Materials with a high compressive strength to weight ratio We know about extremely strong materials such as carbon nanotubes. However, this is only in tension. What are some high strength-to-weight materials (both available and hypothetical) in uniaxial compression?
Such a material would be useful in supertall structures and vacuum airships.
Of my research, PVC pipe seems to be a good bet at 100MPa and only 1.3 g/cm^3. 
However, this is still inadequate for a vacuum airship.
Beryllium is also a likely candidate as it is 1400MPa and only 1.85 c/cm^3, making it doable in a vacuum airship. However, it is very toxic to work with (I don't know if bulk beryllium is safe) and expensive.
 A: Relying on the claims I made in my comment about the relation between some of the moduli and the speed of pressure waves, I tried searching the web for tables of the speed of sound in solid materials, and found (as I thought I recalled) that diamond is clear winner amung traditional materials with $v_s = 12,000\text{ m/s}$.
However, WolframAlpha says the highest is $18,350\text{ m/s}$ for "carbon" without saying what bulk material. I woud assume nanotubes or graphene.
A: Compressive strength to density ratio is not the most critical factor for vacuum balloons, as the most dangerous failure mode for vacuum balloons is buckling. The elasticity modulus to density squared ratio is more important. This issue is considered in my US patent application 20070001053 (11/517915) (written together with my coauthor) - http://akhmeteli.org/wp-content/uploads/2011/08/vacuum_balloons_cip.pdf . It is shown that no homogeneous shell can be both light enough to float in air and strong enough to withstand atmospheric pressure. However, finite element analysis shows that spherical sandwich structures made of commercially available materials can meet these requirements. The face sheets can be made of beryllium, boron carbide, or some other materials; the core can be made of aluminum honeycomb or some other materials. However, manufacturing of such vacuum balloons is not easy and has not been done yet.
