Issue 48

F. V. Antunes et alii, Frattura ed Integrità Strutturale, 48 (2019) 676-692; DOI: 10.3221/IGF-ESIS.48.64 681 m = a m .(  -  ) 2 +b m .(  -  )+c m a m = m am .(y/W)+b am m am = 8.4471x10 -9 .  2 - 4.6826x10 -8 .  -1.5894x10 -4 b am = 1.2076 x10 -8  2 - 1.9219 x10 -7 .  -1.3955 x10 -4 b m = m bm .(y/W)+b bm m bm = -2.3457 x10 -6  2 - 2.9335x10 -4  + 4.1625 x10 -2 b bm = 9.6990 x10 -7  2 - 2.4575x10 -4  - 2.8677 x10 -4 c m = m cm .(y/W)+b cm m cm = -2.7508 x10 -4  2 + 4.3628x10 -2  - 3.7856 x10 -2 b cm = -1.5507x10 -4  2 - 9.5592 x10 -4  + 1.0744 b = a b .(  -  ) 2 +b b .(  -  )+c b a b = a ab .  2 +b ab .  +c ab a ab = 4.9864 x10 -8 .(y/W)+1.4723x10 -8 b ab = -2.9412 x10 -6 .(y/W) - 1.9886x10 -6 c ab = -2.6726 x10 -4 .(y/W)+ 3.0499x10 -5 b b = a bb .  2 +b bb .  +c bb a bb = -4.3391x10 -5 .(y/W) 2 +2.1155 x10 -6 .(y/W)-5.3868 x10 -6 b bb =2.1858 x10 -3 .(y/W) 2 -9.6565x10 -4 .(y/W)+5.8686 x10 -4 c bb = 5.7786 x10 -2 .(y/W)-2.6952E-03 c b = a cb .  2 +b cb .  +c cb a cb =-6.9194 x10 -4 . (y/W) + 2.6846E-04 b cb =7.8256 x10 -2 .(y/W) + 5.9363E-04 c cb == 2.0543.(y/W) 2 -6.2440 x10 -3 .(y/W)-1.5349x10 -1 The units of  and  are degrees. The parametric region where this solution is valid is:  0, 60º  ; x/W   0.4, 0.75 mm  ; y/W   0, 0.167  . This solution has an average difference of 0.53 % relative to numerical values, with maximum and minimum differences of +2.2 e -3.01, respectively. The average difference was obtained from the absolute values of the differences. Typical variations of geometric factor, Y II , with (  -  ),  , y and x can be seen in Figs. 6a, 6b, 6c and 6d, respectively. Y II has a complex variation with (  -  ) as can be seen in Fig. 6a. Minimum values, close to zero were obtained for  -  0, as could be expected. As for Y I , minimum values do not coincide exactly with  =0, which is a consequence of the complexity of the situation. Y II increase with crack length, x (Fig. 6d). A good fitting to results in Figs. 6b (  ), 6c (y) and 6d (x) was obtained with second order polynomials. The comparison between Figs. 6 indicates that the highest variation of Y II occur with (  -  ), followed by x. The magnitudes of the variations of Y II with x and  are similar. The complex variations of Y II , namely with  , complicated significantly the development of a regression function. Since similar values of K II are expected for symmetrical values of (  ), this parameter was replaced by  . Besides since minimum values occur in general for  0, Y II was studied as a function of  being  the angle for minimum Y II . Considering this change of independent variable and adequate values for  , the results of Fig. 6a modify to results in Fig. 7. All the results fall on the same trend, which indicates that the parameter proposed is adequate. An empirical expression with 54 constants was fitted to the numerical values of Y II : Y II =a.sin(  -  -t) (7) a = a a .(x/W) 2 +b a .(x/W)+c a a a = a aa .  2 +b aa .  +c aa a aa =-1709.472357 .(y/W) 2 +519.545852 (y/W )+ 21.384622 b aa =959.716917.(y/W) 2 -249.236159 (y/W )+ 6.561228 c aa =-98.482311.(y/W) 2 +4.234582(y/W )+ 1.006272 b a = a ba .  2 +b ba .  +c ba a ba =1953.567396.(y/W) 2 -541.35909 (y/W )-19.747265 b ba = -1101.79278.(y/W) 2 +255.808188(y/W )-7.044048 c ba =113.285142.(y/W) 2 -3.803387 (y/W )+ 0.704463