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

Analytical expressions
=======================

In this paragraph, the analytical expressions of the visco-hyper-elastic law are described. We recall that when the shear relaxation modulus is equal to zero, we find the equations of the hyper-elastic law.

Piola-Kirchhoff 2 stress tensor of the visco_hyper_elastic law
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The case of hyper-elastic stresses
-------------------------------------

We will detail the analytical expression of the Piola-Kirchhoff constraints for the hyper-elastic potential of Signorin (:math:`p=2` and :math:`q=1`) in the incompressible case. We therefore have the Piola-Kirchhoff stress tensor 2, representing the stresses measured in the initial configuration, which is written as:

.. math::
 :label: eq-38

    \ stress PKTwo
    =
    2\ frac {\ partial {\ Psi} ^ {\ mathrm {iso}}} {\ partial\ partial\ partial\ ECGDroite}}
    +
    2\ frac {\ partial {\ Psi} ^ {\ mathrm {vol}}}} {\ partial\ partial\ partial\ ECGDroite}

With both potentials:

.. math::
 :label: eq-39

    {\ Psi} ^ {\ mathrm {iso}}
    =
    {C} _ {10}\ left (\ TensTwoInVa {1} {\ EDil} - 3\ right)
    +
    {C} _ {01}\ left (\ TensTwoInVa {2} {\ EDil} - 3\ right)
    +
    {C} _ {20}\ left (\ TensTwoInVa {1} {\ EDil} -3\ right) ^ {2}

    \ text {and}

    {\ Psi} ^ {\ mathrm {vol}}
    =
    \ frac {\ bulkModulus} {2} {\ left (\ JacobTransfor-1\ right)} ^ {2}

To obtain the constraints, it is necessary to derive the potential:

.. math::
 :label: eq-40

    \ left\ {\ begin {array} {ll}
        \ TensTwoCmpco {\ stress PKTwo ^ {\ mathrm {iso}}} {i} {j}
        =
        2\ frac {\ partial {\ Psi} ^ {\ mathrm {iso}}}} {\ partial\ TensTwoinva {1} {\ EDil}}
          \ frac {\ partial\ TensTwoInVa {1} {\ EDil}} {\ partial {\ TensTwoCmpco {\ stress PKTwo} {i} {i} {j}}}}
        +
        2\ frac {\ partial {\ Psi} ^ {\ mathrm {iso}}}} {\ partial\ TensTwoinva {2} {\ EDil}}
          \ frac {\ partial\ TensTwoInVa {2} {\ EDil}} {\ partial {\ TensTwoCmpco {\ stress PKTwo} {i} {j}}}}}\\
        \ TensTwoCmpco {\ stress PKTwo ^ {\ mathrm {vol}}} {i} {j}
        =
        2\ frac {\ partial {\ Psi} ^ {\ mathrm {vol}}} {\ partial\ JacobTransfor}} {\ partial\ JacobTransfor}
          \ frac {\ partial\ JacobTransfor} {\ partial {\ partial {\ tenstwOCmpco {\ stress PKTwo} {i} {j}}}}}
    \ end {array}\ right.


With:

.. math::
 :label: eq-41

    \ left\ {\ begin {array} {ll}
        \ frac {\ partial {\ Psi} ^ {\ mathrm {iso}}}} {\ partial\ TensTwoInva {1} {\ EDil}}
        =
        {C} _ {10} + 2 C_ {20}\ left (\ TensTwoInVa {1} {\ EDil} -3\ right)\\
        \ frac {\ partial {\ Psi} ^ {\ mathrm {iso}}}} {\ partial\ TensTwoinva {2} {\ EDil}}
        =
        {C} _ {01}
    \ end {array}\ right.

As well as the derivatives of the reduced invariants (*cf.* [bib5] _ for the derivatives of the invariants of a tensor):

.. math::
 :label: eq-42

    \ frac {\ partial\ TensTwoInVa {1} {\ EDil}} {\ partial {\ TensTwoCmpco {\ ECGDroite} {i} {i} {j}}}}
    =
    \ left (\ TensTwoInva {3} {\ ECGDroite}\ right) ^ {-\ frac {1} {3}}
    \ left (
        {\ tensTwoCmpco {{\ tensTwoUnit}} {i} {j}}
        -
        \ frac {1} {3} {\ TenStwOcmpco {\ inverse {\ ECGDroite}} {i} {j}}}\ TensTwoInVa {1} {\ ECGDroite} {\}
    \ right)


.. math::
 :label: eq-43

    \ frac {\ partial\ TensTwoInVa {2} {\ EDil}} {\ partial {\ TensTwoCmpco {\ ECGDroite} {i} {i} {j}}}}
    =
    \ left (\ TensTwoInva {3} {\ ECGDroite}\ right) ^ {-\ frac {2} {3}}
    \ left (
        \ tensTwoInva {1} {\ ECGDroite}\ tensTwoImpco {{\ tensTwoUnit}} {i} {j}
        -
        {\ tensTwoCmpco {\ ECGDroite} {i} {j}}
        -
        \ frac {2} {3} {\ TenStwOcmpco {\ inverse {\ ECGDroite}} {i} {j}}}\ TensTwoInVa {2} {\ ECGDroite} {\}
    \ right)

.. math::
 :label: eq-44

    \ frac {\ partial\ JacobTransfor} {\ partial {\ partial {\ TenStwocmpco {\ ECGDroite} {i} {j}}}}}
    =
    \ frac {1} {2}\ left (\ TensTwoInva {3} {\ ECGDroite}\ right) ^ {\ frac {1} {2}} {\ TenStwocmpco {\ inverse {\ Ocmpco {\ inverse {\ ECGDroite}}} {\ inverse {\}}} {i} {j}}

So here is the analytical expression for volume constraints:

.. math::
 :label: eq-45

    \ TensTwoCmpco {\ stress PKTwo ^ {\ mathrm {vol}}} {i} {j}
    =
    \ BulkModulus\ left (\ JacobTransfor-1\ right)\ JacobTransfor {\ TenStwocmpco {\ inverse {\ ECGDroite}} {i} {i} {j}} {j}}

And isochoric constraints:

.. math::
 :label: eq-46

    \ TensTwoCmpco {\ stress PKTwo ^ {\ mathrm {iso}}} {i} {j}
    =
    2\ left\ lbrace
        \ left ({C} _ {10} + 2 C_ {20}\ left (\ TensTwoInVa {1} {\ EDil} -3\ right)\ right)\ right)
        {\ JacobTransfor} ^ {-\ frac {2} {3}}
        \ left (
            \ TensTwoCmpco {{\ TensTwoUnit}} {i} {j}
            -
            \ frac {1} {3} {\ TenStwOcmpco {\ inverse {\ ECGDroite}} {i} {j}}}\ TensTwoInVa {1} {\ ECGDroite} {\}
        \ right)
        +
        {C} _ {01}
        {\ JacobTransfor} ^ {-\ frac {4} {3}}
        \ left (\ TensTwoInVa {1} {\ ECGDroite}\ TensTwoInVa {1} {\ TensTwoInVa {1} {\}\ TensTwoInVa {1} {\}\ TenStwoInVa {1} {\ TensTwoInVa
               -
               {\ tensTwoCmpco {\ ECGDroite} {i} {j}}
               -
               \ frac {2} {3} {\ TenStwOcmpco {\ inverse {\ ECGDroite}} {i} {j}}}\ TensTwoInVa {2} {\ ECGDroite} {\}
        \ right)
    \ right\ rbrace

Case of visco-hyper-elastic stresses
------------------------------------------

By coupling viscosity with hyper-elasticity, the Piola-Kirchhoff 2 stress tensor becomes:

.. math::
 :label: eq-47

    {\ stress PKTwo}
    =
    {\ stress PKTwo} ^ {\ mathrm {iso}}
    +
    {\ stress PKTwo} ^ {\ mathrm {vol}}
    +
    \ sum_ {i=1} ^N\ TensTwo {H} _ {i}


Case of the elastic matrix of the hyperelastic law
-----------------------------------------------------

We will detail the analytical expression of the elastic matrix for the hyper-elastic potential of Signorin (:math:`p=2` and :math:`q=1`) in the incompressible case. So we have:

.. math::
 :label: eq-48

    \ ModulusTangent
    =
    4\ frac {\ partial^ {2} {\ Psi} ^ {\ mathrm {iso}}}}
           {\ partial C\ partial C}
    +
    4\ frac {\ partial^2 {\ Psi} ^ {\ mathrm {vol}}} {\ partial C\ partial C}

It is therefore necessary to derive (twice) the potential. For the isochoric part:

.. math::
: label: eq-49a

    \ TensFourCMPCO {\ modulusTangent^ {\ mathrm {iso}}} {i} {j} {k} {k} {l} {l}
    =
    \ frac {\ partial^2 {\ Psi} ^ {\ mathrm {iso}}}
         {\ partial^2\ tensTwoInVa {1} {\ EDil}}
    \ frac {\ partial^2\ TensTwoInVa {1} {\ EDil}}
         {\ partial\ TensTwoCmpco {\ ECGDroite} {\} {i} {i} {i} {j}\ partial\ TensTwoCmpco {\ ECGDroite} {k} {k} {l}}
    +
    \ frac {\ partial^2 {\ Psi} ^ {\ mathrm {iso}}}
         {\ partial^2\ TensTwoInVa {2} {\ EDil}}
    \ frac {\ partial^2\ TensTwoInVa {2} {\ EDil}}
         {\ partial\ TensTwoCmpco {\ ECGDroite} {\} {i} {i} {i} {j}\ partial\ TensTwoCmpco {\ ECGDroite} {k} {k} {l}}

And for the volume part

.. math::
: label: eq-49b

    \ TensFourCMPCO {\ modulusTangent^ {\ mathrm {vol}}} {i} {j} {k} {k} {l} {l}
    =
    \ frac {\ partial^2 {\ Psi} ^ {\ mathrm {vol}}}}
         {\ partial^2\ JacobTransfor}
    \ frac {\ partial^2\ JacobTransfor}
         {\ partial\ TensTwoCmpco {\ ECGDroite} {\} {i} {i} {i} {j}\ partial\ TensTwoCmpco {\ ECGDroite} {k} {k} {l}}

Material constants are assumed to be constant. So we have:

.. math::
: label: eq-50

    \ frac {\ partial^2 {\ Psi} ^ {\ mathrm {iso}}} {\ partial^2\ TensTwoinva {1} {\ EDil}}} = 2 C_ {20}}
    \ text {and}
    \ frac {\ partial^2 {\ Psi} ^ {\ mathrm {iso}}}} {\ partial^2\ TensTwoinva {2} {\ EDil}}} = 0

We can see that the coefficient :math:`\bulkModulus` is indeed a penalty coefficient and that its choice has an impact on the conditioning of the matrix. Derivatives of reduced invariants:

.. math::
 :label: eq-51

    \ frac {\ partial^2\ TensTwoInVa {1} {\ EDil}}
         {\ partial\ TensTwoCmpco {\ ECGDroite} {\} {i} {i} {i} {j}\ partial\ TensTwoCmpco {\ ECGDroite} {k} {k} {l}}
    =
    \ left (\ TensTwoInva {3} {\ ECGDroite}\ right) ^ {-\ frac {1} {3}}
    \ left (
        \ TensTwoCmpco {\ inverse {\ ECGDroite}} {k} {i} {i}\ TensTwoCmpco {\ inverse {\ ECGDroite}} {l} {j}\ TensTwoInva {1} {\ ECGDroite}} -
        \ TensTwoCmpco {\ inverse {\ ECGDroite}} {i} {i} {j}\ TenStwocmpco {{\ TensTwoUnit}} {k} {l}
        -
        \ TensTwoCmpco {\ inverse {\ ECGDroite}} {k} {k} {l}\ TenStwocmpco {{\ TensTwoUnit}} {i} {j}
        +
        \ frac {1} {3}\ TensTwoCmpco {\ inverse {\ ECGDroite}} {k} {l}\ TensTwoCmpco {\ inverse {\ ECGDroite}} {i} {j} {j}\ TensTwoInVa {1} {\ ECGDroite}}
    \ right)


.. math::
: label: eq-52

    \ frac {\ partial^2\ TensTwoInVa {2} {\ EDil}}
         {\ partial\ TensTwoCmpco {\ ECGDroite} {\} {i} {i} {i} {j}\ partial\ TensTwoCmpco {\ ECGDroite} {k} {k} {l}}
    =
    -\ frac {2} {3}\ left (\ tensTwoInva {3} {\ ECGDroite}\ right) ^ {-\ frac {2} {2} {3}}
        \ TensTwoCmpco {\ inverse {\ ECGDroite}} {k} {l}
        \ left (
            \ tensTwoInva {1} {\ ECGDroite}\ tensTwoImpco {{\ tensTwoUnit}} {i} {j}
            -
            \ TensTwoCmpco {\ ECGDroite} {i} {j} {j}
            -
            \ frac {2} {3}\ TensTwoCmpco {\ inverse {\ ECGDroite}} {i} {j}\ TensTwoInVa {2} {\ ECGDroite}
        \ right)
        +
        \ left (\ TensTwoInva {3} {\ ECGDroite}\ right) ^ {-\ frac {2} {3}}
        \ left (
            \ TensTwoCmpco {{\ TensTwoUnit}} {k} {k} {l}\ TensTwoCmpco {{\ TensTwoUnit}} {i} {j}
            -
            \ TensTwoCmpco {{\ TensTwoUnit}} {i} {i} {k}\ TensTwoCmpco {{\ TensTwoUnit}} {j} {l}
            +
            \ frac {2} {3}\ TensTwoCmpco {\ inverse {\ ECGDroite}} {k} {i}
                        \ TensTwoCmpco {\ inverse {\ ECGDroite}} {l} {j}
                        \ TensTwoInVa {2} {\ ECGDroite}
            -
            \ frac {2} {3}\ TensTwoCmpco {\ inverse {\ ECGDroite}} {i} {j}
                        \ left (
                            \ TensTwoInVa {1} {\ ECGDroite}
                            \ TensTwoCmpco {{\ TensTwoUnit}} {k} {l}
                            -
                            \ TensTwoCmpco {\ ECGDroite} {k} {l}
                        \ right)
        \ right)


.. math::
 :label: eq-54

    \ frac {\ partial^2\ JacobTransfor} {\ partial\ TensTwoCmpco} {\ ECGDroite} {i} {j}\ partial\ TensTwocmpco {\ ECGDroite} {\} {\} {k} {l}}
    =
    \ frac {1} {4}\ left (\ TensTwoInVa {3} {\ ECGDroite}\ right) ^ {\ frac {1} {4}}
    \ left (
        \ TensTwoCmpco {\ inverse {\ ECGDroite}} {k} {l}
        \ TensTwoCmpco {\ inverse {\ ECGDroite}} {i} {j}
        -
        2\ TenStwocmpco {\ inverse {\ ECGDroite}} {k} {i}
          \ TensTwoCmpco {\ inverse {\ ECGDroite}} {l} {j}
    \ right)

Case of the elastic matrix of the visco-hyper-elastic law
-----------------------------

In the case of taking viscosity into account, the analytical expression for the elastic matrix becomes:

.. math::
 :label: eq-55

    \ ModulusTangent
    =
    \ modulusTangent^ {\ mathrm {vol}}
    +
    \ modulusTangent^ {\ mathrm {iso}}
    +
    \ sum_ {i=1} ^N\ frac {\ partial\ TensTwo {H} _ {i}} {\ partial\ ECGDroite}
    =
    \ modulusTangent^ {\ mathrm {vol}}
    +
    \ modulusTangent^ {\ mathrm {iso}}
        \ left (
            1
            +
            \ sum_ {i=1} ^N g_i\ tau_ {i}
                \ frac {\ left (1-\ exp\ left (-\ frac {dt} {\ tau_ {i}}\ right)\ right)\ right)}
                {dt}
        \ right)