Why is an incompressible isotropic elastic medium always a liquid This was a claim made by my professor in my mechanical metallurgy class. I don't see how it is true. I could very well have a solid (which is incompressible) and have isotropic properties. We deal with such types of materials every day.
I have no idea as to how to even begin approaching this problem.
Any help would be appreciated.
 A: A common polycrystalline metal is isotropic if it is not cold worked. The average diameter of typical steel grains is between 0.01 to 0.1 mm.
As they have random orientations, we can use the law of large numbers to consider any macroscopic part isotropic.
If a metal rod is placed in a cylindrical cavity with tight tolerances and compressed, it is as incompressible as a liquid in the same situation.
So, pratically I think that most of the metal objects that surround us can be taken as incompressible and isotropic elastic materials.
A: It sounds like your professor is stringing together some typical idealizations and assumptions but in somewhat questionable way.

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*We often assume that liquids are incompressible because their shear modulus is far, far less than their bulk modulus. But we make the same assumption for elastomers for the same reason. We also often assume that ductile materials are incompressible while plastically deforming for a related reason: their shear deformation is far, far greater than their dilatation. No real material is incompressible.


*We often idealize liquids as being isotropic. But we make the same assumption for amorphous solids and small-grained polycrystalline materials without a prevailing orientation. I believe that all real materials exhibit some degree of short-range order and anisotropy, although for a liquid, any short-range order would average out to zero over time because the atoms are free to rearrange. You can't say this about a frozen supercooled liquid.
So perhaps what your professor meant is this: Of the materials that we tend to idealize as incompressible, the only state that exhibits an temporally average anisotropy of zero at any location is the ideal liquid state.
