2. Ongoing relationship

We assume the hypothesis of small disturbances and an isotropic environment. Maxwell’s model is represented by a purely viscous shock absorber and a serial Hookean spring. This model takes into account volume viscosity and deviatory viscosity. The equations are then simply:

(2.1)\[\begin{split} \ begin {array} {c} {\ dot {\ mathrm {\ mathrm {\ epsilon}}}}} _ {v} (t) =\ frac {1} {\ mathrm {\ eta}} _ {v}}} {\ mathrm {\ sigma}}} _ {\ mathrm {\ sigma}}} _ {\ mathrm}} (t) +\ frac {1} {K}}\ dot {\ mathrm}} {\ sigma}} _ {\ mathrm {m}}} (t)\\\ dot {{e}} _ {\ mathit {ij}}} (t) =\ frac {1} {{\ mathrm {\ eta}}} {\ eta}}} (t) _ {\ mathit {ij}}} (t) =\ frac {1} {2G} {\ mathrm {\ eta}}}} (t) =\ frac {1} {2G} {\ mathrm {\ eta}}}} _ {d}}} _ {d}} {s}} _ {\ mathit {ij}} (t) +\ frac {1} {2G}}\ dot {{s}} _ {\ mathit {ij}}}} (t)\ end {array}\end{split}\]

with:

\({\mathrm{\epsilon }}_{v}\): volume deformation

\({\mathrm{\sigma }}_{m}\): the average effective stress \(\mathrm{\sigma }=\frac{1}{3}\text{Tr}\left(\mathrm{\sigma }\right)I\)

\({e}_{\mathit{ij}}\): the coefficients of the deviatory stress tensor \({e}_{\mathit{ij}}={\mathrm{\epsilon }}_{\mathit{ij}}-\frac{1}{3}{\mathrm{\epsilon }}_{v}{\mathrm{\delta }}_{\mathit{ij}}\)

\({s}_{\mathit{ij}}\): the coefficients of the effective deviatory stress tensor \({s}_{\mathit{ij}}={\mathrm{\sigma }}_{\mathit{ij}}-{\mathrm{\sigma }}_{m}{\mathrm{\delta }}_{\mathit{ij}}\)

\(K\): the isostatic modulus of elasticity

\(G\): the shear modulus

\({\mathrm{\eta }}_{v}\): volume viscosity

\({\mathrm{\eta }}_{d}\): deviatoric viscosity