5. C modeling#

The C modeling is carried out with finite elements of shells from the « COQUE_3D » family on a half-circumference of the tube.

5.1. Characteristics of the mesh#

There are 36 quadratic quadratic grid cells on interval \(0,\pi\), in the \(Ox>0\) half-plane. The cells are built on the linear cells used for the elements of the DKT family, see section B modeling, but they were transformed into « QUAD_9 » with the command « CREA_MAILLAGE (MODI_MAILLE =_F (OPTION = » QUAD8_9 « ) ») « . Then the command « MODI_MAILLAGE (ORIE_NORM_COQUE) » is carried out to ensure the correct orientation of the normals to the shell. There are 432 knots.

5.2. Modelling characteristics and loading conditions#

There are 10 layers in the thickness, each with 3 sub-points for calculating stress fields and internal variables.

The lower edge is affected by total blocking in the axial direction \(Oz\); the upper edge is affected by a uniform connection on the translation in the axial direction in order to simulate kinematics identical to that of a very long tube. The values of the pressures applied are corrected 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 =) ».

5.3. 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 \(10^{-6}\). The precision required to meet the plane stress condition is: « RESI_CPLAN_RELA =1e-06 ». Newton’s method uses the option « MATRICE =” TANGENTE “``. (To draw the curves, a calculation performed with a maximum time step of \(0.1\) is used.)

5.4. 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.

5.4.1. Route 1#

_images/ModelC_deplR.png — Fig. 5.1 Radial displacement#

_images/ModelC_deplZ.png — Fig. 5.2 Axial displacement#

_images/ModelC_siTT_1.png — Fig. 5.3 Circumferential stresses at time 1#

_images/ModelC_siTT_4.png — Fig. 5.4 Circumferential stresses at time 4#

_images/ModelC_vari_1.png — Fig. 5.5 Plastic deformation at instant 1#

_images/ModelC_vari_4.png — Fig. 5.6 Plastic deformation at instant 4#

5.4.2. Route 2#

_images/ModelC_chg2_deplR.png — Fig. 5.7 Radial displacement#

_images/ModelC_chg2_siTT_3.png — Fig. 5.8 Circumferential stresses at time 3#

_images/ModelC_chg2_siTT_5.png — Fig. 5.9 Circumferential stresses at time 5#

_images/ModelC_chg2_vari_3.png — Fig. 5.10 Plastic deformation at instant 3#

_images/ModelC_chg2_vari_5.png — Fig. 5.11 Plastic deformation at instant 5#

5.5. 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.

5.5.1. 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.	10
Instant t = 2.	10
Instant t = 3.	10
Instant t = 4.	35
Instant t = 5.	30

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.	15
Instant t = 2.	16
Instant t = 3.	20
Instant t = 4.	100
Instant t = 5.	80

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

DX displacement on average surface	Tolerance (%)
Instant t = 0.5	3
Instant t = 1.	40
Instant t = 2.	40
Instant t = 3.	50
Instant t = 4.	4000
Instant t = 5.	500

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

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	3
Instant t = 1.	4
Instant t = 2.	4
Instant t = 3.	3
Instant t = 4.	30
Instant t = 5.	30

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

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	6
Instant t = 1.	10
Instant t = 2.	10
Instant t = 3.	10
Instant t = 4.	85
Instant t = 5.	70

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

DZ displacement on average surface	Tolerance (%)
Instant t = 0.5	12
Instant t = 1.	30
Instant t = 2.	30
Instant t = 3.	30
Instant t = 4.	600
Instant t = 5.	5300

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.	0.2
Instant t = 3.	0.3
Instant t = 4.	0.2
Instant t = 5.	0.2 (ABSOLU)

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

Component SIYY on an average area	Tolerance (%)
Instant t = 0.5	0.1
Instant t = 1.	0.2
Instant t = 2.	0.2
Instant t = 3.	1.0
Instant t = 4.	0.2
Instant t = 5.	0.8 (ABSOLU)

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

Component SIYY on an average area	Tolerance (%)
Instant t = 0.5	0.5
Instant t = 1.	0.5
Instant t = 2.	0.5
Instant t = 3.	2.0
Instant t = 4.	0.5
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.	10
Instant t = 2.	10
Instant t = 3.	10
Instant t = 4.	6
Instant t = 5.	6

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.	20
Instant t = 2.	20
Instant t = 3.	20
Instant t = 4.	20
Instant t = 5.	20

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.	45
Instant t = 2.	45
Instant t = 3.	45
Instant t = 4.	25
Instant t = 5.	25

5.5.2. 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.	3
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.	5
Instant t = 3.	7
Instant t = 4.	7
Instant t = 5.	10

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

Coefficient factor of the applied pressure	Tolerance (%)
Instant t = 0.5	3
Instant t = 1.	3
Instant t = 2.	9
Instant t = 3.	10
Instant t = 4.	12
Instant t = 5.	12

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

Component SIYY on an average area	Tolerance (%)
Instant t = 3.	7
Instant t = 5.	8

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

Component SIYY on an average area	Tolerance (%)
Instant t = 3.	10
Instant t = 5.	12

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

Component SIYY on an average area	Tolerance (%)
Instant t = 3.	20
Instant t = 5.	10

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

Component V1 on average area	Tolerance (%)
Instant t = 3.	30
Instant t = 5.	20

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

Component V1 on average area	Tolerance (%)
Instant t = 3.	30
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.	45
Instant t = 5.	30