How to write classical dynamics of solids in tensor form (relation of stiffness and viscosity tensor)? This is a question about dynamics. If I have understood correctly there should be a tensor that describes the dynamics of a (solid?) body (= viscosity ?). I mean, tensor that includes the time dependence. 
I would do it in a following way:
$$
\sigma_{ij} = (C_{ijkl}(t) + \frac{\partial C_{ijkl}(t)}{\partial t} t + \frac{\partial C_{ijkl}(t)\partial C_{ijkl}(t)}{(\partial t)^2} + ...)\varepsilon_{kl}
$$
This is just a Taylor expansion in time for anisotropic elastic solids. Term $\frac{\partial C_{ijkl}(t)}{\partial t}$ would now be the linear viscosity (right?).
What are the next terms called? Or is there any? Elastic solid with viscosity is not something that can be described by elasticy theory anymore? 
The solid is now assumed to be perfectly recoverable: particles return to their original locations after time t (that is finite?). I.e. this solid is not fluid.
In my understanding viscosity of solids it not well defined (or is it?). How would you approximate it with stiffness tensor formalism?
edit:
So, in above should I also include $\varepsilon$ inside the derivative like this (+ fixed typo in taylor expansion):
$$
\sigma_{ij} = (C_{ijkl}\varepsilon_{kl})(t) + \frac{\partial (C_{ijkl}\varepsilon_{kl})(t)}{\partial t} t + \frac{\partial^2 (C_{ijkl}\varepsilon_{kl})(t)}{(\partial t)^2}t^2 + ...
$$
Would this be more general (in a meaningful way)? Also, Can I get the behavior of kelvin-voight and maxwell types with this? And how? For details see answers below.
 A: This is a very good question! The answer, sadly, is far less the simple than the question...
Linear elasticity is, basically, a first-order perturbation thoery. It might be thought of as the definition of a solid. As such, it is universal in the deepest theoretical-physics meaning. 
Irreversible deformation, however, is much less universal and depends intimately on the dissipation mechanism that carries the deformation. Even without going into tensor calculus, in the 1D case there are two fundamentally different types of visco-elastic behaviour: Maxwell type and Kelvin-Voigt type. And this is just linear visco-ealsticity. There are also elasto-plasticity and visco-plasticity, and more. Another complication that arrises is that some materials leave the linear elasticity regime well before they start to deform irreversibly. In short - it's a mess.
There are tens of models that incorporate viscous deformation into solid mecahnics, and there are more references on this subject that you could possibly hope to read. two books that I liked are this one, that gives a nice overview of visco-elasticity, and this one that is a nice introductory text to dislocation-mediated plasticity in crystals. But as I said, there are uncountably many others...
