1. Reference problem#

1.1. Geometry#

The biaxial test is carried out on a single isoparametric finite element of square shape QUAD8, a mesh group called \(\mathrm{BLOC}\). The length of each edge is \(\mathrm{1m}\). The different sides of this square are mesh groups named \(\mathrm{HAUT}\), \(\mathrm{BAS}\), \(\mathrm{DROIT}\), and \(\mathrm{GAUCHE}\). The group of elements \(\mathrm{COTE}\) also contains the mesh groups \(\mathrm{DROIT}\) and \(\mathrm{GAUCHE}\); the mesh group \(\mathrm{APPUI}\) the mesh group \(\mathrm{BAS}\).

1.2. Material properties of Hostun sand#

The elastic properties are:

  • isotropic compressibility module: \(K=148\mathrm{MPa}\)

  • shear modulus: \(\mu =68\mathrm{MPa}\)

The anelastic properties (Hujeux model) come from K.Hamadi’s thesis thesis [2] and correspond to low-density Hostun sand:

  • power of the nonlinear elastic law: \({n}_{e}=0.\) (linear elastic)

  • \(\beta =30.\)

  • \(d=2.5\)

  • \(b=0.2\)

  • friction angle: \(\phi =33°\)

  • characteristic angle: \(\Psi =33°\)

  • critical pressure: \({P}_{\mathrm{CO}}=-400\mathrm{kPa}\)

  • reference pressure: \({P}_{\mathrm{ref}}=-1000\mathrm{kPa}\)

  • elastic radius of the isotropic mechanism: \({r}_{\mathrm{ela}}^{s}={10}^{-4}\)

  • elastic radius of the deviatory mechanism: \({r}_{\mathrm{ela}}^{d}=0.01\)

  • \({a}_{\mathrm{mon}}=0.017\)

  • \({a}_{\mathrm{cyc}}=0.0001\)

  • \({c}_{\mathrm{mon}}=0.08\)

  • \({c}_{\mathrm{cyc}}=0.04\)

  • \({r}_{\mathrm{hys}}=0.05\)

  • \({r}_{\mathrm{mob}}=0.9\)

  • \({x}_{m}=1.\)

  • \(\mathrm{dila}=1.\)

1.3. Boundary conditions and loads#

The biaxial test presented here is carried out using D_ PLAN modeling. The trips normal to the study plan are therefore zero. A vertical displacement is imposed on the test piece while maintaining the lateral pressure constant in the study design. 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 drained case, which is simpler, because it does not involve the influence of the interstitial pressure of the fluid. Pure mechanical modeling is then chosen.

In the model under consideration, the square element represents a quarter of the sample. The boundary conditions are therefore as follows:

Symmetry conditions:

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

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

Lateral pressure conditions:

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

Loading conditions:

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

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

Charging is carried out in two phases:

  • An isotropic stress state, \({P}_{o}=100\mathrm{kPa}\), is initially assigned to mesh \(\mathrm{BLOC}\);

  • A vertical displacement is imposed on the group of elements \(\mathrm{HAUT}\) and varies between \(t=0.\) and \(t=10.\) from \({u}_{y}=0.\) and \({u}_{y}=-0.2\) (total vertical deformation of \(20\text{\%}\)).

1.4. Results#

The solutions are post-treated at point \(C\), in terms of stress \({\sigma }_{\mathrm{yy}}\), total volume deformation \({\varepsilon }_{v}\) and isotropic work-hardening coefficients \(({r}_{\text{ela}}^{\text{iso},m}+{r}_{\text{iso}}^{m})\) and deviatory \(({r}_{\text{ela}}^{d,m}+{r}_{\text{dev}}^{m})\).

Validation is carried out by comparison with the GEFDYN solutions provided by Ecole Centrale Paris ( < http://www.mssmat.ecp.fr/-GEFDYN ,016->` http://www.mssmat.ecp.fr/-GEFDYN ,016-`_).

The calculation of elementary option PDIL_ELGA is also carried out for this softening problem. This calculation option makes it possible to estimate the maximum value to be allocated to the regularization parameter A1 of media with a second expansion gradient [R5.04.03]. This value is a function of the material parameters, the stress state and the values of the internal variables at the time of calculation [1].

1.5. Bibliographical references#

[1] Foucault A. « Modeling the cyclical behavior of earthen structures integrating regulation techniques «  . Doctoral thesis, École Centrale Paris, École Centrale Paris, Châtenay Malabry, France, 2010.

[2] Hamadi K. « Modeling bifurcations and instabilities in geomaterials ». PhD thesis, École Centrale Paris, Châtenay Malabry, France, 2006