numero25

A. Spagnoli et alii Frattura ed Integrità Strutturale, 25 (2013) 94-101; DOI: 10.3221/IGF-ESIS.25.14 96 The condition for self-balanced stress is that the function   f x / d is periodic (with material microstructure period d ). For the sake of simplicity, we assume     f x / d cos 2 x d   (this could be regarded as a first order approximation through Fourier series of a general periodic function). Now, since the above   f x / d is a even function with respect to x , we have that the value of the SIFs (i.e. referring to the projected crack) are (using Buckner’s superposition principle, which is based on the stresses in the uncracked body along the crack lines):         l l l I a a a 0 2 2 2 2 2 2 0 0 0 l l l II a a a 0 2 2 2 2 2 2 0 0 0 f x d cos 2 x d 2 l l l l K 2 dx 2 dx 2 dx l J d l x l x l x f x d cos 2 x d 2 l l l l K 2 dx 2 dx 2 dx l J d l x l x l x                                                               (4) where 0 J is the zero order Bessel function. In such a self-balanced microstress field, it can be reasonably assumed that the crack symmetrically kinks (due to the mixed mode of fracture) with respect to the y -axis and at each material microstructure semi-period, i.e. at each reversal in the microstress spatial courses ( a = d/ 2 in Fig. 2). In the case of a singly-kinked crack (of projected crack length 2l , as is reported in Fig. 1), the SIFs at the tips of the inclined part of the crack can be expressed through the SIFs I K and II K of a straight crack of length equal to the projected length of the kinked crack [2-7] :         I 11 I 12 II II 21 I 22 II k a , b a K a , b a K k a , b a K a , b a K         (5) where ij a are coefficients depending on the slant angle  (positive counter-clockwise for tip coordinate x > 0) and the length ratio b a between the deflected leading segment and the horizontal trailing segment. If a geometry different from that of an infinite plate with a central crack is examined, the geometric factor of the SIFs defined with respect to the projected crack would be different from the unity, but the expression in Eq. 5 would not change. Figure 2 : Mixed-mode crack growth in the self-balanced microstress field. The coefficients ij a for b a  (and, with good approximation, also for b a 0.3  ) are [2]:         3 2 1 2 11 12 1 2 1 2 21 22 a cos a 2sin cos a sin cos a cos 2 cos                  (6) l    x y 0 d/2 x d 2 1 n a  ~ a  ~  x a  ~ a  ~                  3 ) / ( ~ dxf a  ) / ( ~ dxf a 