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eISSN: 2574-9927

Material Science & Engineering International Journal

Review Article Volume 5 Issue 2

A mean stress model of fatigue life of metal materials under multiaxial loading

Xiangqiao Yan

Center for composite materials and structure, Harbin institute of technology, China

Correspondence: Xiangqiao Yan, Center for composite materials and structure, Harbin institute of technology, 150080, China

Received: July 15, 2020 | Published: April 26, 2021

Citation: Yan X. A mean stress model of fatigue life of metal materials under multiaxial loading. Material Sci & Eng. 2021;5(2):60-69. DOI: 10.15406/mseij.2021.05.00157

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Abstract

Due to the complexity of multiaxial fatigue damage of metal materials, up to date, it is still a challenging task to establish a multiaxial fatigue model with influence of different mean stress. In this paper, a linear mean stress model is presented on basis of the multiaxial model of fatigue life of metal materials by Liu and Yan. By using the experimental data of fatigue life of metal materials reported in the literature, the model is systematically validated

Keywords: multiaxial fatigue, mean stress, metallic materials

Introduction

Many critical mechanical components experience multiaxial cyclic loading during their service life, such as railroad wheels, crankshafts, axles, and turbine blades, etc. Different from the uniaxial fatigue problem, the multiaxial fatigue problem is more complex due to the complex stress states. In recent decades, a significant amount of research has been devoted to acquire a better understanding of the failure mechanisms under multiaxial loading, including theoretical model study (e.g., a stress invariant method,1,5–10 and a critical plane method,11–13 fatigue tests of metallic materials under multiaxial loading (e.g., fatigue test for engineering steels by Gough3 and fatigue of wrought high-tensile alloy steel by Frith.4 Due to the complexity of multiaxial fatigue damage of metal materials, up to date, it is still a challenging task to establish a multiaxial fatigue model with influence of different mean stress. In this paper, a linear mean stress model is presented on basis of the multiaxial model of fatigue life of metal materials by Liu & Yan1,2 By using the experimental data of fatigue life of metal materials reported in the literature, the model is systematically validated.

A multiaxial mean stress model

In this section, the Marin’s mean stress model proposed by Liu and Yan1 is first described briefly. Then a linear mean stress model is given. After the reader reads this paper, the reason that the linear mean stress model is presented by the author is naturally understood.

A Marin’s mean stress model

A model for fatigue life prediction under multiaxial stress states can be expressed mathematically as follows:

F( σ ea , σ em ,ρ,N, c 1 , c 2 ,...)=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaamOrai aacIcacqaHdpWCdaWgaaqaaKqzadGaamyzaiaadggaaKqbagqaaiaa cYcacqaHdpWCdaWgaaqaaKqzadGaamyzaiaad2gaaKqbagqaaiaacY cacqaHbpGCcaGGSaGaamOtaiaacYcacaWGJbWaaSbaaeaajugWaiaa igdaaKqbagqaaiaacYcacaWGJbWaaSbaaeaajugWaiaaikdaaKqbag qaaiaacYcacaGGUaGaaiOlaiaac6cacaGGPaGaeyypa0JaaGimaaaa @551E@ (1)

where σ e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHdpWCdaWgaa WcbaGaamyzaaqabaaaaa@385C@ is a mechanical parameter which is a measure of stress states in multiaxial loading, and here, the von Mises equivalent stress is adopted ; σ ea MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aaSbaaeaajugWaiaadwgacaWGHbaajuaGbeaaaaa@3BF4@  and σ em MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaqcfaOaeq4Wdm 3aaSbaaeaajugWaiaadwgacaWGTbaajuaGbeaaaaa@3C00@ are the amplitude and the mean value of the equivalent stress, respectively. Multiaxial parameter ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHbpGCaa a@37D1@ is defined as follows:

ρ= σ 11,a σ e,a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHbpGCcq GH9aqpdaWcaaqaaiabeo8aZnaaBaaabaqcLbmacaaIXaGaaGymaiaa cYcacaWGHbaajuaGbeaaaeaacqaHdpWCdaWgaaqaaKqzadGaamyzai aacYcacaWGHbaajuaGbeaaaaaaaa@45B3@  (2)

where σ 11,a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaaGymaiaaigdacaGGSaGaamyyaaqcfayabaaaaa@3CBD@  is the amplitude of the first invariant of stress tensor. It is evident that, for the axial and shear fatigue loading, the value of the multiaxial parameter ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHbpGCaa a@37D1@  is equal to 1 and 0, respectively c 1 , c 2 ,... MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGJbWcda WgaaqcfayaaKqzadGaaGymaaqcfayabaGaaiilaiaadogadaWgaaqa aKqzadGaaGOmaaqcfayabaGaaiilaiaac6cacaGGUaGaaiOlaaaa@4121@ . in the Eq. (1) are material coefficients, which are varied with the multiaxial parameter ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHbpGCaa a@37D1@ . Based on the previous wide investigations (e.g., Tao and Xia14 and Marin’s general equation15) of engineering rules suitable for predicting the mean stress effect under the bending loading, it is assumed here that the model of multiaxial fatigue life has the following form:

log σ ea (1 α ρ σ em σ ea ) n ρ = A ρ logN+ C ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgacqaHdpWCdaWgaaqaaKqzadGaamyzaiaadggaaKqbagqa aiaacIcacaaIXaGaeyOeI0IaeqySde2aaSbaaeaajugWaiabeg8aYb qcfayabaWaaSaaaeaacqaHdpWCdaWgaaqaaKqzadGaamyzaiaad2ga aKqbagqaaaqaaiabeo8aZnaaBaaabaqcLbmacaWGLbGaamyyaaqcfa yabaaaaiaacMcadaahaaqabeaacaWGUbWaaSbaaeaajugWaiabeg8a Ybqcfayabaaaaiabg2da9iaadgeadaWgaaqaaKqzadGaeqyWdihaju aGbeaaciGGSbGaai4BaiaacEgacaWGobGaey4kaSIaam4qamaaBaaa baqcLbmacqaHbpGCaKqbagqaaaaa@648F@ (3)

where A ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGbbWaaS baaeaajugWaiabeg8aYbqcfayabaaaaa@3A74@ C ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGdbWaaS baaeaajugWaiabeg8aYbqcfayabaaaaa@3A76@ α ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqyda WgaaqaaKqzadGaeqyWdihajuaGbeaaaaa@3B4D@ n ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbWaaS baaeaajugWaiabeg8aYbqcfayabaaaaa@3AA1@ are material parameters dependent on the multiaxial parameter. For the axial and shear fatigue loading, Eq. (3) can be simplified as, respectively:

log σ a (1 α 1 σ m σ a ) n 1 = A 1 logN+ C 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgacqaHdpWCdaWgaaqaaKqzadGaamyyaaqcfayabaGaaiik aiaaigdacqGHsislcqaHXoqylmaaBaaajuaGbaqcLbmacaaIXaaaju aGbeaadaWcaaqaaiabeo8aZnaaBaaabaqcLbmacaWGTbaajuaGbeaa aeaacqaHdpWCdaWgaaqaaKqzadGaamyyaaqcfayabaaaaiaacMcada ahaaqabeaacaWGUbWaaSbaaeaajugWaiaaigdaaKqbagqaaaaacqGH 9aqpcaWGbbWaaSbaaeaajugWaiaaigdaaKqbagqaaiGacYgacaGGVb Gaai4zaiaad6eacqGHRaWkcaWGdbWaaSbaaeaajugWaiaaigdaaKqb agqaaaaa@5E56@  (4)

And

log 3 τ a (1 α 0 τ m τ a ) n 0 = A 0 logN+ C 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgadaGcaaqaaiaaiodaaeqaaiabes8a0naaBaaabaqcLbma caWGHbaajuaGbeaacaGGOaGaaGymaiabgkHiTiabeg7aHnaaBaaaba qcLbmacaaIWaaajuaGbeaadaWcaaqaaiabes8a0naaBaaabaqcLbma caWGTbaajuaGbeaaaeaacqaHepaDdaWgaaqaaKqzadGaamyyaaqcfa yabaaaaiaacMcadaahaaqabeaacaWGUbWaaSbaaeaajugWaiaaicda aKqbagqaaaaacqGH9aqpcaWGbbWaaSbaaeaajugWaiaaicdaaKqbag qaaiGacYgacaGGVbGaai4zaiaad6eacqGHRaWkcaWGdbWaaSbaaeaa jugWaiaaicdaaKqbagqaaaaa@5E8C@  (5)

In the absence of mean stress, the Eq. (3) is simplified as:

log σ ea = A ρ logN+ C ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgacqaHdpWCdaWgaaqaaKqzadGaamyzaiaadggaaKqbagqa aiabg2da9iaadgeadaWgaaqaaKqzadGaeqyWdihajuaGbeaaciGGSb Gaai4BaiaacEgacaWGobGaey4kaSIaam4qamaaBaaabaqcLbmacqaH bpGCaKqbagqaaaaa@4CA4@  (6)

Thus under the axial and shear fatigue loading, the Eqs (4) and (5) can be written as respectively:

log σ a = A 1 logN+ C 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgacqaHdpWCdaWgaaqaaKqzadGaamyyaaqcfayabaGaeyyp a0JaamyqamaaBaaabaqcLbmacaaIXaaajuaGbeaaciGGSbGaai4Bai aacEgacaWGobGaey4kaSIaam4qaSWaaSbaaKqbagaajugWaiaaigda aKqbagqaaaaa@4A49@  (7)

And

log 3 τ a = A 0 logN+ C 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgadaGcaaqaaiaaiodaaeqaaiabes8a0TWaaSbaaKqbagaa jugWaiaadggaaKqbagqaaiabg2da9iaadgeadaWgaaqaaKqzadGaaG imaaqcfayabaGaciiBaiaac+gacaGGNbGaamOtaiabgUcaRiaadoea daWgaaqaaKqzadGaaGimaaqcfayabaaaaa@4B16@  (8)

Eq. (7) is the well known S-N curve equation. Eq. (8) is the variant form of the S-N curve equation under the shear fatigue condition, which can be written as:

log τ a = A 0 ' logN+ C 0 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgacqaHepaDdaWgaaqaaKqzadGaamyyaaqcfayabaGaeyyp a0JaamyqamaaDaaabaqcLbmacaaIWaaajuaGbaGaai4jaaaaciGGSb Gaai4BaiaacEgacaWGobGaey4kaSIaam4qamaaDaaabaqcLbmacaaI WaaajuaGbaGaai4jaaaaaaa@4B08@  (9)

in which A 0 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGbbWaa0 baaeaajugWaiaaicdaaKqbagaacaGGNaaaaaaa@3A1A@  and C 0 ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGdbWaa0 baaeaajugWaiaaicdaaKqbagaacaGGNaaaaaaa@3A1C@ are related with A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGbbWaaS baaeaajugWaiaaicdaaKqbagqaaaaa@396E@  and C 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGdbWaaS baaeaajugWaiaaicdaaKqbagqaaaaa@3970@  through the following relationships:

A 0 ' = A 0 ,  C 0 ' = C 0 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGbbWaa0 baaeaajugWaiaaicdaaKqbagaacaGGNaaaaiabg2da9iaadgeadaWg aaqaaKqzadGaaGimaaqcfayabaGaaiilaabaaaaaaaaapeGaaiiOa8 aacaWGdbWcdaqhaaqcfayaaKqzadGaaGimaaqcfayaaKqzadGaai4j aaaajuaGcqGH9aqpcaWGdbWcdaWgaaqcfayaaKqzadGaaGimaaqcfa yabaGaeyOeI0YaaOaaaeaacaaIZaaabeaaaaa@4D98@ (10)

In view of the complexity of fatigue life analysis in the multiaxial stress states, and also taking into account that the existing literature has accumulated a large number of fatigue experimental data under the axial and shear loading, from the point of application, it is assumed that the material parameters in Eq. (3) can be obtained by interpolating the material parameters in Eqs (4) and (5), i.e.:

A ρ = A 1 ρ+ A 0 (1ρ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGbbWaaS baaeaajugWaiabeg8aYbqcfayabaGaeyypa0JaamyqaSWaaSbaaKqb agaajugWaiaaigdaaKqbagqaaKqzadGaeyyXICDcfaOaeqyWdiNaey 4kaSIaamyqamaaBaaabaqcLbmacaaIWaaajuaGbeaacqGHflY1caGG OaGaaGymaiabgkHiTiabeg8aYjaacMcaaaa@5081@  (11)

C ρ = C 1 ρ+ C 0 (1ρ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGdbWcda WgaaqcfayaaKqzadGaeqyWdihajuaGbeaacqGH9aqpcaWGdbWcdaWg aaqcfayaaKqzadGaaGymaaqcfayabaGaeyyXICTaeqyWdiNaey4kaS Iaam4qaSWaaSbaaKqbagaajugWaiaaicdaaKqbagqaaiabgwSixlaa cIcacaaIXaGaeyOeI0IaeqyWdiNaaiykaaaa@4FFD@  (12)

α ρ = α 1 ρ+ α 0 (1ρ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqyda WgaaqaaKqzadGaeqyWdihajuaGbeaacqGH9aqpcqaHXoqylmaaBaaa juaGbaqcLbmacaaIXaaajuaGbeaacqGHflY1cqaHbpGCcqGHRaWkcq aHXoqylmaaBaaajuaGbaqcLbmacaaIWaaajuaGbeaacqGHflY1caGG OaGaaGymaiabgkHiTiabeg8aYjaacMcaaaa@51E9@  (13)

n ρ = n 1 ρ+ n 0 (1ρ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbWaaS baaeaajugWaiabeg8aYbqcfayabaGaeyypa0JaamOBaSWaaSbaaKqb agaajugWaiaaigdaaKqbagqaaiabgwSixlabeg8aYjabgUcaRiaad6 gadaWgaaqaaKqzadGaaGimaaqcfayabaGaeyyXICTaaiikaiaaigda cqGHsislcqaHbpGCcaGGPaaaaa@4F4C@  (14)

In this way, multiaxial fatigue life can be directly estimated from the stress invariant parameter and the multiaxial S–N curve with the attention on mean stress effect.

A linear mean stress model

According to the Marin’s mean stress model of multiaxial loading, i.e., Eq.(3), a linear mean stress model presented in this paper is as follows:

log σ ea (β ρ γ ρ σ em σ e0 )= A ρ logN+ C ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgacqaHdpWClmaaBaaajuaGbaqcLbmacaWGLbGaamyyaaqc fayabaGaaiikaiabek7aInaaBeaabaqcLbmacqaHbpGCaKqbagqaai abgkHiTiabeo7aNnaaBaaabaqcLbmacqaHbpGCaKqbagqaamaalaaa baGaeq4Wdm3aaSbaaeaajugWaiaadwgacaWGTbaajuaGbeaaaeaacq aHdpWCdaWgaaqaaKqzadGaamyzaiaaicdaaKqbagqaaaaacaGGPaGa eyypa0JaamyqamaaBaaabaqcLbmacqaHbpGCaKqbagqaaiGacYgaca GGVbGaai4zaiaad6eacqGHRaWkcaWGdbWaaSbaaeaajugWaiabeg8a Ybqcfayabaaaaa@64D6@  (15)

where

σ eo = σ o ρ ( 3 τ o ) (1ρ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaamyzaiaad+gaaKqbagqaaiabg2da9iabeo8aZTWa aSbaaKqbagaajugWaiaad+gaaKqbagqaaSWaaWbaaKqbagqabaqcLb macqaHbpGCaaqcfaOaaiikamaakaaabaGaaG4maaqabaGaeqiXdq3a aSbaaeaajugWaiaad+gaaKqbagqaaiaacMcalmaaCaaajuaGbeqaaK qzadGaaiikaiaaigdacqGHsislcqaHbpGCcaGGPaaaaaaa@535F@  (16)

in which σ o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGVbaajuaGbeaaaaa@3B3E@  and τ o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDda WgaaqaaKqzadGaam4Baaqcfayabaaaaa@3AA7@  are, respectively, a tensile fatigue limit and a shear fatigue limit. Under the axial and shear fatigue loading, Eq (15) can be written as

log σ a (β 1 γ 1 σ m σ 0 )= A 1 logN+ C 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgacqaHdpWCdaWgaaqaaKqzadGaamyyaaqcfayabaGaaiik aiabek7aITWaaSraaKqbagaajugWaiaaigdaaKqbagqaaiabgkHiTi abeo7aNnaaBaaabaqcLbmacaaIXaaajuaGbeaadaWcaaqaaiabeo8a ZnaaBaaabaqcLbmacaWGTbaajuaGbeaaaeaacqaHdpWCdaWgaaqaaK qzadGaaGimaaqcfayabaaaaiaacMcacqGH9aqpcaWGbbWaaSbaaeaa jugWaiaaigdaaKqbagqaaiGacYgacaGGVbGaai4zaiaad6eacqGHRa WkcaWGdbWcdaWgaaqcfayaaKqzadGaaGymaaqcfayabaaaaa@5E9D@  (17)

and

log 3 τ a (β 0 γ 0 τ m τ 0 )= A 0 logN+ C 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgadaGcaaqaaiaaiodaaeqaaiabes8a0TWaaSbaaKqbagaa jugWaiaadggaaKqbagqaaiaacIcacqaHYoGydaWgbaqaaKqzadGaaG imaaqcfayabaGaeyOeI0Iaeq4SdC2aaSbaaeaajugWaiaaicdaaKqb agqaamaalaaabaGaeqiXdq3aaSbaaeaajugWaiaad2gaaKqbagqaaa qaaiabes8a0naaBaaabaqcLbmacaaIWaaajuaGbeaaaaGaaiykaiab g2da9iaadgeadaWgaaqaaKqzadGaaGimaaqcfayabaGaciiBaiaac+ gacaGGNbGaamOtaiabgUcaRiaadoeadaWgaaqaaKqzadGaaGimaaqc fayabaaaaa@5ED3@ (18)

in which β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHYoGylm aaBeaajuaGbaqcLbmacaaIXaaajuaGbeaaaaa@3AE4@ , γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZoWzda WgaaqaaKqzadGaaGymaaqcfayabaaaaa@3A50@ , β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHYoGyda WgbaqaaKqzadGaaGimaaqcfayabaaaaa@3A4A@  and γ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZoWzda WgaaqaaKqzadGaaGimaaqcfayabaaaaa@3A4F@  are material constants determined by fitting experimental data of fatigue life.

By the way, material parameters and in Eq (15) can be determined by using the interpolation formulas similar to formulas (13) to (14). Eq (16) is a multiaxial fatigue limit prediction equation recently proposed by Liu & Yan2

Experimental verifications and discussions

Based on the experimental data of fatigue life of metallic materials from literature, in this section, fatigue life analysis will be carried out by using fatigue life prediction equations with the linear mean stress effect and the Marin’s mean stress effect. The comparison of two results will be given. In order to quantitatively evaluate the accuracy of the fatigue life prediction, the following error indexes are defined:

ER1= ( N cal N exp )×100 N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGfbGaam OuaiaaigdacqGH9aqpdaWcaaqaaiaacIcacaWGobWaaSbaaeaajugW aiaadogacaWGHbGaamiBaaqcfayabaGaeyOeI0IaamOtamaaBaaaba qcLbmaciGGLbGaaiiEaiaacchaaKqbagqaaiaacMcacqGHxdaTcaaI XaGaaGimaiaaicdaaeaacaWGobWaaSbaaeaajugWaiGacwgacaGG4b GaaiiCaaqcfayabaaaaaaa@5094@  (19)

ER2= ( N cal ' N exp )×100 N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGfbGaam OuaiaaikdacqGH9aqpdaWcaaqaaiaacIcacaWGobWaa0baaeaajugW aiaadogacaWGHbGaamiBaaqcfayaaiaacEcaaaGaeyOeI0IaamOtam aaBaaabaqcLbmaciGGLbGaaiiEaiaacchaaKqbagqaaiaacMcacqGH xdaTcaaIXaGaaGimaiaaicdaaeaacaWGobWaaSbaaeaajugWaiGacw gacaGG4bGaaiiCaaqcfayabaaaaaaa@5141@ (20)

MER1= 1 n i=1 n (ER1) i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGnbGaam yraiaadkfacaaIXaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOBaaaa daaeWbqaaiaacIcacaWGfbGaamOuaiaaigdacaGGPaWaaSbaaeaaju gWaiaadMgaaKqbagqaaaqaaKqzadGaamyAaiabg2da9iaaigdaaKqb agaajugWaiaad6gaaKqbakabggHiLdaaaa@4BC7@  (21)

MER2= 1 n i=1 n (ER2) i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGnbGaam yraiaadkfacaaIYaGaeyypa0ZaaSaaaeaacaaIXaaabaGaamOBaaaa daaeWbqaaiaacIcacaWGfbGaamOuaiaaikdacaGGPaWaaSbaaeaaju gWaiaadMgaaKqbagqaaaqaaKqzadGaamyAaiabg2da9iaaigdaaKqb agaajugWaiaad6gaaKqbakabggHiLdaaaa@4BC9@  (22)

where n is the number of experimental cases, N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGobWaaS baaeaajugWaiGacwgacaGG4bGaaiiCaaqcfayabaaaaa@3B9C@ is experimental fatigue life; N cal MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGobWaaS baaeaajugWaiaadogacaWGHbGaamiBaaqcfayabaaaaa@3B80@ and N cal ' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGobWaa0 baaeaajugWaiaadogacaWGHbGaamiBaaqcfayaaiaacEcaaaaaaa@3C2C@ are computed fatigue lives by the Marin’s mean stress model and the linear mean stress model, respectively. In this section, the fatigue life prediction of three metallic materials, Cast iron, 18G2A steel, and S355J0 alloy steel under uniaxial and multiaxial loading are carried out. For the sake of clear discussions, they are described in the forms of examples, respectively. The material constants determined by using the experimental data of fatigue life of these materials are listed in Table 1 & Table 2. Note here material constants, and, can be evaluated respectively by means of Eqs (7) and (8). For example, for 18G2A steel, let N f MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGobWaaS baaeaajugWaiaadAgaaKqbagqaaaaa@39AC@ =2000000 (cycles), the following two values can be obtained.

σ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaaGimaaqcfayabaaaaa@3A6B@ =286 (MPa), τ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDda WgaaqaaKqzadGaaGimaaqcfayabaaaaa@3A6D@ =188 (MPa)

Materials

A 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGbbWaaS baaeaajugWaiaaigdaaKqbagqaaaaa@396F@   C 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGdbWcda WgaaqcfayaaKqzadGaaGymaaqcfayabaaaaa@3A0A@  

R-square

α 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqylm aaBaaajuaGbaqcLbmacaaIXaaajuaGbeaaaaa@3AE1@   n 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbWaaS baaeaajugWaiaaigdaaKqbagqaaaaa@399C@   β 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHYoGylmaaBa aabaqcLbmacaaIXaaaleqaaaaa@3944@   γ 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZoWzlm aaBaaajuaGbaqcLbmacaaIXaaajuaGbeaaaaa@3AE9@  

Cast iron

-0.140

2.858

0.840

-0.499

1.210

1.663

-0.039

18G2A steel

-0.131

3.286

0.932

-12.13

0.139

1.161

-0.279

S355J0 alloy steel

-0.177

3.535

0.893

-4.316

0.215

1.059

-0.343

Table 1 Material constants and R-square of fitted S-N equation under the axial fatigue loading

Materials

A 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGbbWcda WgaaqcfayaaKqzadGaaGimaaqcfayabaaaaa@3A07@   C 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGdbWcda WgaaqcfayaaKqzadGaaGimaaqcfayabaaaaa@3A09@  

R-square

α 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqylm aaBaaajuaGbaqcLbmacaaIWaaajuaGbeaaaaa@3AE0@   n 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbWaaS baaeaajugWaiaaigdaaKqbagqaaaaa@399C@   β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHYoGylm aaBaaajuaGbaqcLbmacaaIWaaajuaGbeaaaaa@3AE2@   γ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZoWzlm aaBaaajuaGbaqcLbmacaaIWaaajuaGbeaaaaa@3AE8@  

Cast iron

-0.074

2.887

0.965

-0.604

0.700

1.357

-0.044

18G2A steel

-0.074

2.984

0.884

-27.233

0.133

1.223

-0.379

S355J0 alloy steel

-0.076

2.994

0.670

-138.02

0.079

1.290

-0.251

Table 2 Material constants and R-square of fitted S-N equation under the shear fatigue loading

Example 1: Cast iron

For cast iron reported by Berto, Lazzarin & Tovo16 under zero mean stress, predicted results of fatigue life are given in Table 3. Error ranges of fatigue life are, respectively, [-47, 80](%), [-41, 66](%) and [-43,28](%), for the axial, shear and multiaxial fatigue loading, which shows that the fatigue lives predicted by means of Eqs (6), (7) and (8) are very satisfactory. Note here that there is an early failure case with a relative error 746%, see Table 3, which should be deleted. Under the axial fatigue loading with mean stress , the error ranges of fatigue life,  and,are,respectively, [-28,44] (%) and [-29,45] (%), see Table 4, which illustrates that fatigue lives predicted by means of the linear mean stress model,i.e., Eq (17),almost are same as those by means of the Marin’s mean stress model, i.e., Eq (4). Under the shear fatigue loading with mean stress, the error ranges of fatigue life,  and,are, respectively, [-34,30] (%) and [-34,23] (%), see Table 4, from which it can be seen that accuracy of fatigue lives predicted by using the two mean stress models is very high, and that the fatigue lives predicted by the linear mean stress model, i.e., Eq. (18), are a little better than those by the Marin’s mean stress model, i.e., Eq. (5). Under the multiaxial fatigue loading with mean stress, the error ranges of fatigue life,  and, are, respectively, [-72, 30] (%) and [-62, -16] (%), see Table 4, which shows that that accuracy of fatigue lives predicted by using the two mean stress models is satisfactory (Table 3).

σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B30@   (MPa)

σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGTbaajuaGbeaaaaa@3B3C@  (MPa)

τ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDlm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B32@  (MPa)

τ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDda WgaaqaaKqzadGaamyBaaqcfayabaaaaa@3AA5@  (MPa)

N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGobWaaS baaeaajugWaiGacwgacaGG4bGaaiiCaaqcfayabaaaaa@3B9C@  (Cycles)

ER(%)

160

160

130

130

108.5

108.5

0

 

0

 

0

 

0

 

0

 

0

 

0

 

0

 

0

 

140

130

120

110

100

100

90

90

90

80

80

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 0

0

 

0

 

0

 

0

 

0

 

0

 

220

200

200

180

180

180

160

140

140

140

130

120

110

100

100

90

90

90

80

80

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

 0

40,194

86,632

159,000

193,903

408,667

1,091,520

10,000

38,500

45,750

338,500

350,000

285,000

980,000

3,698,000

5,055,500

16,400

28,500

46,800

103,000

229,000

297,500

68,500

520,000

602,000

1,361,500

1,998,000

15

-47

28

5

80

-32

36

28

8

-39

-41

-28

3

66

21

-43

-34

-16

-14

-5

-27

746

12

-4

28

-13

Table 3 Comparison of experimental data and predicted results of fatigue life of Cast iron with zero mean stress

σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B30@  (MPa)

σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGTbaajuaGbeaaaaa@3B3C@  (MPa)

τ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDlm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B32@  (MPa)

τ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaWgaa WcbaGaamyBaaqabaaaaa@3866@  (MPa)

N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGobWaaS baaeaajugWaiGacwgacaGG4bGaaiiCaaqcfayabaaaaa@3B9C@  (Cycles)

ER1 (%)

ER2 (%)

100

80

70

60

0

 

0

 

0

 

0

 

0

 

0

 

0

 

80

90

60

70

70

100

60

100

80

70

60

0

 

0

 

0

 

0

 

0

 

0

 

0

 

80

90

60

70

70

100

60

0

 

0

 

0

 

0

 

150

140

140

130

120

110

110

80

90

60

70

70

100

60

0

 

0

 

0

 

0

 

150

140

140

130

120

110

110

80

90

60

70

70

100

60

46,121

136,081

709,456

1,389,28

21,000

61,890

73,000

286,830

597,350

1,663,00

1,720,00

43,525

20,535

1,410,00

180,000

173,000

9000

1,040,00

-13

44

-28

10

30

12

-5

-34

-6

9

5

-5

-54

-4

18

23

-72

30

-10

45

-29

7

23

9

-7

-34

-4

15

11

-16

-41

-62

-29

-26

-49

-48

Table 4 Comparison of experimental data and predicted results of fatigue life of Cast iron with non-zero mean stress

Example 2: 18G2A steel

For 18G2A steel reported by Gasiak & Pawliczek17, under zero mean stress, predicted results of fatigue life are given in Table 5. Error ranges of fatigue life are, respectively, [-28, 105] (%), [-34, 81] (%) and [-54,-13] (%), for the axial, shear and multiaxial fatigue loading, which shows that the fatigue lives predicted by means of Eqs (6), (7) and (8) are very satisfactory. Under the axial fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [-45,92] (%) and [-40,92] (%), see Table 6, which illustrates that fatigue lives predicted by the linear mean stress model, i.e., Eq (17) almost are same as those by means of the Marin’s mean stress model, i.e., Eq (4). Under the shear fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [-46,53](%) and [-54,46] (%), see Table 6, from which it can be seen that accuracy of fatigue lives predicted by using the Marin’s mean stress model, i.e., Eq (5) almost is same as that by means of the linear mean stress model, i.e., Eq (18). Under the multiaxial fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [-49,172](%) and[-42,170](%), see Table 7, which shows that the fatigue lives predicted by using the linear mean stress model, i.e., Eq (15), are a little better than those by the Marin’s mean stress model, i.e., Eq (3) (Table 4 -Table 7).

σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B30@  (MPa)

σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGTbaajuaGbeaaaaa@3B3C@  (MPa)

τ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDlm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B32@  (MPa)

τ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaWgaa WcbaGaamyBaaqabaaaaa@3866@  (MPa)

N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGobWaaS baaeaajugWaiGacwgacaGG4bGaaiiCaaqcfayabaaaaa@3B9C@  

ER(%)

399.6

399.8

399.8

364.1

365.3

367

275

275.4

275.5

0

0

0

0

0

0

0

0

0

199.7

199.7

199.5

180.2

180.2

180

164.5

164.6

164.4

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

223

222.6

222.7

195.7

195.4

196.2

183.9

184

183

199.7

199.7

199.5

180.2

180.2

180

164.5

164.6

164.4

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

128630

153881

168309

327016

375852

415067

1306495

2869140

3364694

224794

278939

310708

637980

785106

874370

2296066

3095310

4173870

168497

192588

226543

609639

615487

737954

1303741

1407201

2101434

22

2

-7

-3

-17

-28

105

-8

-21

-10

-26

-34

81

50

27

15

-15

-32

-13

-24

-35

-35

-36

-46

-27

-32

-54

Table 5 Comparison of experimental data and predicted results of fatigue life of 18G2A steel with zero mean stress

σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B30@  (MPa)

σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGTbaajuaGbeaaaaa@3B3C@  (MPa)

τ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDlm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B32@  (MPa)

τ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDda WgaaqaaKqzadGaamyBaaqcfayabaaaaa@3AA5@  (MPa)

N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGobWaaS baaeaajugWaiGacwgacaGG4bGaaiiCaaqcfayabaaaaa@3B9C@  

ER1(%)

ER2(%)

369.5

369.6

369.1

298.3

298.5

298.5

268.8

268.9

268.9

247.3

247.1

290.2

288.6

265.7

274.3

255

254.5

235.3

235.4

0

0

0

0

0

0

0

0

0

0

0

0

0

123.2

123.2

123

99.4

99.5

99.5

89.6

89.6

89.6

82.4

82.4

290.2

288.6

265.7

274.3

255

254.5

235.3

235.4

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

175

174.5

151.6

151.3

151.3

138.4

138.4

138.2

137.8

135.4

135.2

131.4

131.2

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

58.3

58.2

50.5

50.4

50.4

46.1

46.1

138.2

137.8

135.4

135.2

131.4

131.2

57172

65725

72614

131431

168583

178962

578316

671481

712823

976665

1356769

95759

133071

157011

173047

241679

422196

887342

1126890

157758

229202

551271

640123

689753

2176479

3187921

356608

526773

710704

860208

1123272

1501895

-13

-24

-31

92

49

40

-4

-17

-22

7

-22

31

-2

56

11

39

-20

-31

-45

-25

-46

46

29

20

25

-15

53

8

1

-15

-5

-27

-24

-34

-40

92

49

41

2

-12

-17

19

-14

14

-14

54

5

45

-16

-20

-37

-35

-54

47

30

21

37

-6

46

4

2

-14

3

-21

Table 6 Comparison of experimental data and predicted results of fatigue life of 18G2A steel with non-zero mean stress (under uniaxial fatigue loading)

σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B30@  (MPa)

σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGTbaajuaGbeaaaaa@3B3C@  (MPa)

τ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDlm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B32@  (MPa)

τ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDda WgaaqaaKqzadGaamyBaaqcfayabaaaaa@3AA5@  (MPa)

N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobWaaSbaaS qaaiGacwgacaGG4bGaaiiCaaqabaaaaa@395D@  

ER1(%)

ER2(%)

146.5

146.7

146.5

129.8

129.8

121.4

121.4

118.5

118.3

139.4

139.4

129.2

129.5

123.2

123.2

120.9

120.9

118

48.8

48.9

48.8

43.3

43.3

40.5

40.5

39.5

39.4

139.4

139.4

129.2

129.5

123.2

123.2

120.9

120.9

118

146.5

146.7

146.5

129.8

129.8

121.4

121.4

118.5

118.3

139.4

139.4

129.2

129.5

123.2

123.2

120.9

120.9

118

48.8

48.9

48.8

43.3

43.3

40.5

40.5

39.5

39.4

139.4

139.4

129.2

129.5

123.2

123.2

120.9

120.9

118

87304

93327

121935

382788

417127

1408351

1505671

2317350

3056739

115525

138497

211846

233041

257221

366179

681921

794444

1165727

172

151

95

100

83

4

-3

-20

-39

3

-14

17

4

53

7

-31

-41

-49

170

149

94

117

100

18

11

-8

-29

-15

-29

11

-1

58

11

-26

-36

-43

Table 7 Comparison of experimental data and predicted results of fatigue life of 18G2A steel with non-zero mean stress (under multiaxial fatigue loading)

Example 3: S355J0 alloy steel

For S355J0 alloy steel reported by Gasiak & Pawliczek18 under zero mean stress, predicted results of fatigue life are given in Table 8. Error ranges of fatigue life are, respectively, [-25,17] (%) and [-33,81] (%), for the axial and shear fatigue loading, which shows that the fatigue lives predicted by means of Eqs (7) and (8) are very accurate. Under multiaxial fatigue loading, error range of fatigue life, [-68,-13](%), is satisfactory. Under the axial fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [-57, 45] (%) and [-51, 39] (%), see Table 9, which illustrates that accuracy of fatigue lives predicted by using the two mean stress models is very high, and that the fatigue lives predicted by the linear mean stress model, i.e., Eq (17), are a little better than those by the Marin’s mean stress model, i.e., Eq (4). Under the shear fatigue loading with mean stress, the error ranges of fatigue life, and, are, respectively, [-42, 79] (%) and [-48, 86] (%), see Table 9, from which it can be seen that accuracy of fatigue life predicted by using the two mean stress models is very high, and that the fatigue lives predicted by the Marin’s mean stress model, i.e., Eq (5), are a little better than those by the linear mean stress model, i.e., Eq (18). Under the multiaxial fatigue loading with mean stress, the error range and the mean error of fatigue lives predicted by using the linear mean stress model, i.e., Eq (15), are, respectively, [-38,99] (%) and 8%, see Table 9, which shows that that accuracy of fatigue lives predicted by using the linear mean stress model is very high. However, the error range and the mean error of fatigue lives predicted by means of the Marin’s mean stress model, i.e., Eq (3), are, respectively, [-89,-68] (%) and -84%, see Table 9, which shows that the predicted by the Marin’s mean stress model is very poor. The comparison of the two results is shown In Figure 1. What is the reason that, under the multiaxial fatigue loading with mean stress, the accuracy of fatigue lives of S355J0 alloy steel predicted by the Marin’s mean stress model is very poor? For this, the author observes the material constants listed in Table 1 & Table 2 and finds that, for S355J0 alloy steel, is much less than, (=0.03127, see Table 10), which is the reason that the accuracy of fatigue lives predicted by using the Marin’s mean stress model, i.e., Eq. (3), is very poor. In order to further illustrate it, in the following form of Marin’s mean stress model:

log σ ea (1α σ em σ ea ) n = A ρ logN+ C ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGciGGSbGaai 4BaiaacEgacqaHdpWClmaaBaaajuaGbaqcLbmacaWGLbGaamyyaaqc fayabaGaaiikaiaaigdacqGHsislcqaHXoqydaWcaaqaaiabeo8aZn aaBaaabaqcLbmacaWGLbGaamyBaaqcfayabaaabaGaeq4Wdm3aaSba aeaajugWaiaadwgacaWGHbaajuaGbeaaaaGaaiykamaaCaaabeqaaK qzadGaamOBaaaajuaGcqGH9aqpcaWGbbWaaSbaaeaajugWaiabeg8a YbqcfayabaGaciiBaiaac+gacaGGNbGaamOtaiabgUcaRiaadoeada WgaaqaaKqzadGaeqyWdihajuaGbeaaaaa@5FAA@  (23)

σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B30@  (MPa)

σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGTbaajuaGbeaaaaa@3B3C@  (MPa)

τ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDlm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B32@  (MPa)

τ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDda WgaaqaaKqzadGaamyBaaqcfayabaaaaa@3AA5@  (MPa)

N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobWaaSbaaS qaaiGacwgacaGG4bGaaiiCaaqabaaaaa@395D@  (Cycles)

ER(%)

400

365

275

0

0

0

200

183

167

0

0

0

0

0

0

0

0

0

0

0

0

224

196

184

200

183

167

0

0

0

0

0

0

0

0

0

153000

403000

1295000

300000

632000

3150000

190000

739000

2117000

17

-25

14

-33

81

-17

-13

-55

-68

Table 8 Comparison of experimental data and predicted results of fatigue life of S355J0 alloy steel with zero mean stress

σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B30@  (MPa)

σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGTbaajuaGbeaaaaa@3B3C@  (MPa)

τ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDlm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B32@  (MPa)

τ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacqaHepaDdaWgaa WcbaGaamyBaaqabaaaaa@3866@  (MPa)

N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobWaaSbaaS qaaiGacwgacaGG4bGaaiiCaaqabaaaaa@395D@
(Cycles)

ER1(%)

ER2(%)

247

269

368

236

257

269

275

290

0

0

0

0

0

0

118

121

130

121

124

130

140

82

90

123

236

257

269

275

290

0

0

0

0

0

0

39

40

43

121

124

130

140

0

0

0

0

0

0

0

0

139

151

175

133

136

139

118

121

130

121

124

130

140

0

0

0

0

0

0

0

0

46

50

58

133

136

139

39

40

43

121

124

130

140

1011000

713000

72000

1083000

246000

153000

171000

131000

1043000

670000

158000

1130000

850000

525000

2356000

1516000

378000

803000

366000

210000

117000

-9

-20

35

-57

17

45

15

11

79

-6

-42

-6

-6

14

-89

-86

-68

-92

-86

-84

-84

7

-11

21

-51

19

39

6

-6

86

-7

-48

0

-4

13

-28

-9

99

-38

8

21

8

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

 

Marin’s mean stress model

Linear mean stress model

Materials

α 1 / α 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqylm aaBaaajuaGbaqcLbmacaaIXaaajuaGbeaacaGGVaGaeqySde2cdaWg aaqcfayaaKqzadGaaGimaaqcfayabaaaaa@4063@   n 1 / n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbWcda WgaaqcfayaaKqzadGaaGymaaqcfayabaGaai4laiaad6gadaWgaaqa aKqzadGaaGimaaqcfayabaaaaa@3E72@   β 1 / β 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHYoGylm aaBaaajuaGbaqcLbmacaaIXaaajuaGbeaacaGGVaGaeqOSdi2cdaWg aaqcfayaaKqzadGaaGimaaqcfayabaaaaa@4067@   γ 1 / γ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZoWzlm aaBaaajuaGbaqcLbmacaaIXaaajuaGbeaacaGGVaGaeq4SdC2aaSba aeaajugWaiaaicdaaKqbagqaaaaa@3FDA@  

Cast iron

0.826159

1.726845

1.225497

0.88636

18G2A steel

0.445415

1.045113

0.949305

0.736148

S355J0 alloy steel

0.031271

2.697616

0.820739

1.366534

Table 10 Comparison of material constants used to describe the effect of mean stress on fatigue life

Figure 1 Comparison of fatigue lives predicted by the linear mean stress model and the Marin’s mean stress model under multiaxial fatigue loading with non-zero mean stress.

by letting, and, , fatigue lives predicted are listed in 11, in which the error ranges are, respectively, [-60,16](%) and [-37,188] (%), and the mean errors are, respectively, -21(%) and 38(%) (Table 12). Comparison of detail errors are given in Table 12, from which it can be seen that the accuracy of fatigue lives predicted by using the linear mean stress model, i.e., Eq (15), is much better than that by using the Marin’s mean stress model, Eq (3).

σ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B30@  (MPa)

σ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWClm aaBaaajuaGbaqcLbmacaWGTbaajuaGbeaaaaa@3B3C@  (MPa)

τ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDlm aaBaaajuaGbaqcLbmacaWGHbaajuaGbeaaaaa@3B32@  (MPa)

τ m MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHepaDda WgaaqaaKqzadGaamyBaaqcfayabaaaaa@3AA5@  (MPa)

N exp MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGobWaaSbaaS qaaiGacwgacaGG4bGaaiiCaaqabaaaaa@395D@  

α= α 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqycq GH9aqpcqaHXoqydaWgaaqaaKqzadGaaGymaaqcfayabaaaaa@3CED@ , n= n 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbGaey ypa0JaamOBaSWaaSbaaKqbagaajugWaiaaigdaaKqbagqaaaaa@3C2E@

ER1(%)

α= α 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqycq GH9aqpcqaHXoqydaWgaaqaaKqzadGaaGimaaqcfayabaaaaa@3CEC@   n= n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbGaey ypa0JaamOBaSWaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaaa@3C2D@
ER1(%)

118

121

130

121

124

130

140

39

40

43

121

124

130

140

118

121

130

121

124

130

140

39

40

43

121

124

130

140

2356000

1516000

378000

803000

366000

210000

117000

-60

-49

16

-52

-12

5

5

-1

26

188

-37

15

38

38

Table 11 Comparison of experimental data and predicted results of fatigue life of S355J0 alloy steel under multiaxial fatigue loading with non-zero mean stress

Marin’s mean stress model

Linear mean stress model

α= α 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqycq GH9aqpcqaHXoqydaWgaaqaaKqzadGaaGymaaqcfayabaaaaa@3CED@ , n= n 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaacaWGUbGaeyypa0 JaamOBamaaBaaaleaacaaIXaaabeaaaaa@3956@

α= α 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqycq GH9aqpcqaHXoqydaWgaaqaaKqzadGaaGimaaqcfayabaaaaa@3CEC@ , n= n 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbGaey ypa0JaamOBaSWaaSbaaKqbagaajugWaiaaicdaaKqbagqaaaaa@3C2D@

α= α ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHXoqycq GH9aqpcqaHXoqydaWgaaqaaKqzadGaeqyWdihajuaGbeaaaaa@3DF2@ , n= n ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcaWGUbGaey ypa0JaamOBaSWaaSbaaKqbagaajugWaiabeg8aYbqcfayabaaaaa@3D33@

Error range (%)

Mean error (%)

Error range (%)

Mean error (%)

Error range (%)

Mean error (%)

Error range (%)

Mean error (%)

[-60,16]

-21

[-37,188]

38

[-89,-68]

-84

[-38,99]

8

Table 12 General summary of the analysis of variance (ANOVA), mean and coefficient of variation (CV), of the productive and nutritional characteristics of the different proportions of oats and vetch
F tests: ***; P<0.001, **; P<0.01, *; P<0.05, ns; Not significant

Concluding remarks

On basis of the present investigation, it can be seen that:

  1. 1、For some metal materials, for example, Cast Iron, an accuracy of fatigue lives predicted by means of the linear mean stress model and the Marin’s mean stress model is very high and the two models almost have same accuracy.
  2. 2、For S355J0 alloy steel, under the axial and pure shear fatigue loading, an accuracy of fatigue lives predicted by means of the linear mean stress model and the Marin’s mean stress model is very high. But under the multiaxial fatigue loading, an accuracy of fatigue lives predicted by using the Marin’s mean stress model is very poor, while on the contrary, fatigue lives predicted by means of the linear mean stress model are very satisfactory. On basis of the two points above, the author considers that the linear mean stress model presented in this paper is more suitable than the Marin’s mean stress model in performing the fatigue life assessment. Up to date, as well known to us, the Marin’s mean stress model has been widely used to perform the fatigue life assessment of metal materials with mean stress effect.

Why is the linear mean stress model, which, in the mathematical formula, has no an advantage over the Marin’s mean stress model, presented in this paper? The reason is on basis of the following two aspects:

  1. By using the Marin’s mean stress model, the author ever attempted to propose a multiaxial fatigue limit model, and found that numerical computation difficulty occurred sometimes in determining material constants in the Marin’s mean stress model.
  2. When non-proportional loading fatigue is carried out, Itoh’s formula19 is usually used to reveal the relationship of the amplitude of the equivalent non-proportional stress, σ ean MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaamyzaiaadggacaWGUbaajuaGbeaaaaa@3C74@ , and the amplitude of the effective proportional stress, σ ea MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaamyzaiaadggaaKqbagqaaaaa@3B81@  , i.e.,

σ ean = σ ea (1+βF) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaamyzaiaadggacaWGUbaajuaGbeaacqGH9aqpcqaH dpWCdaWgaaqaaKqzadGaamyzaiaadggaaKqbagqaaiaacIcacaaIXa Gaey4kaSIaeqOSdiMaamOraiaacMcaaaa@484C@ σ ean / σ ea =λ+γF MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaamyzaiaadggacaWGUbaajuaGbeaacaGGVaGaeq4W dm3aaSbaaeaajugWaiaadwgacaWGHbaajuaGbeaacqGH9aqpcqaH7o aBcqGHRaWkcqaHZoWzcaWGgbaaaa@48A5@  (24)

where β MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHYoGyaa a@37B2@  is material sensitivity parameter to load-path non-proportionality defined on σ 3 τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCca aJuaYaaOaaaeaacaaIZaaabeaacqaHepaDaaa@3B28@  stress plane; F is usually called a non-proportional loading factor which expresses the severity of non proportional loading. The author finds that the prediction results of fatigue life obtained by using Itoh’s formula sometimes are not satisfactory. Thus the following attempt is made:

σ ean / σ ea =λ+γF MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaamyzaiaadggacaWGUbaajuaGbeaacaGGVaGaeq4W dm3aaSbaaeaajugWaiaadwgacaWGHbaajuaGbeaacqGH9aqpcqaH7o aBcqGHRaWkcqaHZoWzcaWGgbaaaa@48A5@  (25)

where λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaH7oaBaa a@37C5@ , γ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHZoWzaa a@37B8@  are constants to be determined by linear fitting of σ ean MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9 Ff0dmeaabaqaciGacaGaaeqabaWaaeaaeaaakeaajuaGcqaHdpWCda WgaaqaaKqzadGaamyzaiaadggacaWGUbaajuaGbeaaaaa@3C74@  and F. Obviously, the Itoh’s formula (24) has mechanical advantage over the linear formula (25). But verification by experimental data of fatigue life of metallic materials shows that the accuracy of fatigue lives predicted by means of the linear formula (25) is higher than that by the Itoh’s formula (24), which will be reported in another paper.

Acknowledgments

None.

Conflicts of interest

The authors declare that there is no conflict of interest.

References

  1. Liu BW, Yan XQ. A new model of multiaxial fatigue life prediction with influence of different mean stresses. International Journal of Damage Mechanics. 2019;28(9):1323–1343.
  2. Liu BW, Yan XQ. A multiaxial fatigue limit prediction equation for metallic materials. ASME Journal of Pressure Vessel Technology. 2020.
  3. Gough, HJ. Engineering steels under combined cyclic and static stresses. Proceedings of the Institution of Mechanical Engineers. 1949;160:417–440.
  4. Frith, PH. Fatigue of wrought high-tensile alloy steel. Proceedings of the Institution of Mechanical Engineers. 1956;462–499.
  5. Cristofori A, Susmel L, Tovo R. A stress invariant based criterion to estimate fatigue damage under multiaxial loading. International Journal of Fatigue. 2008;30:1646–1658.
  6. Papadopoulos IV, Davoli P, Gorla C, et al. A comparative study of multiaxial highcycle fatigue criteria for metals. Int J Fatigue. 1997;19:219–235.
  7. Sines G. Behaviour of metals under complex static and alternating stresses. In: Metal Fatigue, McGraw Hill, New York. 1959:145–169.
  8. Crossland B. Effect of large hydroscopic pressures on the torsional fatigue strength of an alloy steel. Proceedings of international conference on fatigue of metals; London, New York. 1956. p. 138–49.
  9. Li B, De Freitas M. A procedure for fast evaluation of high cycle fatigue under multiaxial random loading. J Mech Design. 2002;124:558–563.
  10. Wang CH, Brown MW. Multiaxial random load fatigue: life prediction techniques and experiments. Fourth international conference on biaxial/multiaxial fatigue, May 31–June 3, Paris. 1994.
  11. Susmel L, Lazzarin P. A bi-parametric modified Wöhler curve for high cycle multiaxial fatigue assessment. Fatigue and Fracture of Engineering Materials and Structures. 2002;25:63–78.
  12. Papadopoulos IV, Davoli P, Gorla C, et al. A comparative study of multiaxial high-cycle fatigue criteria for metals. Int J Fatigue. 1997;19:219–235.
  13. Papadopoulos IV. Long life fatigue under multiaxial loading. Int J Fatigue. 2001;23:839–849.
  14. Tao G, Xia Z. Mean stress/strain effect on fatigue behavior of an epoxy resin. International Journal of Fatigue. 2007;29(12):2180–2190.
  15. Marin J. Interpretation of fatigue strength for combined stresses. In: Proceedings of international conference on fatigue of metals. Institution of mechanical engineers; London. 1956. p. 184–194.
  16. Berto F, Lazzarin P, Tovo R, Multiaxial fatigue strength of severely notched cast iron specimens. International Journal of Fatigue. 2014;67:15–27.  
  17. Gasiak G, Pawliczek R. The mean loading effect under cyclic bending and torsion of 18G2A steel. In: M de Freitas, editor. Proceedings of 6th International Conference on Biaxial/ Multiaxial Fatigue and Fracture; Lisbon, Portugal. 2001. p. 213–222.
  18. Gasiak G, Pawliczek R. Application of an energy model for fatigue life prediction of construction steels under bending, torsion and synchronous bending and torsion. Int J Fatigue. 2003;25(12):1339–1346.  
  19. Itoh T, Sakane M, Ohnami M, et al. Nonproportional low cycle fatigue criterion for type 304 stainless steel. Transact ASME J Engng Mater Technol. 1995;117(3):285–292.
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