6. D modeling

The D modeling is carried out with finite elements of shells from the « COQUE_SOLIDE » family on a half-circumference of the tube.

6.1. Characteristics of the mesh

There are 36 linear quadrangle cells on the interval \(0,\pi\), in the \(Ox>0\) half-plane, but one cell along the axis of the tube \(Oz\). The cells are built on the quadratic cells used for reference modeling, see section Modeling A, but they have been transformed into « HEXA8 ». The command « MODI_MAILLAGE (ORIE_PEAU) » is carried out to ensure the correct orientation of the normals to the shell.

There are 5 sub-points in the thickness for calculating stress fields and internal variables.

6.2. Loading conditions

The lower face is affected by total blocking in the axial direction \(Oz\); the upper face is affected by a uniform connection in the axial direction to simulate kinematics identical to that of a very long tube.

The pressures applied are natively corrected in this shell model to respect the exact metric of the internal and external walls, so that they are identical to those applied in modeling A. The keyword factor used in « AFFE_CHAR_MECA » is « FORCE_COQUE =_F (PRES =) ».

6.3. Characteristics of nonlinear static modeling

The time step chosen is \(0.5\), fixed for all non-linear calculation steps; however, a cut is authorized in case of convergence failure, the criterion « RESI_GLOB_RELA » being set to \(10^{-6}\). Newton’s method uses the option « MATRICE =” TANGENTE “``. To calculate the limit pressure, load control is used by controlling the load by the radial movement of a node in the external wall. (To draw the curves, a calculation performed with a maximum time step of \(0.1\) is used.)

6.4. Post-treatment

The post-processing of the radial displacement is done at any time in a node on the \(Oy=0\) axis. The post-processing of constraints and internal variables is done on a mesh and an integration point with its family of sub-points, close to the \(Ox\) axis.

6.4.1. Route 1

Fig. 6.1 Radial displacement

Fig. 6.2 Axial displacement

Fig. 6.3 Circumferential stresses at time 1

Fig. 6.4 Circumferential stresses at time 4

Fig. 6.5 Plastic deformation at instant 1

Fig. 6.6 Plastic deformation at instant 4

6.4.2. Route 2

Fig. 6.7 Radial displacement

Fig. 6.8 Circumferential stresses at time 3

Fig. 6.9 Circumferential stresses at time 5

Fig. 6.10 Plastic deformation at instant 3

Fig. 6.11 Plastic deformation at instant 5

6.5. Tested sizes and results

The quantities tested are confronted with a non-regression criterion and also with their reference values produced by modeling A, see Modeling A.

6.5.1. Route 1

• Radial displacement on the mean surface (tube \(h/R = 1/20\))

DX displacement on average surface	Tolerance (%)
Instant t = 0.5	1
Instant t = 1.	2
Instant t = 2.	1
Instant t = 3.	2
Instant t = 4.	2
Instant t = 5.	1

• Radial displacement on the mean surface (tube \(h/R = 1/10\))

DX displacement on average surface	Tolerance (%)
Instant t = 0.5	2
Instant t = 1.	3
Instant t = 2.	3
Instant t = 3.	5
Instant t = 4.	2
Instant t = 5.	2

• Radial displacement on the mean surface (tube \(h/R = 1/5\))

DX displacement on average surface	Tolerance (%)
Instant t = 0.5	6
Instant t = 1.	10
Instant t = 2.	10
Instant t = 3.	12
Instant t = 4.	100
Instant t = 5.	20

• Uniform axial displacement \(h/R = 1/20\) tube

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	1
Instant t = 1.	1
Instant t = 2.	1
Instant t = 3.	1
Instant t = 4.	3
Instant t = 5.	3

• Uniform axial displacement \(h/R = 1/10\) tube

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	1
Instant t = 1.	1
Instant t = 2.	1
Instant t = 3.	1
Instant t = 4.	3
Instant t = 5.	2

• Uniform axial displacement \(h/R = 1/5\) tube

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	2
Instant t = 1.	3
Instant t = 2.	4
Instant t = 3.	4
Instant t = 4.	1
Instant t = 5.	20

• Average circumferential surface stress tube \(h/R = 1/20\)

Component SIYY on an average area	Tolerance (%)
Instant t = 0.5	0.1
Instant t = 1.	0.2
Instant t = 2.	1.0
Instant t = 3.	4.0
Instant t = 4.	0.2
Instant t = 5.	2.0 (ABSOLU)

• Average circumferential surface stress tube \(h/R = 1/10\)

Component SIYY on an average area	Tolerance (%)
Instant t = 0.5	0.5
Instant t = 1.	1.0
Instant t = 2.	0.2
Instant t = 3.	1.0
Instant t = 4.	0.1
Instant t = 5.	0.6 (ABSOLU)

• Average circumferential surface stress tube \(h/R = 1/5\)

Component SIYY on an average area	Tolerance (%)
Instant t = 0.5	2.0
Instant t = 1.	4.0
Instant t = 2.	3.0
Instant t = 3.	1.0
Instant t = 4.	1.0
Instant t = 5.	3.0 (ABSOLU)

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/20\))

Component V1 on average area	Tolerance (%)
Instant t = 0.5
Instant t = 1.	0.5
Instant t = 2.	0.5
Instant t = 3.	0.5
Instant t = 4.	2.0
Instant t = 5.	2.0

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/10\))

Component V1 on average area	Tolerance (%)
Instant t = 0.5
Instant t = 1.	0.5
Instant t = 2.	0.5
Instant t = 3.	0.5
Instant t = 4.	2.0
Instant t = 5.	2.0

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/5\))

Component V1 on average area	Tolerance (%)
Instant t = 0.5
Instant t = 1.	5.0
Instant t = 2.	5.0
Instant t = 3.	5.0
Instant t = 4.	5.0
Instant t = 5.	5.0

6.5.2. Path 2 monotonous internal pressure to ruin

• Radial displacement on the mean surface (tube \(h/R = 1/20\))

Coefficient factor of the applied pressure	Tolerance (%)
Instant t = 0.5	1
Instant t = 1.	1
Instant t = 2.	1
Instant t = 3.	5
Instant t = 4.	6
Instant t = 5.	7

• Radial displacement on the mean surface (tube \(h/R = 1/10\))

Coefficient factor of the applied pressure	Tolerance (%)
Instant t = 0.5	2
Instant t = 1.	2
Instant t = 2.	1
Instant t = 3.	5
Instant t = 4.	5
Instant t = 5.	6

• Radial displacement on the mean surface (tube \(h/R = 1/5\))

Coefficient factor of the applied pressure	Tolerance (%)
Instant t = 0.5	5
Instant t = 1.	5
Instant t = 2.	2
Instant t = 3.	5
Instant t = 4.	4
Instant t = 5.	5

• Average circumferential surface stress tube \(h/R = 1/20\)

Component SIYY on an average area	Tolerance (%)
Instant t = 3.	5
Instant t = 5.	8

• Average circumferential surface stress tube \(h/R = 1/10\)

Component SIYY on an average area	Tolerance (%)
Instant t = 3.	8
Instant t = 5.	10

• Average circumferential surface stress tube \(h/R = 1/5\)

Component SIYY on an average area	Tolerance (%)
Instant t = 3.	15
Instant t = 5.	16

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/20\))

Component V1 on average area	Tolerance (%)
Instant t = 3.	20
Instant t = 5.	15

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/10\))

Component V1 on average area	Tolerance (%)
Instant t = 3.	35
Instant t = 5.	20

• Cumulative plastic deformation on the average surface (tube \(h/R = 1/5\))

Component V1 on average area	Tolerance (%)
Instant t = 3.	50
Instant t = 5.	35