Ongoing relationship
=================

We assume the hypothesis of small disturbances and we split the deformation tensor into an elastic part, a thermal part, an anelastic part (known) and a viscous part. The equations are then:

:math:`\begin{array}{c}{\mathrm{\epsilon }}_{\text{tot}}={\mathrm{\epsilon }}_{e}+{\mathrm{\epsilon }}_{\text{th}}+{\mathrm{\epsilon }}_{a}+{\mathrm{\epsilon }}_{v}\\ \mathrm{\sigma }=\mathrm{A}\left(T\right){\mathrm{\epsilon }}_{e}\\ {\dot{\mathrm{\epsilon }}}_{v}=g\left({\sigma }_{\text{eq}},\lambda ,T\right)\frac{3}{2}\frac{\stackrel{~}{\mathrm{\sigma }}}{{\sigma }_{\text{eq}}}\end{array}`

with:

:math:`\lambda`: cumulative viscous deformation :math:`\dot{\lambda }=\sqrt{\frac{2}{3}{\dot{\mathrm{\epsilon }}}_{v}\mathrm{:}{\dot{\mathrm{\epsilon }}}_{v}}`

:math:`\stackrel{~}{\mathrm{\sigma }}`: :math:`\stackrel{~}{\mathrm{\sigma }}=\mathrm{\sigma }-\frac{1}{3}\text{Tr}\left(\mathrm{\sigma }\right)\mathrm{Id}` constraint deviator

:math:`{\sigma }_{\text{eq}}`: equivalent stress :math:`{\sigma }_{\text{eq}}=\sqrt{\frac{3}{2}\stackrel{~}{\mathrm{\sigma }}\mathrm{:}\stackrel{~}{\mathrm{\sigma }}}`

:math:`\mathrm{A}\left(T\right)`: elasticity tensor

The power density associated with the dissipation of viscoelastic energy is written as:

:math:`{D}_{\mathit{intr}}=\stackrel{~}{\mathrm{\sigma }}\cdot {\dot{\mathrm{\epsilon }}}_{v}=\dot{\lambda }\mathrm{.}{\sigma }_{\text{eq}}`