2. Continuous model

2.1. Behavioral equations

Maxwell’s model is simply written:

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

where \({\sigma }^{v}\) designates the viscous stress tensor and \({\dot{\epsilon }}^{v}\) 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:

(2.2)\[ {\ dot {\ mathrm {\ sigma}}}} ^ {v}} ^ {v} +\ frac {1} {\ mathrm {\ tau}} {\ mathrm {\ sigma}}} ^ {v} =k\ dot {\ dot {\ mathrm {\ epsilon}}\]

The initial state by default corresponds to zero viscous stress.

2.2. Definition of energies

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

(2.3)\[ \ text {VISCELAS} =\ frac {1} {2k} {\ mathrm {\ sigma}} ^ {v}\ mathrm {:} {\ mathrm {\ sigma}}} ^ {v}\]

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

\[\]

: label: eq-4

text {VISCDISS} =frac {1} {kmathrm {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}} ^ {