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CHAPTER 8:
FAILURE
Principles of Fracture Mechanics
8.1 What is the magnitude of the maximum stress that exists at the tip of an internal crack having a radius of
curvature of 1.9 × 10
4
mm (7.5 × 10
6
in.) and a crack length of 3.8 × 10
2
mm (1.5 × 10
3
in.) when a tensile stress of
140 MPa (20,000 psi) is applied?
Solution
Equation 8.1 is employed to solve this problemi.e.,
Va
l
ue
s
f
o
r
σ
0
(140 MPa), 2a (3.8 × 10
2
mm), and ρ
t
(1.9 × 10
4
mm) are provided in the problem statement.
The
r
e
f
o
r
e, we
s
o
l
ve
f
o
r
σ
m
a
s
f
o
ll
ow
s:
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8.2 Estimate the theoretical fracture strength of a brittle material if it is known that fracture occurs by the
propagation of an elliptically shaped surface crack of length 0.5 mm (0.02 in.) and a tip radius of curvature of 5 × 10
3
mm (2 × 10
4
in.), when a stress of 1035 MPa (150,000 psi) is applied.
Solution
I
n o
r
de
r
t
o e
sti
ma
t
e
t
he
t
heo
r
e
ti
ca
l
fr
ac
t
u
r
e
str
eng
t
h o
f
t
h
is
ma
t
e
ri
a
l
it
is
nece
ss
a
r
y
t
o ca
l
cu
l
a
t
e
σ
m
u
si
ng
Equation 8.1 given that σ
0
= 1035 MPa, a = 0.5 mm, and ρ
t
= 5 × 10
−3
mm. Thus,
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8.3 If the specific surface energy for aluminum oxide is 0.90 J/m
2
, then using data in Table 12.5,
compute the critical stress required for the propagation of an internal crack of length 0.40 mm.
Solution
We may determine the critical stress required for the propagation of an internal crack in aluminum oxide
using Equation 8.3. Taking the value of 393 GPa (Table 12.5) as the modulus of elasticity, and realizing that values for
γ
s
(0.90 J/m
2
) and 2a (0.40 mm) are given in the problem statement, leads to
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8.4 An MgO component must not fail when a tensile stress of 13.5 MPa (1960 psi) is applied. Determine
the maximum allowable surface crack length if the surface energy of MgO is 1.0 J/m
2
. Data found in Table 12.5
may
prove helpful.
Solution
The maximum allowable surface crack length for MgO may be determined using a rearranged form of
Equation 8.3. Taking 225 GPa as the modulus of elasticity (Table 12.5), and realizing that values of σ
c
(13.5 MPa)
and
γ
s
(
1.0
J/
m
2
)
a
r
e g
i
ven
i
n
t
he p
r
ob
l
em
st
a
t
emen
t
, we
s
o
l
ve
f
o
r
a
, a
s
f
o
ll
ow
s:
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