PSI - Issue 75

Francesco Collini et al. / Procedia Structural Integrity 75 (2025) 375–381 F. Collini et al. / Structural Integrity Procedia 00 (2025) 000–000

377

3

• e i , i = 1 , 2 , 3 are dimensionless coe ffi cients relevant to mode I, II, and III local stresses, respectively, dependent on the V-notch opening angle 2 α and on Poisson’s ratio ν (Lazzarin and Zambardi (2001); Visentin et al. (2023)); • E is the Young’s modulus of the base material; • K v i , i = 1 , 2 , 3 are the Notch-Stress Intensity Factors (N-SIFs), relevant to mode I, II, and III local stresses, respectively (Gross and Mendelson (1972)); • γ i = 1 − λ i , i = 1 , 2 , 3 are the stress singularity exponents relevant to mode I, II, and III local stresses, respec tively; λ i , i = 1 , 2 , 3 are the first eigenvalues of the stress singularity Equations (Williams (1952); Gross and Mendelson (1972)) for mode I, II, and III local stresses; • R c , i , i = 1 , 2 , 3 are the radii of the structural volume relevant to mode 1, 2, and 3 local stresses that can be estimated by comparing the averaged SED in threshold condition, ∆ w th , of the defect-free material and that of the sharply notched material as: R c , i =    c w e i E ∆ K v i , th 2 ∆ w th     1 2 γ i  (2) By considering the engineering expressions of the K v 1 as (Atzori et al. (2005)): (4) where: α γ 1 is a shape factor accounting for the notch geometry-remote loading condition system, and a is the reference dimension of the notch, or alternatively by exploiting the definition of the a e ff (Atzori et al. (2003, 2005)). Recently, Collini et al. (2025) derived the following generalized equation by including Eq. (4) for each loading mode in Eq. (1): K v K v 1 = α γ 1 σ g √ π a 1 = σ g √ π a γ 1 e ff γ 1 (3)

√ E ∆ w th  3 √ c w 2 ( 1 − δ j , i ) γ i c , i

γ i c , i

i = 1 R

(5)

∆ σ g , th =

√ π   3

j = 1   3

 e j  a

γ j e ff τ, j 

2

γ j e ff σ, j

i = 1 R

+ λ σ g a

= 

    ∆ K a v

E ∆ w th  3 √ c w

γ i c , i

i = 1 R

v eq ∆ w , th

∆ K v

eq ∆ w , th

(6)

3  j = 1    3  i = 1

   e j  a

,

 π a v

=

γ j e ff τ, j 

( 1 − δ j , i ) γ i

2

R 2

γ j e ff σ, j

+ λ σ g a

eq ∆ w

=

eq ∆ w

c , i

which has the same form as the characteristic notch fracture mechanics equation (Atzori et al. (2005)), in which ∆ K v eq ∆ w , th and a v eq ∆ w represent the equivalent N-SIF and the equivalent crack size, respectively. These take into account all three loading modes, material properties through R c , i , notch size, the geometry of the components through a e ff , i , and the biaxiality ratio of the remote multiaxial loads through λ σ g = ∆ τ g ∆ σ g . By using the definition of the equivalent remote stress range as proposed in Collini et al. (2025):

1 √ c w 

∆ w th · 2 E

(7)

∆ σ eq ∆ w , 0 =

1 + 2 λ 2

σ g (1 + ν )

it was possibile to defined an a v

0 , eq ∆ w and introduce an El Haddad-like correction to Eq. 6 as follows: ∆ σ g , th ∆ σ eq ∆ w , 0 = 1  a v eq ∆ w a v 0 , eq ∆ w + 1

(8)