PSI - Issue 80

372 Yohei Sonobe et al. / Procedia Structural Integrity 80 (2026) 368–377 Y. Sonobe et al. / Structural Integrity Procedia 00 (2023) 000–000 5 Based on the assumptions above, discretizing the governing integral equation Eq. (12) yields the following system of linear equations for the unknown weights W j : N e j = 1 W j Ω j B ( η, n ) σ γ xx ( y i , z i ,η,ζ ) + B ( − η, n ) σ γ xx ( y i , z i , − η,ζ ) d η d ζ = − p 0 y i a n + a − a B ( η, n ) ∞ ζ thresh σ γ xx ( y i , z i ,η,ζ ) d ζ d η (13) In Eq. (13), the left-hand side is the sum of the unknown weights W j multiplied by their corresponding influ ence coefficients. In the right-hand side, the first term is the internal pressure which varies with coordinate variable y i , and the second term is a correction term to account for the influence W ( η,ζ ) = p 0 in the deep crack face region ( ζ > ζ thresh ). Once this system of linear equations is solved for the weights W j , the value of the weight function at the crack tip, W ( a ,ζ ) , is determined. Finally, by substituting W ( a ,ζ ) into Eq. (10) K I ( ζ ) can be obtained. 3. Numerical results and discussion To generalize the results and elucidate the pure surface effect, the analysis is presented in terms of a normalized mode I SIF, F I ( n ) . This factor is obtained by normalizing the calculated 3D SIF, K I , with the

Table 1. Reference solutions for crack opening displacement δ u ∞ ( η, n ) and mode-I SIF K ∞ I ( n ) for a 2D line crack in an infinite plate under n -th order power-law pressure p 0 ( η/ a ) n in plane strain condition.

E δ u ∞ ( η, n ) 4(1 − ν 2 ) a 2 − η 2 p 0

K ∞ I ( n ) p 0 √ π a

n

0 1 2 3 4 5

1

1

1/2 1/2 3/8 3/8

( η/ a ) / 2

2 6 2 ( η/ a ) 8 + 8( η/ a )

1 + 2( η/ a ) 1 + 2( η/ a ) 3 + 4( η/ a ) 2 3 + 4( η/ a ) 2

4 40 4 ( η/ a ) 48

5/16

+ 8( η/ a )

O

0 . 5

1 . 0

O

0 . 5

1 . 0

O

0 . 5

1 . 0

z/a

z/a

z/a

0 . 5

0 . 5

0 . 5

1 . 0

1 . 0

1 . 0

z o /a

z o /a

z o /a

Crack front

Crack front

Crack front

y/a

y/a

y/a

(a) N e =1200

(b) N e =4800

(c) N e =19200

Fig. 2. Mesh configurations used for the convergence study, showing refinement near the free surface vertex ( y / a = 1 , z / a = 0 ). (a) Base mesh ( N e = 1200 ). (b) Fine mesh ( N e = 4800 ). (c) Extra-fine mesh ( N e = 19200 ).