Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 379 together with Eqns. (20) and (21). (3) Pair of longitudinal forces with compensating moment The third schematic, shown in Fig. 2(s3), corresponds to the end loading with N 0 ≠ 0, V 0 =0 and M 0 =0. In this case, equilibrium is guaranteed by a compensating counter-clockwise moment equal to N 0 ( h 1 + h 2 )/2 applied to the lower beam. In this case, the compliance coefficients are 0 2 3(1 ) 2 2 N w c k     (23) and 0 2 3 (1 ) 2 N x s xz d E F G      (24) where F  is defined in Eqn. (20) together with Eqn. (21). The beam model proposed in this paper leads to the simple explicit expressions given by Eqns. (18) to (24) for the compliance coefficients describing both root rotations and root displacements as functions of the model parameters. The model parameters characterize the specimen material and geometry (namely, the orthotropy ratios and the relative thickness of the two layers), in addition to the correction factor k w introduced in Eqn. (1), which is the only a priori unknown model parameter. Analogous root rotation and displacement compliance coefficients have been derived under the assumption of Euler- Bernoulli beam theory, as detailed in Appendix A. They are listed below: 0 1/4 3/2 1/4 3/4 (1 ) 3 2 ( ) V EB EB w c k     , 0 0 2 (1 ) 3 2 M V EB EB EB w c d k      , 0 2 (1 ) 3 2 2 N EB EB w c k     (25a) 0 1/4 3/4 5/2 1/4 (1 ) 2 3 ( ) M EB EB w d k     , 0 3/4 5/2 3/4 1/4 (1 ) 3 2 ( ) N EB EB w d k     (25b) where the upper/sub-script EB has been used to differentiate with respect to the previous results. These coefficients could be obtained from Eqns. (18)-(24) by imposing infinite shear stiffness, which implies ˆ 0 a  and then ˆ 0 d  . Suitable values for k w or EB w k are here chosen by reproducing well-established values for the compliance coefficients in the literature. In particular, the root rotation coefficients determined in [3] using accurate 2D finite element analyses, shown in Table 10 in appendix B, are considered. Details about the matching procedure are given in Appendix B. Tabs. 1-9 show the values of k w or EB w k obtained by matching the three coefficients in isotropic and orthotropic specimens. The tables also define the values of the root rotation and displacement coefficients obtained a posteriori using the matched k w or EB w k in Eqns. (18)-(24) and (25). In the next two sections some numerical results are shown and discussed, and an application of these results is presented with reference to an exemplary fracture problem. N UMERICAL RESULTS AND DISCUSSION n this section, some sets of numerical results obtained by employing the analytical expressions for the root rotations and displacements derived in this paper and listed in Tabs. 1-9 are presented. This has three aims. The first is to validate the model through comparison with the results of other methods in the literature. The second is to highlight the effects of the choice of the matching compliance coefficient. The third is to discuss the effects of the geometrical and material properties of the specimen on the compliance coefficients. I