PSI - Issue 12

Venanzio Giannella et al. / Procedia Structural Integrity 12 (2018) 479–491 V. Giannella/ Structural Integrity Procedia 00 (2018) 000 – 000

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A complete benchmark among the three approaches has been presented by Giannella et al. (2017a), considering a complex industrial application with allowance for load spectrum effects. A further benchmark between FD and LC approaches on a crack propagating in an aeroengine turbine stage has been presented by Citarella et al. (2016b). Both FD and FL approaches predict the same SIFs along the initial crack front, but their crack propagation simulations end up with different CGRs since the displacements or loads, applied on the submodel boundary, are kept as fixed along the whole crack-growth while the DBEM subdomain stiffness gradually decreases, asking, in principle, for a continuous update of the submodel boundary conditions (the impact of such approximation on crack growth is different between FD and FL approaches). The adopted DBEM modelling cannot allow for the spatial variability of the material properties caused by the temperature gradients in the submodel. Consequently, uniform thermomechanical properties (Tab. 1), evaluated at the submodel average temperature, have been considered for the DBEM analysis. When solving the fracture problemwith the FD/FL approaches, the forced hypothesis of uniform thermomechanical properties in the DBEM subdomain produces a non-negligible impact on accuracy of SIFs and CGRs. This is due to the submodel non-negligible temperature gradients that, in combination with the temperature variability of material properties, do not allow for a correct assessment of the stress field. On the contrary, the LC approach can circumvent such drawback because less sensitive to property variations in the subdomain. As a matter of fact, its accuracy is mainly driven by the accuracy of stress assessment along the crack path; such stresses are obtained in the FEM environment with a full allowance for temperature gradients and then exported to the DBEM crack faces (by preliminary transformation into tractions). Namely, the DBEM boundary conditions (tractions applied on the crack faces) correctly incorporate the variations of material properties vs. temperature, because in the FEM environment the spatial temperature variations are rigorously enforced. Moreover, using the LC approach, the submodel size can be kept lower than for FD/FL approaches (Citarella et al., 2016b) with a consequent reduced temperature spatial variation. This is beneficial to the crack growth accuracy because, in the adopted implementation, the Paris’ law (Eq. 1) calibration parameters cannot allow for such variations and are defined at the submodel average temperature. In summary the LC approach turns out to exhibit both computational and accuracy advantages vs. FD and FL and, therefore, only the LC approach is selected to tackle the fracture problem presented in this paper. da⁄dN = C∆K n (1) 3.2. FEM-DBEM LC approach The FEM-DBEM LC approach is based on the application of the superposition principle to fracture mechanics problems and is here described (Fig. 7) with reference to a thermal-stress crack problem (see also Wilson, 1979): • starting from an original uncracked domain (A), undergoing some loading, a crack can be opened (B) and loaded with tractions corresponding to those calculated over the dashed line of the virtual crack in (A); • the new configuration (B), equivalent to the previous one (A), can be transformed by using the superposition principle, splitting the boundary conditions as provided in (C) and (D) (Eq. (2)); (C) represents the original problem to solve (a crack is opened in the uncracked loaded domain (A) and the original stresses are consistently redistributed), whereas (D), after the tractions sign inversion, turns into the equivalent problem (E) that will be effectively worked out; namely, SIFs for case (C) turn out to be equal (Eq. (3)) to those calculated for the simpler problem (E). In conclusion, using the boundary conditions retrieved from the considered thermal-stress problem (Fig. 5), a pure stress analysis for a fracture problem can be solved, in which the boundary conditions are tractions applied on crack faces, equal in magnitude but opposite in sign to those calculated over the dashed line in Fig. 7 (position A). K a = K b = 0 = K c + K d (2) K c = − K d = K e (3)