Environmental Engineering Reference
In-Depth Information
formulations by either the universal slopes equation (Equation [1.12a]) or
the characteristic slopes equation (Equation [1.12b]):
Δ
2
S
012
0 6
6
[1.12a ]
35
u
N
−
012
012
2
0 6
0 6
N
=
3.
012
12
+
0
0
.
ε
ε
ε
5
+
ε
,
ε
E
f
Δ
ε
σ
f
b
,
[1.12b ]
()
()
c
2
N
2
2
N
=
ε
f
ε
f
2
E
where
S
u
is ultimate tensile strength,
f
true frac-
ture stress, and
b
and
c
are material constants. In terms of the characteristic
slopes (Equation [1.12b]) the value of fatigue life at which the transition
from low cycle (plastic) to high cycle (elastic) occurs is given by
ε
f
is true fracture strain,
σ
1
⎛
⎜
⎞
⎟
E
bc
ε
f
2
.
[1.12c ]
N
⎛
=
tr
⎝
σ
f
Fatigue crack growth rate (FCGR, d
a
/d
N
) is determined by measuring the
extension of a pre-crack using visual, potential drop, unloading compliance
or other techniques over the elapsed number of load cycles from stress con-
trol tests conducted on either compact tension (CT) or three-point bend
specimens and is related to the range of stress intensity factor (
K
). Typical
crack extension curves at two different starting stress ranges (
Δ
) versus
number of cycles are shown in Fig. 1.8a and the slopes of the curve yields
d
a
/d
N
. The plot on logarithmic scale of (d
a
/d
N
) versus
Δ
σ
K
( Fig. 1.8b ) clearly
reveals three stages. Stage I is associated with crack blunting with very little
crack growth, while crack growth in stage II can be related using Paris' law:
Δ
d
d
a
,
[1.13 ]
()
p
A
K
=
N
where
p
is the Paris parameter/constant with values ranging from 2 to 4;
this covers the majority of the crack growth event before entering the fi nal
stage (stage III) where plastic fracture occurs as crack length reaches a crit-
ical value (
a
f
) corresponding to the plane strain fracture toughness (
K
IC
)
value:
2
K
IC
a
=
.
[1.14 ]
f
22
πσ
Y
2
max
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