Materials for engineering, 3rd Edition - (Malestrom)

2.4 Statistics of brittle fracture
Determination of mechanical properties 45
Since brittle solids fail by the propagation of pre-existing cracks under tension,
measurement of their tensile strength will be subject to considerable
experimental scatter. Ceramics and glasses contain a distribution of crack
lengths, so it follows that there will be a statistical variation in their measured
strengths. Furthermore, a large component will fail at a lower stress than a
small one, since it is more likely that it will contain one of the larger cracks
– so there is a volume dependence of the strength. Statistical models of
strength are therefore needed in order to give a true prediction of the mechanical
properties of brittle solids.
Weibull has defined the survival probability, P s (V), as the fraction of
identical samples (each of volume V) which survive loading to a tensile
stress σ, proposing the relation:
⎧
P V
V
S ( ) = exp –
⎛ σ ⎞
⎨
Vo
⎝ σ
⎩
o ⎠
m
⎫
⎬
⎭
[2.9]
where σ o ,V o and m are constants, m being known as the Weibull modulus.
The lower the value of m, the greater the variability of strength. The values
of these constants are found by experiment: if the stress varies with position
(as in the case of a bend test), then equation [2.9] can be integrated over the
volume to give the appropriate design load.
2.5 Hardness testing
Hardness is not a well-defined property of materials and the tests employed
assess differing combinations of the elastic, yielding and work-hardening
characteristics. All the tests are essentially simple and rapid to carry out and
are virtually non-destructive, so they are well-suited as a means of quality
control. The hardness of materials has been assessed by a wide variety of
tests, but we will confine ourselves to discussing two types of measurement
– the resistance to indentation and the height of rebound of a ball or hammer
dropped from a given distance.
2.5.1 Indentation hardness tests
There are two types of indentation hardness test. The first type (Brinell and
Vickers) measures the size of the impression left by an indenter of prescribed
geometry under a known load whereas the second type (Rockwell) measures
the depth of penetration of an indenter under specified conditions.