1. Reference problem#
1.1. Geometry of the problem#
It is a column with height \(\mathit{LZ}=5m\), length \(\mathit{LX}=1m\), and width \(\mathit{LY}=1m\). In \(Z=\frac{\mathit{LZ}}{2}\), this column has an interface-type discontinuity. The column is thus entirely crossed by the discontinuity.
The geometry of the column is shown in the figure.
Figure 1.1-a : Problem geometry
1.2. Material properties#
The parameters given in the Table correspond to the parameters used for modeling in the hydromechanical coupled case. The coupling law used is” LIQU_SATU “. The parameters specific to this coupling law are given but have no influence on the solution (because we chose to take a uniformly zero pore pressure throughout the domain). Only the elastic parameters have an influence on the solution of the pseudo-coupled problem. The contact used is of the “MORTAR” type. The associated cohesive law is” CZM_LIN_MIX “.
Liquid (water) |
Viscosity \({\mu }_{w}(\mathit{en}\mathit{Pa.s})\) Compressibility module \(\frac{1}{{K}_{w}}(\mathit{en}{\mathit{Pa}}^{\text{-1}})\) Liquid density \({\rho }_{w}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})\) |
\({5.10}^{\text{-10}}\) \(1\) |
Elastic parameters |
Young’s modulus \(E(\mathit{en}\mathit{MPa})\) Poisson’s ratio \(\nu\) Thermal expansion coefficient \(\alpha (\mathit{en}{K}^{\text{-1}})\) |
\(0\) \(0\) |
Coupling parameters |
Biot coefficient \(b\) Initial homogenized density \({r}_{0}(\mathit{en}\mathit{kg}\mathrm{/}{m}^{3})\) Intrinsic permeability \({K}^{\text{int}}(\mathit{en}{m}^{2})\) |
\(\mathrm{2,5}\) \({\mathrm{1,01937}}^{\text{-19}}\) |
Parameters of the cohesive law |
Critical constraint \({\mathrm{\sigma }}_{c}(\mathit{en}\mathit{MPa})\) Cohesive energy \({G}_{c}(\mathit{en}\mathit{Pa}\mathrm{.}m)\) Increase coefficient \(r\) |
\(900\) \(10\) |
Table 1.2-1 : Material Properties
On the other hand, the forces associated with gravity (in the equation for the conservation of momentum) are neglected. The reference pore pressure is taken to be zero \({p}_{1}^{\text{ref}}=0\mathit{MPa}\) and the porosity of the material is \(\varphi =\mathrm{0,15}\).
1.3. Boundary conditions#
The following Dirichlet conditions apply:
on the [ABCD] face, movements are blocked in all directions (\({u}_{\text{x}}=0\), \({u}_{\text{y}}=0\) and \({u}_{\text{z}}=0\)),
on the [EFGH] side, the movements following \(y\) are blocked \({u}_{\text{y}}=0\) and the movements following \(x\) and \(z\) are imposed for each moment of calculation: \({u}_{\text{x}}=f(t)\), \({u}_{\text{z}}=g(t)\).
8 different loads are carried out. The values of the movements imposed on the upper face for each of the 8 calculation moments are summarized in the table below:
Moment |
\({u}_{\text{x}}=0\) |
|
1 |
\(0\) |
|
2 |
\(0\) |
|
3 |
\(0\) |
|
4 |
\(0\) |
|
5 |
\(0\) |
|
6 |
\(0\) |
|
7 |
\(0.001\) |
|
8 |
\(0\) |
|
Table 1.3-1: Displacements imposed on the upper side