Fig. 5 Experimental pressure-load characteristic of indentations formed in work-hardened mild steel by
a hard spherical indenter. Yield stress of steel, Y = 77 kg/mm
2
. Ball diameter is 10 mm. The broken line is
the theoretical result for elastic deformation. OL, elastic region; LM, elastic-plastic region; and MN, fully
plastic region. Source: Ref 6
Fully Cold Worked versus Fully Annealed Metals. The above discussion has considered ideal plastic metals,
which, by definition, do not work harden during deformation and show a constant yield stress when the linear
strain is increased as in a tensile or compressive test. Metals that have been sufficiently cold worked would
behave in approximately this manner. However, fully annealed, and, in reality, most metals will have a
tendency to work harden during deformation and will be characterized by a continuously increasing yield stress
with increasing strain. The relation P
m
= CY (C ≈ 3), which applies to fully cold worked metals, also holds for
fully annealed metals provided Y denotes the yield stress corresponding to the strain produced during testing,
which is higher than the initial yield stress. The value of C has the same approximate value of 3 as for ideal
plastic metals. Also, P
m
is found to be a function of d/D for fully annealed metals, and therefore, geometrically
similar indentations would give identical hardness values, as for ideal plastic metals. It may be noted here that
the value of C ≈ 3 obtained through slip line analysis as already mentioned has also been recently confirmed
through finite element analysis (Ref 12, 13).
Tabor (Ref 6) showed that the strain, ε, in the plastic region of an indentation is proportional to the ratio d/D
and empirically determined the proportionality constant to be approximately 20 for many metals, thus arriving
at the relation, ε = 20d/D. Combining this equation with the relationship Y = bε
x
, where b and x are constants for
a given metal and x is the strain hardening coefficient, Tabor has shown that W = c
1
d
1
n
= c
2
d
2
n
= c
3
d
3
n
and so on
for indentations made with indenters of different diameters D
1
, D
2
, D
3
… This is Meyer's law, mentioned earlier
and first derived by Meyer empirically. This relation has been shown to hold fairly well for many materials.
The value of n in the Meyer equation is roughly related to the strain hardening coefficient x by the relation n = x
+ 2.
Conical and Pyramidal Indenters (Ref 6). Shortly after the introduction of Brinell hardness testing, Ludwik (Ref
14) proposed hardness testing using a conical indenter and defined the hardness as the mean pressure over the
surface of the indentation. Thus, for an indent with an included angle of 90°, the Ludwik hardness number is
given by:
HL = 4W /
(Eq 13)
where d is the diameter of the impression. This concept is similar to that of Brinell and therefore has no real
physical significance. The true pressure, P, between the indenter and the indentation is given by the ratio of
load to projected area, that is, 4W/πd
2
(similar to Meyer's concept for a spherical indenter), which means that
the Ludwik hardness number is 1/ times the mean yield pressure P. Experiments have shown that Ludwik
hardness is practically independent of the load for a given indenter, though it depends on the cone angle. It is
observed that the yield pressure increases as the cone semi-angle decreases, and the effect may be partially
explained as due to friction between the indenter and the indentation (Ref 15).
The diamond pyramidal indenter was first introduced by Smith and Sandland (Ref 16) and was later developed
by Vickers-Armstrong, Ltd. The indenter is in the form of a square pyramid with opposite faces making an
included angle of 136° with each other. The origin of this value of the angle is traced to the Brinell hardness
testing practice. It is customary in Brinell hardness testing to select loads so that the indentation diameter lies
between 0.25D and 0.5D where D is the indenter diameter. Thus, an average of the two diameters, 0.375D, was
chosen for the indentation diameter as shown in Fig. 6, which also shows the origin of the included angle 136°
of the Vickers indenter. The geometry of the indenter is such that projected area of the indentation is 0.927
times the area of the contact surface. Since Vickers hardness, HV, is defined as the load divided by the surface
area of the indentation, the yield pressure, P, is related to the Vickers hardness number by the relation HV =
0.927P.