.. include:: ../../../../../_cache/math_styles.rst

Viscous formulation
=====================

Piola-Kirchhoff stress tensor 2
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~


The viscosity is modelled by introducing a Prony series on the isochoric part of the second Piola-Kirchhoff tensor :math:`{\stressPKTwo}^{\mathrm{iso}}`. The Piola-Kirchhoff stress tensor for the visco-elastic part is written as the finite sum of :math:`N` tensors :math:`\tensTwo{H}_{i}`

.. math::
 :label: eq-34

    {\ stress PKTwo} ^\ mathrm {visc}
    =
    \ sum_ {i=1} ^N\ TensTwo {H} _ {i}

With,

.. math::
 :label: eq-35

     \ TensTwo {H} _ {i} |_ {t+\ Delta t}
     =
     \ exp\ left (-\ frac {dt} {\ tau_ {i}}\ right)\ left. \ tensTwo {H} _ {i}\ right\ green_ {t}
     +
     g_i\ tau_ {i}\ left (1-\ exp\ left (-\ frac {dt} {\ tau_ {i}}\ right)\ right)\ right)\ frac {\ left ({\ left ({\ stress PKTwo}} ^\\ stress} ^\ mathrm {iso}\ left ({\ stress}} ^\ mathrm {iso}\ green_ {t+\ Delta t}
     -
     {\ stress PKTwo} ^\ mathrm {iso}\ green_ {t}\ right)} {dt}


The Prony series depends on two lists of parameters :math:`{g_i}` and :math:`{\tau_i}`, which respectively present the long-term shear relaxation module and the relaxation time
correspondent.


Lagrangian elasticity tensor
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The elastic stiffness tensor ("tangent" matrix for Newton's problem) is given by deriving the stress tensor:

.. math::
 :label: eq-36

    \ ModulusTangent
    =
    \ sum_ {i=1} ^N\ frac {\ partial\ TensTwo {H} _ {i}} {\ partial\ ECGDroite}