Introduction to Fracture Mechanics phần 2 pptx

Irwin introduced, for this purpose, the energy release
rate
G which is defined as
.


d
dW
P
W
P
, and thus
G
, can be evaluated for different loading
G
11
W
P
, and thus
G
, can be evaluated for different loading
conditions. This definition is valid for both linear and
nonlinear elastic deformation of the body.
G
is a function
of the load (or displacement) and crack length. It is
independent of the boundary conditions, in particular
whether the loading is fixed-displacement or fixed-load.
The Griffith criterion for fracture initiation in an ideally
brittle solid can be re-phrased in terms of
G such that

.2
'
2
S
a





We define the compliance C (inverse of the stiffness) of
G
12
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
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.
Modes of Fracture
Definitions
Mode I (tensile opening mode): The crack faces
separate in a direction normal to the plane of the crack.
The displacements are symmetric with respect to
the
x – z and x – y planes.
14
Mode II (in-plane sliding mode): The crack faces
are mutually sheared in a direction normal to the
crack front. The displacements are symmetric with
respect to the
x – y plane and anti-symmetric with
respect to the
x – z plane.
Modes of Fracture
Definitions
Mode III (tearing or anti-plane shear mode): The
crack faces are sheared parallel to the crack front.
The displacements are antisymmetric with respect
to the
x
–
y
and
x
–
z

planes.
15
to the
x
–
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.
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
16
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).
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.
Plane Crack Problem
Equilibrium Equations
The equilibrium equations (no body forces) are
,0
1








rrr
rrr
rr





18
,0
2
1







rrr
rr





where and are the polar coordinates as shown
previously.
r

Plane Crack Problem
Strain-Displacement
The strain-displacement relations for polar coordinates
are:
.
1
1









u

u
u
r



,
1
,










u
r
r
u
r
u
rr
rr
19
.
1

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

11112
2
2
22
2
2
2



















rrrrrrrrr
rrrr
rr


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

rrrr
E
.
2



G
G


20
For the case of plane strain ( ):
0

zz



,12








rrrr
G

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

