Stress-Life (S-N) Approach

S-N Curve

The S-N curve, first developed by Wöhler, defines a relationship between stress and number of cycles to failure. Typically, the S-N curve (and other fatigue properties) of a material is obtained from experiment; through fully reversed rotating bending tests. Due to the large amount of scatter that usually accompanies test results, statistical characterization of the data should also be provided (certainty of survival is used to modify the S-N curve according to the standard error of the curve and a higher reliability level requires a larger certainty of survival).


Figure 1. S-N Data from Testing
When S-N testing data is presented in a log-log plot of alternating nominal stress amplitude S a or range S R versus cycles to failure N , the relationship between S and N can be described by straight line segments. Normally, a one or two segment idealization is used.


Figure 2. One Segment S-N Curves in Log-Log Scale
(1)
S = S 1 ( N f ) b 1

for segment 1

Where, S is the nominal stress range, N f are the fatigue cycles to failure, b l is the first fatigue strength exponent, and S I is the fatigue strength coefficient.

The S-N approach is based on elastic cyclic loading, inferring that the S-N curve should be confined, on the life axis, to numbers greater than 1000 cycles. This ensures that no significant plasticity is occurring. This is commonly referred to as high-cycle fatigue.

S-N curve data is provided for a given material using the Materials module.

Cycle Counting

Cycle counting is used to extract discrete simple "equivalent" constant amplitude cycles from a random loading sequence. One way to understand "cycle counting" is as a changing stress-strain versus time signal. Cycle counting will count the number of stress-strain hysteresis loops and keep track of their range/mean or maximum/minimum values.

Rainflow cycle counting is the most widely used cycle counting method. It requires that the stress time history be rearranged so that it contains only the peaks and valleys and it starts either with the highest peak or the lowest valley (whichever is greater in absolute magnitude). Then, three consecutive stress points ( S 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaIXaaabeaaaaa@37B5@ , S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaIXaaabeaaaaa@37B5@ , and S 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaIXaaabeaaaaa@37B5@ ) will define two consecutive ranges as Δ S 1 = | S 1 S 2 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqqHuoarcaWGtbWaaSbaaSqaaiaaigdaaeqaaOGaeyypa0ZaaqWa aeaacaWGtbWaaSbaaSqaaiaaigdaaeqaaOGaeyOeI0Iaam4uamaaBa aaleaacaaIYaaabeaaaOGaay5bSlaawIa7aaaa@44AE@ and Δ S 2 = | S 2 S 3 | MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqqHuoarcaWGtbWaaSbaaSqaaiaaikdaaeqaaOGaeyypa0ZaaqWa aeaacaWGtbWaaSbaaSqaaiaaikdaaeqaaOGaeyOeI0Iaam4uamaaBa aaleaacaaIZaaabeaaaOGaay5bSlaawIa7aaaa@44B1@ |. A cycle from S 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaIXaaabeaaaaa@37B5@ to S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaaIXaaabeaaaaa@37B5@ is only extracted if Δ S 1 Δ S 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacqqHuoarcaWGtbWaaSbaaSqaaiaaigdaaeqaaOGaeyizImQaeuiL dqKaam4uamaaBaaaleaacaaIYaaabeaaaaa@40E1@ . Once a cycle is extracted, the two points forming the cycle are discarded and the remaining points are connected to each other. This procedure is repeated until the remaining data points are exhausted.


Figure 3. Rainflow Cycle Counting Method to Determine Cycles

Equivalent Nominal Stress

Since S-N theory deals with uniaxial stress, the stress components need to be resolved into one combined value for each calculation point, at each time step, and then used as equivalent nominal stress applied on the S-N curve.

Various stress combination types are available with the default being "Absolute maximum principle stress". "Absolute maximum principle stress" is recommended for brittle materials, while "Signed von Mises stress" is recommended for ductile material. The sign on the signed parameters is taken from the sign of the Maximum Absolute Principal value.

Mean Stress Correction

Generally, S-N 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 take into account the effect of non-zero mean stresses.

The Gerber parabola and the Goodman line in Haigh's coordinates are widely used when considering mean stress influence, and can be expressed as:

Gerber:(2)
S e = S r ( 1 ( S m S u ) 2 )
Goodman:(3)
S e = S r ( 1 S m S u )
Where,
S m
Mean stress given by S m = ( S m a x + S m i n ) / 2
S r
Stress Range given by S r = S m a x S m i n
S e
Stress range after mean stress correction (for a stress range S r and a mean stress S m )
S u
Ultimate strength

The Gerber method treats positive and negative mean stress correction in the same way that mean stress always accelerates fatigue failure, while the Goodman method ignores the negative means stress. Both methods give conservative result for compressive means stress. The Goodman method is recommended for brittle material while the Gerber method is recommended for ductile material. For the Goodman method, if the tensile means stress is greater than UTS, the damage will be greater than 1.0. For the Gerber method, if the mean stress is greater than UTS, the damage will be greater than 1.0, with either tensile or compressive.

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


Figure 4. Haigh Diagram and Mean Stress Correction Methods

Parameters affecting mean stress influence can be defined using the mean stress correction in the Fatigue Module dialog and the Assign Material dialog.

GERBER2:

Improves the Gerber method by ignoring the effect of negative mean stress.

SODERBERG:

Is slightly different from GOODMAN; the mean stress is normalized by yield stress instead of ultimate tensile stress.(4)
S e = S a ( 1 S m S y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGLbaabeaakiabg2da9maalaaabaGaam4uamaaBaaaleaa caWGHbaabeaaaOqaamaabmaabaGaaGymaiabgkHiTmaalaaabaGaam 4uamaaBaaaleaacaWGTbaabeaaaOqaaiaadofadaWgaaWcbaGaamyE aaqabaaaaaGccaGLOaGaayzkaaaaaaaa@4246@
Where,
S e MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGLbaabeaaaaa@37E5@
Equivalent stress amplitude
S a MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGHbaabeaaaaa@37E1@
Stress amplitude
S m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGTbaabeaaaaa@37ED@
Mean stress
S y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWG5baabeaaaaa@37F9@
Yield stress

FKM:

If only one slope field is specified for mean stress correction, the corresponding Mean Stress Sensitivity value ( M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C9@ ) for Mean Stress Correction is set equal to Slope in Regime 2 (Figure 5). Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ( R = S min / S max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGsbGaeyypa0Jaam4uamaaBaaaleaaciGGTbGaaiyAaiaac6ga aeqaaOGaai4laiaadofadaWgaaWcbaGaciyBaiaacggacaGG4baabe aaaaa@431F@ ) values. The Corrected value is then used to choose the S-N curve for the damage and life calculation stage.


Figure 5.
Note: The FKM equations below illustrate the calculation of Corrected Stress Amplitude ( S e A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaaaaa@38AC@ ). The actual value of stress used in the Damage calculations is the Corrected stress range (which is 2 S e A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgw SixlaadofadaqhaaWcbaGaamyzaaqaaiaadgeaaaaaaa@3BB2@ ). These equations apply for SN curves input by the user (by default, any user-defined SN curve is expected to be input for a stress ratio of R=1.0).

There are 2 available options for FKM correction in HyperLife. They are activated by setting FKM MSS to 1 slope/4 slopes in the Assign Material dialog.

If only one slope is defined and if mean stress correction on an SN module is set to FKM:
Regime 1 (R > 1.0)
S e A = S a ( 1 M ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9iaadofadaWgaaWcbaGa amyyaaqabaGcdaqadaqaaiaaigdacqGHsislcaWGnbaacaGLOaGaay zkaaaaaa@3FB3@
Regime 2 (-∞ ≤ R ≤ 0.0)
S e A = S a + M * S m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9iaadofadaWgaaWcbaGa amyyaaqabaGccqGHRaWkcaWGnbGaaiOkaiaadofadaWgaaWcbaGaam yBaaqabaaaaa@4008@
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@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9maabmaabaGaaGymaiab gUcaRiaad2eaaiaawIcacaGLPaaadaWcaaqaaiaadofadaWgaaWcba GaamyyaaqabaGccqGHRaWkdaqadaqaamaaliaabaGaamytaaqaaiaa iodaaaaacaGLOaGaayzkaaGaam4uamaaBaaaleaacaWGTbaabeaaaO qaaiaaigdacqGHRaWkdaWccaqaaiaad2eaaeaacaaIZaaaaaaaaaa@4902@
Regime 4 (R ≥ 0.5)
S e A = 3 S a ( 1 + M ) 2 3 + M MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaakiabg2da9maalaaabaGaaG4maiaa dofadaWgaaWcbaGaamyyaaqabaGcdaqadaqaaiaaigdacqGHRaWkca WGnbaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaGcbaGaaG4m aiabgUcaRiaad2eaaaaaaa@43D9@
Where,
S e A MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaGaamyqaaaaaaa@38AC@
Stress amplitude after mean stress correction (Endurance stress)
S m
Mean stress
S a
Stress amplitude
M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytaaaa@36C8@
Slope entered for region 2

If all four slopes are specified for mean stress correction, the corresponding Mean Stress Sensitivity values are slopes for controlling all four regimes. Based on FKM-Guidelines, the Haigh diagram is divided into four regimes based on the Stress ratio ( R = S min / S max MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKf MBHbqefqvATv2CG4uz3bIuV1wyUbqedmvETj2BSbqefm0B1jxALjhi ov2DaebbnrfifHhDYfgasaacH8srps0lbbf9q8WrFfeuY=Hhbbf9v8 qqaqFr0xc9pk0xbba9q8WqFfea0=yr0RYxir=Jbba9q8aq0=yq=He9 q8qqQ8frFve9Fve9Ff0dmeaacaGacmGadaWaaiqacaabaiaafaaake aacaWGsbGaeyypa0Jaam4uamaaBaaaleaaciGGTbGaaiyAaiaac6ga aeqaaOGaai4laiaadofadaWgaaWcbaGaciyBaiaacggacaGG4baabe aaaaa@431F@ ) values. The Corrected value is then used to choose the S-N curve for the damage and life calculation stage.

If four slopes are defined and mean stress correction is set to FKM:
Regime 1 (R > 1.0)
S e = ( S a + M 1 S m ) ( ( 1 M 2 ) / ( 1 M 1 ) ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaaaaOGaeyypa0ZaaeWaaeaacaWGtbWaaSbaaSqa aiaadggaaeqaaOGaey4kaSIaamytamaaBaaaleaacaaIXaaabeaaki aadofadaWgaaWcbaGaamyBaaqabaaakiaawIcacaGLPaaadaqadaqa amaalyaabaWaaeWaaeaacaaIXaGaeyOeI0IaamytamaaBaaaleaaca aIYaaabeaaaOGaayjkaiaawMcaaaqaamaabmaabaGaaGymaiabgkHi Tiaad2eadaWgaaWcbaGaaGymaaqabaaakiaawIcacaGLPaaaaaaaca GLOaGaayzkaaaaaa@4CA0@
Regime 2 (-∞ ≤ R ≤ 0.0)
S e = S a + M 2 S m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaaaaOGaeyypa0Jaam4uamaaBaaaleaacaWGHbaa beaakiabgUcaRiaad2eadaWgaaWcbaGaaGOmaaqabaGccaWGtbWaaS baaSqaaiaad2gaaeqaaaaa@3F86@
Regime 3 (0.0 < R < 0.5)
S e = ( 1 + M 2 ) S a + M 3 S m 1 + M 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaDa aaleaacaWGLbaabaaaaOGaeyypa0ZaaeWaaeaacaaIXaGaey4kaSIa amytamaaBaaaleaacaaIYaaabeaaaOGaayjkaiaawMcaamaalaaaba Gaam4uamaaBaaaleaacaWGHbaabeaakiabgUcaRiaad2eadaWgaaWc baGaaG4maaqabaGccaWGtbWaaSbaaSqaaiaad2gaaeqaaaGcbaGaaG ymaiabgUcaRiaad2eadaWgaaWcbaGaaG4maaqabaaaaaaa@47E3@
Regime 4 (R ≥ 0.5)
S e = ( ( 1 + 3 M 3 ) S a M 4 ( 1 + 3 M 3 ) S m ) ( ( 1 + M 2 ) / ( ( 1 3 M 4 ) ( 1 + M 3 ) ) ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGLbaabeaakiabg2da9maabmaabaWaaeWaaeaacaaIXaGa ey4kaSIaaG4maiaad2eadaWgaaWcbaGaaG4maaqabaaakiaawIcaca GLPaaacaWGtbWaaSbaaSqaaiaadggaaeqaaOGaeyOeI0Iaamytamaa BaaaleaacaaI0aaabeaakmaabmaabaGaaGymaiabgUcaRiaaiodaca WGnbWaaSbaaSqaaiaaiodaaeqaaaGccaGLOaGaayzkaaGaam4uamaa BaaaleaacaWGTbaabeaaaOGaayjkaiaawMcaamaabmaabaWaaeWaae aacaaIXaGaey4kaSIaamytamaaBaaaleaacaaIYaaabeaaaOGaayjk aiaawMcaaiaac+cadaqadaqaamaabmaabaGaaGymaiabgkHiTiaaio dacaWGnbWaaSbaaSqaaiaaisdaaeqaaaGccaGLOaGaayzkaaWaaeWa aeaacaaIXaGaey4kaSIaamytamaaBaaaleaacaaIZaaabeaaaOGaay jkaiaawMcaaaGaayjkaiaawMcaaaGaayjkaiaawMcaaaaa@5FC2@
Where,
S e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaam4uamaaBa aaleaacaWGLbaabeaaaaa@37E4@
Fully reversed fatigue strength (Endurance stress)
S m
Mean stress
S a
Stress amplitude
M i MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamytamaaBa aaleaacaWGPbaabeaaaaa@37E3@
Slopes at each region

Damage Accumulation Model

Palmgren-Miner's linear damage summation rule is used. Failure is predicted when:(5)
D i = n i N i f 1.0
Where,
N i f
Materials fatigue life (number of cycles to failure) from its S-N curve at a combination of stress amplitude and means stress level i .
n i
Number of stress cycles at load level i .
D i
Cumulative damage under n i load cycle.

The linear damage summation rule does not take into account the effect of the load sequence on the accumulation of damage, due to cyclic fatigue loading. However, it has been proved to work well for many applications.