a model for cutting ability

[Cliff Stamp]

... edge width vs. carbide size.

Typical sharpened edges, high polish, go down to about 0.1 microns, this is the same scale as the carbides in the high finish steels like AEB-L, 52100, etc. . The high carbide volume P/M steels have carbides of about 5-10 microns, the individual carbides are fairly small, but there is often so much of them that you get clusters of macro carbides which are several times the size of the average carbide. The carbides in coarse steels like 440C, D2 or materials like Talonite are 50 microns as they are left from the as cast structure. However as long as the abrasive can cut the carbides the edge can form much smaller than the carbide size.

If you take a piece of knotty wood and plane it, as long as the plane is really sharp and set at a low angle it should cut the knots right along with the wood. However if the wood is really knotty you will eventually get to the point where there is actually no wood to hold the knots in place so fine corners will not be overly durable. Imagine a bow for example which was full of knots in the tips and what would happen if you drew it.

Consider as well if the knots started to approach the hardness of the plane. It would then become very likely for them to be torn out rather than be cut. A similar situation happens in the very high carbide steels when edges are taken to very low angles and the alloy carbides are very hard. There isn't enough steel to hold the carbides in the edge and the carbides are so hard they will require much higher interaction forces to cut them which means it is more likely for them to be torn out. In short, they don't get as sharp and they won't stay as sharp initially.

However, eventually the edge will thicken to the point where it is now large enough to hold the carbides and the edge will be very stable and resist blunting strongly. As they are wearing to this point they also tend to be very aggressive when slicing because the chunks of carbide which get torn out leave an aggressive slicing finish. You tend to get oscillating behavior where the edge wears, becomes aggressive, and repeats until it stabilizes.

-Cliff

 

[db]

From Cliff’s own forum.
http://www.cutleryscience.com/forum/read.php?5,26
John Juranitch : Sharpening Secrets of a Pro
Posted by: cliffstamp (IP Logged)
Date: August 16, 2007 12:27PM
This article which appeared in Popular Science Feb. 1977 can be seen on the following page :
[users.ameritech.net]
Juranitch is responsible for introducing many people to sharpening and the above article has some very strong points such as :
"Edge design starts with a decision on how much taper to build in, and is determined by what you plan to do with the edge. The rule is to taper it back
just short of the point at which it will collapse when worked most severely."
This is the fundamental rule behind most geometrical constraints on blades.
---
I’m very happy I was able to help cliff come to this conclusion in this thread.

 

[cds4byu]

Now for a harder question, can you model cutting ability? Under a specific amount of force can you predict how deep a blade cuts and how this changes with repeated cuts?

CA(x)=CA_0/(1+b*x^c)

Where CA(x) is the cutting ability after a given amount of cuts in the test medium (x). CA_0 is the initial cutting ability with no blunting, which is a function of the stiffness of the media and the shape of the blade and initial sharpness. The constants b and c relate how and at what rate the blade blunts. These are the same coefficients as noted in the above linked posts as that part of the equation is of course the equation which I previously used to model sharpness.

Can you give me units for CA(x) and CA_0? Are you measuring (cut distance)/(pound of applied force), which was implied earlier when you said you wanted to predict "how deep a blade cuts under a specific amount of force"?

Note in the above I modeled the force in a simple manner :

F_w+F_e

How does this total force relate to the depth of cut?

Once you have these two coefficients you can also calculate the ratio F_e/F_w which then tells you the fractional dependence of the edge retention on the cutting lifetime. This for example is very high for hemp but very low for soft woods. This shows that you can cut soft wood very well with a blunt blade because most of the force is due to wedging, but trying to cut hemp rope with a dull blade isn't productive. This is why for example makers trying hype blades will do something like cut up a coffee can and then slice wood showing the "superior" edge retention of the knife. In fact because the F_e/F_w ratio is so low for that media the test is nothing but hype. So as a side note, this also lets you quantify hype.

Could you provide some guidance on how I would determine F_e/F_w for a medium, as opposed to for a blade?

Thanks,

Carl

 

[db]

Another very interesting quote from cliff from his forum considering his views on using a CATRA in the FFD2 threads.

http://www.cutleryscience.com/forum/read.php?3,14
Posted by: cliffstamp (IP Logged)
Date: March 07, 2007 09:01PM
I discussed the ways people can measure sharpness in the article on this website. At somepoint I will either develop specifically a machine similar to the
CATRA sharpness testor or just buy one we can passaround to standardize it.
...
and
...
> Also,
> how many runs should I do in order to reach a
> particular confidence in my data?
With three runs you are generally fairly confident in the existance of a trend or not. There is little gains above that because is is 1/sqrt(n) behavior.
You will also hit a wall with systematic errors fast so anything about 5 is not practical, unless it is just machine driven so there is no actual effort.

 

[Cliff Stamp]

Can you give me units for CA(x) and CA_0?

You would measure cutting ability by either the force applied to reach a given depth (and or vertical travel) or the depth under a given force, so choose units accordingly. If you want to be complicated you can vary both at the same time but it would seem uncessary unless you had to.

How does this total force relate to the depth of cut?

The total force provides the work done which would equal the resistive work against the blade when traveling through the media.

... how I would determine F_e/F_w for a medium, as opposed to for a blade?

Those are the forces on the blade by the media. For example when you press a sharp knife into a piece of hemp rope you will notice it takes a small amount of force to start the cut, this is F_e, as the blade moves into the material the force will dramatically increase, the rest of it is just the wedging forces, F_w.

You can reduce this by putting the cord under tension so it pulls itself apart and does not wedge on the blade in which case F_w drops off dramatically. I noticed this doing rope cutting years ago that it made a difference how close to the edge of the cuts I would make the slices in terms of how much force was required.

You can see a much more dramatic difference by taking say a Mora 2000 and ER Fulcrum and taking both to razor sharpness and then carving woods. The F_e will be the same for both as they will both easily make the initial cuts into the woods, but F_w for the Fulcrum will be many times greater than for the Mora because of the much greater cross section.

This difference in F_w is why thinner blades will have longer cutting lifetimes even if the edge retention is actually lower because F_e has to increase dramatically to compensate for the already much lower F_w and all that people really see is F_t. Mike Swaim made this point on rec.knives about ten years ago, he was one of the first to clearly note the difference between being sharp and cutting well.

The ER Fulcrum for example will never cut well (requires heavy force on all but the shallowest media) as no matter how sharp the edge, the cross section is actually significantly thicker than a splitting maul. It is more of a scraper prybar than a knife.

-Cliff

 

[cds4byu]

You would measure cutting ability by either the force applied to reach a given depth (and or vertical travel) or the depth under a given force, so choose units accordingly. If you want to be complicated you can vary both at the same time but it would seem uncessary unless you had to.

Do you have data that shows force is proportional to depth of cut? This is the first time that I have seen such a model. Most people who model cutting talk about a force necessary to initiate cutting, then cutting proceed to whatever depth at a relatively constant force. Certainly this is what the CATRA REST force data show.

Carl

 

[Cliff Stamp]

...force is proportional to depth of cut?

Once the entire blade is in the material the force would be expected to level thus you would only measure up to that point to gauge cutting ability by depth at a given force, how far an axe head travels into wood under a given swing is such an example of cutting ability being measured by depth of cut under a given force. The force in this case again doing work on the axe which is then counteracted by the wood doing the same amount of work against the axe.

-Cliff

 

[cds4byu]

Once the entire blade is in the material the force would be expected to level thus you would only measure up to that point to gauge cutting ability by depth at a given force, how far an axe head travels into wood under a given swing is such an example of cutting ability being measured by depth of cut under a given force. The force in this case again doing work on the axe which is then counteracted by the wood doing the same amount of work against the axe.

-Cliff

I disagree. The axe does not move into a substance by force, but by energy. The kinetic energy in the axe head at the swing goes into cutting energy in the wood.

Again, I ask, do you have any data that shows depth of cut is proportional to force? I've not seen any.

Carl

 

[Cliff Stamp]

The axe does not move into a substance by force, but by energy.

The fundamental cause of movement is of course force, this is just newtons law and the basis for all analysis of movement. Since the axe does not travel forever with its velocity as imparted by the user then it means a force must act on it to provide the necessary acceleration, the wood provides that force obviously.

The kinetic energy in the axe head at the swing goes into cutting energy in the wood.

Yes you can look at it from a work perspective, this of course does not mean forces are not involved as forces are what are doing the work obviously. I noted this viewpoint in the above. As you exert force on an axe you do work against it, when the axe moves into the wood the wood exerts a force against it doing the necessary work to equal what was done on the axe by the user.

Again it is simply a force vs cut depth relationship. A much stronger man can exert a much greater force on the swing and thus drive the axe much deeper into the wood.

I ask, do you have any data that shows depth of cut is proportional to force?

I have lots of data which shows that that knives with greater cutting ability will cut deeper into a media at a given force, that is a fairly basic concept. Cut a piece of wood exerting maximum force with your wrist, an Opinel can remove a much larger shaving than a Cold Steel Voyager. The force is the same but the cut depth massively different. In this case, the knives will equalize in cut depth so that the opposition force of the wood is equal to the applied force from the wrist (well more of a torque balance actually in most cases).

-Cliff

 

[hardheart]

is this clsoe to what you guys are talking about? T.A., Martin, and McCallion H. "Knife Slicing of Wood Across the Grain." Wood Science and Technology 30 (1996): 397-410.



A graph of load versus crosshead displacement for three specimens cut at 50 mm/min
is shown in Fig. 7. The three curves are quite similar given the natural variability of the
wood samples. As the blade penetrates the wood the cutting force increases linearly and
continues to rise after the head of the blade is fully embedded in the block. It attains
a maximum value of 135 kN/m at a depth of about 6 mm. Thereafter a steady state is
reached and the cutting force remains relatively constant for the rest of the cut. Minor
fluctuations in the loads reflect noise in the signal due to microcracking in the wood.
When the blade first contacts the body, there is no apparent jump in the load. This
indicates that the wood fibres fracture almost immediately at a very low load. Most of
the energy going into the cutting operation is, therefore, associated with compressing
and shearing the wood around the rigid indentor.



Figure 13 shows the cutting forces recorded for wet and dry wood specimens sliced
with a 1 mm thick steel blade. The dry wood yields an average cutting force of 84 kN/m;
a reduction of 34% over that measured with blade 1 (see Table 3). The theoretical model
predicts a 50% reduction in the cutting force for a blade of half the thickness. However,
this does not occur in practice, because the energy required to fracture the fibres
remains constant and the volume of material being deformed increases in a non-linear
manner, as the blade thickness increases. A large fluctuation in the steady state load
occurs at a crosshead displacement of about 15 mm, where a ball of wood is torn from
the block. At this point, more energy is dissipated and the cutting force rises. The steady
state cutting force in the wet wood gradually increases from 32 kN/m to 50 kN/m as the
blade penetrates the block. This increase occurs because the wood fibres return to their
original orientations after shearing past the wedge and compress the web behind the
blade. This effect adds a friction load to the cutting force. The compressive stress on the
web is estimated to be 500 kPa, based on the experimental results and a friction
coefficient of 0.236. With improved blade designs and less permanent fibre damage, this
effect becomes more of a problem. Therefore, the ideal blade should not have a web
behind it.



Lastly we examine the effect of the blade angle on the theoretical cutting force when
using a frictionless blade. Figure 14 shows the effect of the blade angle for two particular
cases: (a) a blade of constant thickness, T, and (b) a blade of constant depth, D. In case
(a), the surface area of the blade tends to infinity as the blade angle tends to zero.
Therefore, the cutting force solution also tends to infinity for small blade angles. As the
blade angle increases two effects come into play; the volume of material being deformed
decreases, and the rotation of the fibres near the blade surface increases. These two
effects cause the theoretical cutting load to decrease. As the wedge opens up, the cutting
operation tends towards crushing, which is more likely to induce fracture along the
grain and an inferior surface finish on the cut. The theoretical model does not provide
a suitable representation of the stress distribution around the blade, when crushing
occurs.
In case (b) the cutting load rises from zero as the blade angle increases and then falls
away again for the same reasons as those just mentioned. This figure suggests that
a blade angle of greater than 30 ~ would reduce the cutting force. This finding is
supported by experimental work by Donohue (1994), which shows a decreasing cutting
force for a constant blade thickness as the blade angle is increased from 15 ~ to 45 ~
A predicted cutting force of 205 kN/m is indicated by a line at 13 = 30 ~ This is somewhat
higher than that recorded, but is in the correct range considering the unknown material
properties for the Pinus radiata samples. A full finite element analysis for an
orthotropic material with finite strain should be developed, in conjunction with the
experimental determination of the material constants, to obtain an accurate
understanding of the stresses in the cutting region. A fracture mechanics approach
could then be used to predict crack growth across and along the fibres.

 

[Cliff Stamp]

[ref quote]

As the blade penetrates the wood the cutting force increases linearly and
continues to rise after the head of the blade is fully embedded in the block.

Yes, as noted previously, however the responce will not always be linear in general, it depends on the shape of the blade. Thanks for the reference. You can see this easily if you just put a piece of carboard on a scale and press a knife into it and watch the force responce on the scale.

-Cliff

 

[cds4byu]

is this clsoe to what you guys are talking about? T.A., Martin, and McCallion H. "Knife Slicing of Wood Across the Grain." Wood Science and Technology 30 (1996): 397-410.

Hardheart, could you send me a copy of this paper? Thanks for the data.

There are a couple of things we need to be careful about when we use this data relative to edge retention.

A graph of load versus crosshead displacement for three specimens cut at 50 mm/min
is shown in Fig. 7. The three curves are quite similar given the natural variability of the
wood samples. As the blade penetrates the wood the cutting force increases linearly and
continues to rise after the head of the blade is fully embedded in the block. It attains
a maximum value of 135 kN/m at a depth of about 6 mm. Thereafter a steady state is
reached and the cutting force remains relatively constant for the rest of the cut.

First, this data is not data at constant force, which is what Cliff has proposed for modeling cutting performance. It's data at constant push cutting velocity.

Second, this data shows the effect of what Cliff calls F_w. There is a cutting force required to shear the fibers, and a frictional/wedging force required to move the cut material out of the way. The linear increase is due to the increased width of the blade as you get farther back from the cutting edge. The constant force is caused by the blade being fully engaged. Once the blade is fully engaged, the force is no longer proportional to depth of cut.

Further, as you carefully read this paper, the paper is not about edge geometry, but about "wedge geometry" It's looking at how the gross geometry of the blade affects cutting forces, not about how the edge geometry affects cutting forces. I agree that both edge geometry and blade geometry have an effect on cutting forces, particularly in rigid media. But what I term sharpness is dependent only on edge geometry. You can see this because there is no gross geometry difference between a sharp and a dull knife blade. When you sharpen a knife (as opposed to reprofiling a blade), you only modify the edge geometry.

Doing a cross cut in the middle of a block of wood is probably among the most extreme examples of cases where blade geometry, rather than edge geometry, dominates the forces. In CATRA ERT testing, there are two mechanisms that cause the wedging force to be close to zero. First, the media is sheet media, so immediately upon cutting through a thin sheet, the media falls away. This minimizes the wedging force on the cut end. Secondly, the blade is free to move away from the stack of media, which minimizes the wedging force on the uncut end.

In the rope cut testing DiamondBlade did, the blade was free to move laterally to minimize wedging force. Further, the rope was free to move away from the edge one fiber at a time, so the wedging force was very small.

For more flexible media, like meat and skin (which are more relevant for hunting knife usage), the data shows up differently. The image below, of cutting force in chicken breast, comes from Zhou, D., Claffee, M., Lee, K., and McMurray, G., "Cutting,'by Pressing and Slicing', Applied to the Robotic Cut of Bio-materials, Part II: Force during Slicing and Pressing Cuts", Proceedings of the 2006 IEEE International Conference on Robotics and Automation , 2256-2261. In this image, the forces are compressive, and so are negative. Therefore, high forces show up at the bottom of the graph. As the blade is pushed into the meat the force rises, until the meat fractures. The force then stays constant until the blade leaves the meat. The paper describes in detail how the fracture force varies with edge geometry. CATRA REST values behave very similar to the plot in this paper, and have been demonstrated to increase as blades dull. Therefore, I believe that CATRA REST values are a much better indicator of blade sharpness than is total cutting force in a rigid material.

For a non-rigid material, I have yet to see any data that says cutting depth is determined by cutting force.

Carl

 

[hardheart]

http://www.mediafire.com/?5oihuynyusn

 

[cds4byu]

Thanks, Hardheart. I already had the Atkins paper, but not the other.

Carl

 

[Cliff Stamp]

The linear increase is due to the increased width of the blade as you get farther back from the cutting edge. The constant force is caused by the blade being fully engaged. Once the blade is fully engaged, the force is no longer proportional to depth of cut.

Yes, that would be exactly what I said.

Further, as you carefully read this paper, the paper is not about edge geometry, but about "wedge geometry" It's looking at how the gross geometry of the blade affects cutting forces, not about how the edge geometry affects cutting forces.

Again, I was speaking of the entire geometry clearly.

When you sharpen a knife (as opposed to reprofiling a blade), you only modify the edge geometry.

I would not put this in the definition of sharpness, it should be independent of edge thickness (gross thickness, where the edge meets the primary grind) and angle. There are many reasons why including these aspects into sharpness is very troubling. If you did for example, there is no way one of the FFD2 blades would ever be as "sharp" as an Opinel for cutting woods.

For a non-rigid material, I have yet to see any data that says cutting depth is determined by cutting force.

Again, as I clearly noted, if the material is not rigid there are obviously no wedging forces.

-Cliff

 

[cds4byu]

Again, as I clearly noted, if the material is not rigid there are obviously no wedging forces.

-Cliff

But you use CATRA data (on a non-rigid medium) to talk about cutting performance, and quote cutting performance as "depth cut under a given force". But it is only the wedging force that increases with the depth of cut. Therefore, if you use cut depth at a given force to quantify cutting performance, you're primarily measuring wedge affects, which have little, if anything, to do with edge retention.

Carl

BTW, CATRA ERT data doesn't measure depth cut at a given force. It measures depth of cut for a given slicing distance, under a given applied force. This is a fundamentally different quantity, and needs to be handled differently in calculations.

 

[Cliff Stamp]

But you use CATRA data (on a non-rigid medium) to talk about cutting performance, and quote cutting performance as "depth cut under a given force".

Go back to the first post in this thread. It starts off with :

"cutting ability is basically inversely proportional to the force on the blade"

This basically means the better a blade cuts then the less force you need to exert to make a given cut.

The post continues by modeling this force with two components, a constant term, which represents wedging, and a variable term, which represents the force against the edge. The latter is variable because as the blade blunts it will take more force against the edge to push it through the material but the wedging force is constant. The 1 in the equation represents the constant wedging force.

[CATRA]

It measures depth of cut for a given slicing distance, under a given applied force.

Yes, the same reasoning would apply which should be kind of obvious as the model represents it exactly as noted in the fits.

-Cliff

 

[jdm61]

Typical sharpened edges, high polish, go down to about 0.1 microns, this is the same scale as the carbides in the high finish steels like AEB-L, 52100, etc. . The high carbide volume P/M steels have carbides of about 5-10 microns, the individual carbides are fairly small, but there is often so much of them that you get clusters of macro carbides which are several times the size of the average carbide. The carbides in coarse steels like 440C, D2 or materials like Talonite are 50 microns as they are left from the as cast structure. However as long as the abrasive can cut the carbides the edge can form much smaller than the carbide size.

If you take a piece of knotty wood and plane it, as long as the plane is really sharp and set at a low angle it should cut the knots right along with the wood. However if the wood is really knotty you will eventually get to the point where there is actually no wood to hold the knots in place so fine corners will not be overly durable. Imagine a bow for example which was full of knots in the tips and what would happen if you drew it.

Consider as well if the knots started to approach the hardness of the plane. It would then become very likely for them to be torn out rather than be cut. A similar situation happens in the very high carbide steels when edges are taken to very low angles and the alloy carbides are very hard. There isn't enough steel to hold the carbides in the edge and the carbides are so hard they will require much higher interaction forces to cut them which means it is more likely for them to be torn out. In short, they don't get as sharp and they won't stay as sharp initially.

However, eventually the edge will thicken to the point where it is now large enough to hold the carbides and the edge will be very stable and resist blunting strongly. As they are wearing to this point they also tend to be very aggressive when slicing because the chunks of carbide which get torn out leave an aggressive slicing finish. You tend to get oscillating behavior where the edge wears, becomes aggressive, and repeats until it stabilizes.

-Cliff

Interesting stuff about carbide size. Is that why some stainless knives have a rough finish on the micro bevels when you sharpen the a 440C or ATS 34 knife? To me it looks like little vertical grinder scratches. I don't see than when I sharpen one of my W2 blades, which is pretty fine grain stuff. I tend to flat grind my knives down to around .020 on hunters and maybe .030 on big knives than convex the edge and 'roll" the convex portion back a bit flatter both on the grinder and when I hand sand it.

 

[Cliff Stamp]

The grindability is low due to the high volume of carbides and as well in general the large aus-grain with standard heat treatments, both of which lower the surface finish. In comparison, W2 has a very low carbide volume, high grindability and very fine aus-grain. Nice blade steel.

-Cliff

 

[cds4byu]

Go back to the first post in this thread. It starts off with :

"cutting ability is basically inversely proportional to the force on the blade"

This basically means the better a blade cuts then the less force you need to exert to make a given cut.

Yes, the same reasoning would apply which should be kind of obvious as the model represents it exactly as noted in the fits.

-Cliff

Cliff,

Perhaps its because you are trying to talk to people who don't have PhD's in physics. Perhaps it's because you want to have one model that applies to every situation. I really don't know the reason, and I shouldn't speculate because I don't know; and it's not really relevant.

But your explanations on this topic are imprecise. You draw no distinction between rigid media and non-rigid media. You don't put units on your terms. You assume an inverse relationship without any proof. And you justify it all by saying "it fits the data".

I don't mind if you say you have an empirical relationship that fits the data. But to claim you have a physics-based model, when you have the same model regardless of the cut measurement, seems to me to be dramatically overstretching things.

As a physicist, I think you're aware of dimensional analysis. None of your equations have units, and if you put units on them, I think you'd find that there were constants missing, and the constants would have different units for different cases, and none of them would seem to make physical sense.

As an example of what I believe to be oversimplification, you refer to the CATRA test by saying the work done by the knife in cutting is proportional to the amount of media cut. Then you imply that the work done by the knife is equal to the normal force times the distance cut. That's true in push cutting, but not in slicing. The work done in slicing is equal to the normal force times the distance cut, PLUS the slicing force times the distance sliced. And I'm sure the slicing force is _not_ constant, because I've looked at the edge geometry.

If your model doesn't account for the slicing force as well as the normal force, it's missing part of the work. And to just hand wave it away works fine, if you only want an empirical model. But if you want a physics-based model (which you claim you have), you need to include _all_ of the phenomena, then explicitly which phenomena you assume to be unimportant.

Here are some references of people who have made physics-based models of cutting and/or measured forces involved in cutting. All of them are more careful than your model. None of them claims to be as comprehensive as your model. All of them provide careful data that shows how well their model fits. And, as far as I can see, none of their models simplify to your model, although they have key aspects related to your model.

Marsot, J, Claudon, L. and Jacqmin, M, "Assessment of knife sharpness my means of a cutting force measuring system", Applied Ergonomics 38 (2007), pp. 83-39.

Dowgiallo, A, "Cutting force of fibrous materials", Journal of food Engineering 66(2005), pp. 57-61.

D. Zhou, M. Claffee, K.-M. Lee and G. McMurray, "Cutting, ‘by Pressing
and Slicing’, Applied to the Robotic Cut of Bio-materials, Part I: Modeling of Stress Distribution", Proceedings of the IEEE
International Conference on Robotics and Automation (ICRA06), Orlando,
FL, May 2006, pp. 2896-2901.

D. Zhou, M. Claffee, K.-M. Lee and G. McMurray, "Cutting, ‘by Pressing
and Slicing’, Applied to the Robotic Cut of Bio-materials, Part II: Force
during Slicing and Pressing Cuts", Proceedings of the IEEE
International Conference on Robotics and Automation (ICRA06), Orlando,
FL, May 2006, pp. 2256-2261.

McGorry, R.W., Dowd, P.C., and Dempsey, P.G., "The effect of blade finish and blade edge angle on forces used in meat cutting operations", Applied Ergonomics 36 (2005), 71-77.

McGorry, R.W., Dowd, P.C., and Dempsey, P.G., "Cutting moments and grip forces in meat cutting operations and the effect of knife sharpness", Applied Ergonomics 34 (2003), 375-382.

Atkins, A.G., Xu, X., and Jeronimids, G., "Cutting, by 'pressing and slicing,' of thin floppy slices of materials illustrated by experiments on cheddar cheese and salami", Journal of Materials Science 39(2004), pp. 2761-2766.

I have pdf copies of these articles, and would be happy to email them to you. Copyright laws prohibit me from posting them.

I don't think your model is without merit. I do think it's not carefully done, and that it would be much more persuasive if you'd work it over carefully and provide some data that shows both where it matches and where it doesn't (because no model is perfect).

Please don't take this as a personal attack. I'm glad to see that you are putting out ideas about testing and edge retention. I'd just be much more convinced if your work were a little bit more detailed and carefully presented. When work is presented in detail, then it can be critically evaluated. When work is presented in broad generalities, it can only be broadly criticized, which often appears to be in the form of personal attacks.

Thanks,

Carl