Substitute the known expression for σ
xx
, σ
yy
and σ
xy
in the Mode I crack problem (derived last time) and
find that:







2
sin1
2
cos
2
1



r
K
I









r
K
I
Substitute in to the Mises yield condition:
Plane strain:
   
2
2
2
2
2cos121sin
2
3
2
ys
I
r
K












These expressions can be used to solve for the radius
of the plastic zone r
p
as a function of θ:
Plane strain:
     




















cos121sin


















cossin
2
3
1
4
1
2
2
ys
I
p
K
r

p
p
For θ = 45˚,
,
3
1


10


 
8.2
1
381.0

stressplaner
strainplaner
p
p
3
Extent of the plastic zone is significantly larger for the
plane stress case.
Plastic Zone Shape
Plane stress/plane strain
11
Plastic Zone Shape
Plane stress/plane strain
12
Plastic Zone Size





ys
I
p
K
r

Similar analyses can be done to determine the plastic
zone size and shape for Mode II and Mode III loading.
Specimen Thickness Effects
Plane stress/plane strain
14
Thickness B
Meaning of ς
Recall the Strain Energy Release Rate ς .
What does it physically represent? It is the rate of
decrease of the total potential energy with respect to
crack length (per unit thickness of crack front),i.e.
15


a
PE



