1. Reference problem#

1.1. Geometry#

Axisymmetric cylinder (models \(A\) and \(E\)) or rectangular plate (modeling \(B\)) or straight pipe (models \(C\) and \(D\)), or beam (modeling \(F\)), or parallelepiped \(\mathrm{3D}\)

_images/100002000000015200000125878F662AF179A927.png

(models \(G\), \(H\) and \(J\)).

Cylinder geometry \((\mathrm{mm})\):

  • \(a=1\)

  • \(b=2\)

  • \(H=4\)

1.2. Property of the materials#

For all models:

Young’s module: \(E=200000\mathrm{MPa}\)

Tangent module: \({E}_{t}=50000\mathrm{MPa}\)

Poisson’s ratio: \(\nu =0.3\)

\({\sigma }_{0}=400\mathrm{MPa}\)

\(s=1.0{E}^{-2}°{C}^{-1}\)

Coefficient of thermal expansion: \(\alpha =1.0{E}^{-5}°{C}^{-1}\)

Volume heat: \({C}^{p}=0{\mathrm{J.mm}}^{-3}\mathrm{.}°{C}^{-1}\)

Thermal conductivity: \(\lambda =1.0{E}^{-3}{\mathrm{W.mm}}^{-1}\mathrm{.}°{C}^{-1}\)

For the isotropic material declared orthotropic, it comes:

E_L= E_T= E_N= \(E\)

nu_LT = nu_LN= nu_TN= \(\nu\)

G_LT= G_LN= G_TN= \(75000\mathrm{MPa}\)

ALPHA_L = ALPHA_T = ALPHA_N = \(\alpha\)

For models from \(A\) to \(G\):

\({\sigma }_{y}(T)={\sigma }_{0}(1-\mathrm{s.}(T-{T}_{0}))\)

For models \(H\) and \(I\):

\({\sigma }_{y}(T)\mathrm{=}{\sigma }_{0}\) \((s=0)\)

_images/100002000000018A0000012A94701AA69F070EC8.png

Figure: Material tensile curve

1.3. Boundary conditions and loads#

  • Models \(A\), \(E\) and \(K\): \(\mathrm{uz}=0\) on the sides \(\mathrm{AB}\) and \(\mathrm{CD}\) (Fixed \(\mathrm{Oz}\) axis)

  • Models \(B\) and \(I\): \(\mathrm{uy}=0\) on the sides \(\mathrm{AB}\) and \(\mathrm{CD}\), \(\mathrm{ux}=0\) in \(A\)

  • Models \(C,D\) and \(F\): embedding in \(A\), \(\mathrm{Uy}=0\) in \(C\)

  • Modelings \(G\), \(H\), \(L\), and \(M\): \(\mathrm{uy}=0\) on the sides \(\mathrm{AB}\) and \(\mathrm{CD}\), \(\mathrm{ux}=\mathrm{uz}=0\) (node \(\mathrm{N3}\)), \(\mathrm{uz}=0\) (node): on the sides and \(\mathrm{N4}\)

  • \(T(t)=\gamma t+\mathrm{T0}\) with: \(\gamma =1°C/s\) and \(\mathrm{T0}=0°C\).

  • Models \(H\) and \(I\): Initial deformation fields: \(\varepsilon =\alpha (T-{T}_{0})\mathrm{Id}\)

  • Modeling \(J\): Field of imposed deformations: \({\varepsilon }_{\mathrm{yy}}=0\)