08 Elastic-Plastic Fracture Mechanics

(1)

OPen INteractive Structural Lab

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

_y ^I

YS

u K

 E

   

1 2

2

I y

YS

r K

 

 

  

 

4

YS

   

G

(5)

_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)

_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

Estimation of CTOD from the strip-yield model.

4 4 2

2 2 4

y I I

YS YS

r K K

u K K

E E E

 ^ ^  ^  ^ 

8 ln sec 2

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 ,

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

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

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.

 For load control,

 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)

 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

Nonlinear energy release rate.

*

P

d dU

J dA dA

  

      

d dU

J dA dA

_

  

      

0 0

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)

 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

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)

¹⁵

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

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)

 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

Two arbitrary contours Γ¹and 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 ^x^j

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.

 I_n : an integration constant that depends on n.

 𝜎_𝑖𝑗 and _𝑖𝑗 𝜀 : dimensionless functions of n and θ.

20

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

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)

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

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

 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

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)

 Therefore

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

25

0 0

1 1

2 2 2

P P

P

P P

P

dP Pd P

b b

 

   





    



   

(26)

An edge-cracked plate in bending Example

 Ω = Ω_nc (angular displacement under no crack)+ Ω_c (angular displacement when cracked)

 If the crack is deep, Ω_c >> Ω_nc.

 The energy absorbed by the plate

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

26

U  

0^

M d 

(27)



If the material properties are fixed, dimensional analysis leads to

 Integration by parts

 If the crack is relatively deep Ω_nc should be entirely elastic, while Ω_c may contain both elastic and plastic contributions.

27

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 

₀^^p

Md 

^p

(28)

 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

c

M

0

2

^c

J Md

c

b

 



^U^c ^



⁰^^c^Md^^c

(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

① ②

④ ③

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

yy

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 J_Icis 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 J_Ic 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:

 Δ_T : the total remote displacement, C_m (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

J  J T T

&

R app R

J  J T T

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, C_m=  and for displacement control, C_m= 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

(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 a_o.

(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 a_o.

(37)

Computing J for a Growing Crack

³⁷

 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 U_D(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

(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