7. Summary of results#

The A, B, C, D and E models show that method XFEM makes it possible to find the asymptotic field of the theory for a crack opening in \(I\) mode. It can be seen that the field of displacement is accurately represented since, in particular, we find the analytical values of the stress intensity factors.

In paragraph [19], we restored the evolution of the field of movement as a function of the angle of inclination, over a large angular range. We thus demonstrate that the calculated displacement field remains invariant in the local frame of reference linked to the crack: the asymptotic field « follows » the movement of the crack, in accordance with the theory. Therefore, the geometry of the domain in the crack frame of reference, the regularity and the « directionality » of the mesh, have no influence on the accuracy of the calculations of the test case, with linear elements.

In addition, the appearance of a transition zone between the Dirichlet and Neumann boundary conditions is not observed. For example, in C modeling, the test on the \(\mathit{L2}\) displacement error standard is performed on the corner undergoing both a Neumann loading and a Dirichlet condition. The displacement calculated by Aster « sticks » to the analytical solution. Should we also ensure that the same applies to the field of constraints? Developments in the Code_aster should set up a calculation of the energy standard to confirm these observations.

However, there are 2 limitations to the validation presented above:

  • On the one hand, the « infinity » inclination of the crack is not possible in our model. In the C modeling, the angle of inclination is between \(\mathrm{-}135°\) and \(135°\) because of the Dirichlet boundary condition on one of the edges of the domain. Given the symmetry of the problem, continuing this study on the rest of the trigonometric circle does not seem relevant.

  • On the other hand, B modeling shows that validation is possible with quadratic elements. However, there is a significant deterioration in packaging.