3. Modeling A

3.1. Characteristics of modeling

Meshes:

1. of type HEXA20 for 3D models_ INCO_UPG and 3D_ INCO_UP

2. type HEXA8 for 3D modeling_ INCO_UPO

Following axis \(z\):

• total thickness \(e=0.01\mathrm{mm}\)

• 2 layers of elements

For the purposes of stripping into a middle node, we define node \(\mathrm{NOEUMI}=A+(0.0.e/4)\) where the deformations and stresses are the same as in \(A\).

Conditions limits :

DDL_IMPO = GROUP_NO =' FACSUP ', DZ = 0

GROUP_NO =” FACINF “, DZ = 0 sides \(\mathit{AEFD}\) (\(z\mathrm{=}0\) and \(z\mathrm{=}0.01\)) GROUP_NO =” FACEAB “, DX = 0 sides \(\mathit{AB}\)

FACE_IMPO = GROUP_MA =” FACEEF “, DNOR = 0 sides \(\mathit{EF}\)

PRES_REP = GROUP_MA =” FACEAE “, PRES = 60 sided \(\mathit{AE}\)

3.2. Characteristics of meshes

Mesh 1 HEXA20:

Number of knots: 1501 knots

Number of meshes: 240 HEXA20

Mesh 2 HEXA8:

Number of knots: 429 knots

Number of meshes: 240 HEXA8

3.3. Tested sizes and results

Results at point \(A\):

• first column 3D_ INCO_UPG without imposing \(\mathrm{GONF}=0\)

• second column 3D_ INCO_UPG by imposing \(\mathrm{GONF}=0\)

• third column 3D_ INCO_UP

• fourth column 3D_ INCO_UPO

Identification	Reference type	Reference value	Tolerance
			1	2	3	4
\(v\)	“ANALYTIQUE”	10-5	10-5	10-5	10-5	10-5
\({\sigma }_{\mathrm{xx}}\)	“ANALYTIQUE”		0.10%	0.10%	0.10%	0.15%
\({\sigma }_{\mathrm{yy}}\)	“ANALYTIQUE”	—60.	0.50%	0.50%	0.50%	9.0%
\({\sigma }_{\mathrm{zz}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	5.0%
\({\sigma }_{\mathrm{xy}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	50.00%
\({\varepsilon }_{\mathrm{xx}}\)	“ANALYTIQUE”	10-4	10-5	10-5	10-5	10-5
\({\varepsilon }_{\mathrm{yy}}\)	“ANALYTIQUE”	—6. 10-4	0.50%	0.50%	0.50%	7.0%
\({\varepsilon }_{\mathrm{zz}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	0.50%
\({\varepsilon }_{\mathrm{xy}}\)	“ANALYTIQUE”		10-5	10-5	10-5	3,10-5
\({\varepsilon }_{\mathrm{eq}}\) - INVA_2	“ANALYTIQUE”	6.92 10-4	10-5	10-5	10-5	5,10-2
\({\varepsilon }_{\mathrm{eq}}\) - PRIN_1	“ANALYTIQUE”	—6. 10-4	0.50%	0.50%	0.50%	7.0%
\({\varepsilon }_{\mathrm{eq}}\) - PRIN_2	“ANALYTIQUE”		0.50%	0.50%	0.50%	10-5
\({\varepsilon }_{\mathrm{eq}}\) - PRIN_3	“ANALYTIQUE”	10-4	10-5	10-5	10-5	5,10-2
\({\sigma }_{\mathrm{eq}}\) - VMIS	“ANALYTIQUE”	138.56	0.50%	0.50%	0.50%	5.00%
\({\sigma }_{\mathrm{eq}}\) - TRESCA	“ANALYTIQUE”		0.50%	0.50%	0.50%	5.00%
\({\sigma }_{\mathrm{eq}}\) - PRIN_1	“ANALYTIQUE”	-60.	0.50%	0.50%	0.50%	10.0%
\({\sigma }_{\mathrm{eq}}\) - PRIN_2	“ANALYTIQUE”		0.50%	0.50%	0.50%	5.00%
\({\sigma }_{\mathrm{eq}}\) - PRIN_3	“ANALYTIQUE”		0.50%	0.50%	0.50%	1.50%
\({\sigma }_{\mathrm{eq}}\) - VMIS	“ANALYTIQUE”	138.56	0.50%	0.50%	0.50%	5.0%

Results at point \(F\):

• first column 3D_ INCO_UPG without imposing \(\mathrm{GONF}=0\)

• second column 3D_ INCO_UPG by imposing \(\mathrm{GONF}=0\)

• third column 3D_ INCO_UP

• fourth column 3D_ INCO_UPO

Identification	Reference type	Reference value	Tolerance
			1	2	3	4
\(u\)	“ANALYTIQUE”	—2.12 10-5	0.10%	0.10%	0.10%	0.10%
\(v\)	“ANALYTIQUE”	2.12 10-5	0.10%	0.10%	0.10%	0.10%
\({\sigma }_{\mathrm{xx}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	3.0%
\({\sigma }_{\mathrm{yy}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	6.0%
\({\sigma }_{\mathrm{zz}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	1.0%
\({\sigma }_{\mathrm{xy}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	2.0%
\({\varepsilon }_{\mathrm{xx}}\)	“ANALYTIQUE”		10-5	10-5	10-5	10-5
\({\varepsilon }_{\mathrm{yy}}\)	“ANALYTIQUE”		10-5	10-5	10-5	10-5
\({\varepsilon }_{\mathrm{xy}}\)	“ANALYTIQUE”	1.5 10-4	0.50%	0.50%	0.50%	2.0%

Checking the transition to the nodes for the middle nodes (only for the result obtained without imposing GONF = 0) - value at node \(\mathrm{NOEUMI}\):

Identification	Reference type	Reference value	Tolerance ( \(\text{\%}\) )
\({\sigma }_{\mathrm{xx}}\)	“ANALYTIQUE”		0.50%
\({\sigma }_{\mathrm{yy}}\)	“ANALYTIQUE”	—60.	0.50%
\({\sigma }_{\mathrm{zz}}\)	“ANALYTIQUE”		0.50%
\({\varepsilon }_{\mathrm{xx}}\)	“ANALYTIQUE”	10-4	0.50%
\({\varepsilon }_{\mathrm{yy}}\)	“ANALYTIQUE”	—6. 10-4	0.50%

3.4. notes

Apart from 3D_ INCO_UPO, very good results are obtained regardless of the formulation adopted since for all the quantities examined, the difference between the solution obtained with the code and the analytical solution is less than \(\text{0.2 \%}\). We can see that the difference between the solutions obtained by imposing or not imposing condition \(\mathit{tr}(\varepsilon )\mathrm{=}0\) is insignificant.

The results obtained with 3D_ INCO_UPOsont less accurate. This is due to the fact that the elements used are linear. To find a more accurate result, the mesh would have to be refined.