Definition of Brinell hardness I'm trying to understand the concept of Brinell scale. In Brinell test, a steel ball is pressed against the material under examination and the diameter of the indentation left by the ball is measured. Then the Brinell hardness is calculated as follows:
$$\operatorname{BHN}=\frac{2P}{\pi D \left(D-\sqrt{D^2-d^2}\right)}$$
where $P$ is the pressure, $D$ is the diameter of the ball and $d$ is the diameter of the indentation.
Now, this is a quite involved formula; I'm curious as to why it is defined this way. All sources I've read regarding this concept just state the formula, without much reasoning. So why is useful to define it using the above formula?
This also brings me to another question: don't the dimensions of the material sample affect the result? Does it not matter if we use samples of different thicknesses? Not that I have a particular reason to believe it does matter, but I'm just wondering.
 A: I haven't been able to find the details of how John Brinell developed his hardness test, but here is my guess, based on how I would proceed if given the task:


*

*He wanted an easily replicatable test.  An extremely hard sphere (originally hardened steel, but now tungsten carbide) pressed into the material being tested seemed to fit the bill.  Previously, the depth of impressions made in cylinders of a material pressed into each other under standardized conditions had been used as a measure of hardness.

*A series of experiments on samples of identical material using spheres of different sizes and constant pressure $Pc$ would produce a curve of data points relating depth d of the indentation to diameter of the sphere.  Curve fitting, perhaps guided by a bit of physical intuition, would produce an approximate relation :
$$D(D−√(D^2−d^2))= constant$$


*Data plotted from a further series of experiments on identical material using spheres of fixed size $Do$ but varying pressure $P$ turned out to show an approximately straight-line relationship between $P$ and $D(D−√(D^2−d^2))$.

*Another series of tests would be done to confirm that the relationships in #2 and #3 still hold (approximately) for materials known to have different hardness (e.g., steel from a single batch heat treated identically except tempered differently).
"Hardness" per se does not have an a priori meaning.  The formula used in calculating Brinell hardness is one of several empirical definitions of hardness. Other definitions of hardness such as the Vickers test and Rockwell hardness are closely related to but not exactly the same as Brinell hardness; they are not even quite proportional to each other.  
So the bottom line answer is: the formula used in the Brinell test summarizes a set of empirical relations, and gives reasonably consistent results over a useful range of hardnesses, ball sizes, and pressures.
