3. Modeling A: Linear mesh, classical formulation#
3.1. Characteristics of modeling#
Figure 3.1‑a: Section of the mesh in the plane of the crack
The items are all of order 1.
The advantage of this modeling is to serve as a basis for more advanced formulations, and thus to be able to observe the contribution and improvements of other methods.
3.2. Characteristics of the mesh#
Number of knots: 11310
Number of meshes: 14453
Type of meshes |
Number of meshes |
POI1 |
4 |
SEG2 |
39 |
TRIA3 |
360 |
QUAD4 |
930 |
PENTA6 |
5440 |
HEXA8 |
7680 |
Table 3.2-1: Characteristics of meshes
3.3. note#
The calculation of stress intensity factors is done using POST_K1_K2_K3 (method for extrapolating movements on the lips of the crack) [bib2].
3.4. Test values and results of modeling A#
The POST_K1_K2_K3 procedure makes it possible to identify the values of the stress intensity factors to the nearest coefficient. It should be noted that this method identifies the stress intensity factor \(\mathrm{K1}\) (respectively \(\mathrm{K2}\), \(\mathrm{K3}\)) from the displacement jump using a least squares method.
3.4.1. Results in the case of tensile (K1) and torsional (K3) loading#
Identification |
Reference |
Aster |
% difference |
|
\(\mathrm{K1}\) at node \({\mathrm{PFON}}_{\mathrm{FIN}}\) |
5.35 106 |
5.35 106 |
4.52 106 |
15 |
\(\mathrm{K3}\) at node \({\mathrm{PFON}}_{\mathrm{FIN}}\) |
-11.22 106 |
-9.54 106 |
-9.54 106 |
15 |
The values of \(\mathrm{K1}\) and \(\mathrm{K3}\) must be the same [Figure 4.2-a] for all the nodes at the bottom of the crack because we have an axisymmetric configuration. Here, we’re only testing values at node \({\mathrm{PFON}}_{\mathrm{FIN}}\).
3.4.2. Results in the case of non-contact flexural loading (K1)#
Identification |
Reference |
Aster |
% difference |
|
\(\mathrm{K1}\) at node \(A\) |
11.71 106 |
9.18 106 |
11.71 106 |
22 |
The value of \(\mathrm{K1}\) is compared to the reference solution only at the point of maximum openness (node \(A\)) because it is the only analytical value available in the literature.
3.4.3. Results in the case of flexural loading (K1) with contact#
Identification |
Reference |
Aster |
% difference |
\(\mathrm{K1}\) at node \(A\) |
10.17 106 |
8.38 106 |
20 |
The result obtained is compared to that obtained by*Code_Aster* without taking contact into account (non-regression). This consideration is carried out using the active constraints method.
3.4.4. Changes in K1, K2, K3 along the crack bottom#
Figure 4.2-a: \(\mathrm{K1}\), \(\mathrm{K2}\) and \(\mathrm{K3}\) along the crack bottom (in \({\mathrm{MPa.m}}^{1/2}\))
Figure 4.2-b: \(\mathrm{K1}\) along the crack bottom (in \({\mathrm{MPa.m}}^{1/2}\))
Feedback on the results:
The [Figure 4.2-a] shows the evolution of stress intensity factors along the crack bottom of the axisymmetric crack of depth \(100\mathrm{mm}\) subjected to traction and twisting. Axisymmetric results are well observed (with the exception of calculation errors). In addition, we note that the crack is not stressed in \(\mathrm{II}\) mode.
On the [Figure 4.2-b], we highlight the consideration of contact. On the crack half when opening, \(\mathrm{K1}\) has lower values including contact, because contact rigidifies the structure. Out of the closed half, \(\mathrm{K1}\) sucks.
In fact, the contact does not take place over the entire upper half of the crack [Figure 4.2‑c] but on a slightly smaller area. On the [Figure 4.2-c] the red zone represents the contact zone and the blue zone represents the non-contact zone.
Figure 4.2-c: Contact