1. Reference problem#

1.1. Geometry#

Figure 1.1-a: Geometry of the reference problem

It is a cylinder with height \(H\mathrm{=}\mathrm{20mm}\), inner radius \(\mathit{Rint}\mathrm{=}\mathrm{4.118mm}\), and outer radius \(\mathit{Rext}\mathrm{=}\mathrm{4.746mm}\).

1.2. Material properties#

Material properties are described by the following parameters:

Thermal properties:

\(\rho {C}_{p}\mathrm{=}2000000{\mathit{J.m}}^{\mathrm{-}3}\mathrm{.}°{C}^{\mathrm{-}1}\)
\(\lambda \mathrm{=}9999.9{\mathit{W.m}}^{\mathrm{-}1}\mathrm{.}°{C}^{\mathrm{-}1}\)

Metallurgical properties:

\(\mathit{TDEQ}\mathrm{=}809°C\)
\(K\mathrm{=}{\mathrm{1.135.10}}^{\mathrm{-}2}\)
\(N\mathrm{=}2.187\)
\(\mathit{T1C}\mathrm{=}831°C\)
\(\mathit{T2C}\mathrm{=}0.°C\)
\({\mathit{QSR}}_{K}\mathrm{=}14614\)
\(\mathit{AC}\mathrm{=}{\mathrm{1.58.10}}^{\mathrm{-}4}\)
\(M\mathrm{=}4.7\)
\(\mathit{T1R}\mathrm{=}\mathrm{949,1}°C\)
\(\mathit{T2R}\mathrm{=}0.°C\)
\(\mathit{AR}\mathrm{=}\mathrm{-}5.725\)
\(\mathit{BR}\mathrm{=}0.05\)

Thermo-elastic mechanical properties:

Young’s modulus: \(E\mathrm{=}80000\mathit{MPa}\)
Poisson’s ratio: \(\mathit{NU}\mathrm{=}0.35\)
Same expansion coefficient for hot and cold phases \({F}_{\mathit{ALPHA}}=\mathrm{8,0}\times {10}^{-6}°{C}^{-1}\) and \({C}_{\mathit{ALPHA}}=\mathrm{8,0}\times {10}^{-6}°{C}^{-1}\)

Mechanical Properties of the Law META_LEMA_ANI

Parameters related to viscosity, phase \(\alpha\) pure

F1_A = 2.39

F1_M = 0.

F1_N = 4.39

F1_Q = 19922.8

Viscosity parameters, blend \(\alpha +\beta\)

F2_A = 0.22

F2_M = 0.77 E-4

F2_N = 2.96

F2_Q = 21023.7

Parameters related to viscosity, phase \(\beta\) pure

C_A = 9.36

C_M = 0.99 E-4

C_N = 6.11

C_Q = 6219

Coefficient of the anisotropy matrix in plane \((r,\theta ,z)\), phase a

F_ MRR_RR = 0.4414

F_ MTT_TT = 0.714

F_ MZZ_ZZ = 1

F_ MRT_RT = 0.75

F_ MRZ_RZ = 0.75

F_ MTZ_TZ = 0.75

Coefficient of the anisotropy matrix in plane \((r,\theta ,z)\), phase b

C_ MRR_RR = 1

C_ MTT_TT = 1

C_ MZZ_ZZ = 1

C_ MRT_RT = 0.75

C_ MRZ_RZ = 0.75

C_ MTZ_TZ = 0.75

1.3. Boundary conditions and loads#

Thermal part:

A uniform temperature is imposed all over the tube:

Time (\(s\))	Temperature (\(°C\))
-1.

36.1
	799.7
	838.67
	876.52
49.2	894.5

Mechanical part:

The lower part of the cylinder (\(\text{FACE\_INF}\)) is blocked when moving as follows \(z\): \(\mathit{UZ}(x,y\mathrm{,0})\mathrm{=}0\)

The entire upper part of the cylinder (\(\text{FACE\_SUP}\)) has a uniform \(z\) displacement

Pressure is imposed on the inside face of the tube (\(\text{FACE\_INT}\)):

Time (\(s\))	Pressure (\(\mathit{MPa}\))
-1.0

36.1	6.74
49.2	6.74

We take into account the background effect on the upper part of the tube (FACE_SUP):

Time (\(s\))	Pressure (\(\mathit{MPa}\))
-1.0

36.1	6.74*coeff
49.2	6.74*coeff

With \(\mathit{coef}\mathrm{=}(\mathit{Rint}\mathrm{\times }\mathit{Rint})\mathrm{/}\mathrm{[}(\mathit{Rext}\mathrm{\times }\mathit{Rext})\mathrm{-}(\mathit{Rint}\mathrm{\times }\mathit{Rint})\mathrm{]}\)