PSI - Issue 42

Dennis Domladovac et al. / Procedia Structural Integrity 42 (2022) 382–389 Domladovac et al. / Structural Integrity Procedia 00 (2019) 000–000

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2.2. Evaluation method

Used in this research, the J-Integral according to Rice (1968) is a fracture mechanics concept to evaluate the strain energy release rate. One assumption of the J-Integral evaluation is that the crack tip is the only inhomogeneity in the adhesive layer. Therefore, our research should have a closer look to a fracture process, where this assumption is not true, caused by the gap in the adhesive layer. For this, the J-Integral evaluation was based on J = Γ w d y − ⃗ t · ∂⃗ u ∂ x d s (1) as without a gap. Here w is the strain-energy density, Γ a arbitrary path clockwise around the notch-tip, ⃗ t the traction vector on the path Γ , ⃗ u the displacement vector and d s an element of the arc length in Γ . x and y are the coordinates according Fig. 1, where x is the crack propagation direction. The calculation of the J-Integral for the DCB-specimen in situ with was found by Paris and Paris (1988). With the force F and angles θ 1 and θ 2 measured on the load introduction and the adhesive with b . A requirement for this approach is the unloaded end of the specimen on the opposite of the crack, checked with the measurement of the specimen’s end rotation in this study. Another possibility to calculate the J-Integral for the DCB-specimen with the deflection curve values is J = 2 x t x end t y ( x ) φ ( x ) d x (3) and was presented by Schrader et al. (2022). This equation is based on Eq. 1 for a path around the adhesive layer with the assumption of a neglectable contribution in y -direction. The product of traction y-component t y ( x ) and the corresponding slope φ ( x ) of the adherends are integrated over the adhesive layer. From the end of the adhesive layer x end to the crack tip position x t in beam direction. Due to this aim of J-Integral evaluation and the localisation of the gap the adherend’s deformation has been measured. With these measurements and the assumption of the Euler-Bernoulli beam theory the deflection curve and the derived quantity can be computed. The expectation is to localise gaps by regions of an constant transverse force. For this purpose the transverse force has to be calculated from the third derivative of the deflection curve. The DIC delivers the deflection curve w ( x ) directly. Di ff erentiating this three times with respect to beam axis x and with the assumption of the Euler-Bernoulli beam theory leads to the transverse force Q ( x ), by usage of a constant EI z from the analytical value: Q ( x ) = − EI z d 3 w d x 3 . (4) The BFSM performed with fibre optics provides the strain of the adherend’s marginal fibre ε ( x ).With the distance to the neutral axis c the curvature κ ( x ) of the adherend can be computed κ ( x ) = ε ( x ) / c and thereby the bending moment M b ( x ) = − κ ( x ) EI z . Based on this measurement only one derivative is required to calculate the transverse force Q ( x ): Q ( x ) = − EI z c d ε ( x ) d x (5) By di ff erentiating the transverse force with respect to x , one obtains the line-load q y ( x ) and thus the adhesive traction t y ( x ) = q y ( x ) / b with the adhesive width b . Furthermore, by integrating the curvature with respect to x the slope φ ( x ) is received as φ ( x ) = x x end κ ( ¯ x ) d ¯ x + φ ( x end ) (6) at a time with the position of the adhesive end x end and the load introduction x 0 . The constant of integration φ 0 is directly measured as end rotation. With the calculated traction t y ( x ) and the slope φ ( x ) the J-Integral in Eq. 3 can be calculated. The acquired data was evaluated using MATLAB ® and filtered with the build-in LOWESS filter. J = F ( θ 1 + θ 2 ) b (2)