4. B modeling#
4.1. Characteristics of B modeling#
Plane modeling “AXIS_THMS”. Mechanical law “ELAS_ORTH”. Coupling “LIQU_SATU”.
\(20\mathrm{\times }20\) \(\mathit{Q4}\) elements of equal width. Coupling terms are also anisotropic.
This is the same modeling as before but using the orthotropy modules (although the test piece is considered to be isotropic and the characteristics remain the same in each direction). In theory, the results should be exactly the same.
However, the model differs slightly here and the analytical solution indicated above is no longer entirely accurate. In fact, instead of the relationship used in isotropics:
\(d\varphi \text{=}(b-\varphi )(d{\varepsilon }_{V}\text{-}3{\alpha }_{0}\text{dT}\text{+}\frac{{\text{dp}}_{w}}{{K}_{s}})\)
The relationship used in this case becomes tensor (cf. doc R7.01.11) and is:
\(d\phi \text{=}B\mathrm{:}d\epsilon \text{-}\phi d{\epsilon }_{V}\text{-}3{\alpha }_{\phi }\text{dT}\text{+}\frac{{\text{dp}}_{\text{gz}}\text{-}{S}_{\text{lq}}{\text{dp}}_{c}}{{M}_{\phi }}\)
with
\(\frac{1}{{M}_{\varphi }}\text{=}(B\text{-}\varphi \delta )\mathrm{:}{S}_{0}^{S}\mathrm{:}\delta\)
where \({S}_{0}^{S}\) is the skeletal flexibility matrix, a function of the Young’s modulus of the solid matrix \({E}^{S}\) and the Poisson’s ratio of the solid matrix \({\nu }^{S}\).
Moreover, porosity cannot be integrated analytically here and is therefore explicitly integrated (porosity taken at the previous time). The resolution is therefore less precise here.
The objective of this modeling is to quantify the difference obtained by using this modeling. We also use the secant matrix in order to test the possibility of this prediction in the orthotropic case.
4.2. B modeling results#
The same results as before are tested first over 10 time steps as for modeling A.
Result at the final instant \(3600s\) :
\(N°\) NODE |
\(\mathrm{COOR}\text{\_}X\) |
|
Reference \(\mathrm{PRE1}(\mathrm{MPa})\) |
Aster \(\mathrm{PRE1}(\mathrm{MPa})\) |
Differences \((\text{\%})\) |
Tolerance \((\text{\%})\) |
||
\(1\) |
0 |
0 |
|
|
|
|
||
\(2\) |
0 |
|
|
|
|
|
The results obtained here are as accurate as in the isotropic case.
To be sure, we test the same case but with 15 steps of time:
Result at the final instant \(3600s\) :
\(N°\) NODE |
\(\mathrm{COOR}\text{\_}X\) |
|
Reference \(\mathrm{PRE1}(\mathrm{MPa})\) |
Aster \(\mathrm{PRE1}(\mathrm{MPa})\) |
Differences \((\text{\%})\) |
Tolerance \((\text{\%})\) |
||
\(1\) |
0 |
0 |
|
|
|
|
||
\(2\) |
0 |
|
|
|
|
|
We converge perfectly towards the analytical solution.