Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 374 A practical application of the solutions to the analysis of an exemplary fracture specimen is shown in the fifth section, where the two formulations developed in the paper are compared to show the limits of the simplified model based on Euler-Bernoulli beam theory. The sixth section concludes the paper with a final discussion on the differences between the proposed model and others in the literature and on the limit of applicability of the present formulation. M ODEL FORMULATION he two-layered system under consideration is shown in Fig. 1b and consists of two semi-infinite beams ( x ≥ 0) which have constant rectangular cross sections, unit width and height h i (i=1,2). The beams are made of the same linearly elastic and orthotropic material with principal material directions x , y , z and longitudinal Young's modulus E x , transverse Young's modulus E z , in plane shear modulus G xz and Poisson ratios  xz and  zx . The beams are connected by a continuous bond which is assumed to be perfect in the longitudinal x -direction, while it ensures a partial composite action in the transverse z -direction. As a consequence, interlayer uplifts can occur at the interface, but no slips between the layers at their interface are possible. Such a connection is modeled by continuously distributed tangential and normal reactions; the latter are related to the relative deflection between the two beams according to a linearly elastic relationship with k the proportionality constant. For our convenience, this connection modulus is expressed in terms of the average transverse stiffness of the beams as 1 2 2 z w k E k h h   , (1) where an arbitrary correction factor, say k w , is introduced to take into account that the beam model is only an approximation. In Kanninen's original formulation, where E z = E x and h 1 = h 2 , the correction factor was assumed as k w =1. Recently, with reference to the same Kanninen's symmetrical DCB specimen, Thouless [10] proposed alternative values for k w to obtain a reasonably good approximation for the energy release rate under 2D Linear Elastic Fracture Mechanics (LEFM) conditions. As detailed in the third section and Appendix B, k w is here determined so that the results found according to the proposed model match well-established root rotation compliance coefficients in the literature. Finally, the system is subject to the self-equilibrated loading condition shown in Fig. 1b: each beam is loaded at the end x =0 by point forces acting in the axial direction, N 0 , and in the direction normal to the beam axis, V 0 , and by a bending moment, M 0 , in addition to a compensating moment, N 0 ( h 1 + h 2 )/2, applied at the same end of the lower beam. These loading conditions are part of the elementary crack tip loadings which are used in the literature to fully describe the local crack tip fields and the energy release rate in beam-type structures subjected to arbitrary end loadings [2, 25]. Under the assumption of small strains, displacements and rotations and assuming Timoshenko beams to approximately account for the effects of the shear deformations, the problem of the equilibrium of the two-layered system under consideration is governed by three differential equations for each beam arm, with a total of eight unknowns: p t and p n , respectively the shear and normal tractions between the two layers at their common interface, and  i , w i and u i , respectively the rotation, the deflection along the z -direction and the axial displacement along the x -direction for the i -th beam ( i =1,2). The differential equations are     '' 1 '' ' 1 3 '' ' ( 1) 0 ( 1) 0 1 0 12 2 i x i i t i s xz i i i n i x i i s xz i i i t E h G h E h u p w p h G h w p                           (2) where, hereinafter, primes denote differentiation with respect to x and  s =5/6 is the shear correction factor. In the reference systems of Fig. 1b, the rotations are positive if counter-clockwise, whereas the deflections are positive if downward and the axial displacements are positive if rightward. Furthermore, the continuously distributed normal tractions p n are defined as T