1. Reference problem#

1.1. Geometry#

A torsional test is reproduced on a hollow cylinder subjected to internal \({P}_{i}\) and external confinement pressures \({P}_{e}\), to a vertical force \(F\) and to a torsional moment \(M\). This test is schematized by a cubic sample (a material point) subjected to confinement stresses \({\sigma }_{z}\), \({\sigma }_{r}\) and \({\sigma }_{\theta }\), and to a shear deformation \({\epsilon }_{z\theta }=M(t)\) (Figure).

If \({P}_{i}={P}_{e}={\sigma }_{0}\) and \(F=\frac{2{\sigma }_{0}}{\pi }\), the following confinement values are obtained:

  • \({\sigma }_{\theta }={\sigma }_{r}={\sigma }_{0}\)

  • \({\sigma }_{z}=3{\sigma }_{0}\)

We then have the following values of the main constraints:

  • \({\sigma }_{1}=2{\sigma }_{0}+\mid {\sigma }_{z\theta }\mid\)

  • \({\sigma }_{2}={\sigma }_{0}\)

  • \({\sigma }_{3}=2{\sigma }_{0}-\mid {\sigma }_{z\theta }\mid\)

But above all, the main axes rotate by an angle \(\alpha\) equal to [:ref:` 1 < 1 >`]:

\(\alpha (t)=\frac{1}{2}\mathrm{arctan}\left(\frac{2{\sigma }_{z\theta }(t)}{{\sigma }_{z}-{\sigma }_{\theta }}\right)=\frac{1}{2}\mathrm{arctan}\left(\frac{{\sigma }_{z\theta }(t)}{{\sigma }_{0}}\right)\)

_images/100000000000019B000000C842D8CF86D4113104.png

Figure 1.1-a: Schematic diagram of the torsional test on a hollow cylinder

1.2. Properties of the sample#

The elastic parameters are:

  • the isotropic compressibility module: \(K=\mathrm{516,2}\mathit{MPa}\)

  • the shear modulus: \(\mu =\mathrm{238,2}\mathit{MPa}\)

The parameters of the Mohr-Coulomb law are:

  • the angle of friction: \(\phi =33°\)

  • the characteristic angle: \(\Psi =27°\)

  • cohesion: \({c}_{0}=1\mathit{kPa}\)

1.3. Boundary conditions and loads#

The torsional test presented here is carried out on a hardware point with the SIMU_POINT_MAT command.

We are taking a \({\sigma }_{0}=-50\mathit{kPa}\) lockdown pressure. The constraints imposed are therefore as follows:

  • \({\sigma }_{\mathit{xx}}={\sigma }_{\mathit{yy}}=-50\mathit{kPa}\)

  • \({\sigma }_{\mathit{zz}}=-150\mathit{kPa}\)

The imposed shear deformation \({\epsilon }_{\mathit{yz}}\) varies linearly between \(t=0\) and \(100\mathit{sec}\) from \(0\) to \(\mathrm{0,01}\text{\%}\) in \(N=10\) steps of time.