wimms
09-07-2016, 06:24 AM
Hi guys
I was looking into making my own Brinell hardness tester. Partly because I could not find Lee hardness tester for reasonable price here, partly because I do not like the need to use microscope. When I read up on Brinell, my engineering mind immediately asked - why do we need to measure the indent diameter visually, when we could measure indent depth with a dial indicator and compute the diameter? Surely dial indicator is cheaper than microscope and can be made much more convenient to use.
This thought kept nagging. It was more of curiosity rather than necessity to solve for me, but along the way I learned something I would like to share. Nothing earth shattering, just perhaps a perspective that might be useful to some other engineering minds.
Good for background: http://www.hegewald-peschke.com/interesting-facts/guidelines-to-hardness-testing.html
I wanted to find a reliable way to determine Brinell without microscope using dial indicator, ideally a direct measurement. I had this feeling that there is a way to do that. So I searched for what has been done before. CabineTree is obvious example, but it seems odd, not exactly Brinell method. First thing that ringed a bell is that there exists a Rockwell test method with Brinell loads and penetrators. I like Rockwell method - it makes sense and does not need optical measuring. I picked up on that and tried to figure out how to measure Brinell without expensive lab equipment.
The usual Brinell formula in literature is: HB=2P/(Pi*D*(D - sqrt(D^2 - d^2)))
I searched how to calculate indent diameter d from measured depth so it could be plugged into above Brinell formula. Couldn't even think of a search term so tried to derive myself using Pythagorean theorem from known radius and height. Eventually I found what the term is and found several more helpful formulas.
The indentation left by the ball is called Spherical Cap.
http://mathworld.wolfram.com/SphericalCap.html
If you look at how they derive radius of the base circle: (R-h)^2 + a^2 = R^2, thats how pythagorean theorem is used. Wolfram is nice site, it can solve math formulas for you. I asked it to solve for h.
http://www.wolframalpha.com/input/?i=solve+%28R-h%29^2%2Ba^2%3DR^2+for+h
h = R - Sqrt[R^2 - a^2]
And then tried to use diameters only, calling them D and d for base circle:
http://www.wolframalpha.com/input/?i=solve+%28D%2F2-h%29^2%2B%28d%2F2%29^2%3D%28D%2F2%29^2+for+h
h = 1/2(D-sqrt(D^2-d^2)) Hmm, that looks familiar..
Brinell is defined as a ratio HB=P/A where P is force and A is area of contact of the spherical cap.
The objective of the Briness measurement is to find the area of this cap, the rest is math.
Spherical Cap has area A=Pi*D*h, where D is ball diameter and h is cap height, in our case - depth of indentation. By substituting A in HB=P/A we get HB=P/(Pi*D*h)
And now it clicked, the term (D-sqrt(D^2-d^2)) in Brinell formula - it is h derived from known D and measured d (1/2 factor moves up to 2P).
Turns out that actually HB=P/(Pi*D*h) is a natural direct formula for HB=P/A. So apparently measuring h directly is very natural approach, and in every sense equivalent to measuring indent diameter. So the choice of measuring d or h is dictated only by which gives you more accuracy. I liked that and was encouraged to pursue depth measurement.
What about direct reading of h from metric dial indicator to represent HB value, is it possible?
My thinking was to normalize relationships of force to indentation area so that measurement of h would represent HB. To get closer, I wanted at least to have ratios to be multiples of 10, not complex fractions. Need to pick some load weight anyway, so what if we pick it so that we can isolate h from the formula?
HB=P/(Pi*D*h) is equivalent to HB=P/(Pi*D)/h and we can notice that the term P/(Pi*D) is constant for the device.
If we pick the loading weight to be P=Pi*D, then P/(Pi*D)=1, and HB=1/h
For eg. if D=3mm, then P=Pi*3=9.42478kg (20.7781 lb)
then P/(Pi*D)= 1 kg/mm and HB=1.0 kg/mm / h(mm) = Brinell number (kg/mm^2)
Brinell lookup table for h would look like this:
0.01mm = 100 HB
0.02mm = 50 HB
0.04mm = 25
0.10mm = 10
0.20mm = 5
So,
1) we can make a measurement of indentation depth,
2) we do not need to optically measure the indentation diameter d
3) we need just to do a 1/h calculation
The result would be Brinell hardness.
It is not direct reading though, it's inverse. But simple to calculate, often without a calculator. I have no idea how to handle inversion with simple mechanical means, so I failed to achieve my goal to get direct reading, but its not the destination that matters, right?
What I found instead is that for every Brinell tester there is a device specific constant, knowing which finding Brinell hardness is a simple division: HB=C/h
I didn't know that, and this is pretty cool.
The Lee Precision hardness tester uses a 4mm ball and 27.2155kg (60 lb) force (HBS 4/27/30)
The constant it has is: P/(Pi*D) = 2.16574, so for Lee tester, computing Brinell hardness from its indentation depth(in mm) would be HB=2.16574/h
Lets see if that makes sense. Lets take few diameter reading values from Lee lookup table, say 0.038, 0.058 and 0.079. First lets compute the diameter d in mm: 0.038"=0.9652mm,
then find the depth: h=(D-sqrt(D^2-d^2))/2=(4-sqrt(4^2-0.9652^2))/2=0.0590988587mm,
then apply the formula HB=2.16574/h=2.16574/0.0590988587 = 36.646 and now lookup up what Lee table says.
So doing the same for all the samples from Lee table we get:
0.038 - h=0.0591 - HB=36.646 Lee: 36.6
0.058 - h=0.1406 - HB=15.405 Lee: 15.4
0.079 - h=0.2699 - HB= 8.026 Lee: 8.0
The correspondence is exact!
Looks like it doesn't even matter what exact load weight you are using. If you know it you can construct your lookup table, or simply calculate directly as C/h.
Standards are guidelines, it doesn't mean that if you deviate you'll get your readings off by million. But they exist for a reason, so lets look at how well does Lee's tester align with BHN testing standards.
Brinell method recommends to use degree of loading P/D^2=30. It is in units of kgf/mm^2 ie. units of pressure. That is to say that standard recommends you apply minimum amount of pressure to an area to get meaningful hardness reading. It is said to be increasingly important as hardness of material goes up, and that HB values are comparable with different ball/load combinations only when the degree of loading is the same. Yet there are standard Brinell test loads down to HBW 1/1 which means 1mm ball and 1kg load, giving loading degree of 1. Loading of 30 is there probably because most frequently measured metal is steel and it does need more force to yield than soft metals. For soft metals lower pressure is in order.
Lee's degree of loading P/D^2 is 27.2155/4^2=1.7 - a far cry from 30, and not matching any of the BHN methods.
Method of picking load P=Pi*D for 3mm ball gives P/D^2 ratio of 1.27. Even less, but does it matter?
Whether and how much this affects accuracy of HB testing of lead I'm not sure. It has been shown that Lee tester correlates quite well with many other lead HB testers, which all have this ratio quite different for many reasons. It seems that for alloys so soft as we are dealing with this doesn't matter much and main load selection criteria is ability to measure indentation with good accuracy within the range of hardnesses we deal with.
Basically we are pretty free to choose both the ball size and the loading force. It is easy to pick a suitable loading for your measurement tool, or simply by knowing what the load is, you can still compute the BHN pretty easily.
So what is the practical use for the above?
The simplest application is to just take the dial indicator and use it directly as indentation punch. If you look at the tip of it, what you see is a ball there, of 2.5mm diameter. Suspend a dial indicator in a holder, put your sample under it winding up the dial few mm, zero the reading, and then apply appropriate load through the probe shaft that goes through the dial. After removing the load, take the reading, simple C/h calculation and done.
I took my 0.01mm dial indicator and measured the ball at ~2.5mm, it is protruding ~0.71mm
P=Pi*D = Pi*2.5 = 7.8534kg for the device constant (P/Pi*D) to be 1.0
Check for estimated measurement range, as its limited by reasonable depth h at different HB values:
h(5HB)=0.2mm
h(25HB)=0.04mm
h(50HB)=0.02mm
Well within range of protruding ball of the probe tip although getting close to resolution of the indicator.
So I made quick test. I have a 0.01mm dial indicator, I picked some lead muffins, stacked them, weighted them to be 3.003kg, thus P/Pi*D constant=0.382353. Then I setup a dial indicator touching another muffin on solid base, zeroed it, and simply held the stack of lead muffins on top of the dial's through-pin with my hands. To avoid rocking induced error due to uneven shape of the measured muffin, I zeroed my dial while applying slight preload with my finger, so that after removing the main load and applying similar preload I could exclude the error from any play. I measured residual dent depth of 0.04mm.
So, HB=0.382353/0.04=9.558. This is range scrap and I happen to know it's ~99%Pb-1%Sb.
Even simpler method would be just to take a loose ball of known diameter, put on the sample, and rest a known weight on it, then measure the indentation depth with whatever accuracy you can, find the constant P/(Pi*D) and calculate C/h.
You see, Brinell hardness measurement is pretty simple afterall, and there are very many ways to do it.
Perhaps this can help someone to get a better feel on how BHN testers work, perhaps it inspires someone to build their own hardness tester and have confidence in it, perhaps someone can modify Lee tester to add a dial indicator and not need microscope anymore.
PS.
I didn't want to clutter already long post with this, but it should be mentioned. There is a reason why lab grade BHN testers are using optical readout. It may make you think that Lee's tester is therefore more accurate, but it is not true, it is a gimmick giving you false sense of accuracy, in actual use it is nothing but inconvenience. CarbineTree and Saeco is a living proof of that. But why and when do you need optical readout?
There is a big deal around material pile-up or sink-in that is still being researched. Mostly in context of steal hardness, it may matter somewhat for harder lead alloys. In nutshell, it is possible that pile-up forms around the indenter ball crater which increases the contact area and can cause overestimation of HB if not accounted for.
see Fig.1 http://www.sciencedirect.com/science/article/pii/S0020768305003203
The bad thing is that for alloys that exhibit some elastic rebound the residual indentation does not represent the actual contact shape during loading - no matter if you measure depth or diameter, you're still screwed.
For purely plastic materials, rebound doesn't occur, but the pile-up might. The optical measurement of indent diameter can account for that, for depth based measurements it needs to be accounted for by compensating factor. Thats probably the only reason why for unknown materials optical measuring of the diameter is more all-encompassing method. In actual measurement accuracy, optical method is much more involved than you might think. It is the main source of BHN measurement errors in the whole industry. But the industry is concerned mainly with BHN range 100-700 and different materials, so alot of what matters there doesn't for lead alloys.
But I did look for possible errors coming from pile-up around the indentation and ways to deal with that.
Literature often mentions that amount of pile-up has good correlation with material’s work-hardening exponent n. Many have attempted to find a formula to compute the pileup, often matching to empirical data.
Some examples of function f(n) that try to predict amount of pile-up or sink-in based on exponent n:
f(n) = -0.85768*n + 0.35768
f(n) = 5/2*((2-n)/(4+n))-1
f(n) = 1/2*((2+n)/2)^(2*(1-n)/n)-1
f(n) = 1.276-1.748n+2.451n2-1.469n3-1
Plotting them together gives an idea of the curves:
http://http://www.wolframalpha.com/input/?i=plot++5%2F2*%28%282-n%29%2F%284%2Bn%29%29-1+and+-0.85768*n+%2B+0.35768+and+1%2F2*%28%282%2Bn%29%2F2 %29^%282*%281-n%29%2Fn%29-1+and+1.276-1.748n%2B2.451n2-1.469n3-1+for+n+from+0+to+1
To compensate for pile-up you compute corrected hc = h*(1+f(n)). You might also measure the pileup directly.
Thus if there is a need, formula with correction would be: HB = P/(Pi*D*h*(f(n)+1))
f(n) is generally constant for material, so we can move it to the constant side and compensate for it while we pick P:
if P=Pi*D*(f(n)+1), then P/(Pi*D*(f(n)+1))=const=1, and we still have HB=1/h (mm)
The big trouble is finding this exponent n for your alloy. For Pb alloys I have seen figures 0.165 and 0.25 but finding a reliable source has been difficult. Maybe I simply don't know what to look for.
You find the compensation factor by plugging n into any formula above and simply calculate it, For eg.
f(n) for Pb is -0.85768*0.25 + 0.35768 = 0.14326, thus compensation factor is 1.14326
Note that different formulas above give quite notably different results, so you need to pick a formula with some reasoning. Anyway, it doesn't really matter, as the formula tries to be predictive, while in the end what you need is a constant that applies to your alloy. It can be found experimentally.
Load P selection would compensate for pile-up like so:
if D=3mm, then P=Pi*3*1.14326=10.7749716514kg (23.75 lb)
This would provide correction of measured indentation depth in mm so that Brinell hardness is 1/h for lead based alloys.
So it appears that depth based Brinell tester is also very easy to calibrate. Essentially, calibration boils down to finding the correction factor f(n) for the alloy, and it is applied by adjusting the main load weight. By measuring a reference material and adjusting the load weight so that measured HB is right, we can find the f(n) correction factor without ever even knowing it.
As a note, you may notice that Lee's tester did match calculations exactly without any compensation factor. So we can see that Lee's tester does not compensate for pileup/sinkin of alloys and it is your responsibility to measure the dent diameter so that you find exactly the actual contact diameter, even if it extends higher (or sinks lower) from the material surface.
I was looking into making my own Brinell hardness tester. Partly because I could not find Lee hardness tester for reasonable price here, partly because I do not like the need to use microscope. When I read up on Brinell, my engineering mind immediately asked - why do we need to measure the indent diameter visually, when we could measure indent depth with a dial indicator and compute the diameter? Surely dial indicator is cheaper than microscope and can be made much more convenient to use.
This thought kept nagging. It was more of curiosity rather than necessity to solve for me, but along the way I learned something I would like to share. Nothing earth shattering, just perhaps a perspective that might be useful to some other engineering minds.
Good for background: http://www.hegewald-peschke.com/interesting-facts/guidelines-to-hardness-testing.html
I wanted to find a reliable way to determine Brinell without microscope using dial indicator, ideally a direct measurement. I had this feeling that there is a way to do that. So I searched for what has been done before. CabineTree is obvious example, but it seems odd, not exactly Brinell method. First thing that ringed a bell is that there exists a Rockwell test method with Brinell loads and penetrators. I like Rockwell method - it makes sense and does not need optical measuring. I picked up on that and tried to figure out how to measure Brinell without expensive lab equipment.
The usual Brinell formula in literature is: HB=2P/(Pi*D*(D - sqrt(D^2 - d^2)))
I searched how to calculate indent diameter d from measured depth so it could be plugged into above Brinell formula. Couldn't even think of a search term so tried to derive myself using Pythagorean theorem from known radius and height. Eventually I found what the term is and found several more helpful formulas.
The indentation left by the ball is called Spherical Cap.
http://mathworld.wolfram.com/SphericalCap.html
If you look at how they derive radius of the base circle: (R-h)^2 + a^2 = R^2, thats how pythagorean theorem is used. Wolfram is nice site, it can solve math formulas for you. I asked it to solve for h.
http://www.wolframalpha.com/input/?i=solve+%28R-h%29^2%2Ba^2%3DR^2+for+h
h = R - Sqrt[R^2 - a^2]
And then tried to use diameters only, calling them D and d for base circle:
http://www.wolframalpha.com/input/?i=solve+%28D%2F2-h%29^2%2B%28d%2F2%29^2%3D%28D%2F2%29^2+for+h
h = 1/2(D-sqrt(D^2-d^2)) Hmm, that looks familiar..
Brinell is defined as a ratio HB=P/A where P is force and A is area of contact of the spherical cap.
The objective of the Briness measurement is to find the area of this cap, the rest is math.
Spherical Cap has area A=Pi*D*h, where D is ball diameter and h is cap height, in our case - depth of indentation. By substituting A in HB=P/A we get HB=P/(Pi*D*h)
And now it clicked, the term (D-sqrt(D^2-d^2)) in Brinell formula - it is h derived from known D and measured d (1/2 factor moves up to 2P).
Turns out that actually HB=P/(Pi*D*h) is a natural direct formula for HB=P/A. So apparently measuring h directly is very natural approach, and in every sense equivalent to measuring indent diameter. So the choice of measuring d or h is dictated only by which gives you more accuracy. I liked that and was encouraged to pursue depth measurement.
What about direct reading of h from metric dial indicator to represent HB value, is it possible?
My thinking was to normalize relationships of force to indentation area so that measurement of h would represent HB. To get closer, I wanted at least to have ratios to be multiples of 10, not complex fractions. Need to pick some load weight anyway, so what if we pick it so that we can isolate h from the formula?
HB=P/(Pi*D*h) is equivalent to HB=P/(Pi*D)/h and we can notice that the term P/(Pi*D) is constant for the device.
If we pick the loading weight to be P=Pi*D, then P/(Pi*D)=1, and HB=1/h
For eg. if D=3mm, then P=Pi*3=9.42478kg (20.7781 lb)
then P/(Pi*D)= 1 kg/mm and HB=1.0 kg/mm / h(mm) = Brinell number (kg/mm^2)
Brinell lookup table for h would look like this:
0.01mm = 100 HB
0.02mm = 50 HB
0.04mm = 25
0.10mm = 10
0.20mm = 5
So,
1) we can make a measurement of indentation depth,
2) we do not need to optically measure the indentation diameter d
3) we need just to do a 1/h calculation
The result would be Brinell hardness.
It is not direct reading though, it's inverse. But simple to calculate, often without a calculator. I have no idea how to handle inversion with simple mechanical means, so I failed to achieve my goal to get direct reading, but its not the destination that matters, right?
What I found instead is that for every Brinell tester there is a device specific constant, knowing which finding Brinell hardness is a simple division: HB=C/h
I didn't know that, and this is pretty cool.
The Lee Precision hardness tester uses a 4mm ball and 27.2155kg (60 lb) force (HBS 4/27/30)
The constant it has is: P/(Pi*D) = 2.16574, so for Lee tester, computing Brinell hardness from its indentation depth(in mm) would be HB=2.16574/h
Lets see if that makes sense. Lets take few diameter reading values from Lee lookup table, say 0.038, 0.058 and 0.079. First lets compute the diameter d in mm: 0.038"=0.9652mm,
then find the depth: h=(D-sqrt(D^2-d^2))/2=(4-sqrt(4^2-0.9652^2))/2=0.0590988587mm,
then apply the formula HB=2.16574/h=2.16574/0.0590988587 = 36.646 and now lookup up what Lee table says.
So doing the same for all the samples from Lee table we get:
0.038 - h=0.0591 - HB=36.646 Lee: 36.6
0.058 - h=0.1406 - HB=15.405 Lee: 15.4
0.079 - h=0.2699 - HB= 8.026 Lee: 8.0
The correspondence is exact!
Looks like it doesn't even matter what exact load weight you are using. If you know it you can construct your lookup table, or simply calculate directly as C/h.
Standards are guidelines, it doesn't mean that if you deviate you'll get your readings off by million. But they exist for a reason, so lets look at how well does Lee's tester align with BHN testing standards.
Brinell method recommends to use degree of loading P/D^2=30. It is in units of kgf/mm^2 ie. units of pressure. That is to say that standard recommends you apply minimum amount of pressure to an area to get meaningful hardness reading. It is said to be increasingly important as hardness of material goes up, and that HB values are comparable with different ball/load combinations only when the degree of loading is the same. Yet there are standard Brinell test loads down to HBW 1/1 which means 1mm ball and 1kg load, giving loading degree of 1. Loading of 30 is there probably because most frequently measured metal is steel and it does need more force to yield than soft metals. For soft metals lower pressure is in order.
Lee's degree of loading P/D^2 is 27.2155/4^2=1.7 - a far cry from 30, and not matching any of the BHN methods.
Method of picking load P=Pi*D for 3mm ball gives P/D^2 ratio of 1.27. Even less, but does it matter?
Whether and how much this affects accuracy of HB testing of lead I'm not sure. It has been shown that Lee tester correlates quite well with many other lead HB testers, which all have this ratio quite different for many reasons. It seems that for alloys so soft as we are dealing with this doesn't matter much and main load selection criteria is ability to measure indentation with good accuracy within the range of hardnesses we deal with.
Basically we are pretty free to choose both the ball size and the loading force. It is easy to pick a suitable loading for your measurement tool, or simply by knowing what the load is, you can still compute the BHN pretty easily.
So what is the practical use for the above?
The simplest application is to just take the dial indicator and use it directly as indentation punch. If you look at the tip of it, what you see is a ball there, of 2.5mm diameter. Suspend a dial indicator in a holder, put your sample under it winding up the dial few mm, zero the reading, and then apply appropriate load through the probe shaft that goes through the dial. After removing the load, take the reading, simple C/h calculation and done.
I took my 0.01mm dial indicator and measured the ball at ~2.5mm, it is protruding ~0.71mm
P=Pi*D = Pi*2.5 = 7.8534kg for the device constant (P/Pi*D) to be 1.0
Check for estimated measurement range, as its limited by reasonable depth h at different HB values:
h(5HB)=0.2mm
h(25HB)=0.04mm
h(50HB)=0.02mm
Well within range of protruding ball of the probe tip although getting close to resolution of the indicator.
So I made quick test. I have a 0.01mm dial indicator, I picked some lead muffins, stacked them, weighted them to be 3.003kg, thus P/Pi*D constant=0.382353. Then I setup a dial indicator touching another muffin on solid base, zeroed it, and simply held the stack of lead muffins on top of the dial's through-pin with my hands. To avoid rocking induced error due to uneven shape of the measured muffin, I zeroed my dial while applying slight preload with my finger, so that after removing the main load and applying similar preload I could exclude the error from any play. I measured residual dent depth of 0.04mm.
So, HB=0.382353/0.04=9.558. This is range scrap and I happen to know it's ~99%Pb-1%Sb.
Even simpler method would be just to take a loose ball of known diameter, put on the sample, and rest a known weight on it, then measure the indentation depth with whatever accuracy you can, find the constant P/(Pi*D) and calculate C/h.
You see, Brinell hardness measurement is pretty simple afterall, and there are very many ways to do it.
Perhaps this can help someone to get a better feel on how BHN testers work, perhaps it inspires someone to build their own hardness tester and have confidence in it, perhaps someone can modify Lee tester to add a dial indicator and not need microscope anymore.
PS.
I didn't want to clutter already long post with this, but it should be mentioned. There is a reason why lab grade BHN testers are using optical readout. It may make you think that Lee's tester is therefore more accurate, but it is not true, it is a gimmick giving you false sense of accuracy, in actual use it is nothing but inconvenience. CarbineTree and Saeco is a living proof of that. But why and when do you need optical readout?
There is a big deal around material pile-up or sink-in that is still being researched. Mostly in context of steal hardness, it may matter somewhat for harder lead alloys. In nutshell, it is possible that pile-up forms around the indenter ball crater which increases the contact area and can cause overestimation of HB if not accounted for.
see Fig.1 http://www.sciencedirect.com/science/article/pii/S0020768305003203
The bad thing is that for alloys that exhibit some elastic rebound the residual indentation does not represent the actual contact shape during loading - no matter if you measure depth or diameter, you're still screwed.
For purely plastic materials, rebound doesn't occur, but the pile-up might. The optical measurement of indent diameter can account for that, for depth based measurements it needs to be accounted for by compensating factor. Thats probably the only reason why for unknown materials optical measuring of the diameter is more all-encompassing method. In actual measurement accuracy, optical method is much more involved than you might think. It is the main source of BHN measurement errors in the whole industry. But the industry is concerned mainly with BHN range 100-700 and different materials, so alot of what matters there doesn't for lead alloys.
But I did look for possible errors coming from pile-up around the indentation and ways to deal with that.
Literature often mentions that amount of pile-up has good correlation with material’s work-hardening exponent n. Many have attempted to find a formula to compute the pileup, often matching to empirical data.
Some examples of function f(n) that try to predict amount of pile-up or sink-in based on exponent n:
f(n) = -0.85768*n + 0.35768
f(n) = 5/2*((2-n)/(4+n))-1
f(n) = 1/2*((2+n)/2)^(2*(1-n)/n)-1
f(n) = 1.276-1.748n+2.451n2-1.469n3-1
Plotting them together gives an idea of the curves:
http://http://www.wolframalpha.com/input/?i=plot++5%2F2*%28%282-n%29%2F%284%2Bn%29%29-1+and+-0.85768*n+%2B+0.35768+and+1%2F2*%28%282%2Bn%29%2F2 %29^%282*%281-n%29%2Fn%29-1+and+1.276-1.748n%2B2.451n2-1.469n3-1+for+n+from+0+to+1
To compensate for pile-up you compute corrected hc = h*(1+f(n)). You might also measure the pileup directly.
Thus if there is a need, formula with correction would be: HB = P/(Pi*D*h*(f(n)+1))
f(n) is generally constant for material, so we can move it to the constant side and compensate for it while we pick P:
if P=Pi*D*(f(n)+1), then P/(Pi*D*(f(n)+1))=const=1, and we still have HB=1/h (mm)
The big trouble is finding this exponent n for your alloy. For Pb alloys I have seen figures 0.165 and 0.25 but finding a reliable source has been difficult. Maybe I simply don't know what to look for.
You find the compensation factor by plugging n into any formula above and simply calculate it, For eg.
f(n) for Pb is -0.85768*0.25 + 0.35768 = 0.14326, thus compensation factor is 1.14326
Note that different formulas above give quite notably different results, so you need to pick a formula with some reasoning. Anyway, it doesn't really matter, as the formula tries to be predictive, while in the end what you need is a constant that applies to your alloy. It can be found experimentally.
Load P selection would compensate for pile-up like so:
if D=3mm, then P=Pi*3*1.14326=10.7749716514kg (23.75 lb)
This would provide correction of measured indentation depth in mm so that Brinell hardness is 1/h for lead based alloys.
So it appears that depth based Brinell tester is also very easy to calibrate. Essentially, calibration boils down to finding the correction factor f(n) for the alloy, and it is applied by adjusting the main load weight. By measuring a reference material and adjusting the load weight so that measured HB is right, we can find the f(n) correction factor without ever even knowing it.
As a note, you may notice that Lee's tester did match calculations exactly without any compensation factor. So we can see that Lee's tester does not compensate for pileup/sinkin of alloys and it is your responsibility to measure the dent diameter so that you find exactly the actual contact diameter, even if it extends higher (or sinks lower) from the material surface.