12.2 Static Sword Properties
12.2.1 Sword Bending: Purely Elastic
|How do parameters of your sword
influence how much it bends elastically?
That necessitates to consider the basic "beam-bending" experiment
as shown when I
first asked this question.
Assessing static sword properties means to try to change the shape of the sword by using forces acting on it. You could, for instance, try to make it longer by pulling at it. That sounds a bit weird but that is how we assessed basic static properties of materials in chapter 3. Remember the tensile test? That gave us Young's modulus, among other things. Doing a tensile test in reverse - pushing instead of pulling - was also mentioned and we will deal with that in the next sub-chapter.
|Here I look at a
static load applied
"sideways" or a classic bending experiment. Fix the hilt of your
sword to something unmovable and then apply forces at right angles to the blade
in the tip region. If we avoid immediate fracture by not using forces that are
too large, the blade will bend. If we avoid permanent bending or plastic
deformation by keeping the force below some limit, we will experience elastic deformation - the sword will assume its old
shape again upon removing the force - and that's what we go for here.
In other words: we go for the classical "bending beam" situation. That was already discussed by Leonardo da Vinci and Galileo Galilei - but it is still the night mare of fledgling engineering students.
How you could do the experiment with an actual sword I have shown you before. Below is therefore a picture showing only the abstract essentials needed for an in-depth analys
|Imagine that pink thing to be your sword seen
sideways. What you would want to know is how much it bends for a given force.
You can express the magnitude of bending - the distance indicated by the double
arrow in the picture - for a given force in millimeters (or inch), so we get a
number for one property of a sword now.
Measuring that number is not too difficult. But calculating that number is a daunting task. "Beam bending" is the first thing that all students have to master who are forced to attend "Technical Mechanics" lectures, and it will have been an unforgettable experience to the surviving physicists and engineers. You not only had to understand something called "area moment of inertia", you actually had to understand and be able to solve a fourth-order differential equation, requiring proficiency in differentiation and integration and a fearless mind. The links, by the way, lead you to the science modules dealing with the topic in question without avoiding equations.
This is bad news already but it will get worse! It is tough to calculate the bending of a beam with a constant cross-section but you just about can forget it completely for a variable cross-section, like for your sword. For a computer it is no problem at all, but if you try to do it with pencil and paper only, you will run into your limits rather quickly.
|Now the good news: The result of
rather tricky calculations is simple and easy to grasp. Let's see what we can
deduce for a "sword" with a rectangular cross-section; i.e. a
thickness of d and a width of b as indicated in the
picture above. The thickness might be in the order of 5 mm, and the width might
be around 40 mm, for example. If you apply some force - for example by putting
a weight on the tip region - it will bend down a distance
zmax; we may call that the deflection. What might we guess about that quantity? And what is the truth?
|Maybe you realize by now that
mathematical equations are great inventions. All of the above and much more can
be contained in just
one line of
precise symbols! The thing to remember is that the "bending property"
of a sword is particularly sensitive to the length of the blade and its thickness. Or, to be more precise, to the average thickness since we never have a rectangular
cross-section. If one wants to be really precise, the exact shape of the
cross-section has to be taken into account by calculating its
area momentum. That might
be a daunting task but the result is exceedingly simple: Just one number!. The
larger, the smaller the bending.
What we got so far is tricky enough - and we have not yet considered that the cross-section (and therefore the area momentum) typically changes as we go along the length of a blade. Not to mention that for curved blades we no longer have a "simple" one-dimensional problem. How do we deal with that?
all - besides pointing out the obvious: The basic rules as outlined
above are still valid in principle for
every blade. If your sword blade tapers form hilt to tip, just conceive it as a
sequence of pieces with constant but diminishing cross-sections. Each sequence
bends according to the rules above, and the total shape is given by connecting
the pieces. This can be seen nicely in
where the bending of the thin regions is more severe than in the thicker
regions closer to the hilt.
If your blade is curved, it will usually not bend very much anyway since it is not very thin at any place. There is no curved equivalent of the small sword shown in the bending picture. Otherwise - see above.
|What we are going to look at next is how the bending experiment and what we learned from it relates to our "classical" tensile test experiment.|
© H. Föll (Iron, Steel and Swords script)