Benchmark solution
=====================


Reference solution for linear water pressure
--------------------------------------------------------


.. image:: images/Object_23.svg
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,

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.. _RefImage_Object_24.svg:


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


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In this writing,

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designates the Biot module and

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.

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

.. image:: images/Object_29.svg
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The boundary conditions on the side walls and the fact that the stress state is homogeneous give:


.. image:: images/Object_30.svg
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So we finally have to solve the two equations:


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And we get:


.. image:: images/Object_32.svg
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In our case,


.. image:: images/Object_33.svg
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Development of the analytical solution CJS
-------------------------------------------

We always have:

for deformations:

.. image:: images/Object_34.svg
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for constraints:

.. image:: images/Object_35.svg
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**Elastic phase:**

By simply writing the elastic law, it comes:


.. image:: images/Object_36.svg
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Moreover, we also know that during this phase

.. image:: images/Object_38.svg
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(:math:`\mathrm{=}\mathit{trace}(\sigma )`) remains constant because

.. image:: images/Object_39.svg
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. For the components of the diverter, we deduce:


.. image:: images/Object_40.svg
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and

.. image:: images/Object_41.svg
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either:

.. image:: images/Object_42.svg
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and

.. image:: images/Object_43.svg
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Therefore:

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So when we reach the criterion

.. image:: images/Object_45.svg
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, we have:


.. image:: images/Object_46.svg
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That is to say, the transition between the elastic and perfectly plastic states occurs for an axial deformation equal to:


.. image:: images/Object_47.svg
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The corresponding stress state is noted:


.. image:: images/Object_48.svg
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and

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**Plastic phase:**

We note

.. image:: images/Object_50.svg
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The deviator of the inverse of the tensor s

In general, we have the following quantities:


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.. _RefImage_Object_52.svg:


.. image:: images/Object_53.svg
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.. _RefImage_Object_53.svg:


.. image:: images/Object_54.svg
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.. _RefImage_Object_54.svg:


.. image:: images/Object_55.svg
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.. _RefImage_Object_55.svg:


.. image:: images/Object_56.svg
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either:

.. image:: images/Object_57.svg
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and

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Therefore:

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From this we deduce:


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and

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in addition:

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and

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As we have:

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So the tensor

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is written:


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and

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He then comes to

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:


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According to the elastic law, we also have:


.. image:: images/Object_71.svg
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and

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where:


.. image:: images/Object_73.svg
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.. image:: images/Object_74.svg
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and with:

.. image:: images/Object_75.svg
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and

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or, according to the above, we have for

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:


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And for

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:


.. image:: images/Object_80.svg
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From this we deduce that the deviatory load function is written as:


.. image:: images/Object_81.svg
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Taking into account the fact that

.. image:: images/Object_82.svg
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, we then find for the plastic multiplier:


.. image:: images/Object_83.svg
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What happens with the formulas of

.. image:: images/Object_84.svg
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and

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previous ones:


.. image:: images/Object_86.svg
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The analytical expression of constraints is finally concluded:

By posing:


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We have:


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.. image:: images/Object_89.svg
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Benchmark results
----------------------

Constraints

.. image:: images/Object_90.svg
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,

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and

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at points :math:`A`, :math:`B`, and :math:`C`.


Uncertainty about the solution
---------------------------

Exact analytics solution for CJS1.