Biomedical Engineering Reference
In-Depth Information
methods, for instance in the case of results from three-point and four-point
bending tests.
Good structural ceramic materials should have high parameters of the
Weibull distribution: m is of particular interest as a high value of m means
high reproducibility of the strength values and high reliability of the
material (the strength distribution if narrow). Traditional ceramics have m
values of around 5. In high-quality advanced structural ceramics, m is
typically between 10 and 15. Making the defect size distribution narrow and/
or ensuring some flaw tolerance in the material are means of improving the
Weibull modulus m.
Without additional information, the Weibull distribution (equation 4.11) is
difficult to use because it contains many unknown variables. In practice, when
testing normalised samples with constant volume, and assuming that
σ u is
negligible with respect to characteristic strength (which for most ceramics is
possible), the so-called two-parameter Weibull distribution may be used:
m
s
s 0
P f ¼
1
exp
½
4
:
13
This is possible to evaluate and therefore is frequently used. Figure 4.6 shows
examples of two different two-parameter Weibull distributions for ceramic
materials, which illustrate the effect of
σ 0 and m on the shape of probability
function. The first derivative of P f with respect to strength,
, gives the failure
probability density (Fig. 4.6(b)) and describes the frequency of the strength
values.
The two-parameter Weibull distribution (equation 4.13) can be linearised
as:
σ
1
¼
m ln s
m ln s 0
½
:
ln ln
4
14
1
P f
￿ ￿ ￿ ￿ ￿ ￿
and in this form its results are typically plotted as shown in Fig. 4.7.
Correct Weibull analysis is important in characterising the strength of
ceramics. International standards (e.g. EN 843-5: 2006) require the testing of
sets of at least 30 specimens in order to calculate the Weibull parameters
with sufficient confidence.
4.3
Fracture origins
As already mentioned, fractures occur when the local stress at particular
defects reaches a critical value. The defect becomes unstable and a crack
begins to grow. In the absence of plastic deformation, this process does not
stop and leads to catastrophic failure. The most important feature in this
respect is therefore the critical flaw at the place where the fracture begins,
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