Mean Stress Correction

Use mean stress correction to account for the effect of non-zero mean stresses.

Generally, fatigue curves are obtained from standard experiments with fully reversed cyclic loading. However, the real fatigue loading could not be fully-reversed, and the normal mean stresses have significant effect on fatigue performance of components. Tensile normal mean stresses are detrimental and compressive normal mean stresses are beneficial, in terms of fatigue strength. Mean stress correction is used to account for the effect of non-zero mean stresses.

Depending on the material, stress state, environment, and strain amplitude, fatigue life will usually be dominated either by microcrack growth along shear planes or along tensile planes. Critical plane mean stress correction methods incorporate the dominant parameters governing either type of crack growth. Due to the different possible failure modes, shear or tensile dominant, no single mean stress correction method should be expected to correlate test data for all materials in all life regimes. There is no consensus yet as to the best method to use for multiaxial fatigue life estimates. For stress-based mean stress correction method, Goodman and FKM models are available for tensile crack. Findley model is available for shear crack. For strain-based mean stress correction method, Morrow and Smith,Watson and Topper are available for tensile crack. Brown-Miller and Fatemi-Socie are available for shear crack. If multiple models are defined, SimSolid selects the model which leads to maximum damage from all the available damage values.

Goodman Model

Use the Goodman model to assess damage caused by tensile crack growth at a critical plane.

(1)
S e = S a 1 S m S U MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGtbWaaSbaaSqaaiaadwgaaeqaaO Gaeyypa0ZaaSaaaeaacaWGtbWaaSbaaSqaaiaadggaaeqaaaGcbaWa aeWaaeaacaaIXaGaeyOeI0YaaSaaaeaacaWGtbWaaSbaaSqaaiaad2 gaaeqaaaGcbaGaam4uamaaBaaaleaacaWGvbaabeaaaaaakiaawIca caGLPaaaaaGaeyyTH8laaa@4004@
Where:
  • S m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGtbWaaSbaaSqaaiaad2gaaeqaaa aa@33AE@ is the Mean stress given by S m = S m a x + S m i n / 2 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGtbWaaSbaaSqaaiaad2gaaeqaaO Gaeyypa0ZaaSGbaeaadaqadaqaaiaadofadaWgaaWcbaGaamyBaiaa dggacaWG4baabeaakiabgUcaRiaadofadaWgaaWcbaGaamyBaiaadM gacaWGUbaabeaaaOGaayjkaiaawMcaaaqaaiaaikdaaaaaaa@3FBE@
  • S a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGtbWaaSbaaSqaaiaadggaaeqaaa aa@33A2@ is the Stress amplitude
  • S e MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGtbWaaSbaaSqaaiaadwgaaeqaaa aa@33A6@ is the stress amplitude after mean stress correction
  • S u MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaebbnrfifHhDYfgasaacH8srps0l bbf9q8WrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbba9q8WqFfea0=yr0R Yxir=Jbba9q8aq0=yq=He9q8qqQ8frFve9Fve9Ff0dmeaabaqaciGa caGaaeqabaqaaeaadaaakeaacaWGtbWaaSbaaSqaaiaadwhaaeqaaa aa@33B6@ is the ultimate strength

The Goodman method treats positive mean stress correction in the way that mean stress always accelerates fatigue failure, while it ignores the negative mean stress. This method gives conservative result for compressive mean stress.

A Haigh diagram characterizes different combinations of stress amplitude and mean stress for a given number of cycles to failure.


Figure 1. Goodman Haigh diagram

Findley Model

The Findley criterion is often applied for the case of finite long-life fatigue. The equation for each plane is as follows:(2)
Δ τ 2 + k σ n = τ f * ( N f ) b MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq qHuoarcqaHepaDaeaacaaIYaaaaiabgUcaRiaadUgacqaHdpWCdaWg aaWcbaGaamOBaaqabaGccqGH9aqpcqaHepaDdaqhaaWcbaGaamOzaa qaaiaacQcaaaGccaGGOaGaamOtamaaBaaaleaacaWGMbaabeaakiaa cMcadaahaaWcbeqaaiaadkgaaaaaaa@47A4@
Where: τ f * MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aa0 baaSqaaiaadAgaaeaacaGGQaaaaaaa@397E@ is computed from the shear fatigue strength coefficient, τ f ' MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aa0 baaSqaaiaadAgaaeaacaGGNaaaaaaa@397C@ , using: (3)
τ f * = 1 + k 2 τ f ' MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aa0 baaSqaaiaadAgaaeaacaGGQaaaaOGaeyypa0ZaaOaaaeaacaaIXaGa ey4kaSIaam4AamaaCaaaleqabaGaaGOmaaaaaeqaaOGaeqiXdq3aa0 baaSqaaiaadAgaaeaacaGGNaaaaaaa@41A7@
The correction factor 1 + k 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOaaaeaaca aIXaGaey4kaSIaam4AamaaCaaaleqabaGaaGOmaaaaaeqaaaaa@397A@ typically has a set value of about 1.04.
Note: τ f * MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aa0 baaSqaaiaadAgaaeaacaGGQaaaaaaa@397E@ must be defined based on amplitude. If τ f ' MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqiXdq3aa0 baaSqaaiaadAgaaeaacaGGNaaaaaaa@397C@ is not defined by the user, SimSolid calculates it using the following equation:(4)(30)
τ f ' =Cf*0.5*SRI1 Where, Cf= 2 1+ k 1+ k 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHep aDdaqhaaWcbaGaamOzaaqaaiaacEcaaaGccqGH9aqpcaWGdbGaamOz aiaacQcacaaIWaGaaiOlaiaaiwdacaGGQaGaam4uaiaadkfacaWGjb GaaGymaaqaaiaabEfacaqGObGaaeyzaiaabkhacaqGLbGaaiilaaqa aiaadoeacaWGMbGaeyypa0ZaaSaaaeaacaaIYaaabaGaaGymaiabgU caRmaalaaabaGaam4AaaqaamaakaaabaGaaGymaiabgUcaRiaadUga daahaaWcbeqaaiaaikdaaaaabeaaaaaaaaaaaa@51E8@
The constant k is determined experimentally by performing fatigue tests involving two or more stress states. For ductile materials, k typically varies between 0.2 and 0.3.

FKM

Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio (R=Smin/Smax) values. The Corrected value is then used to choose the SN curve for the damage and life calculation stage.

The FKM equations below illustrate the calculation of Corrected Stress Amplitude ( S e A MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaaaaa@38A9@ ). The actual value of stress used in the Damage calculations is the Corrected stress Amplitude (which is 2 S e A MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgw SixlaadofadaqhaaWcbaGaamyzaaqaaiaadgeaaaaaaa@3BAF@ ). These equations apply for SN curves that you input.

Regime 1 (R>1.0): S e A = S a 1 M MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9iaadofadaWgaaWcbaGa amyyaaqabaGcdaqadaqaaiaaigdacqGHsislcaWGnbaacaGLOaGaay zkaaaaaa@3FB0@

Regime 2 (-∞≤R≤0.0): S e A = S a + M * S m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9iaadofadaWgaaWcbaGa amyyaaqabaGccqGHRaWkcaWGnbGaaiOkaiaadofadaWgaaWcbaGaam yBaaqabaaaaa@4005@

Regime 3 (0.0<R<0.5): S e A = 1 + M S a + M 3 S m 1 + M 3 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9maabmaabaGaaGymaiab gUcaRiaad2eaaiaawIcacaGLPaaadaWcaaqaaiaadofadaWgaaWcba GaamyyaaqabaGccqGHRaWkdaqadaqaamaaliaabaGaamytaaqaaiaa iodaaaaacaGLOaGaayzkaaGaam4uamaaBaaaleaacaWGTbaabeaaaO qaaiaaigdacqGHRaWkdaWccaqaaiaad2eaaeaacaaIZaaaaaaaaaa@48FF@

Regime 4 (R≥0.5): S e A = 3 S a 1 + M 2 3 + M MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9maalaaabaGaaG4maiaa dofadaWgaaWcbaGaamyyaaqabaGcdaqadaqaaiaaigdacqGHRaWkca WGnbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGcbaGaaG4m aiabgUcaRiaad2eaaaaaaa@43D6@

Where S e A MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaaaaa@38A9@ is the stress amplitude after mean stress correction (Endurance stress), S m MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGTbaabeaaaaa@37EA@ is the mean stress, S a MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGHbaabeaaaaa@37DE@ is the stress amplitude, and M is the mean stress sensitivity.


Figure 2.

Morrow

Morrow is the first to consider the effect of mean stress through introducing the mean stress σ 0 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaaicdaaeqaaaaa@389D@ in fatigue strength coefficient by:

(5)
ε a e = σ ' f σ 0 E 2 N f b MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aa0 baaSqaaiaadggaaeaacaWGLbaaaOGaeyypa0ZaaSaaaeaadaqadaqa aiabeo8aZjaacEcadaWgaaWcbaGaamOzaaqabaGccqGHsislcqaHdp WCdaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaaaeaacaWGfbaa amaabmaabaGaaGOmaiaad6eadaWgaaWcbaGaamOzaaqabaaakiaawI cacaGLPaaadaahaaWcbeqaaiaadkgaaaaaaa@4987@

Thus the entire fatigue life formula becomes:

(6)
ε a = σ f ' σ 0 E 2 N f b + ε f ' 2 N f c MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aaS baaSqaaiaadggaaeqaaOGaeyypa0JaaGPaVpaalaaabaWaaeWaaeaa cqaHdpWCdaqhaaWcbaGaamOzaaqaaiaacEcaaaGccqGHsislcqaHdp WCdaWgaaWcbaGaaGimaaqabaaakiaawIcacaGLPaaaaeaacaWGfbaa amaabmaabaGaaGOmaiaad6eadaWgaaWcbaGaamOzaaqabaaakiaawI cacaGLPaaadaahaaWcbeqaaiaadkgaaaGccaaMc8Uaey4kaSIaaGPa Vlabew7aLnaaDaaaleaacaWGMbaabaGaai4jaaaakmaabmaabaGaaG Omaiaad6eadaWgaaWcbaGaamOzaaqabaaakiaawIcacaGLPaaadaah aaWcbeqaaiaadogaaaaaaa@56EC@

Morrow's equation is consistent with the observation that mean stress effects are significant at low value of plastic strain and of little effect at high plastic strain.

MORROW2 : Improves the Morrow method by ignoring the effect of negative mean stress.

Smith, Watson, and Topper

Smith, Watson, and Topper proposed a different method to account for the effect of mean stress by considering the maximum stress during one cycle (for convenience, this method is called SWT in the following). In this case, the damage parameter is modified as the product of the maximum stress and strain amplitude in one cycle.

(7)
ε a S W T σ max = ε a σ a = σ a σ ' f E 2 N f b + ε ' f 2 N f c MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeqyTdu2aa0 baaSqaaiaadggaaeaacaWGtbGaam4vaiaadsfaaaGccqaHdpWCdaWg aaWcbaGaciyBaiaacggacaGG4baabeaakiabg2da9iabew7aLnaaBa aaleaacaWGHbaabeaakiabeo8aZnaaBaaaleaacaWGHbaabeaakiab g2da9iabeo8aZnaaBaaaleaacaWGHbaabeaakmaabmaabaWaaSaaae aacqaHdpWCcaGGNaWaaSbaaSqaaiaadAgaaeqaaaGcbaGaamyraaaa daqadaqaaiaaikdacaWGobWaaSbaaSqaaiaadAgaaeqaaaGccaGLOa GaayzkaaWaaWbaaSqabeaacaWGIbaaaOGaey4kaSIaeqyTduMaai4j amaaBaaaleaacaWGMbaabeaakmaabmaabaGaaGOmaiaad6eadaWgaa WcbaGaamOzaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaadoga aaaakiaawIcacaGLPaaaaaa@5F95@

The SWT method will predict that no damage will occur when the maximum stress is zero or negative, which is not consistent with reality.

When comparing the two methods, the SWT method predicted conservative life for loads predominantly tensile, whereas, the Morrow approach provides more realistic results when the load is predominantly compressive.

Fatemi-Socie

This model is for shear crack growth. During shear loading, the irregularly shaped crack surface results in frictional forces that will reduce crack tip stresses, thus hindering crack growth and increasing the fatigue life. Tensile stresses and strains will separate the crack surfaces and reduce frictional forces. Fractographic evidence for this behavior has been obtained. Fractographs from objects that have failed by pure torsion show extensive rubbing and are relatively featureless in contrast to tension test fractographs where individual slip bands are observed on the fracture surface.


Figure 3. Fatemi-Socie model

To demonstrate the effect of maximum stress, tests with the six tension-torsion loading histories were conducted. They were designed to have the same maximum shear strain amplitudes. The cyclic normal strain is also constant for the six loading histories. The experiments resulted in nearly the same maximum shear strain amplitudes, equivalent stress and strain amplitudes and plastic work. The major difference between the loading histories is the normal stress across the plane of maximum shear strain.

The loading history and normal stress are shown in the figure at the top of each crack growth curve. Higher maximum stresses lead to faster growth rates and lower fatigue lives. The maximum stress has a lesser influence on the initiation of a crack if crack initiation is defined on the order of 10 mm, which is the size of the smaller grains in this material.

These observations lead to the following model that may be interpreted as the cyclic shear strain modified by the normal stress to include the crack closure effects.(8)
Δ γ 2 1 + k σ n , max σ y = τ f ' G 2 N f b γ + γ f ' 2 N f c γ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq qHuoarcqaHZoWzaeaacaaIYaaaamaabmaabaGaaGymaiabgUcaRiaa dUgadaWcaaqaaiabeo8aZnaaBaaaleaacaWGUbGaaiilaiGac2gaca GGHbGaaiiEaaqabaaakeaacqaHdpWCdaWgaaWcbaGaamyEaaqabaaa aaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacqaHepaDdaqhaaWcba GaamOzaaqaaiaacEcaaaaakeaacaWGhbaaamaabmaabaGaaGOmaiaa d6eadaWgaaWcbaGaamOzaaqabaaakiaawIcacaGLPaaadaahaaWcbe qaaiaadkgadaWgaaadbaGaeq4SdCgabeaaaaGccqGHRaWkcqaHZoWz daqhaaWcbaGaamOzaaqaaiaacEcaaaGcdaqadaqaaiaaikdacaWGob WaaSbaaSqaaiaadAgaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaa caWGJbWaaSbaaWqaaiabeo7aNbqabaaaaaaa@5F71@

The sensitivity of a material to normal stress is reflected in the value k / σ y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4Aaiaac+ cacqaHdpWCcaWG5baaaa@3A58@ . Where, σ y MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4WdmNaam yEaaaa@38B5@ is stress where a significant total strain of 0.002 is used in SimSolid. If test data from multiple stress states is not available, k = 0.3. This model not only explains the difference between tension and torsion loading but also can be used to describe mean stress and non-proportional hardening effects. Critical plane models that include only strain terms cannot reflect the effect of mean stress or strain path dependent on hardening.

The transition fatigue life, 2Nt, is selected because the elastic and plastic strains contribute equally to the fatigue damage. You can obtain it from the uniaxial fatigue constants.

(9)
2 N f = E ε f ' σ f ' 1 b c MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaad6 eadaWgaaWcbaGaamOzaaqabaGccqGH9aqpdaqadaqaamaalaaabaGa amyraiabew7aLnaaDaaaleaacaWGMbaabaGaai4jaaaaaOqaaiabeo 8aZnaaDaaaleaacaWGMbaabaGaai4jaaaaaaaakiaawIcacaGLPaaa daahaaWcbeqaamaabmaabaWaaSaaaeaacaaIXaaabaGaamOyaiabgk HiTiaadogaaaaacaGLOaGaayzkaaaaaaaa@484E@

Employ the Fatemi-Socie model to determine the shear strain constants.

(10)
Δ γ 2 1 + k σ n , max σ y = τ f ' G 2 N f b γ + γ f ' 2 N f c γ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq qHuoarcqaHZoWzaeaacaaIYaaaamaabmaabaGaaGymaiabgUcaRiaa dUgadaWcaaqaaiabeo8aZnaaBaaaleaacaWGUbGaaiilaiGac2gaca GGHbGaaiiEaaqabaaakeaacqaHdpWCdaWgaaWcbaGaamyEaaqabaaa aaGccaGLOaGaayzkaaGaeyypa0ZaaSaaaeaacqaHepaDdaqhaaWcba GaamOzaaqaaiaacEcaaaaakeaacaWGhbaaamaabmaabaGaaGOmaiaa d6eadaWgaaWcbaGaamOzaaqabaaakiaawIcacaGLPaaadaahaaWcbe qaaiaadkgadaWgaaadbaGaeq4SdCgabeaaaaGccqGHRaWkcqaHZoWz daqhaaWcbaGaamOzaaqaaiaacEcaaaGcdaqadaqaaiaaikdacaWGob WaaSbaaSqaaiaadAgaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaa caWGJbWaaSbaaWqaaiabeo7aNbqabaaaaaaa@5F71@

First, note the exponents should be the same for shear and tension.

(11)
b γ = b c γ = c MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaWGIb WaaSbaaSqaaiabeo7aNbqabaGccqGH9aqpcaWGIbaabaGaam4yamaa BaaaleaacqaHZoWzaeqaaOGaeyypa0Jaam4yaaaaaa@3F5E@

Shear modulus is directly computed from the tensile modulus.

(12)
G = E 2 ( 1 + ν ) MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4raiabg2 da9maalaaabaGaamyraaqaaiaaikdacaGGOaGaaGymaiabgUcaRiab e27aUjaacMcaaaaaaa@3E09@
You can estimate yield strength from the uniaxial cyclic stress strain curve.(13)
σ y = K ' ( 0.002 ) n ' = σ f ' ε f ' b c ( 0.002 ) b c MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaadMhaaeqaaOGaeyypa0Jaam4samaaCaaaleqabaGaai4j aaaakiaacIcacaaIWaGaaiOlaiaaicdacaaIWaGaaGOmaiaacMcada ahaaWcbeqaaiaad6gadaahaaadbeqaaiaacEcaaaaaaOGaeyypa0Za aSaaaeaacqaHdpWCdaqhaaWcbaGaamOzaaqaaiaacEcaaaaakeaacq aH1oqzdaqhaaWcbaGaamOzaaqaaiaacEcadaWcaaqaaiaadkgaaeaa caWGJbaaaaaaaaGccaGGOaGaaGimaiaac6cacaaIWaGaaGimaiaaik dacaGGPaWaaWbaaSqabeaadaWcaaqaaiaadkgaaeaacaWGJbaaaaaa aaa@5394@

Normal stresses and strains are computed from the transition fatigue life and uniaxial properties.

(14)
Δ ε p 2 = ε f ' ( 2 N t ) c MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq qHuoarcqaH1oqzdaWgaaWcbaGaamiCaaqabaaakeaacaaIYaaaaiab g2da9iabew7aLnaaDaaaleaacaWGMbaabaGaai4jaaaakiaacIcaca aIYaGaamOtamaaBaaaleaacaWG0baabeaakiaacMcadaahaaWcbeqa aiaadogaaaaaaa@449E@
(15)
Δ ε e 2 = σ f ' E ( 2 N t ) b MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq qHuoarcqaH1oqzdaWgaaWcbaGaamyzaaqabaaakeaacaaIYaaaaiab g2da9maalaaabaGaeq4Wdm3aa0baaSqaaiaadAgaaeaacaGGNaaaaa GcbaGaamyraaaacaGGOaGaaGOmaiaad6eadaWgaaWcbaGaamiDaaqa baGccaGGPaWaaWbaaSqabeaacaWGIbaaaaaa@4587@
(16)
σ n , max = Δ σ 4 = E Δ ε e 4 MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaad6gacaGGSaGaciyBaiaacggacaGG4baabeaakiabg2da 9maalaaabaGaeuiLdqKaeq4WdmhabaGaaGinaaaacqGH9aqpdaWcaa qaaiaadweacqqHuoarcqaH1oqzdaWgaaWcbaGaamyzaaqabaaakeaa caaI0aaaaaaa@482B@

Substituting the appropriate the value of elastic and plastic Poisson’s ratio gives:

(17)
Δ γ e 2 = 1.3 Δ ε e 2 Δ γ p 2 = 1.5 Δ ε p 2 MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaWcaa qaaiabfs5aejabeo7aNnaaBaaaleaacaWGLbaabeaaaOqaaiaaikda aaGaeyypa0JaaGymaiaac6cacaaIZaWaaSaaaeaacqqHuoarcqaH1o qzdaWgaaWcbaGaamyzaaqabaaakeaacaaIYaaaaaqaamaalaaabaGa euiLdqKaeq4SdC2aaSbaaSqaaiaadchaaeqaaaGcbaGaaGOmaaaacq GH9aqpcaaIXaGaaiOlaiaaiwdadaWcaaqaaiabfs5aejabew7aLnaa BaaaleaacaWGWbaabeaaaOqaaiaaikdaaaaaaaa@5057@

Separating the elastic and plastic parts of the total strain results in these expressions for the shear strain life constants:

(18)
τ f ' = 1.3 Δ ε e 2 1 + k σ n , max σ y G 2 N t b γ γ f ' = 1.5 Δ ε p 2 1 + k σ n , max σ y G 2 N t c γ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacqaHep aDdaqhaaWcbaGaamOzaaqaaiaacEcaaaGccqGH9aqpdaWcaaqaaiaa igdacaGGUaGaaG4maiabfs5aejabew7aLnaaBaaaleaacaWGLbaabe aaaOqaaiaaikdaaaWaaeWaaeaacaaIXaGaey4kaSIaam4Aamaalaaa baGaeq4Wdm3aaSbaaSqaaiaad6gacaGGSaGaciyBaiaacggacaGG4b aabeaaaOqaaiabeo8aZnaaBaaaleaacaWG5baabeaaaaaakiaawIca caGLPaaadaWcaaqaaiaadEeaaeaadaqadaqaaiaaikdacaWGobWaaS baaSqaaiaadshaaeqaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaWG IbWaaSbaaWqaaiabeo7aNbqabaaaaaaaaOqaaiabeo7aNnaaDaaale aacaWGMbaabaGaai4jaaaakiabg2da9maalaaabaGaaGymaiaac6ca caaI1aGaeuiLdqKaeqyTdu2aaSbaaSqaaiaadchaaeqaaaGcbaGaaG OmaaaadaqadaqaaiaaigdacqGHRaWkcaWGRbWaaSaaaeaacqaHdpWC daWgaaWcbaGaamOBaiaacYcaciGGTbGaaiyyaiaacIhaaeqaaaGcba Gaeq4Wdm3aaSbaaSqaaiaadMhaaeqaaaaaaOGaayjkaiaawMcaamaa laaabaGaam4raaqaamaabmaabaGaaGOmaiaad6eadaWgaaWcbaGaam iDaaqabaaakiaawIcacaGLPaaadaahaaWcbeqaaiaadogadaWgaaad baGaeq4SdCgabeaaaaaaaaaaaa@789B@

Brown-Miller

This model is for shear crack growth. Brown and Miller conducted combined tension and torsion tests with a constant shear strain range. The normal strain range on the plane of maximum shear strain will change with the ratio of applied tension and torsion strains. Based on the data shown below for a constant shear strain amplitude, Brown and Miller concluded that two strain parameters are needed to describe the fatigue process because the combined action of shear and normal strain reduces fatigue life.


Figure 4. Fatigue Life versus Normal Strain Amplitude

Influence of Normal Strain Amplitude

Analogous to the shear and normal stress proposed by Findley for high cycle fatigue, they proposed that both the cyclic shear and normal strain on the plane of maximum shear must be considered. Cyclic shear strains will help to nucleate cracks and the normal strain will assist in their growth. They proposed a simple formulation of the theory:

(19)
Δ γ ^ 2 = Δ γ max 2 + S Δ ε n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq qHuoarcuaHZoWzgaqcaaqaaiaaikdaaaGaeyypa0ZaaSaaaeaacqqH uoarcqaHZoWzdaWgaaWcbaGaciyBaiaacggacaGG4baabeaaaOqaai aaikdaaaGaey4kaSIaam4uaiabfs5aejabew7aLnaaBaaaleaacaWG Ubaabeaaaaa@47AC@

Where Δ γ ^ MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKafq 4SdCMbaKaaaaa@3911@

is the equivalent shear strain range and S is a material dependent parameter that represents the influence of the normal strain on material microcrack growth and is determined by correlating axial and torsion data. Here, Δ γ max MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq 4SdC2aaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3C01@ is taken as the maximum shear strain range and Δ ε n MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq yTdu2aaSbaaSqaaiaad6gaaeqaaaaa@3A20@ is the normal strain range on the plane experiencing the shear strain range Δ γ max MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeuiLdqKaeq 4SdC2aaSbaaSqaaiGac2gacaGGHbGaaiiEaaqabaaaaa@3C01@ . Considering elastic and plastic strains separately with the appropriate values of Poisson's ratio results in:
(20)
Δ γ max 2 + S Δ ε n = A σ f ' E ( 2 N f ) b + B ε f ' ( 2 N f ) c MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq qHuoarcqaHZoWzdaWgaaWcbaGaciyBaiaacggacaGG4baabeaaaOqa aiaaikdaaaGaey4kaSIaam4uaiabfs5aejabew7aLnaaBaaaleaaca WGUbaabeaakiabg2da9iaadgeadaWcaaqaaiabeo8aZnaaDaaaleaa caWGMbaabaGaai4jaaaaaOqaaiaadweaaaGaaiikaiaaikdacaWGob WaaSbaaSqaaiaadAgaaeqaaOGaaiykamaaCaaaleqabaGaamOyaaaa kiabgUcaRiaadkeacqaH1oqzdaqhaaWcbaGaamOzaaqaaiaacEcaaa GccaGGOaGaaGOmaiaad6eadaWgaaWcbaGaamOzaaqabaGccaGGPaWa aWbaaSqabeaacaWGJbaaaaaa@585F@

Where:

A = 1.3+0.7S

B = 1.5+0.5S

Mean stress effects are included using Morrow's mean stress approach of subtracting the mean stress from the fatigue strength coefficient. The mean stress on the maximum shear strain amplitude plane, σ n MathType@MTEF@5@5@+= feaahqart1ev3aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeq4Wdm3aaS baaSqaaiaad6gaaeqaaaaa@38D5@ , is one half of the axial mean stress leading to:

(21)
Δ γ max 2 + S Δ ε n = A σ f ' 2 σ n , m e a n E ( 2 N f ) b + B ε f ' ( 2 N f ) c MathType@MTEF@5@5@+= feaahqart1ev3aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaacq qHuoarcqaHZoWzdaWgaaWcbaGaciyBaiaacggacaGG4baabeaaaOqa aiaaikdaaaGaey4kaSIaam4uaiabfs5aejabew7aLnaaBaaaleaaca WGUbaabeaakiabg2da9iaadgeadaWcaaqaaiabeo8aZnaaDaaaleaa caWGMbaabaGaai4jaaaakiabgkHiTiaaikdacqaHdpWCdaWgaaWcba GaamOBaiaacYcacaWGTbGaamyzaiaadggacaWGUbaabeaaaOqaaiaa dweaaaGaaiikaiaaikdacaWGobWaaSbaaSqaaiaadAgaaeqaaOGaai ykamaaCaaaleqabaGaamOyaaaakiabgUcaRiaadkeacqaH1oqzdaqh aaWcbaGaamOzaaqaaiaacEcaaaGccaGGOaGaaGOmaiaad6eadaWgaa WcbaGaamOzaaqabaGccaGGPaWaaWbaaSqabeaacaWGJbaaaaaa@6159@