2. Benchmark solution

2.1. Reference solution for linear water pressure

,

designating the stresses, deformations and water pressures obtained in phase one has:

In this writing,

designates the Biot module and

.

The conditions at the limits of zero flow and the conservation of the water body give

The boundary conditions on the side walls and the fact that the stress state is homogeneous give:

So we finally have to solve the two equations:

And we get:

In our case,

2.2. Development of the analytical solution CJS

We always have:

for deformations:

for constraints:

Elastic phase:

By simply writing the elastic law, it comes:

Moreover, we also know that during this phase

(\(\mathrm{=}\mathit{trace}(\sigma )\)) remains constant because

. For the components of the diverter, we deduce:

and

either:

and

Therefore:

So when we reach the criterion

, we have:

That is to say, the transition between the elastic and perfectly plastic states occurs for an axial deformation equal to:

The corresponding stress state is noted:

and

Plastic phase:

We note

The deviator of the inverse of the tensor s

In general, we have the following quantities:

either:

and

Therefore:

From this we deduce:

and

in addition:

and

As we have:

So the tensor

is written:

and

He then comes to

:

According to the elastic law, we also have:

and

where:

and with:

and

or, according to the above, we have for

:

And for

:

From this we deduce that the deviatory load function is written as:

Taking into account the fact that

, we then find for the plastic multiplier:

What happens with the formulas of

and

previous ones:

The analytical expression of constraints is finally concluded:

By posing:

We have:

2.3. Benchmark results

Constraints

,

and

at points \(A\), \(B\), and \(C\).

2.4. Uncertainty about the solution

Exact analytics solution for CJS1.