### 12.2.6 Static Properties of Composite Swords

A Little Bit of Theory: Plastic Deformations of Composite Swords
Here we look a bit more closely on how composites perform in static mechanical testing. First we look at how a composite made from ideal materials performs in a classic tensile test experiment. This is a good exercise to get a first idea of what we are up to. Next I will generalize to real composite swords made from not-so-ideal materials.
We start with comparing the stress-strain diagrams of a hard ("red") and soft ("blue") ideal steel. They are shown below.
 Stress - strain diagram for a hard steel, deforming only elastically, and soft steel, deforming elastically and plastically The rectangles sybolize shape change
We apply stress until a certain elongation (=strain) is reached that shall be the same for both steels. It is indicated by the yellow dot on both strain axes. The red steel can do this strain at a stress (red dot on the stress axis) that is still in the purely elastic deformation region. The blue steel deforms plastically after its yield stress is reached (pink dot at the stress axis; at the deviation from the straight line). It is evident that the red steel needs more stress (red dot on its stress axis) than than the blue steel (blue dot on its stress axis) for the strain chosen.
The red or blue rectangles symbolize the specimen lengths at the various points in question.
When we now release the stress, the red steel goes right back to its original length while the blue steel is plastically, i.e. permanently deformed and stays a bit longer (green dot at its strain axis). If I want to return it to its original length I have to "push it down", i.e. apply compressive stress as indicated by the purple dot on the negative stress axis of the blue steel.
Now let's do a tensile test with a composite of the two steels as schematically shown in the next figure. Just weld the two pieces together somehow, making an "ideal" weld, of course. Let's go through what happens step by step.
 Tensile test for a composite of soft and hard steel
1. In the beginning of the test both steels are elongated elastically and the stress-strain curve is identical to the first part of both individual curves above since both steels behave identically - they have the same Young's modulus, after all.
2. As soon as the yield stress of the blue steel is reached, it will now deform plastically. That takes less stress than the elastic deformation, so the total stress the machine needs to apply can go down a bit. Upon further elongation the stress-strain curve gets a bit flatter and deviates from the straight line as shown.
3. The pre-set strain is now reached at a somewhat lower stress level indicated by the lower yellow dot.
4. Releasing the stress, the blue steel would like to stay a bit elongated as indicated by the dark blue dot on the strain axis. That would, however, leave the red steel elastically stretched and "under stress" as indicated by the red dot on the stress axis.
5. The red steel is fully attached to the blue steel and acts like a tensed spring. It pulls the blue steel "down", putting it under compressive stress.
6. The final state will be some slight elongation of the red steel, retaining some tensile stress, and some compression and compressive stress in the blue steel. The "winner" with the lowest stress will be the steel that occupies a larger volume in the composite.
In the schematic picture above the relation between red and blue is about 50 : 50 so the effect will be noticeable. The red part will be a bit too long, the blue part a bit too short compared to what they would like to be at zero stress value. The whole contraption thus would bend somewhat to the left.
We can immediately transfer these insights to the bending situation of a one-material sword. As discussed before, plastic deformation starts on the outside because that's were stresses are highest. In a way, it's not so different from a composite with a soft and a hard part. Here the outer layer is the soft part. In the beginning the plastically deformed part of the blade will be just a small part of the whole blade. The plastic deformation effects will be hardly noticeable in the "stress - strain" curve, which, in the bending case, would be the force vs. deflection curve. You would not notice the slight bending of the blade after the test because the large part that was only elastically deformed will have "pulled back" the small plastically deformed parts on the outside.
But the outer layers now contain a "build-in stress". It is compressive on the side that experienced tensile stress during the experiment and tensile on the other side.
You may not notice that. However, if you now bend the blade again and in the other direction (maybe because you suspect that it might no longer be perfectly straight), the outer layers will reach the yield stress earlier because their built-in stress from the earlier experiment must be added to the outside stress. Plastic deformation then starts earlier and a larger part of the blade will be affected. The still far larger part that was only elastically deformed will pull the blade back to being (almost) straight again
You get the point. Repeated bending back-and-forth that appears to be fully elastic in the beginning, might slowly built up structural changes in your blade that will weaken it. One bending experiment too much then might damage the blade since it remains bend. Worse, your blade might now fail you in battle when large stresses are encountered.
How about bending a composite sword? The katana type (soft inside, hard shell) can now experience several modes of deformation, e.g.:
1. Stresses everywhere are below the yield stresses of both materials. Purely elastic behavior, in other words.
2. The yield stress in the outer (hard) layer is reached but no yet in the softer interior
3. The yield stress in both materials are reached
4. The yield stress in the softer inner materia is reached but not in the outer hard shell.
his calls for a systematic approach and none better than a good picture:

 Deformation modes in a composite (soft core, hard outside) blade. Identical stresses (=fixed bending) is assumed but the yield stresses of the two steels are somewhat different
We look at one half of a section of a blade edge-on. The stress axis, in other words, would be the edge of the blade and we see the right-hand side of the blade. The soft core (light blue) is somewhat thicker then the hard outer shell (yellow). The blade is under stress (consider it bend) and the red line shows how the stress increases from zero at the neutral line smack in the center of the blade to the outside. The blue and red dashed lines denote the yield stresses of the two steel.
In this example we have the same stress distribution in all four cases but somewhat different yield stresses of the two steels. What do we see?
Case 1: In both steels the maximum stress is below the yield stress. Only elastic deformation takes place.
Case 2: Stresses are larger then the yield stress in the outer layer of the hard steel but still below the yield stress of the blue steel. Some plastic deformation in the outer layer occurs.
Case 3: In both steels the yield stress is reached in some parts. Plastic deformation occurs in the outer layer and deeper inside.
Case 4: Only the soft steel experiences stresses higher than its yield stress. Plastic deformation occurs inside the blade.
This gives some impression of what could happen if your two steels are somewhat different, everything else being the same. Now let's look at what happens to a given blade as the stress is increased. You just increase the bending to do that as schematically shown by the black curve on top of each diagram.

 Tensile test for a composite of soft and hard steel
Once more and unavoidably, we start with elastic bending only (case 1) but then - surprise! - comes plastic deformation of the soft core (case 2) before the outer shell is affected! More bending introduces plastic deformation in both steels (case 3) and finally we have continuous plastic deformation of most of the body of the blade (case 4).
In any case, we have more parts of the sword plastically deformed than we would have in a one-steel sword
It should be clear from what I pointed out before that the only elastically stressed parts will try to "undo" the plastic deformation of the other parts as soon as the outside stress is relieved. It should be equally clear that in any case you now have tensile and compressive stresses inside your blade that change its behavior upon encountering new stress.
We may draw a first conclusion: We can make the simple composite blade discussed above from steels with all kinds of yield stresses and in many different geometries. Many different responses to bending stresses are thus possible for those blades. It should thus be no surprise that some smiths, some schools, some regions, and so on should have been particularly famous because they happened to produce a particularly good sword. Small details matter! The precise geometry, the materials used, the way the forging and heat treatments were done, and so on, made the difference. And I'm still talking ideal materials here! Ulfberht swords come to mind, for example.
I went through this in some detail to demonstrate a major points

 A sword made completely from the ideal hard steel will have a superior elastic limit and will always suffer less plastic deformation than a composite sword for a given stress
I have used the term "elastic limit" in the sense of how far you could bend a blade before "something happens". It is a property that is easy to understand in principle but a bit tricky if you look at it in detail. In purely scientific terms the elastic limit is reached as soon as parts of the blade experience plastic deformation. That means - again in purely scientific terms - that the elastic limit is nothing but the yield stress of the "softest" component. In practical terms, however, the situation is more complex for the following reasons:
1. As pointed out above, a composite blade made from ideal steel may undergo considerable plastic deformation in parts but bend back to being almost straight again when the stress is relieved. The "bender" of the sword who does the test would be inclined to assign an elastic limit to the blade that is well above the one given by the lowest yield stress.
2. For a composite blade made form real and inhomogeneous steel, the "scientific" elastic limit is reached as soon as some particular weak part somewhere in he blade "gives", i.e. deforms plastically. A human tester wouldn't notice that and once more assign an elastic limit far larger than the "scientific" one.
We have a problem now. A properly measured elastic limit for composite blade tends to be lower than what a human tester would come up with by just bending the sword. But nobody is interested in the scientific number, firstly because it just doesn't do justice to a real sword and, secondly, because it almost never exists. Quantitative (and by necessity more or less destructive tests) are almost never done with real swords. In what follows I will use the term "elastic limit" therefore in the less precise but more human context.
If we now look at pattern welded swords, nothing changes in principle. The major difference is that plastic deformation now will occur here and there in the twisted striped rods, depending on the local geometry and the resulting stress distribution.
We now have arrived at a major "theoretical" result for ideal steels:
 The elastic limit behavior of composite blades is worse than that of the hard and better than that of the soft steel Same thing for the plastic behavior,
"Worse" means that the elastic limit is lower and that more volume deforms plastically upon bending.
We have just hit upon what's known as "law of averages" for composites. It simply states that a specific property of a composite is some kind of average of the properties of its constituents. Not your usual average, and not necessarily easy to calculate, mind you. I have given you a detailed example of how that works for Young's modulus. The consequence is that a composite property can never be outside the range found by its constituents.
The law of averages is not a real law, just a guideline, though. It is not overly helpful when it comes to "digital" properties that cannot be easily expressed in numbers. If you combine a ductile and a brittle material, the plastic deformation or sudden fracture properties cannot be an average of the individual properties . What numbers would you average? Nevertheless, the composite in question would be less prone to sudden fracture but possibly bend a lot instead.