1005
Technical Committee 105 /
Comité technique 105
particles
d
i
, and
i
*
is the maximum volume that the particles
d
i
can fill without any alteration of the mixture. Therefore the
relationship
i
/
i
*
describes the proportion of voids that is
filled by particles
i
with respect to the available void space for
these particles (Figure 2).
Figure 2. Schematic drawing of the relationship
i
/
i
*.
Regarding stresses, a relationship
i
/
i
*
close to 1 indicates
that particles
i
are filling well the void space reserved for those
particles and therefore it is highly probably that the stress chains
within the granular material flow along these particles. On the
other hand, a relationship
i
/
i
*
close to zero indicates that
particles
i
are in a loose state within the granular material and
therefore doesn’t support significant level of stress.
Regarding crushing, the relationship
i
/
i
*
take action with
two opposite trends: (i) when
i
/
i
*
is close to 1 particles take
high level of stresses that increase the probability of crushing
but these particles are well confined by other particles that
reduces the probability of crushing; on the other hand when
i
/
i
*
is close to zero, particles are with low level of stress but
have a reduced number of contact points with other particles.
These two opposite trends suggest that there is a particular value
of
i
/
i
*
for which the probability of crushing is maximum,
this can be set as the compacity relationship for maximum
crushing (
i
/
i
*
)
mc
.
Figure 3. Probability of crushing of particles depending on the
relationship
i
/
i
*.
A function that can describe the probability of crushing of
particles depending the on the relationship
i
/
i
*
is presented
in equation 14 (Ocampo 2009), Figure 3:
*
*
1
4 ) (
i
i
i
i
f
i
p
(14)
Parameter
is directly related with the compacity
relationship for maximum crushing (
i
/
i
*
)
mc
as follows:
mc i
i
)
ln( )2 ln(
*
(15)
2.3 Probability of crushing of particles depending on its
strength
The effect of the particles size on the strength of particles can be
captured using the Weibull relationship. Furthermore as the
model presented in this paper focuses on cyclic loading, the
strength of the particles depends on the number of loading
cycles through a Wohler law. Equation 16 represent the
crushing probability of any particle that is the result of including
the Wohler law into the Weibull relationship:
m b
i
b
i
f
c
f
i
dN
d p
)
(
exp 1 ) (
1
(16)
Where
m
is the Weibull Law parameter,
is the applied
stress,
1
is the strength of a particle having unit diameter and
for one loading cycle,
N
is the number of cycles,
b
f
is the slope
of the Wohler fatigue law, and
b
c
is the fracture parameter
proposed by Lee (1992).
2.4 Combined probability of crushing
The crushing probabilities depending on the compacity and on
the strength correspond to independent process; as a result the
combined probability of crushing can be assessed as the product
of both probabilities:
) (
) (
i
f
f
f
dp
p p
i
(17)
2.5 Grain size distribution of crushed particles
When original particles break, they produce a set of smaller sub
particles modifying the grain size distribution of the mixture.
Afterwards, for another loading cycle, these sub particles can
break again. This process is repetitive for different loading
cycles and can be described by a Markov process. In such type
of process an initial particle having
d
i
size (state 1) breaks and
produce a set of particles having sizes
d
j
<
d
i
(states 2 to 11), this
process is described in Figure 4.
Figure 4. Diagram of state transition in a Markovian process.
In a Markovian process the transition between states is
described by the Transition probability Matrix
as follows:
nn
ni
ii
n
p
p p
p
p
,
,
,
,1
1,1
0 0
0
.
(18)
In this matrix components
p
ii
are the probability of no failure
of particle
i
that is obtained from equation 17 as
p
ii
=1-
p
fi
.
Elements
p
ij
for
j
<
i
represent the transition probability of a
particle having an initial size
i
to a size
j
. Transition
probabilities of particles during crushing was studied
experimentally by Ocampo 2009 using particles with different
colour for each initial size. Figure 5 represent the fitting of the
experimental results using a beta function.
