Issue 37

S. Vantadori et alii, Frattura ed Integrità Strutturale, 37 (2016) 215-220; DOI: 10.3221/IGF-ESIS.37.28 218 Eq. (6) is deduced by employing the Castigliano theorem in the manner suggested by Paris [10], being  the crack kinking angle (Fig. 2(b)) and 0 1 3.0 a a  . Note that, as is shown in Fig. 2(b), the kinked crack path consists of the two segment, named 1 a and 2 a . If the value of 2 a obtained from Eq. 6 is negative, it means that the effective crack length is 1 0 a aa  with 0 1 3.0 a a  . Such a length is obtained from the following equation by employing an iterative procedure: ]} ) [(] [ { ) ( ) ( 0 0 1 0 1 0 4 2 6 0 0 2 cos cos 2 cos 2 sin 2 cos ) ( 6        Va W a a V a a Va BWC S E u       (7) Finally, the critical stress-intensity factor, S IC K , is computed through Eqs 4 and 5, by considering a straight crack having length equal to the projected length of the effective kinked crack:         0 1 2 1 0 2 1 0 2 max 3.0 when cos with cos 2 3 a a W a a a f a a a BW P K S IC             (8a) or     0 1 1 0 1 0 2 max 3.0 when cos with cos 2 3 a a W a a f a a BW P K S IC           (8b) R ESULTS AND DISCUSSION alues of fracture toughness, S IC K , of the analysed cadaveric femur diaphysis of a 24-month-old bovine are computed with respect to the osteons alignment: perpendicular or parallel to the loading direction. The former specimens are extracted from anterior (FA-1 and FA-2 in Tab. 1), posterior (FP in Tab. 1), medial (FM-1 and FM-2 in Tab. 1) and lateral (FL in Tab. 1) cortical bone, whereas the latter specimens from posterior cortical bone (FP- long1 and FP-long2 in Tab. 1). All specimens exhibit a non-linear slow crack growth before the peak load is reached. It can be observed that the fracture toughness values for specimens characterised by different osteons alignment with respect to the loading direction are significantly different. As a matter of fact, when the osteons alignment is perpendicular to the loading direction, crack grows under Mixed Mode (Fig. 3(a)), and a higher resistance to fracture is observed. When the osteons alignment is parallel to the loading direction, crack grows under Mode I (Fig. 3(b)), and a lower resistance to fracture is observed. For transversal specimens, the average value of S IC K ( m MPa 15.0 87.3  ) is in the range of the cortical bone fracture values [16]. Such a value is then compared with those determined by Libonati et al. [13] according to ASTM standards [14], following the LEFM and considering cracks under pure Mode I loading. They found m MPa K S IC 1.06.5   from SE(B) type and m MPa K S IC 6.08.5   from C(T) type. The difference of such results with respect to those here obtained is due to the fact that the reduction of fracture toughness for a kinked crack as compared with the straight counterpart has not been taken into account by Libonati et al. in Ref. [13]. The present study highlights that the value of the near-tip stress-intensity factor of a kinked crack can be considerably lower than that for a straight crack of the same length, and that has to be taken into account for prevention, diagnosis and treatment of bone traumas. V