PSI - Issue 13

Masayuki Arai et al. / Procedia Structural Integrity 13 (2018) 131–136 Author name / Structural Integrity Procedia 00 (2018) 000 – 000

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to introduce a fruitful technology in the repairing process: namely, the stop-hole and press-fitted inclusion methods. The main issue is to determine the appropriate size and position of the machined hole for the existing crack. Thus, it is required to calculate precisely the stress intensity factor at the tip of crack near the inclusion. The mathematical approach for solving this problem is the Kolosov-Muskhelishvili complex potential formulation (Woo and Chan (1992)), where a mapping function is used to transform the crack geometry onto a unit circle. A solution is found either by solving an integral equation or using series expansions of complex potential functions and the mapping function. Another approach is the continuous distributed dislocation formulation (Hill et al. (1996)), in which a singular integral equation is solved numerically by distributing edge dislocations along the crack locus. In this study, the latter method was selected to derive the stress intensity factor at the crack tip near the inclusion. The fundamental solution, which was obtained by Dundurs and Mura (1964), was applied to solve the problem of the interaction between the inclusion and the crack. The problem treated here has already been solved by many researchers. The novelty of this study is to develop a new algorithm that the crack tip progresses automatically in the direction of a zero stress intensity factor K II identified by solving a singular integral equation. This algorithm would provide the capability to precisely predict a crack propagation path, and to quickly determine the size and location of an inclusion. In this study, it is refered to as the “probing method.” 2. Formulation of the problem Consider the problem of a curved crack in an infinite elastic plate with an inclusion, which is subjected to a uniform tensile stress p at far field. Here, it is assumed that the circular inclusion with radius of + ( ≪ ) shall be press fitted in the circular hole with radius of in an infinite elastic plate. The material constants are denoted by the shear modulus , Kolosov’s constant and Poisson’s ratio , and low subscript corresponds to 1 for elastic plate and 2 for inclusion, respectively. Such a problem can be solved by superimposing three auxiliary problems. The first problem (problem A) represents the stress field in an elastic plate with an inclusion and edge dislocations distributed continuously along a crack line. The tractions along the crack line are expressed as ̅ . The second problem (problem B) represents the stress field in an elastic plate with an inclusion, but without a crack, subjected to a uniform tensile stress p . The tractions in this case are expressed as ̃ . The third problem (problem C) represents the internal stress field generated by press-fitted inclusion. The tractions are also expressed as ̂ . Consequently, the superposition of the tractions derived from all auxiliary problems has to be zero: ̅ + ̃ + ̂ = 0 in order to impose the condition of free traction on a crack plane.

Fig. 1 Decomposition of crack problem in an elastic plate with press-fitted inclusion Figure 2 indicates the geometry of the problem, including an edge dislocation near the inclusion on an elastic plate. Here, the edge dislocations are continuously distributed along the virtual crack line. The global coordinate ( , ) is set at the center of the hole. This figure also includes two inclined virtual cracks indicated by solid and dotted lines. The dotted line lies parallel to the ̂ axis, which is obtained by rotating the axis through an angle counterclockwise ( ≤ ̂ ≤ ), and the solid line lies parallel to the ̂ axis, which is obtained by rotating the axis through an angle counterclockwise ( ≤ ̂ ≤ ). The stress components on the line ≤ ̂ ≤ and ̂ = ℎ , which are observed from the local coordinate ( ̂ , ̂ ) , are expressed by,