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

cmstewart@utep.edu

me.utep.edu/cmstewart/