What materials have the greatest toughness, compressive strength, and tensile strength?

This depends greatly upon your application.

Cured concrete has very high compressive strength and low tensile strength. (but depends a lot upon the shape too)

Carbon fiber and carbon nanotubes have high total tensile strength, but their compressive strength depends upon their alignment.

Steel can be a good balance of both compressive and tensile, espeically depending upon the type of steel, but may not exceed either of the above singly.

Really hard to answer without knowing more about the constraints of your project.

bye for now!

EDIT

Sorry did not see comment about vacumn in center.

Do a search on submarine hull materials and you'll get good results. Such as...

"Don't think so. It says hull is made out of 'carbon fibre', not CFRP.

While carbon fiber is most notably used to reinforce composite materials, particularly the class of materials known as carbon fiber reinforced plastics, there are other uses. Reinforced carbon-carbon (RCC) consists of carbon fiber-reinforced graphite, for example.

Most modern rowing shells are made of carbon fiber, which significantly lowers the weight of the boat.

Anyway, I think they are talking Aramid, Kevlar or Twaron.

"Deepglider development is in the design phase and is being carried out by T. J. Osse, the designer of Seaglider. The principal challenge is to develop a hull that is strong enough to withstand 6000 dbar pressure while being light enough to be neutrally buoyant in seawater. Three hull materials were considered: ceramic (alumina), structural amorphous (‘liquid’) metal, and composite. None of these materials is widely used for pressure hulls, but each shares the characteristic that hulls fabricated from them can be made buoyant, in contrast to using conventional metals (e.g. aluminum, steel, titanium). Vendors for hulls in these materials are comparatively few. Performance, price, and availability considerations have led to the selection of carbon or boron fiber. We are in the process of identifying vendors who have the facility to fabricate a composite hull."

https://www.onr.navy.mil/sci_tech/32...3/omeriks1.... "

from: http://www.defencetalk.com/forums/showthread.php?t...

Compressive Strength Of Materials

iron and steel Steel is often used to absorb energy in car impacts because it is tough concrete for tensile strength

titanium

answer short and quickly guys...steel.....steel is the answer