Issue 39

M. Shariati et alii, Frattura ed Integrità Strutturale, 39 (2017) 166-180; DOI: 10.3221/IGF-ESIS.39.17 172 The Newmark family includes many widely used methods. The average acceleration method is one of them for structural dynamics applications, which is unconditionally stable. In this method, J and ] are equal to 0.5 and 0.25, respectively. We choose the mean acceleration method in this study. I NTERACTION INTEGRAL AND SIF COMPUTATIONS he general form of domain integral for axisymmetric problems, introduced by Moran and Shih [24] is as follows: T c I r dA r 1 ¬ : ¬ ª º ³ ’ ’ ˜ ˜ ¬ ¼ q P P q (4-1) where, c r is the crack tip radial coordinate, r q r z e , q , lj lj ij i l P W u , G V is the energy-momentum tensor and W is summation of strain and kinetic energy. In this work, the interaction integral method is used to compute the mode I stress intensity factor ( K I ). By superimposing the actual and auxiliary fields on the domain integral, in the absence of thermal strains the general axisymmetric form of interaction integral ( MI ) in local Cartesian coordinate systems on crack tip (Fig. 2) for FGMs is obtained as below. aux aux aux aux aux aux r r ir i r ir i r ij ij r ij ij c A aux aux iz r iz i r z aux aux aux aux aux aux r r ijkl r kl ij i i i r ij j i r ir i r u u q MI u u q r r r r u u q u u C u B u u u r r , , , * 1, , , , , , , , 1 T T T T V V V H V V V H V V H H U V V V V ½ § · ° ° ¨ ¸ ® ¾ ¨ ¸ ° ° © ¹ ¯ ¿ § § · ¨ ¸ ¨ ¸ © ¹ ³ ^ ` aux ir i r u r q dA , / V · ¨ ¸ ¨ ¸ © ¹ (4-2) where q is a weight function varying from unity at the crack tip to zero on boundary of domain A * . For a stationary crack in axisymmetric state, the relation between the M-integral and the SIF is identical to its relation in plane strain state [25]. Figure 2 : Local (r, z) coordinate system. tip aux aux tip MI K K K K E 2 I I II II 2 1 Q (4-3) T