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the ultimate failure of the material to the applied stress. Up to this point the distributions
have been phenomenological in that the failure rates have been chosen to accurately
describe a given data set. Subsequently we show a dynamical basis for the Weibull
distribution as well as others.
It is worth studying the general case of the rate
]
η
g
(
t
)
=
r
0
[
1
+
r
1
t
(3.233)
in terms of three parameters. In the long-time limit
t
1
/
r
1
the general rate becomes
equivalent to the mortality risk (
3.230
) with
η
=
β
−
.
1
(3.234)
By expanding the general form of the rate (
3.233
) in a Taylor series,
)
η
1
η
1
r
1
t
)
η
,
g
(
t
)
=
r
0
(
r
1
t
+
≈
r
0
(
r
1
t
and comparing it with (
3.230
), we obtain for the lumped parameter
r
0
r
1
η
+
α
=
1
.
(3.235)
The border between where the Weibull distribution has a maximum at
t
>
0(for
η>
0
)
and the region where it diverges at the origin is at
η
=
0
.
The emergence of a maximum
at
t
.
The adoption of (
3.233
) serves the purpose of turning this divergent behavior into a
distribution density with a finite value at
t
>
0 is clearly depicted in Figure
3.9
, which illustrates some cases with
β>
1
=
0:
exp
1
[1
r
0
r
1
1
r
1
t
]
1
+
η
−
(
t
)
=
−
+
.
(3.236)
1
+
η
/β
, where the lumped parameter is
givenby(
3.235
) and from left to right
λ
is 0.5, 1.0, 1.5 and 3.0, respectively.
1
Figure 3.9.
The four curves depicted have values of
β
=
2and
λ
=
1
/α
