1. Reference problem#
The mechanical problem treats the loading with imposed displacement (case of relaxation) of an axisymmetric specimen under anisothermal conditions.
1.1. Geometry#
The geometry of the cylindrical test piece is defined by:
radius: r = 3 mm;
length: l = 60 mm.
Since the plane (x, z) is a plane of symmetry, only half of the test piece is modelled.
The test piece is subjected to a uniform temperature field that may vary over time.
1.2. Material properties#
The characteristics are as follows:
Keyword ELAS:
\(\mathit{YOUNG}\mathrm{=}150000.0\mathit{MPa}\)
\(\mathrm{NU}=0.30\)
Keyword VENDOCHAB:
\({S}_{\mathit{VP}}=\mathrm{0,}\)
\(\alpha =0(\mathit{SEDVP1}),\)
\(\beta =0(\mathit{SEDVP2}),\)
\({N}_{\mathit{VP}}=f(T),\)
\({M}_{\mathit{VP}}=f(T)\)
\({K}_{\mathit{VP}}=f(T)\)
\({A}_{D}=f(T),\)
T (°C) |
\({N}_{\mathit{vp}}\) |
|
|
|
|
|
900 |
12.2 |
10.5 |
10.5 |
2110 |
6.3 |
3191.62 |
1000 |
10.8 |
9.8 |
9.8 |
1450 |
5.2 |
2511.35 |
1025 |
10.45 |
9.625 |
9.625 |
1285 |
4.925 |
2341.30 |
In the case where the coefficient \({K}_{D}\) is only dependent on the temperature (modeling c and d).
T (°C) |
900 |
1000 |
1025 |
\({K}_{D}\) |
15 |
15 |
15 |
In the case where the coefficient \({K}_{D}\) is chosen depending on the temperature and on \({\sigma }_{0}\) (modeling A and B).
T (°C) |
0 MPa |
100 MPa |
200 |
200 MPa |
900 |
14,355 |
14,855 |
15,355 |
|
1000 |
14.5 |
15 |
15.5 |
|
1025 |
14.5363 |
15.0363 |
15.5363 |
15.5363 |
1050 |
14.5725 |
15.0725 |
15.5725 |
15.5725 |
Table 1.2-1: K_D dependent on temperature and stress |
1.3. Boundary conditions and loads#
1.3.1. Thermal#
The temperature field is homogeneous but not stationary. Its variation is as follows: T is constant and is equal to 1000° C. from t=0s to t=200000s (55.55h); then T increases linearly to reach 1025° C. at t=2000000s (555.55 h); T is then constant at 1025° C.
Image 1.3-1: Thermal loading
1.3.2. Mechanics#
The plane of symmetry (z, x) imposes a blocking of UY at y=0mm. The mechanical load is UY=0.1 mm in z=30mm. Charging is not instantaneous due to convergence problems. We chose to impose the constraint or the displacement linearly over time in order to reach the final load in 0.1 seconds.
1.4. Initial conditions#
Zero stresses and deformations.