Chapter 4 Epfm

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Chapter 4Chapter 41



OverviewOverview

2



In case of LEFM (Linear elastic fracture mechanics)

is not valid, CTOD and the J integral method will be

adopted – especially for ductile materials. For ductile materials, the crack faces moves prior to

fracture and creates a blunt crack tip, this crack

o enin can be used as a measure of the tou hness of

the material. This parameter is known as Crack Tip Opening

Displacement or commonly in the form of

abbreviation CTOD – this parameter developed byWells.

The displacement at the original crack tip and the

90° intercept. These are equivalent if the crack

blunts in a semicircle.

3



Evolution of Crack BluntingEvolution of Crack Blunting

As crack opens under applied load

4Crack advances at critical

value of CTOD

FIGURE 3.2: Estimation of CTOD from the

displacement of the effective crack in the Irwin

plastic zone correction.



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

effective length of the crack by ry - half the diameter of the

plastic zone. Hence the opening of the crack at it’s real tip can be

approximated from the calculated elastic displacements of the

virtual (extended) crack evaluated at a point some r from the

virtual crack tip as shown in Figure 4.1.

5

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



The CTOD is given by double the displacement uyy in the tensile direction,for plane stress this is given by the equation:

where for plane stress

Evaluating this at r y from the crack tip θ = 180o;

( )

π µ 22

1 y I yy

r K u

+=

−+

=

221

222

2 θ θ

π µ cossin K

r K u I yy

ν

ν

+

−=

1

3K

K

Substituting for the plastic zone size;

Hence6

( )ν µ

+=

12 E

2

2

1

=

ys

I y

K r

σ π ( )

ys

I yy

K u

πσ µ 22

1 2

+=gives

and substituting

ys

I

yy

K

E u

πσ δ

24

2 ==

K



CTOD,

However from Chapter 2 (LEFM);E K G I

2

=

Or according to Dugdale model;

ys

I yy

K

E u

πσ δ

24

2 ==

7

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

ys ys m

R

m

G

σ σ δ ==

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 the

mouth of the crack using a clip gauge. This procedure is detailed in the ASTM E1820 Standard

provides for CTOD measurements on both compact andSENB specimens.

The machine notch is precracked produced undercarefully controlled fatigue loading conditions (the rateof stress intensity factor is between 0.5 and 2.5 MPa√ms-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.

8



CTOD analysis using ASTM standardsCTOD analysis using ASTM standards

l o a

d Pc

fracture

(a) (b) (c)

PiPu Pm

fracture

Pi

9

M outh opening

Figure (a) - Fracture surface does not exhibit tearing and the final fracture occurs

under increasing load.

Figure (b) - Fracture surface exhibits tearing and the final fracture occurs under

increasing load.

Figure (c) - Fracture surface exhibits tearing and the final fracture occurs under

decreasing load.

Three main categories of fracture surface



Experimental determination of CTOD

Experimental CTOD can be expressed as;

δ = δel + δpl

The plastic displacement at the crack

mouth, V is related to the plastic CTOD,

δpl through similar triangles construction:

W

P

a

r p ( W - a )

z

∆p

Vp

10



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

0.48.

( )1K I ν 22

−

( )

( ) zaaW r

V aW r

E m

K

p

p p

ys

plel

I

++−

−+=+=

σ δ δ δ

2

11

⋅=

W

a f

2

3

Bw

PS K I

( )

( ) zaaW r

V aW r

p

p

pl++−

−=

p δ

E 2 ysel σ =



The plastic component Vp is obtained from the

load-displacement curve by constructing a line

parallel to the elastic loading line as

illustrated

Experimental determination of CTODExperimental determination of CTOD cont’dcont’d

Vp

Load

Mouth opening displacement

ve

12

12



EXAMPLE

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 δc 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 crack

opening displacement, δc when E = 210 Gpa, ν = 0.3, σYS = 800 MPa

13



Solution

pl C += el ( )E ys

I el

21K σ

ν δ 22

−=

⋅=W

a f

2

3

Bw

PS K I

For SENB specimen KI can be calculated as follows;

Where P is estimated about 31.6 kN from the graph.

14

For = 0. , can e ca cu ate using t e formu a for SENBspecimen or can be obtained from the table in section A 3.5 of ASTM

E399.

W

f

2

3

22

1

1212

7293315219913

−

+

+

−

−−

=

wa

wa

w

a

w

a

w

a

w

a

w

a

W

a f

....

W

; hence = 2.66

W

a f

( )( )

( ) ( ) m MPa .K I 7.4766206.003.0

25.0106.31

2

3

3

=⋅

×

=

And K I can be calculated as;



( )( )( )

006.010210800

3.07.73

22

=×

−=2

14elδ

Solution cont’d

The elastic component of the crack mouth displacement Vp

is;

The plastic component of the crackmouth displacement Vp is

determined from the test record P –

( ) mm p 286.003.003.04.0

10103.006.04.0 3

=+×

××−=−

δ

mm pl el C 292.0286.0006.0 =+=+= δ δ δ

The critical crack opening displacement δc is;

y raw ng a ne rom e

maximum load parallel to the linear

portion of the curve. We have Vp =

1mm. δp is determined as;

15



JJ IntegralsIntegrals The J-integral represents a way to calculate the strain

energy release rate, or work (energy) per unit fracture

surface area, in a material.

The theoretical concept of J-integral was developed in1967 by Cherepanov and in 1968 by Jim Rice

independently, who showed that an energetic contour

ath inte ral called J was inde endent of the ath

around a crack.

16

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 specimens

containing cracks of different lengths and plotting the

area under the graph U for each specimen as a function

of the crack length and thus evaluating dU/dA and henceJ.

It is the rate of energy absorbed per unit area as the

JJ IntegralsIntegrals Cont’dCont’d

crack grows; it is not however the energy release ratebecause the plastic energy is not recoverable as it would

be in the elastic case.

There are also specific specimen geometries (deeply

double notched and notched three point bendingspecimens) that allow J to be measured from a single

specimen.

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

17



Plotting two curves for specimens differing only in thelength of the crack, a and a + ∆a, the energy required to

grow the crack is the difference in the areas under the two

graphs shaded in the Figure 4.5.

Since the area decreases as the crack grows dU/da is

negative and J = ‐dU/da at unit thickness.

Hence;

JJ IntegralsIntegrals Cont’dCont’d

18

Figure 4.5 : Energy release rate for nonlinear deformation

Where U is the

potential energy of

the system and A

the area of the

crack.



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 JIC materialvalues;

JIC Determination – A Procedure for the Determination of

Techniques. EPFM - On the Determination of Elastic-Plastic Fracture

Material Parameters: A Comparison of Different TestMethods

ASTM-E1820 - Standard Test Method for Measurement ofFracture Toughness.

19



Calculation of J for SENB specimen.

Where;

J el = elastic component of J J pl = plastic component of J

pl el J J J +=

( )K J el

22 1 ν −=

Where;

η pl = 1.9 if the load-line displacement is used for Apl .

η pl = 3.667 – 2.199 (a/w) + 0.437 (a/w)2 if the CTOD is

used for Apl .

Validation;

• where20

( )a w B

AJ

pl pl

pl −

= η

Y

Q J

aw B σ 25≥−

, 2

ult YS Y

σ σ σ

+=



21



22



Minimum four specimenswith the same size of

fatigue crack, a/W > 0.5.

Each specimen is loadedto a different point on the

load displacement curve

and then unloaded.

Crack is marked (steelspecimens - heat tinting)

to enable measurement of

stable crack growth.

23

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