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 are:

  • power of the nonlinear elastic law: \({n}_{e}=\mathrm{0,4}\);

  • \(\beta =24\);

  • \(d=\mathrm{2,5}\);

  • \(b=\mathrm{0,2}\);

  • friction angle: \(\varphi \mathrm{=}33°\);

  • expansion angle: \(\psi \mathrm{=}33°\);

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

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

  • elastic radius of isotropic mechanisms: \({r}_{\mathrm{éla}}^{s}=\mathrm{0,001}\);

  • elastic radius of deviatory mechanisms: \({r}_{\mathrm{éla}}^{d}=\mathrm{0,005}\);

  • \({a}_{\mathrm{mon}}=\mathrm{0,008}\);

  • \({a}_{\mathrm{cyc}}=\mathrm{0,0001}\);

  • \({c}_{\mathrm{mon}}=\mathrm{0,2}\);

  • \({c}_{\mathrm{cyc}}=\mathrm{0,1}\);

  • \({r}_{\mathrm{hys}}=\mathrm{0,05}\);

  • \({r}_{\mathrm{mob}}=\mathrm{0,9}\);

  • \({x}_{m}=1\);

  • \(\mathrm{dila}=1\);

The hydraulic properties are:

  • Biot coefficient: \(B=1\);

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

  • viscosity: \(\nu =\mathrm{0,001}\);

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

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

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 (\({u}_{y}=0\)), left side (\({u}_{x}=0\)) and bottom (\({u}_{z}=0\)) faces.

1.3.2. Loading#

Phase 1: sample consolidation up to confinement pressure \({p}_{0}\)

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

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, a loading in displacement with an amplitude \(\Delta u\) equal to \(\mathrm{0,02}m\) is applied to the upper face, so as to obtain a homogeneous deformation of the sample of \(2\text{\%}\).

1.4. Results#

The solutions are post-treated at point \(C\), in terms of equivalent Von Mises stress \(Q\) (\(\mathrm{=}\sqrt{\frac{1}{2}({\sigma }^{d}\mathrm{:}{\sigma }^{d})}\)), effective isotropic pressure \(P\) (\(\mathrm{=}\frac{\text{trace}(\sigma \text{'})}{3}\)), plastic volume deformation \({\varepsilon }_{v}^{p}\), and isotropic work hardening coefficients \(({r}_{\text{iso}}^{m}+{r}_{\text{ela}}^{\text{iso},m})\) and \(({r}_{\text{iso}}^{c}+{r}_{\text{ela}}^{\text{iso},c})\) and and deviation \(({r}_{d}^{m}+{r}_{\text{ela}}^{d,m})\).

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