PSI - Issue 5

M. Dabiri et al. / Procedia Structural Integrity 5 (2017) 385–392 M. Dabiri et al. / Structural Integrity Procedia 00 (2017) 000 – 000

4

388

This equation can be rewritten in the following form for cyclic loading defining K ε = ε

σ S

e and K σ =

∆ε∆σ = (K t ∆ S) 2 E

(2)

It was proposed by Topper et al. (Topper, et al., 1969) that a fatigue notch factor, K f , should be used instead of K t to take the effect of the size of the notch into account when the material component is subjected to cyclic loading. This value was proposed by Peterson (Peterson, 1974) as = 1 + 1+ −1 (3) where R is the notch root radius and a is the material characteristic length, which is given as a=0.0254 ( 2070 S u ) 1.8 (4) where S u is in MPa and a in mm. Other empirical equations to obtain the fatigue notch factor were proposed by Neuber (Neuber, 1958) and Siebel et al. (Siebel & Stieler, 1955) with fairly small quantitative differences, especially for steels with high strength levels. The combination of Neuber’s rule and CSSC in the Ramberg -Osgood form yields the following equation for the cyclic loading

1 n' ⁄

2

(∆σ ) 2

= (K f ∆ S)

∆σ 2K' )

E +2∆σ (

(5)

E

Another approximation procedure that can be used in a manner similar to Neuber’s rule is the SED method, as described by Glinka (Molski & Glinka, 1981). By setting the equation proposed by Glinka for cyclic loading, its intercept with the cyclic stress-strain curve gives

1 n' ⁄

2

(∆σ ) 2

= (K t ∆ S)

4∆σ n ' +1 (

∆σ 2K' )

E +

(6)

E

4.2. Modified versions of Neuber and SED rules to account for plane strain conditions

The circumferentially notched shaft investigated turned out to be a rather extreme case of constraint, which dictates the plane strain condition. This restricts the plastic flow because of the occurrence of biaxial stress state at the notch root under axial loading. It has been shown that the empirical methods, especially Neuber’s rule, make conservative predictions by overestimation of the notch strain (Jones, et al., 1998). The level of conservatism decreases when changing the constraint effect from plane strain to plane stress state. The SED method, however, shows better results under plane strain conditions but still acceptable results under plane stress state (Zeng & Fatemi, 2001). To improve the accuracy of the approximation methods under plane strain conditions, several proposals were made. Of those, Dowling et al. (Dowling, et al., 1979) proposed replacing the uniaxial stress-strain curve ( − ) with the “biaxial plane strain” stress -strain curve ( ′ − ′ ) ′ = ( √(1− 1 + 2 ) ) (7) ′ = ( 1− 2 √(1− + 2 ) ) (8) where is the generalized Poisson’s coefficient as follows