Issue 53

I. Monetto et alii, Frattura ed Integrità Strutturale, 53 (2020) 372-393; DOI: 10.3221/IGF-ESIS.53.29 388 [13] Massabò, R., Ustinov, K., Barbieri, L., Berggreen, C. (2019). Fracture mechanics solutions for interfacial cracks between compressible thin layers and substrates, Coatings, 9 (3), art. no. 152. DOI: 10.3390/coatings9030152. [14] Barbieri, L., Massabò, R., Berggreen, C. (2018). The effects of shear and near tip deformations on interface fracture of symmetric sandwich beams, Eng. Fracture Mech., 201, pp. 298-321. DOI: 10.1016/j.engfracmech.2018.06.039. [15] Cotterell, B., Chen, Z. (2000). Buckling and cracking of thin films on compliant substrates under compression, Int. J. Fract., 104, pp. 169-79. DOI: 10.1023/A:1007628800620. [16] Kanninen, M.F. (1973). An augmented double cantilever beam model for studying crack propagation and arrest, Int. J. Fracture, 9, pp. 83-92. [17] Kanninen, M.F. (1974). A dynamic analysis of unstable crack propagation and arrest in the DCB test specimen, Int. J. Fracture, 10, pp. 415-430. [18]Williams, J.G. (1989). End corrections for orthotropic DCB specimens, J. Compos. Sci. Technol., 35, pp. 367-376. [19] Andrews, M.G., Massabò, R. (2008). Delamination in flat sheet geometries in the presence of material imperfections and thickness variations, Composites Part B, 39, pp. 139-150. DOI: 10.1016/j.compositesb.2007.02.017. [20] Massabò, R., Cavicchi, A. (2012). Interaction effects of multiple damage mechanisms in composite sandwich beams subjected to time dependent loading, Int. J. Solids Struct., 49, pp. 720-738. DOI: 10.1016/j.ijsolstr.2011.11.012. [21] Campi, F., Monetto, I. (2013). Analytical solutions of two-layer beams with interlayer slip and bi-linear interface law, Int. J. Solids Struct., 50, pp. 687-698. [22] Monetto, I. (2015). Analytical solutions of three-layer beams with interlayer slip and step-wise linear interface law, Compos. Struct., 120, pp. 543-551. [23] Monetto, I., Campi, F. (2017). Numerical analysis of two-layer beams with interlayer slip and step-wise linear interface law, Eng. Struct., 144, pp. 201-209. [24] Monetto, I. (2019). The effects of an interlayer debond on the flexural behavior of three-layer beams, Coatings, 9, art. no. 258. [25] Hutchinson, J.W., Suo, Z. (1991). Mixed mode cracking in layered materials, Adv. Appl. Mech., 29, pp. 63–191. A PPENDIX A. S IMPLIFIED FORMULATION ACCORDING TO E ULER -B ERNOULLI BEAM THEORY his Appendix presents a simplified formulation based on modeling the two semi-infinite beams shown in Fig. 1b as linearly elastic and orthotropic Euler-Bernoulli beams. According to Euler-Bernoulli beam theory, the shear strains within the two layers are neglected. This implies that, firstly, the rotations are related to the deflections of the beams ( i =1,2) as ' i i w    (A.1) secondly, infinite shear stiffnesses are assumed so that ˆ 0 a  and then ˆ ˆ4 0 d e    . In this case, the general solution of Eqn. (7) modifies as     1 2 3 4 exp( ) cos( ) sin( ) exp( ) cos( ) sin( ) n p x C x C x x C x C x            (A.2) where 4 ˆ / 2 e   (A.3) with ˆ e given by Eqn. (8). In analogy to the procedure described in the second section, the shear tractions follow from Eqn. (11), the internal forces are given by Eqn. (12), and the rotations and axial displacements are derived by Eqns. (13a-b); whereas Eqn. (13c) for the deflections is substituted by the following 15 d i i i w x C       (A.4) The remaining Eqn. (14) for the relative rotation and deflection and Eqn. (15) for the arbitrary constants are still valid. Following the procedure described in the second section yields explicit expressions of the compliance coefficients T