1. Reference problem#

1.1. Geometry#

_images/100000000000051400000456D838525F86D79D41.png

The 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 \(\mathrm{HAUT}\), \(\mathrm{BAS}\),, \(\mathrm{DEVANT}\), \(\mathrm{ARRIERE}\), \(\mathrm{DROIT}\), and \(\mathrm{GAUCHE}\). The group of elements \(\mathit{SYM}\) also contains the groups of cells \(\mathrm{BAS}\), \(\mathrm{DEVANT}\) and \(\mathrm{GAUCHE}\); the group of elements \(\mathrm{COTE}\) the groups of cells \(\mathrm{ARRIERE}\) and \(\mathrm{DROIT}\).

1.2. Material properties#

The elastic properties are:

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

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

The anelastic properties (Hujeux) come from the document provided by École Centrale Paris [1]:

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

      • \(\beta =24\)

      • \(d=2.5\)

      • \(b=0.2\)

      • friction angle: \(\varphi =33°\)

      • angle of expansion: \(\psi =33°\)

      • critical pressure: \({P}_{\mathit{c0}}=-1000\mathit{kPa}\)

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

      • elastic radius of isotropic mechanisms: \({r}_{\mathrm{éla}}^{s}={10}^{-3}\)

      • elastic radius of deviatory mechanisms: \({r}_{\mathrm{éla}}^{d}=5.{10}^{-3}\)

      • \({a}_{\mathrm{mon}}={10}^{-4}\)

      • \({a}_{\mathrm{cyc}}=0.008\)

      • \({c}_{\mathrm{mon}}=0.2\)

      • \({c}_{\mathrm{cyc}}=0.1\)

      • \({r}_{\mathrm{hys}}=0.05\)

      • \({r}_{\mathrm{mon}}=0.9\)

      • \({x}_{m}=1\)

      • \(\text{dila}=1\)

1.3. Boundary conditions and loads#

An isotropic compression test consists in imposing on the test piece an equal load variation on each face of the sample.

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 \(\mathrm{DEVANT}\)

      • Loading conditions:

        • \({P}_{n}=1\) on mesh groups \(\mathrm{COTE}\) and \(\mathrm{HAUT}\)

Charging is carried out in three phases:

      • isotropic compression loading between \(t=-10\) and \(t=0\) where the pressure on mesh groups \(\mathrm{COTE}\) and \(\mathrm{HAUT}\) varies between \(p=-100\mathrm{kPa}\) and \(p=-300\mathrm{kPa}\).

      • isotropic tensile loading between \(t=0\) and \(t=10\), where the pressure varies between \(p=-300\mathrm{kPa}\) and \(p=-100\mathrm{kPa}\).

      • isotropic compression loading between \(t=10\) and \(t=20\) where the pressure varies between \(p=-100\mathrm{kPa}\) and \(p=-340\mathrm{kPa}\).