Biomedical Engineering Reference
In-Depth Information
4.3
Crack tip separation modes: I, opening mode; II, sliding mode; III,
tearing mode.
2. Cracks have infinitely sharp tips: cracks are considered to be flat (2D)
structures with ideally sharp tips, i.e. tip radius of curvature r
0.
3. Crack borders are free of traction forces: however, in reality traction
forces exist in many cases.
→
According to linear elastic fracture mechanics, the stress near the crack tip
is:
K
s
r
y
¼
1
=
2
f
ðyÞ
½
4
:
3
ð
2
p
r
Þ
where r and
are polar coordinates and K is a constant called the stress
intensity factor. By using elasticity theory it can be shown that:
θ
1
=
2
K
¼
Y
sðp
a
Þ
½
4
:
4
where
is the applied stress, a is half the crack length and Y is a constant
that depends on the crack opening mode and the geometry of the specimen
(the so-called geometric factor). The geometric factor Y is for small notch-
like or semicircular cracks, approximately 1.12. For a circular crack
embedded in a homogeneous uniaxial tensile stress field, it is 2/
σ
.
The three modes of crack tip separation are shown in Fig. 4.3. Generally,
the most critical and most important is mode I, known as the opening mode.
The stress intensity factor for this situation is denoted K
I
(notations of K
II
and K
III
are used for other modes). The critical value of K
I
, at which the
crack begins to propagate, is denoted as K
IC
. This value is an important
material parameter and is known as the fracture toughness.
When the value of the fracture toughness K
IC
(using equation 4.4) is
known, the critical strength
π
σ
f
of a material with certain types of defects
(with size of a
c
) or critical crack length (critical size of defect) can be
