Issue 33

J. Fan et alii, Frattura ed Integrità Strutturale, 33 (2015) 463-470; DOI: 10.3221/IGF-ESIS.33.51 468       2 2 2 2 0 1 2 ( , ) 4 8 ln cos tan 1 2 2 1 x H u x a b a b a a x a a a b a b x a                                              (18b) 0 2 2 2 2 sin ( , ) 2 tan 2 cos cos cos cos cos 2 2 2 2 2 a H u x a a b a b x x a x a b b b b b                                                                  (19b) Fig.2 shows the variations of the dimensionless partial derivatives   , / u x a a   which is used for calculating the weight function h ( x , a ). Through Fig.2 (a), it is clearly found that for the values of a / b ≤0.5, the dimensionless derivatives, separately, obtained by Eq. (17b) and (18b) agree quite well with those from the exact u ( x , a )-solutions presented in Eq. (19b). For 0.7≤ a / b ≤0.9, the predicted values of   , / u x a a   by Eq. (17b) can give lower errors than Eq. (18b) due to the consideration of the higher order of   2 1 x a  . From Fig.2 (b), it is obviously observed that once the values of a / b >0.9, great differences are presented between the partial derivatives determined by Eq. (17b) and (18b) to (19b). Figure 2 : Dimensionless partial derivatives of collinear cracks (a) 0≤ a / b ≤0.8; (b) 0.95≥ a / b ≥0.9. Calculations of weight functions and stress intensity factors To calculate the weight function through Eq. (2), the only unknown information is K ( a ) (1) . To make the problem easier, it is better to choose the uniformly distributed stress field as the reference load system (1) since the basic assumption relevant to the crack shape is most suited for this type of loading condition. Therefore, the formula for computing the weight function of a finite width plate is in the form given below: 0 ( , ) ( , ) ( ) u x a H h x a a f a a      (20) For a finite width plate with a central through crack, the correction function, dependent on the crack length 2 a and the width of the finite plate 2 b , is given as [1]: 2 3 ( ) 1 0.128 0.288 1.525 f a p p p     (21) where p = a / b is defined as the ratio between the crack length 2 a to the width of the plate 2 b . Substituting (21) into (15), the partial derivative u ( x , a ) for the center crack is solved as the function of a and x :