage model.

• (3). Compared with the VF, PHC, and VKG damage models, the proposed EBDP model is able to supply more accurate fatigue lifetime estimations for the selected two materials experiencing multiaxial variable amplitude loads.

Author Contributions

Conceptualization, Z.-Q.T.; methodology, Z.-Q.T.; software, Z.-Q.T.; validation, Z.-Q.T.; formal analysis, L.-X.L.; investigation, Z.-L.Z.; resources, Z.-Q.T.; data curation, Z.-Q.T.; writing—original draft preparation, Z.-Q.T.; writing—review and editing, H.C.; visualization, Z.-Q.T.; supervision, Z.-Q.T.; project administration, X.P.; funding acquisition, Z.-Q.T. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Acknowledgments

The researchers would like to thank Drew V. Nelson of Stanford University for providing experimental data from M.W. Brown of the University of Sheffield and T.E. Langlais of Seagate Technology.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1. Schematic diagram of strain paths: (a) proportional loading path; (b) 45° non-proportional loading path; and (c) 90° non-proportional loading path.

Figure 2. Varying characteristics of the stress and strain on the critical plane. (a) Phase angle [Forumla omitted. See PDF.]. (b) Phase angle[Forumla omitted. See PDF.]. (c) Phase angle. (d) Largest normal stress excursion on the critical plane.

Figure 3. (a) Varying characteristics of the maximum shear stress amplitude [Forumla omitted. See PDF.], maximum normal strain excursion [Forumla omitted. See PDF.], and largest shear strain amplitude [Forumla omitted. See PDF.] on the maximum shear plane; and (b) varying characteristics of the axial equivalent stress modification factor [Forumla omitted. See PDF.], [Forumla omitted. See PDF.] damage parameter, and Shang–Wang damage parameter [Forumla omitted. See PDF.] on the critical plane.

Figure 4. Flowchart of fatigue life prediction under variable amplitude multiaxial loading.

Figure 5. Generic material plane Δ and definition of angles Φ and θ.

Figure 6. (a) Thin-walled tubular specimen subjected to combined tension and torsion loading; and (b) material plane [Forumla omitted. See PDF.] perpendicular to component surface and having normal vector [Forumla omitted. See PDF.] at angle [Forumla omitted. See PDF.] to [Forumla omitted. See PDF.]-axis.

Figure 7. (a) Strain-time variable amplitude axial and torsional loading histories, (b) strain loading paths in [Forumla omitted. See PDF.] strain space, (c) equivalent relative strain histories counted by Wang–Brown’s reversal counting method, and (d) the normal stress excursion [Forumla omitted. See PDF.] with regard to the J-J′ reversal.

Figure 8. Loading paths for En15R steel utilized in this investigation [44]. (a) Path A. (b) Path B. (c) Path C. (d) Path D. (e) Path E. (f) Path F. (g) Path G. (h) Path H.

Figure 9. Loading paths for the 7050-T7451 aluminum alloy utilized in this investigation [46]. (a) Path A. (b) Path B. (c) Path C. (d) Path D. (e) Path E. (f) Path F. (g) Path G.

Figure 10. The comparison of experimental and predicted fatigue lives based on the proposed EBDP model with the VF, PHC, and VKG model for (a) the 7050-T7451 aluminum alloy; and (b) En15R steel.

Figure 11. The distribution of logarithmic error in the prediction of multiaxial fatigue life by the (a) EBDP model; (b) VF model; (c) PHC model; and (d) VKG model.

Figure 12. The normal curve of the relative error for the examined fatigue life prediction model.

Figure 13. Comparisons of axial equivalent stress modification factors for all counted reversals of a selected loading path by (a) the 7050-T7451 aluminum alloy; and (b) En15R steel.

Table 1

Experimental and computed results of smooth thin-walled tube specimens with 7050-T7451 aluminum alloy under constant amplitude axial–torsional cyclic loading [42].

Spe. No.	Δεx/2 (%)	Δσx/2 (MPa)	Δγxy/2 (%)	Δτxy/2 (MPa)	φ (Deg)	Δεeq/2 (%)	Nf (Cycle)	EBDP	VF
A74	0.493	292.82	0.849	158.28	0	0.7	1460	1446	1743
A159	0.545	338.80	0.905	178.43	45	0.7	521	589	817
A122	0.703	416.21	1.208	230.55	90	0.7	203	267	187

Table 2

Fatigue properties of the investigated materials.

Material	Ref.	$\begin{array}{c}\mathit{E}\\ \left(\mathit{G}\mathit{P}\mathit{a}\right)\end{array}$	$\begin{array}{c}\mathit{G}\\ \left(\mathit{G}\mathit{P}\mathit{a}\right)\end{array}$	${\mathit{\nu }}_{\mathit{e}}$	$\begin{array}{c}{{\mathit{\sigma }}^{\mathbf{\prime }}}_f\\ \left(\mathit{M}\mathit{P}\mathit{a}\right)\end{array}$	${{\mathit{\epsilon }}^{\mathbf{\prime }}}_{\mathit{f}}$	$\mathit{b}$	$\mathit{c}$	$\begin{array}{c}{{\mathit{\tau }}^{\mathbf{\prime }}}_{\mathit{f}}\\ \left(\mathit{M}\mathit{P}\mathit{a}\right)\end{array}$	${{\mathit{\gamma }}^{\mathbf{\prime }}}_{\mathit{f}}$	${\mathit{b}}_{\mathit{0}}$	${\mathit{c}}_{\mathit{0}}$
En15R steel	[44]	205	80	0.28	1114	0.259	−0.097	−0.515	870.3	0.518	−0.097	−0.515
7050-T7451 aluminum alloy	[46]	70	27	0.33	602.7	0.587	−0.0457	−0.8206	399.28	0.6088	−0.0755	−0.6021

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