Elasticity_ Barber

W = 1 2
and hence
∫ ∆S
0
σ yy (s)δ(s)ds = K2 I (κ + 1)
4πµ
∫ ∆S
0
13.4 Method of images 213
√
(∆S − s)
s
ds = K2 I ∆S(κ + 1)
8µ
(13.52)
G = − ∂U
∂S = ∂W
∂S = K2 I (κ + 1)
. (13.53)
8µ
A similar calculation can be performed for a crack loaded in shear, causing a
mode II stress intensity factor K II . The two deformation modes are orthogonal
to each other, so that the energy release rate for a crack loaded in both modes
I and II is given by
G = (K2 I + K2 II )(κ + 1)
. (13.54)
8µ
This expression can be written in terms of E, ν, using (13.18). We obtain
G = (K2 I + K2 II )
E
= (K2 I + K2 II )(1 − ν2 )
E
(plane stress) (13.55)
(plane strain). (13.56)
As long as the process zone is small in the sense defined in §13.3.1, a
component containing a crack loaded in tension (mode I) will fracture when
K I = K Ic or G = G c ,
where K Ic , G c are interrelated material properties, the former being known as
the fracture toughness. The critical energy release rate G c seems to imply a
single failure criterion under combined mode I and mode II loading, but this
is illusory, since experiments show that the value of G c varies with the mode
mixity ratio K II /K I .
13.4 Method of images
The method described in §13.3.2 applies strictly to the case of a crack in an
infinite body, but it will provide a reasonable approximation if the length of
the crack is small in comparison with the shortest distance to the boundary
of a finite body or to some other geometric feature such as another crack or
an interface to a different material. However, the same methodology could be
applied to other problems if we could obtain the solution for a dislocation
located at an arbitrary point in the uncracked body.
Closed-form solutions exist for bodies of a variety of shapes, including
the traction-free half plane, the infinite body with a traction-free circular
hole and the infinite body containing a circular inclusion of a different elastic