Elastic-Plastic Fracture Mechanics
Professor S. Suresh Elastic Plastic Fracture
Previously, we have analyzed problems in which the plastic zone was small compared to the specimen dimensions (small scale yielding). In today’s lecture we present techniques for analyzing situations in which there can be large scale yielding, and determine expressions for the stress components inside the
plastic zone. We will begin with a discussion of the J integral.
SMA � c 2000 MIT Fatigue and Fracture 1
nentoa ora fSld n Structures and Solids of Journal International SMA Mechanics The mechanics. KeyJ.mechanics. Rice,R. Reference: significance fracturein elastic-plastic crackItenormous has tip. Sanders, (Related works: Eshelby, (Related works: J � Integral c J 2000 MIT
integral is a line integral (path-independent) around the ora fApidMechanics Applied of Journal
, 1968. rgesi oi tt Physics State Solid in Progress
Derivation , 1960; Cherepanov, Applied of Journal , 1969) Fatigue and Fracture 1956; 2 SMA around Consider crackthe path the shown tip below: J � Integral c 2000 MIT Derivation Continued Fatigue and Fracture 3 normal unit vector,normal i.e. SMA u T d S a We will theuse following variables: deformation-induced blunting of crack tip. We consider straina small weanalysis; any neglect J s = crack length. = corresponding displacement vector. = a curve linking lowerthe and upper crack surfaces. = traction vectordefined relation an outward in on curve to this = an element of arc on this curve. � Integral c 2000 MIT T � n � � . Derivation Continued Fatigue and Fracture 4 SMA We use the depicted below:schematically The (reversible)non-linear elasticity). stress-strainresponse is J � Integral c 2000 MIT J 2 deformation theory of plasticity (equivalentplasticity of deformationto theory Derivation Continued Fatigue and Fracture 5 The totalmechanical potential energy of the crackedbody is SMA potential the energy of applied the loading. represents This strainthethe of stored sum potential energy and For comparable (i.e. forcracks). comparable monotonic loading, stationary (i.e. giveplasticity) of thatresults are (incremental theory theory Not appropriate forNot appropriate situations where significant J � proportional Integral c 2000 MIT loading u J M 2 � deformation theory and deformation theory u e + u Derivation app Continued unloading Fatigue and Fracture J 2 flow occurs. 6 Derivation
J Integral Continued
Z Z
u � w d A , T � u d s
M
A S In the previous integral:
w � strain energy density (per unit volume); recall that
@ w
� � :
ij
@ �
ij
A d an element of cross section A within S .
We now evaluate the derivative of the mechanical potential
u
energy, M , with respect to crack length.
SMA � c 2000 MIT Fatigue and Fracture 7 Derivation
J Integral Continued
� �
Z
u d @ u
M
, � w y d , T � d s
a d @ x
S
� J
J represents the rate of change of net potential energy with respect to crack advance (per unit thickness of crack front) for a
non-linear elastic solid. J also can be thought of as the energy
flow into the crack tip. Thus, J is a measure of the singularity strength at the crack tip for the case of elastic-plastic material response.
SMA � c 2000 MIT Fatigue and Fracture 8 Derivation
J Integral Continued
For the special case of a linear elastic solid,
d � � d U
PE M
J � G � , � ,
d a d a
2
, �
K
2
� 1 , �
E
K
This relationship can be used to infer an equivalent I c value
J from I c measurements in high toughness, ductile solids in which
K valid I c testing will require unreasonably large test specimens.
SMA � c 2000 MIT Fatigue and Fracture 9 SMA Consider twodifferent around paths crackthe tip: J � Integral c 2000 MIT Derivation Continued Fatigue and Fracture 10 Derivation
J Integral Continued
J � J
S S
along 1 along 2
The J Integral is independent of the path around the crack tip.
S
If 2 is in elastic material,
2
, �
K
2
J � 1 , �
S
2
E
SMA � c 2000 MIT Fatigue and Fracture 11 HRR field
We now consider the Hutchinson, Rice, Rosengren (HRR) singular crack tip fields for elastoplastic material response. (Recall Williams solution assumes linear elastic material behavior). Assume: Pure power law material response:
� �
n
� �
� �
� �
0 0
� � n
= material constant, 0 = reference yield strength, = strain
� � � � �E
hardening exponent, 0 reference yield strain 0 .
For linear elastic material n � 1 , for perfectly plastic response
n � 1 .
SMA � c 2000 MIT Fatigue and Fracture 12 Continued HRR field
With these assumptions, the crack tip fields (HRR field) can be derived. (Ref: J.W. Hutchinson, JMPS, 1968 and J.R. Rice and G.F. Rosengren, JMPS, 1968.)
1
� �
n +1
J
� � � � ~ � � ; n �
ij 0 ij
�� � I r
0 0 n
n
� �
n +1
J
� � �� � ~ � � ; n �
ij 0 ij
�� � I r
0 0 n
n
� �
n +1
J
1
n +1
u � �� r u ~ � � ; n �
i 0 i
�� � I r
0 0 n
I n
The function n has a weak dependence on .
SMA � c 2000 MIT Fatigue and Fracture 13 CTOD
�
The variation in crack tip opening displacement t or (CTOD) for different material response is depicted below:
The crack tip opening displacement depends on distance from the crack tip. We need an operational definition for CTOD.
SMA � c 2000 MIT Fatigue and Fracture 14 CTOD
�
The definition of t is somewhat arbitrary since the opening displacement varies as the crack tip is approached. A commonly
�
used operational definition is based on the 45 construction depicted below (see C.F. Shih, JMPS, 1982).
SMA � c 2000 MIT Fatigue and Fracture 15 CTOD
J
� � d
t n
�
0
d n � �E
n is a strong function of , and a weak function of 0 . Plane Strain:
d � 0 : 3 , 0 : 65 65 : �0 ! 1 �
n for n
Plane Stress:
d � 5 : 0 , 07 : 1 07 : �1 ! 1 �
n for n
SMA � c 2000 MIT Fatigue and Fracture 16 CTOD
Presuming dominance of HRR fields
J J
� � d �
t n
� �
0 0 For Small Scale Yielding (SSY)
2
, �
K
I
2
J � 1 , �
E
, �
2 2
K 1 , �
I
� � d
t n
E �
0
SMA � c 2000 MIT Fatigue and Fracture 17 CTOD
Importance/Applications of CTOD:
� Critical CTOD as a measure of toughness.
� Exp. measure of driving force.
� Multiaxial fracture characterization.
� K J
Specimen size requirements for I c and I c testing.
SMA � c 2000 MIT Fatigue and Fracture 18 J-Dominance
Just as for the K field, there is a domain of validity for the HRR
( J -based) fields.
SMA � c 2000 MIT Fatigue and Fracture 19 J-Dominance
Under plane strain and small scale yielding conditions, it has been found that:
1
� r
0 p
4
For J dominance the uncracked ligament size b must be greater
� 25 � J �� than 25 times the CTOD or 0 . The variation in stress ahead of the crack is depicted on the following page:
SMA � c 2000 MIT Fatigue and Fracture 20 J-Dominance
SMA � c 2000 MIT Fatigue and Fracture 21 Kand J-Dominance
� � 350
Consider a low strength steel with 0 MPa,
p
K � 250 E � 210
I c MPa m and GPa. What are the Minimum
K J specimen size requirements for valid I c and I c measurements?
SMA � c 2000 MIT Fatigue and Fracture 22 K-Dominance
K
ASTM standard E399 (1974) for I c testing:
� �
2
K
I c
a; b; t � 5 : 2
�
0
� K
Substitute the known values for 0 and I c . Find that
a; b; t � 28 : 1 m ! � � 50 inches �
SMA � c 2000 MIT Fatigue and Fracture 23 J-Dominance
J
For I c testing, the condition requires that for a deeply cracked compact tension or bend specimen:
, �
2
J K 1 , �
I c I c
b � 25 � 25
� E �
0 0
b � 0 02 : m
Specimen size requirements for J testing are much less severe
than for K testing.
SMA � c 2000 MIT Fatigue and Fracture 24 J-Dominance
J K
The measured I c value may be converted to equivalent I c value. The validity of this approach has been verified by extensive testing.
SMA � c 2000 MIT Fatigue and Fracture 24 J-Dominance
Example: notched bar loaded axially (induces bending and stretching)
SMA � c 2000 MIT Fatigue and Fracture 25 J-Dominance
SMA � c 2000 MIT Fatigue and Fracture 26