PSI - Issue 80

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

373

6 Y. Sonobe et al. / Structural Integrity Procedia 00 (2023) 000–000 corresponding SIF from a 2D plane strain solution, K ∞ I ( n ) , subjected to the same pressure distribution as follows: F I ( n ) = K I K ∞ I ( n ) (14) K ∞ I ( n ) and δ u ∞ ( η, n ) were calculated for each pressure order ( n = 0 , 1 , . . . , 5 ) and are summarized in Table 1. The numerical analysis was conducted with a domain cutoff depth of ζ thresh / a = 10 . This value ensures that the relative error of the weight function, W i , in the deepest elements remains less than 0.1% with respect to the plane strain solution for all tested parameter combinations ( ν, n ). Three mesh configurations, shown in Fig. 2 were used to ensure the mesh convergence of the solution. The base mesh (Fig. 2(a)) employs a non-uniform division strategy. To capture sharp gradients in the weight function, the smallest elements are concentrated near the surface vertex ( a , 0) . This refinement was achieved using a geometric progression with a common ratio of 1.2 in both the y and z directions, resulting in 30 × 40 divisions ( N e = 1200 ). The minimum element sizes were ∆ y / a = 8 . 5 × 10 − 4 and ∆ z / a = 1 . 34 × 10 − 3 . In addition to the base mesh, a fine mesh ( N e = 4800 ) and an extra-fine mesh ( N e = 19200 ) were prepared by subdividing the base elements by factors of two and four in each direction, respectively (Figs. 2(b) and (c)). 3.1. Validity and convergence To evaluate the validity and convergence of the proposed method, F I ( n ) was calculated for ν = 0 . 5 , where the free-surface effect can be most apparent. Figure 3 shows the convergence behavior of the SIF in panels (a) through (f) for cases where the order of the internal pressure distribution was varied from n = 0 to n = 5 . The ordinate represents F I ( n ) , while the abscissa shows the normalized depth from the free surface, z 0 / a . Each panel compares the results for the three distinct mesh densities: base ( N e = 1200 ), fine ( N e = 4800 ), and extra-fine ( N e = 19200 ). The results demonstrate good convergence to a single solution curve with increase of N e . This robust convergence is a direct result of the proposed formulation. By design, the fundamental density function incorporates the analytical 2D plane-strain solution for the crack opening displacement. Consequently, the numerically calculated weight function W i becomes a smooth, well-behaved function, ensuring stable and accurate solutions that are insensitive to mesh refinement. Furthermore, the analysis accurately captures the theoretical behavior near the free surface ( z 0 / a 0 ), where F I ( n ) approaches zero. While minor discrepancies between the meshes are visible near the peak of F I ( n ) , the solution is considered well-converged with the extra-fine mesh ( N e = 19200 ). Therefore, all subsequent analyses in this study used this mesh configuration. 3.2. Distribution of F I ( n ) with depth, z o / a The calculated distributions of F I ( n ) , shown in Figures 4(a)-(f), illustrate several key phenomena of a 3D crack. At the free surface ( z o / a = 0 ), F I ( n ) vanishes, consistent with the theory of a weak vertex singularity (Benthem, 1977). The SIF then reaches a peak value just below the surface. This peak arises from the transition in the stress state through the thickness, as elucidated by studies like Kwon and Sun (2000). Near the free surface, the state approaches plane stress conditions, which allows for a larger COD. However, the region immediately at the crack tip is dominated by the high constraint of a plane strain state. The SIF peak occurs in the transition zone where the tendency for a larger COD, driven by the surface’s lower constraint, interacts with the rapid onset of the interior’s high-constraint plane strain field. This effect becomes more pronounced with a larger Poisson’s ratio ν . Finally, deep within the material ( z o / a 1 ), the SIF asymptotically approaches unity as the 3D solution reverts to the 2D plane strain solution. Beyond the effect of ν , the specific shape of the SIF distribution is also sensitive to the exponent, n , and its parity. For even-ordered pressure distributions ( n = 0 , 2 , 4 ), the symmetric crack opening allows the free-surface effect to penetrate deeper into the material. As n increases, the applied load becomes more concentrated near the crack front, which causes the SIF peak to both decrease in magnitude and shift closer to the surface.