Fracture
Mechanics
Crack Tip Plasticity
Crack Tip Plasticity
Presented by
Calvin M. Stewart, PhD
MECH 5390-6390
Fall 2020
Outline
• Introduction
• Irwin’s Plastic Zone Size
• Crack Tip Opening Displacement
• Dugdale Strip Yield Model
• Crack Tip Opening Displacement
• Classic Yield Criteria
Introduction
Introduction
• In the previous chapter, elastic stress field equations for a sharp crack, equations,
were obtained.
• These equations result in infinite stresses at the crack tip, i.e. there is a stress
singularity.
• However, real materials have an atomic structure, and the minimum finite tip
radius is about the interatomic distance. This limits the stresses at the crack tip.
• More importantly, structural materials deform plastically above the yield stress
and so in reality there will be a plastic zone surrounding the crack tip.
Introduction
• Along the x axis at θ=0,
• The Plastic zone size, ry can be
approximated by substituting
in the yield strength
Introduction
• From this figure the assumption is inaccurate, since part of the stress
distribution (shown hatched in the figure) is simply cut off above σys.
• Also, there is no a priori reason why the plastic zone should be
circular.
• It is extremely difficult to give a proper description of plastic zone size
and shape.
• The Irwin and Dugdale approaches assume a plastic zone shape and
provide a better approximation of the size.
• The classic Yield criteria assume a plastic zone size and provide a
better approximation of shape.
Irwin’s Plastic Zone Size
Irwin’s Plastic Zone Size
• Irwin made the following assumptions
•
•
•
•
The plastic zone is circular
Only the situation along the x-axis (y=0 and θ=0°)
The material is elastic-perfectly plasticity
A plane stress state is considered.
Irwin’s Plastic Zone Size
We must introduce the
effective crack length, aeff
aeff = a + an
aeff = effective crack length
a = physical crack length
Δan = increment in the
notational crack length
Irwin’s Plastic Zone Size
B
• aeff behaves as part of the crack. The
stress distribution ahead of aeff begins to
match the elastic conditions such that,
A
=
B
A
The area gained by translating the
elastic distribution (B) is equal to the
area lost by truncating the opening
stress at the level of the flow stress (A)
Α =  ys  a
ry
B=
0
KI
dr −  ys  ry
2 r
Fracture Toughness, Kic
aeff
a
The areas are equivalent to each other
Note,
Irwin’s Plastic Zone Size
=
A
ry
 ys  a = 
0
ry
 ys  a = 
K I =  ys 2 r
B
B
  ( a + a )
2 r
 ys ( a + ry ) =
y
K eff =   ( a + a )
Fracture Toughness, Kic
KI
dr −  ys  ry
2 r
0
( a + r ) =
Effective SIF, Keff
dr −  ys  ry
2  ( a + a )
2
2  ( a + a )
 ys 2
KI
( a + ry ) =  2 2 ry
ys
A
aeff
ry
a
ry
( a + r ) =
y
ry 2 ry
The size of the plastic zone can be determined
Irwin’s Plastic Zone Size
Thus, Irwin’s plastic zone size is twice
the First Approximation
( a + r ) = 2r
y
y
1  KI
2ry = 
   ys
2

 KI
 = 0.318 

  ys
• ry = nominal crack tip;
• 2ry = diameter of plastic zone
• Plastic zone size and crack tip
singularity characterized by K.
• Common Design Problem



2
aeff
a
• If Ki is applied to a for a variety of materials
find the ry
The size of the plastic zone can be determined
Irwin’s Plastic Zone Size
Crack Tip Opening Displacement (CTOD)
Crack Tip Opening Displacement (CTOD)
• Well’s discovered that many structural
steels used in fracture experiments
exhibit Blunting
• The extent of Blunting increased in
proportion to the toughness of the
material
• Recalling from earlier that crack tip
displacement in the elastic field…
u=2
K r
 
 
  
cos   1 + sin 2   − cos 2   
E  2
 2 
2
 2 
v=2
K r
 
 
  
sin   1 + sin 2   − cos 2   
E  2
 2 
2
 2 
SIF, K
CTOD, δ
Remember,
 E

E =  E
1 − 2
Crack Tip Opening Displacement
0
• Set K=Kic
• At θ=90°
KI
u=2
E
v=2
KI
E
r
 
2  
2   
cos   1 + sin   − cos    = 0
2
 2 
2
 2 
K
r
 
 
  
sin   1 + sin 2   − cos 2    = 4 I
2
E
 2 
2
 2 
0
1
plane stress
plane strain
r
2
Thus the displacement behind the crack tip, v can be measured taking
CTOD, δ
r = ry
Such that,
KI
v=4
E
ry
2
Remember,
 E

E =  R
1 − 2
Crack Tip Opening Displacement
For Elastic-Plastic Materials
1
ry =
2
 KI

  ys



2
ry
2
=
1 KI
2  ys
CTOD, δ
Into v gives,
2 KI 2
v=
 E  ys
2v = 
v
4 K I2
CTOD =  =
 E  ys
plane stress
plane strain
Crack Tip Opening Displacement
• Irwin’s approach Crack Tip Opening Displacement is given as
4 K I2
CTOD =  =
 E  ys
• Later, this expression will be compared to the expression from
Dudgale’s approach.
Dugdale’s Strip Yielding Model
Dugdale’s Strip Yielding Model
• Also called the “Strip Yield Model”
• Dugdale Assumed
1. All plastic deformation concentrates on a strip in front of the crack
2. The notional crack carries the yield stress
3. Superposition / complex functions employed
2a
an
aeff = 2a + 2an
 ys
where Δan is the plastic zone length.
Dugdale’s Approach
• To make it easier to draw, let focus on one half of
the crack in the center-cracked plate
• Superposition – We approximate that the
solution is due to the addition of a remote
tensile stress (elastic) and local compressive
stresses (plastic)
K IA  K IB = K IC + K ID
Plastic Zone
Approximation
Local Crack Closing Stress
Remote Elastic Stress
A
B
C
D
Dugdale’s Approach
• For C
• It can be solve similar to a crack line loading, P
KI =
a+x
a−x
P
a
• Replacing the load with a distributed internal pressure
a
KI =
P
 a −a
a
KI =
P
a 
0
a+x
dx
a−x
a+x
a−x
2P
+
dx =
a−x
a+x
a
a

0
a
a −x
2
2
dx
Dugdale’s Approach
• Note For C
• That the stress are only applied from a  x  a + an
K IC =
a +an
2P
 ( a + an
a + an

)
( a + a )
a
n
2
−x
dx
2
a +an




2
P
a
+

a


(
)
x


n
K IC =
arcsin




a + an  
 ( a + an ) 


a

• Setting,
P = − ys
K = −2 ys
C
I
Gives,
 a 
a + an 
arccos 

 
 a + an  
 ys
Dugdale’s Approach
• For, the remove elastic stress
• at aeff = a + an
D
K ID =   ( a + an )
• Now together,
K IA  K IC + K ID
• It should be noted that a finite value of σy must exist after the crack tip
of “A” due to yielding, such that ,
K K +K =0
A
I
C
I
D
I
−2 ys
 a 
arccos 
 = − 

 a + an 
 
cos 
 2
 ys

a
 =
 a + an
Dugdale’s Approach
• Given,
 
cos 
 2
 ys

a
 =
 a + an
• Manipulated to
 
sec 
 2
 ys
• Note:
• If,

an
 = 1 +
a

−1) E2 n x 2 n
(
x 5x
sec ( x ) = 1 + +
+ ... = 
2 24
2n !
n=
2
x

2
4

an
1  
 1+
= 1+ 
a
2  2 ys
n
2

 + HOT

where E2n is the Euler number
Dugdale’s Approach
an
1  
1+
= 1+ 
a
2  2 ys
• If,
• Then,
2

 + HOT

 

    ys
2 ys 2
  a   KI
an =
= 
8 ys
8   ys
2
2
2

 + HOT

Drop the HOT
  a   KI
an =
= 
8 ys
8   ys
2
2



2
Dugdale’s Approach
• Alternative Solution using Westergaard stress function can be used to
generate Δan
D ( z ) =

 a + an 
1− 

z


2
• Δan carries yield strength, stress reduction can be accounted for by
superimposed point loads
( a + an ) − b 2
C ( z ) =
2
2
2
 z − ( a + an ) ( z − b )
2 Pz
2
where
P =  ys  db
Dugdale’s Approach
• Add these two functions and note that the crack tip singularity cannot
exist due to plasticity
 D ( z ) − C ( z ) = 0
• Manipulation gives,
z
z − ( a + an )
2 ys
2
−
2 ys
z

z − ( a + an )
 a 
−
arccos 
=0

 a + an 
2
2
 a 
arccos 
=0
 a + an 
  a   KI
an =
= 
8 ys
8   ys
2
2



2
Plastic Zone Comparison
Thus, Dugdale plastic zone size is
2
 KI 
  KI 
an = 
 = 0.393 


8   ys 
  ys 
2
This is somewhat larger than the diameter
of the plastic zone according to Irwin.
Irwin’s analysis gives a plastic zone
diameter 2ry,
1  KI
2ry = 
   ys
2

 KI
 = 0.318 

  ys



2
Dugdale’s Strip Yielding Model
Crack Tip Opening Displacement (CTOD)
Crack Tip Open Displacement
• An important aspect of the Dugdale approach in terms of stress
functions is that it enables a basic expression for the CTOD to be
calculated. The crack flank displacement, v, in the region between a
and a + Δan is
• With y=0, in the strip yielding model.
Crack Tip Open Displacement
• The solution of equation (3.18) is fairly difficult but can be furnished
as
2v = 
8 ys a    
CTOD =  =
ln sec 
 E    2  ys
 
 
 
IN the case for LEFM conditions,

 ys
1
 2 a K I 2
CTOD 
=
E  ys E  ys
CTOD, δ
CTOD Comparison
Thus, Dugdale CTOD
 2 a K I 2
CTOD 
=
E  ys E  ys
• The Irwin CTOD is slight larger at
K I2
4 K I2
CTOD =  =
= 1.27
 E  ys
E  ys
Classic Yield Criteria
Classic Yield Criteria
• In the Classic yield criteria the shape is determined from a first order
approximation to the size.
• Classical yield criteria such as
• Tresca
• Von Mises
• Etc.
• Can be employed.
Classic Yield Criteria
• The Von Mises yield criterion states that yielding will occur when
• Where  1 ,  2 ,  3 are the principal stresses
• Consider the stress field equations of a center-crack plate in principal
stress as follows
Plane stress
3 = 0
Plane strain
 3 =  ( 1 +  2 )
Classic Yield Criteria
• Substituting in the Von Mises furnishes,
• Rearraning to solve for r, and
normalizing by ry.
Classic Yield Criteria
• Along the x-axis (θ = 0 °) the plane strain value of r(θ) is much less
than the plane stress value. Assuming ν= 1/3,
• Similar derivations of plastic zone shapes can be obtained for mode II
and mode III loading. Results of such derivations are given in
reference 3 of the bibliography to this chapter.
Classic Yield Criteria
Summary
➢ The plastic zone size is the area ahead of the crack tip where plastic deformation
occurs.
➢ The Irwin and Dugdale approximated the plastic zone size but assume a shape
(circular, strip yielding)
➢ They are only valid when the size of the zone is small compared with the crack length and
when the influence of the boundaries can be ignored (for instance if the width of the
specimen is large).
➢ If these conditions are not met, the overall stress state is more complex than that predicted
using the near-field equations, and the equilibrium balance is more difficult to formulate.
➢ Yield Criterion approximate the plastic zone shape but assume a size.
➢ The Plastic Zone Size and Shape is greatly influenced by thickness (Plane Stress
versus Plane Strain)
Homework 6
• None
References
• Janssen, M., Zuidema, J., and Wanhill, R., 2005, Fracture Mechanics, 2nd
Edition, Spon Press
• Anderson, T. L., 2005, Fracture Mechanics: Fundamentals and Applications,
CRC Press.
• Sanford, R.J., Principles of Fracture Mechanics, Prentice Hall
• Hertzberg, R. W., Vinci, R. P., and Hertzberg, J. L., Deformation and Fracture
Mechanics of Engineering Materials, 5th Edition, Wiley.
• https://www.fracturemechanics.org/
Calvin M. Stewart
Associate Professor
Department of Mechanical Engineering
CONTACT
INFORMATION
The University of Texas at El Paso
500 W. University Ave, Suite A126, El
Paso, TX 79968-0521
Ph: 915-747-6179
cmstewart@utep.edu
me.utep.edu/cmstewart/