2. Benchmark solution

2.1. Calculation method

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):

\(\text{Div}(\sigma )=0\)

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

\(\sigma =\sigma \text{'}-\mathit{bp}1\)

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

\(\nu\) being zero, boundary conditions and loading make the problem one-dimensional according to \(y\). Only \({ϵ}_{\mathit{yy}}\) is non-zero and therefore in each block:

\(\sigma =E{ϵ}_{\mathit{yy}}{e}_{y}\otimes {e}_{y}-{\mathit{bp}}_{i}1\)

with \({p}_{i}\) the pore pressure imposed in the current block.

So in each block, the total stress tensor is written in the form:

\(\sigma ={\sigma }_{\mathit{xx}}{e}_{x}\otimes {e}_{x}+{\sigma }_{\mathit{yy}}{e}_{y}\otimes {e}_{y}\)

However, in each block, the boundary conditions at the level of the lips of the interfaces that delimit the blocks are written \(\sigma \mathrm{.}n=-Pn\), so:

\(\begin{array}{c}({\sigma }_{\mathit{xx}}{e}_{x}\otimes {e}_{x}+{\sigma }_{\mathit{yy}}{e}_{y}\otimes {e}_{y})\mathrm{.}n=-Pn\\ {\sigma }_{\mathit{xx}}({e}_{x}\mathrm{.}n){e}_{x}+{\sigma }_{\mathit{yy}}({e}_{y}\mathrm{.}n){e}_{y}=-Pn\\ ({\sigma }_{\mathit{xx}}({e}_{x}\mathrm{.}n){e}_{x}+{\sigma }_{\mathit{yy}}({e}_{y}\mathrm{.}n){e}_{y})\mathrm{.}{e}_{y}=-P({e}_{y}\mathrm{.}n)\\ {\sigma }_{\mathit{yy}}({e}_{y}\mathrm{.}n)=-P({e}_{y}\mathrm{.}n)\\ {\sigma }_{\mathit{yy}}=-P\text{car}{e}_{y}\mathrm{.}n\ne 0\end{array}\)

Finally in each block

\({\sigma }_{\mathit{yy}}=E{ϵ}_{\mathit{yy}}-{\mathit{bp}}_{i}=-P\) or \({ϵ}_{\mathit{yy}}=\frac{-P+{\mathit{bp}}_{i}}{E}\)

with \({p}_{i}\) the pore pressure imposed in the current block.

2.2. Reference quantities and results

In particular, we are interested in movements in the \(y\) direction in each block:

• in the lower block, based on boundary conditions, \({u}_{y}(y)=\frac{-P+{\mathit{bp}}_{1}}{E}\ast (\frac{L}{2}+y)\)

• in the lower block, the movements are symmetric with respect to the \((\mathit{Ox})\) axis. In fact, this block is symmetric with respect to the axis \((\mathit{Ox})\) and the boundary conditions and the loads applied to it also follow this symmetry (mechanical pressure distributed over the crack lips and embedment of the point \(A\)). Vertical displacements can therefore be written as: \({u}_{y}(y)=\frac{-P+{\mathit{bp}}_{2}}{E}\ast y\)

• in the upper block, based on boundary conditions, \({u}_{y}(y)=\frac{-P+{\mathit{bp}}_{3}}{E}\ast (\frac{-L}{2}+y)\)

2.3. Uncertainty about the solution

None, it is an analytical solution.