Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 373 arbitrary end loadings. They determined the compliance coefficients describing root rotations due to elementary, self- equilibrated crack tip loading systems for a wide range of thickness-wise positions of the crack and orthotropy ratios. The results obtained through rigorous 2D finite element analyses were validated by comparison with results obtained using other methods for different geometries in both isotropic [2] and orthotropic [8-9] materials. Recently, the accuracy of the numerical compliance coefficients has been demonstrated in [10] and by the analytical solutions in [11], for symmetric orthotropic strips, and in [12-13] for thin layers on half-planes. Root rotation coefficients can also be obtained from expressions of the energy release rate in terms of crack tip forces using the methodology proposed in [3], which has been recently applied in [10] to Double Cantilever Beam (DCB) specimens and in [14] to sandwich specimens under general end loadings. The methodology does not allow to derive root displacements. Little attention has been placed on deriving the root displacements, which do not directly enter into the energy release rate and mode mixity and have been shown to play a role only in special problems. As examples, they need to be included in compliance-based analyses of the strain energy release rate, in the analysis of displacement controlled fracture tests, such as the wedge test, and in the analysis of problems where contact occurs along the crack surfaces, as in the inverted four point bending test. They are also important in modeling buckling and post-buckling of the detached layers using structural theories [5-6,15]. Kanninen [16] was among the firsts to include the effects of the transverse elasticity of the intact portion ahead the crack tip in the study of crack propagation and arrest in a symmetrical DCB specimen. Because of the symmetry, half of the specimen was treated as a finite length beam which is partly free and partly supported by a Winkler foundation with stiffness modulus dependent on the beam elastic and geometrical parameters. This model was firstly based on the Euler- Bernoulli beam theory to be used for the analysis of stationary cracks. Subsequently, the model was extended using the Timoshenko beam theory and a generalized elastic foundation depending on two parameters [17-18]. Recently Thouless [10] proposed an approach which links to the original Kanninen's [16] formulation and uses the solution of an elastic and isotropic Euler-Bernoulli beam on an elastic foundation in order to find the relationships between the compliance coefficients describing root rotations and root displacements for a symmetric DCB specimen subjected to transverse point forces. These relationships were furthermore used to derive the root displacements also for orthotropic specimens. In this paper, a structural model for the closed form derivation of the crack tip compliance coefficients in orthotropic specimens under general end loadings, as shown in Fig. 1a, is presented. The main goal of the work is to formulate a simple but sufficiently accurate model for practical applications, which is able to provide the root displacement coefficients. The model is based on shear deformable beam theory and builds on the Kanninen's original formulation, which is here extended in order to consider new end-loading conditions. In particular, the problem of the equilibrium of a two-layered system with elastic interlayer uplift is analysed, as shown in Fig. 1b. The solution procedure previously followed by the authors to solve the problem of multi-layered systems with imperfect interfaces [19-24] is employed and leads to explicit expressions for the root rotation and displacement compliance coefficients under different crack tip loading conditions. Figure 1: (a) Edge cracked specimen subjected to end loadings. (b) Elastically connected semi-infinite beams: model geometry and self- equilibrated elementary end loading conditions. The paper is organized as follows. In the next section the equilibrium, compatibility and constitutive equations governing the model formulation are detailed and rearranged into two coupled linear differential equations for the shear and normal interfacial tractions. Details on the solution procedure leading to exact expressions for the interfacial tractions and all other static and kinematic variables are provided. In the third section explicit expressions for the relative rotation and displacement compliance coefficients under three elementary end loading conditions are derived analytically and presented for Timoshenko and Euler-Bernoulli layers. The simplified formulation based on Euler-Bernoulli beam theory is detailed in Appendix A. The novel expressions are employed on varying elastic moduli and geometrical parameters and the related numerical results are shown and discussed in the fourth section after validation through comparison with results in the literature. All numerical results shown are listed in table form in Appendix B, where the matching procedure is described.