Why isn't viscosity considered a property of solids?
The concept of deformations of solids under loading has a long history. It just isn't called 'viscosity' - instead this is just part of the mechanical properties of solids. 

Relevant places to start include:

Wikipedia on Plasticity

Wikipedia on Deformation

The Non-Destructive Testing org on Elastic/Plastic Deformation

In the purely elastic regime, if you pull on a steel bar it will extend under the load, and will return to its original dimensions when the load is removed. If you apply too large of a load, it will plastically deform, so that the dimensions will not be the same when the load is removed. Under some circumstances even fairly small loads over long times can result in permanent deformation. 

Physically, these can be a result of:


Dislocation formation and movement in crystals (e.g. slip bands)
Grain boundary motion
Creep (point defect formation and diffusion)
Crazing in plastics (formation of open volumes)
And a number of other, less common, mechanisms.


These topics are typically covered in metallurgy and materials science courses/textbooks. Again, even in amorphous solids, one would not tend to call it viscosity - that is reserved for liquids.