PSI - Issue 39

R. Yarullin et al. / Procedia Structural Integrity 39 (2022) 364–378 Author name / Structural Integrity Procedia 00 (2021) 000–000

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3.3. Stress Intensity Factors evaluation The SIFs are computed at mid-side nodes along the crack front by using the M-integral formulation. In the 3-D domain version, the M-integral integration takes place over a volume, according to the Eq. (1): ( 1 , 2 ) = ∫ � ( 1 ) ( 2 ) 1 + ( 2 ) ( 1 ) 1 − ( 1 , 2 ) 1 � = 1,2,3 = 1,2 (1) Where is the Kronecker delta and is a function that is equal to 1 at the stress intensity evaluation point and zero on the outer boundary of the domain of integration. To evaluate the M -integral numerically, two solutions are assumed and superposed: solution (1) is the finite element solution for the problem of interest; solution (2) is an auxiliary solution. The domain of integration is a cylinder that encloses a portion of a crack front, as illustrated in Fig. 8 (Warzynek et al. 2005).

Fig. 8. Domain of integration for computing the 3-D M-integral.

The mixed mode SIFs, computed according to the M -integral formulation, are inserted in Eq. (2) and the equivalent SIF is calculated. The equivalent SIF formulation implemented in FRANC3D gives the user the possibility to calibrate the formula based on his own requirements, by inserting proper values for and . In this study, those two values were calibrated based on the preliminary experimental data available. The value was set to 1.155 whereas was set to 0.75 . = � 2 + ( ) 2 + ( ) 2 (2) The propagation law is a simple Paris, Eq. (3), with material constants reported in Tab. 1, consistent with units of and . = � � (3) The growth direction is based on the classical Maximum Tangential Stress (MTS) theory: it asserts that the crack will grow towards the location of the maximum tangential tensile stress. This criterion seeks out a mode I crack path. Therefore, the kink angle formulation used is: = 2 −1 ⎝ ⎜ ⎛ 1−�1+8� � 2 4� � ⎠ ⎟ ⎞ (4)