Introduction to Fracture Mechanics
1
Introduction to Fracture Mechanics
From Suresh: Fatigue of Materials
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INTRODUCTION
Importance of Fracture Mechanics :
 All real materials contain defects: understand
the influence of these defects on the strength of
the material.
Defect
-
tolerant design
philosophy.
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the material.
Defect
-
tolerant design
philosophy.
 Relevance for Fatigue: understand the initiation
and growth of fatigue cracks.
We will use two approaches, an energy-based
approach and a more rigorous mechanics approach.
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Key Idea : Griffith (1921) postulated that for unit
crack extension to occur under the influence of the
applied stress, the
decrease
in potential energy of
Griffith Fracture Theory

E

5
= E Plane stress
The surface energy of the crack system is
2
1
'
v
E
E


sS
aBW

4

Plane strain
where γ
S
is the free surface energy per unit surface area.
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The total system energy is then given by
.4
'
22
SSP
aB
E

'2
a
E
S
f




7
a

As the second derivative, d
2
U/da
2
is negative, the
above equilibrium condition gives rise to
unstable
crack propagation. This applies for brittle materials;
it must be modified for ductile materials such as metals.
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Orowan (1952) extended Griffith’s brittle fracture
concept to metals by simply adding a term representing
plastic energy dissipation. The resultant expression
for fracture initiation is
,
)('2E
p
s

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The total mechanical potential energy of a cracked
elastic body is given by the general expression
FP
wW



where is the stored elastic strain energy and is

F
w
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the work done by the external forces.

F
Irwin (1956) proposed an approach for the
characterization of the driving force for fracture in
cracked bodies, which is conceptually equivalent to
that of the Griffith model.
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Irwin introduced, for this purpose, the energy release
rate
G which is defined as
.


d
dW
P




We define the compliance C (inverse of the stiffness) of
G
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a cracked solid as C=u / F. It can be shown that
.
2
2
da
dC
B
F

Thus measurements of compliance as a function of crack
length allow the energy release rate to be evaluated.
G
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Modes of Fracture
The three basic modes of separation of the crack
surfaces (
modes of fracture) are depicted below:
13
Combinations of modes (mixed-mode loading) are
also possible.
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Modes of Fracture
Definitions
Mode I (tensile opening mode): The crack faces

–
y
and
x
–
z
planes.
The crack face displacements in modes II and III
find an analogy to the motion of edge dislocations
and screw dislocations, respectively.
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Plane Crack Problem
The preceding analysis considered fracture from an
energy standpoint. We now carry out a linear elastic
stress analysis of the cracked body, which will
allow us to formulate critical conditions for the
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allow us to formulate critical conditions for the
growth of flaws more precisely. An analysis of this
type falls within the field of
Linear Elastic
Fracture Mechanics
(LEFM).
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We consider a semi-infinite crack in an infinite plate
of an isotropic and homogeneous solid as shown below:
17
Our goal is to develop expressions for the stresses,
strains and displacements around the crack tip.
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
rrr
rr





where and are the polar coordinates as shown
previously.
r

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Plane Crack Problem
Strain-Displacement
The strain-displacement relations for polar coordinates
are:
.
1
1









u

2
1













r
u
r
u
u
r
r
r





The strain compatibility equation in polar coordinates is:
.0








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Plane Crack Problem
Hooke’s Law
Hooke’s Law (for plane stress, ):
0

zz

,
rr
E






,






G
.
2




rrr
G
G




,12
rr
G








.2



rr

r
r
rr
rr










.
1















rrrr
The boundary conditions for this plane crack problem
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The boundary conditions for this plane crack problem
are: for
0





r
.




These conditions express the fact that the crack is
traction-free (no loads applied to crack face).
Note: there is no condition on .
rr

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A choice of the Airy stress function for the present
crack problem should be such that
x has a singularity
at the crack tip, and is single-valued. We try a solution
of the form:





rArAp 






.2sin2cos
2
2
2
1




rBrBq
This form leads to the following expression for x:













.
2
sin
sin
22
2







B
A
r
Note that we have expressed x as a symmetric part and
an anti-symmetric part. The symmetric part provides
the
Mode I solution while the anti-symmetric part
provides the
Mode II solution. We will derive the
Mode I solution here.
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2
2
r




rx
rr
r
1
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









2sin2sin1
11
 BAr
Apply the boundary conditions:


,0cos
11



BA
