Poisson Effect Versus Neutral Plane - A Theory

[tom sawyer]

Recently, Tim Baker reported on a couple of simple experiments he did to demonstrate the practical significance of something we were talking about on another site, namely the Poisson Effect.  I thought I'd share them and throw out a theory on what was going on.

First, let me briefly summarize the Poisson Effect.  When something is stretched it gets narrower, and when something is compressed it gets wider.  This goes for wood. When you bend a bow limb, the back tries to get narrower at the same time the belly is trying to get wider.  The result is that the limb edges curl towards the back.  So a rectangular cross section when unbraced, becomes something of a "smile" at full draw.  The effect is more pronounced with a wider limb.  Tim bent a piece of wood with rectangular cross section, and showed the curl was visible to the naked eye using a shadow cast across the limb.  So right there, I think the wheels must have been turning in his head about his absolute insistence that a rectangular cross-section is superior.

Second, he took the piee of wood and cut off the inner edges so make a "v" shaped belly (leaving the back flat).  When he bent this he saw the the edges of the limb now curled down towards the belly, opposite the direction of the Poisson Effect.  He didn't speculate about why this was, but I think he reallized this was a significant observation.  I've thought about this and have an explanation that helps me answer a question I've posed to myself before.  Namely, how does a neutral plane behave when a limb cross-section isn't symmetrical?

A neutral plane (NP) is that imaginary line where half the limb mass is on one side, half on the other.  Its a plane, which means it has to be flat right?  But an assymmetrical cross-section is not going to have a flat NP.  And yet I think it wants to be flat, and when you bend the limb it tries to get that way.  So if you design a cross-section that has a "smile" (rounder belly than back) or a "frown" (flatter belly than back), when you bend that limb it is going to try and move to straighten this situation out.  How much, depends no how far from flat the NP is to begin with and how far you bend the limb.

Taken together, I think its possible to make the NP "frown" just enough (by rounding belly slightly more than back) to counter the effect of the Poisson Effect, which would mean no net change in the cross-sectional shape during the draw.  I would suppose that this is the true "best" cross-section for equal distribution of forces across a bow limb.

What do you think?

[tom sawyer]

Why would this matter, and how would exactly negating the Poisson Effect have any advantage you ask?

If there is no net movement in the cross-sectoin, then there is less energy lost to returning this mass to its initial position.  Increased effiency would be the result.  And the two counter-acting forces, would presumably produce slightly greater stored energy in the limb.  Granted, these increases would probably be small.

[duffontap]

A neutral plane (NP) is that imaginary line where half the limb mass is on one side, half on the other. 

The neutrality of the plane refers to the zero-tension, zero-compression position within the limb and not the equal division of mass.  Unless there were a perfect balance of tension and compression strength in a stave, the NP wouldn't correspond to the physical center, or mass center of the limb. 

Also, I would think that this position would 'warp' its way though a character stave rather than being a flat plane but I could be wrong. 

             J. D. Duff

[Pat B]

Lennie, I will have to reread this a few times for it to totally sink in but I think I get the gist.     Let me throw in a monkey wrench...for woods that are weaker in compression, wouldn't a belly that has more curvature than the back suffer excessive strain down the center of the belly(crown) especially if the back is also crowned? Would this be a good candidate for trapezoidal cross section?     Pat

[duffontap]

Lennie,

Would Tim say that the rectangular section still has the energy-storage advantage even if it is more subject to the poison effect?  I do think you're onto something. 

      J.  D.

By the way, if you make my head explode, the blood is on your hands.

[Justin Snyder]

Interesting Lennie. I would be interested to see you put this into practical use.   
I think you may be reading into it a little.  I think it is wood being crushed between the back and the belly.  If the energy is too great for the back it explodes.  If the energy is to great for the belly it frets.  The energy on the bow that does neither is stored in the wood in the center of the bow, and tries to escape out the sides.  Because it is being pushed toward the belly it pushes the belly wood out more. The belly wood being pushed out is still trying to escape out the belly, so it curls toward the belly and outside edge.  By rounding it will not have a straight edge where the human eye can see the movement. Just my thoughts. Justin

[SimonUK]

For me it's simpler than that. The bow with a v cross section frowns because it has no belly at the sides to stop them curling towards the belly. After all, you're stretching the back and it wants to be as short as possible.

[tom sawyer]

JD and Pat, for a selfbow the position of the NP coincides with the center of mass.  A single material has only one elasticity, so it stretches and compresses equally well.  The "strength in compression" and "strength in tension" numbers (which are different for a given material) have nothing to do with the neutral plane, they only refer to when the back or belly will break.  When you say "a wood is stronger in tension than compression", it doesn't mean the back is doing more work than the belly.

With a crowned back, you'd make your belly either flat or slightly roudned but not as much as the back.  That would make the cross section "frown", and it would go away and be flat at full draw.  Theoretically.

Using different materials in one limb, like a bamboo backing that is less elastic than wood, shifts the NP towards the backjing but it doesn't change the fact that the NP is not planar and wants to get that way.  Come to think of it, theres a good reason to try and keep the cross-seciotn from "smilling" or frowning on a bamboo-backed bow, because this could lead to splintering when the backing gets mashed together by the Poisson Effect.

Can't you feel your head getting bigger?

[duffontap]

JD and Pat, for a selfbow the position of the NP coincides with the center of mass.  A single material has only one elasticity, so it stretches and compresses equally well.  The "strength in compression" and "strength in tension" numbers (which are different for a given material) have nothing to do with the neutral plane, they only refer to when the back or belly will break.  When you say "a wood is stronger in tension than compression", it doesn't mean the back is doing more work than the belly.

As I understand it:
1.  Wood is not completely homogeneous (consider Yew with sapwood  or Osage with wide variety of latewood ring widths).
2.  The same piece of wood does respond differently to compression and tension stresses. 

        J. D. Duff

[Badger]

Lennie, couple of ways you could look at it, the tension or work you are putting into the string and being transferred to the limb is causing this effect, wouldn't the reverse affect just transfer the energy back to the arrow when you rleased the string?
   The other possible way to look at it is also the reason I prefer slightly oval shaped limbs, is that when the affect starts to take place it is simply amatter of the wood kind of giving up and folding, using more mass in the limb than you actually need for the same amount of work. I find I can control mass better on oval shaped limbs than flat limbs, the poisen affect may be a reason for this. Steve

[Pat B]

Damn, I'll be reading and rereading all night!   Takes my pea brain a long time to absorb and comprehend. Give me a few days and I'll get back to you.
   This is interesting and I actually think I understand what you(or Tim) are saying...but I still have to think about it for a while.     Pat
   

[tom sawyer]

Steve, I agree that rounding edges minimizes the consequences of the Poisson effect since there is less mass moving out there.  As for the bend transferring energy back, whenever you move something it has to be moved back so the Poisson bend going back uses a bit of energy.  If it didn't move at all, you wouldn't be wasting any energy.  I don't know if this is why your rounded-edge bows do better, it might also be better aerodynamics of the limb.

JD, a yew bow with sapwood back would be an exception.  And wood isn't perfectly homogeneous, but it is reasonably so.  Most bows have a fairly smooth thickness taper as judged by the feathering of rings on the belly.  If density were all over the place, you wouldn't see this.  And wood does respond differently to compresion and tension, but that doesn't mean it produces more spring power on the tension side than the compression side.  Elasticity is elasticity, whether you stretch or compress.  A metal spring is as hard to stretch as it is to compress, even though the metal might break in tension before compression.

Simon, you might be correct in your description but the back side wants to be skinnier so I would have predicted that the sides would still curl up.  In fact I would have thought they'd curl up farther since there's no belly wood preventing movement.  It seems to me that all the parts of the limb are trying to get to a place where there is eual mass on both the tension and compressoin sides.  Hence the idea of the NP wanting to be flat (in a 2-D sense).

[marvin]

Lennie,

The NP refers to a "place" where the tension and compression forces are theoretically equal correct?

So if I take a perfectly rectangluar cross section your saying that the NP is physicaly/geographicly positioned dead center because I have the same amount of wood(mass) doing tension work as compression work?

If I remove material(mass) from either side of the original location of the nuetral plane then I have a situation where more wood(mass) is doing more tension or compression which would have to shift the physical/geographical location of the NP correct?

Looking for clarification before I jump in and reveal my ignorance further

[SimonUK]

Simon, you might be correct in your description but the back side wants to be skinnier so I would have predicted that the sides would still curl up.  In fact I would have thought they'd curl up farther since there's no belly wood preventing movement.  It seems to me that all the parts of the limb are trying to get to a place where there is eual mass on both the tension and compressoin sides.  Hence the idea of the NP wanting to be flat (in a 2-D sense).

I think it's wrong to say that the neutral plane 'wants' to be any particular shape. It is what it is, depending on what the wood immediately around it wants to do.

[SimonUK]

Sorry, just to clarify what I mean on my first post. I was saying that the stretched back wants to be shorter - it does this, not by being skinnier, but by 'cutting the corner' on the curve you produce by bending the bow. i.e. it tries to form a less severe curve. But I see your point... generally things do get skinnier when stretched.