Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 392  EB w k 0 V EB d 0 M EB d 0 N EB d 0 V EB c 0 N EB c 1 0.812 5.436 16.154 8.077 3.659 2.718 0.8 0.790 4.990 13.214 6.607 3.769 2.495 0.6 0.714 4.791 10.851 5.426 4.231 2.396 0.4 0.577 4.998 9.071 4.536 5.507 2.499 Table 7: Numerical results according to the Euler-Bernoulli beam based model and 0 M EB d based matching for orthotropic materials with  =  =1.  EB w k 0 V EB d 0 M EB d 0 N EB d 0 V EB c 0 N EB c 1 0.812 5.436 16.153 8.077 3.659 2.718 0.8 0.816 4.911 13.109 6.555 3.680 2.456 0.6 0.833 4.435 10.440 5.220 3.768 2.218 0.4 0.885 4.035 8.151 4.075 3.995 2.018 Table 8: Numerical results according to the Euler-Bernoulli beam based model and 0 V EB d based matching for orthotropic materials with  =  =1. The third case of matching based on the root rotation compliance coefficient 0 V EB d under transverse force deserves more attention. Neglecting the effects of shear deformations in fracture specimens approximated as Euler-Bernoulli beams leads to unacceptable underestimates of the energy release rate. In order to overcome this limit, as suggested in [10], it appears more suitable to base the matching of 0 V EB d on a modified root rotation coefficient obtained by modifying the exact 2D results 0 0 1 2 V V a a  to account for the effects of the shear deformations so that they are correctly modelled in the energy release rate. This is done by equating   0 0 0 0 0 1 1 2 1 2 (1 ) V V EB V EB V V EB s xz d a a a a                    (B.4) where the second term on the right hand side of Eqn. (B.4) comes directly from the expression of the energy release rate defined in Eqn. (6) in [3] in terms of modified crack tip stress resultants. For symmetric specimens this is equivalent to perform matching on the energy release rate. Combining Eqn. (B.4) and the second of Eqn. (25a) gives   0 0 0 4 2 1 2 (1 ) 3 2 V EB w V EB V EB a a k      . (B.5) This modification implies that, in the case for  =1, the correction factors obtained by matching any of the three root compliance coefficients coincide. The rescaling laws presented in [3] for the root compliance coefficients 1 2 K K K d a a   show that the correction factors obtained using Eqns. (B.3-5) are independent of  . The root compliance coefficients in Eqn. (25) for  ≠ 1 can then be easily obtained by rescaling the values corresponding to  =1 by   1/4 for 0 M EB d and 0 N EB d , by   1/2 for 0 0 V M EB EB d c  and 0 N EB c and by   3/4 for 0 V EB c . Tabs. 7 to 9 report results for  =1 and  =1.  EB w k 0 V EB d 0 M EB d 0 N EB d 0 V EB c 0 N EB c 1 0.818 5.416 16.124 8.062 3.639 2.708 0.8 1.009 4.417 12.432 6.216 3.138 2.208 0.6 1.317 3.527 9.310 4.655 2.672 1.763 0.4 1.868 2.777 6.762 3.381 2.281 1.389 Table 9: Numerical results according to the Euler-Bernoulli beam based model and 0 N EB d based matching for orthotropic materials with  =  =1.