Recently there has been some discussion of the effect of edge angle on edge retention and cutting lifetime :
http://forums.swordforum.com/showthr...5&pagenumber=2
I proposed the use of the following model to describe the behavior of the edge in regards to extent of blunting through a media :
y(x)=a*x^b+c
Further to this would be how to model the cutting ability :
c(x)=D/(E+F*y(x,c=0))
This comes from just assuming that cutting ability is inversely proportional to the total of the forces on the edge which are a sum of any constant force such as the wedging and then the force due to blunting. Of course all the constants can be absorbed to produce :
c(x)=c_0/(1+y(x,c=0))
where c_0 is the initial sharpness. Now as an application, I have much data gathered on hemp and rope for which this works but to eliminate any issues with "human influences" I'll start off with some Buck knives CATRA data as it is machine gathered and from an independent source which has already been published in 2001.
http://www.bladeforums.com/forums/sh...d.php?t=127499
From the following graph :
The 420HC blade with the "Edge 2000" profile radically outperforms the BG-42 blade with the more obtuse edge profile until the blades have seriously degraded. The "Edge 2000" process reduces the angle to 14.5 degrees per side and flattens the sharpening by using a hard cardboard wheel to replace a cloth wheel, so the previous more obtuse bevels were also heavier convex at the edge. Now of course when the profiles are made similar the BG-42 blade pulls ahead :
Now as to the model :
From the numerical quanties produced you can determine both the type of blunting, which comes from the power coefficient, and the rate of blunting, which comes from the multiplicative constant. Now I don't want to go into too much numerical detail because I actually digitized that data from the graph but even a light overview shows a fairly strong behavior.
With application of the above two models to enough data it would be possible to both correlate the parameters to the physical properties such as edge angle and thickness and then as well look at the dependance on hardness, wear resistance, carbide size and so on. This would then allow prediction of stability and edge retention based on such characteristics.
Note as well that it also proposes another viewpoint in consideration of performance. For example while BG-42 is outperformed by 420HC when the angles are 20 vs 14.5, when they are equal it reverses. This implies that at some angle between 14.5 and 20 the BG-42 blade would have equal long term edge holding at a more obtuse angle. This increased angle would also give it better geometrical durability so it implies that properties which give better edge retention such as hardness and wear resistance could actually
enhance durability by allowing thicker edge profiles at a given cutting lifetime.
As one final note, the way that CATRA presents the results isn't overly informative because the general question that people want to know is something like "How much more material can I cut with BG-42 over 420HC.". In order to answer this from the CATRA graphs you have to use horizontal intersection asymptotes. In this case this gives a nonlinear function of the amount of media cut. However if you take the two curves produced from the model, calculate the percentage cutting ability and its inverse function, and calculate the the intersection of the origional functions :
ybg(x)= 10.9/(1.+0.44*x**0.97) #BG-42;
y420(x)=10.4/(1.+0.37*x**1.34) #420HC;
y21(x)=((10.4/ybg(x)-1)/0.37)**(1/1.34) #intersect;
yrbg(x)=1/(1.+0.44*x**0.97) #relative BG42;
yribg(x)=((1/x-1)/0.44)**(1/0.97) #its inverse;
eval(x)=(yribg(x/100)/y21(yribg(x/100))-1)*100 #desired result;
you can produce the following graph :
The x-axis is the reduction in cutting ability of the knives and the y-axis shows how much more material the BG-42 blade can cut over the 420HC blade. When the blades are cutting about half of optimal, point (1), the BG-42 blade will have cut about 20% more material - not overly impressive. However, when the blades are used down to about 25% of optimal, point (2), the BG-42 blade will have cut over 60% more material.
So if you like to keep your blades very sharp and they are always cutting very close to optimal, that CATRA data doesn't show much of an advantage to BG-42. Since BG-42 it also more expensive and 420HC is tougher, more ductile, easier to grind and more corrosion resistant - it seems obvious that 420HC is the clear winner. However if you are willing to use your blades to very blunt states then BG-42 will have a large advantage.
Note clearly here that it would not be advisble here to generalize on every 420HC and BG-42 blade from that one comparison but to instead use it to comment on Buck Knives only. However it does raise some general points of interest about edge retention in general as influenced by steel properties and geometry. I also wanted to show how numerically you can analyze such edge retention data to determine material properties as well as how to present edge retention data in a manner which is visually obvious.
As noted, of personal interest would be to correlate the numerical quantities to the geometrical/material properties. If you increase hardness for example does it have a stronger effect on the multiplicative constant or the power constant, similar for wear resistance and carbide size. To be clear - there is
NO NEED to have identical blades or steels to determine these relationships. All influences can be modeled at the same time to determine their influence as well as the correlation between them.
It of course much easier to do the math if the blades are only different in one aspect, but the statistics exists to handle many different variations at once. I have worked with models with are far more complex than the above where you have to model not only the dependance of a function on physical parameters but how those parameters are influenced by the temperature and pressure of the enviroment. So if all you have is a random assortment of knives in all different shapes and sizes and steels then go right ahead and generate some data. You can then determine all the coefficients and then model them by appropriate functions.
-Cliff
