Environmental Engineering Reference
In-Depth Information
Crack Propagation Model
In the fracture-mechanics method, the stress parameter that governs the rate at which
fatigue cracks grow is the
stress intensity factor
, defined as follows for tensile loading:
K
=
S Q
p
a
(12-14)
where
K =
stress intensity factor (psi-in
0.5
)
S =
tensile stress normal to the crack (psi)
Q =
crack shape factor
a =
crack radius for a circular crack or length of the semi-minor axis of an el-
liptical crack (in)
A stress cycle that is completely in compression (
i.e., S
max
< 0) is usually assumed to
cause neither crack growth nor retardation. Referring to Figure 12-1, this explains why fa-
tigue is usually not the design driver on the downwind surfaces of HAWT rotor blades, where
the steady thrust loads cause compressive stress. Values of the crack shape factor in Equation
12-14 for some common crack configurations are as follows:
-- Through-crack in an infinitely-wide plate
Q
= 1.00
-- Internal circular or “penny-shaped” crack
Q
= 4/p
2
= 0.405
-- Surface semi-circular crack
Q
» 4.8/p
2
= 0.486
-- Internal elliptical crack
Q
» 1/f
2
-- Surface semi-elliptical crack
Q
» 1.2/f
2
p /2
1 - [1 - (
a
/
c
)
2
]sin
2
f
=
q
d
q
(12-15a)
0
where
c
= length of the semi-major axis of the crack (in)
Evaluating j for a range of ratios
a/c
from 0 (
i.e.
, a linear crack) to 1.0 (a circular crack),
we can derive the following convenient empirical equation for a surface crack:
Q
» 1.200 - 0.714(
a
/
c
)
0 £
a
/
c
£ 1.0
(12-15b)
Fatigue crack propagation models have been developed that are based on either the max-
imum stress intensity in a given layer in the spectrum,
K
max
, or the stress-intensity range, D
K
=
K
max
-
K
min
, or both. The following three crack growth models, based on
K
max
and the ratio
R
f
of minimum-to-maximum stress, were considered during the development of structural
design methods for the Mod-2 HAWT:
(a) A
retardation model
, which accounts for any beneficial effects of load excursions during
and between load spectra. Excursions are known to cause local plastic “wakes” that close
crack tips and slow crack growth. This model, which considers all cycles to be damaging,
has the general mathematical form
m
S
max
R
OL
S
maxmax
da
/
dn
µ (1 -
R
f
)
k
K
l
max
S
max
³ 0;
R
OL
³ 1.0
(12-16a)
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