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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.

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Odd thought that might make this easier to find: the speed of longitudinal pressure waves (sound) is the bulk modulous/density (which is close to what you want). Perhaps you can find a tabulation of the speed of sound in colids and simple look up the ration you want for the strong candidates. Added: Ah...it is the P-wave modulous over density that you want, which implies that the speed of sound is a good figure of merit here. –  dmckee Jan 22 '13 at 2:14
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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.

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There are ways around buckling such as using hollow tubes, or in extreme cases fractal-shaped tubes. But making that would be very difficult, not to mention machining very hard materials. –  Kevin Kostlan Jan 23 '13 at 16:06
    
I am not saying that the structure we describe is the only viable design, but its viability is confirmed by computations. I have not seen (satisfactory) results of such computations for any other structures. –  akhmeteli Jan 25 '13 at 1:05
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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.

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there is a reference for this figure (18,350) at knowledgedoor.com/2/elements_handbook/speed_of_sound.html -"See p. 194 in Michael de Podesta. Understanding the Properties of Matter, 2nd edition. London: Taylor & Francis, 2002." But I did not look at the reference. –  akhmeteli Nov 10 '13 at 14:25
    
I forgot to say that this figure (18,350) is for diamond. –  akhmeteli Nov 10 '13 at 15:36
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