1. Reference problem#

1.1. Geometry#

The triaxial test is carried out on a single isoparametric finite element of cubic shape \(\mathrm{CUB8}\). The length of each edge is 1. The different facets of this cube are mesh groups named \(\mathrm{HAUT}\), \(\mathrm{BAS}\),, \(\mathrm{DEVANT}\), \(\mathrm{DERRIERE}\), \(\mathrm{DROIT}\), and \(\mathrm{GAUCHE}\). The mesh group SYM also contains the mesh groups \(\mathrm{BAS}\), \(\mathrm{DEVANT}\) and \(\mathrm{GAUCHE}\); the group of elements \(\mathrm{COTE}\) the mesh groups \(\mathrm{DERRIERE}\) and \(\mathrm{DROIT}\).

 _images/10000000000005B30000047DFCD6776F6FFADCA9.jpg

Table 1.1-1 : sample mesh

1.2. Material properties#

The elastic properties are:

  • isotropic compressibility module: \(K=\mathrm{516,2}\mathit{MPa}\)

  • shear modulus: \(\mu =\mathrm{238,2}\mathit{MPa}\)

The hydraulic properties are:

  • Biot coefficient: \(b=1\)

  • fluid compressibility module: \({K}_{l}=1000\mathit{GPa}\)

The parameters of the Mohr-Coulomb law are:

  • friction angle: \(\varphi =33°\)

  • angle of expansion: \(\psi =27°\)

  • cohesion: \({c}_{0}=1\mathit{kPa}\)

1.3. Boundary conditions and loads#

A triaxial test consists in imposing on the specimen a variation in vertical load while maintaining the lateral pressure constant. It can be drained (the interstitial pressure of the fluid does not vary during the test) or non-drained (the valve is closed: the interstitial pressure of the fluid changes in the sample). Here we are interested in the undrained case.

In the model under consideration, the cubic element represents one eighth of the sample. The boundary conditions are therefore as follows:

  • Symmetry conditions:

  • \({u}_{z}=0\) on mesh group \(\mathrm{BAS}\)

  • \({u}_{x}=0\) on mesh group \(\mathrm{GAUCHE}\)

  • \({u}_{y}=0\) on mesh group \(\mathrm{DEVANT}\)

  • Lateral pressure conditions:

  • \({P}_{n}=1\) on mesh group \(\mathrm{COTE}\)

  • Loading conditions:

  • \({P}_{n}=1\) on mesh group \(\mathrm{HAUT}\) (phase 1)

  • \({u}_{z}=-1\) on mesh group \(\mathrm{HAUT}\) (phase 2)

Charging is carried out in two phases:

**Initialization*. Isotropic loading between \(t\in \left[-2;0\right]\mathit{secondes}\): the \(P\) pressure on the cell groups \(\mathit{COTE}\) and \(\mathit{HAUT}\) varies from \(0\) to \(P={P}_{0}=50\mathit{kPa}\), the isotropic preconsolidation pressure in the initial state;

  • triaxial test properly: displacement imposed on the group of elements \(\mathrm{HAUT}\) with \(t\) varying between \(t\in \left[0-12\right]\mathit{secondes}\) and \({u}_{z}\) varying between \({u}_{z}\in \left[0;-\mathrm{0,12}\right]\mathit{mm}\). The total vertical deformation \({\epsilon }_{\mathit{zz}}\) is \(\mathrm{0,012}\text{\%}\);

\({\epsilon }_{1}\)

_images/Forme1.gif

\({\Sigma }_{3}=P\left(t\right)\) \({\Sigma }_{2}=P\left(t\right)\)

Table 1.3-1 : Description of the triaxial test

1.4. Results#

The solutions given by a real coupled hydromechanical calculation are compared to those given by SIMU_POINT_MAT, which solves the following purely mechanic problem:

\(\Sigma =\sigma +bp=C\mathrm{:}\epsilon -\frac{b{K}_{l}}{3}\mathit{trace}\left(\epsilon \right)=P\left(t\right)\)

The solutions are post-treated at point \(C\) for the terms of effective horizontal stress \({\sigma }_{\mathit{xx}}\), hydraulic pressure \(p\), as well as those of plastic volume deformation \({\epsilon }_{v}^{p}\) and plastic deviatoric deformation \(\mid {\epsilon }_{d}^{p}\mid =\sqrt{\frac{3}{2}\left(\epsilon -\frac{{\epsilon }_{v}^{p}}{3}I\right)\mathrm{:}\left(\epsilon -\frac{{\epsilon }_{v}^{p}}{3}I\right)}\).