PSI - Issue 18

Roberta Massabò et al. / Procedia Structural Integrity 18 (2019) 484–489 Author name / Structural Integrity Procedia 00 (2019) 000–000

487

4

account for the effects of crack tip shear on the near tip deformations and are important for short cracks,   1/ 4 ,2 2 / 4 / 3 / / (1 0.208 / ) cr D IIC P hE h a h a    G with 3 2 / E E   (see Li et al, 2004, Andrews and Massabò, 2007, Barbieri et al., 2018 for a discussion on these effects). The response diagrams are compared with the experimental results on two Graphite/epoxy [0] 24 laminates tested in (Madhukar and Drzal, 1992). The geometry is defined by : L = 50 mm, 2h = 3.4 mm, 0 25 a  mm, b = 25 mm (width). The material of the beam in the first diagram is graphite/epoxy AS-4/828 with 2 139 16.7 E   GPa and 23 6 G  GPa and Mode II fracture energy 1.04 0.17 IIC   G N/mm, calculated in (Madhukar and Drzal, 1992) using the compliance method. The material of the beam in the second diagram is a graphite/epoxy AS-4C/828 with 2 158 5.06 E   GPa and 23 6 G  GPa and 1.15 0.13 IIC   G N/mm. Two theoretical curves are shown in each diagrams. The curves in Fig. 3a have been obtained using the average values of the elastic constants and energy release rate, red dashed lines Model (a), and the maximum values, black solid lines Model (b). In Fig. 3b the red curve corresponds to the average values, Model (a), and the black curve has been obtained using the average value of energy release rate and the maximum value of the Young modulus, Model (b). The experimental results, under displacement control show a load drop in the critical load at the onset of propagation; this is due to an unstable propagation of the crack which grows catastrophically and arrests near the mid span. The homogenized model, which is under crack-length control, is able to capture the snap-back instability and follow the virtual branch where crack growth is associated to a reduction of the load-point displacement. Crack propagation is modelled also in the region beyond the mid-span to show that the curve stably approaches the limiting solution (dotted line) corresponding to two fully delaminated layers (see Lundsgaard-Larsen et al., 2012). Fig. 4 highlights the capability of the homogenized model to analyze multiple delamination fracture and reproduce the effects of the interaction between delaminations. The dimensionless diagrams depict the critical load for the propagation of the cracks in the cantilever beam in Fig. 2b versus load-point deflection. The diagram (a) refers to a homogeneous isotropic beam where (2) (1) 2 h h  , (3) (1) 1 h h  and the initial crack lengths are 0 / 5.5 U a h  and 0 / 6 L a h  . The diagram (b) to a homogeneous isotropic beam where ( 2) (1) (3) (1) 1 / 3 h h h h   and the initial crack lengths are 0 / 5 U a h  and 0 / 4 L a h  . The delaminations are assumed to propagate collinearly when the energy release rate, calculated using the compliance method, approaches the critical value, II IIC  G G . The results of the homogenized model are compared with the results of the discrete-layer cohesive-crack model with spring-contact in (Andrews et al, 2006; Andrews and Massabò, 2008). A local snap-through instability is observed in the diagram (a) when the upper crack starts to propagate in A and approaches the lower crack tip, in B. Then the load to propagate the crack must be increased, due to a shielding phenomenon, up to point C where the two cracks propagate together unstably. In the diagram (b) the lower crack, which is shorter, starts to propagate at the maximum load, point A; crack propagation is unstable and characterized by a snap-back instability up to point B. Then there is a sudden drop in the load, to point C, caused by a sudden amplification discontinuity. After point C the only the lower crack propagates. Table 1 – 2D elasticity solution for Energy Release Rate and mode mixity angle in the sandwich beam in Fig. 5, with / 2 h H  Sandwich beam 2D Elasticity Source Three isotropic layers 2 2 2 (1) Fig. 5 highlights the capability of the homogenized model to analyze mode II dominant fracture in layered beams. Results (red curves) are presented for the critical load for propagation of a debond in a sandwich beam with three isotropic layers (geometry and material properties in the caption) using the homogenized model and accurate 2D elasticity solutions (Table 1). The results are also compared with those assuming the material to be homogeneous and the delamination at the same thickness-wise location. The effects of the layup on the critical load for crack propagation are significant and well captured by the homogenized model. The reduced accuracy of the solutions of the homogenized model when applied to the homogeneous beam depends mainly on the crack tip conditions which have (2) (1) 2 2 1 / 4 E E  2 E h 2 0.265 1 0.484  0.739 0.621 a h h a h a h a P                                    G ; 1 tan (  / ) 83.1  II G G I    Barbieri et al, 2018