45
Crack Tip Plasticity Fracture Mechanics
Crack Tip Plasticity
Fracture Mechanics
Crack Tip PlasticityPresented 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
eff na a a= +
aeff = effective crack length a = physical crack lengthΔan = increment in the notational crack length
• aeff behaves as part of the crack. The stress distribution ahead of aeff begins to match the elastic conditions such that,
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)
The areas are equivalent to each other
effa
B
AA B=
0 2
y
ys
r
Iys y
a
Kdr r
r
=
= −
Α
B
Fracture Toughness, Kic
a
Irwin’s Plastic Zone Size
The size of the plastic zone can be determined
A B=
( )
( )( )
( )( )
( )
0
0
2
2
2
2
2
2
22
y
y
r
Iys ys y
r
ys ys y
ys y y
y y
ys
Iy y
ys
Ka dr r
r
a aa dr r
r
a aa r r
a aa r r
Ka r r
= −
+ = −
+ + =
+ + =
+ =
( ) 2y y ya r r r + =
Note,
Effective SIF, Keff
effa
B
A
a
( )effK a a = +
2I ysK r =
Fracture Toughness, Kic
Irwin’s Plastic Zone Size
Thus, Irwin’s plastic zone size is twice the First Approximation
• ry = nominal crack tip;
• 2ry = diameter of plastic zone
• Plastic zone size and crack tip singularity characterized by K.
• Common Design Problem• If Ki is applied to a for a variety of materials
find the ryThe size of the plastic zone can be determined
effa
a
( ) 2y ya r r + =
Irwin’s Plastic Zone Size
2 2
12 0.318I I
y
ys ys
K Kr
= =
Irwin’s Plastic Zone SizeCrack 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…
2 2
2 2
2 cos 1 sin cos2 2 2 2
2 sin 1 sin cos2 2 2 2
K ru
E
K rv
E
= + −
= + −
CTOD, δ
SIF, K
Crack Tip Opening Displacement
• Set K=Kic
• At θ=90°
2 2
2 2
2 cos 1 sin cos 02 2 2 2
2 sin 1 sin cos 42 2 2 2 2
I
I I
K ru
E
K Kr rv
E E
= + − =
= + − =
2
1
E plane stress
E Eplane strain
= −
1 0
0
Remember,
Thus the displacement behind the crack tip, v can be measured taking
Such that,
42
y
yI
r r
rKv
E
=
=
CTOD, δ
Crack Tip Opening Displacement
Remember,
For Elastic-Plastic Materials
Into v gives,
2
1
2
Iy
ys
Kr
=
1
2 2
y I
ys
r K
=
22 I
ys
Kv
E =
2v =
CTOD, δ
24 I
ys
KCTOD
E
= =
2
1
E plane stress
E Rplane strain
= −
v
Crack Tip Opening Displacement
• Irwin’s approach Crack Tip Opening Displacement is given as
• Later, this expression will be compared to the expression from Dudgale’s approach.
24 I
ys
KCTOD
E
= =
Dugdale’s Strip Yielding Model
Dugdale’s Strip Yielding Model
• Also called the “Strip Yield Model”
• Dugdale Assumed1. 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
where Δan is the plastic zone length.
2a2 2eff na a a= +
na
ys
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)
A B C D
I I I IK K K K = +
Plastic Zone Approximation Local Crack Closing Stress Remote Elastic Stress
A B C D
Dugdale’s Approach• For
• It can be solve similar to a crack line loading, P
• Replacing the load with a distributed internal pressure
C
I
P a xK
a xa
+=
−
2 20 0
2
a
I
a
a a
I
P a xK dx
a xa
P a x a x P aK dx dx
a x a xa a a x
−
+=
−
+ −= + =
− + −
Dugdale’s Approach• Note For
• That the stress are only applied from
• Setting, Gives,
C
na x a a +
( ) ( )
( )
( )
2 2
2
2arcsin
n
n
a a
C nI
an n
a a
C n
I
nn a
a aPK dx
a a a a x
P a a xK
a aa a
+
+
+ =
+ + −
+ =
+ +
ysP = −
ys
2 arccosC nI ys
n
a a aK
a a
+ = −
+
Dugdale’s Approach• For, the remove elastic stress
• at
• Now together,
• It should be noted that a finite value of σy must exist after the crack tip of “A” due to yielding, such that ,
eff na a a= +
D
( )D
I nK a a = +
A C D
I I IK K K +
0A C D
I I IK K K + =2
arccosys
n
a
a a
− = −
+ cos
2 ys n
a
a a
= +
Dugdale’s Approach• Given,
• Manipulated to
• Note:
• If,
cos2 ys n
a
a a
= +
sec 12
n
ys
a
a
= +
( )( ) 22 4
215sec 1 ...
2 24 2 !
n n
n
n
E xx xx
n
=
−= + + + = where E2n is the Euler number
2
11 1
2 2 2
n
ys
ax HOT
a
+ = + +
Dugdale’s Approach
• If,
• Then,
Drop the HOT
2
11 1
2 2
n
ys
aHOT
a
+ = + +
2 2ys
ys
22 2
8 8
In
ys ys
Kaa HOT
= = +
22 2
8 8
I
n
ys ys
Kaa
= =
Dugdale’s Approach• Alternative Solution using Westergaard stress function can be used to
generate Δan
• Δan carries yield strength, stress reduction can be accounted for by superimposed point loads
( )2
1
D
n
za a
z
=
+ −
( )( )
( ) ( )
2 2
2 22
2 n
C
n
Pz a a bz
z a a z b
+ − =
− + − ysP db= where
Dugdale’s Approach• Add these two functions and note that the crack tip singularity cannot
exist due to plasticity
• Manipulation gives,
( ) ( ) 0D Cz z − =
( ) ( )2 22
2arccos 0
ys
nn n
z z a
a az a a z a a
− =
+ − + − +
2arccos 0
ys
n
a
a a
− =
+
22 2
8 8
I
n
ys ys
Kaa
= =
Thus, Dugdale plastic zone size is
This is somewhat larger than the diameter of the plastic zone according to Irwin.
Irwin’s analysis gives a plastic zone diameter 2ry,
2 2
12 0.318I I
y
ys ys
K Kr
= =
2 2
0.3938
I In
ys ys
K Ka
= =
Plastic Zone Comparison
Dugdale’s Strip Yielding ModelCrack 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
8ln sec
2
ys
ys
aCTOD
E
= =
IN the case for LEFM conditions,
1ys
22
I
ys ys
KaCTOD
E E
=
2v =
CTOD, δ
Thus, Dugdale CTOD
• The Irwin CTOD is slight larger at
CTOD Comparison
22
I
ys ys
KaCTOD
E E
=
2 24
1.27I I
ys ys
K KCTOD
E E
= = =
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 are the principal stresses
• Consider the stress field equations of a center-crack plate in principal stress as follows
1 2 3, ,
3 0 =
Plane stress
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/
CONTACT INFORMATION
Calvin M. Stewart
Associate Professor
Department of Mechanical Engineering
The University of Texas at El Paso
500 W. University Ave, Suite A126, El Paso, TX 79968-0521
Ph: 915-747-6179
me.utep.edu/cmstewart/
