Issue 51

R. Massabò et alii, Frattura ed Integrità Strutturale, 51 (2020) 275-287; DOI: 10.3221/IGF-ESIS.51.22

red dashed lines Model (a), and the maximum values, black solid lines Model (b). The experimental results, obtained 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 [18,25], 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. Fig. 5b shows the critical load for crack propagation in an End Loaded Split (ELS) specimen. The material is a unidirectional E-glass-epoxy laminate with elastic constants given in the caption of Fig. 5b. This is a pure mode II problem. Local effects are similar to those of the ENF specimen and described in Fig. 3. The critical load for crack propagation obtained using the homogenized model is / cr IIC L P hE  G   2 / 3 / h a . The solution coincides with predictions using discrete models and elementary beam theory. An accurate solution of the ELS specimen based on dimensional analysis, orthotropy rescaling and 2D finite element analyses has been obtained in [26] and has been recently confirmed by the analytical solution in [27]. For the E-glass-epoxy examined here, the exact solution is / cr IIC L P hE  G   1/4 2 / 3 / (1 ( ) / ) II h a Y h a     , with 1/4 0.33, =2.56 and ( ) 0.4 II Y        . The 2D solution accounts for the near tip deformations which are not accounted for in the homogenized solution since continuity conditions at the crack tip are imposed on global quantities. The relative error on the critical load for crack propagation is 4%, 2%,1% for / 10, 20, 40 a h  , respectively.

(a) (b) Figure 6 : (a) Critical load for crack propagation versus load point displacement in the ELS specimen tested in [28] . Geometry and materials: L = 130 mm, 2 h = 5.98 mm, 0 60 a  mm, b = 20 mm. Cross-ply carbon/epoxy laminate with layup [(0 2 /90) 7 /0 2 //0 2 /(90/0 2 ) 7 ] and 144 L E  GPa, 10 T E  GPa, 4.2 LT TT G G   GPa, 0.25 LT   and 0.3 TT   . Mode II fracture energy 0.5, 0.45 IIC  G N/mm. (b) Comparison of theoretical results using the homogenized model and 2D finite element models. ] The diagram in Fig. 6 shows the critical load for crack propagation versus load point displacement in the End Load Split (ELS) specimen tested in [28]. The material is a cross ply laminate with 46 unidirectional carbon plies of thickness 0.13 mm, layup [(0 2 /90) 7 /0 2 //0 2 /(90/0 2 ) 7 ] and elastic constants given in the caption of Fig. 6. The specimen has total length 160 T L  mm , width 20 B  mm and initial crack length 0 60 a  mm. The theoretical results have been obtained by assuming that a part of the specimen, of length 30 mm, has been clamped so that the free length is 130 L  mm (this is necessary to match 2D finite element results to the slope of the experimental linear branch; using T L would produce unacceptably large Crack propagation in End-Load Split cross-ply laminate [(0 2 /90) 7 /0 2 //0 2 /(90/0 2 ) 7

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