PSI - Issue 13

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

135

5

= 2 1 1 + 1 √ 2 − ̂ (2 +1) = 2 1 1 + 1 √ 2 − ̂ (2 +1)

{ = − 2 + 1 √ 2 1 − 1 ̂ 1 (1) = − 2 + 1 √ 2 1 − 1 ̂ 1 (1) {

(9)

3. Probing method A crack path simulation is performed by the following step-by-step procedure: (1) Crack tip progresses with a small size toward the deflection angles ( + 1) = ( ) − ( + ∆ ) and ( + 1) = ( ) − ( + ∆ ) at an incremental calculation step + 1 , where ( ) and ( ) are the crack tip angles at the previous step , is the probing angle, and { = 1 ( = 1) = ( ≥ 2) (2) Stress intensity factors ΙΙ ( ( ) − ( + ∆ )) and ΙΙ ( ( ) − ( + ∆ )) are calculated by changing the probing angle ( + ∆ ) , and the minimum stress intensity factors are then searched via a sub-incremental calculation at a fixed step of + 1 . { ΙΙ min = ΙΙ ( ( ) − ( + ∆ )) ΙΙ min = ΙΙ ( ( ) − ( + ∆ )) where the following minimum condition must be satisfied: { | ΙΙ ( ( + 1))| + | ΙΙ ( ( + 1))| < | ΙΙ min | + | ΙΙmin | − < | ΙΙ ( ( + 1))| − | ΙΙ ( ( + 1))| < The direction of the crack tips is determined and the angle parameters are reset as: ( + 1) = ( ) + ( + 1), ( + 1) = ( ) + ( + 1) and ( ) = ( ) = 0 After that, the calculation returns to the first step (1). 4. Crack path simulation approaching the inclusion The simultaneous singular integral equation was converted to an integration interval of [−1,1] with the following variable conversion: { ̂ ( ) = + 2 + − 2 sin 2 ( = 1,2, … , ) ̂ ( ) = + 2 + − 2 sin 2 ( = 1,2, … , ) This provides a system of ( 2 (2 + 1) × 2 (2 + 1) ) algebraic equations in the unknown functions ̂ ( ) , ̂ ( ) . It is a straightforward task to numerically solve the system of equations and repeat the calculation according to the probing algorithm. The program was developed in the C language. As an example of the crack path simulation, the interaction problem shown in Fig. 4 was considered. The combination of material constants and inclusion size is listed in Table. 1. The hole radius was assumed to be 5.0 mm and initial straight crack length was 10 mm as shown in Fig. 4. As a loading condition, the tensile stress p of 100 MPa was enforced for this interaction model. All simulations were also conducted under a plane stress condition. Fig. 4 also includes the results of crack path simulation. Influence of difference ε in radius between inclusion and hole on the crack path was examined in Fig. 4 (a). This result revealed that the crack tip is attracted to the inclusion by a tensile