dissertation global and local fracture properties of metal matrix ...

Section 7
1) The Poisson distribution which corresponds to the homogeneous particle distribution
[83]
r
µ
P(
r)
= exp( −µ
) , (7.9)
r!
where P(r) is the probability, µ the mean value of the number of particles per quadrat, and r a
variable (the number of particles per quadrat);
2) The negative binomial distribution which corresponds to the clustered particle
distribution [83]
85
r
⎛ ( k + r −1)!
⎞⎛
p ⎞ ⎛ 1 ⎞
P( r)
= ⎜ ⎟⎜
⎟ ⎜ ⎟ , (7.10)
⎝ ( k −1)!
r!
⎠⎝1
+ p ⎠ ⎝1
+ p ⎠
where k and p are parameters determined according to [83].
7.4. The effect of ECAP on the particle distribution, the global and local fracture
properties in MMCs with severe clustered particle distribution
7.4.1. The effect of ECAP on the microstructure
The uniformity of the particle distribution increases with the number of ECAP passes (Fig.
7.4). For the as-fabricated MMC, clusters elongated in the extrusion direction are observed.
The clusters have a size of up to 40 µm in the direction perpendicular to the extrusion
direction. Large particle free zones, having a size of up to 200 µm in the extrusion direction
and up to 40 µm in the perpendicular direction, are located between particle clusters (Fig.
7.4a). From Fig. 7.5a, it is seen that the histogram corresponding to the particle distribution
for the as-fabricated material seems to follow the negative binomial distribution indicating a
clustering. After 4 ECAP passes, particle free zones become smaller and the particle clusters
start to scatter (Fig. 7.4b). For this material condition, a deviation from the negative binomial
distribution to the Poisson distribution can be noted in the histogram (Fig. 7.5b). Almost
complete declustering and an absence of particle free zones are observed for the MMC after 7
ECAP passes (Fig. 7.4c). The histogram of the particle distribution follows the Poisson
distribution confirming the particle declustering, as well (Fig. 7.5c). Thus, these observations
prove the validity of proposed Eq. 7.8.
k