riddle me this

[Rob]

Is a hollow ground edge (say a plane iron ground at 25) ever so slightly weaker than an edge ground dead flat at the same angle?  The theory being that the concave nature of the grind would leave less supporting metal behind the edge?

I appreciate for practical purposes it probably makes no difference, this is about a theoretical debate only.

Is there less metal behind the edge of a hollow ground edge than a straight ground edge?

[jeffs55]

The answer would seem to be elementary for the reasons you have stated. It cannot be stronger or even as strong with all else being equal since it has less metal to support the cutting edge. So sayeth Sherlock Holmes when I forwarded this to him. How are things on Baker St today?

[Jeff Farris]

Rob,

It's a complicated dance between the thickness of the tool and the diameter of the grindstone.

In the example you've just mentioned...a plane iron...on the Tormek 250mm wheel you would need some pretty sophisticated measuring equipment to measure the "hollow" cut in the bevel. I'm not enough of a mathematician to calculate it, but I would wager if you either measured or calculated, the depth of hollow on a standard plane iron would be significantly less than .001". I'm not sure how much "weaker" that is.

[Ken S]

Rob,

I can think of two ways to measure this:

1) Using geometry with the length of the bevel and the diameter.  The math is beyond me,

2) Eyeballing the bevel against a straightedge.

In the real world, I think the argument probably began using smaller diameter grinding wheels, maybe six or seven inches.  A smaller diameter would produce a more hollow grind. I have read that turning skews should be "dead flat".  I'm not sure how "undead" a 250mm wheel would be.  I suspect we are not too far away from determining how many angels could dance on the head of a pin.....

Ken

[Rob]

I accept all of that guys.  This was never about a practical, real issue of actually caring whether or not a hollow grind would be weaker in real life.  It was about a theoretical debate. On a different woodworking forum (nothing to do with sharpening) I had contended that a hollow grind was (albeit marginally I fully accept) weaker than a straight grind for any given bevel angle.  The reason is that by grinding hollow the arc of the circumference of the grinding wheel has nibbled into metal that would have been left in place by a straight grind.  That lessened mass of metal is what causes the increased weakness (small I know).  He contended the exact opposite ie that the wheel grinding process leaves MORE metal behind the edge than a straight grind, thus the edge was strengthened.  He presented a thoughtful diagram and it seemed plausible (still does if truth be known).  Even though I know I'm right intuitively, I can't seem to articulate it objectively and its driving me nuts.

[Jeff Farris]

To my mind, you're right. Theoretically weaker, but practically immeasurable when the foundation of the argument is a 10" wheel...except on very thick tools like mortising chisels and the like.

I take it from your last sentence you don't want to call him a dunderhead and move on. 

Funny, Ken...when a related subject came up on another forum, I used exactly the same metaphor.

[Rob]

LOL...thanks for cheering me up guys :-)

[Ken S]

Jeff, I guess great minds really do think alike......

Ken

ps.  Glad we cheered you up, Rob.

[Rob]

Isn't that funny.  And whats more I've not heard that angels dancing expression before.  Its a good one:-)

So further to the original riddle then chaps:

If you set the angle master to 25 degrees (really carefully touching the stone in the two necessary places as well as the blade etc and having dialled in the correct wheel diameter).  You then grind to the finished angle, the angle of the very tip ie where the bevel meets the flat underside - should that be 25 degrees.  Or, should the 25 degrees be measured with one line = the whole bevel right to the heel and the other line, the flat underside?  This measurement would be different if the hollow was sufficiently large by perhaps 1 or 2 degrees?

I ground a fresh 25 degree whole bevel on a chisel yesterday, set at 25 deg and I've just measured the resultant grind and its at 30 degrees?? (probably 29)  I did need to regrade the stone several times during grinding to keep it cutting.  Do you think that additional abrasion of the stone reduced its diameter and thereby caused the angle to step up from 25 to 30?  Or, is this perhaps related to the above?

[Jeff Farris]

Doubt that it is the stone getting smaller that caused the change, it's the tool getting shorter that caused the change. Think about it...pivoting on the Universal Support as it does, if you keep grinding, eventually you'll get to a square edge. On anything I'm doing where the angle is actually critical, I always reset the angle once I get the lion's share of the grinding done.

And the angle is measured to the chord between the two points, generally speaking.

[Rhino]

I can see the other angle too (please pardon the pun, I cannot resist.

To use very crude graphics: A  )____ blade shape should have more material than a /___ bade shape even though both has the same angle at the very edge.

I don't really have side to defend in this debate, just making trouble.

[Herman Trivilino]

Rob, this drawing illustrates the concept.



Simply put, as you move from the cutting edge of the tool towards its body, the steel gets thicker faster on the curved bevel.

As Jeff points out, though the effect is small.  The drawing is not to scale for a typical bevel created with the Tormek grindstone.  If it were, the effect would be too small to see.

Look at it this way.  For a bevel width of 5 mm, and a grindstone radius of 125 mm (half of the 250 mm diameter of a new Tormek SG-250 grindstone) the bevel width is only 2% of the radius of curavature.  In the drawing I made I'd guess that the bevel width is more like 20% or 30% of the radius.

[Herman Trivilino]

1) Using geometry with the length of the bevel and the diameter.  The math is beyond me,

Suppose you lay a straight-edge on that curved bevel and you can see daylight between the straight edge and the surface of the bevel.  Let's call the maximum distance between the straight edge and the curved bevel d, the depth of the curve.

Let's say the grindstone radius is r and the bevel width is w.  Then



For example, suppose r = 125 mm and w = 5 mm.  Then using the above formula we get a depth d of about 0.025 mm.  This is smaller than the width of most human hairs, although perhaps a very fine one could be inserted in that gap!

[Rob]

Brilliant Herman :-)  I knew you would be able to extract the mathematical formula for this.  Thanks.  Oddly I did grind a 25 degree bevel and I did offer the hollow ground bevel to a straight edge (engineers square) and I then placed white paper behind that setup and you could plainly see the half moon of light shining through the gap in the chord.

The debate I was having on this other forum was that because it was hollow ground there is LESS metal behind the edge and therefore constitutes a weaker edge.  The other chap argued there was MORE metal and therefore a hollow grind gives a stronger edge.  Both understood that the actual practical difference is so negligible as to be academic, but by this time heels were dug in, hackles were raised etc and it was about who was right and who wrong rather than whether it actually mattered or not :-)

In the end it appeared the way the question is phrased can have an impact on the answer because if the chord is ground to 25 degrees, then a straight ground bevel will naturally be 25 degrees.  The chord of the hollow grind will also, but the very tip of the edge won't be quite the same angle (because a small amount of metal will have been deleted by the arc of the circumference of the grindstone, reducing the angle a tad.

So if one fixes the angle at the very tip of the edge of both hollow and straight grinds and then compares them, the hollow grind will have more metal BECAUSE the chord will be at a different angle than the straight ground bevel.  If the chord of both are at 25 then the tip of the hollow will actually be less and one might argue that the edge is weaker.

Did that make any sense :-)

Honestly I do appreciate this is semantics....I'm being a total pedant and dancing angels are about to hove into view....but this was about honour, not engineering :-)   

[Ken S]

At the risk of seeming very worldly, one of my teachers suggested adding three degrees to the traditional grinding angle when using a Tormek to compensate for  the hollow grinding effect of the wheel.  Twenty eight degrees instead of twenty five.

Rob, I would suggest you set the angle to twenty eight degrees, thirty one minutes, and fourteen seconds to approximately compensate for your time zone.

Ken