Issue 48

J.P.S.M.B. Ribeiro et alii, Frattura ed Integrità Strutturale, 48 (2019) 332-347; DOI: 10.3221/IGF-ESIS.48.32 338 and DLJ specimens were tested in conventional tension, while the ENF and SLB specimens were tested using a three-point bending (3PB) setup. All tests were carried at room temperature and 1 mm/min of velocity. The manual a measurement in the DCB, ENF and SLB tests was done by approximating the crack tip to the nearest 1/8 of mm in the scale, which was made possible by the resolution of the images (0.02 mm/pixel). The 3PB setup is depicted in Fig. 4 for the case of a SLB specimen. Fracture toughness estimation The Linear Elastic Fracture Mechanics (LEFM) approach is a simple approach to estimate G for strength prediction purposes [30]. However, difficulties arise when dealing with materials (e.g., adhesives) with ductility. On the other hand, typically failure in adhesive bonds takes place under mixed-mode due to the different properties of the joints’ components, the applied load and joint architecture, which demands the knowledge of both G IC and G IIC , and also the use of mixed- mode criteria [22]. In this work, the CBBM data reduction method was selected to estimate G IC and G IIC from the DCB and ENF tests, respectively, and G I and G II from the SLB tests. The CBBM provides the fracture measurements only from the experimental compliance ( C ) measured during the tests [31]. This method procedure includes an equivalent crack length ( a eq ), which is defined from the P -  curve. Moreover, it accounts for the Fracture Process Zone (FPZ), which generates around the crack tip due to the materials’ plasticity, otherwise neglected in the analysis when considering the measured value a . For the DCB specimen, G IC is calculated as eq IC f 2 2 2 2 xy 2 6 1 5 a P G G B h h E           (1) where E f is a corrected flexural modulus to account for stress concentrations at the crack tip and stiffness inconsistency between specimens, and G xy is the shear modulus of the adherends. Full derivation can be found in the study of Constante et al. [32]. Applied to the ENF test, G IIC can be obtained by the following expression eq f 2 2 IIC 2 3 9 . 16 P a G B E h  (2) A detailed description of the method can be found in reference [31]. The CBBM applied to the SLB specimens, detailed by Fernández et al. [33], is based on the beam theory of Szekrényes and Uj [34]. Application of the Irwin-Kies expression gives the total energy release rate ( G T ) which, after equation splitting according to Szekrényes and Uj [34], provides G I and G II f f 2 2 2 2 2 eq eq I II 2 3 2 2 3 xy 12 9 3 and . 16 10 16 P a P a P G B E h G B h B E h    G (3) N UMERICAL WORK Models’ construction umerical simulations were undertaken in Abaqus ® to simulate the SLJ and DLJ, in order to validate the experimentally estimated fracture envelopes further in this work. The two-dimensional models included geometrical non-linearities. The models were based on plane-strain elements (CPE4 of Abaqus ® ) for the adherends with elastic-plastic continuum formulation and cohesive elements (COH2D4 of Abaqus ® ) for the adhesive layer. In Fig. 5 it is possible to find a typical mesh for the DLJ with L O =25 mm. It should be stressed that, for all DLJ models, horizontal symmetry was applied to reduce the computational effort. The mesh refinement described next always assured mesh convergence. The elements’ size at the adhesive layer’s edges was 0.2 mm × 0.2 mm, thus, only a single row of CZM elements was used to populate the adhesive layer thickness. This corresponds to using the continuum CZM approach, in which the CZM laws should reproduce the constitutive behaviour of the full adhesive layer, including the t A -dependent stiffness. For all the models, a total of 8 elements was considered in the adherends through-thickness, whereas between 40 and 160 solid elements were introduced length-wise in the adhesive layer length (between the smallest and largest L O ). To speed up the simulations, although without compromising the analysis results, the FE mesh was graded horizontally and vertically (this effect is visible in Fig. 5). All models were fixed at one edge while a vertical restraint and tensile displacement N