Hi J,
You have brought up some interesting points. Yes a deeper analysis could be done. But I'm sure that my analysis has debunked the myth that hollow shapes are necessarily worse. Also I'm satisfied that primary angles should be close to the microbevel angle (the 25/30 degree double bevel combo is probably wasteful in terms of support.)
It's true that my conclusions are not based on an analysis of the stresses in the metal and resulting flex. This is only a concern when comparing the double bevel to the hollow grind/flat hone. And this ended up being the important comparison. For really thin blades (0.080") the two geometries are within thousandths of an inch and I don't think there would be a substantial performance difference. But a properly sized grinding wheel could match any primary bevel angle. For thicker blades you can use a larger wheel to make the hollow and so that method wins again. So over all the hollow grind/flat hone method can always win in a geometry comparison against the double bevel. But the "other factors" affected my conclusions. I think I'll reword my conclusion.
My idea was to look at which way to sharpen a blade of given thickness. Thicker blades seem to give better performance and manufactures of high quality planes use thicker blades than the 0.080" Stanley blades. So I presume that the increased contact height of a thicker blade is not a serious penalty to pay for the increased stiffness of the thicker blade. The thickness of the blade is so important that Steve Knight's
1/4" blades apparently give incredible performance even without a cap iron to pretension the blade and with a larger contact height as he uses single or double bevels. When comparing different blade thicknesses it is not the flex of the bevel that counts, it is the flex along the length of the blade.
This stuff is complicated. In fact modeling it will always be insufficient. Experimental evidence would be the only way to ensure all things are accounted for.
In theory the minimum clearance angle is the bed angle. That is like using a chisle to chop a mortise. But if you did this then after the first few shavings are made and the blade edge crumbles and dulls then the cutting edge will be above the bevel surface and you will not produce another shaving because you will not be able to push down the wide bevel to compress the wood enough to get the cutting edge into the wood.
Because the plane blade extends below the sole, it is slowly worn down just like in honing but here the wood is the abrasive. Brent Beach suspects that when this worn area gets large enough the blade feels dull as you drag it across the wood with great pressure. So a large clearance angle would reduce the frequency of sharpening. I guess it is a trade off between edge toughness due to included angle, supportive geometry, sharpening frequency and desired wood surface finish. For a particular wood and task (face or eng grain) there will be an optimum clearance angle for your assessment of the correct balance of the tradeoffs. There probably isn't one optimal clearance angle for all woods/tasks. I think I will add something about this to my page. Thanks. I did the analysis at 15 degrees clearance because that seems to be about what people like to use.
Leonard Lee writes "Anything between 30 and 35 degrees is quite acceptable. If you go much lower than 30 degrees you encourage blade chatter; if you go much higher than 35 degrees any wear dulls the blade much faster, but, more significantly, you reduce the relief angle unacceptably, particularly on planes with a 45 degree bed. Bevel-down planes as a group represent an exception to the rule that blades should be sharpened at the lowest angle consistent with edge retention, because the bevel angle has no bearing on the cutting angle. A sturdy edge is wanted, and a basic grind angle between 30 to 35 degrees will give you good blade stability and the least chatter."
It is too bad Lee doesn't say what he thinks the relief angle is for. He has implied it is involved with more than how quickly the blade dulls as Beach has suggested.
Thanks for your thoughts!
Peter