Elastic-Plastic Fracture Mechanics
Introduction
•
When does one need to use LEFM and EPFM?
•
What is the concept of small-scale and large-scale yielding?
Background Knowledge
•
Theory of Plasticity (Yield criteria, Hardening rules)
•
Concept of K, G and K-dominated regions
•
Plastic zone size due to Irwin and Dugdal
Contents of this Chapter
•
The basics of the two criteria used in EPFM: COD (CTOD), and J-Integral (with H-R-R)
•
Concept of K- and J-dominated regions, plastic zones
•
Measurement methods of COD and J-integral
•
Effect of Geometry
LEFM and EPFM
LEFM
• In LEFM, the crack tip stress and displacement field can be uniquely characterized by K, the
stress intensity factor. It is neither the magnitude of stress or strain, but a unique parameter that
describes the effect of loading at the crack tip region and the resistance of the material. K filed is
valid for a small region around the crack tip. It depends on both the values of stress and crack size.
We noted that when a far field stress acts on an edge crack of width “a” then
for mode I, plane strain case
R

|

S
|

T
xx
yy
xy
U
|

V
|
W
KI
2r

3 O
L
1

sin(
)
sin(
)
M
2
2 P
M
P
 M

3 P
cos
1  sin(
) sin(
)
2 M
2
2 P
M
P

3
M
P
sin(
) sin(
)
M
P
2
N 2
Q
 zz  0 for plane stress;  zz   ( xx   yy ) for plane strain
u UK
R

S
V
u
T W2
x
y
I

 O
L
cos
(
k

1

2
sin
(
))
r M2
2 P
M
P
 P
2 M
sin ( k  1  2 cos ( ))
M
2
2 P
N
Q
2
2
LEFM cont.
For  =0
R

|S

|T

xx
yy
xy
Singularity dominated region
U
|V
K

|W 2r
For  =
I

2
1O
L
M
P
1
M
P
M
0P
N
Q
, all  ij  0
LEFM concepts are valid if the plastic zone is much smaller than the singularity zones.
Irwin estimates rp 
1 KI 2
(
)
2  ys
1 KI 2
)
8  ys
Dugdale strip yield model: rp  (
ASTM: a,B, W-a
 2.5
(
KI
 ys
)2 ,
i.e. rp 
1
50
of specimen dimension.
EPFM
In EPFM, the crack tip undergoes significant plasticity as seen in the following diagram.
Ideal elastic brittle behavior
cleavage fracture
Load ratio, P/Py
sharp tip
P: Applied load
Py: Yield load
1.0
Fracture
Displacement, u
Blunt tip
Limited plasticity at crack
tip, still cleavage fracture
Load ratio, P/Py
•
1.0
Fracture
Displacement, u
large scale
blunting
Large scale plasticity
fibrous rapture/ductile
failure
Load ratio, P/Py
Void formation & coalescence
failure due to fibrous tearing
Fracture
1.0
Displacement, u
Load ratio, P/Py
Blunt tip
1.0
Fracture
Displacement, u
EPFM cont.
• EPFM applies to elastoc-rate-independent materials, generally in the large-scale plastic
deformation.
•
Two parameters are generally used:
(a) Crack opening displacement (COD) or crack tip opening displacement (CTOD).
(b) J-integral.
•
Both these parameters give geometry independent measure of fracture toughness.
y
Sharp crack
x


Blunting crack
ds
EPFM cont.
• Wells discovered that Kic measurements in structural steels required very large thicknesses for
LEFM condition.
--- Crack face moved away prior to fracture.
--- Plastic deformation blunted the sharp crack.
  2u y
Note: k 
Sharp crack

3
and E  2  (1   )
1
  CTOD 
Blunting crack
K I2

  2ys E
4
4 G
  ys
2
K
since G  I
E
• Irwin showed that crack tip plasticity makes the crack behave as if it were longer, say from size a to a + rp
rp 
From Table 2.2,
Set  =  ,
uy
KI

2
uy 
KI
1
(
)2
2
 ys
r


sin( )[ k  1  2 cos 2 ( )]
2
2
2
k 1
KI
2
ry
2
-----plane stress
 
a  ry
CTOD and strain-energy release rate
• Equation
  CTOD 
4 G
relates CTOD ( ) to G for small-scale yielding. Wells proved that
  ys

Can valid even for large scale yielding, and is later shown to be related to J.
•

can also be analyzed using Dugdales strip yield model. If “  ” is the opening at the end of the strip.
Consider an infinite plate with a image crack subject to a

 ys
 
  2u y 
8 ys a
E
lin sec(


)
2  ys

Expanding in an infinite series,

If
8 ys a 1   2 1   4
[ ( 
) 
( 
) ...]
E 2 2  ys
12 2  ys
K 2I

G
 0 (   ys ), then  =

, and
 ys
 ys E  ys

K I2
1 
 2
[1  ( 
) ]
can be given as: 
 ys E
6 2  ys
In general,   G , m = 1.0 for plane stress; m = 2.0 for plane strain
m ys
Alternative definition of CTOD

Blunting crack

Sharp crack
Blunting crack
Displacement at the original crack tip
Displacement at 900 line intersection, suggested by Rice
CTOD measurement using three-point bend specimen
displacement
Vp
z
a
p
W
P
 pl 
 expanding
r p( W
- a)
rp (W  a)Vp
rp (W  a)  a  z
'
'
'
Elastic-plastic analysis of three-point bend specimen
m ys E

rp (W  a)Vp
loa d
   el   pl 
K I2
V,P
rp (W  a)  a  z
Where  pl is rotational factor, which equates 0.44 for SENT specimen.
• Specified by ASTM E1290-89
p
e
Mouth opening
--- can be done by both compact tension, and SENT specimen
• Cross section can be rectangular or W=2B; square W=B
 el
K I2 (1   2 )

2 ys E
KI is given by
P
a
KI 
 f( )
W
B W
 pl 
rp (W  a)Vp
rp (W  a)  a  z
loa d
CTOD analysis using ASTM standards
Pc
Pi
fracture
(a)
Pu
Pm
Pi
fracture
(b)
(c)
Mouth opening
Figure (a). Fracture mechanism is purely cleavage, and critical CTOD  c <0.2mm, stable crack growth,
(lower transition).
 i --- CTOD corresponding to initiation of stable crack growth.
 u --- Stable crack growth prior to fracture.(upper transition of fracture steels).
Figure (c)  i and then  m---CTOD at the maximum load plateau (case of raising R-curve).
Figure (b).
More on CTOD
K I2
J
 COD 
or 
 yE
y
The derivative is based on Dugdale’s strip yield model. For
Strain hardening materials, based on HRR singular field.
n
n 1
1


J
n 1
ui   y 
 r ui  , n 
  y y I n 
By setting =0 and n the strain hardening index based on
n 1
 y 3   e   ij

 
y 2  y   y


*Definition of COD is arbitrary since   u y  x,0   u y  x,0 
1
n1
A function   x 
as the tip is approached
*Based on another definition, COD is the distance between upper
and lower crack faces between two 45o lines from the tip. With this
J
Definition

d
COD
n
y
Where d n  d n  ,  y , n  ranging from 0.3 to 0.8 as n is varied from
3 to 13 (Shih, 1981)
*Condition of quasi-static fracture can be stated as the
Reaches a critical value  COD . The major advantage is that this
provides the missing length scale in relating microscopic failure
processes to macroscopic fracture toughness.
*In fatigue loading,  tip continues to vary with load and is a
function of:
2

k
I
(a) Load variation
 
2 y
(b) Roughness of fracture surface (mechanisms related)
(c) Corrosion
(d) Failure of nearby zones altering the local stiffness response
3.2 J-contour Integral
• By idealizing elastic-plastic deformation as non-linear elastic, Rice proposed J-integral, for regions
beyond LEFM.
• In loading path elastic-plastic can be modeled as non-linear elastic but not in unloading part.
• Also J-integral uses deformation plasticity. It states that the stress state can be determined knowing
the initial and final configuration. The plastic strain is loading-path independent. True in proportional
load, i.e. d 1  d 2  d 3  d 4  d 5  d 6  k
1
2
3
4
5
6
• under the above conditions, J-integral characterizes the crack tip stress and crack tip strain and
energy release rate uniquely.
• J-integral is numerically equivalent to G for linear elastic material. It is a path-independent integral.
• When the above conditions are not satisfied, J becomes path dependent and does not relates to any
physical quantities
3.2 J-contour Integral, cont.
y
x

ds
Consider an arbitrary path ( ) around the crack tip. J-integral is defined as
z
u
J  (wdy  Ti i ds),
xi

w
z
 ij
0
 ij d ij
where w is strain energy density, Ti is component of traction vector normal to contour.
It can be shown that J is path independent and represents energy release rate for a material where  ij
is a monotonically increasing with  ij
ui
Proof: Consider a closed contour: J *  ( wdy  Ti
ds )

x
i
*
*
z

Using divergence theorem: J * 
z
(
A*
ui
w


 ij
)dxdy
x
x i
x
A
*
Evaluation of J Integral ---1
 w
ui 

J  

( ij
)  dxdy
x 
 x x j
A* 
*

 ij
w
w  ij
Evaluate


  ij
x
 ij x
x
Note  ij 
w
is only valid if such a potential function w exists
 ij
w
1


  ij [
(ui , j ) 
(uij,,ij )]
x
2
x
x
ui
1

 u j
  ij [
(
)
(
)]
2
x j x
x i x
Since  ij   ji
Again,
Recall
  ij
 ij
x j
 ij
 ui
(
)
x j x
 0 (equilibrium)
leads to
(equilibrium) leads to
ui


(
) 
(
x j
x
x j
ij
ui
)
x
Evaluation of J Integral ---2
Hence, J *  0. Thus for any closed contour J *  0.
Now consider
2
4
1
3
J  J1  J 2  J 3  J 4  0
1
Recall
2
J* 
3
z
4
( wdy  t ii

w
ds)
x
On crack face, ti i  0, dy  0 (no traction and y-displacement), thus
J3  J 4  0, leaving behind J1   J2
Thus any counter-clockwise path around the crack tip will yield
J; J is path independent.
Evaluation of J Integral ---3
ti
y
a
x
2D body bounded by '
'
In the absence of body force, potential energy 

z z
z z
t i i ui ds
wdA 
A'
 ''
Suppose the crack has a vertical extension, then
d

da
A'
dui
dw
dA   i
ds
da
da
'

Note the integration is now over '
(1)
Evaluation of J Integral ---4
d
 x 
 
x
 
 
since
 1
da a a x a x
a
d
w w
w ui
 (

)dA  t ii (

)ds
da
a x
a
x
'
'
Noting that
z
z
(2)

A
w
w  i j
 ui

  ij
(
)
a
 ij a
x j a
Using principle of virtual work,
eq.(1), we have
d
0
da
for equilibrium, then from
z
z
zt z
 ij
A'
ui

(
) dA 
x j a
t ii
'
ui
ds
a
du
Thus, d   ii i ds  dw dA
da
'
dx
A'
dx
Using divergence theorem and multiplying by -1

d

da
z
( wn x  t ii

'
dui
)ds 
dx
z
wdy  t ii
'
w
ds
x
Evaluation of J Integral ---5
Therefore, J is energy release rate
elastic material
In general   U  F and J   
d
,
da
for linear or non-linear
A
Load
u
  Potential energy; U=strain energy stored; F=work done by
external force and A is the crack area.
p
dU *  dU
d
-dP
a
Displacement
p
  U  P  U *  Complementary strain energy =  dP
0
Evaluation of J-Integral
dU *
For Load Control
J
da p
dU
J 
For Displacement Control
da

The Difference in the two cases is 1 .dp.d 
2
both load Displacement controls are same
p
D


J   .dp  
.dp
a 0
a p
0
p
dU and hence J for
or


p
J    pd    
.d 
a 0
a 

K I2
J=G and is more general description of energy release rate J  '
E
More on J Dominance
J integral provides a unique measure of the strength of the singular
fields in nonlinear fracture. However there are a few important
Limitations, (Hutchinson, 1993)
(1) Deformation theory of plasticity should be valid with small strain
behavior with monotonic loading
(2) If finite strain effects dominate and microscopic failures occur, then
this region should be much smaller compared to J dominated region
1
Again based on the HRR singularity

 n1 I
J
 ij   y 
  ij  , n 
  y y I n r 
Based on the condition (2), we would
like to evaluate the inner radius ro of J
dominance. Let R be the radius
ro
where the J solutions are satisfied
within 10% of complete solution.
R
FEM shows that ro 3 COD
•However we need ro should be greater than the forces zone
(e.g. grain size in intergranular fracture, mean spacing of voids)
•Numerical simulations show that HRR singular solutions hold
good for about 20-25% of plastic zone in mode I under SSY
• Hence we need a large crack size (a/w >0.5) . Then finite strain
region is 3 COD , minimum ligament size for valis JIC is
25 J IC
b
y
• For J Controlled growth elastic unloading/non proportional loading
should be well within the region of J dominance
dJ J

and a
da R
R
• Note that near tip strain distribution for a growing crack has a
logarithmic singularity which is weaker then 1/r singularity for a
stationary crack
Williams solution to fracture problem
Williams in 1957 proposed Airy’s stress function
  R  r   
As a solution to the biharmonic equation
2
2

1

1

 4  0 where  2  2 
 2 2
r r r r 
For the crack problem the boundary conditions are
    r  0 for   
Note  will have singularity at the crack tip but is single valued
  r 2 p  r ,   q  r , 
Note that both p and q satisfy Laplace equations such that
2 p  2 q  0
Now, for the present problem.
p  A1r  cos   Az r  sin 
q  B1r 2 cos    2   Az r  2 sin    2 
Then
  r 2  A1 cos   B1 cos    2  
 r 2  A2 sin   B2 sin    2  
Consider only mode I solution with
  r 2  A1 cos   B1 cos    2  
 2
   2     1   2  r   A1 cos   B1 cos    2  
r
 1  
 r   

r  r  
    1 r    A1 sin      2  B1 sin    2  
Williams Singularity…3
Applying boundary conditions,
 A1 +B1  cos   0
 A1     2  B1  sin   0
Case (i)
cos   0
or,
Case (ii)
sin   0
2Z  1
, Z=0,1,2...
2

B1  
A
2 1

Z
B1   A1
Since the problem is linear, any linear combination of the above two will also be
acceptable.
  Z2 with Z=  ...  3, 2, 1,0,1,2,3...
Thus
Though all values are mathematically fine, from the physics point of view, since
 ij  r  and  ij  r 
Williams Singularity…4
U  12  ij ij  r 2 
=
2
0

R
 ij ij rdrd
1
r0 2
2A
R
0
r0
 
r
2  1
drd
Since U should be provided for any annular rising behavior r0 and R ,
U   as r0  0,   1 (  1 makes ˆ ij  0)
Also, ui  r r 1 needs  >  1. Thus
 =- 12 ,0, 12 ,1, 23 , 2... with  = Z2 where Z=-1,0,
positive number.
The most dominant singular form
 =- 12 and B1 
A1
3
Williams Singularity…4
Now   r A1 cos 2  13 cos 32 
3
2
 
   r  ...
+ r 2 ...
5
2
and
 
 
 ij  A1r   ij I    ij r 0   ij r
where  
1
2
1
2
 indicates the order of  
 
Note the second term in  ij   ij r 0 is a non-singular
and non-vanishing term. However, higher order vanish as r  0
K
with A1  I
2
KI
 ij 
 ij I    T  ix jx (no sum on x)
2 r
Williams Singularity…5
Now
  r A1 cos 2  13 cos 32 
3
2
 
   r  ...
  r 2 ...
5
2
and
 
 
 ij  A1r   ij I    ij r 0  ij r
where  
1
2
1
2
 indicates the order of  
 
Note the second term in  ij   ij r 0 is a non-singular
and non-vanishing term. However, higher order vanish as r  0
K
with A1  I
2
KI
 ij 
 ij I    T  ix jx (no sum on x)
2 r
Williams Singularity…6
For in-plane stress components,
I
I
  xx  xy 
K I   xx    xy     T 0 
 I




I


2 r   yx    yy     0 0 
yy 
 yx
Second-term is generally termed as "T-stress" or
"T-tensor" with  xx  T
For brittle crack of length 2a in x-z plane
y

with  yy &  xx
applied
K I   yy  a and

T= yy   xx
x
2a
z
HRR Singularity…1
Hutchinson, Rice and Rosenbren have evaluated the character of crack tip
in power-law hardening materials.
Suppose the material is represented by Ramberg-Osgood model,
 
 

  
0  0
0 
n
 0  Reference value of stress=yield strength

 0  0 , strain at yield
E
  dimensionless constant
n  strain-hardening exponent
Note if elastic strains are negligible, then
 

  
 
y
 y
n
 ij 3   eq 
 

y
2   ij 
n 1
ˆ ij
3
;  eq 
ˆ ij
y
2
HRR Singularity…2
Then
 4  f  ,  0 , r , n, 
  C1   r s   2   r t
(similar to Williams expression)
  k   0 r s    
Applying the appropriate boundary conditions

1
 n 1
0 
n
 n 1
EJ

2


I
r
 0 n 
 ij   0 
 ij 
EJ


E  0 2  I n r 
I n  Integration constant
 ,   Dimensionless functions of n and 
 ij  n, 
 ij  n, 