Jawge,
I've tried to put a simple overview together. Here's the short (!) explanation.
As for tension-compression ratios: look at the wood database and compare Modulus of rupture (tension strength) with crushing strength (compression strength). As of elasticity, that's given by the modulus of elasticity. The lower, the more elastic the wood is.
This database has thought me that we have an equivalent to osage orange in Europe, namely common pear. Hawthorn (Crataegus laevigata) is probably very close to that (my preferred bow wood).
I'm trying to add an image, which says more than my explanation (Hope this link works now).
https://www.milieuinfo.be/dms/d/d/workspace/SpacesStore/d05ff317-a04e-4cbc-960e-148d65813f0b/Trapping-toasting-thinning.jpgTo avoid set, we need to find the balance at which the belly wood is never compressed beyond the proportional limit (the amount of stress at which the wood starts to surrender, either in tension or in compression; typically this is in compression). The closer the belly surface is to the neutral plane (NP), the less compression each fiber is exposed to. So we want to bring the neutral plane as close as necessary towards the belly.
We can achieve this in at least three ways, abbreviated TTT, for trapping, toasting, thinning:
1) thin the limbs. This brings the NP closer to both back and belly. The trade-off is a weaker bow limb. This can be mitigated by making it wider, at the cost of a higher bow mass. Since bow mass matters most at the outer limbs, the solution is to have a wide but thin working upper bow limb, and a narrow stiff outer limb for minimal mass. The maximum width is mostly constrained by the stave properties.
2) Heat-treat the belly. Heat-treating or toasting makes the belly wood stronger, so less belly is required to counter the tension load. Hence, the neutral plane shifts downwards.
3) Trap the back, which results in more belly relative to back surface. As a result, the belly is now stronger relative to the back, which also shifts the neutral plane towards the belly. Trapping reduces total mass, increasing cast.
In all three cases, the belly is less strained, as a result of which the bow can be drawn farther before taking set than in the reference condition. In the latter two, the bow can be made a bit thicker for the same draw length before taking set. Since a change in a bow limb from 1 cm to 1.1 cm gives you a 33% increase in draw weight but only 10% increase in mass (and less if trapped), the advantage of a trapped back –provided your wood can take it- is self-explanatory.
Under what circumstances is a crowned/trapped back favorable?
1) when you cannot make the limbs broad enough. There’s plenty of bow wood that never gets 2 or 3” wide. And a crowned stave is easier to find than a flat stave.
2) When you have very tension-strong wood (relative to compression strength). This is why Elm bows perform best when crowned.
3) When you are a rookie and tillering mistakes are easily made and hard to mend (the thinner the limb, the more careful you need to tiller to avoid hinges)
Under what circumstances is crowning unfavorable?
1) when you have tension-weak wood (relative to compression; e.g., Eastern red cedar and other true Juniper species, Sassafras). This is why Juniper is often backed with sinew, to provide a tension-strong back to compensate for the compression-strong juniper (as far as I’ve seen, relative to its elasticity the wood that is strongest in compression).
2) when you decide to have broad thin limbs anyway. Trapping thin limbs yields only minor mass reductions.
