1. Reference problem

1.1. Geometry

The structure is a 2D plate of dimensions \(\mathrm{LX}=10m\) and \(\mathrm{LY}=30m\), with a straight and horizontal opening crack of length \(a=5m\), located halfway up (see [Figure 1.1‑a]).

To obtain the reference solution, the problem is treated by the classical finite element method and the crack is meshed. On the other hand, for the validation of the modal calculation with the X- FEM method, the crack is not meshed, and the geometry is in fact a healthy plate without cracks. The crack will then be introduced by level sets directly into the command file using the DEFI_FISS_XFEM [U4.82.08] operator.

Figure ‑ 1.1-a : Geometry of the cracked plate

1.2. Material properties

Young’s module: \(E=205000\mathrm{MPa}\) (unless otherwise stated)

Poisson’s ratio: \(\nu =0.0\)

Density: \(\rho =7800\mathrm{kg}/{m}^{3}\)

1.3. Boundary conditions and loads

The plate is embedded on its lower edge and subjected to uniform pressure \(P=10\mathrm{Mpa}\) on its upper edge in order to open the crack (see Figure3.2-A).