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Chapter 9
Fracture mechanics:
Show figures 9.1 through 9.6.
Materials that would fail by ductile failure are obviously preferable for most
engineering applications, but this does not occur for ceramics.
Stress concentration:
Statics analysis shows that when microscopic cracks occur in a solid material, the stress at the crack tip is
concentrated, or in other words magnified above the level of the applied stress.
Show figure 9.7.
The
maximum stress occurs at the crack tip according to:
⎥
⎥
⎦
⎤
⎢
⎢
⎣
⎡
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
+
=
2
/
1
0
2
1
t
m
a
ρ
σσ
Here
σ
m
is the maximum stress at the crack tip,
σ
0
is the nominal applied tensile stress,
ρ
t
is the radius of
curvature at the crack tip, and a is the length of a surface crack, or half the length of an internal crack.
For a
long enough crack, this reduces to:
2
/
1
0
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
c
m
a
The ratio
σ
m
/
σ
0
is often referred to as the stress concentration factor, K
t
2
/
1
2
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
c
t
a
K
For a variety of other geometries, the stress concentration factor is given in figure 9.8.
Omit Griffith theory of brittle fracture, stress analysis of cracks, and problem 9.7.
Fracture toughness:
Due to the disastrous nature of fracture, much effort has been expended to understand fracture mechanics.
From a combination of fundamental and empirical reasons, brittle fracture will occur when the fracture
toughness (K
c
) of a material is exceeded, where
( )
a
w
a
Y
K
c
C
πσ
/
=
Here Y(a/w) is a geometrical factor that depends on the crack dimensions, where a is the crack length and w is
the specimen thickness;
σ
c
is the critical stress for crack propagation, and subsequent failure; and a is again the
length of a surface crack of half the length of an internal crack.
If a
→
0 or w
→∞
, then Y
→
1, ,but only for the
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View Full Document geometry of figure 9.11.
As the sample thickness is increased, the fracture toughness declines, until the plane
strain region is obtained, as illustrated in figure 9.14.
The fracture toughness (K
IC
) is the critical value of the stress intensity factor at a crack tip needed to produce
catastrophic failure under simple uniaxial loading.
The subscript I stands for Mode I loading (unixial),
illustrated in figure 9.9a while the subscript C stands for critical.
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This note was uploaded on 04/23/2008 for the course ES 260 taught by Professor Rasmussen during the Spring '08 term at Clarkson University .
 Spring '08
 Rasmussen

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