relation betweenGand the change ofCwill be derived for a plate with an edge crack, whichis loaded by a forceFin pointP. When the crack length increases fromatoa+da, twoextreme situations can be considered :•fixed grips: pointPis not allowed to move (u= 0),•constant load: forceFis kept constant.PFuFua+daaPFig. 4.4 :Edge crack in a plate loaded by a forceF.

28
a
a
a
+
da
a
+
da
F
F
u
u
dU
i
dU
i
dU
e
Fig. 4.5 :
Internal and external energy for fixed grips (left) and constant load (right).
4.4.1
Fixed grips
Using the fixed grips approach, it is obvious that
dU
e
= 0. The force
F
will decrease (
dF <
0)
upon crack growth and the change of the internally stored elastic energy can be expressed in
the displacement
u
and the change
dF
. The energy release rate can be calculated according
to its definition.
fixed grips :
dU
e
= 0
dU
i
=
U
i
(
a
+
da
)
−
U
i
(
a
)
(
<
0)
=
1
2
(
F
+
dF
)
u
−
1
2
Fu
=
1
2
udF
Griffith’s energy balance
G
=
−
1
2
B
u
dF
da
=
1
2
B
u
2
C
2
dC
da
=
1
2
B
F
2
dC
da
4.4.2
Constant load
With the constant load approach, the load will supply external work, when the crack propa-
gates and the point
P
moves over a distance
du
. Also the elastic energy will diminish and this
change can be expressed in
F
and
du
. The energy release rate can be calculated according to
its definition.
constant load
dU
e
=
U
e
(
a
+
da
)
−
U
e
(
a
) =
Fdu
dU
i
=
U
i
(
a
+
da
)
−
U
i
(
a
)
(
>
0)
=
1
2
F
(
u
+
du
)
−
1
2
Fu
=
1
2
Fdu

30
a
F
2
h
u
B
F
u
Fig. 4.7 :
Beam with a central crack loaded in Mode I.
u
=
Fa
3
3
EI
=
4
Fa
3
EBh
3
→
C
=
Δu
F
=
2
u
F
=
8
a
3
EBh
3
→
dC
da
=
24
a
2
EBh
3
→
G
=
1
B
bracketleftbigg
1
2
F
2
dC
da
bracketrightbigg
=
12
F
2
a
2
EB
2
h
3
[J m
−
2
]
G
c
= 2
γ
→
F
c
=
B
a
radicalBig
1
6
γEh
3
The question arises for which beam geometry, the energy release rate will be constant, so no
function of the crack length.
It is assumed that the height of the beam is an exponential
function of
a
. Applying again formulas from linear elastic beam theory, it can be derived for
which shape
G
is no function of
a
.
a
h
Fig. 4.8 :
Cracked beam with variable cross-section.
C
=
Δu
F
=
2
u
F
=
8

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- Fall '19
- Fracture mechanics