Chemistry Reference
In-Depth Information
The equations cited above are for an ideal semi-infinite plate, with no bound-
ary effects. Application to real specimens requires calibration factors, so that the
fracture toughness of
Eq. (4-81)
, at the critical point is given by:
1
=
2
K
c
5σ
f
Γ½πa
(4-82)
where
(gamma) is a calibration factor which is itself a function of specimen
geometry and crack size. The
Γ
values have been tabulated (mainly for metals)
for a variety of shapes
[17]
. The independent variable in
Eq. (4-82)
is the crack
depth,
a
. To measure
K
c
, sharp cracks of various known depths are made in spe-
cimens with fixed geometry and plots of
Γ
are linear with slope
K
c
. Instrumented impact tests yield values for the specimen fracture energy,
U
f
.
With such data, the critical strain energy release rate can be calculated accord-
ing to
[18]
:
a
versus
σ
f
Γ
O
U
f
BDφ
G
c
5
(4-83)
where
B
and
D
are the specimen depth and width, respectively, and the calibration
factor
φ
is a function of the specimen geometry and the ratio of the crack depth
and specimen width. Here again, the independent variable is the crack depth,
a
,
as manifested in parameter
φ
.
The
total work of
crack formation equals
G
c
3
the
crack area.
πa
]
1/2
[
YG
c
]
1/2
, or when
Catastrophic failure is predicted to occur when
σ
[
5
[
YG
c
]
1/2
.
K
c
and
G
c
are the parameters used in
linear elastic fracture
mechanics
(LEFM). Both factors are implicitly defined to this point for plane
stress conditions. To understand the term
plane stress
, imagine that the applied
stress is resolved into three components along Cartesian coordinates; plane
stress occurs when one component is equal to 0 (the stress in the direction nor-
mal to the plane of the specimen). Such conditions are most likely to occur
when the specimen is thin.
This reference to specimen thickness leads to a consideration of the question
of why a polymer that is able to yield will be less brittle in thin than in thicker
sections. Polycarbonate is an example of such behavior. Recall that yielding
occurs at constant volume (tensile specimens neck down on extension). In thin
objects the surfaces are load-free and can be drawn inward as a yield zone grows
ahead of a crack tip. In a thick specimen the material surrounding the yield zone
is at a lower stress than that in the crack region. It is not free to be drawn into the
yield zone and acts as a restraint on plastic flow of the region near the crack tip.
As a consequence, fracture occurs with a lower level of energy absorption in a
thick specimen. The crack tip in a thin specimen will be in a state of plane stress
while the corresponding condition in a thick specimen will be plane strain. Plane
strain is the more dangerous condition.
The parameters that apply to plane strain fracture are
G
Ic
and
K
Ic
, where the
subscript I indicates that
K
c
5
the crack opening is due to tensile forces.
K
Ic
is
