5. G modeling#
5.1. Characteristics of modeling#
The geometry and the load are identical to the F model: a quarter of the structure is modeled, pressure is applied on the upper side. In this modeling, the initial crack is meshed. Compared to the F model, cohesive zones are introduced in the extension of the crack. This extension is represented by level-sets, so that the discontinuity is taken into account by modeling XFEM, as for modeling F. The cohesive law CZM_LIN_MIX is introduced into this model XFEM by the command DEFI_CONTACT.
Loading: Unit pressure distributed on the face of the block opposite the plane of the lip:
\(P=1\mathrm{MPa}\) in the \(Z=1250\mathrm{mm}\) plan.
The cohesive parameters are chosen so that this has the effect of opening a few cohesive elements in the vicinity of the initial crack bottom:
To not have a complete rupture, but simply a de-cohesion close to the initial crack point, we take \({G}_{c}>{G}_{\mathit{max}}\), with \({G}_{\mathit{max}}\) the maximum local \(G\) for the long front, while maintaining the same order of magnitude for both values. In our case, \({G}_{c}=2.5\times {10}^{-4}{\mathit{N.mm}}^{-1}\) versus \({G}_{\mathit{max}}=7.2\times {10}^{-5}{\mathit{N.mm}}^{-1}\).
To still observe de-cohesion in the vicinity of the tip, the characteristic size of the cohesive zone \({l}_{c}=\frac{E{G}_{c}}{(1-{\nu }^{2}){\sigma }_{c}^{2}}\) is chosen so as to cover a few elements while remaining small compared to the size of the structure \(h\le {l}_{c}\le a\). In this test case, to reduce the calculation time, we took \({l}_{c}=14\mathit{mm}\), which leads to \({\sigma }_{c}=2\mathit{MPa}\), to be compared with typical element sizes \(h=1\mathit{mm}\) on the short side of the ellipse, and \(h=2\mathit{mm}\) on the long side.
5.2. Characteristics of the mesh#
Number of knots: 4522
Number of meshes and types: 22300 TETRA4
5.3. Tested sizes and results#
The values tested are the \(\mathrm{K1}\) equivalent stress intensity factors along the crack bottom, calculated by CALC_G. In order to obtain a regular result, only 5 points distributed uniformly along the crack base are post-treated (NB_POINT_FOND =5). We test the values at the end points of the front \(A\) (\(s\mathrm{=}0\)) and \(B\) (\(s\mathrm{=}\mathrm{26,7}\)).
The smoothing “LAGRANGE” is used for \(\theta\) and “LAGRANGE_NO_NO” for \(G\).
Identification |
Reference type |
Reference value |
Tolerance |
CALC_G_ XFEM: \(\mathit{K1}(A)\) |
“ANALYTIQUE” |
0.000 |
|
CALC_G_ XFEM **: **: \(\mathrm{K1}(B)\) |
“ANALYTIQUE” |
4.068 |
|
5.4. note#
For this test, there is a discrepancy of a few percent. Precision can be improved by choosing a smaller cohesive zone and further refining the mesh. This step was not done here so that the modeling could be completed in less than a minute.