HRR Integral
Recall
with I n

1
n 1
EJ 
   n, 
2
  0 I n r 
is the integration constant.
n
 0  EJ  n 1
 ij 

   n, 
E   02 I n r 
 ij   0 
n
 
 

  
0  0
Recall the material equation,
0 
Now the singularity, unlike K I varies as a function of n and state of stress.
HRR Integral
 EJ 
Recall  ij   0 

2


I
r
0 n 

1
n 1
  n, 
 ij 
 0 
EJ 


E   02 I n r 
n
n 1
  n, 
with I n is the integration constant, as shown in the Figure below.
n
 
 
Recall the material equation,

  
0  0
0 
Now the singularity, unlike K I , varies as a function of n and state of stress.
HRR Integral, cont.
Angular variation of dimensionless stress for n=3 and n=13.
HRR Integral, cont.
Note the singularity is of the strenth  1r  n1 . For the specific case of n=1 (linearly
elastic), we have 1r singularity.
1
Note also that the HRR singularity still assumes that the strain is infinitesimal, i.e.,
 ij  12  ui , j  u j ,i  , and not the finite strain Eij  12  ui , j  u j ,i  uk ,iuk , j . Near the tip where
the strain is finite, (typically when ij  0.1), one needs to use the strain measure E.
Some consequences of HRR singularity
In elastic-plastic materials, the singular field is given by
J
 ij  k1  
r
J
 
 ij  k2  
r
1
n 1
1
n 1
(with n=1 it is LEFM)
 stress is still infinite at r  0.
 the crack tip were to be blunt then  xx  0 at r  0 since it is now a free surface.
This is not the case in HRR field.
 HRR is based on small strain theory and is not thus applicable in a region very
close to the crack tip.
HRR Integral, cont.
Large-strain crack tip finite element results of McMeeking and Parks.
Blunting causes the stresses to deviate from the HRR solution close to the crack tip.
HRR Integral, cont.
Some additional notes on CTOD and J-integral
CTOD is based on Irwin’s formulation
K 
rp  21  I 
  YS 
uy 
2
r
K 1
KI p
2
2
4 K I2
4 G
  2u y 

  YS E   YS
Or based on strip yield (Dugdale’s model)
    

ln sec 
 2  ys  
E
 
 
G

8 ys a
 ys
CTOD is generally determined by three point bend test.
HRR Integral, cont.
J-Integral
J   ( wdy  ti

u x
ds )
x
with
 ij
w    ij ij dV and ti   ij n j
0
• Derived for non-linear elastic material and hence is not valid directly for elasto-
plastic material during unloading part.
• Assumes deformation theory, and hence needs proportional loading.
• Assumes small strain formulation (no large strain, no large rotation admissible).
• J is path independent when there is no traction or displacement in the crack faces.
• J cannot be evaluated at the tip where there is singularity since the theory assumes
that inside the domain the region should be analytic.
HRR Integral, cont.
Large Strain Zone
HRR singularity still predicts infinite stresses near the crack tip. But when the crack
blunts, the singularity reduces. In fact at  xx  0 at r  0 for a blunt crack. The following
is a comparison when you consider the finite strain and crack blunting. In the figure,
FEM results are used as the basis for comparison.
The peak occurs at
x 0
and
J
decreases as x  1. This
corresponds to
approximately twice the width
of CTOD. Hence within this
region, HRR singularity is not
valid.
Large-strain crack tip finite element results of McMeeking and Parks.
Blunting causes the stresses to deviate from the HRR solution close to the crack tip.
HRR Integral, cont.
Evaluation of J
Early works of Landes and Begley were based on the definition
J 
1  U 


B  a  
Obtain P   curve for different crack sizes a1  a2  a3  a4 .
Compute
dU
from the previous results.
da
Plot J vs.  (see Fig.3-13 below).
The method is not useful there are too many specimens and tests.
HRR Integral, cont.
HRR Integral, cont.
Evaluation of J using DENT (Double edge notched tension)
Note that in this case,
dA=2 da =-2 db
  
0  a  dP
P
P
J=
1
2
  
  12  
 dP

b
P
0
P
  el    pl  
   
 
  dP

b

b
P 
P 
0 
P
K12 1   pl 
 / 2 
 dP
E
b  P
0 
P
1
2
Let the deep notch  P  b H ( Pb )
( A) where H represents a function. Now,
HRR Integral, cont.
  P 
/ P P
P

  H ( b )  H ( b ). b
 b  P
  
From (A)  P   H / ( Pb )
 P b
1
  
   
 P    P  P  P  
 b  P b 
 P b 
and finally (integrating by parts)


K12 1  P
J

2
P
d


P


P
P
E / 2b  0

HRR Integral, cont.
Evaluation of J using Edge Cracked Plate in bending
It can be shown that
K I2 2
J / 
E
b
P
 M d
0
In general for any configuration
K I2 PU P
J / 
E
Bb
where  P is a dimensionless constant.
P
HRR Integral, cont.
Relationship between J and CTOD
In LEFM J  G . However in the case of small scale yielding (SSY), J  m YS G
where m is a dimensionless constant that depends on the state of stress and the
material properties.
Contour along the boundary of the strip yield zone ahead of a crack tip.
HRR Integral, cont.
Note that in the figure  is a contour along the strip yield zone. If the cohesive zone
extent is considerably larger than the cohesive zone width, i.e.,    then dy  0;
and the traction on the crack face ny  1, nx  nz  0leads to t   yy. Then,
J   wdy  ti
ui
ds
x
Define a new coordinate system with  = -x . For a fixed value of  ,
 yy   yy (  )
uy  uy ( )
If the cohesive zone length  is small compared to other dimensions, then

J  2  yy (  )
0
Since
u y (  )

d
 2u 
y
  2u y (    ); d  
d
  

J    yy ( ) d
0
HRR Integral, cont.
For the strip yield model, since  yy ( )   ys
J   ys
The above relationship is valid for plane stress with elastic-perfectly plastic materials.
More rigorous analysis based on HRR Integral
Shih showed that when HRR integral can be assumed to be valid, then

dn J
0
where d n is a dimensionless constant. For non-hardening materials, d n  1. Thus in
summary, there is a unique relationship between J and CTOD. Both the quantities are
valid for elastic-plastic materials under certain assumptions.
HRR Integral, cont.
3.4 Crack Growth Resistant Curves
Schematic J resistance curve for a ductile material.
HRR Integral, cont.
Both J and CTOD increase as the crack begins to grow and display a rising R-curve.
R represents the resistance of the material to crack growth. Initially there is a stable
crack growth and then it becomes unstable.
Note that J IC and critical CTOD indicate the initiation of crack growth. The initiation
cannot be precisely defined in practice. For generic materials, R-curve may provide a
better indication of the toughness of the material rather than just J IC . If we define a
tearing modulus, TR then
TR 
E J R
 02 a
Stability and R-curve for elastic materials
We know from the Griffith theory, in perfectly brittle elastic materials crack
extension starts when G  2 w f where w f is the surface energy per side of the crack
face. Since the crack size a extends after initiation, the strain energy release rate (or
crack driving force) G increases leading to an unstable crack growth in ideally
elastic brittle material. However if the material is not ideal elastic, then the
resistance to crack growth R increases as shown in the following figure.
HRR Integral, cont.
Schematic driving force R-curve diagrams
The left figure shows a material with a flat R-curve, while the right shows that of rising
R-curve. In the flat type, while the crack is stable at stress level 1 , it becomes unstable
at a stress of  2. On the rising curve, the crack continues to grow from a0 to ac with
stress levels  1 to  4. Thus the conditions for stable crack growth is
G  R and
dG dR

da da
HRR Integral, cont.
Unstable crack growth occurs when
dG dR

da da
Some notes on R-curve for elastic materials:
 For a flat R-curve, one can define a critical value of Gcritical  Gc.
 For a rising R-curve, it is difficult to estimate the precise value of G at which
crack starts to grow. This provides no information on the growth process or the
R-curve.
HRR Integral, cont.
Reasons for R-curve
Resistance to fracture may arise from the energy needed to create new surfaces,
additional inelastic irrecoverable dissipative energy needed. R  2  WI
For a flat R-curve, one can define a critical value of
If R  2 and  is a constant, then R-curve is flat (only in ideal brittle material).
For ductile fracture in metals, plastic zone size increases with crack growth and
hence R-curve rises. For infinitely large specimen, plastic zone size eventually
reaches a steady state; R-curve then reaches a steady state.
Falling R-curve is possible if strain-rate effect reduces the size of plastic zone with
crack growth.
State of stress alters the shape of R-curve. Plane stress has steeper R-curve
compared to plane strain. R-curve may be affected by free boundary effects.
Displacement control produces more stable than load control.
HRR Integral, cont.
Schematic J resistance curve for a ductile material.
The driving force is expressed in terms of applied tearing modulus
Tapp 
E J
( ) 
 02 a
Condition for stable crack growth
J  J R and Tapp  TR
HRR Integral, cont.
Unstable crack growth when
Tapp  TR
P 2a 2
2 Pa 3
J G 
and  
BEI
3EI
J
2P 2a
J
2a 2
|p 
|a 
a
BEI
P
BEI

2 Pa 2

2a 3
|p 
|a 
a
EI
P
BEI
hence
Tapp
2P 2a
4a 3
2a 3 1
 2 {1 
( m 
)}
 0 BI
EI
3EI
HRR Integral, cont.
J-controlled fracture
J-controlled fracture established crack tip conditions for J as well as CTOD.
HRR Integral, cont.
L---characteristic length scale; uncrack ligment in SSY, K and J characterize
crack tip. r is where 1
singularity. For monotonic, quasistatic loading J-dominated
r
zone occurs in the plastic zone. Well inside the plastic zone (J-controlled), elastic
singularity is not valid. However inside the plastic zone, HRR solution is valid and the
1
stresses  n1 . Now finite strain occurs within 2 from tip. Here HRR field is not
r
valid.
In SSY, K characterizes crack tip, but 1
singularity is not valid upto the tip.
r
Here J is valid for elastic-plasticity, but the
HRR Integral, cont.
In elastic-plastic conditions, J is still approximately valid. Here K-field is not valid.
 As plastic zoone increases in size, K-dominated zone disappears but J-dominated
zone persists.
 Here K has no meaning; j-integral and CTOD are valid parameters.
Large scale yielding
Size of finite strain zone increases.
 J is not unique.
 Single parameter fracture mechanics does not exist.
 Critical J exhibit a size and geometry dependence.
HRR Integral, cont.
J-controlled crack growth
Note that in elastic unloading, non-linear elastic modeling (J-integral) is not valid. For
J-controlled fracture, non proportional plasticity must be embedded within J-controlled
region.
HRR Integral, cont.
SSY controlled crack growth