4. F modeling#
4.1. Characteristics of modeling#
In this modeling, the crack is not meshed. We use the X- FEM method.
Given the symmetries of the problem, it is possible to represent only one eighth of the structure (as is done in modeling A). However, with the X- FEM method, it is not possible to represent a crack that is located in a plane of symmetry (on the edge of the modeled domain). In this modeling, we therefore model a quarter of the structure, that is to say a portion of \(90°\) of the ellipse.
The mesh is composed of cells HEXA8, uniformly distributed along the axes \(X\) and \(Y\) and distributed in geometric progression along the axis \(Z\) so that in the plane \(Z\mathrm{=}0\), the cells are approximately cubes with sides \(10\mathit{mm}\).
Symmetry conditions are applied to the faces in \(X\mathrm{=}0\) and \(Y\mathrm{=}0\). The rigid mode along the \(Z\) axis is blocked by blocking the movement along \(Z\) of the point located in \((\mathrm{0,0},\mathrm{-}1250\mathit{mm})\).
Loading: Unit pressure distributed on the two normal sides of the block:
\(P\mathrm{=}1\mathit{MPa}\) in the \(Z\mathrm{=}\mathrm{\pm }1250\mathit{mm}\) plans.
4.2. Characteristics of the mesh#
Number of knots: 21000
Number of meshes and types: 13000 PENTA6 and 12500 HEXA8 (linear mesh)
Figure 4.2-1: initial mesh, overview
Figure 4.2-2: initial mesh, zoom in the middle plane
As this initial mesh is much too rough for a precise calculation of stress intensity factors along the bottom of the crack, a procedure for automatic refinement of cells close to the bottom of the crack is used, as recommended in the documentation [U2.05.02].
The desired mesh size is \(b\mathrm{/}9\). This will induce 5 successive refinements. The mesh size of the mesh thus refined is then \(h\mathrm{=}\mathrm{0,39}\mathit{mm}\).
The refined mesh (the one on which the mechanical calculation is performed) has the following characteristics:
26484 knots
7720 TETRA4, 10650 PYRAM5 and 20080 HEXA8
This mesh induces 99 points along the crack bottom and, taking into account the blocking conditions, 118404 equations in the system to be solved.
Figure 4.2-3: refined mesh, zoom in on the area near the crack
4.3. Tested sizes and results#
The values tested are the stress intensity factors \(\mathrm{K1}\) along the crack bottom, calculated either by CALC_G_XFEM or by POST_K1_K2_K3.
For CALC_G_XFEM, the integration crown is worth \(2h\mathrm{-}4h\). The smoothing by default (Legendre) is used.
For POST_K1_K2_K3, the maximum curvilinear abscissa is \(4h\). In order to reduce the calculation times of POST_K1_K2_K3, only 20 points distributed uniformly along the crack bottom are post-processed.
Note that the calculation time for CALC_G’s Legendre smoothing is insensitive to this number.
The values are tested at points \(A\) (\(s\mathrm{=}0\)) and \(B\) (\(s=\mathrm{26,7}\)).
Identification |
Reference type |
Reference value |
Tolerance |
CALC_G_ XFEM: \(\mathit{K1}(A)\) |
“ANALYTIQUE” |
0.000 |
|
CALC_G_ XFEM **: **: \(\mathit{K1}(B)\) |
“ANALYTIQUE” |
4.068 |
|
POST_K1_K2_K3: \(\mathit{K1}(A)\) |
“ANALYTIQUE” |
0.000 |
|
POST_K1_K2_K3: \(\mathit{K1}(B)\) |
“ANALYTIQUE” |
4.068 |
|
For the operator CALC_G_XFEM, smoothing types LAGRANGE do not make it possible to obtain easily usable results; smoothing of type LEGENDRE is therefore preferred.