2. Benchmark solution#
2.1. Calculation method used#
The reference solution is that obtained by a direct temporal resolution of the transitory problem. The DYNA_VIBRA operator is used to identify the displacement fields, with a Newmark time integration diagram.
The evolution of stress intensity factors as a function of time is then calculated by interpolation of the movement jumps (operator POST_K1_K2_K3).
2.2. Benchmark results — Modeling A#
For modeling A, the plate is stressed by a force in the \((O,x,y)\) plane and the movements in the \(z\) direction are blocked. The reference result, calculated by direct temporal resolution on a 2D mesh, is plotted in the following figure. The horizontal displacement of the top of the plate and the stress intensity factor oscillate with a frequency corresponding to the first natural mode of the structure.
2.3. Benchmark results — B modeling#
The evolution of the three stress intensity factors is plotted in the following figure for the node located in the middle of the crack bottom. The oscillations of the stress intensity factors show the preponderant contribution of the first mode of flexure of the plate in the \(x\) direction and of the first mode of flexure in the \(z\) direction.
2.4. Benchmark results — C modeling#
The C modeling represents a 2D non-meshed crack (X- FEM). The other characteristics are identical to modeling A. For modeling C, the reference solution is the solution calculated by code_aster from a direct temporal resolution of the transitory problem. The commands used to generate this solution are provided in the command file as comments.
2.5. Benchmark results — D modeling#
The D model represents a non-meshed 3D crack (X- FEM). The other characteristics are identical to modeling B. For modeling D, the reference solution is the solution calculated by code_aster from a direct temporal resolution of the transitory problem. The commands used to generate this solution are provided in the command file as comments.
2.6. Uncertainty about the solution#
The explicit direct resolution of the transitory problem can be considered accurate. The uncertainty concerning the identification of stress intensity factors by interpolation of movement jumps is of the order of 5%.
2.7. Bibliographical references#
GALENNE, S. DI DOMIZIO: Theta method in fracture mechanics: development of the bilinear form g in 3D and application to the case of low-frequency dynamics, Note EDF HT-65/05/024/A, 2005