A novel model for low-cycle multiaxial fatigue life prediction based on the critical plane-damage parameter

Abstract

Multiaxial fatigue of the components is a very complex behavior. This analyzes the multiaxial fatigue failure mechanism, reviews and compares the advantages and disadvantages of the classic model. The fatigue failure mechanism and fatigue life under multiaxial loading are derived through theoretical analysis and formulas, and finally verified with the results of multiaxial fatigue tests. The model of multiaxial fatigue life for low-cycle fatigue life prediction model not only improves the prediction accuracy of the classic model, but also considers the effects of non-proportional additional hardening phenomena and fatigue failure modes. The model is proved to be effective in low-cycle fatigue life prediction under different loading paths and types for different materials. Compared with the other three classical models, the proposed model has higher life prediction accuracy and good engineering applicability.

Introduction

Metal fatigue damage is a greatly common phenomenon. Most engineering components are in service under multiaxial load conditions which lead to the threat of fatigue failure to varying degrees. Multiaxial fatigue of the components is a very complex behavior. For example, the aero-engine works in extremely harsh conditions, and in such environments, the premature fatigue failure of its components is often caused by the airplane crashes. Therefore, it is extremely important to reliably predict the fatigue life of the components.^1–3 Currently, there is no unified multiaxial fatigue failure criterion which can apply different materials, structures, and loading conditions. One of the researchers’ main goals is to find the effective damage parameter and establish an accurate multiaxial fatigue life model. The multiaxial fatigue life models currently proposed are mainly divided into three categories: equivalent stress–strain method, energy method, and critical plane method. In early days, when dealing with multiaxial fatigue problems under complex stress conditions, the static strength theory was used equivalently for multiaxial problems into uniaxial conditions, and then uniaxial fatigue theory was used to deal with complex multiaxial fatigue problems. Among them, Von Mises criterion and Tresca criterion are effective in addressing multiaxial fatigue problems under proportional loading. However, many of the actual engineering structures and components are operated under non-proportional multiaxial loading. The fatigue behavior under non-proportional loading conditions is far different from uniaxial or multiaxial proportional loading. Therefore, many scholars are doing more theoretical and experimental research on non-proportional multiaxial fatigue life prediction.^4,5 The energy method occupies an important position in the field of multiaxial fatigue. Many researchers believe that multiaxial fatigue damage is closely related to cyclic plastic deformation and plastic strain energy of materials.^6,7 The critical plane method is considered to be the most effective method for solving multiaxial fatigue problems.^8–11

According to the fatigue failure mechanism of the material, Socie ⁸ roughly divided the failure form into the shear-type and tensile-type. Brown and Miller (BM) ⁹ defined the maximum shear strain plane as the critical plane for the first time, and the maximum shear strain and normal strain on the critical plane are considered as the key damage parameters. This model established the damage parameter of the maximum shear strain and the maximum normal strain on the maximum shear strain critical plane, but the effect of the mean stress is not mentioned. Wang and Brown ¹⁰ revised the BM model with the mean stress but ignored the effect of the cyclic hardening effect on fatigue life. Fatemi and Socie (FS) ¹¹ proposed normal stress on the maximum shear strain plane instead of normal strain as the damage parameter, and the model predicted the fatigue life better when it considers mean stress and cyclic hardening effects. Smith, Watson, and Topper (SWT) ¹² defined the maximum normal strain plane as the critical plane, and the fatigue life model which used the maximum normal strain range and the maximum normal stress as the damage parameter. The maximum normal stress and strain play an important role in tensile-type’s multiaxial fatigue processes.^8,12 The SWT model has better life prediction effect for the tensile crack failure mode, but has a worse prediction effect for pure torsion and multiaxial fatigue. Wu et al. ¹³ modified the SWT parameters and proposed a model with better prediction. Chen-Xu-Huang (CXH) ¹⁴ model, which is the sum of the normal and shear strain energy on different critical planes is used as the multiaxial damage parameter, and the different failure modes are considered. However, the CXH model did not effectively distinguish the contribution of the normal with shear strain energy to the damage, so the prediction result tended to be conservative. The Liu and Mahadevan ¹⁵ and Shang model ¹⁶ was also similar to the above models. Li ¹⁷ proposed the MKBM model which considers the maximum normal stress to revise the KBM model. The model solved the effects of the non-proportional loading path problem very well.

Based on the critical plane, a multiaxial fatigue damage parameter and a life prediction model are proposed, and the additional hardening phenomenon and fatigue failure mode are considered. At the same time, three classical multiaxial models, namely, FS, SWT, and CXH are compared and evaluated by multiaxial fatigue test data of four materials. The results show that the new model has better life prediction ability.

The critical plane method

The stress and strain on the surface of a thin-walled tubular specimen subjected to combined tension and torsion loading can be expressed as ¹⁰

$$\sigma =\left(\begin{array}{ccc}{\sigma }_x\hfill & {\tau }_{\mathrm{xy}}\hfill & 0\hfill \\ \multicolumn{1}{c}{{\tau }_{\mathrm{xy}}}& 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill \end{array}\right),\epsilon =\left(\begin{array}{ccc}{\epsilon }_x\hfill & 1/2{\gamma }_{\mathrm{xy}}\hfill & 0\hfill \\ \multicolumn{1}{c}{1/2{\gamma }_{\mathrm{xy}}}& -{v_{\mathrm{ef}}}_{\mathrm{f}}{\epsilon }_x\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -{v_{\mathrm{ef}}}_{\mathrm{f}}{\epsilon }_x\hfill \end{array}\right)$$ (1)

$${v_{\mathrm{ef}}}_{\mathrm{f}}=\frac{v_{\mathrm{e}}{\epsilon }_{\mathrm{e}}+v_{\mathrm{p}}{\epsilon }_{\mathrm{p}}}{{\epsilon }_{\mathrm{e}}+{\epsilon }_{\mathrm{p}}}$$ (2)

where ${\sigma }_x$ and ${\tau }_{\mathrm{xy}}$ are the normal stress and shear stress, respectively; ${\epsilon }_x$ and ${\gamma }_{\mathrm{xy}}$ are the normal strain and shear strain, respectively; ${v_{\mathrm{ef}}}_{\mathrm{f}}$ represents the effective Poisson’s ratio; $v_{\mathrm{e}}$ is the elastic Poisson ratio; $v_{\mathrm{p}}$ is the plastic Poisson’s ratio, the value is usually 0.5 for metal materials; ${\epsilon }_{\mathrm{e}}$ and ${\epsilon }_{\mathrm{p}}$ are the elastic phase strain and plastic phase strain, respectively; and $\theta$ is the angle between the specimen axis and normal direction of an element. ¹⁸ The stress–strain can be expressed as follows

$${\epsilon }_{\theta }=\frac{1-v_{\mathrm{eff}}}{2}{\epsilon }_x+\frac{1+v_{\mathrm{eff}}}{2}{\epsilon }_x\cos 2\theta +\frac{{\gamma }_{\mathrm{xy}}}{2}\sin 2\theta$$ (3)

$$\frac{{\gamma }_{\theta }}{2}=\frac{1+v_{\mathrm{eff}}}{2}{\epsilon }_x\sin 2\theta -\frac{{\gamma }_{\mathrm{xy}}}{2}\cos 2\theta$$ (4)

$${\sigma }_{\theta }=\frac{{\sigma }_x}{2}+\frac{{\sigma }_x}{2}\cos 2\theta +\frac{{\tau }_{\mathrm{xy}}}{2}\sin 2\theta$$ (5)

The stress–strain response on the critical plane of tensile-type and shear-type is obtained using equations (3)–(5). In particular, the maximum normal strain range $\mathrm{\Delta }{\epsilon }_{\mathrm{n}}$ and the maximum shear strain range $\mathrm{\Delta }{\gamma }_{\max }$ on each plane can be calculated under sinusoidal loading by a premise mathematical solution ¹⁹

$$\mathrm{\Delta }{\gamma }_{\max }=\mathrm{\Delta }\epsilon \sqrt{{[\frac{\mathrm{\Delta }\gamma }{\mathrm{\Delta }\epsilon }\cos 2\theta \cos \phi -(1+v_{\mathrm{eff}})\sin 2\theta ]}^2+{\left(\frac{\mathrm{\Delta }\gamma }{\mathrm{\Delta }\epsilon }\cos 2\theta \sin \phi \right)}^2}$$ (6)

$$\mathrm{\Delta }{\epsilon }_{\mathrm{n}}=\frac{1}{2}\mathrm{\Delta }\epsilon \sqrt{{[2(1+v_{\mathrm{eff}}){\cos }^2\theta -2v_{\mathrm{eff}}+\frac{\mathrm{\Delta }\gamma }{\mathrm{\Delta }\epsilon }\sin 2\theta \cos \phi ]}^2+{\left(\frac{\mathrm{\Delta }\gamma }{\mathrm{\Delta }\epsilon }\sin 2\theta \sin \phi \right)}^2}$$ (7)

where $\mathrm{\Delta }\epsilon$ is the axial strain range; $\mathrm{\Delta }\gamma$ is the shear strain range and $\phi$ is the phase angle between tensional strain and torsional strain.

Multiaxial fatigue damage model

FS model

Fatemi and Socie ¹¹ established a multiaxial fatigue life model for the shear failure mode, which allows crack initiation and propagation to be considered. The shear strain amplitude and the maximum normal stress combination are used as a damage parameter on the maximum shear strain plane, and the model is mostly used for the multiaxial fatigue life prediction dominated by shear-type failure; the fatigue life can be calculated as

$$\frac{\mathrm{\Delta }{\gamma }_1}{2}(1+k\frac{{\sigma }_{\mathrm{n}}^{\max }}{{\sigma }_{\mathrm{y}}})=\frac{{\tau }_{\mathrm{f}}^{\prime }}{G}(2N_{\mathrm{f}}{)}^{b_0}+{\gamma }_{\mathrm{f}}^{\prime }(2N_{\mathrm{f}}{)}^{c_0}$$ (8)

where $\mathrm{\Delta }{\gamma }_1$ is the maximum shear strain range on the maximum shear strain plane; k is the material constant; ²⁰ ${\sigma }_{\mathrm{n}}^{\max }$ is the maximum normal stress on the maximum shear strain plane; ${\sigma }_{\mathrm{y}}$ is the yield stress; ${\tau }_{\mathrm{f}}^{\prime }$ and ${\gamma }_{\mathrm{f}}^{\prime }$ are the shear fatigue strength coefficient and the shear fatigue ductility coefficient, respectively; b₀ and c₀ are the shear fatigue strength exponent and the shear fatigue ductility exponent, respectively; G is the shear modulus.

SWT model

The SWT ¹² model was originally proposed to consider the effect of the mean stress on the fatigue life under uniaxial loading, and it can only be used for multiaxial fatigue life prediction guided by tensile crack failure. The maximum normal strain plane is defined as the critical plane, and the maximum normal strain amplitude and the maximum normal stress are used as the damage parameters. The fatigue life can be calculated as

$$\frac{\mathrm{\Delta }{\epsilon }_1}{2}{\sigma }_{\mathrm{n},\max }=\frac{{\sigma }_{\mathrm{f}}^{\prime 2}}{E}(2N_{\mathrm{f}}{)}^{2b}+{\sigma }_{\mathrm{f}}^{\prime }{\epsilon }_{\mathrm{f}}^{\prime }(2N_{\mathrm{f}}{)}^{b+c}$$ (9)

where $\mathrm{\Delta }{\epsilon }_1$ is the maximum normal strain on the maximum strain plane; ${\sigma }_{\mathrm{n},\max }$ is the maximum normal stress on the maximum strain plane; ${\sigma }_{\mathrm{f}}^{\prime }$ and ${\epsilon }_{\mathrm{f}}^{\prime }$ are the fatigue strength coefficient and the fatigue ductility coefficient, respectively; b and c are the fatigue strength exponent and the fatigue ductility exponent, respectively; and E is the Young’s modulus.

CXH model

Chen et al. ¹⁴ suggested that the damage parameter should consider two typical failure modes. Accordingly, they proposed a critical plane-energy model for tensile-type failure, in which the damage parameter consists in the summation of normal and shear strain energy calculated either on the critical maximum tensile strain plane or on the critical maximum shear strain plane according to, respectively, the tensile or shear type of multiaxial fatigue failure. So the life prediction model for the tensile-type failure is obtained as

$$\frac{\mathrm{\Delta }{\epsilon }_1^{\max }}{2}\frac{\mathrm{\Delta }{\sigma }_1}{2}+\frac{\mathrm{\Delta }{\tau }_1}{2}\frac{\mathrm{\Delta }{\gamma }_1}{2}=\frac{{\sigma }_{\mathrm{f}}^{\prime 2}}{E}(2N_{\mathrm{f}}{)}^{2b}+{\sigma }_{\mathrm{f}}^{\prime }{\epsilon }_{\mathrm{f}}^{\prime }(2N_{\mathrm{f}}{)}^{b+c}$$ (10)

where $\mathrm{\Delta }{\epsilon }_1^{\max }$ is the maximum tensile strain range; and $\mathrm{\Delta }{\sigma }_1$ , $\mathrm{\Delta }{\tau }_1$ and $\mathrm{\Delta }{\gamma }_1$ are the normal stress, shear stress range, and shear strain range that occurs on the maximum tensile strain plane, respectively.

Meanwhile, the maximum shear strain plane was defined as the critical plane for the shear-type failure. Therefore, the life prediction model is given as

$$\frac{\mathrm{\Delta }{\tau }_1}{2}\frac{\mathrm{\Delta }{\gamma }_{\max }}{2}+\frac{\mathrm{\Delta }{\epsilon }_1}{2}\frac{\mathrm{\Delta }{\sigma }_1}{2}=\frac{{\tau }_{\mathrm{f}}^{\prime 2}}{G}(2N_{\mathrm{f}}{)}^{2b_0}+{\tau }_{\mathrm{f}}^{\prime }{\gamma }_{\mathrm{f}}^{\prime }(2N_{\mathrm{f}}{)}^{b_0+c_0}$$ (11)

where $\mathrm{\Delta }{\gamma }_{\max }$ is the maximum shear strain range; $\mathrm{\Delta }{\tau }_1$ , $\mathrm{\Delta }{\epsilon }_1$ , and $\mathrm{\Delta }{\sigma }_1$ are the shear stress, normal strain range, and normal stress range on the maximum shear strain plane, respectively.

The proposed model

Usually, the selection of different critical planes is based on different fatigue failure modes in the multiaxial fatigue life prediction. The maximum shear strain plane is generally defined as the critical plane for the shear-type failure, such as the FS model. And the maximum normal strain plane is determined as the critical plane for the tensile-type failure, like the SWT model. Although the CXH model considers two typical failure modes, it ignores the effect of mean stress. Li et al.^21,22 believe that the CXH model cannot effectively reflect the difference between the normal and shear strain energy to fatigue damage, so the prediction result tends to be too conservative. The proposed new damage parameters can make up for the deficiency of the CXH model.

Based on the above analysis, for the shear-type failure, the maximum shear strain plane is defined as the critical plane. The main damage parameter consists of the shear stress and the shear strain on the critical plane, and the second control parameter includes the maximum normal stress and normal strain range on the same plane. The fatigue life can be expressed as

$$\frac{\mathrm{\Delta }{\tau }_1}{2}\frac{\mathrm{\Delta }{\gamma }_1}{2}(1+\beta \frac{\sqrt{{\sigma }_{\mathrm{n},\max }·E·\mathrm{\Delta }{\epsilon }_{\mathrm{n}}}}{{\tau }_{\mathrm{f}}^{\prime }})=\frac{{\tau }_{\mathrm{f}}^{\prime 2}}{G}(2N_{\mathrm{f}}{)}^{2b_0}+{\tau }_{\mathrm{f}}^{\prime }{\gamma }_{\mathrm{f}}^{\prime }(2N_{\mathrm{f}}{)}^{b_0+c_0}$$ (12)

where $\mathrm{\Delta }{\tau }_1$ and $\mathrm{\Delta }{\gamma }_1$ are the shear stress and shear strain range on the maximum shear strain plane; ${\sigma }_{\mathrm{n},\max }$ and $\mathrm{\Delta }{\epsilon }_{\mathrm{n}}$ are the maximum stress and normal strain range on the maximum shear strain plane; and $\beta$ is the constant related to uniaxial tensile fatigue of material, which revises uniaxial fatigue data to purely torsional fatigue data level.

The constant $\beta$ usually takes 1 under proportional loading, but the additional hardening phenomenon is serious under non-proportional loading. When the loading path is uniaxial tension compression or proportional loading without mean stress

$$\mathrm{\Delta }{\epsilon }_{\mathrm{n}}=(1-\upsilon )\frac{\mathrm{\Delta }\epsilon }{2}$$ (13)

$${\sigma }_{\mathrm{n},\max }=\frac{{\sigma }_{\mathrm{a}}}{2}$$ (14)

$$\frac{\mathrm{\Delta }\epsilon }{2}=\frac{{\sigma }_{\mathrm{f}}^{\prime }}{E}(2N_{\mathrm{f}}{)}^b+{\epsilon }_{\mathrm{f}}^{\prime }(2N_{\mathrm{f}}{)}^c$$ (15)

Putting equations (13)–(15) into equation (12), $\beta$ can be expressed as

$$\begin{array}{c}\beta =[\frac{\frac{{{\tau }_{\mathrm{f}}^{\prime }}^2}{G}{\left(2N_{\mathrm{f}}\right)}^{2b_0}+{\tau }_{\mathrm{f}}^{\prime }{\gamma }_{\mathrm{f}}^{\prime }{\left(2N_{\mathrm{f}}\right)}^{b_0+c_0}}{\frac{\mathrm{\Delta }{\tau }_1}{2}\frac{\mathrm{\Delta }{\gamma }_1}{2}}-1]\times \frac{{\tau }_{\mathrm{f}}^{\prime }}{\sqrt{(1-{\upsilon }_{\mathrm{e}})\frac{{{\sigma }_{\mathrm{f}}^{\prime }}^2}{2}{(2N_{\mathrm{f}})}^b+(1-{\upsilon }_{\mathrm{p}})\frac{{\sigma }_{\mathrm{f}}^{\prime }E{\epsilon }_{\mathrm{f}}^{\prime }}{2}{(2N_{\mathrm{f}})}^c}}\hfill \end{array}$$ (16)

In terms of the critical plane-damage parameter concept, the maximum normal strain plane is taken as the critical plane for the tensile-type. The combination of the normal strain and the normal stress is determined the main damage parameter on the critical plane, and the maximum shear stress and shear strain range on the same plane are the second control parameters. The fatigue life can be obtained as

$$\frac{\mathrm{\Delta }{\sigma }_1}{2}\frac{\mathrm{\Delta }{\epsilon }_1}{2}(1+{\beta }_0\frac{\sqrt{{\tau }_{\max }·G·\mathrm{\Delta }{\gamma }_{\mathrm{n}}}}{{\sigma }_{\mathrm{f}}^{\prime }})=\frac{{\sigma }_{\mathrm{f}}^{\prime 2}}{E}(2N_{\mathrm{f}}{)}^{2b}+{\sigma }_{\mathrm{f}}^{\prime }{\epsilon }_{\mathrm{f}}^{\prime }(2N_{\mathrm{f}}{)}^{b+c}$$ (17)

where $\mathrm{\Delta }{\sigma }_1$ and $\mathrm{\Delta }{\epsilon }_1$ are the normal stress and normal strain range on the maximum normal strain plane; ${\tau }_{\max }$ and $\mathrm{\Delta }{\gamma }_{\mathrm{n}}$ are the maximum shear stress and shear strain range on the maximum normal strain plane; and ${\beta }_0$ is the constant related to uniaxial torsion fatigue of material. It can be referenced as $\beta$ .

When the loading path is uniaxial torsion or proportional loading without mean stress

$$\frac{\mathrm{\Delta }{\gamma }_{\mathrm{n}}}{2}=(1+{\upsilon }_{\mathrm{e}})\frac{{\tau }_{\mathrm{f}}^{\prime }}{G}(2N_{\mathrm{f}}{)}^{b_0}+(1+{\upsilon }_{\mathrm{p}}){\gamma }_{\mathrm{f}}^{\prime }(2N_{\mathrm{f}}{)}^{c_0}$$ (18)

$${\tau }_{\max }=\frac{{\tau }_{\mathrm{a}}}{2}$$ (19)

Putting equations (18) and (19) into equation (17), ${\beta }_0$ can be expressed as

$${\beta }_0=[\frac{\frac{{\sigma }_{\mathrm{f}}^{\prime 2}}{E}{(2N_{\mathrm{f}})}^{2b}+{\sigma }_{\mathrm{f}}^{\prime }{\epsilon }_{\mathrm{f}}^{\prime }{(2N_{\mathrm{f}})}^{b+c}}{\frac{\mathrm{\Delta }{\sigma }_1}{2}\frac{\mathrm{\Delta }{\epsilon }_1}{2}}-1]·\frac{{\sigma }_{\mathrm{f}}^{\prime }}{\sqrt{(1+{\upsilon }_{\mathrm{e}})\frac{{{\tau }_{\mathrm{f}}^{\prime }}^2}{2}{(2N_{\mathrm{f}})}^{b_0}+(1+{\upsilon }_{\mathrm{p}})\frac{{\tau }_{\mathrm{f}}^{\prime }G{\gamma }_{\mathrm{f}}^{\prime }}{2}{(2N_{\mathrm{f}})}^{c_0}}}$$ (20)

Experimental verifications

The experimental data of four materials are used to evaluate the validity of the model and the other three models. These four materials are 16MnR, Pure Ti, Ti alloy BT9, and AISI 304. All fatigue tests are multiaxial loading tests of sine waves under strain control, and the test specimens are all thin-walled tubular specimens. The loading paths and types come from references.^8,23,24Figure 1 shows the five kinds of loading paths used in this article. The material constants used are summarized in Table 1. In addition, the corresponding optimal prediction criteria are listed in Table 2. Table 3 shows the material constants of the FS model and the proposed model.

Figure 1.

Loading paths.

Table 1.

Fatigue properties of materials.

Materials	E/GPa	G/GPa	${\sigma }_{\mathrm{y}}$ /MPa	${\sigma }_{\mathrm{f}}^{\prime }$ /MPa	${\epsilon }_{\mathrm{f}}^{\prime }$	${\tau }_{\mathrm{f}}^{\prime }$ /MPa	${\gamma }_{\mathrm{f}}^{\prime }$	b	c	b 0	c 0	Loadingpaths
16MnR 23	212.5	81.1	351.4	966.4	0.842	345	0.712	−0.101	−0.618	−0.057	−0.512	a, c
Pure Ti 24	112	40	475	647	0.548	485	0.417	−0.033	−0.646	−0.069	−0.523	a, b, c
Ti alloyBT9 24	118	43	910	1180	0.278	881	0.180	−0.025	−0.665	−0.082	−0.470	a, b, c
AISI 304 8	183	82.8	325	1000	0.171	709	0.413	−0.114	−0.402	−0.121	−0.353	a, c, d, e

Table 2.

Failure modes of materials, the optimal prediction criteria CXH model, and the proposed model.

	16MnR	Pure Ti	Ti alloy BT9	AISI 304
Main failure modes	Mixed	Shear	Shear	Tensile
CXH	Shear	Shear	Shear	Tensile
The proposed	Shear	Shear	Shear	Tensile

CXH: Chen-Xu-Huang.

Table 3.

The material constants k, $\beta$ , and ${\beta }_0$ .

	16MnR	Pure Ti	Ti alloy BT9	AISI 304
K	1.1	0.8	0.8	2
$\beta$	1	0.1	0.1
${\beta }_0$				2.5

The comparative results between the experimental life and the predicted fatigue life for different models are depicted in Figure 2. The experimental verifications above show that the proposed multiaxial fatigue model, FS model, and CXH model give acceptable results for the four kinds of materials. The three models are suitable for life prediction, but most of the SWT model prediction results are outside the factor of 2, even if the tensile-type failure material AISI 304 has a poor prediction result. The SWT model is not good for prediction of multiaxial loading. Figure 3 shows the probability density function (PDF) of prediction errors prediction ability of the four models. It can obtain the new model having smaller mean prediction errors, so the proposed model has stable prediction accuracy. The CXH model considers two different forms of fatigue failure, so the prediction result is better than the FS model. However, the FS model prediction requires a large amount of uniaxial fatigue test data and the calculation is too complex, which is not conducive to engineering applicability. The prediction standard deviation of CXH is small, but the prediction error is large. In sum, the prediction error accuracy of the proposed model is high, and the model comprehensively considers additional hardening phenomenon and the failure mode of the materials.

Figure 2.

Comparison between the predicted life and the experimental life. (a) The proposed model. (b) FS model. (c) SWT model. (d) CXH model.

Figure 3.

Probability density function (PDF) of prediction errors.

Note: $\mathrm{Error}=\lg N_{\mathrm{e}}-\lg N_{\mathrm{p}}$ , ²⁵ $N_{\mathrm{e}}$ and $N_{\mathrm{p}}$ are experimental data and predicted data, respectively.

Conclusion

This proposes a multiaxial fatigue life model for low-cycle fatigue life prediction based on the critical plane-damage parameter concept. The proposed model not only improves the prediction accuracy of the classic model, but also considers the effects of non-proportional additional hardening phenomena and fatigue failure modes. The model is proved to be effective in low-cycle fatigue life prediction under different loading paths and types for different materials. Compared with the other three models, the proposed model has higher life prediction accuracy and good engineering applicability.

Author biographies

Jianhui Liu is an Associate Professor at LanZhou University of Technology. His main research direction is mechanical strength theory.

Xin Lv is a Master’s student from LanZhou University of Technology. He is mainly engaged in metal multiaxial fatigue damage research.

Yaobing Wei is a Professor at LanZhou University of Technology. His main research direction is mechanical strength theory.

Xuemei Pan is a Master’s student from LanZhou University of Technology. She is mainly engaged in metal multiaxial fatigue damage research.

Yifan Jin is a Master’s student from LanZhou University of Technology. He is mainly engaged in composite material multiaxial fatigue damage research.

Youliang Wang is an Associate Professor at LanZhou University of Technology. His main research direction is precision machining.