4. Modeling B: Quadratic mesh, classical formulation#

4.1. Characteristics of modeling#

Figure 5.1-a: Mesh and Torus

A torus is created around the crack. Torus elements are quadratic elements. The elements outside the torus are linear. In addition, BARSOUM elements (middle knots moved to a quarter) are used for cells that have an edge belonging to the crack bottom [bib3].

The advantage of using a BARSOUM mesh is to obtain more accurate results.

4.2. Characteristics of the mesh#

Number of knots: 20030

Number of meshes: 16449

Type of meshes	Number of meshes
POI1	2000
SEG3	39
TRIA3	360
QUAD4	610
QUAD8	320
PENTA6	4800
PENTA15	640
HEXA8	5760
HEXA20	1920

Table 5.2-1: Characteristics of meshes

The middle nodes of the edges of the elements touching the bottom of the crack are moved to a quarter of these edges, to obtain better precision.

4.3. Test values and B modeling results#

Results in the case of tensile (\(\mathrm{K1}\)) and torsional (\(\mathrm{K3}\)) loading ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ ~~~~~~~~~

Identification	Reference	Aster	% difference
\(\mathrm{K1}\) at node \({\mathrm{PFON}}_{\mathrm{FIN}}\)	5,35 106	5,35 106	5,35 106	5,7
\(\mathrm{K3}\) at node \({\mathrm{PFON}}_{\mathrm{FIN}}\)	-11,22 106	-10,80 106	-10,80 106	3.8

The values of \(\mathrm{K1}\) and \(\mathrm{K3}\) must be the same [Figure 6.2‑a] for all the nodes at the bottom of the crack because we have an axisymmetric configuration. Here, we’re only testing values at the last node of the crack (\({\mathrm{PFON}}_{\mathrm{FIN}}\)).

4.3.1. Results in the case of non-contact flexural loading (\(\mathrm{K1}\))#

Identification	Reference	Aster	% difference
\(\mathrm{K1}\) at node \(A\)	11.71 106	11.71 106	10.29 106	12

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.

4.3.2. Results in the case of flexural loading (\(\mathrm{K1}\)) with contact#

Identification	Reference	Aster	% difference
\(\mathrm{K1}\) at node \(A\)	10.59 106	10.59 106	10.59 106	13

The result obtained is compared to that obtained by the Aster calculation without taking into account contact (non-regression). The contact resolution method is that of active constraints.

4.4. Changes in K1, K2, K3 along the crack bottom#

Figure 4.4-a: \(\mathrm{K1}\), \(\mathrm{K2}\) and \(\mathrm{K3}\) along the crack bottom (in \({\mathrm{MPa.m}}^{1/2}\))

Figure 4.4-b: \(\mathrm{K1}\) along the crack bottom (in \({\mathrm{MPa.m}}^{1/2}\))