The general method used to compare knives for edge retention is to cut a specific material until a given amount of blunting is reached with both knives with the ratio of the material cut determing the performance advantage as "At the end of the test Knife 1 had cut 25 (5) % more material than Knife 2." An alternate method is to cut a given amount of material and make similar statements about the sharpness of both knives at that point, "Knife 1 was 15 (5)% sharper than Knife 2 after the test." The sharpness ratio in general is not as meaningful as the material ratio because usually cutting tools are sharpened after they reach a given state of blunting not after they cut a specific amount of material. However custom knifemaker Phil Wilson noted early in his edge retention tests that the rate of blunting slowed significantly as knives continued to be used, he attributed this to the blunting taking place by different means during early and late use 1. Due to this nonlinear response it is necessary to specify any point of comparison as : "Cutting until the sharpness was reduced to 10% of optimal, Knife 1 cut 25 (5) % more material than Knife 2."
The general problem is to determine the cutting advantage at each stage of the test. With smooth functions for the sharpness of two knives as functions of the amount of material cut, S1(x), S2(x), the cutting advanage of the second knife after x amount of material is cut is given by :
CA21(x)= InvS2( S1(x) )/x
Here CA(21x) is the ratio of material cut comparing blade 2 to blade 1, InvS2 is just the inverse function of S2(x). One straightforward approach to this problem is to model the experimental data to determine the necessary sharpness curves, determine the inverse function numerically and then calculate the cutting advantage with the above forumla. A model for the behavior of sharpness during use has been developed and successfully applied2 and numerical inverses are a known problem and the above function is then just basic math.
The model however introduces a dependancy into the calculation which is not necessary. A more optimal approach is to determine the CA(x) directly from the raw data. The problem then is at any given x find the horizontal intersects of the two data sets, S1(x) and S2(x) which are no longer smooth curves but discreet values with significant noise as with all physical measurements3. This is illustrated in the image in the right which is data collected by Paul Hansen during the CPM-D2 vs D2 group evaluation 4. After 60 cuts into the rope the CPM-D2 blade is slightly sharper. Examining the other data set it is apparent that the D2 blade has a similar amount of blunting between 20 and 40 cuts. The exact values can be determined by linear interpolation.
In general if the noise is really high there horizontal intersects will have multiple solutions. The method used by the author is to calculate all such solutions and perform an average. A montecarlo simulation is performed to allow these intersects to be bounded by a meaningful uncertainties. This algorithm is implemented in FORTRAN 77 code which is freely available by email request. Such analysis was performed on the above graphical data. The output is as follows :
Number of data 4 Number of Monte Carlo Runs 100 Sharpness Advantage 0.00 100.00 12.00 68.00 14.00 0.68 0.16 20.00 33.00 10.00 45.00 12.00 1.36 0.55 40.00 34.00 4.00 28.00 5.00 0.82 0.18 60.00 30.00 2.00 22.00 2.00 0.73 0.08 Sharpness Advantage avg 1.10 +/- 0.39 Cutting Advantage 0.00 100.00 12.00 68.00 14.00 0.00 0.00 20.00 33.00 10.00 45.00 12.00 1.46 0.47 40.00 34.00 4.00 28.00 5.00 0.82 0.40 60.00 30.00 2.00 22.00 2.00 0.63 0.48 Cutting Advantage avg 0.98 +/- 0.45
Here no cutting advantage is exhibited by either knife either in regard to the sharpness ratio during the cutting or the amount of material the blades have to cut to reach a given sharpness.
As the loss of sharpness during cutting is a nonlinear responce, edge retention comparisons have to take this into account by using a multi-point comparison method. This can be done by modeling the data and inverting the functions or finding the horizontal intersects on the data themsevles. The superior performance is then characterized by the sharpness advantage SA(x) and cutting advantage CA(x) functions.
1 : Phil Wilson, private communication, 1999.
2 : Cliff Stamp, Edge retention modeling, http://www.cutleryscience.com,2007
3 : Cliff Stamp, Physical Consistency, http://www.cutleryscience.com,2007
4 : Paul Hansen, CPM-D2 (Spyderco Military) compared against wrought D2 (Mel Sorg), http://www.cutleryscience.com,2007
|Written: Sept. 2007||Updated: Aug. 2007||Copyright (c) 2007 : Cliff Stamp|