1. Reference problem#

1.1. Geometry#

For modeling A, the shear test is carried out on a single isoparametric finite element of square shape QUAD8, a cell 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 \(\mathrm{BAS}\).

_images/Shape1.gif

For modeling B the test is carried out on a single isoparametric finite element of cubic shape HEXA8. The length of each edge is 1. The different facets of this cube are mesh groups named \(\mathrm{HAUT}\), \(\mathrm{BAS}\),, \(\mathrm{DEVANT}\), \(\mathrm{ARRIERE}\), \(\mathrm{DROIT}\), and \(\mathrm{GAUCHE}\).

_images/100000000000051400000456D838525F86D79D41.png

1.2. Material properties of Hostun sand#

The elastic properties are:

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

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

The equivalent orthotropic elastic properties are:

  • \({E}_{L}={E}_{T}={E}_{N}=176.906250\mathit{MPa}\)

  • \({G}_{L}={G}_{T}={G}_{N}=68\mathit{MPa}\)

  • \({\mathrm{\nu }}_{L}={\mathrm{\nu }}_{T}={\mathrm{\nu }}_{N}=0.30078125\)

The anelastic properties (Hujeux model) come from K.Hamadi’s thesis thesis [:ref:` **2 <**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: \({P}_{\mathit{CO}}=-400\mathit{kPa}\)

  • 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 shear test presented here is carried out using D_ PLAN modeling (A modeling) and 3D modeling (B modeling).

For modeling A, the movements normal to the study plan are zero. A vertical displacement is imposed on the specimen while maintaining the lateral pressure constant in the study design. 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:

  • \({P}_{n}=100\mathit{kPa}\) on mesh group \(\mathrm{COTE}\)

Loading conditions:

  • \({P}_{n}=100\mathit{kPa}\) on mesh group \(\mathrm{HAUT}\)

  • \({u}_{x}=-1\mathit{mm}\) 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 horizontal displacement is imposed on the group of elements \(\mathrm{HAUT}\) and varies between \(t=0.\) and \(t=10.\) from \({u}_{y}=0.\) and \({u}_{x}=-1\mathit{mm}\).

For modeling B, an embedment is imposed on the specimen on the face \(\mathrm{BAS}\) and a horizontal movement along X on the face \(\mathrm{HAUT}\), while maintaining the constant lateral pressure in the lateral faces. The boundary conditions are therefore as follows:

The embedding condition:

  • \({u}_{x}={u}_{y}={u}_{z}=0\) on mesh group \(\mathit{BAS}\)

Lateral pressure conditions:

  • \({P}_{n}=100\mathit{kPa}\) on mesh groups \(\mathrm{DEVANT}\), \(\mathrm{ARRIERE}\),, \(\mathrm{DROIT}\), and \(\mathrm{GAUCHE}\).

Loading conditions:

  • \({P}_{n}=100\mathit{kPa}\) on the \(\mathrm{HAUT}\) mesh group applied from the first moment of calculation and maintained thereafter

  • \({u}_{x}=-\mathrm{0,2}m\) on the \(\mathrm{HAUT}\) mesh group applied progressively by a ramp function

Charging is therefore carried out in two phases:

  • An isotropic stress state, \({P}_{o}=100\mathrm{kPa}\), is initially assigned to cells \(\mathrm{HAUT}\), \(\mathrm{DEVANT}\), \(\mathrm{ARRIERE}\), \(\mathrm{DROIT}\) and \(\mathrm{GAUCHE}\). ;

  • A horizontal displacement according to X is imposed on the group of elements \(\mathrm{HAUT}\) and varies between \(t=0.\) and \(t=10.\) from \({u}_{x}=0.\) and \({u}_{x}=-0.2m\).

1.4. Results#

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

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