If Im re-grinding an edge with a belt grinder at a given angle, and I have the option of either 1) pressing the side of the edge bevel into a somewhat slack belt to create a convex edge or 2) installing a platen behind the belt to create a flat edge bevel, which option should I use to create the most efficient cutting edge? I assume from your explanation that installing the platen to create the flat bevel would yield the more efficient cutting edge.
Assuming the edge would not fold with the flat profile then yes it would generally be more efficient. The presupposition is the critical part and where the discussion has to get a little complicated. The main constraint on an edge when cutting through a media is can it sustain its shape under the influence of the lateral forces. Is it stiff enough to withstand the attempts of the media to bend it otherwise will just buckle during the cut. Thus from a basic perspective you want to determine the absolute minimal geometry which allows the knife to keep its shape.
Now before I discuss this in detail I want to point out that there are a few other issues. If for example you just focused on that aspect and ignored the static and dynamic balance points the knife could cut horribly even though you have crafted a geometry which is extremely efficient at going through the media. The blade could have no ability for you to actually provide force to the cut or could induce severe vibrations on impact. However, these are usually not in opposition with cutting profiles constraints just additional factors which need to be considered.
So you have decided that you want to generate the most efficient geometry by balancing the edge grind to exactly match the force responce upon it. Well what does this mean exactly anyway. Consider what would happen if you took a decent machete and ground a 5 degree edge on it and used it for some wood work. The edge would buckle almost immediately. Make a note of the maximum depth of the damage. Repeat this with a 7 degree edge, 9, 11, 13, ..., until the damage stops. Now consider the results, you have shown that the angle which is needed to keep the edge stable is obviously dependent on the height of the blade from the edge.
For example the five degree damage depth may be extensive but at some height the blade will be thick enough even though ground that acute that it won't bend any more. So it is obvious that at a specific height that five degrees is functional. It should then also become immediately obvious that the optimal profile is one which starts at the minimum which keeps the very edge stable but quickly is reduced after that because as you move back from the edge you need less angle to keep it stable. Once you do some experimenting you will discover that your new profile is actually stronger than you predicted from your earlier experiments. This is because once an edge starts to bend the forces on it are magnified because of the way they load the edge, essentially the leverage increases. So you will find you can actually reduce the angles more than the first step wise analysis would predict.
Now this is where many people would note, hey wait a minute, doesn't this show that convex edges are superior. Yes it does - for that media and that method of cutting. But note clearly what it says, it doesn't say all convex edges are superior, it describes one very specific geometry which minimizes the stable cross section. Now take that same cross section and look at it critically. Consider the difference in performance between that bevel and two appropriate flat bevels. The performance would be very similar, in fact very difficult to tell apart.
Now further consider a convex bevel which was slightly off due to an improper optomization and a multiple bevel flat which was closer to the optimal curvature responce. Which would have superior performance. This should make it obvious that matching the cross section requirements is what is critical and the curvature to get there is a small refinement. Yes, it will be a refinement, because convex bevels offer essentially an additional degree of freedom, they reduce to flat bevels when the curvature is zero. However the main focus for a user has to be first and foremost on the cross section this is why you never say use convex bevels you always say minimize the cross section.
Now returning to earlier, I said that bevel just deduced is specific to that media. Consider the same with a small utility knife and cut thick pink styrofoam insulation. You will find a radically different result. The angle sensitivity above the edge quickly reduces to zero. In fact with a little thought and imagination you try negative angles (concave grinds) and find the stability limit is actuall less than zero. So the new optimal profile isn't convex any more but a hollow/convex hybrid. You then go back to your chopping blade and wonder if it is the same and see if you go high enough on the blade can you invert the curvature and still keep stability - yes you can. This is why hardwood felling axes have primary hollow grinds.
Thus you can predict that for some media which are fairly soft, but still very abrasive, the optimal profile would be deep hollow primary and a barely existing flat edge bevel. It isn't then difficult to realize how pretty much any geometry would be optimal for some media because you would expect all of them to have unique force responce curves. This as I noted in the above isn't the only part, you also have to match the weight, length and balance requirements of the user, as well consider that you are generally not cutting just one thing so you usually have to make some sort of compromise.
As a specific example, my large chopping blades typically are similar to 0.050"/8:0.020"/15 degrees. They drift around this a little, but the relief grind is about eight degrees and the edge thickness is about 0.050" (this is too thick but grinding down the primary on large chopper takes a lot of time and the effect is minimal due to the nonlinearity of the force responce). The actual edge apex angle is about 15 degrees, usually slightly less and is about 0.0150-0.020" thick. Now this is a bevel I freehand sharpen so it is one smooth curvature Fikes style usually, but again, it isn't the curvature which gives the performance it is the cross section. Which is why I never advocate curvature but cross section as a goal. As I noted, the optimal curvature will change from media to media, but the fundamental goal of cross section minimization always stays the same.
-Cliff
