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is not the time but the size of the crack. The probability that a crack in a material occurs
in the interval (
t
.
The WD was subsequently developed as the pdf for the minimum extreme values
among measurements modeled by an initial distribution bounded by a smallest value.
In material science, crack initiation dominates the phenomenon of fracture in brittle
solids, which initiated the development of models in which defects, point sources of
stress concentration, are statistically distributed in a unit volume of a solid. As Bury
[
3
] points out, each of these defects determines the local strength of the material and
consequently the overall strength of the material is equal to the minimum local strength.
A chain is no stronger than its weakest link. The asymptotic model of specimen strength
is the WD and is given by [
3
]
,
t
+
dt
)isgivenby
ψ
W
(
t
)
dt
exp
β
s
σ
F
W
(
s
)
=
1
−
W
(
s
)
=
1
−
−
,
(5.69)
where
s
is the stress,
β
is a dimensionless shape parameter and
σ
is a scale parameter
with the dimensions of stress.
The modern statistical modeling of the strength of materials is based on the WD, but
it had not previously been linked to the fractional calculus and the asymptotic form of
the MLF.
5.3
Fractional stochastic equations
5.3.1
The fractional Langevin equation
We now generalize the fractional differential equation to include a random force
ξ(
t
)
and in this way obtain a fractional Langevin equation
t
−
α
0
D
t
[
)
]+
λ
α
Q
Q
(
t
(
t
)
=
Q
(
0
)
+
ξ(
t
).
(5.70)
(
1
−
α)
Here we have introduced two distinct kinds of complexity into the dynamical web. The
fractional relaxation takes into account the complicated webbing of the interactions
within the process of interest and the random force intended to represent how this com-
plex web interacts with its environment. Note that on the face of it this approach to
modeling complexity stacks one kind of complexity on another without attempting to
find a common cause. It remains to be seen whether such a common cause exists.
The solution to the fractional Langevin equation is obtained in the same way as the
simpler deterministic fractional dynamical equation, by using Laplace transforms. The
transformed equation is
u
α
λ
α
+
ξ(
Q
(
0
)
u
)
Q
(
s
)
=
u
α
+
u
α
,
(5.71)
λ
α
+
u
where we need to know the statistical properties of the random force in order to properly
interpret the inverted Laplace transform. Note the difference between the
u
-dependences
of the two coefficients of the rhs of (
5.71
). The inverse Laplace transform of the first
