PSI - Issue 12

Venanzio Giannella et al. / Procedia Structural Integrity 12 (2018) 479–491 V. Giannella/ Structural Integrity Procedia 00 (2018) 000 – 000

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For mixed-mode 3D problems, the J -integral is related to the three basic fracture modes through the components J I , J II and J III : J = J I + J II + J III (5) In Rigby et al. (1998), a decomposition method through which the integrals J I , J II and J III in elastic problems are directly calculated from J is presented. Firstly, J is divided into two components: J = J S + J (6) J and J AS are obtained from symmetric and anti-symmetric elastic fields around the crack plane, respectively. As the mode I elastic fields are symmetric to the crack plane, the following relationship holds: J S = J and J AS = J + J (7) J II and J III integrals can be calculated from J by making an additional analysis on the anti-symmetric fields. Then, when the J -integral is calculated as sum of three separated contributions of mode I, II and III, the Stress Intensity Factors i can be obtained as: J = J I + J II + J III = 1 ′ ( 2 + 2 ) + 2 1 2 (8) , where is the shear modulus and ′ = (Young’s modulus) for plane stress, or ′ = (1 − 2 ) ⁄ for plane strain. The method for deriving the three separate K values from J can be found in Rigby (1998). Once the three separate K , K and K values are obtained, the crack-growth is carried on by calculating an effective K value (BEASY 2016) to be inserted in Eq. 1. K eff = √(K I + |K III |) 2 + 2K II 2 (9) 4. FEM-DBEM results A semi-circular part-trough crack, with radius equal to 0.015 inches, is inserted in the DBEM submodel (Fig. 9a), in the most critical point as pointed out by a preliminary multiaxial variable-amplitude fatigue analysis. The chosen initial crack size is sufficiently large to reach SIFs higher than the K ℎ along the whole crack front, in correspondence of the main cycle, whereas its initial orientation is chosen considering the direction of the maximum principal stresses in that area, again evaluated in correspondence of the baseline cycle with the highest SIFs as provided by the implementation of the rainflow algorithm. The area surrounding the crack insertion point as well as the crack faces have been step-by-step remeshed along the propagation (Fig. 9b), using fully quadratic quadrilateral (9 nodes) / triangular (6 nodes) elements. During the remeshing process, several rings of internal points (Fig. 9c) are positioned along the crack front in order to allow the J -integral evaluation. The DBEM model containing the initial crack comprises 2341 quadratic elements. Some springs with fictitious stiffness have been applied on few cut surface elements (Fig. 9a) in order to provide the rigid body constraints (the load applied on the crack faces is self-equilibrated). Using the LC approach, accurate SIFs can be computed even with coarse meshes as shown in Giannella et al. (2017b) or Citarella et al. (2016b). SIFs calculated along the initial crack front are shown in Fig. 10 for all the considered load cases (Fig. 3).