PSI - Issue 80

Yohei Sonobe et al. / Procedia Structural Integrity 80 (2026) 368–377 Y. Sonobe et al. / Structural Integrity Procedia 00 (2023) 000–000

376

9

3.3. Maximum normalized SIFs, F max I max o / a Figures 5 and 6 quantitatively illustrate the effects of Poisson’s ratio, ν , and the pressure distribution exponent, n , on the maximum normalized SIF, F max I ( n ) , and its depth, z max o / a . A clear, overarching trend is evident for the effect of ν . For all pressure orders, an increase in ν monoton ically increases both the peak value of F max I ( n ) and its depth from the surface. This behavior is governed by the interplay between two competing mechanisms: the SIF-reducing effect of the vertex singularity at the free surface and the SIF-enhancing effect of the three-dimensional constraint within the material’s interior. As ν increases, the constraint effect becomes more dominant, resulting in a higher and deeper SIF peak. Conversely, the exponent of the pressure distribution, n , has the opposite effect. An increase in n leads to a decrease in the peak SIF and brings its depth closer to the surface. This is because higher-order distributions concentrate the load near the crack front. This localization of load suppresses the overall crack opening and shifts the SIF peak toward the surface. Two additional behaviors are noteworthy. The uniform pressure case ( n = 0 ) produces a peak at a significantly deeper location than any other order, a result attributed to the widespread crack opening it generates. Furthermore, as the exponent n becomes larger, both the peak value of SIF and its location in depth seem to converge toward stable, asymptotic values. 4. Conclusions In this study, we developed the body force method (BFM) based high-precision technique to analyze the mode I stress intensity factor for a deep surface crack under various power-law pressures ( n = 0 , 1 , . . . , 5 ). The method’s key innovation is a novel fundamental density function derived from the exact 2D plane strain solution for the crack opening displacement. This approach yielded robust and well-converged solutions, enabling a clear investigation of the problem’s underlying physics. The key finding of this study is the quantitative elucidation of the competition governing the SIF dis tribution: the SIF-reducing vertex singularity at the free surface versus the SIF-enhancing effect of the 3D internal constraint away from the surface. Specifically, increasing Poisson’s ratio ν strengthens the internal constraint, leading to a higher and deeper SIF peak. In contrast, increasing the pressure order n localizes the load, which lowers the peak and shifts the peak SIF location much closer to the free surface. ( n ) , and their depths, z

1 . 25

1 . 25

n =0 n =2 n =4

n =1 n =3 n =5

1 . 20

1 . 20

1 . 15

1 . 15

F max I ( n )

F max I ( n )

1 . 10

1 . 10

1 . 05

1 . 05

1 . 00

1 . 00

0 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

0 . 0

0 . 1

0 . 2

0 . 3

0 . 4

0 . 5

ν

ν

(a) Even-order pressure

(b) Odd-order pressure

Fig. 5. Influence of Poisson’s ratio ν on the maximum normalized mode I SIF, F max I distributions ( n = 0 , 2 , 4 ). (b) Results for odd-order pressure distributions ( n = 1 , 3 , 5 ).

( n ) . (a) Results for even-order pressure