PSI - Issue 51

Abdelhak Nehila et al. / Procedia Structural Integrity 51 (2023) 152–159 A.Nehila and W. Li / Structural Integrity Procedia 00 (2022) 000–000

156

5

where Δ K th represents the threshold value and Δ σ f is the fatigue limit range (Δ σ f = 2 σ for R = −1). Eq. (3) is aimed at a metal plate with an infinite length and a central edge crack. Therefore, in order to extend the model for an edge notched specimen the surface correction factor, α edge was introduced with α edge = 1.122 as described by Atzori et al (2003). Thus, Eq. (3) can be rewritten as follows: 2 2 th th edge f f Δ Δ 1 1 ' ( ) ( ) Δ 1.258 Δ K K a π σ π σ α = = ⋅ （ 4 ） where σ f = 500 MPa and Δ K th = 7.82 MPam 1/2 under R = −1. The crack size a ’ is obtained, a ’ = 15.48 μ m. For the stress intensity factor of notched specimens Δ K , the traditional method is supposed to be applied on a piece of metallic plate with the unnotched side is considered infinitely long. Janssen et al. (2004) have given the general calculation formula of the stress intensity factor range is given as follows: 2 edge t { {1 exp[ ( 1)]}} a K a d K d α σ π Δ = Δ + − − − （ 5 ） where a is the crack length, d is the depth of notch, d = 3 mm and K t is the stress concentration factor. Eq. （ 5 ） is extended to calculate the stress intensity factor of a notched specimen with finite dimensions, and the corresponding finite dimension correction factor is introduced. Eq. (5) is rewritten as: 2 t edge f 2 f Δ Δ { {1 xp[ ( 1)]}} a K K a d e d α α σ π α = + − − − （ 6 ） With taking α f as the correction factor for an edge crack with length of ( a + d ) in finite dimension specimen. For a specimen with a circular v-shaped notch, the expression of α f is shown as follows: 1/2 f 2 3 edge 1 1 1 3 1 5 1 11 1 ( ) 2 2 8 14 15 ξ ξ ξ α ξ ξ ξ α − = + + − + （ 7 ） where （ 8 ） Where W is the cross-sectional diameter of the unnotched part, W = 12 mm. Thus the correction factor can be calculated, α f = 1.0069. Considering El Haddad’s idea for the small crack correction, the original crack length a is replaced by the effective crack length a + a ’, and Eq. (6) can be rewritten as follows: 2 t edge f 2 f ' Δ Δ { ' {1 exp[ ( 1)]}} a a K K σ π a a d d α α α + = + + − − − （ 9 ） According to EI Haddad’s et al. model (1979), when the realistic crack size a approaches to zero, then we should consider that the stress intensity factor range Δ K with small crack correction (Eq. (9)) approaches the fatigue limit of notched specimen. Therefore, the stress intensity factor range Δ K is equal to the threshold value and the following relation is obtained according to Eqs (2) and (9) : 2 t edge f th edge f 2 f ' Δ Δ { ' {1 exp[ ( 1)]}} Δ Δ ' a a K K α α σ π a a d K α σ π a d α + = + + − − − = = （ 10 ） Taking a = 0, and with mathematical transformation the following equation is given as: 2 ( W W d a − × + ' ) ξ =

Δ 1 σ σ α = Δ

1

（ 11 ）

2

'

K

d a

a

f

f

t 2

{1 {1 exp[

(

1)]}}

+ − −

−

'

d α

f

In order to consider the effect of stress concentration on the S-N curve the classical fatigue research analysis introduce an important parameter called ‘notch factor, K f ’. The definition of K f is given by Bannantine (1990) as follows: