1. Reference problem#
1.1. Geometry#
Cylinder geometry \((m)\):
Inner radius |
\({R}_{i}=19.5\) |
Outside radius |
\({R}_{e}=20.5\) |
Point \(F\) |
|
Thickness |
\(h=1.0\) |
Height |
\(L=10.0\) |
1.2. Material properties#
The material is homogeneous, isotropic, linear, thermoelastic, and the initial state is pristine.
Young’s module |
\(E=2.0E-5{\mathrm{N.mm}}^{-2}\) |
Poisson’s Ratio |
\(\nu =0.3\) |
Volume density |
\(\rho =8.0{E}^{-6}{\mathrm{kg.mm}}^{-3}\) |
Expansion coefficient |
\(\alpha =1.0{E}^{-5}°{C}^{-1}\) |
1.3. Boundary conditions and loads#
Imposed travel:
\(\Omega =1.0{s}^{-1}\) along the \(\mathrm{OZ}\) axis
Imposed loading:
gravity, \(g\mathrm{=}10.0{\mathit{m.s}}^{\mathrm{-}2}\) along the \(\mathrm{OZ}\) axis
traction force on the upper side: \(\mathrm{-}160.0{E}^{\mathrm{-}4}N\) equivalent to a force distributed over \(\mathrm{CD}\) of \(-8.0{E}^{-4}{\mathrm{N.mm}}^{-1}\)
Thermal expansion: \(T(\rho )-{T}_{\mathrm{ref}}(\rho )=\frac{({T}_{s}+{T}_{i})}{2}+\frac{({T}_{s}-{T}_{i})\mathrm{.}(r-R)}{h}\) with:
case 1: \({T}_{s}=0.5°C,{T}_{i}=-0.5°C,{T}_{\mathrm{ref}}=0.0°C\)
case 2: \({T}_{s}=0.1°C,{T}_{i}=0.1°C,{T}_{\mathrm{ref}}=0.0°C\)
These temperature fields are calculated with THER_LINEAIRE, using a stationary calculation on the same mesh, but with a PLAN model in order to have an affine solution in thickness.
Since the boundary conditions in movement (and rotation) are different depending on the modeling in question, they will be described later (in the paragraphs relating to the models).