Issue 48

M. K. Hussain et alii, Frattura ed Integrità Strutturale, 48 (2019) 599-610; DOI: 10.3221/IGF-ESIS.48.58 602 Similarly, the notch sliding displacement (NSD) can be obtained as [29]                                                 1 1 1 1 1 1 1 1 2 1 2 2 3 cos 2 cos 2 sin 2 sin 2 1 sin II II II II II II II II II II B u u u u r G B r B C r B C r G (9) where 1 C , 2 C and 3 C are constants which depend upon eigenvalues,  and  . Thus, using Eqn. (8) the coefficient 1 A can be calculated and 1 B can be calculated from the NSD (Eqn. (9)) under all loading conditions (pure mode I, pure mode II and mixed mode (I/II)). Assuming the isoparametric quadratic quadrilateral elements are deployed at the notch tip, the FE displacement along notch flank nodes 1–2–4 (Fig. (2)) can be written with r being the distance from the notch tip as [29]      2 FE v Ar Br (10) where the constants A and B are constants and can be obtained from the FE displacements using the following equation                     1 2 2 2 2 2 4 4 4 FE FE A r r v B r r v (11) where 2 FE v and 4 FE v are FE displacements at nodes 2 and 4, respectively, and 2 r and 4 r are distances of nodes 2 and 4, respectively, from notch tip 1. The FE NOD can be expressed as    2 2 2 FE v Ar Br (12) Figure 2: A notch flank finite elements around a notch tip. The residual  between the analytical NOD (Eqn. (8)) and FE NOD (Eqn. (12)) can be written as             1 2 2 2 1 1 2 2 2 I FE v v AC r Ar Br (13)