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:


.. math::
 :label: eq-1

    \ 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}


with:

:math:`{\mathrm{\epsilon }}_{v}`: volume deformation

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

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

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

:math:`K`: the isostatic modulus of elasticity

:math:`G`: the shear modulus

:math:`{\mathrm{\eta }}_{v}`: volume viscosity

:math:`{\mathrm{\eta }}_{d}`: deviatoric viscosity