Introduction to Fracture Mechanics phần 5 docx
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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
K
6
(plane strain); σ
3
= 0 (plane stress).
2
cos
2
2
3
r
K
I
2
sin1
2
cos
2
2
r
K
I
Substitute in to the Mises yield condition:
Plane strain:
2
2
2
2
2cos121sin
2
3
2
ys
I
r
K
7
2
2
r
Plane stress:
22
2
2cossin
2
3
1
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
2
3
4
1
2
2
2
I
p
K
r
8
2
4
ys
p
Plane stress:
cossin
2
3
1
4
1
2
2
ys
I
p
K
r
Check: We note that the Plane Stress case reduces to
our first order estimate for
θ = 0.
Also note that (K
I
/ σ
ys
)
2
has dimensions of length.
Next we will compare the extent of the plastic zone
9
Next we will compare the extent of the plastic zone
in the two situations, plane stress and plane strain,
for two cases,
θ = 0 and θ = 45˚.
For θ = 0,
,
3
1
9
1
stressplaner
strainplaner
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
Engineering Formulae
For Plane Stress:
2
1
ys
I
p
K
r
For Plane Strain:
13
For Plane Strain:
2
3
1
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
What is the connection between ς and K?
