Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 386 crack tip shear force. As an example, without loss of generality, in an isotropic specimen with  =0.3 from Tabs. 1 and 7 of Appendix B we have 0 1.559 V c  and 0 0 3.659 2.347 V V EB c c   . In order to clarify this relevant difference and understand which model leads to the most accurate estimate for the root displacement due to crack tip shear, the finite element method has been used and a numerical analysis has been carried out through the commercial finite element code ANSYS 2019R1. A mesh consisting of plane stress four-node quadrilateral elements has been used and convergence checked by varying the mesh in size and number of elements. The corresponding root rotation and displacement compliance coefficients have been determined following the method described in [3]. The root rotation compliance coefficients due to bending moment and transverse force listed in Table 10 of Appendix B have been successfully reproduced. Furthermore, the root displacement compliance coefficient due to shear has been calculated, that is 0 1.537 V c  . This result shows straightforwardly that the Timoshenko beam based model leads to more accurate estimates for the root displacement compliance coefficients than the Euler-Bernoulli beam based model which yields a 138% error on the prediction. In order to account for the effects of shear deformations in fracture specimens approximated as Euler-Bernoulli beams, the simplified model requires that the particular matching procedure based on Eqn. (B.4) is used, which has the effect of decreasing the correction factor of the transverse elasticity to include the effects of the shear deformations. However, this leads to accurate estimates for the root rotations but to eccessive overestimates for the root displacements. On the other hand, it is worthwhile noting that the terms on the right hand side of Eqn. (29) have different relevance in the evaluation of the maximum relative deflection in the DCB specimen under consideration. In particular, the fourth term related to root displacements under shear is in general little to not relevant compared to the others for many geometries and orthotropy ratios. To better understand this point, the contributions due to bending moment, shear deformations and root rotations and displacements are conveniently evaluated as percentage of the total maximum relative deflection for degenerate orthotropic specimens (  =1) with  =1,0.5,0.05,0.025. In the case of medium/long delaminations ( a / h ≥ 6) the term 0 V c has no relevance and is smaller than 1% of the total even for very low  =0.025. For medium/short length delaminations (e.g. a / h =3), the first term related to bending moment induced by the applied loads is the most relevant for high  =1,0.5 but decreases from about 55% of the total for  =1 to 26% for  =0.025; the contribution of root rotations becomes the most relevant for low  =0.05,0.025 and increases from 40% for  =1 to 58% for  =0.025; the shear deformations term increases from 5% for  =1 to 12% for  =0.025; once again, the contribution of 0 V c has very little relevance and is smaller than 1% for  =1,0.5 and equals to 3% for  =0.05 and 4% for  =0.025. C ONCLUSIONS his paper deals with the analytical derivation of root compliance coefficients which can be used to define root rotations and root displacements at the crack tip cross section of orthotropic cracked beams. The work is limited to homogeneous and orthotropic materials with the axes of material symmetry parallel to the reference axes. Explicit and simple expressions have been derived for use in practical applications under general loading conditions at the crack tip. Accounting for root effects is important to accurately define energy release rate, mode mixity and displacements fields in the specimens. The derivation builds on and extends one-dimensional formulations in the literature and describes the intact part of the specimen as two beams joined by an elastic Winkler type bond. The effects of both shear deformations and transverse elasticity are taken into account; in addition, the continuity condition imposed on the relative sliding displacements of the two beams at their interface to describe the continuity of the intact portion ahead of the crack tip makes the axial and bending problems coupled. This allows to extend Thouless' [10] treatment to problems where the geometry is not symmetrical and where root displacements may be relevant, and to loading by axial forces. The novel analytical expressions for the root coefficients depend on the elastic constants and the geometrical parameters and one a priori unknown parameter, k w , which defines the connection modulus and describes the effects of the transverse elasticity. The parameter is obtained through matching of well established results in the literature for the root rotation compliance coefficients. For fixed geometry and material properties, obtaining an unique value for the parameter k w derived by matching the root rotations due to the different elementary loadings would imply an accuracy of the one-dimensional model in predicting the displacement field comparable to that of 2D elasticity. The results show that this is true in isotropic and orthotropic T