8. Modeling I

X- FEM method with CALC_G_XFEM and POST_K1_K2_K3 axi-symmetric.

8.1. Characteristics of modeling

This modeling makes it possible to test the calculation of \(\mathrm{K1}\) using POST_K1_K2_K3 and CALC_G_XFEM (option CALC_K_G) on a non-meshed axi-symmetric crack (method X- FEM).

Two types of loads are considered. The first is simple traction applied to the top and bottom edges of the plate. The second is simple traction applied to the top and bottom edges of the plate and a rotation of \(150\mathit{trs}\mathrm{/}\mathit{min}\) around the symmetric axis.

8.2. Characteristics of the mesh

Number of knots: 20301

Number of meshes and type: 20000 QUA4 and 600 SE2 (linear mesh)

8.3. Tested sizes and results

We test the values of \(G\) and \({K}_{I}\) calculated by the command CALC_G_XFEM option “CALC_K_G” and by the command POST_K1_K2_K3, as well as the .value of \(G\) calculated by the command CALC_G_XFEM (option “CALC_G”).

Loading 1: simple pull applied to the top and bottom edges of the plate

Identification	Reference type	Reference value	Tolerance ( \(\text{\%}\) )
\(G\) (CALC_G_XFEMoption “CALC_K_G”)	“ANALYTIQUE”	11.59	2.1%
\(\mathit{K1}\) (CALC_G_XFEMoption “CALC_K_G”)	“ANALYTIQUE”	1.60E+06	6.0%
\(G\) (CALC_G_XFEMoption “CALC_K_G”)	“AUTRE_ASTER”	11.78	0.4%
\(\mathit{K1}\) (CALC_G_XFEMoption “CALC_K_G”)	“AUTRE_ASTER”	1,64E+06	2,0%
\(G\) (CALC_G_XFEMoption “CALC_G”)	“ANALYTIQUE”	23.17	2.1%
\(G\) (POST_K1_K2_K3)	“ANALYTIQUE”	11.59	6.0%
\(\mathit{K1}\) (POST_K1_K2_K3)	“ANALYTIQUE”	1.60E+06	6.0%

Loading 2: simple traction applied to the top and bottom edges of the plate and a rotation of 150 \(\mathrm{trs}/\mathrm{min}\) around the symmetric axis

Identification	Reference type	Reference value	Tolerance ( \(\text{\%}\) )
\(G\) (CALC_G_XFEMoption “CALC_K_G”)	“AUTRE_ASTER”	2136.52	0.3%
\(\mathit{K1}\) (CALC_G_XFEMoption “CALC_K_G”)	“AUTRE_ASTER”	2,191E+07	2.5%
\(G\) (CALC_G_XFEMoption “CALC_G”)	“AUTRE_ASTER”	4273.04	0.3%
\(G\) (POST_K1_K2_K3)	“AUTRE_ASTER”	2136.52	4.0%
\(\mathit{K1}\) (POST_K1_K2_K3)	“AUTRE_ASTER”	2,191E+07	2,5%

8.4. notes

In this test case, the \(a\mathrm{/}W\) ratio between the crack size \(a\) and the width \(W\) is \(\mathrm{0,2}\). Edge effects therefore contribute to the difference between the numerical solution for a finite edge and the reference solution for a continuous environment.