1. Reference problem#

1.1. Geometry#

This is a cubic sample in \(1/8\) representation using a \(\text{HEXA20}\) element.

_images/Object_1.svg

1.2. Material properties#

The elastic properties are:

  • isotropic compressibility module: \(K=516200\mathrm{kPa}\);

  • shear modulus: \(\mu =238200\mathrm{kPa}\);

  • density [1] _

: \({\rho }_{s}=2500\mathrm{kg}/{m}^{3}\).

The anelastic properties of the cyclic Hujeux law come from the document referenced at the following internet address: < http://www.mssmat.ecp.fr/IMG /pdf/resp_loph40.pdf >` http://www.mssmat.ecp.fr/IMG /pdf/resp_loph40.pdf`_ < http://www.mssmat.ecp.fr/IMG /pdf/resp_loph40.pdf >``_. These are the parameters relating to Hostun sand:

  • power of the nonlinear elastic law: \({n}_{e}=0.4\);

  • \(\beta =24\);

  • \(d=2.5\);

  • \(b=0.2\);

  • friction angle: \(\varphi =33°\);

  • characteristic angle: \(\psi =33°\);

  • critical pressure: \({P}_{\mathrm{c0}}=-1\mathrm{MPa}\);

  • reference pressure: \({P}_{\mathrm{ref}}=-1\mathrm{MPa}\);

  • elastic radius of isotropic mechanisms: \({r}_{\mathrm{ela}}^{s}=0.001\);

  • elastic radius of deviatory mechanisms: \({r}_{\mathrm{ela}}^{d}=0.005\);

  • \({a}_{\mathrm{mon}}=0.008\);

  • \({a}_{\mathrm{cyc}}=0.0001\);

  • \({c}_{\mathrm{mon}}=0.2\);

  • \({c}_{\mathrm{cyc}}=0.1\);

  • \({r}_{\mathrm{hys}}=0.05\);

  • \({r}_{\mathrm{mob}}=0.9\);

  • \({x}_{m}=1\);

  • \(\mathrm{Dila}=1\).

The hydraulic properties are:

  • Biot coefficient: \(B=1\);

  • the density of water: \({\rho }_{e}=1000\mathrm{kg}/{m}^{3}\);

  • viscosity: \(\nu =0.001\);

  • intrinsic permeability: \({K}^{\text{int}}={1.E}^{-8}{m}^{3}/\mathrm{kg}/s\);

  • the compressibility module of water: \({K}_{e}={1.E}^{+12}\mathrm{Pa}\).

1.3. Boundary conditions and loads#

1.3.1. Boundary conditions#

These are the symmetry conditions on the element, which represents \(1/8\) of the sample. The movements are blocked on the front panels (

_images/Object_2.svg

), left lateral (

_images/Object_3.svg

) and lower (

_images/Object_4.svg

).

1.3.2. Loading#

Phase 1: consolidation of the sample up to the confinement pressure p0

The sample is brought to a homogeneous state of isostatic actual stresses \({\sigma }_{\mathrm{xx}}^{0}={\sigma }_{\mathrm{yy}}^{0}={\sigma }_{\mathrm{zz}}^{0}={\sigma }^{0}=-30\mathrm{kPa}\), by imposing pressure \({\sigma }_{0}\) on the rear, right lateral, and upper faces of the element, while maintaining zero water pressures \(\text{PRE1}\) in the sample.

Phase 2: undrained triaxial load

To obtain the non-drained conditions, zero hydraulic flows are imposed on all faces.

By maintaining a pressure equal to \({\sigma }_{0}\) on the rear and right lateral faces, an alternating pressure loading with an amplitude \(\Delta \sigma\) equal to \(15\mathit{kPa}\) is applied to the upper face, so as to obtain a variation in the vertical stress \({\sigma }_{\mathrm{zz}}\) in the sample included in the interval \(\mathrm{-}45\mathit{kPa}\mathrm{\le }{\sigma }_{\mathit{zz}}\mathrm{\le }\mathrm{-}15\mathit{kPa}\) (with the minus sign convention for compression).

1.4. Results#

The solutions are post-treated at point \(C\), in terms of effective isotropic pressure \(P\) (\(\mathrm{=}\text{tr}(\sigma \text{'})\mathrm{/}3\)), plastic volume deformation \({\varepsilon }_{v}^{p}\) and isotropic work hardening coefficients \(({r}_{\mathrm{iso}}^{m}+{r}_{\mathrm{ela}}^{\mathrm{iso}})\) and \(({r}_{\mathit{iso}}^{c}+{r}_{\mathit{ela}}^{\mathit{iso}})\) and deviations \(({r}_{d}^{m}+{r}_{\mathrm{ela}}^{d})\) and \(({r}_{d}^{c}+{r}_{\mathrm{ela}}^{d})\).

Validation is carried out by comparison with solutions GEFDYNfournies by Ecole Centrale Paris.