2. 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:

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

with:

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

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

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

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

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

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