PSI - Issue 47

Robert Eriksson et al. / Procedia Structural Integrity 47 (2023) 227–237 R. Eriksson, A. Azeez / Structural Integrity Procedia 00 (2023) 000–000

234

8

In the same way, a “residual stress intensity factor”, K res , is introduced as

K res = K ρ 1 + K ρ 2

(9)

which subjects the crack to an internal load which must be added to any externally applied load. Note that K ρ 1 < 0 and K ρ 2 > 0. In summary: • For the LCF cycle, K res = 0 since P 1 = P 2 = ⇒ σ ys1 = σ 2 = ⇒ K ρ 1 = − K ρ 2 . • For the LUCF cycle, K res 0 since K ρ 1 K ρ 2 . After the load has been changed to P 2 (or in the case of the LCF cycle, have been hold constant at P 2 = P 1 ), the temperature drops to T 2 . The fracture load, P f , is calculated iteratively from  K int = P f √ BB N W 2 + 0 . 886 + 4 . 64 a + ρ W − a + ρ 2 a + ρ 3 a + ρ

3 2 

4 

a + ρ W  1 − a + ρ W 

W 

W 

W 

+ 14 . 72 

− 5 . 6 

13 . 32 

      

( K int + K res ) κ 4 µ √ ρπ

ϵ pzf = estimated strain σ ysf = | σ flow ( ϵ pzf ) | from constitutive model; see Appendix A K ρ f = − σ ysf  2( a + ρ ) π    β 1  1 − a a + ρ  1 2 + β 2 3  1 − a a + ρ  3 2 K int + K ρ f = 0 criterion for stopping iteration

(10)

7 2   

5 

a a + ρ 

7 

a a + ρ 

5 2

β 3

β 4

1 −

1 −

+

+

This must be solved iteratively since K int is unknown leading to an unknown plastic zone strain, ϵ pzf , and, consequently, an unknown yield stress, σ ysf . The variable K int is only an intermediate result used to find P f . Residual stresses can be accounted for in two ways: either they are added to the load or they modify material strength. Here, the latter approach is chosen and K res is added to the stress intensity used to calculate the average plastic zone strain, ϵ pzf . Note that, for LUCF, K res is typically K res < 0 thus reducing ϵ pzf and consequently reducing σ flow , thereby reducing σ ysf (i.e. plasticity occurs at a lower load when K res < 0). Finally, the critical stress intensity, K f , is calculated from P f at the correct crack length, a , (as oppose to at a + ρ ).

3 2 

4 

13 . 32 

a W 

a W 

a W 

+ 14 . 72 

− 5 . 6 

a W

2 +

2

3

a W −

P f √ BB N W

K f =

0 . 886 + 4 . 64

(11)

 1 − a

W 

4. Comparison to experimental data and discussion

The model was applied to the test data listed in Table 1. It turned out that the influence of the minimum temperature (i.e. T 2 = 20 ◦ Cand T 2 = 50 ◦ C) was negligible, so these were treated as a common dataset. The comparison between experimental results and the model prediction is shown in Fig. 6. For reference, Fig. 6 also includes the Wallin model which is Wallin (2003)    K f = 0 . 15 K Ic +  K Ic ( K 1 − K 2 ) + K 2 if K 2 ≥ K 1 − K Ic then set K 2 = K 1 if K f ≤ K Ic then set K f = K Ic (12)