1. Reference problem#
1.1. Geometry of the problem#
It is a block with height \(\mathit{LZ}=10m\), length \(\mathit{LX}=10m\), and width \(\mathit{LY}=2m\). This block has two cohesive interface discontinuities (non-meshed interface introduced into the model in the form of a level curve (level-set) using the operator DEFI_FISS_XFEM). The first is identified by the normal level-set of equation \({\mathit{lsn}}_{1}=Z-\mathrm{0.5X}-0.2\) and crosses the entire block in the horizontal direction. The second interface is identified by the normal level-set from equation \({\mathit{lsn}}_{2}=Z+\mathrm{0.5X}+0.2\). It connects to the lower lip of the first interface. So the second interface only exists in the part of the block such as \({\mathit{lsn}}_{1}<0\). The junction curve between the two interfaces verifies \({\mathit{lsn}}_{1}={\mathit{lsn}}_{2}=0\) and has the equation \(\{\begin{array}{c}X=-0.4\\ Z=0\end{array}\). The domain is thus divided into 3 blocks, a lower block, an upper block and an intermediate block located between the two interfaces. Points \(A(\mathrm{6,}\mathrm{0,}5.7)\), \(A\text{'}(\mathrm{6,}\mathrm{2,}5.7)\) \(B(\mathrm{6,}\mathrm{0,}\mathrm{4,3})\), \(B\text{'}(\mathrm{6,}\mathrm{2,}4.3)\), \(C(\mathrm{4,}\mathrm{0,}4.7)\) and \(C\text{'}(\mathrm{4,}\mathrm{2,}4.7)\) will be used for the evaluation of the quantities tested.
The geometry of the block is represented 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 hydro-mechanical coupled case. The coupling law used is” LIQU_SATU “. The cohesive model type is” MORTAR “and the cohesive law used 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 \(\alpha (\mathit{en}{K}^{\text{-1}})\) |
\(0.2\) \(0\) |
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 and loads#
2D case
The Dirichlet conditions that are applied are:
the movements following \(x\) are blocked in on the right edge and imposed equal to \(U={10}^{-4}m\) on the left edge of the domain,
the movements following \(y\) are blocked on the lower side and imposed equal to \(-U=-{10}^{-4}m\) on the upper edge of the domain.
3D case
The Dirichlet conditions that are applied are:
the movements following \(x\) are blocked in on the right edge and imposed equal to \(U={10}^{-4}m\) on the left edge of the domain,
the movements following \(z\) are blocked on the lower side and imposed equal to \(-U=-{10}^{-4}m\) on the upper edge of the domain,
the movements following \(y\) are blocked throughout the domain.