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### Transcript of Elastic-Plastic Fracture Mechanics - Elastic-Plastic Fracture Mechanics Introduction...

• Elastic-Plastic Fracture Mechanics

Introduction

• When does one need to use LEFM and EPFM?

• What is the concept of small-scale and large-scale yielding?

Contents of this Chapter

• The basics of the two criteria used in EPFM: COD (CTOD), and J-Integral (with H-R-R)

• Concept of K- and J-dominated regions, plastic zones

• Measurement methods of COD and J-integral

• Effect of Geometry

Background Knowledge

• Theory of Plasticity (Yield criteria, Hardening rules)

• Concept of K, G and K-dominated regions

• Plastic zone size due to Irwin and Dugdal

• LEFM and EPFM

LEFM

• In LEFM, the crack tip stress and displacement field can be uniquely characterized by K, the

stress intensity factor. It is neither the magnitude of stress or strain, but a unique parameter that

describes the effect of loading at the crack tip region and the resistance of the material. K filed is

valid for a small region around the crack tip. It depends on both the values of stress and crack size.

We noted that when a far field stress acts on an edge crack of width “a” then

for mode I, plane strain case

σ

σ

τ π

θ

θ θ

θ θ

θ θ

x x

y y

x y

IK

r

R S |

T|

U V |

W| =

+

L

N

M M M M M MM

O

Q

P P P P P PP

2 2

1 2

3

2

1 2

3

2

2

3

2

c o s

s i n ( ) s i n ( )

s i n ( ) s i n ( )

s i n ( ) s i n ( )

σ σ ν σ σzz zz xx yy= = +0 for plane stress; for plane strain( )

u

u

K r k

k

x

y

I RST UVW

= − +

+ −

L

N

M M MM

O

Q

P P PP

2

2 1 2

2

2 1 2

2

2

2µ π

θ θ

θ θ2

cos ( sin ( ))

sin ( cos ( ))

• LEFM concepts are valid if the plastic zone is much smaller than the singularity zones.

Irwin estimates

Dugdale strip yield model:

r K

p

I

ys

= 1

2

2

π σ ( )

r K

p I

ys

= 1

8

2 ( ) σ

ASTM: a,B, W-a 2.5 , i.e. of specimen dimension.≥ ( ) KI

ys σ

2 r p ≤

1

50

LEFM cont.

Singularity dominated region

σ

σ

τ π

x x

y y

x y

IK

r

R S |

T |

U V |

W |

=

L

N M M M

O

Q P P P2

1

1

0

For =0 θ

For = 2

all ijθ θ

σ, = 0

• EPFM

• In EPFM, the crack tip undergoes significant plasticity as seen in the following diagram.

sharp tip

I d e a l e la s ti c b r i t t l e b e h a v i o r

c l e a v a g e f r a c tu r e

P : A p p l ie d l o a d

P : Y i e ld l o a dy D is p la c e m e n t , u

ra ti

o ,

P /P

y

1 .0

F r ac tu re

B lu n t t i p

L im ite d p la st ic it y a t c r ac k

t ip , s t ill c l e a v ag e f ra c tu r e

D isp la cem en t, u

ra ti

o ,

P /P

y

1 .0

F r ac tu re

• Blunt tip

Void formation & coalescence

failure due to fibrous tearing Disp lacement, u

ra ti

o ,

P /P

y

1 .0 Fracture

l a r g e s c a l e

b l u n t i n g

L a r g e s c a l e p l a s t i c i t y

f i b r o u s r a p t u r e / d u c t i l e

f a i l u r e D is p la c e m e n t , u

ra ti

o , P /P

y

1 .0 F rac tu re

• EPFM cont.

• EPFM applies to elastoc-rate-independent materials, generally in the large-scale plastic

deformation.

• Two parameters are generally used:

(a) Crack opening displacement (COD) or crack tip opening displacement (CTOD).

(b) J-integral.

• Both these parameters give geometry independent measure of fracture toughness.

δSharp crack

Blunting crack

y

x

Γ ds

• EPFM cont.

• Wells discovered that Kic measurements in structural steels required very large thicknesses for

LEFM condition.

--- Crack face moved away prior to fracture.

--- Plastic deformation blunted the sharp crack.

δ Sharp crack

Blunting crack

• Irwin showed that crack tip plasticity makes the crack behave as if it were longer, say from size a to a + rp -----plane stress

From Table 2.2,

Set ,

r K

p I

y s

= 1

2

2

π σ ( )

u K r

ky I= + −

2 2 2 1 2

2

2

µ π θ θ

s in ( ) [ c o s ( ) ]

θ π= u k K r

y I

y =

+ 1

2 2µ π a ry+

θ π=

δ π σ

= =2 4 2

2 u

K

E y

I

y s

Note:

since

k E= − +

= + 3

1 2 1

ν ν

µ ν and ( )

δ π σ

= =CTOD 4 G

ys

G K

E

I= 2

• CTOD and strain-energy release rate

• Equation relates CTOD ( ) to G for small-scale yielding. Wells proved that

Can valid even for large scale yielding, and is later shown to be related to J.

• can also be analyzed using Dugdales strip yield model. If “ ” is the opening at the end of the strip.

δ π σ

= =CTOD 4 G

ys

δ δ

δ δ

δ σ

ys

Consider an infinite plate with a image crack subject to a

Expanding in an infinite series,

σ σ∞ =

δ σ

π π σ

σ = = ⋅2

8 u

a

E y

y s

y s

li n s e c ( 2

)

δ σ

π π σ

σ π σ

σ = ⋅ + ⋅ +

8 1

12

2 4ys

ys ys

a

E [ 1

2 (

2 (

2 ) ) . . . ]

If , and can be given as:

In general,

δ σ

π σ σ

= + ⋅ K

E

I

y s y s

2

2[ 1 1

6 (

2 ) ]

σ σ

σ σ δ σ σys

ys

ysE

G ≈

• Alternative definition of CTOD

δSharp crack

Blunting crack

δBlunting crack

Displacement at the original crack tip Displacement at 900 line intersection, suggested by Rice

CTOD measurement using three-point bend specimen

W

P

a

r p(W-a)

z

Vp

δ p

'

'

'δ p l p p

p

r W a V

r W a a z =

− + +

( )

( )

displacement

expandingδ

• Elastic-plastic analysis of three-point bend specimen

δ δ δ σ

= + = + −

− + +el p l I

ys

p p

p

K

m E

r W a V

r W a a z

2 ( )

( )

Where is rotational factor, which equates 0.44 for SENT specimen.δ pl

• Specified by ASTM E1290-89 --- can be done by both compact tension, and SENT specimen

• Cross section can be rectangular or W=2B; square W=B

KI is given by

δ ν

σe l I

y s

K

E =

−2 21 2

( )

K P

B W f

a

W I = ⋅ ( )

δ p l p p

p

r W a V

r W a a z =

− + +

( )

( )

lo a d

Mouthopening

υ p υ e

V,P

• CTOD analysis using ASTM standards

Figure (a). Fracture mechanism is purely cleavage, and critical CTOD

• More on CTOD

The derivative is based on Dugdale’s strip yield model. For

Strain hardening materials, based on HRR singular field.

By setting =0 and n the strain hardening index based on

*Definition of COD is arbitrary since

A function as the tip is approached *Based on another definition, COD is the distance between upper

and lower crack faces between two 45o lines from the tip. With this

Definition

2

or ICOD y y

K J

E δ

σ σ = =

( ) 11

1 ,

n

n

n i y i

y y n

J u r u n

I αε θ

ασ ε

+ +

  =  

 

1

3

2

n

y ije

y y y

ε σσα ε σ σ

−  

=     ( ) ( ), 0 , 0y yu x u xδ + −= −

( ) 1

1nx +−

COD n

y

J dδ

σ =

• Where ranging from 0.3 to 0.8 as n is varied from

3 to 13 (Shih, 1981)

*Condition of quasi-static fracture can be stated as the

Reaches a critical value .