PSI - Issue 61

Mehmet N. Balci et al. / Procedia Structural Integrity 61 (2024) 331–339 Balci and Yalcin / Structural Integrity Procedia 00 (2019) 000 – 000

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point (singular) elements are used. ½ right half-side of symmetrical problem is modelled. Red line shows the half length of the crack. Singular elements are useful for modelling stress concentrations and crack tips. Quarter-point or singular element which can interpolate the stress distribution in the vicinity of the crack tip at which stress has the 1 r singularity where r is the distance from the crack tip ( ) 1 r a (Stolarski et al., 2018).

Fig. 2. (a) PLANE 77/PLANE 183 element in ANSYS (2016), (b) its triangular counterpart, (c) Quarter point singular element, (d) Finite elements in the local ( ) ,   coordinate system, (e) Finite element mesh for the coating substrate model. In this study, we implemented the Displacement Correlation Technique (DCT) for the computation of the SIFs (Eischen 1987; Yildirim et al. 2005; Dag and Ilhan, 2008). The method is applied by utilizing the quarter-point singular elements depicted in Fig. 2(c). The circular region, , r a around the crack tip is modelled by means of the these elements. Displacement fields for the quarter-point elements in the isoparametric coordinate system shown in Fig. 2(d) are given by Dag and Ilhan (2008): ( ) ( ) 8 1 , , , i i i u N u     = =  ( ) ( ) 8 1 , , , i i i v N v     = =  (12) where i u and i v are nodal displacements, and ( ) , i N   are shape functions. In the DCT, the asymptotic displacement fields are correlated with the displacement fields of the quarter-point elements located on the crack faces. Asymptotic distributions of the displacement components at the tip of the edge crack in the polar coordinate system ( ) , r  shown in Figure 2 are expressed as follows (Eischen 1987; Jin and Noda, 1994): ( ) tip 2 tip 2 tip tip , cos 1 2sin sin 1 2cos , 2 2 2 2 2 2 2 2 I II K K r r u r                        = − + + + +                         (13) ( ) tip 2 tip 2 tip tip , sin 1 2cos cos 1 2sin , 2 2 2 2 2 2 2 2 I II K K r r v r                        = + − − − −                         (14) where I K and II K are respectively mode I and mode II stress intensity factors (SIFs), and the superscript “tip” implies that the corresponding material property has to be computed at the crack tip. The asymptotic fields given by these expressions are exactly same as those valid for a crack positioned in a homogenous medium. The material properties in a continuously graded medium however are variable and their crack-tip values are to be used in the asymptotic fields. Crack opening displacements are obtained as: ( ) ( ) 1 , , , 2 I r v r v r K      + − − = ( ) ( ) 1 , , . 2 II r u r u r K      + − − = (15) The crack opening displacements for the quarter-point elements shown in Fig. 4 are of the forms (Kim and Paulino, 2002): ( ) ( ) ( ) ( )   , , 4 , E C D B r v r v r v v v v R   − − = − − − ( ) ( ) ( ) ( )   , , 4 . E C D B r u r u r u u u u    − − = − − − (16)