3. Modeling A#
3.1. Characteristics of modeling#
This is an X- FEM modeling, in plane deformations, where the interfaces are defined by level functions (normal level sets noted \(\text{LN}\)).
The equations of the level functions for the three horizontal cracks are as follows:
\(\text{LN}1=Y-0.5\) eq 3.1-1
\(\text{LN}1\mathrm{=}Y\mathrm{-}1.5\) eq 3.1-2
\(\text{LN}2=Y-2.5\) eq 3.1-3
\(\text{LN}3=Y-3.5\) eq 3.1-4
3.2. Characteristics of the mesh#
Figure 3.2-a: The modeling mesh A.
The mesh comprises 4 cells of the QUAD4 type, represented in FIG. 3.2-a.
For example, we notice in this figure that node \(\mathrm{N1}\) sees 2 cracks. It must therefore be enriched twice and it then has the kinematic degrees of freedom DX, DY, H1X, H1Y, H2X and H2Y. Since the contact is active, node \(\mathrm{N1}\) also has Lagrange degrees LAGS_C and LAG2_C.
On the other hand, we notice for example that mesh \(\text{M1}\) « sees » 4 cracks. The element associated with this mesh will therefore store the fields of the four cracks, regardless of the degrees of freedom associated with its nodes.
3.3. Tested sizes and results#
The movements at the level of the crack lips are tested. The DX displacement should follow function \(\text{Depl\_X}\) in equation 2.1-1. The displacement DY must follow function \(\text{Depl\_Y}\) in equation 2.1-2. The stair deformation of FIG. 3.4-a is obtained.
Identification |
Reference |
SOMM_ABSpour DX- \(\text{Depl\_X}\) (master side) |
0 |
SOMM_ABSpour DY- \(\text{Depl\_Y}\) (master side) |
0 |
SOMM_ABSpour DX- \(\text{Depl\_X}\) (slave side) |
0 |
SOMM_ABS for DY- \(\text{Depl\_Y}\) (slave side) |
0 |
Table 3.3-1
Figure 3.4-a: Deformed structure.
We test the value of \({E}^{e}\) produced by the POST_ERREUR operator.
Identification |
Reference type |
Reference value |
Tolerance |
Ee |
“ANALYTIQUE” |
2.6 107 |
|
We test the value of \({\parallel u\parallel }_{{L}^{2}}\) produced by the POST_ERREUR operator.