2. Modeling A#

In this D_ PLAN modeling, the plate is cracked over half a length. The crack is described by method XFEM. The crack is enriched geometrically, on a radius \({R}_{\mathit{ENRI}}\mathrm{=}\mathrm{0,1}\).

The elements are linear of type TRIA3.

2.1. Characteristics of the mesh#

The unit square is meshed regularly []. To build the mesh, we rely on a regular grid \(100\mathrm{\times }100\).

NOMBRE OF NOEUDS: 10201

NOMBRE OF MAILLES: 20400

TRIA3: 20000

_images/100000000000023F00000241640D3D6365C7CCA9.png

Figure 2.1-1: Meshing with triangle-elements

2.2. Tested sizes and results#

2.2.1. Tested sizes:#

For this horizontal crack, we test the value of the stress intensity factors \({K}_{I}\) and \({K}_{\mathrm{II}}\) as well as the value of the energy restoration rate \(G\) given by CALC_G.

For method \(G-\mathrm{thêta}\) (command CALC_G), the following theta field crown is chosen:

\({R}_{\mathit{inf}}\mathrm{=}\mathrm{0,1}a\) and \({R}_{\text{sup}}\mathrm{=}\mathrm{0,3}a\) where \(a\) is the length of the crack.

On the other hand, we test the displacement field calculated by Code_Aster. Instead of performing a local test on a few cells by TEST_RESU, we test the displacement field on a large number of cells. An arbitrary test area has been delimited in the field [].

In practice, we compare: \({∥{U}^{\mathit{calc}}\mathrm{-}{U}^{\mathit{ana}}∥}_{{L}_{2}}<\mathit{tolerance}\mathrm{\times }{∥{U}^{\mathit{ana}}∥}_{{L}_{2}}\).

_images/100002010000012300000110D9A38DDE904A9EF8.png

Figure 2.2.1-1: Definition of test GROUP_MA

We are finally testing the energy of the structure, the \({L}^{2}\) travel norm, throughout the field.

2.2.2. Results:#

Stress Intensity Factors Test:

Identification

Reference

Tolerance

CALC_G

K1

1.00

1.0%

K2

0.00

1.0%

G

1.0 10-5

1.0%

Testing the standard_L2 of the error on the displacement field: \({\Vert {U}^{\mathit{calc}}-{U}^{\mathit{ana}}\Vert }_{{L}_{2}}<\mathit{tolerance}\times {\Vert {U}^{\mathit{ana}}\Vert }_{{L}_{2}}\)

Identification

Reference

Tolerance

POST_ELEM

NORME

0.00

0.1%

Structure energy test:

Identification

Reference

Tolerance

POST_ERREUR

REFERENCE

3,50687407712 10-6

0.1%

Testing the \({L}^{2}\) travel standard across the field:

Identification

Reference

Tolerance

POST_ERREUR

REFERENCE

7,6057690825 10-6

0.1%

2.3. Additional results:#

On the [], the displacement field is represented with amplification of the movement jump at the interface. We can see that the crack opens rigorously in \(\mathit{mode}I\), as expected.

_images/1000000000000233000001E0D23F553334A666B1.png

Figure 2.3-1: Field of movement (with offset)