PSI - Issue 53

Francesco Collini et al. / Procedia Structural Integrity 53 (2024) 74–80

77

4

Author name / Structural Integrity Procedia 00 (2023) 000–000

∆ σ g

0.5

10

e 3 e 2 e 1

Plain strain, ν =0 . 3

e i

c w

R c

Fillet radius neglected

0

0

− 1

0

1

0.5

Load ratio , R

γ 3 γ 2 γ 1

γ i =1 − λ i

 

2

1+ R

, − 1 ≤ R ≤ 0

2 α

2 a

(1 − R )2

c w ( R ) =

 

2

1 − R

, 0

(1 − R )2

0

0

90

180

V-notch opening angle , 2 α [ ◦ deg]

∆ σ g

Fig. 2: SED reference system; the functions e i (2 α ) and γ i (2 α ) are approximated with the equations provided in Visentin et al. (2022).

• c w is a coe ffi cient accounting for the mean stress e ff ect; • e i , i = I , II , III are dimensionless coe ffi cients relevant to opening, sliding, and tearing local stresses, respectively, dependent on the V-notch opening angle 2 α and on Poisson’s ratio ν ; • E is the elastic modulus of the material; • γ i = 1 − λ i , i = I , II , III are the stress singularity exponents relevant to opening, sliding, and tearing modes respectively; λ i , i = I , II , III are the first eigenvalues of Williams’ Equation for mode I, II, and III, respectively; • K v i , i = I , II , III are the Notch Stress Intensity Factors (N-SIFs), relevant to opening, sliding, and tearing loading modes, respectively, and can be calculated with the engineering formula Atzori et al. (2005):

i = √ πα γ i

K v

· a γ i · σ

i = I , II , III

(2)

g ,

Let: ∆ w th be the range of averaged SED at threshold conditions in the structural volume and ∆ σ g , th be the threshold range of the gross nominal stress; by substituting Equation 2 into Equation 1, a relationship between ∆ w th , the stress raiser size a and the fatigue limit ∆ σ g , th can be obtained:

√ πα γ i · a



e i E 

2

3  i = 1

γ i · ∆ σ

g , th

∆ w th = c w ·

(3)

R γ i c

from which the fatigue limit in terms of the range of the nominal stress ∆ σ g , th can be determined as:

 2

γ 1 + γ 2 + γ 3 c e 2  α γ 2 a

∆ w th E · R

(4)

∆ σ g , th =

√ π c w  R

e 1  α γ 1 a

γ 1 

γ 2 

e 3  α γ 3 a ,   ∆ K v γ 3  2

2

2( γ 2 + γ 3 ) c

2( γ 3 + γ 1 ) c

2( γ 1 + γ 2 ) c

+ R

+ R

th , eq =  ∆ w th E · R

∆ K v

γ 1 + γ 2 + γ 3 c

th , eq √ c w

(5)

√ π  R ∆ K v th , eq √ c w  π · a v

=

1 γ i γ i a , i = 1 , 2 , 3

2( γ 2 + γ 3 ) c

2( γ 3 + γ 1 ) c

2( γ 1 + γ 2 ) c

2 γ 1 e ff , 1

2 γ 2 e ff , 2

2 γ 3 e ff , 3

a e ff , i = α

e 1 · a

+ R

e 2 · a

+ R

e 3 · a

,  

3  i = 1

i = 1 , 2 , 3 i j k

( γ j + γ k )

R 2 c

2 γ i e ff , i

a v

(6)

· e i · a

,

=

eq =

eq