Crack Tip Opening Displacement (CTOD) and J Integral Methods in Fracture Mechanics, Study notes of Mechanics of Materials

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Chapter 4Chapter 4

1

OverviewOverview

Evolution of Crack BluntingEvolution of Crack Blunting

As crack opens under applied load

Crack advances at criticalvalue of CTOD

FIGURE 3.2:

Estimation of CTOD from the

displacement of the effective crack in the Irwinplastic zone correction.

The effect of a plastic zone at the crack tip is to extend the

effective length of the crack by r

y

half the diameter of the

plastic zone.

Hence the opening of the crack at it’s real tip can beapproximated from the calculated elastic displacements of thevirtual (extended) crack evaluated at a point some r

y

from the

y

virtual crack tip as shown in Figure 4.1.

Figure 4.1 : Additional crack opening as a result of plasticity at crack tip

CTOD,

However from Chapter 2 (LEFM);

E

K

G

I

2

Or according to Dugdale model;

ys I

yy

K

E

u

πσ

δ

2

4

2

=

=

where m is a constant 1 for plane stress and 2 for plane strain (1< m < 3)

ys

ys

m

R

m

G

Where;

R

is work of fracture =

G

C

at fracture

ys

ys C

C

m

R

m

G

σ

σ

δ

CTOD Measurement/TestingCTOD Measurement/Testing^

The main objective of the test is to determine the criticalcrack at the onset of crack extension.

This is done by measuring the displacement at themouth of the crack using a clip gauge.

This procedure is detailed in the ASTM E1820 Standardprovides for CTOD measurements on both compact andSENB specimens. The machine notch is

precracked

produced under

The machine notch is

precracked

produced under

carefully controlled fatigue loading conditions (the rateof stress intensity factor is between 0.5 and 2.5 MPa

m

s-1).

The specimen is then gradually loaded until themaximum load.

Plot of the load versus the clip gauge displacementsituated close to the mouth opening is made.

Experimental determination of CTOD

Experimental CTOD can be expressed as;

δ

=

δ

el

δ

pl

The plastic displacement at the crack mouth, V

p

is related to the plastic CTOD,

δ

through similar triangles construction:

p

δ

pl

through similar triangles construction:

W

P

a^ r

p

(W-a)

z

p

V

p

Experimental determination of CTODExperimental determination of CTOD

cont’dcont’d

Where

r

p

is rotational factor.

Experiments show that

r

p

lies btween 0.33 to

K

I

2

2

z

a

a

W

r

V a W r E m

K

p

p

p

ys

pl

el

I

−

−

=

=

σ

δ

δ

δ

2

W

a

f

3 2

Bw

PS

K

I

z

a

a

W

r

V

a

W

r

p

p

pl

−

−

=

p

E

ys

I

el

E

XAMPLE

A three-point bend specimen with S = 25 cm, W = 6 cm, a = 3 cm, and B = 3 cm isused to determine the critical crack opening displacement

δ

c

of a steel plate

according to British Standard BS 5762. The load-versus crack mouth displacement(P-V) record of the test is shown in figure below. Determine the critical crackopening displacement,

δ

c

when E = 210 Gpa,

ν

= 0.3,

σ

YS

= 800 MPa

Solution

pl

C

δ

δ

δ

=

el

(

)

E

ys

I

el

2

1

K

σ

ν

δ

2

2

−

=

  

 

⋅

=

W

a

f

3 2

Bw

PS

K

I

For SENB specimen K

I

can be calculated as follows;

Where P is estimated about 31.6 kN from the graph. For

= 0.5,

can be calculated using the formula for SENB



 

a

a

For

= 0.5,

can be calculated using the formula for SENB

specimen or can be obtained from the table in section A 3.5 of ASTME399.

  

  

W

a

f

3 2

2

(^12)

1

2

1

2

7

2

93

3

15

2

1

99

1

3

 



−





  

  













−





−

−





=

 

 

a w

a w

a w

a w

a w

a w

a w

W

a

f

.

.

.

.

W

a

; hence

 

 

W

a

f

(

)

(

)

(

)

(

)

m

MPa

K

I

3 2

3

×

And

K

I

can be calculated as;

JJ

IntegralsIntegrals

The

J-integral

represents a way to calculate the strain

energy release rate, or work (energy) per unit fracturesurface area, in a material.

The theoretical concept of J-integral was developed in1967 by Cherepanov and in 1968 by Jim Riceindependently, who showed that an energetic contour path integral (called

J

) was independent of the path

path integral (called

J

) was independent of the path

around a crack.

Figure 4.4 : Line integral around the crack tip – J integral

It can be evaluated experimentally by measuring thestress strain curves for a number of identical specimenscontaining cracks of different lengths and plotting thearea under the graph U for each specimen as a functionof the crack length and thus evaluating dU/dA and henceJ.

It is the rate of energy absorbed per unit area as the crack grows; it is not however the energy release rate

JJ

IntegralsIntegrals

Cont’dCont’d

crack grows; it is not however the energy release rate because the plastic energy is not recoverable as it wouldbe in the elastic case.

There are also specific specimen geometries (deeplydouble notched and notched three point bendingspecimens) that allow J to be measured from a singlespecimen.

These experiments allow J to be plotted as a function ofthe crack extension.

Although J = dU/dA is the same as the definition of the energyrelease rate, G used earlier, the J integral for the plastic casedoes not represent the energy released as the crack growsbecause much of the energy used performs plastic deformation.(LEFM, J = G; EPFM, J = R, resistance to crack growth)

The standardised test method for determining J

IC

material

values;

J

IC

Determination

• A Procedure for the Determination of

Ductile Fracture Toughness Values Using J IntegralDuctile Fracture Toughness Values Using J Integral Techniques

EPFM

• On the Determination of Elastic-Plastic Fracture

Material Parameters: A Comparison of Different TestMethods

ASTM-E

• Standard Test Method for Measurement of

Fracture Toughness.

Calculation of J for SENB specimen.

Where;

J

el

= elastic component of J

J

pl

= plastic component of J

pl

el

J

J

J

= (

)

E

K

J

el

2

2

1

ν

−

=

Where;

η

pl

= 1.9 if the load-line displacement is used for

A

pl

η

pl

= 3.667 – 2.199 (a/w) + 0.437 (a/w)

2

if the CTOD is

used for

A

pl

Validation;

where

E

a

w

B

A

J

pl

pl

pl

−

=

η

Q^ Y

J

a

w

B

σ

25

≥

−

,

2

ult

YS

Y

σ

σ

σ

=