1. Reference problem#
1.1. Geometry and boundary conditions#
Volume element materialized by a cube with a unit side (\(m\)):
Figure 1.1-a: Geometry
The load is such that a state of uniform stress and deformation is obtained in the volume.
The bottlenecks are as follows:
side \(\mathit{ABCD}\): \(\mathit{DZ}\mathrm{=}0\)
side \(\mathit{BCGF}\): \(\mathit{DX}\mathrm{=}0\)
side \(\mathit{ABFE}\): \(\mathit{DY}\mathrm{=}0\)
face \(\mathit{EFGH}\): movement \(\mathit{Uz}(t)\)
Temperature \(T(t)\) is assumed to be uniform across the cube; the reference temperature is \(0°C\).
\(\mathit{Uz}\) and \(T\) vary over time as follows:
instant \(t\) |
0 |
0 |
100 |
200 |
300 |
\(\mathit{Uz}(t)\) |
|
|
|
|
|
\(T(t)\) |
|
|
|
|
A purely mechanical loading is therefore carried out, then heating is carried out by blocking the direction \(\mathit{Uz}\), before cooling. This makes it possible to verify the separation of thermal and mechanical deformations as well as the non-recovery of mechanical properties after heating.
1.2. Material properties#
For the Mazars model, the following parameters were used (value at \(0°C\)):
Elastic behavior:
Thermal characteristics:
Damaging behavior:
\({\epsilon }_{d0}=1.0{\text{10}}^{-4};\phantom{\rule{2em}{0ex}}{A}_{c}=1.15;\phantom{\rule{2em}{0ex}}{A}_{t}=1.0;\phantom{\rule{2em}{0ex}}{B}_{c}=2000.0;\phantom{\rule{2em}{0ex}}{B}_{t}=10000.0;\phantom{\rule{2em}{0ex}}k=0.7\)
It is also considered that \(E\) and \({B}_{c}\) vary with temperature. Their evolution is given in figures [Figure 1.2-a] and [Figure 1.2-b].
Figure 1.2-a: Evolution of Young’s modulus with temperature
Figure 1.2-b: Evolution of \({B}_{C}\) with temperature