7. E modeling

7.1. Characteristics of modeling

Meshing with incompressible 3D elements of type PENTA15 only

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 \(A\).

Boundary conditions:

DDL_IMPO = GROUP_NO =' FACSUP ', DZ = 0

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

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

PRES_REP = GROUP_MA =” FACEAE “, PRES = 60. side \(\mathrm{AE}\)

7.2. Characteristics of the mesh

Number of knots: 1501 knots

Number of meshes: 480 PENTA15

7.3. 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

• 3D fourth column_ INCO_UPOavec linear elements

Identification	Reference type	Reference value	Tolerance
			1	2	3	4
\(u\)	“ANALYTIQUE”		10-5	10-5	10-5	10-5
\(v\)	“ANALYTIQUE”	10-5	0.10%	0.10%	0.10%	0.50%
\({\sigma }_{\mathrm{xx}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	6.0%
\({\sigma }_{\mathrm{yy}}\)	“ANALYTIQUE”	—60.	0.50%	0.50%	0.50%	15.0%
\({\sigma }_{\mathrm{zz}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	5.00%
\({\sigma }_{\mathrm{xy}}\)	“ANALYTIQUE”		0.6	0.6	0.6	03/06/12
\({\varepsilon }_{\mathrm{xx}}\)	“ANALYTIQUE”	10-4	0.50%	0.50%	0.50%	1.0%
\({\varepsilon }_{\mathrm{yy}}\)	“ANALYTIQUE”	—6. 10-4	0.50%	0.50%	0.50%	1.0%
\({\varepsilon }_{\mathrm{zz}}\)	“ANALYTIQUE”		10-5	10-5	10-5	10-5
\({\varepsilon }_{\mathrm{xy}}\)	“ANALYTIQUE”		10-5	10-5	10-5	5,10-5
\({\varepsilon }_{\mathrm{eq}}\) - INVA_2	“ANALYTIQUE”	6.92 10-4	0.50%	0.50%	0.50%	8.0%
\({\varepsilon }_{\mathrm{eq}}\) - PRIN_1	“ANALYTIQUE”	—6. 10-4	0.50%	0.50%	0.50%	8.0%
\({\varepsilon }_{\mathrm{eq}}\) - PRIN_2	“ANALYTIQUE”		10-5	10-5	10-5	10-5
\({\varepsilon }_{\mathrm{eq}}\) - PRIN_3	“ANALYTIQUE”	10-4	0.50%	0.50%	0.50%	8.0%
\({\sigma }_{\mathrm{eq}}\) - VMIS	“ANALYTIQUE”	138.56	0.50%	0.50%	0.50%	10.%
\({\sigma }_{\mathrm{eq}}\) - TRESCA	“ANALYTIQUE”		0.50%	0.50%	0.50%	10.%
\({\sigma }_{\mathrm{eq}}\) - PRIN_1	“ANALYTIQUE”	-60.	0.50%	0.50%	0.50%	12.%
\({\sigma }_{\mathrm{eq}}\) - PRIN_2	“ANALYTIQUE”		0.50%	0.50%	0.50%	5.00%
\({\sigma }_{\mathrm{eq}}\) - PRIN_3	“ANALYTIQUE”		0.50%	0.50%	0.50%	6.0%
\({\sigma }_{\mathrm{eq}}\) - VMIS	“ANALYTIQUE”	138.56	0.50%	0.50%	0.50%	10.%

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

• 3D fourth column_ INCO_UPOavec linear elements

Identification	Reference type	Reference value	Tolerance
			1	2	3	4
\(u\)	“ANALYTIQUE”	—2.12 10-5	0.10%	0.10%	0.10%	0.30%
\(v\)	“ANALYTIQUE”	2.12 10-5	0.10%	0.10%	0.10%	0.30%
\({\sigma }_{\mathrm{xx}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	50.00%
\({\sigma }_{\mathrm{yy}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	6.0%
\({\sigma }_{\mathrm{zz}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	0.50%
\({\sigma }_{\mathrm{xy}}\)	“ANALYTIQUE”		0.50%	0.50%	0.50%	1.50%
\({\varepsilon }_{\mathrm{xx}}\)	“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%	1.50%

Checking the transition to the nodes for the middle nodes (only for the result obtained without imposing \(\mathrm{GONF}=0\)) - 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%

7.4. notes

Except for the 3D_ INCO_UPO modeling, 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 \(\text{0.5 \%}\). We can see that the difference between the solutions obtained by imposing or not imposing condition \(\mathrm{tr}(\varepsilon )=0\) 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.