# 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.

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Unfortunately, gels, plastics and other glasses are amorphous solids. Solids that have fluid like behavior. And this is exactly the point of my question: What do I need to model amorphous solid and why. – Juha Feb 13 '12 at 12:40

## 1 Answer

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...

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Thanks, for the good answer. You bring out exactly the same points that I am wondering: there are lots of models... What I actually wanted to know is that do these models assume some of the tensor components to be zero and the components are different. For example (just quessing here) in Maxwell type behavior, only first order expansion in time is used (no creep, linear viscosity) and in Kelvin-voight type behavior, also higher terms are used... Actually my boss just walked in and said that I also should have $\dot{\varepsilon}$ in my formula. So, can I have the kelvin-voight model with this? – Juha Feb 13 '12 at 12:38
Note that whenever irreversible deformation occurs, $\epsilon_{ij}$ is not even defined any more, as the reference configuration changes. So this whole approach of developing $$\sigma_{ij}=C_{ijkl}\epsilon_{ij}+\partial_t(\dots\epsilon_{ij})+\dots$$ is basically invalid. It seems, though, that what you're looking for is standard isotropic visco-elasticity. So grab the textbook I recommended and read the first few chapters. It's fairly easy. – yohBS Feb 14 '12 at 7:41
Ok, good point about the irreversible deformation... The system I actually want to apply this is fracture (LEFM crack growth). And how I would start with this is: take equilibrium theory and apply it to non-equilibrium phenomena by saying that the dynamic effects are small. (Then wave your hands a lot to get it through PRL.) Anyway I will grab the textbook and accept your answer, for now. If you find something about comparison of the different models, please let me know. Thanks again! – Juha Feb 14 '12 at 11:52
Each time you're making it worse! You want to do fracture? God forbid. References about fracture of amorphous solids: here and here and here and here. – yohBS Feb 14 '12 at 13:34
I think I've read some of those... And I think I sort of disagree with them (location of the crack tip is not the same as the "hole edge" for the continuum equations)... There is something fundamental about fracture that mankind does not understand. I would guess that its the border between discrete and continuum mechanics that is missing. Anyway, this is getting a bit of topic, but if you want to discuss more I am all ears. – Juha Feb 14 '12 at 14:45