numero25

A. Spagnoli et alii, Frattura ed Integrità Strutturale, 25 (2013) 94-101; DOI: 10.3221/IGF-ESIS.25.14 97 Note that the local SIFs in Eqs 5 and 6 are equal to those of an inclined straight crack of projected semi-length l forming an angle 2    with respect to the loading axis of ( )   [2]. It is reasonable to assume that, as the crack propagates following the path in Fig. 2, only its latter deflection influences the stress field near the crack tips (for example, the local SIFs at the crack tip 3 in Fig. 2 are assumed to be equal to those of a singly-kinked crack with an inclined segment corresponding to the segment 2-3, and by taking b a 0.3  , see Eq. 6) [19]. The approximate computation is thus based on the assumption that the near-tip stress field depends on the local crack direction at the crack tip. The local SIFs at the crack tip are assumed to be expressed by Eqs 5 and 6 for deflected (Mode I+II) segments (see segments 1-2 and 1 2   in Fig. 1), despite the fact that the length ratio b a between the leading segment and preceding segment might in general vary from 0 to  during propagation (in Fig. 1, for example, b is the running quantity for the crack propagating along the deflected segment 1-2 so that b a ranges from 0 to a value greater than the unity). Direction of growth of the kinked crack and effective SIF As is mentioned above, the crack might kink at each material microstructure semi-period, namely at each reversal in the microstress spatial courses. The classical criterion of Erdogan and Sih [20] is applied herein to describe the mixed-mode crack propagation. Accordingly the kinking angle  , defined with respect to the general inclined axis of the crack (Fig. 1), is expressed by 2 I I II II k k 1 1 2arctan 8 4 k 4 k                   (7) where the SIFs values appearing in Eq. 7 are those determined according to Eqs 5 and 6. The angle  is positive counter- clockwise and is in the range –  and +  . The sign in Eq. 7 is chosen so as to have the smallest absolute value of  . Once the freshly formed kinked segment develops to a finite length, an equivalent SIF eq k can be calculated according to Eqs 5 and 6. An effective driving force can be determined by applying the coplanar strain energy release rate theory [21], that is, the equivalent SIF eq k is given by 2 2 eq I II k k k   (8) F ATIGUE GROWTH IN NOMINALLY MODE I KINKED CRACKS ow let us restrict our attention to nominally Mode I cracks, i.e. cracks submitted to a remote Mode I fatigue loading ( )    . Hence, for an infinite plate, we have the following SIFs related to the projected crack of semi- length l : ( ) ( ) a a I 0 I 0 ( ) ( ) ( ) ( ) a a II 0 I 0 ( ) ( ) 2 l 2 l K l 1 J K 1 J d d 2 l 2 l K l J K J d d                                                                                             (9) In Eq. 9, we assume that the microstress field (e.g. a a a t P t P max (t ) min (t )            ) is time-varying and proportional to the applied remote loading of period P (constant amplitude fatigue loading). Under remote Mode I loading and a superimposed shear microstress field, cracks propagate ‘on average’ along the x - axis following a zig-zag pattern (the same pattern is followed also in the presence of superimposed normal microstress). In general, it turns out that the crack slanting angle decreases as the crack length increases with respect to the material microstructural length d , namely (l / d)    . Obviously, crack kinking occurs only in the presence of a multiaxial stress field. Therefore, for the simplest case of uniaxial remote and microstress fields, a complanar growth of cracks occurs ( 0   ) and, according to Eq. 8, eq I k K    N