1

6.2 CTOD as yield criterion

CTOD : crack tip opening displacement

Instead, Wells proposed t (CTOD ) as a measure of fracture toughness.

Irwin plastic zone

Initial sharp crack has blunted prior to fracture

Well’s experimental work: attempt to measure KIc for structural steels

ButNon-negligible plastic deformation

blunted crack

LEFM inaccurate : materials too tough !!!

Estimation of t using Irwin model :

Crack length: a + ry

By definition, 2 att y yu r r where uy is the crack opening

sharp crack

2

21 22 2 2 2

Iy

K ru sin - cos

Crack opening uy

12 2

IK r

uy

(see eqs 4.40)

and for plane stress, 3

1

2 1

E

We have

42

2yI

trK

E

From Irwin model, the radius of the plastic zone is2

1

2I

yY

Kr

24 I

tY

K

E

CTOD related uniquely to KI and G.

4t

Y

G

and also,

CTOD appropriate characterizing crack-tip-parameter when LEFM no longer valid.

Can be proved by a unique relationship between CTOD and the J integral.

3

More general criterion than K (valid for LEFM)

6.3 The J contour integral as yield criterion

Derive a criterion for elastic-plastic materials, with typical stress-strain behavior:

A→B : linear

A

BC

B→C : non-linear curve

C→D : non-linear, same slope as A-B

D

non-reversibility: A-B-C ≠ C-D-A

Simplification: non-linear elastic behavior

reversibility: A-B-C = C-D-A

Material behavior is strain history dependent !Non unique solutions for stresses

elastic-plastic law replaced by the non-linear elastic law

A/D

BC

= Deformation theory of Plasticity

Correct only for a monotonic loading

4

Definition of the J-integral

Rice defined a path-independent contour integral J for the analysis of crack

showed that its value = energy release rate in a nonlinear elastic body with a crack

Historically,

J generalizes the concept of G to non-linear materials

Fixed-grips conditions: Dead-load conditions

• For linear materials J = G

0

0U P( )d

• Load-displacement diagram: potential energy

0

0

P*U ( P )dP

P 0

0

U

U*

U* : Complementary energy

U : Elastic strain energy

≠ (in general)

5

J da

• Geometrical interpretation:

Definition of J using the potential energy

dJ

dA

A = a B : for a cracked plate with through crack

loading/unloading for the given body with crack lengths a and a+da

OB and OB’

, :P a

Possible relationship between the load P and the displacement while the crack is moving.

J dA Pd dU We have

dU is the difference between the areas under OB’Pdappears as the area AA’B’B

Thus, J dA J B da AA’B’B = OB’B’

,P a

a daa

d

P

B’

A’

B

A

O

and OB : OA’B’ – OAB

+ OAB – OA’B’

6

• In particular ,

At constant force (dual form):

1 *

P

UJ

B a

At constant displacement:

1 UJ

B a

0

0

1 Pd

B a

0

0

1 P

PdP

B a

Generalization of eqs 3.38a and 3.43a to non-linear elastic materials

Useful expressions for the experimental determination of J

d

0

7

• Experimental determination of the J-integral

Multiple-specimen method (Begley and Landes (1972))

(1) Consider cracked specimens with different crack lengths ai

Procedure

(2) For each specimen, record of the load-displacement P-u curve under fixed-grips

(1) (2)

8

(3) Calculation of the potential energy for given values of displacement u

(4) Negative slopes of the P – a curves determined and plotted versus displacement for different crack lengths :

Critical value JIc of J at the onset of crack extension (material constant)

(3)

= area under the load-displacement curve

(4)

9

Single-specimen method by Rice , Merkle and Corten

Generally, when crack lengths that are important compared with the unbroken ligament dimension.

0

2J Pd

b

b : ligament length

Estimation formula derived by Rice et al. (1973) for specimens in tension and under bending

The case of combined tension and bending treated by Merkle and Corten (1974), modified by Clarke and Landes (1979)

J can be determined directly from the load-displacement curve of a single cracked specimen.

Principle

Writing of a relationship between load, displacement and body’s characteristic lengths using a dimensional analysis.

See for example pp 116-119 of Anderson’s book, third ed.

for the deeply cracked three-point bending or compact tension specimen J is given by,

(application pb3 series7)

Example:

10

J as a path-independent line integral

ii

uJ w dy T ds

x

T : traction vector at a point M on the bounding surface , i.e. i ij jT n

The contour is followed in the counter-clockwise direction.

n : unit outward normal.

0

mn

mn ij ijw d

with strain energy density

w dy dsx

u

T

u : displacement vector at the same point M.T

n

crack

M

x

y

11

Equivalence of the two definitions

Recall for the potential energy (per unit thickness),

t

i iS

a wdS T u ds

X x

Y

y

• 2D solid of unit thickness of area S ,

with a linear crack of length a along OX (fixed)

• Crack faces are traction-free.

Imposed tractions on the part of the contour t

Displacements applied on u

• Total contour of the solid including the crack tip

The tractions and displacements imposed on t and u are independent of a

0

ii

S

d ud d wdS T ds

d a d a d a

O

Proof :

Sa

0

0

it

iu

dT, on

dadu

onda

u

u

t

T

ijij

w

i ij jT n

Change of due to a virtual crack extension

0 S

12

Considering the moving coordinate system x , y (attached to the crack tip),

x X a

d x

da a a x

a x

x X a

Thus,

0

i ii

S

u ud w wdS T ds

da a x a x

However,

ij

ij

w w

a a

ijij a

1

2ji

ijj i

uu

a x x

iij

j

u

a x

since ij ji

iij

j

u

x a

d

da: total derivative/crack length

13

We have,

iij

S S j

uwdS dS

a x a

Thus,

0

i iij ij j

S j

u udS n ds

x a a

0

ii

uT ds

a

The derivative of J reduces to,

0

i ii

S

u ud w wdS T ds

da a x a x

0 0

i i ii i

S

u u uwdS T ds T ds

x a a x

0

ii

S

uwdS T ds

x x

14

Using the Green Theorem, i.e. A

Q PP x,y dx Q x,y dy dx dy

x y

0

ii

S

ud wdS T ds

da x x

0

ii

uwdy T ds

x

J derives from a potential

End of the proof

15

Properties of the J-integral

1) J is zero for any closed contour containing no crack tip.

x

y A

ii

uJ wdy T ds

x

ii

A

uwJ dxd y T ds

x x

Using the Green Theorem, i.e.

Closed contour around A

A

Q PP x,y dx Q x,y dy dx dy

x y

Consider

We have

From the divergence theorem,

iij j

A

uwdxd y n ds

x x

iij j

un ds

x

i

ijA j

udxd y

x x

16

However,

ij

ij

w w

x x

ijij x

1

2ji

ijj i

uu

x x x

iij

j

u

x x

since

iij

j

u

x x

ij ji

The integral becomes,

iij

A j

uwJ dxdy

x x x

Replacing in the integral,

Invoking the equilibrium equation, 0ij

jx

0J

iji i iij ij

j j j

u u u

x x x x x x

i

ijj

u

x x

0

17

2) J is path-independent

x

y

Consider the closed contour21 3 4*

2Γ*

and 0J 1 2 3 4

*J J J J J We have

3

The crack faces are traction free :

0i ij jT n on 4and

dy = 0 along these contours

3 40J J

18

x

y

1 3 2 4 2

Γ*

and

2 2*J J

Note that,

1 20J J J

1 2J J

Any arbitrary (counterclockwise) path around a crack gives the same value of J

J is path-independent

2 followed in the counter-clockwise direction.

19

J can be evaluated when the path is a circle of radius r around the crack tip

crack

r

is followed from to =

We have, ds rd dy r cos d

J integral becomes,

,, cos , i

iu r

J w r T r r dx

When r → 0 only the singular terms remain

For LEFM , we can obtain :2I'

KJ G

E (if mode I loading)

(see eq. 6.58)

20

6.4 HRR theory

0 0 0

n

For uniaxial deformation:

Ramberg-Osgood equation

Power law relationship assumed between plastic strain and stress.

Hutchinson Rice and Rosengren: J characterizes the crack-tip field in a non-linear elastic material.

0 0 E

: dimensionless constant

n : strain-hardening exponent

0 = yield strength

For a linear elastic material n = 1.

material properties

21

1 1

1

n

ijJ

Ar

1

2

n n

ijJ

Ar

Path independence of J ij ij varies as 1/r :

Asymptotic field derived by Hutchinson Rice and Rosengren:

Ai are regular functions that depend on and the previous parameters.

The singularity is recovered when n = 1.

1 r

J defines the amplitude of the HRR field as K does in the linear case.

,, cos , i

iu r

J r w r T r dx

From

0ij ij

fas r

r

The product

1 1 13

n n niu A J r

22

Two singular zones can be identified:

HRR based upon small displacements non applicable.

Small region where crack blunting occurs.

Large deformation

KJ

23

Relationship between J and CTOD

ca

y

iij j

uJ n ds

x

x

t

Consider again the strip-yield problem,

The first term in the J integral vanishes because dy= 0 (slender zone)

Y

Y

iij j

un ds

x

but

yyy y

un ds

x

y

Yu

dxx

yY

uJ dx

x

t

t

Y ydu

Y t (see pb1 Series 8)

24

General unique relationship between J and CTOD:

Y tJ m

m : dimensionless parameter depending on the stress state and materials properties

• The strip-yield model predicts that m=1 (non-hardening material, plane stress condition)

• This relation is more generally derived for hardening materials (n >1) using the HRR displacements near the crack tip, i.e.

1 1 13

n n niu A J r

m becomes a (complicated) function of n

blunted crack 90° tShih proposed this definition for t :

4 Y tG

The proposed definition of t agrees with the one of the Irwin model

Moreover,4

m

in this case

25

6.5 Applications the J-integral

Example 6.5.3

J integral along a specific contour

Example 6.5.2

uy

uy