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


Calculation method used for the reference solution
--------------------------------------------------------

The reference solution is analytical.

**Transition from the initial state to the deformed state:**


.. image:: images/Object_5.svg
    :width: 197
    :height: 45


.. _RefImage_Object_5.svg:

where

:math:`a` is the length of the deformation of the plate following :math:`Y`,

:math:`{a}_{0}` is the initial length of the plate,

:math:`b` is the thickness of the deformed plate,

:math:`{b}_{0}` is the initial thickness of the plate.

Because

.. image:: images/Object_6.svg
    :width: 197
    :height: 45


.. _RefImage_Object_6.svg:

and shell hypotheses, we have

.. image:: images/Object_7.svg
    :width: 197
    :height: 45


.. _RefImage_Object_7.svg:

**Green-Lagrange tensor:**

By definition of the Green-Lagrange tensor, we have

.. image:: images/Object_8.svg
    :width: 197
    :height: 45


.. _RefImage_Object_8.svg:

With

.. image:: images/Object_9.svg
    :width: 197
    :height: 45


.. _RefImage_Object_9.svg:

, so we have

.. image:: images/Object_10.svg
    :width: 197
    :height: 45


.. _RefImage_Object_10.svg:

By substituting, we have

.. image:: images/Object_11.svg
    :width: 197
    :height: 45


.. _RefImage_Object_11.svg:

**Deformation gradient:**

By definition:


.. image:: images/Object_12.svg
    :width: 197
    :height: 45


.. _RefImage_Object_12.svg:

Either

.. image:: images/Object_13.svg
    :width: 197
    :height: 45


.. _RefImage_Object_13.svg:

**Piola-Kirchhoff constraints of the second kind:**

Let :math:`S` be the constraint of :math:`\mathit{PK2}`, in our case,

.. image:: images/Object_14.svg
    :width: 197
    :height: 45


.. _RefImage_Object_14.svg:

**Cauchy constraint**

Let :math:`s` be the Cauchy stress tensor, we have the relationship

.. image:: images/Object_15.svg
    :width: 197
    :height: 45


.. _RefImage_Object_15.svg:

, we deduce then that

.. image:: images/Object_16.svg
    :width: 197
    :height: 45


.. _RefImage_Object_16.svg:


Benchmark results
----------------------

We calculate the :math:`\mathit{DX}` and :math:`\mathit{DY}` displacements at the node :math:`\mathit{NO3}`, the constraints of :math:`\mathit{PK2}` and the Cauchy constraints on the mesh :math:`\mathit{M1}`.


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

Analytical result.


Bibliographical references
---------------------------

Nil.