PSI - Issue 39

9

Daniele Amato et al. / Procedia Structural Integrity 39 (2022) 582–598 Author name / Structural Integrity Procedia 00 (2019) 000–000

590

For those cases where the nonlinearity of the contact algorithm is not effective (only opening stresses are present), the SIFs can be computed according to the superimposition principle, namely by linearly combining the K-factors generated by the tensile and torsional load separately. In Eq. (3), ( ) and ( ) stands for the axial force and the torsional moment related to the -th static step, respectively, whereas , and , are the SIFs generated by a unitary

axial or torsional load, respectively. ( ) = , ( ) + , ( ) ( ) = , ( ) + , ( ) ( ) = , ( ) + , ( )

(3)

Conversely, in those cases where the crack faces are in contact, the SIFs are directly extracted from the stress state at the crack front. In every loading step the loads are considered as the extreme value of a zero-max cycle. Therefore, taken individually, they can be treated as proportional loads and the related criteria for deflection angle calculation can be used. In this study, the Maximum Tangential Stress (MTS) criterion was involved in the kink angle calculation for each static step, separately. The MTS theory asserts that the crack will grow towards the location of the maximum tangential tensile stress (equivalent to maximizing ); this criterion seeks out a mode I crack path [3]. The kink angle formulation used is: ( ) = 2 tan −1 ⎝ ⎜ ⎛ 1−�1+8� ( ) ( ) � 2 4� ( ) ( ) � ⎠ ⎟ ⎞ (4) where, ( ) is a function of and , representing the direction where the tangential stress in the vicinity of the crack tip is maximum and the shear stress vanishes. The approach FRANC3D uses to cope with the deflection angle derivation of an analysis made up of several static steps is the dominant step criterion. This criterion was used to determine both the mission’s deflection angle and the crack propagation rates, at any propagation stage. The dominant step criterion assumes the crack propagation direction due to a mission to be the same as the kink angle of the dominant step. The latter is defined as the static step which leads to the maximum either equivalent SIF or crack growth rate. Since experimental tests were conducted in a thermally controlled environment, at room temperature, there was no need to model a temperature dependent propagation. Therefore, the dominant step was determined as the one with the highest equivalent SIF, formulated in Eq. (5). This ESIF formulation was proposed by Erdogan and Sih accordingly to the maximum tangential stress criterion [3]. ( ) = 1 4 � 3 ( ) 2 + ( ) 2 � − 3 4 � ( ) 2 + 3 ( ) 2 � (5) The procedure for the mission deflection angle derivation works as follow: the combined SIFs, computed from Eq. (3), are inserted to Eq. (4) to find ( ) . The kink angles just calculated are used to compute the equivalent SIFs, ( ) , for a full loading cycle. Among all these thirteen values, the peak value, , ( ∗ ) is taken and the related static step, ∗ , is considered as the most detrimental instant during the mission cycle, which determines both the crack propagation direction and the amount of propagation. Although the equivalent SIF range is already at hand for = ∗ , it is re-evaluated as in Eq. (6). Δ ( ∗ ) = � [ Δ ( ∗ )] 2 + [ Δ ( ∗ )] 2 + [ Δ ( ∗ )] 2 (6)