I realized this is probably a little late now (AS coursework deadline was probably early January!) and I not sure how helpful this will be, but I'll say it anyway.
One of the reasons why yew wood was used to create longbows was that it is a natural composite. When you take a cross-section of a yew tree branch, trunk, you will see 2 different colours. Around the outside, there is a thin section of pale, yellowy wood, called sapwood. This stuff is really elastic, and good in tension. In the middle there is an orangey colour wood called heartwood. This is not very good in tension, but excellent in compression.
The Bowyers (bow makers) realized this and for this reason they would cut the wood so that the sapwood formed the back of the bow, and the heartwood formed the belly. Therefore when the bow was pulled, the back would be stretched, and the belly compressed. Each wood performs the function to which it is best suited. Because the wood occurred this way naturally, this removed the problems of glueing together different types of wood (with different mechanical properties), which is a tricky process, and given that ancient yew war bows were generally required to shoot only about 24 arrows a good long distance before being discarded, the effort was not really worth it. Also glues were expensive - the best for the job coming from places like Hungary (made from some cooked rare fish!).
Quote (from above): "I believe that woods show little, if any, plastic behavior."
Actually, in traditional archery there is a term - "Following the string". When a bow is pulled (especially non-composite bows), inevitably parts of the
heartwood are stretched, because the bulk of the bow is made from belly wood and therefore there are parts of this wood which lie on the other side of the "0 extension" line of the bow. The heartwood doesn't stretch elastically much at all. Once the bow has been pulled, parts of the wood will go into their plastic region. If you were to plot a graph of stress vs strain you would find that the loading path is different to the unloading path, and the unstrung bow is now no longer straight, but bent. It has "followed the string". The area on this graph between the loading and unloading lines represents the loss in energy stored in the limbs of the bow due to the plastic extension of the limbs.
Obviously wood does stretch plastically. Anyone who has ever over-pulled a bow will vouch for the fact that (contrary to popular belief), the bow becomes easier to pull just before it fails (and explodes splinters everywhere!). This would correspond to the decreasing load for increasing extension sections of the stress-strain graph.
Calculations for bows will also be very approximate. I think you methods will be very very rough for several reasons (maybe something to discuss in your report):
- The ultimate strength of a bow is determined by the weakest part of the bow. A yew bow will have a variable area (due to the way it is made - you must 'follow the grain'). There are also knots, and other areas of weaknesses. Small fractures and also notches. As a result of these you will get stress concentration factors coming into play. The failure stress will be the average stress at the weakest point multiplied by its stress concentration factor (SCF). You could approximate this SCF by guessing at the geometry of this weakness and applying formulae, i.e. maybe a notch so deep at the edge or a minimum area, or maybe experimentally. Either way, coming up with the values is not really the point at AS, it is more to do with showing you have thought about all the factors involved, and how you MIGHT find them / use them.
- Also - Yew is a composite. You have 2 materials in this bow with different properties. You could use average properties (and discuss the validity of this), or you could do some more complicated mechanics using redundancy. i.e. each section could be modeled as a rod, with the same external forces applied to them. Mechanically, there are 'too many' rods for this structure to be 'statically determinate', i.e. 1 rod could do the job on its own.
On a non-mechanical note, you may like to add to your report that another reason why english bows were made from yew is due to factors such as material availability. The reasons english / welsh bows are long (and cumbersome) as opposed to short is that no known material was available to bowyers at the time which could stretch that much (i.e. store enough energy) without failing. In the east, they used sinew (a fibrous material made from animal bone) which was exceptionally elastic. They could use short bow backed with either thin sections of wood or bone, and just paste sinew on to it. By making the welsh bow longer, they could store more energy than a short one, without it snapping.
Hope all of that was of some help,
This is a subject that really interests me,
Chris
P.S. When I did my AS coursework 2 years ago I did a similar project to yours for 1 part, and for another part built a device to measure and plot on a computer stored limb energy in a bow over the draw. It was quite sucessfull
You may find it interesting to read "The traditional Bowyer's Bible". This covers an awful lot of ground on making bows and bow performance. Volume 1, chapter "Bow design and Performance" by Tim Baker would be very useful for this topic.
