PSI - Issue 80

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

369

2 Y. Sonobe et al. / Structural Integrity Procedia 00 (2023) 000–000 One source of complexity in 3D crack problems is the stress field singularity at the vertex where the crack front intersects with the free surface. Here, the stress singularity deviates from the classical r − 1 / 2 form associated with 2D cracks. It has been theoretically shown that this results in a weaker order of singularity for the mode I SIF, which is known to approach zero asymptotically at the free surface (Benthem, 1977, 1980; Bažant and Estenssoro, 1979; He et al., 2015). In practical geometries, such as a through-thickness crack in a finite plate, this local vertex singularity is compounded by global geometric factors, such as the interaction between opposing crack face. This interplay results in a characteristic SIF profile that decreases sharply near the free surface before peaking in the plate’s interior (Kwon and Sun, 2000; Shivakumar and Raju, 1990). Consequently, decoupling the contribution of the surface effect, including the vertex singularity, from these global geometric effects remains a significant challenge. The primary objective of the present study is to isolate and quantify the intrinsic influence of a free surface on the SIF distribution, independent of the geometric complexities inherent to finite 3D crack problems. This is achieved by employing an idealized model that eliminates these confounding factors: an infinitely deep surface crack oriented perpendicular to the free surface of a semi-infinite solid. The crack is subjected to a mode I internal pressure that is uniform along its depth and varies across its width according to an arbitrary power law. This loading scheme is chosen because power-law functions can serve as a basis to represent any arbitrary stress profile via a series expansion. The resulting solution for this fundamental loading case can then be extended to general loading scenarios through the principle of superposition (Rice, 1972). The analysis is conducted using a novel numerical method developed based on the body force method (BFM) (Nisitani, 1968). Our approach is composed of two key components: a new fundamental density function derived from the exact crack opening displacement of a 2D crack under plane strain conditions and subjected to the same power-law pressure distribution, and Mindlin’s solution for a point force in a semi infinite solid (Mindlin, 1936). The latter satisfies a traction-free condition in the z = 0 plane automatically, which eliminates the need for surface discretization. A key advantage of this methodology is that it yields robust and accurate solutions while only requiring discretization of the crack face. The numerical stability and accuracy are maintained even for pressure distributions with high-order power-law exponents. Using this method, we systematically investigate the SIF distribution along the crack front, its maximum value, and its asymptotic convergence to the 2D plane strain solution. The results of this study yield a practical guideline for estimating the maximum 3D mode I SIF by scaling the corresponding 2D plane strain solution. The present approach is most effective for finite-thickness components where the crack surface length is small relative to the thickness. 2. Numerical method The analytical model considers a deep surface crack in a homogeneous, isotropic, and elastic semi-infinite body, as shown in Fig. 1(a). The crack lies on the x = 0 plane, and is oriented perpendicular to the free surface at ( z = 0 ), and extends infinitely in the positive z -direction. At the surface, the crack mouth takes a finite length of 2 a along the y -axis. The crack face is subjected to an internal pressure, p ( y , n ) , which is uniform along the crack depth ( z direction) and varies across its surface length ( y -direction) according to a power-law distribution, as shown in Fig. 1(b). This pressure distribution is defined as follows: p ( y , n ) = p 0 y a n (1) Where p 0 is the arbitrary constant and n is a non-negative integer that defines the order of the pressure distribution. The mode I stress intensity factor (SIF) K I for this problem is evaluated using a numerical method based on the body force method (BFM). The detailed formulation of this BFM-based approach is presented in the subsequent section.