Applying Onsager relations to spring and dashpot in series

[muzialis]

∂Hello there,

I am trying to understand the basics of thermodynamics of dissipative systems.
In the attached paper equation (2.13) is derived using the concpt of free-energy and Onsager's relations

It says that, restricting to only one generalized coordinate

∂V / ∂q + ∂D / ∂q' = Q

V being the free energy, D the dissipation function as defined by Onsager's principle, Q the generalized force associated to the generalized coordinate q, and the apex deoting differentiation wit respect to time.

Trying to make sense, I applied to a one-dimensional dissipative sisyem, a dashpot.

Indeed, for a dashpot V = 0 identically, so the equation suggests (D defined as D = 0.5 b q'^2)

b q' = Q

which makes perfect sense (Newtonian viscosity).

If I try to do the same with a spring and dashpot in series, having q1 and q2 as coordinates representing the extension of the spring and the dashpot, I end up with a system ,


∂V / ∂q_1 + ∂D / ∂q'_1 = Q_1
∂V / ∂q_2 + ∂D / ∂q'_2 = Q_2

The first makes perfect sense, expressing the fact the force in the spring equals the applied force.
The second should say the same for the dashpot, but I struggle to understand what Q_2 is. Should be a force conjugate to the displacement of the dashpot, but the dashpot react to an applied rate, not to an applied displacement.

Where am I going wrong?

Thanks

 

[Rap]

Did you ever figure this out? I'm guessing from the title of the link you gave that there is both viscosity and elasticity involved. Elasticity is the reaction to the applied displacement, viscosity to the applied rate of displacement.