The information is there, that you chose to keep ignoring it says something obviously. As I noted in the above, as soon as I found out that there was published work in this matter I sought it out immediately and this is just a hobby for me. -Cliff
Let me try again:
Here is the page I am referring to.
http://www.cutleryscience.com/reviews/model.html
Model the extended cutting ability of knives
Start with the fact that the work done pushing down on the knife has to equal the work done by the knife on the material it is cutting through. This means that the cutting ability, C, which is the depth that will be achieved under a given load is inversely proportional to the total force Ft on the knife by the material during the cut :
C~1/Ft
Since the force on the knife increases as cuts are made because the edge wears this means the cutting ability is dependent on how much material is cut, x, as follows :
C(x) ~1/Ft(x)
with :
Ft(x)=Fw+Fe(x)
Fw is the force it takes the blade to push the material out of the way. It is constant because the gross shape of the blade never changes. The force on the edge, Fe(x) increases because as the edge blunts while cutting it takes more force at the edge to achieve the necessary pressure to cut. The model is now :
C(x)~/(Fw+Fe(x))
Assuming the rate of metal loss from an edge is inversely proportional to the amount of metal loss which would be reasonable based on a few physical principles, this would predict square root behavior for the increase in force. Allowing for some variance from this exact model for a few physical reaons Fe(x) would be expected to be :
Fe=Fi xb
Where the value of b will characterize the type of blunting and Fi is the initial force on the edge at perfect sharpness with no cuts made. Thus the model for cutting ability is :
C(x)~/(Fw+Fi xb)
Replacing the proportionality by an equality and redefining constants :
C(x)=Ci/(1+a xb)
Here Ci is the initial cutting ability, which is dependent on the geometry of the knife and properties of the material being cut. The value of a depends on the ratio of the forces Fe/ Fw and along with b depends on the characteristics of the material being cut and the properties of the steel. In general the main influence of b is on long range blunting where a is more important in the short term. Because there are thus two competing effects it is possible to have comparisons such as the following :
This shows how one steel can have superior edge retention at high sharpness but another steel can have superior long term edge retention. AEB-L is an example of a type I steel and ATS-34 is an example of a type III steel, see the work of
Roman Landes ( Messerklingen und Stahl) for more detail on this issue. That curve was a hypothetical comparison of two knives, for an actual application of the model to some real data on actual knives first some CATRA data which was published by Buck on Bladeforums in 2001:
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 was an enhancement by Buck to increase the intitial cutting ability and cutting lifetime of their knives. The exact defination is given on their website. It basically reduces the angle to 14.5 degrees per side and uses a hard cardboard wheel to replace a cloth wheel so there is less rounding or convexing of the final edge bevel. Note when all blades are given the same enhancement the BG-42 blade no longer has a significant disadvantage early and pulls ahead strongly after significant cutting :
To model this data it was digitized from the above graphs so the parameters will not be noted in detail except to except to state what should be obvious from a casual visual inspection anyway which is that the b value is lower for BG-42 than 420HC, but at the 20 degree profile the a value is much higher which is why it blunts faster early on but catches up late. Note the model well represents the behavior both in the short and late term :
The general question of interest when comparing steels is "How much more material can be cut?" In order to answer this from the above CATRA graphs requires horizontal intersection asymptotes which gives a nonlinear function of the amount of media cut. This is obviously not a trivial method of visual inspection. However from the curves produced from the model with both knives with the 14.5 degree edges the following graph can be calculated by solving for the function intersections and calculating the inverse of the percentage cutting ability :
The x-axis is the reduction in cutting ability 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. 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 when both blades are sharpened frequently to keep them cutting very close to optimal the CATRA data doesn't show much of an advantage to BG-42. However when the knives are used to very blunt states then BG-42 has a large advantage.
Note that this data proposes another consideration of performance. It 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 than the 420HC blade. This increased angle would give it better geometrical durability so it implies that properties which give better edge retention could actually enhance durability by allowing thicker edge profiles at a given cutting lifetime.
The benefit of using the above model is that the parameters can be correlated to properties of the steels and thus predict behavior. As a starting estimate the dependance of b would be expected to be similar to
b(p,h)~1/(p*h*wr)
Where p is the probability that a carbide will tear out on a given cut, h is the size of the hole and wr is the wear resistance. The probability of carbide tear would be related to the strength of the martensite and nature of the carbides (size/amount), grain size, and other properties. Note this is for slicing, for push cutting this would invert :
b(p,h)~p*h/wr
As in push cuts when carbide comes out the performance drops rapidly as the holes just bind up in the material.
Comments can be emailed to cliffstamp[REMOVE]@cutleryscience.com or posted to :
* a model for cutting ability and edge retention
* regarding cutting ability and edge retention as influenced by geometry
Is this not your web page? Is this your Model? I can not understand some of the Grand Canyon leaps you make to support your assumptions (per my previous posts). This is your work and there are no references to the "physical principles" you mention, so I am asking you specifically: expound or give a specific reference to support, not a hand waving statement of "I referenced that previously"..
Yes, Yes, I know: you quote Landes several paragraphs below in the web page above. However, The information in the paragraph in which you reference Landes does not discuss physical principles, rather data on testing. So please don't use this as a blanket reference.
You also have some references at the bottom of this web page. These lead to some of your other web site blogs about this model. On these sites, there is no discussion of the Physics, etc.... which help me understand your model and assumptions.
This is your page, your model. You need to support your model and not pawn it off by hand waving and throwing out names.
TN
:thumbup: 
