Introduction to Fracture Mechanics phần 3 docx

Plane Crack Problem
Airy Stress Function
For the plane problem, the equations of equilibrium are
satisfied when the stress components are expressed by
the
Airy stress function through
x
,
,
11
22
xxx
rr










21
,
,
222
r
r
rr
rr











.
1














x
rr
r
Using these definitions for the stresses and Hooke’s

law, the strains can be expressed in terms of .
x
It can be shown that the compatibility equation, when
expressed in terms of the Airy stress function, satisfies
the biharmonic equation:
 
.
11
,0
2
2
22
2
222











rrrr
The boundary conditions for this plane crack problem
22
The boundary conditions for this plane crack problem
are: for

0





r
.




These conditions express the fact that the crack is
traction-free (no loads applied to crack face).
Note: there is no condition on .
rr

A choice of the Airy stress function for the present
crack problem should be such that
x has a singularity
at the crack tip, and is single-valued. We try a solution
of the form:




,,,
2

rqrpr 

23
Where p and p are harmonic functions of r and θ
(i.e. and
0
2
 p
).0
2
 q
Now consider a separate solution, of
the following form (Williams, 1957):




,




rR
,sincos
21


rArAp 







.2sin2cos
2
2
2
1




rBrBq
This form leads to the following expression for x:










2coscos
11
2
BArx







.
2
sin
sin
2







B
A
r
24






.
2
sin
sin
22

2







B
A
r
Note that we have expressed x as a symmetric part and
an anti-symmetric part. The symmetric part provides
the
Mode I solution while the anti-symmetric part
provides the
Mode II solution. We will derive the
Mode I solution here.
2
2
r

















2coscos12
11
 BAr














rx
rr
r
1
25











2sin2sin1
11
 BAr
Apply the boundary conditions:


,0cos
11



BA




.0sin2
11







BA
The admissible cases are: (i) cos λ π = 0,
hence
where Z is an integer including zero, and thus
B
1
= A
1
λ /(λ + 2) or (ii) sin λπ = 0 and hence λ = Z and
B
1
= A
1
. Since the governing equations are linear, any
linear combination of the admissible solutions provides
a solution, hence
λ
can have any satisfying:
2
12


Z



26

a solution, hence
λ
can have any satisfying:
,
2
Z


Where Z is a positive or negative integer, including zero.
Out of all the possible values of λ, how do we decide
the appropriate value of
λ?
The value of
λ cannot be found from any mathematical
argument. We need to use a physical, based on the total
strain energy around the crack tip. From the expressions
for the stresses, and Therefore the


r
~
.
~


r
27
for the stresses, and Therefore the
strain energy density is given by



r
ij
~
.
~


r
ij
.~
2
1
2


r
ijij

The total strain energy within an annular region,with
inner and outer radii
r
0
and R, respectively, centered at
the crack tip, with unit thickness is
 
.~
2
1
00

12
2
0
2
0



drdrrdrd
R
r
ijij
R
r



28
We assert that the strain energy should be bounded
(Φ < ∞) as r
0
0. Using this physical argument, we
see that λ > -1. (λ = 1 gives ). If λ < 1, the
strain energy will not be bounded.
0

ij

2




Thus the physically admissible values of λ are
,
2
)( ,,2,
2
3
,1,
2
1
,0,
2
1
Z
or


where Z is –1, 0, or a positive integer. Taking the most
dominant singular term (λ = 1/2 and thus B
1
=A
1
/3)
we find that:

29
we find that:
 
















2
5
2
1
2
3
2
3
cos
3
1
2
cos
rrAr



 





~
2/10
2
1
1


rrrA
ijij
I
ijij

The higher order terms, with exponents greater than
zero, vanish as r
 0. We write where
K
I
is the stress intensity factor.
Thus we have that:

2/
1 I
KA 
 

.
~
2
jxix
I
ij
I
ij
r
K




30
The first term is the leading singular term for linear elastic
mode I crack problems. is a function of
θ alone (no r
dependence). The second term, generally referred to as the
‘
T term’, is a non-singular term which can be important in
some situations involving fatigue. is the Kronecker
delta function.
I
ij

~
ij
