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

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Fig. 1. Schematic of the analysis model. (a) A deep surface crack perpendicular to the free surface ( z = 0 ) in a semi-infinite solid. (b) Power-law internal pressure applied to the crack face. (c) Concept of the BFM, where the crack is imitated by embedded body force doublets.

2.1. Governing integral equation based on the BFM In the BFM, the crack is modeled by distributing mode I type standard body force doublets of an unknown density, γ ( η,ζ ) , over the face which is perpendicular to the free surface, as shown in Fig. 1(c). Based on the principle of superposition, the x -component of normal stress, σ xx ( y , z ) , induced at P ( y , z ) by force doublets embedded is given by the following integral: σ xx ( y , z ) = a − a ∞ 0 γ ( η,ζ ) σ γ xx ( y , z ,η,ζ ) d ζ d η (2) Here, γ ( η,ζ ) is the density function to be determined. The kernel, σ γ xx ( y , z ,η,ζ ) , is the fundamental solution that represents the stress at P ( y , z ) induced by a mode I body force doublet of unit magnitude acting at a source point Q ( η,ζ ) . For this fundamental solution, the present study utilizes the formulation derived by Noguchi and Smith (1995). where ν is a Poisson’s ratio, r 1 is the distance between P ( y , z ) and Q ( η,ζ ) , and r 2 is the distance between P ( y , z ) and the mirror image of a source point Q ( η, − ζ ) with respect to the free surface. They are defined as follows: r 1 = ( y − η ) 2 + ( z − ζ ) 2 , r 2 = ( y − η ) 2 + ( z + ζ ) 2 (4) The unknown density function, γ ( η,ζ ) , is determined by enforcing the boundary condition on the crack face. This condition requires that the stress induced by the body force doublets, σ xx ( y , z ) , must balance the given internal pressure, p ( y , n ) , such that σ xx ( y , z ) = − p ( y , n ) . Substituting the expressions for pressure from Eq. (1) and stress from Eq. (2) into this relationship yields the governing singular integral equation: a − a ∞ 0 γ ( η,ζ ) σ γ xx ( y , z ,η,ζ ) d ζ d η = − p 0 y a n (5) σ γ xx ( y , z ,η,ζ ) = 1 − 2 ν 8 π (1 − ν ) 2   1 r 3 1 − 5 − 20 ν + 24 ν 2 r 3 2 + 12(1 − ν )(1 − 2 ν ) r 2 ( r 2 + z + ζ ) 2 + 6 3 z ζ − 2 ν (1 − 2 ν )( z + ζ ) 2 r 5 2   (3)