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# 08 Elastic-Plastic Fracture Mechanics

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## Reference : Fracture Mechanics by T.L. Anderson Lecture Note of Eindhoven University of Technology

2017. 10 by Jang, Beom Seon

Topics in Ship Structures

08 Elastic-Plastic Fracture

Mechanics

(2)

Contents

1. Crack-Tip –Opening displacement 2. The J Contour Integral

3. Relationships Between J and CTOD 4. Crack-Growth Resistance Curves 5. J -Controlled Fracture

6. Crack-Tip Constraint Under Large-Scale Yielding 7. Scaling Model for Cleavage Fracture

## 8. Limitations of Two-Parameter Fracture Mechanics

2

0. INTRODUCTION

(3)

Definition of CTOD

 Structural steels has higher toughness than K

Ic

values characterized by LEFM.

 The crack faces moves apart prior to fracture; plastic deformation blunts an initially sharp crack.

 Crack-Tip-opening Displacement (CTOD) : a measure of fracture toughness.

3

## 1. Crack-Tip-opening displacement

Crack-tip-opening displacement (CTOD). Estimation of CTOD from the displacement of the effective crack in the Irwin plastic zone correction.

(4)

Relationship between CTOD and K

I

and G

:

Irwin plastic zone

 Effective crack length = a+r

y

 The displacement r

y

behind the effective crack tip for plane stress.

##  The Irwin plastic zone correction for plane stress.

 CTOD to be related with K

I

or G .

4

1. Crack-Tip-opening displacement

y

,

r  r   

eff y

a  a r

(3 ) / (1 ), (plane stress)

2(1 )

v v G E

    v

3 1

( 1) 1 4

2 2 2 2 2

2(1 )

y y y

y I I I

v v

r v r r

u K K K

E E

v

   

  

 

  

4

2

2

y I

YS

u K

 E

   

1 2

2

I y

YS

r K

 

 

  

 

4

YS

   

G

(5)

Relationship between CTOD and K

I

and G : Irwin plastic zone

 For plane strain

 The Irwin plastic zone correction for plane strain

 CTOD can be related with The Irwin plastic

5

## 1. Crack-Tip-opening displacement

(3 4 ), (plane strain)

2(1 ) v G E

   v

2

( 1) 4(1 ) 4 4

2 2 2 2 2

22(1 ) (1 )

y y y y

y I I I I

r v r r r

u K K K K

E E E

v v

    

 

   

 

4

2

2 3

I y

YS

u K

 E

 

 

1 2

6

I y

YS

r K

 

 

  

 

4

3

YS

   

G

(6)

Relationship between CTOD and K

I

and G : Strip-yield model

 CTOD in a through crack in an infinite plate subject to a remote tensile stress for plate stress.

 Series expansion of the “ln sec” term gives

6

1. Crack-Tip-opening displacement

Estimation of CTOD from the strip-yield model.

4 4 2

2 2 4

y I I

y I I

YS YS

r K K

u K K

E E E

## 8 ln sec 2

YS

YS

a E

  

 

 

  

 

/

YS

### 0

  

2 I

YS YS

K

 E

 

  G

I2

YS YS

K

m E m

     

G 1 for plane stress 2 for plane strain m

m

2 8

2

y

y I

u K r

E

  

2

2 2 2

2

8 ,

8 ,

YS I

YS YS YS

I

a a K

E E E

here K a

   

   

 

 

(7)

Two most common definitions of CTOD

 Two definitions, (a) and (b) are equivalent if the crack blunts in a semicircle.

 CTOD can be estimated from a similar triangles construction:

 Where r is the rotational factor, a dimensionless constant between 0 and 1.

##  The hinge model is inaccurate when displacements are primarily elastic.

7

1. Crack-Tip-opening displacement

Alternative definitions of CTOD: (a) displacement at the original crack tip and (b) displacement at the intersection of a 90° vertex with the crack flanks.

The hinge model for estimating CTOD from three-point bend specimens.

(8)

Two most common definitions of CTOD

## A typical load ( P ) vs. displacement ( V ) curve from a CTOD

 The shape of the load-displacement curve is similar to a stress-strain curve.

 At a given point on the curve, the displacement is separated into elastic and plastic components by constructing a line parallel to the elastic loading line.

 The dashed line represents the path of unloading for this specimen, assuming the crack does not grow during the test.

 The CTOD in this specimen is estimated by

##  The plastic rotational factor r

p

is approximately 0.44 for typical materials and test specimens.

8

1. Crack-Tip-opening displacement

Determination of the plastic component of the crack-mouth-opening displacement 2

I YS

K m E

   

(9)

Introduction

The J contour integral has enjoyed great success as a fracture characterizing parameter for nonlinear

### materials.

An elastic material : a unique relationship between stress and strain.

An elastic-plastic material : more than one stress value for a given strain if the material is unloaded or cyclically loaded.

Nonlinear elastic behavior may be valid for an elastic- plastic material, provided no unloading occurs.

The deformation theory of plasticity , which relates total strains to stresses in a material, is equivalent to nonlinear elasticity.

The nonlinear energy release rate J could be written as a path independent line integral an energy parameter.

J uniquely characterizes crack-tip stresses and strains in nonlinear materials a stress intensity parameter.

9

## 2. The J Contour Integral

Schematic comparison of the stress strain behavior of elastic- plastic and nonlinear elastic materials.

(10)

Nonlinear Energy Release Rate

 Rice [4] presented J is a path-independent contour integral for the analysis of cracks.

 J is equal to the energy release rate in a nonlinear elastic body that contains a crack.

##  The energy release rate for nonlinear elastic materials.

Π : the potential energy, A : the crack area, U : strain energy stored in the body,

F : the work done by external forces.

 U* the complimentary strain energy

10

2. The J Contour Integral

J d

dA

      U F

U P U

*

     

*

0

U  P  dP Nonlinear energy release rate.

Crack length = a+da Crack length = a

(11)

Nonlinear Energy Release Rate

##  If the plate is in load control, J is given by

 If the crack advances at a fixed displacement, F = 0, and J is given by

 J for load control is equal to J for displacement control.

 J in terms of load

 J in terms of displacement

11

2. The J Contour Integral

Nonlinear energy release rate.

*

P

d dU

J dA dA

  

      

d dU

J dA dA

  

      

0 0

P P

P P

J dP dP

a a

 

   

              

0 0

J Pd P d

a a

 

   

                 U F U

   

* 1 / 2 , * 0, 0

dU   dU  dPd  dU  dU 

*

0

U  

P

 dP

(12)

Nonlinear Energy Release Rate

 More general version of the energy release rate. For the special case of a linear elastic material, J= G . For linear elastic Mode I,

##  Caution when applying J to elastic-plastic material.

 The energy release rate : the potential energy that is released from a structure when the crack grows in an elastic material.

 However, much of the strain energy absorbed by an elastic-plastic material is not recovered. A growing crack in an elastic-plastic material leaves a plastic wake.

 Thus, the energy release rate concept has a somewhat different interpretation for elastic-plastic materials.

 J indicates the difference in energy absorbed by specimens with neighboring crack sizes.

12

## 2. The J Contour Integral

2

K

I

J  E

plastic wake

J dU

dA

 

    

(13)

J as a Path-Independent Line Integral

 Consider an arbitrary counterclockwise path (Γ) around the tip of a crack

 The strain energy density is defined as

13

2. The J Contour Integral

Arbitrary contour around the tip of a crack.

 The traction is a stress vector at a given point on the contour.

## where n

j

are the components of the unit vector normal to Γ .

 J : a path-independent integral

(14)

The J Contour Integral - Proof

Evaluate𝐽along Γ

Using divergence theorem, the line integral can be converted into an areal integral.

Using the chain rule and the definition of strain energy density, the first term in square bracket in Eq. (a). Here, w = strain energy density.

Applying the strain-displacement relationship and ij=ji

Invoking the equilibrium condition

Thus, J=0 for any closed contour

14

## 2. The J Contour Integral

Γ: closed contour

𝐴: Area enclosed by Γ

(a) “To be proved in the next slide.”

0

* *

*

ij i

### 0

A A

j

u

w w w

J dxdy dxdy

x x  x x x

          

                         

(15)

The J Contour Integral - Proof

15

2. The J Contour Integral

1 2

cos( 90) sin( 90)

s s s

n s n s y x

 

      

       

n i j

i j i j

A

C

B

θ

cos sin

s s s

x y

 

    

   

r i j

i j

xi

 s

yj

1 1 2 2

* * *

1 2 1 2

* *

1 1

*

i i ij j i i i i i

i i i i

i i i i

A

i

u u u u

J wdy T ds wdy n ds wdy n ds n ds

x x x x

u u w u u

wdy dy dx dxdy dxdy dxdy

x x x x x y x

w

x x

  

   

   

     

                      

 

      

     

                            

 

 

 

  

 

* 2 *

2

i i i

i ij

A A

j

u u w u

dxdy dxdy

x x  x  x x  x 

                  

                     

   

 

 

2 1

1 2

R C

F F

dxdy F dx F dy

x y

      

   

 

 

1

,

2

x  x x  y

 Using divergence theorem, the line integral can be converted into an areal

## integral.

(16)

The J Contour Integral - Proof

 Consider now two arbitrary contours.

 If Γ

1

and Γ

2

are connected by segments along the crack face (Γ

3

and Γ

4

), a closed contour is formed.

 The total J along the closed contour is equal to the sum of contributions from each segment:

##  On the crack face, T

i

=dy = 0.

 Thus, J

1

= −J

3

.

“Therefore, any arbitrary (counterclockwise) path around a crack will yield the same value of J; J is path-independent.”

16

2. The J Contour Integral

Two arbitrary contours Γ1 and Γ2 around the tip of a crack.

2

2 4 i

u

i

### 0

J J wdy T ds

x

  

         

(17)

J as a Nonlinear Elastic Energy Release Rate

Consider a two-dimensional cracked body bounded by the curve Γ.

Under quasi-static conditions and in the absence of body forces, the potential energy is

Consider the change in potential energy resulting from a virtual extension of the crack:

When the crack grows, the coordinate axis moves. Thus a derivative with respect to crack length can be written as

(b)를 (a)에 적용

By applying the same assumptions

17

## 2. The J Contour Integral

(b)

(a)

(18)

J as a Nonlinear Elastic Energy Release Rate

 Invoking the principle of virtual work gives

 Thus,

 Applying the divergence theorem and multiplying both sides by –1 leads to

Therefore, the J contour integral is equal to the energy release rate for a linear or nonlinear elastic material under quasi-static conditions.

18

## 2. The J Contour Integral

 

2 1

1 2

R C

F F

dxdy F dx F dy

x y

      

   

 

 

F2 w F, 1 0

A x

wdxdy wdy wn ds

x

     

 

n    s yi xj

x y

y x

n n

s s

 

   

 

n i j i j

(19)

J as a Stress Intensity Parameter

Ramberg-Osgood eq. : Inelastic stress-strain relationship for uniaxial deformation

In order to remain path independent, stress–strain must vary as 1/r near the crack tip.

At distances very close to the crack tip, well within the plastic zone, elastic strains are small in comparison to the total strain, and the stress strain behavior reduces to a simple power law.

k1 and k2 are proportionality constants, which are defined more precisely below. For a linear elastic material, n = 1.

19

## 2. The J Contour Integral

(20)

J as a Stress Intensity Parameter

The actual stress and strain distributions are obtained by applying the appropriate boundary conditions HRR singularity named after Hutchinson, Rice, and Rosengren.

The J integral defines the amplitude of the HRR singularity.

In : an integration constant that depends on n.

 𝜎𝑖𝑗 and 𝑖𝑗 𝜀 : dimensionless functions of n and θ.

20

2. The J Contour Integral

Effect of the strain-hardening exponent on the

HRR integration constant. Angular variation of dimensionless stress for n = 3 and n = 13

plane stress

plane strain

(21)

## The Large Strain Zone

 The HRR singularity predict infinite stresses as r → 0, The large strains at the crack tip cause the crack to blunt, which reduces the stress triaxiality locally. The blunted crack tip is a free surface; thus 

xx

must vanish at r = 0.

 HRR singularity does not consider the effect of the blunted crack tip on the stress fields, nor the large strains that are present near the crack tip.

21

## 2. The J Contour Integral

Large-strain crack-tip finite element results.

 

yy/

0

reaches a peak when x

0

/J is unity  twice the CTOD.

 The HRR singularity is invalid within this region, where the stresses are influenced by large strains and crack blunting.

 However, as long as there is a region surrounding the crack tip, the J

## integral uniquely characterizes crack-

tip conditions, and a critical value of

J is a size independent measure of

fracture toughness.

(22)

Laboratory Measurement of J

 Linear Elastic : J = G, G is uniquely related to the stress intensity factor.

 Nonlinear : The principle of superposition no longer applies, J is not proportional to the applied load.

 One option for determining J is to apply the line integral  J integral in test panels by attaching an array of strain gages in a contour around the crack tip.

##  This method can be applied to finite element analysis.

22

2. The J Contour Integral

Schematic of early experimental measurements of J, performed by Landes and Begley.

 A series of test specimens of the same size, geometry, and material and introduced cracks of various lengths.

 Multiple specimens must be tested

and analyzed to determine J.

(23)

## Laboratory Measurement of J

 J directly from the load displacement curve of a single specimen.

 Double-edge-notched tension panel of unit thickness.

 dA = 2da = −2 db

 Assuming an isotropic material that obeys a

Ramberg- Osgood stress-strain law.

From the dimensional analysis,

 Φ is a dimensionless function. For fixed material properties,

23

## 2. The J Contour Integral

(24)

Laboratory Measurement of J

If plastic deformation is confined to the ligament between the crack tips, we can assume that b is the only length dimension that influences Δp.

If plastic deformation is confined to the ligament between the crack tips b is the only length dimension that influences Δp.

Taking a partial derivative with respect to the ligament length

Integrating by part

24

## 2. The J Contour Integral

 

0 0 0 0

0 0 0

1 1 1

2 2 2

1 1 1

2 2 2

2 2 2

P P

P P P P

P P

P P P P

P b

P

P P P P P P P

dP P dP P dP dP

b b P b

P dP P P Pd Pd P

b b b

 

   

        

   

  

fg dx fg f gdx

 

0 0

P P

PdP P P Pd P

  

 

P 0

P

Pd P

0 P

PdP

(25)

Laboratory Measurement of J

 Therefore

 Unit thickness is assumed at the beginning of this derivation  The J integral has units of energy/area.

25

2. The J Contour Integral

0 0

1 1

2 2 2

P P

P

P P

P

dP Pd P

b b



  
(26)

Laboratory Measurement of J

An edge-cracked plate in bending Example

Ω = Ωnc (angular displacement under no crack)+ Ωc (angular displacement when cracked)

If the crack is deep, Ωc >> Ωnc.

The energy absorbed by the plate

##  J for the cracked plate in bending can be written as

26

2. The J Contour Integral

U  

0

M d 

(27)

Laboratory Measurement of J

If the material properties are fixed, dimensional analysis leads to

Integration by parts

If the crack is relatively deep Ωnc should be entirely elastic, while Ωc may contain both elastic and plastic contributions.

27

2. The J Contour Integral

2 3 2 2

0 0 0

2 0 0 0

2 2 1

2 2 2

2

c

M M M

c M

M M

c c c

M M M

J dM F dM F MdM

b b b b b b

M M

F M F dM M dM Md

b b b b b

       

            

       

              

  

  

fg dx fg f gdx

 

2 3

2

c M

M M

b F b b

      

     

 

c

M

0 0

M c

cM cdM MdM

0

2

c

J Md

c

b

  

( )

( )

0 0

2

c el

2

p

c el p

J Md Md

b b

 

          or J  K E

I2

  b 2 

0p

Md 

p
(28)

Laboratory Measurement of J

 General Expression

##  In general, the J integral for a variety of configurations can be written in the following form

 η : dimensionless constant. Note the above Eq. contains the actual thickness, while the above derivations assumed a unit thickness for convenience.

 For a deeply cracked plate in pure bending, η = 2, it can be separated into elastic and plastic components.

28

2. The J Contour Integral

c

M

0

2

c

J Md

c

b

 

Uc

0cMdc
(29)

General

##  For linear elastic conditions, the relationship between CTOD and G is given by

 Sinc J= G for linear elastic material behavior, in the limit of small-scale yielding,

 where m is a dimensionless constant that depends on the stress state and material properties. It can be shown that it applies well beyond the validity limits of LEFM.

 Consider, for example, a strip-yield zone ahead of a crack tip,

29

## 3. Relationships between J and CTOD

Contour along the boundary of the strip-yield zone ahead of a crack tip

 If the damage zone is long and slender (𝜌 >> δ) , the first term in the J contour integral vanishes because dy = 0.

-

m=1 for plane stress m=2 for plane strain

0 0

𝜌

(30)

General

 Since the only surface tractions within 𝜌 are in the y direction

##  Define a new coordinate system with the origin at the tip of the strip-yield zone (𝑋 = 𝜌 − 𝑥)

 Since the strip-yield model assumes 

yy

= 

YS.

 Thus the strip-yield model, which assumes plane stress conditions and a non-hardening material, predicts that m = 1 for both linear elastic and elastic plastic conditions

30

3. Relationships between J and CTOD

1 11 1 12 2 13 3

2 21 1 22 2 23 3 22

3 31 1 32 2 33 3

0

0

yy

T n n n

T n n n

T n n n

   

𝜌

(31)

## General

Many materials with high toughness do not fail catastrophically at a particular value of J or CTOD. Rather, these materials display a rising R curve, where J and CTOD increase with crack growth.

In the initial stages, there is a small amount of apparent crack growth due to blunting. As J increases, the material at the crack tip fails locally and the crack advances further.

Because the R curve is rising, the initial crack growth is usually stable, but an instability can be encountered later.

One measure of fracture toughness JIcis defined near the initiation of stable crack growth.

31

## 4. Crack-Growth Resistance Curves

Schematic J resistance curve for a ductile material

The definition of JIc is somewhat arbitrary.

The slope of the R curve is indicative of the relative stability of the crack growth;

### a material with a steep R curve is less likely to experience unstable crack propagation.

For J resistance curves, the slope is usually quantified by a dimensionless tearing modulus:

(32)

## Stable and Unstable Crack Growth

Instability occurs when the driving force curve is tangent to the R curve.

Load control is usually less stable than displacement control.

In most structures, between the extremes of load control and

### displacement control. The intermediate case can be represented by a spring in series with the structure, where remote displacement is fixed

Driving force can be expressed in terms of an applied tearing modulus:

ΔT : the total remote displacement, Cm (the system compliance), Δ(line displacement)

32

## 4. Crack-Growth Resistance Curves

The slope of the driving force curve for a fixed ΔT is

Stable crack growth

Unstable crack growth

Schematic J driving force/R curve diagram which compares load control and displacement control.

&

R app R

JJ TT

&

R app R

JJ TT

Cm

(33)

Stable and Unstable Crack Growth

##  For most structure

,

If the structure is held at a fixed remote displacement T assuming & J depends only on load and crack length.

For load control, Cm=and for displacement control, Cm= 0 , T=.

33

4. Crack-Growth Resistance Curves

T m 0

P a

d da dP C dP

a P

 

   

      

P a

J J

dJ da dP

a P

 

   

   

T P a T

J J J P

a a P a

   

       

        

       

1

T

m

P a a

J J J

a a P a C P

 

    

         

             

          

Cm

1

P

m

P a

C dP P

a P a

 

  

     

       

T P

J J

a a

 

   

     

   

(34)

Stable and Unstable Crack Growth

 The point of instability in a material with a rising R curve depends on the size and geometry of the cracked structure; a critical value of J at instability is not a material property if J increases with crack

## growth.

 However, It is usually assumed that the R curve, including the J

IC

value, is a material property, independent of the configuration. This is a reasonable assumption, within certain limitations.

34

4. Crack-Growth Resistance Curves

(35)

Computing J for a Growing Crack

The cross-hatched area represents the energy that would be released if the material were elastic.

In an elastic-plastic material, only the elastic portion of this energy is released; the

### remainder is dissipated in a plastic wake.

The energy absorbed during crack growth in an nonlinear elastic-plastic material exhibits a history dependence.

35

## 4. Crack-Growth Resistance Curves

The geometry dependence of a J resistance curve is influenced by the way in which J is calculated.

The equations derived in Section 3.2.5 are based on the pseudo energy release rate definition of J and are valid only for a stationary crack.

There are various ways to compute J for a growing crack.

The deformation J : used to obtain experimental J resistance curves.

.

Schematic load-displacement curve for a specimen with a crack that grows to a1 from an initial length ao.

(36)

## Computing J for a Growing Crack

The cross-hatched area represents the energy that would be released if the material were elastic.

In an elastic-plastic material, only the elastic portion of this energy is released; the

### remainder is dissipated in a plastic wake.

The energy absorbed during crack growth in an nonlinear elastic-plastic material exhibits a history dependence.

36

## 4. Crack-Growth Resistance Curves

The geometry dependence of a J resistance curve is influenced by the way in which J is calculated.

The equations derived in Section 3.2.5 are based on the pseudo energy release rate definition of J and are valid only for a stationary crack.

There are various ways to compute J for a growing crack; the deformation J and the far-field J,

The deformation J

used to obtain experimental J resistance curves.

Schematic load-displacement curve for a specimen with a crack that grows to a1 from an initial length ao.

(37)

## Computing J for a Growing Crack

37

4. Crack-Growth Resistance Curves

The deformation J - continued

the J integral for a nonlinear elastic body with a growing crack is given by

### where b is the current ligament length.

The calculation of UD(p) is usually performed incrementally, since the deformation theory load displacement curve depends on the crack size.

A far-field J

For a deeply cracked bend specimen, J contour integral in a rigid, perfectly plastic material

By the deformation theory

The J integral obtained from a contour integration is path-dependent when a crack is growing in an elastic-plastic material, however, and tends to zero as the contour shrinks to the crack tip.

(38)

## Stable and Unstable Crack Growth

 Ex 3.2) Derive an expression for the applied tearing modulus in the double cantilever beam (DCB) specimen with a spring in series

38

4. Crack-Growth Resistance Curves

(39)

Stationary Cracks

Small-scale yielding,

K uniquely characterizes crack-tip conditions, despite the fact that the 1/ 𝑟 singularity does not exist all the way to the crack tip.

Similarly, J uniquely characterizes crack-tip conditions even though the deformation plasticity and small

### strain assumptions are invalid within the finite strain region.

Elastic-plastic conditions

J is still approximately valid, but there is no longer a K field.

As the plastic zone increases in size (relative to L), the K-dominated zone disappears, but the J-dominated zone persists in some geometries.

The J integral is an appropriate fracture criterion

39

## 5. J-Controlled Fracture

2 2

small-scale yielding

Elastic-plastic conditions

(40)

Stationary Cracks

 Large-scale yielding

 the size of the finite strain zone becomes significant relative to L , and there is no longer a region uniquely characterized by J .

 Single-parameter fracture

mechanics is invalid in large-scale yielding, and critical J values exhibit a size and geometry dependence.

##  For a given material,5 dimensional analysis leads to the following

functional relationship for the

stress distribution within this region:

40

5. J-Controlled Fracture

Large-scale yielding

(41)

J-Controlled Crack Growth

 J- controlled conditions exist at the tip of a stationary crack (loaded monotonically and quasistatically), provided the large strain region is small compared to the in-plane dimensions of the cracked body

41

## 5. J-Controlled Fracture

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