# Meyer's law

Meyer's law is an empirical relation between the size of a hardness test indentation and the load required to leave the indentation. The formula was devised by Prof. Eugene Meyer of the Materials Testing Laboratory at the Imperial School of Technology, Charlottenburg, Germany, circa 1908.

## Equation

It takes the form:

$P\,=\,kd^{n}$

where

• P = pressure in megapascals
• k = resistance of the material to initial penetration
• n = Meyer's index, a measure of the effect of the deformation on the hardness of the material
• d = chordal diameter (diameter of the indentation)

n usually lies between the values of 2, for fully strain hardened materials, and 2.5, for fully annealed materials. It is roughly related to the strain hardening coefficient in the equation for the true stress-true strain curve by adding 2. Note, however, that below approximately d = 0.5 mm (0.020 in) the value of n can surpass 3. Because of this Meyer's law is often restricted to values of d greater than 0.5 mm up to the diameter of the indenter.

The variables k and n are also dependent on the size of the indenter. Despite this, it has been found that the values can be related using the equation:

$P=k_{1}d_{1}^{n_{1}}=k_{2}d_{2}^{n_{2}}=k_{3}d_{3}^{n_{3}}=...$

Meyer's law is often used to relate hardness values based on the fact that if the weight is halved and the diameter of the indenter is quartered. For instance, the hardness value for a test load of 3000 kg and a 10 mm indenter is the same for a test load of 750 kg and a 5 mm diameter indenter. This relationship isn't perfect, but its percent error is relatively small.

A modified form of this equation was put forth by Onitsch:

$P\,=\,1.854kd^{n-2}$