s
failure
¼ s
0
e
bp
(1.6)
where, s
0,
is the strength found by extrapolating the date to zero porosity, and, b, is a
constant determined from the experimental data plotted as in, s
failure,
versus poros-
ity, p. The value of, b, has been found to vary considerably for the same ceramic
material depending on the shape, size and distribution of the porosity [139, 140]. The
relationship is known as the Ryshkewitch-Duckworth [139–142] equation and was
shown by Knud sen [143, 144] to be based on the increased average stress caused by
the reduction in load-bearing area resulting from the porosity.
However, Knudsen did not consider factors that control strength at zero
porosity. Carniglia [138] considered that the flaw should be enclosed within
a volume l, i.e., the Saint Venant volume, such that at the periphery of this
volume the stress re-distributing effect of the flaw was negligible. This volume
was then considered in relation to the spacing, L, between general porosity. Carni-
glia showed that the Ryshkewitch-Duckworth equation was applicable only when
l L. This is, when the Griffith flaw that initiates fracture is larger than the pores
which form the general porosity, and when the spacing between pores is small
compared to the size of the Griffith flaw.
Under these conditions, the average flaw stress acting on the Griffith flaw
has increased by the reduction in load-bearing area and can be considered as
uniform. When l L, a local stress model, such as the Griffith model for an
elliptical crack, can be used. That is the stress magnification produced by a single
flaw, with no interference from other flaws that cause failure. The case when l < L
cannot be treated theoretically in a general manner, because each Griffith flaw
is c lose to anot her stress re-distributing fl aw and therefore the average stress
around each flaw is highly variable. In practice, l L would occur quite fre-
quently f or ceramics. Most cer amics contain a number of smaller pores and
frequently more serious Griffith flaws. If the samples tested have constant Griffith
fracture initiating flaws and the general porosity is variable, t hen the strength
data should fit the Ryshkewitch-Duckworth e quation. Carniglia [138] developed
a more complex equation that showed the Ryshkewitch-Duckworth equation to
be an approximation. If the general porosity remains constant but the size of
the Griffith flaws is variable, then the fracture strength values, s
failure
, should
yield a straight line when plotted against
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ðg
1
= CÞ
p
. This approach should w ork
because, g
1,
and, E, are determined for the material containing the same general
porosity. Flaws can be classified as gross macroscopic, microscopic and sub-
microscopic [145 ]. This means this classification encompasses size only. A flaw
is considered to be gross if it is readily visible to the unaided eye, so that the origin
of failure can be viewed, i.e., large surface cracks, inclusions at the surface, etc.
A microscopic flaw is not easily identifiable and is usually a small crack, void or
a small inclusion. A sub-microscopic flaw is identified using a scanning electron
microscope. Flaws in ceramics result from several causes, i.e. pores might be
caused by differential firing shrinkage on a small scale of size, burn-out of organic
matter (dextrin and fillers), gaseous evolution caused by a reaction on firing,
diffusion of gases or some other mechanism, etc. Differential shrinkage on a
56 M.J. Jackson and M.P. Hitchiner