PSI - Issue 75

Andrew Halfpenny et al. / Procedia Structural Integrity 75 (2025) 234–244 Author name / Structural Integrity Procedia (2025)

238

5

3 2

′ =1+( −1)( +∆ − ∗

2 + +∆ )

(10)

Fig. 2a shows the distribution of the residual compressive stress ahead of the crack tip produced by a fatigue cycle. 0 1⃗⃗⃗⃗⃗⃗⃗⃗⃗ represents portion of the fatigue cycle that drives the crack propagation (crack opening), while 1 2⃗⃗⃗⃗⃗⃗⃗⃗⃗ is the portion of the fatigue cycle that produces a residual compressive stress zone of length ahead of the crack tip (crack closure). The residual compressive stress reaches its maximum (absolute) value at a distance .

(a)

(b)

Fig. 2. Distribution of the residual compressive stress ahead of the crack tip produced by a fatigue cycle (a) and relationship between the linear elastic stress, s, and the elastic- plastic stress, σ, during loading and unloading (b).

In the simpler uniaxial case, the combination of the Neuber rule and the Ramberg-Osgood cyclic plasticity model at the basis of the multiaxial cyclic crack tip plasticity model used in the retardation model included on the ‘Total Life’ method, leads to the relationships shown in Fig. 2b. In the case of a bidimensional stress field, instead, the Ramberg-Osgood cyclic plasticity model can be expressed as follows: = − +( − )( ′ ) 1 ′ ; = − +( − )( ′ ) 1 ′ (11) where and are the Poisson’s ratios for elastic and plastic strain, respectively, and =√ 2 − + 2 . The crack tip residual stress is defined as ( ) = −Δ , where is the crack tip opening stress and is determined by considering Neuber’s rule ( = and = ) and numerically solving Eq. 11. Δ , instead, is calculated by considering Neuber’s rule ( ∆ ∆ = ∆ ∆ and ∆ ∆ = ∆ ∆ ) and numerically solving the following equations: Δ = Δ − Δ +2( Δ − Δ Δ )( Δ 2 ′ ) 1 ′ ; Δ = Δ − Δ +2( Δ − Δ Δ )( Δ 2 ′ ) 1 ′ (12)