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


Calculation method
-----------------

We rely on the result of [:ref:`bib1 <bib1>`]. The state of plane deformations makes it possible to write the uniform displacement field in the cube very easily:


.. math::
 :label: eq-1

    \ {\ begin {array} {c} {u} {u} _ {1} = {a} _ {1}\ mathrm {.} {x} _ {1}\\ {u} _ {2} =w\ mathrm {.} {x} _ {2}\\ {u} _ {3} =0\ end {array}

with :math:`w` the vertical (negative) displacement of the upper face and :math:`{a}_{1}` an arbitrary constant.

The incompressibility condition makes it possible to write:


.. math::
 :label: eq-2

    {a} _ {1}\ mathrm {=}\ frac {\ mathrm {-} w} {1+w}

And we find the relationship between the applied force :math:`F` and the displacement :math:`w` of the upper face:


.. math::
 :label: eq-3

    F=\ mathrm {2S.} \ frac {w\ mathrm {.} (2+w)\ mathrm {.} (1+ {(1+w)} ^ {2})} {{(1+w)}} {{(1+w)}} ^ {3}}\ mathrm {.} (\ frac {\ partial\ Psi} {\ partial {J} _ {1}}} +\ frac {\ partial\ Psi} {\ partial {J}} _ {2}})

:math:`S` is the area, :math:`\Psi` is the deformation potential, and :math:`{J}_{1}`, :math:`{J}_{2}` are the invariants of the Green-Lagrange tensor. The deformation potential used by ELAS_HYPER is as follows:


.. math::
: label: eq-4

    \ PSI = {C} _ {10}\ mathrm {.} ({J} _ {1} -3) + {C} _ {01}\ mathrm {.} ({J} _ {2} -3) + {C} _ {20}\ mathrm {.} {({J} _ {1} -3)}} ^ {2} + {\ Psi} _ {\ text {vol}}

:math:`{\Psi }_{\text{vol}}` is the potential corresponding to incompressibility. It depends on the invariants :math:`{J}_{1}` and :math:`{J}_{2}` and on :math:`{C}_{10}`, :math:`{C}_{01}` and :math:`{C}_{20}` which are the material characteristics. As in addition :math:`S=1` we get:


.. math::
: label: eq-5

    F=2. \ frac {w\ mathrm {.} (2+w)\ mathrm {.} (1+ {(1+w)} ^ {2})} {{(1+w)}} {{(1+w)}} ^ {3}}\ mathrm {.} \ left [{C} _ {10} + {C} _ {01} +2. {C} _ {20}\ mathrm {.} \ frac {{w} ^ {2}\ mathrm {.} {\ left (2+w\ right)} ^ {2}}} {{\ left (1+w\ right)} ^ {2}}}\ right]

Solving this nonlinear equation in :math:`w` is simply done by dichotomy for :math:`w<0`.