Transcript Diapositive 1

6.2 CTOD as yield criterion
CTOD : crack tip opening displacement
Well’s experimental work: attempt to measure KIc for structural steels
But
Initial sharp crack has blunted prior to fracture
Non-negligible plastic deformation
blunted crack
sharp crack
Irwin plastic zone
LEFM inaccurate : materials too tough !!!
Instead, Wells proposed dt (CTOD ) as a measure of fracture toughness.
Estimation of dt using Irwin model :
Crack length: a + ry
By definition, dt  2 u y at r  ry
where uy is the crack opening
1
Crack opening uy
uy 

KI
2
r


sin    1 - 2 cos 2 
2
2
2
KI
r
   1
2
2
We have  
E
2 1   
4K
 dt  2 I
E

uy

(see eqs 4.40)
and for plane stress,  
3
1 
ry
2
1
From Irwin model, the radius of the plastic zone is r y 
2
2
4 KI
dt 
 Y E
and also, dt 
4 G
 Y
 KI 



 Y
2
CTOD related uniquely to KI and G.
CTOD appropriate characterizing crack-tip-parameter when LEFM no longer valid.
Can be proved by a unique relationship between CTOD and the J integral.
2
6.3 The J contour integral as yield criterion
 More general criterion than K (valid for LEFM)
 Derive a criterion for elastic-plastic materials, with typical stress-strain behavior:
C
B
A→B : linear
B→C : non-linear curve
C→D : non-linear, same slope as A-B
non-reversibility: A-B-C ≠ C-D-A
A
Material behavior is strain history dependent !
Non unique solutions for stresses
D
 Simplification: non-linear elastic behavior
C
B
elastic-plastic law replaced by the non-linear elastic law
reversibility: A-B-C = C-D-A
Correct only for a monotonic loading
A/D
= Deformation theory of Plasticity
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Definition of the J-integral
 Historically,
Rice defined a path-independent contour integral J for the analysis of crack
showed that its value = energy release rate in a nonlinear elastic body with a crack
 J generalizes the concept of G to non-linear materials
• For linear materials J = G
P0
• Load-displacement diagram: potential energy 
U : Elastic strain energy
U*
≠ (in general)
U
U* : Complementary energy

0
Fixed-grips conditions:
0
  U   P(  )d 
0

Dead-load conditions
P0
  U   ( P )dP
*
0
4
 Definition of J using the potential energy  :
J 
d
dA
A = a B : for a cracked plate with through crack
• Geometrical interpretation:
OB and OB’ :
P  , a 
B
P
J da
loading/unloading for the given body with
crack lengths a and a+da
B’
a
a  da
O
P  , a  :
A
A’
  d

Possible relationship between the load P
and the displacement  while the crack
is moving.
We have J dA  Pd   dU
dU is the difference between the areas under OB’ and OB : OA’B’ – OAB
Pd appears as the area AA’B’B
Thus, J dA  J B da  AA’B’B + OAB – OA’B’ = OB’B’
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• In particular ,
d

0
At constant displacement:
1  U 
J  

B  a 
1 0  P 
     d
B 0  a 

At constant force (dual form):
1  U * 
J 

B  a 
P
1 P0    
 
 dP
B 0  a  P
Generalization of eqs 3.38a and 3.43a to non-linear elastic materials
Useful expressions for the experimental determination of J
6
• Experimental determination of the J-integral :
 Multiple-specimen method (Begley and Landes (1972)) :
Procedure
(1) Consider cracked specimens with different crack lengths ai
(2) For each specimen, record of the load-displacement P-u curve under fixed-grips
(1)
(2)
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(3) Calculation of the potential energy  for given values of
displacement u
= area under the load-displacement curve
(4) Negative slopes of the P – a curves determined and plotted versus displacement
for different crack lengths :
Critical value JIc of J at the onset of crack extension (material constant)
(3)
8
(4)
 Single-specimen method by Rice , Merkle and Corten :
J can be determined directly from the load-displacement curve of a single cracked specimen.
Generally, when crack lengths that are important compared with the unbroken
ligament dimension.
Estimation formula derived by Rice et al. (1973) for specimens in tension and under
bending
The case of combined tension and bending treated by Merkle and Corten (1974),
modified by Clarke and Landes (1979)
Principle
Writing of a relationship between load, displacement and body’s characteristic
lengths using a dimensional analysis.
See for example pp 116-119 of Anderson’s book, third ed.
Example:
for the deeply cracked three-point bending or compact tension specimen J is given by,
2 d
J   Pd d
b 0
b : ligament length
(application pb3 series7)
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 J as a path-independent line integral
u


J    w dy  Ti i ds 
x 


mn
with w  mn    ij d ij
strain energy density
0
u 

ds 
  w dy  T 
x 

T : traction vector at a point M on the bounding surface  , i.e. Ti  ij n j
u : displacement vector at the same point M.
T
n : unit outward normal.
The contour  is followed in the counter-clockwise direction.
n
M
y
crack
x
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Equivalence of the two definitions
Y
T
• 2D solid of unit thickness of area S ,
t
with a linear crack of length a along OX (fixed)
y
• Crack faces are traction-free.
O
• Total contour of the solid 0 including the crack tip:
x
X
a
Imposed tractions on the part of the contour t
 0  S
S
Displacements applied on u
u
Proof :
u
Recall for the potential energy (per unit thickness),
  a    wdS   Ti ui ds
S
t
Ti  ij n j
ij 
w
 ij
The tractions and displacements imposed on t and u are independent of a
dTi
 0 , on t
da
dui
 0 on u
da
du
d
dw
 
dS   Ti i ds
da S da
da
0
Change of  due to a virtual
crack extension
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Considering the moving coordinate system x , y (attached to the crack tip),
x  X a
d
: total derivative/crack length
da
d     x 
   
da  a  x  a  X

 
 x a

 

a x
Thus,
w w
  ui  ui 
d
  

dS

T

 i
 ds
da S   a  x 

a

x

0 
However,
  ij
 w  w   ij

  ij
 a   ij  a
a
  ij
  1   ui  u j  
  ij

 

 a  2   x j  xi  
since  ij   ji
  ui
   ui 
  ij
a  xj
 x j  a 
12
Thus,
w
   ui 
dS


dS

 ij


 x j  a 
S a
S
We have,
  ij
S
   ui 
dS 


 x j  a 
  ij
0
 ui
n j ds
a
  Ti
0
 ui
ds
a
The derivative of J reduces to,
w w
  ui  ui 
d
  

dS

T

ds
 i


da S   a  x 
x 
0   a
w
  ui 
  ui  ui 
   
dS

T
ds

T

 i
 i


 ds

x

a

a

x



S
0 
0 
  w
  ui  
    
dS   Ti 
 ds 
 S   x 

x
 
0 

13
 Q  P 

Using the Green Theorem, i.e.  P  x, y  dx  Q  x, y  dy   
 dx dy
y

A  x

w
  ui 
d
  
dS

T
 i
 ds
da S   x 

x

0 

  ui  
   wdy  Ti 
 ds 

x

 
0 
J derives from a potential
End of the proof
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Properties of the J-integral
1) J is zero for any closed contour containing no crack tip.
Closed contour around A
y
A
x

u


Consider J     wdy  Ti i ds 
x 

Using the Green Theorem, i.e.
We have J

 Q  P 
P
x,
y
dx

Q
x,
y
dy







 
 dx dy
y

A  x
u
u
w
w
dxd y   Ti i ds  
dxd y   ij i n j ds
x
x
A x

A x


From the divergence theorem,
 ij

ui
  ui 
n j ds  
 ij
 dxd y
x

x

x

A
j
15
The integral becomes,
J
 w   ui  
 
 ij
  dxdy
x  
A  x x j 
However,
  ij
 w  w   ij

  ij
 x   ij  x
x
  ij
  1   ui  u j  
  ij

 

 x  2   x j  xi  
since  ij   ji
  ui
   ui 
  ij
 x  xj
 x j   x 
Invoking the equilibrium equation,
   ui 
 ij

 x j   x 
 ij
 xj
0
0
 ij  ui
   ui 

  ij
 xj  x
 x j   x 
  ij
   ui 
 x j   x 
Replacing in the integral,
J  0
16
2) J is path-independent
y
3
x
4
Γ*2
Consider the closed contour   1  3  *   4
2
We have J   J   J *  J   J 
1
2
3
4
and
J  0
The crack faces are traction free :
Ti  ij n j  0
on  3 and  4
dy = 0 along these contours
J
3
J
4
0
17
y
3
4
Note that,
J
*2
Γ*2
 J
J
and
J
x
  1  3   2   4
2

1
J
1
J
J
2
0
2 followed in the counterclockwise direction.
2
Any arbitrary (counterclockwise) path around a crack gives the same value of J
J is path-independent
18
J can be evaluated when the path is a circle of radius r around the crack tip

crack
r
 is followed from    to  = 
We have,
ds  rd 
dy  r cos  d 
J integral becomes,
 u  r ,  

J    w  r ,  cos  Ti  r ,  i
r d

x 
 

When r → 0 only the singular terms remain
K I2
For LEFM , we can obtain : J  G 
E'
(if mode I loading)
(see eq. 6.58)
19
6.4 HRR theory
Hutchinson Rice and Rosengren:
J characterizes the crack-tip field in a
non-linear elastic material.
 For uniaxial deformation:
  



  
 0 0
 0 
n
Ramberg-Osgood equation
 0 = yield strength
0  0 E
 : dimensionless constant
n : strain-hardening exponent
material properties
Power law relationship assumed between plastic strain and stress.
For a linear elastic material n = 1.
20
 Asymptotic field derived by Hutchinson Rice and Rosengren:
J
ij  A 2  
r
n  n 1
1  n 1
J
ij  A1  
r
n n1 1 n 1
ui  A3 J   r  
Ai are regular functions that depend on  and the previous parameters.
The 1
r singularity is recovered when n = 1.
The product ij ij varies as 1/r :
Path independence of J
From
 u  r ,  

J  r   w  r ,  cos  Ti  r ,  i
d

x 
 

ij ij 
f  
r
as r  0
J defines the amplitude of the HRR field as K does in the linear case.
21
Two singular zones can be identified:
K
J
Small region where crack blunting occurs.
Large deformation
HRR based upon small displacements non applicable.
22
 Relationship between J and CTOD
Consider again the strip-yield problem,

y
dt
a
Y

Y
x
c

The first term in the J integral vanishes because dy= 0
J    ijn j

(slender zone)
u i
ds
x
u y
u
ui
ds   Y y dx
ds   yy n y
but ijn j
x
x
x
J   Y

u y
x
dt
dx
  Y du y
d t
 Y d t
(see pb1 Series 8)
23
General unique relationship between J and CTOD:
J  m Y dt
m : dimensionless parameter depending on the stress state and materials properties
• The strip-yield model predicts that m=1 (non-hardening material, plane stress condition)
• This relation is more generally derived for hardening materials (n >1) using the HRR
displacements near the crack tip, i.e.
n n1 1 n 1
ui  A3 J   r  
Shih proposed this definition for dt :
blunted crack
90°
dt
m becomes a (complicated) function of n
The proposed definition of dt agrees with the one of the Irwin model

Moreover, G  Y dt
4
m

in this case
4
24
6.5 Applications the J-integral
 J integral along a specific contour
 Example 6.5.2
uy
uy
 Example 6.5.3
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