2. Benchmark solution#

2.1. Calculation method#

It is an analytical solution. Taking into account the boundary conditions, displacements can be obtained from the analytical resolution of the equation for the conservation of momentum.

Neglecting gravity, the equation is written (in total constraints):

\(\text{Div}(\mathrm{\sigma })+{r}_{0}g=0\)

In the case of coupled modeling, the total stress tensor is written as:

\(\sigma =\sigma \text{'}-{p}_{1}1\)

\(\sigma \text{'}\) is the stress tensor in the skeleton and \({p}_{1}\) is the pore pressure in the massif. Since the Poisson module \(\nu\) is zero, and being elastic in the case, we have \(\sigma \text{'}=Eϵ\).

But \(\forall y{p}_{1}=0\) so finally \(E\ast \text{Div}(\mathrm{ϵ})+{r}_{0}g=0\)

In the 2D case, \(\nu\) being zero, the boundary conditions and the loading make the problem one-dimensional according to \(y\). Only \({ϵ}_{\mathit{yy}}\) is non-zero and:

\(E\ast \frac{\partial {\mathrm{ϵ}}_{\mathit{yy}}}{\partial y}-{r}_{0}g=0\) or \(\frac{{\partial }^{2}{u}_{y}}{\partial {y}^{2}}=\frac{{r}_{0}}{E}g\) and therefore finally \({u}_{y}(y)=\frac{{r}_{0}g}{2E}{y}^{2}+\mathit{Ay}+B\)

with \(A\) and \(B\) integration constants to be determined. The problem is solved separately in each of the two sub-blocks. In order to determine the integration constants, we use the boundary condition on displacements \({u}_{y}=0\) at the lower and upper ends of the column and the Neumann condition \({\mathrm{\sigma }}_{\mathit{yy}}=-p\) at the interface level are used.

Finally, we get the displacement in the direction \(y\) on both sides of the interface:

  • movements over the interface in \(y=\frac{{\mathit{LY}}^{\text{+}}}{2}\) are spelled \({u}_{y}\left(\frac{{\mathit{LY}}^{\text{+}}}{2}\right)=\frac{p}{E}\left(\frac{\mathit{LY}}{2}\right)-\frac{{r}_{0}g}{2E}{\left(\frac{\mathit{LY}}{2}\right)}^{2}\)

  • movements under the interface in \(y=\frac{{\mathit{LY}}^{\text{-}}}{2}\) are spelled \({u}_{y}\left(\frac{{\mathit{LY}}^{\text{-}}}{2}\right)=-\frac{p}{E}\left(\frac{\mathit{LY}}{2}\right)-\frac{{r}_{0}g}{2E}{\left(\frac{\mathit{LY}}{2}\right)}^{2}\)

In the 3D case, the reference solution is exactly the same:

  • movements over the interface in \(z=\frac{{\mathit{LZ}}^{\text{+}}}{2}\) are spelled \({u}_{z}\left(\frac{{\mathit{LZ}}^{\text{+}}}{2}\right)=\frac{p}{E}\left(\frac{\mathit{LZ}}{2}\right)-\frac{{r}_{0}g}{2E}{\left(\frac{\mathit{LZ}}{2}\right)}^{2}\)

  • movements under the interface in \(z=\frac{{\mathit{LZ}}^{\text{-}}}{2}\) are spelled \({u}_{z}\left(\frac{{\mathit{LZ}}^{\text{-}}}{2}\right)=-\frac{p}{E}\left(\frac{\mathit{LZ}}{2}\right)-\frac{{r}_{0}g}{2E}{\left(\frac{\mathit{LZ}}{2}\right)}^{2}\)

  • the moves following \(x\) and \(y\) are void.

2.2. Reference quantities and results#

We test the value of the movements on the lower and upper lips of the interface:

Vertical movements

Horizontal movements

Lower lip

-4.323558729E-03

0

Upper lip

4.297130927-03

0

2.3. Uncertainties about the solution#

None the solution is analytical.

2.4. Bibliographical references#

  1. R7.02.12 (eXtended Finite Element Method) reference documentation.