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

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

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





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

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

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

