Continuous model
==============


Behavioral equations
-------------------------

Maxwell's model is simply written:

.. math::
 :label: eq-1

    {\ mathrm {\ sigma}} ^ {v} =k (\ mathrm {\ epsilon}} - {\ mathrm {\ epsilon}} ^ {v})\ text {;} {\ mathrm {\ sigma}} =k\ mathrm {\ sigma}} = k\ mathrm {\ epsilon}}} {\ mathrm {\ sigma}}} _ {sigma}} _ {sigma}} _ {v} =k\ mathrm {\ epsilon}}}

where :math:`{\sigma }^{v}` designates the viscous stress tensor and :math:`{\dot{\epsilon }}^{v}` the viscous deformation tensor associated with the damper. We can eliminate the viscous deformation to obtain the equation for the evolution of the viscous stress:

.. math::
 :label: eq-2

    {\ dot {\ mathrm {\ sigma}}}} ^ {v}} ^ {v} +\ frac {1} {\ mathrm {\ tau}} {\ mathrm {\ sigma}}} ^ {v} =k\ dot {\ dot {\ mathrm {\ epsilon}}

The initial state by default corresponds to zero viscous stress.


Definition of energies
-----------------------

The energy volume stored in the spring of the viscous branch is equal to:

.. math::
 :label: eq-3

    \ text {VISCELAS} =\ frac {1} {2k} {\ mathrm {\ sigma}} ^ {v}\ mathrm {:} {\ mathrm {\ sigma}}} ^ {v}

As for the energy volume dissipated by viscosity, it is written as:

.. math::
: label: eq-4

    \ text {VISCDISS} =\ frac {1} {k\ mathrm {\ tau}} {\ int} _ {0} ^ {t} {\ mathrm {\ sigma}}} ^ {v}\ mathrm {::} {\ mathrm {\ tau}} {\ mathrm {\ tau}} {\ int} _ {0} ^ {v}} ^ {sigma}} ^ {v}} ^ {sigma}} ^ {v}} ^ {sigma}} ^ {sigma}} ^ {v}} ^ {sigma}} ^ {sigma}} ^ {v}} ^ {sigma}} ^ {sigma}} ^ {