8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 1/11

Nuclear Engineering and Design 239 (2009) 327–337

Contents lists available at ScienceDirect

Nuclear Engineering and Design

  j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / n u c e n g d e s

Estimation of crack opening area for leak before break analysis

of nuclear reactor system

B. Ghosh ∗, S.K. Bandyopadhyay, H.G. Lele, A.K. Ghosh

Reactor Safety Division, Bhabha Atomic Research Centre, Mumbai 85, India

a r t i c l e i n f o

 Article history:

Received 9 July 2007

Received in revised form 8 August 2008

Accepted 11 August 2008

a b s t r a c t

The crack opening area has been estimated by various models for leak before break analysis of nuclearreactor systems. A number of linear elastic fracture mechanics models have been employed. A model has

been derived based on Forman et al. [Forman, R.G., Hickman, J.C., Shivakumar, V., 1985. Stress intensity

factors for circumferential through cracks in hollow cylinders subjected to combined tension and bending

loads. Eng. Frac. Mech. 21, 563–571] model for stress intensity factors. The linear elastic fracture models

entail the use of plasticity corrections. The convergence behaviour of the iterative implementation for

plasticity correction has been studied. An elasto-plastic fracture mechanics model has also been adopted

for comparative study undertaken here. The range of applicability of the models has been reviewed.

© 2008 Published by Elsevier B.V.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

2. Crack opening area (COA) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

2.1. Linear elastic COA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328

2.1.1. Tada–Paris (Paris and Tada, 1983) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3292.1.2. Zahoor (1985) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

2.1.3. Zahoor (1989) (LEFM) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

2.1.4. Klecker et al. (1986) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

2.1.5. Lacire et al. (1999) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

2.1.6. Forman et al. (1985) (present) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332

2.2. Plasticity correction for elastic models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

2.2.1. Irwin plastic zone correction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

2.2.2. Plastic zone size through Dugdale approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333

2.3. Elasto-plastic COA models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

2.3.1. Zahoor (1989) (EPFM) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334

3. Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

1. Introduction

In the initial stage of failure of a high pressure pipeline, a stable through-wall crack forms before it leads to criticality and consequent

catastrophic break down. The leak before break (LBB) methodology works on the feasibility of detection the leakage from stable through-

wallcrack and deploymentof corrective engineeringsafeguardstrategy on timeto avoid subsequent catastrophic break. This alsoeliminates

or reduces the requirements for pipe-whip restraint and jet impingement shields in piping systems of nuclear power plants.

Theevaluationof thecrack opening area (COA) playsa nodal role in thedemarcation of leakage size crack corresponding tothe detectable

leak flow rate from flawed piping system. The simple single-parameter models for COA due to Kastner et al. (1981), Wuthrich (1983) and

Paris and Tada (1983) suffer from oneor othertypesof geometrical or physicallimitationsin their applicability. In thecurrent study, COA has

∗ Corresponding author.

E-mail addresses: brindaban.ghosh@irsn.fr , bghosh@gmail.com(B. Ghosh).

0029-5493/$ – see front matter © 2008 Published by Elsevier B.V.

doi:10.1016/j.nucengdes.2008.08.025

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 2/11

328 B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337 

now been estimated from more generalized two-parameter models based on thin shell theory. Sanders (1960) derived a path independent

integral expression for energy release rate. Sanders derived analytical solutions of the energy release rate for long circumferential crack

under axial tension (Sanders, 1982) and combined tension and bending (Sanders, 1983). These transcendental expressions for energy

release rate were further simplified by Klecker et al. (1986) and Lacire et al. (1999) f or convenience in use for evaluation of crack opening

area. Nicholson and Simmonds (1980) and Nicholson et al. (1983) have obtained the asymptotic expressions of energy release rates for

axial and circumferential through-wall short cracks, respectively. Subsequently, Forman et al. (1985) obtained a uniformly valid blended

model to getexpression for stress intensity factors under axial tension and bending for circumferential cracks. In the present work, Forman

et al. (1985) transcendental model for stress intensity factor has been simplified in algebraic form. Consequently, the crack opening area

has been calculated based on Klecker et al. (1986), Lacire et al. (1999) and the present models of stress intensity factors for axial tension

and bending. For internal pressure Takahashi (2002) has suggested an equivalent axial load approach. This approach has been compared

with the Rooke and Cartwright (1976) model for stress intensity factor for internal pressure. Zahoor’s (1989) elasto-plastic model has been

implemented for this comparison exercise. The prediction of this model has been compared with those of linear elastic model mentioned

above with appropriate plasticity correction. The relative merits of different models have been assessed.

2. Crack opening area (COA) models

We will discuss about two types of models for COA calculations, (1) linear elastic models which require plasticity corrections and (2)

elasto-plastic models.

 2.1. Linear elastic COA models

The estimates are based on methods of linear elastic fracture mechanics (LEFM), including the effect of shell correction, which require

knowledge of solutions of stress intensity factors (SIFs), K , for each problem.

For internal pressure loading, K  solutions are readily available for both circumferential and longitudinal cracks as functions of a single

geometric parameter, = a/√ 

Rt , relating crack size to pipe geometry (see Rooke and Cartwright, 1976). Consequently, the crack opening

area formulas are also developed as function of this single parameter.

For the case of tension and bending of the pipe section containing axial and circumferential cracks, SIFs are expressed as

K I  = F (, aeff ) √ 

aeff 

where the bulging or Folias factor, F , is a function of the thin shell parameter

2 = 

12(1− 2)a2

eff 

Rt 

and aeff  is the effective semi-crack length (see Section 2.2), which in turn depends on SIF.

Under tensile loading , the crack opening area, At , is obtained by energy method (Castigliano’s theorem) as follows:

 At  =1

t 

∂U t 

∂ t = 2

 0

∂

∂ t 

K 2t 

E

R dÂ

where U t  is the total strain energy in the cracked pipe.

Since1

2Rt 

∂U t 

∂Â= G = K 2t 

E, we obtain, U t = 2Rt 

 0

K 2t 

EdÂ

Thus, we can write,

 At  = R2  t 

EI t  with I t (Â) = 4

 0

ÂF 2t  (Â) dÂ

COA and Griffith configuration: In the case of an infinite flat plate with a central crack of length 2 a, the SIF is given by,

K I  =  √ 

a

Using this relation, the crack opening area is given by,

 Ao = 4 

E

 a

0

 x d x = 2 

Ea2

Taking into account the bulging effect for cracks in shallow shells through Folias factor, as discussed above, the COA for cracks on thin

shells can be expressed as

 A = 4 

E

 a

0

F 2() x d x = 4 

Ea2 1

2

 

0

M2() d = Ao˛()

where ˛ is known as bulging factor for COA.

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 3/11

B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337  329

COA from COD: An alternative method to evaluate the COA, A, is to use crack opening displacement (COD) at the center of the crack and

to calculate COA assuming an elliptic crack opening profile by

 A =

2aı

where a represents the semi-crack length and ı is COD at the center (mouth) of the crack (or CMOD). Thus for Griffith configuration the

CMOD is given by, ıo = 4a /E.

  2.1.1. Tada–Paris (Paris and Tada, 1983) modelFor the case of tension and bending of a pipe section containing a circumferential crack, the SIFs are not formulated as functions of a

single parameter and no simple formula is readily available for the evaluation of SIFs.

Therefore, Paris and Tada (1983) have specifically selected a typical value of the mean radius to thickness ratio, R/t = 10 and formulated

SIFs based on this value. For smaller R/t ratios, the degree of overestimate would increase.

Under Tension and Bending : Under these loadings, the SIFs are expressed in the following forms

K t  =  t 

 (RÂ)F t (Â) and K b =  b

 (RÂ)F b(Â)

where subscripts t and b represent tension and bending, respectively.

The nominal stresses due to tension and bending are defined by

 t  =P 

2Rt and  b =

M

R2t 

where P and M are axial tensile force and bending moment

F t (Â) and F b(Â) which are nondimensional functions accounting for bulging effect of shells on SIF are calculated from Sander’s (1982,1983) approximate formulas for R/t = 10 as

F t (Â) = 1+ 7.5

Â

3/2

− 15

Â

5/2

+ 33

Â

7/2

and F b(Â) = 1+ 6.8

Â

3/2

− 13.6

Â

5/2

+ 20

Â

7/2

From the definition of I t  (see Section 2.1), we get

I t (Â) = 2Â2

1+

Â

1.5

8.6−13.3

Â

+24

Â

2+

Â

3

22.5− 75

Â

+ 205.7

Â

2

− 247.5

Â

3

+ 242

Â

4

Similar to the relation between At  and I t , the cracking opening area due to bending stress is expressed as

 Ab = R2  bE

I b(Â)

The crack opening area for bending, Ab, however cannot be obtained as readily because the “crack absent stress distribution” is notuniform along the crack (direct application of energy method is difficult). For this case of bending stress which varies from  b to  b cos Â,

I b is approximated as

I b(Â) = 3+ cos Â

4I t (Â)

Then, the crack opening area can be taken as the sum of individual areas due to tensile and bending loadings as

 A =  At + Ab

where At  and Ab are the crack opening areas due to tensile and bending loads, respectively.

Note that, once the normal stresses due to axial tension and bending moment are given, here, the crack opening area is independent of 

shell thickness.

Under internal pressure: For the internal pressure loading, SIF solutions are readily available for both circumferential and longitudinal

cracks as functions of a single geometric parameter,

=a/√ 

Rt , relating crack size and pipe geometry. In this case, the SIF is expressed as

K  p =   p√ 

aF  p(), where = a√ Rt 

= Â

 R

t 

Contrary to the cases of axial tension and bending loads, the geometric factor F  p() for this case is a nondimensional function of single

parameter as mentioned earlier.

In the case of circumferential cracks, for a pipe subjected to internal pressure, p, the membrane stress,   p, is estimated by

  p =R

2t  p

In this case, the following formula approximates the curve of  F  p() presented in Rooke and Cartwright (1976).

F  p() =

(1+ 0.32252)1/2

0 ≤ ≤ 1

0.9+ 0.25 1 ≤ ≤ 5

Accuracy of the formula is within a few percent over the range specified above.

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 4/11

330 B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337 

The crack opening area, A p, can be obtained by the method in the previous discussion,

 A p = 2

 0

∂

∂  p

K 2 p

E

R d = 2Rt 

  pE

I  p

where

I  p() =

2(1+ 0.161252) 0 ≤ ≤ 10.02+ 2(0.81+ 0.30 + 0.031252) 1 ≤ ≤ 5

and  ̨p =

1+ 0.161252

for0 ≤ ≤ 10.02

2+ 0.81+ 0.30 + 0.031252 for1 ≤ ≤ 5

In case of longitudinal cracks, for a pipe subjected to internal pressure, p, the hoop stress,  h, is estimated by

 h =R

t p

The crack opening area, Ah, can be obtained by the method in the previous discussion,

 Ah = 2Rt  hE

I h()

where = a/√ 

Rt 

I h()=2(1+ 0.6252) 0 ≤ ≤ 1

0.14+ 2

(0.36+ 0.72 + 0.4052

) 1 ≤ ≤ 5

and ˛h=

1

2

I h()=

1+ 0.6252 0 ≤ ≤ 10.14

2 + 0.36+ 0.72 + 0.4052

1 ≤ ≤ 5

 2.1.2. Zahoor (1985) model

Zahoor (1985) gives SIF relations for circumferential through-wall cracks under various loading conditions.

Defining a quantity k that depends only on the pipe geometry

k =

⎧⎪⎨⎪⎩

1

8

R

t − 1

4

1/4

for 5 ≤ R

t ≤ 10

2

5

R

t − 3

1/4

for 10 ≤ R

t ≤ 20

 Axial tension: The K I  for remotely applied load, P , may be expressed as

K t =√ 

a t F t R

t ,

Â

with F t 

=1+

k5.3303 Â

1.5

+18.773 Â

4.24

Remotely applied bending moment : K I  for remotely applied bending moment, M, may be expressed as

K b =√ 

a bF b

R

t ,

Â

with F b = 1+ k

4.5967

Â

1.5

+ 2.6422

Â

4.24

Internal pressure: The K I  solution for a circumferential through crack in a pipe subjected to internal pressure, p, was taken from Rooke

and Cartwright (1976) and curve fitted, giving

K  p =√ 

a  pF  p() with F  p =

1+ 0.15013/2 for ≤ 2

0.8875+ 0.2625 for 2 ≤ ≤ 5and = a√ 

Rt 

where   p is the membrane stress in the axial direction.

The pressure loading K I  expression is estimated to have better than 1% accuracy for R/t = 10.

In order to calculate the crack opening areas as in the case of Tada–Paris method, that is,

 At  = R2  t 

EI t 

where

I t (Â) = 4

 0

ÂF 2t  (Â) dÂ

from Zahoor (1986) relation for SIF under tension, we obtain

I t  = 2Â2

1+ 4k

1.523

Â

1.5

+ 3.0085

Â

4.24+ 2k2

5.682

Â

2.25

+ 33.63

Â

8.48

+ 25.857

Â

5.74

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 5/11

B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337  331

 2.1.3. Zahoor (1989) (LEFM) model

Solutions for crack mouth opening displacement (CMOD), ı, under elastic loading have been obtained for various geometrical configu-

rations and loading conditions. The following equations were derived by Zahoor (1989) based on the finite element analysis by Kumar and

German (1988).

ı = 4

E(˛t  t + ˛b b)RmÂ

where ˛t  and ˛b are geometrical factors for CMOD and are approximated by

˛t = 1+ k

4.55

Â

1.5

+ 47

Â

3

and ˛b = 1+ k

6.071

Â

1.5

+ 24.15

Â

2.94

with a common coefficient, k, expressing the dependence on Rm/t , given by

k =

⎧⎪⎨⎪⎩

0.125Rm

t − 0.25

0.25

for 5 ≤ Rm

t ≤ 10

0.4Rm

t − 3

0.25

for 10 ≤ Rm

t ≤ 20

Assuming elliptic crack,

 A =

2 (RmÂ)ı =2

E (RmÂ)2(˛t  t + ˛b b)(RmÂ)2 =

E R2m( t I t +  bI b)

thus, I t = 2Â2˛t  and I b = 2Â2˛band ˛t  and ˛b are actually obtained from the relations,

F 2t  =1

4Â

dI t 

dÂ= 1+ k

7.9625

Â

1.5

+ 117.5

Â

3

and F 2b= 1

4Â

dI bdÂ

= 1+ k

10.62425

Â

1.5

+ 59.6505

Â

2.94

 Applicability: 0 < Â/≤0.55 and 5≤Rm/t ≤20.

 2.1.4. Klecker et al. (1986) model

Klecker et al. (1986) extended the Tada–Paris model by including the effect of  Rm/t  for the range between 5 and 15. The resulting

geometrical factors for SIF, obtained by them, for a circumferential crack is given by,

F t = 1+ At 

Â

1.5

+ Bt 

Â

2.5

+ C t 

Â

3.5

and F b = 1+ Ab

Â

1.5

+ Bb

Â

2.5

+ C b

Â

3.5

These are applicable for the range of: 5< Rm/t < 15 and 0 < Â/ < 0.611.

The coefficients of these polynomials are given as

 At = −2.02917 + 1.67763

Rm

t 

− 0.07987

Rm

t 

2

+ 0.00176

Rm

t 

3

Bt = 7.09987 − 4.42394

Rm

t 

+ 0.21036

Rm

t 

2

− 0.00463

Rm

t 

3

C t  = −7.79661+ 5.16676

Rm

t 

− 0.24577

Rm

t 

2

+ 0.00541

Rm

t 

3

 AB = −3.26543 + 1.52784

Rm

t  − 0.072698

Rm

t  2

+ 0.0016011

Rm

t  3

Bb = 11.36322 − 3.91412

Rmt 

+ 0.18619

Rmt 

2

− 0.004099

Rmt 

3

C b = −3.18609 + 3.84763

Rm

t 

− 0.18304

Rm

t 

2

+ 0.00403

Rm

t 

3

Paris and Tada (1983) utilized the SIF solutions of  Sanders (1982) to obtain a polynomial equation for COA; Bhandari et al. (1992)

proposed an extension to this solution to include the effect of Rm/t . This can be obtained by the following formula with the coefficients ( At ,

Bt  and C t ) given by Klecker et al. (1986) as a functions of Rm/t .

˛t  = 1+

Â

3/2

8

7 At +

8

9Bt 

Â

+ 8

11C t 

Â

2+

Â

3

2

5 A2

t  +2

3 At Bt 

Â

+ 2

7(B2

t  + 2 At C t )

Â

2

+ 1

2Bt C t 

Â

3

+ 2

9C 2t 

Â

4

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 6/11

332 B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337 

 2.1.5. Lacire et al. (1999) model

Lacire et al. (1999) analyzed circumferential through-wall cracks on pipes with finite element method using shell elements for this

cylinders (Rm/t > 10) and solid elements for thicker cylinders. The geometrical factors for SIF obtained by them are given by,

F t =  At + Bt 

Â

+ C t 

Â

2

+Dt 

Â

3

+ Et 

Â

4

andF b

1+ (t/2Rm)=

 Ab + Bb

Â

+ C b

Â

2

+ Db

Â

3

+ Eb

Â

4

These are applicable for the range of: 1.5 < Rm/t < 80.5 and 0 < Â/ < 0.611.The coefficients of these polynomials are given as

 At  = 1 Ab = 0.65133− 0.5774ς − 0.3427ς 2 − 0.0681ς 3

Bt  = −1.040− 3.1831ς − 4.83ς 2 + 2.369ς 3 Bb = 1.879+ 4.7951ς + 2.343ς 2 − 0.6197ς 3

C t = 16.71+ 23.10ς + 50.82ς 2 + 18.02ς 3 C b = −9.779− 38.14ς − 6.611ς 2 + 3.972ς 3

Dt  = −25.82− 12.051ς − 87.24ς 2 − 30.39ς 3 Db = 34.56+ 129.9ς + 50.55ς 2 + 3.374ς 3

Et  = 24.70− 54.18ς + 18.09ς 2 + 6.745ς 3 Eb = −30.82− 147.6ς − 78.38ς 2 − 15.54ς 3

with ς = log(t/Rm)

Takahashi (2002) has used SIF solutions by Lacire et al. (1999) to estimate COA for a wide range of Rm/t values. Based on their model, an

alternative equation with a larger range of applicability can be derived by,

˛t  = A2t  +

4 At Bt 

3 Â

+ B2t  + 2 At C t 

2 Â

2

+ 4 At Dt + 4Bt C t 

5 Â

3

+ C 2t  + 2 At Et + 2Bt Dt 

3 Â

3

+ 4Bt Et + 4C t Dt 

7 Â

5

+D2t  + 2C t Et 

4

Â

6

+ 4Dt Et 

9

Â

7

+ E2t 

5

Â

8

For a situation where uncracked body stress is notuniform as in the case of a pipe subjected to a bending moment, no exact relationship

between the SIF and COA is available. For circumferential cracks under a bending moment, however, Paris and Tada (1983) proposed the

following equation derived using engineering judgment

˛b =3+ cos Â

4˛t 

This equation can be used to extend the expressions for the tensile case to the bending case.

 2.1.6. Forman et al. (1985) ( present) model

The present model is based Forman et al. (1985) models for F t  and F b. Due to Forman et al. (1985)

F t = 

Ht 

(2Â)and F b =

 Hb

(2Â)

where

Ht  =Â2

ε

 g (Â)+

C 2 − 2

√ 2

and Hb =

Â2

ε

2√ 

2

sin Â

Â(Â)

2

+

C 2 − 2

√ 2

with

ε2 = t 

R

1 12(1− 2)

, = Â

2ε= 1

2

RÂ√ Rt 

4 

12(1 − 2)

C =

⎧⎪⎨⎪⎩

1+ 16

2 − 0.02933 for ≤ 12√ 

2

1/2

+

0.179

0.885

for ≥ 1

(Â) = 1+ 1

2

 +  cot2  − cot Â√ 2cot

( − Â)/

√ 2+ 2cot Â

and g (Â) = 2√ 

2

1+ 1√ 

2Â

1− Â cot Â

cot

( − Â)/√ 

2+√ 

2cot Â

2

Considering their limiting values as Â→0, F t  and F b can be represented as

F 2t  =1

2ε

⎛⎝

2εC 2 − 2√ 

2Â + 2√ 

2Â

1+ 1√ 

2Â

1− Â cot Â

cot( − Â)/√ 

2+√ 

2cot Â

2⎞⎠

and F 2b =1

2ε Âh(Â)+ 2εC 2 − 2

√ 2Â

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 7/11

B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337  333

where

h(Â) = 2√ 

2

sin Â

Â(Â)

2

= 2√ 

2

sin Â

Â+√ 

2

4Â

 sin  + cos Â

 cot  − 1

cot

( − Â)/√ 

2+√ 

2cot Â

2

The functions, g and h, can be redefined to put F t  and F b in similar functional form as

F 2t  =1

2ε

2√ 2Â( g 2 − 1)+ 2εC 2

and F 2b =1

2ε

2√ 2Â(h2 − 1)+ 2εC 2

and modified functions g and h are:

 g (Â) = 1+ 1√ 2Â

1− Â cot Â

cot

( − Â)/√ 

2+√ 

2cot Âand h(Â) = sin Â

Â+ 1

2Â

 sin  + cos Â( cot  − 1)√ 2cot

( − Â)/

√ 2+ 2cot Â

To facilitate the easy determination of COA, the polynomial approximation for C , g and h are given by:

C =

⎧⎪⎨⎪⎩

1+

64ε2Â2 − 0.0036875

ε3Â3 for  ≤ 2ε√ 

2

εÂ

1/2

+

0.358ε

Â

0.885

for  ≥ 2ε

 g  = 1.00133− 0.10977

Â

+ 3.18598

Â

2

− 4.67544

Â

3

+ 11.97015

Â

4

and h = 1.00071 − 0.05984

Â

+0.85086

Â

2

− 1.41999

Â

3

+ 5.4903

Â

4

  2.2. Plasticity correction for elastic models

Inglis (1913) showed that the stresses at the crack tips are so large that they cause anelastic deformation in the front of the crack tip.

The anelastic deformation such as plastic flow in metals, is mostly irreversible and if the stresses are released the body will not recuperate

its original configuration near the crack tip. The energy that causes the anelastic behaviour is eventually converted into heat energy and is

lost to the surrounding. Thus, the anelastic deformation dissipates mechanical energy into heat energy. Quite often, the flow of material

makes the crack tip blunt which in turn decreases the magnitude of stress components. Thus many potential catastrophic failures may be

avoided just by local plastic deformation at the crack tip. In metals, plastic deformation occurs which is generally caused by nucleation and

motion of dislocations. Flow of materials near the crack tip is also facilitated by the formation of voids under tensile field.

Stress field in the vicinity of a crack tip are determined assuming that the body remains elastic even at the crack tip. Knowing the stress

field, we may invoke one of the two universally accepted yield criteria, von Mises or Tresca, for evaluating the size and shape of the plastic

zone. Rigorous analysis is prohibitively complex because two sets of constitutive relations should be used, one for plastic deformation

inside the plastic zone and another for elastic deformation outside the zone. To complicate the situation further, the surface that separates

the two zones is not known a priori and the analysis may require iterative solutions. Therefore, approximate methods of analysis would be

adopted as follows.

 2.2.1. Irwin plastic zone correction

Irwin (1958) suggested a solution for the plastic zone size through a model which accounts for absence of high stresses within the yield

zone. Consider a plane stress problem. The plastic zone size extends beyond the tip of the actual crack The tip of the effective crack is

located somewhere inside the plastic zone. Note that the material beyond the tip of the effective crack will not be able to sustain stress

beyond yield stress. Moreover the crack faces of the effective crack are not actually separated between points actual and effective crack

tips, and therefore tensile stress equal to yield stress acts over the span between actual and effective crack tips for elastic-perfectly plastic

material. Irwin proposed that the distance between actual and effective crack tips shouldbe chosen such that loads nottaken beyond point

of effective crack tip is equal to load sustained over the distance between actual and effective crack tips.

Thus, the effect of yielding near the crack tip may be incorporated through the effective (plastic zone corrected) crack size which is

calculated from the iterative relation

aeff = a+ K I 

2 2 y

Since K I  also depends on aeff  (see Section 2.1), the correction needs to be implemented in an iterative manner. The procedure here is

same as recommended in Paris and Tada (1983).

 2.2.2. Plastic zone size through Dugdale approach

Dugdale (1960) determined the plastic zone size in front of the crack tip through a different approach. He considered effective crack to

be of length a + r  p0, where r  p0 is plastic zone size.

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 8/11

334 B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337 

 Table 1

Geometric parameters for test cases.

Test cases OD (cm) t (cm) Rm (cm) Rm/t 

1 40.64 3.096 18.770 6.063

2 16.83 1.097 7.866 7.171

3 21.91 1.27 10.320 8.126

4 21.91 1.509 10.200 6.760

5 27.30 1.824 12.738 6.984

6 21.91 1.031 10.939 10.6117 35.56 1.509 17.025 11.283

8 50.80 2.062 24.369 11.818

9 71.12 2.062 34,029 16.503

In Dugdale approach singularity at the tip of effective crack is nullified by a uniform pressure equal to yield stress (for plane stress)

over the plastic zone. In fact, to determine r  p0 the criterion employed is that r  p0 is the length on which pressure   y exactly nullifies the

singularity in elastic stress field distribution.

  2.3. Elasto-plastic COA models

The only model we will discuss about in this category is Zahoor (1989) Elasto-Plastic Fracture Mechanic (EPFM) model.

 2.3.1. Zahoor (1989) (EPFM) modelBased on finite-element analysis by Kumar and German (1988), Zahoor (1989) has recommended the following model for the case with

combined tension and bending, in which the crack opening is correlated as:

ı = 4RmÂ1

E( t  f t +  b f b)+ ˛εoRmÂ

P 

P o

n

h

where

 t =P 

2Rmt and  b =

4

RmM

R4o − R4

i

P o =1

2

⎡⎣−

RmP 2oMo

+

RmP 2oMo

2

+ 4P 2o

1/2⎤⎦

with load ratio, = M/RmP 

The reference limit load for axial tension and bending moment are given by,

P o = 2 oRmt 

− Â − 2sin−1

sin Â

2

and Mo = 4 oR2

mt 

cos

Â

2− 1

2sin Â

˛, n,  o and εo are constants in the Ramberg–Osgood (1943) stress–strain relation,

ε

εo=  

 o+ ˛

 

 o

n

The  o and εo are reference stress and reference strain, respectively, and are related to each other by the relation

 oεo

= E

where  o is 0.2% offset yield strength.

The f t  and f b are defined as follows:

 f t =  f Â

1+ k

4.55

 f Â

Â

1.5

+ 47

 f Â

Â

3

and f b =  f Â

1+ k

6.071

 f Â

Â

1.5

+ 24.15

 f Â

Â

2.94

 Table 2

Loadings and mechanical properties.

 p (bar) 50

M (N m) 2.62×105

E (GPa) 1.88

  y (MPa) 1.0

 u (MPa) 2.27

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 9/11

B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337  335

Fig. 1. (a–f) Comparison of linear elastic COA models.

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 10/11

336 B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337 

Fig. 2. (a–f) Comparison of linear elasto-plastic COA models.

8/3/2019 Ghosh COA

http://slidepdf.com/reader/full/ghosh-coa 11/11

B. Ghosh et al. / Nuclear Engineering and Design 239 (2009) 327–337  337

with a common coefficient, k expressing the dependence on Rm/t given by

k =

⎧⎪⎨⎪⎩

0.125Rm

t − 0.25

0.25

for 5 ≤ Rm

t ≤ 10

0.4Rm

t − 3

0.25

for 10 ≤ Rm

t ≤ 20

and f Â = 1+ 1

ˇ

n− 1

n+ 1

( t F t +  bF b)2

 2o

1+ (P/P o)2

with F t  and F b are same as those of Zahoor (1985).

Graphical representation and tabulated values of  h(n, , (Â/), (R/t ) = 10) has been provided by Zahoor (1989). Linear interpolation ispermissible as prescribed by him.

 Applicability: R/t = 10, 0.0625≤Â/≤0.5.

3. Results and discussion

The crack opening areas have been calculated for range of pipe sizes of the secondary side of an Indian PHWRs. Table 1 f urnishes a list

of the of the relevant geometric parameters and Table 2 presents the representative loading conditions.

Fig. 1a–f shows the COA predictions by different LEFM models due to Paris and Tada (1983), Zahoor (1985), Klecker et al. (1986), Lacire et

al. (1999) and the present model developed from Forman et al. (1985). Fig. 2a–f presents the predictions of some of the above models with

plasticitycorrection as suggestedby Paris and Tada (1983) and Zahoor’s (1989) LEPFMmodel. Fig.1a–f reveals thatTada–Paris overestimates

COAfor Rm/t < 10 and under-estimates for Rm/t > 10. Zahoor-LEFM (1989) always overpredicts the COA, whereas Klecker et al. (1986)predicts

lowest value of COA in almost all cases. Judging from the point of view of law of mean, Zahoor (1985), Lacire et al. (1999) and present model

based on Forman et al. (1985) yield reasonable estimates.

Tada–Paris model has been derived for R/t ratio equal to 10 and for 0 < Â <100◦. Estimation of the formula is expected to yield a slightlyover estimate results for R/t near 10. For smaller R/t ratio, the degree of over estimate would increase. Maricchiolo and Milella (1989) have

observed that, for larger crack length, the discrepancy between experimental and calculated areas increases even when R/t is near 10.

In Zahoor (1985) model, (1) K I  expressions for tension and bending are recommended for use between 0 < Â/ < 0.55; (2) K I  for cases are

slightly conservative relative to the solution due to Sanders (1983); (3) for R/t = 10, the accuracy relative to Sanders (1983) is better than

2%; for R/t ≈5, the relative accuracy is up to 4%; (3) the results for R/t between 15 and 20 are estimated to be conservative by as much as

10% for large crack lengths.

Zahoor (1989) LEPM model has (1) better than 3% accuracy for R/t = 10 as compared to the results in Kumar and German (1988), (2)

solution conservative by as much as 7% for R/t near 5 and as much as 10% for Â/ > 0.4 and R/t between 15 and 20.

With plasticity correction the effective values of semi-crack angle does not converge for any value of individual loadings and their

combination as expiated by Giles and Brust (1991). To aid convergence in all cases discussed in Fig. 2a–f, we have lowered the bending

moment by an order of magnitude. As revealed by Fig. 2a–f, with plasticity correction, the relative prediction of Tada–Paris model is similar

among the LEFM models. Klecker et al. (1986) model behaves in reverse fashion in this case with respect to Tada–Paris model. Lacire et al.

(1999) and Zahoor (1989) gives tenable prediction.

References

Bhandari, S., Faidy, C., Acker, D., 1992. Computation of leak areas of circumferential cracks in piping for application in demonstrating the leak before break behavior. Nucl.Eng. Des. 135, 141–149.

Dugdale, D.S., 1960. Yielding of steel sheets containing slits. J. Mech. Phys. Solids 8, 100–108.Forman, R.G., Hickman, J.C., Shivakumar, V., 1985. Stress intensity factors for circumferential through cracks in hollow cylinders subjected to combined tension and bending

loads. Eng. Frac. Mech. 21, 563–571.Giles, P., Brust, F.W., 1991. Approximate fracture methods for pipes. Part I. Theory. Nucl. Eng. Des. 127, 1–17.Inglis, C.E., 1913. Stress in a plate due to the presence of cracks and sharp corners. Trans. Inst. Naval Architects 55, 219–241.Irwin, G.R., 1958. Fracture. In: Flugge, S. (Ed.), Handbuch der Physik, vol. VI. Springer-Verlag, Berlin, pp. 551–590.Kastner, W., Rohrich, E., Schmitt, W., Steinbuch, R., 1981. Critical crack sizes in ductile piping. Int. J. PVP 9, 197–219.Klecker, R., Brust, F.W., Wilkowski, G., 1986. NRC Leak-Before-Break (LBB.NRC) Analysis Method for Circumferentially Through-Wall Cracked Pipes under Axial plus Bending

Loads, NUREG/CR-4572.Kumar, V., German, M., 1988. Elasto-Plastic Fracture Analysis of Through-Wall and Surface Flaws in Cylinders, EPRI NP-5596.Lacire, M.H., Chapuliot, S., Marie, S., 1999. Stress intensity factors of through wall cracks in plates and tubes with circumferential cracks. ASME PVP 388, 13–21.Maricchiolo, C., Milella, P.P., 1989. Prediction of leak areas and experimental verification on carbon and stainless steel pipes. Nucl. Eng. Des. 111, 47–54.Nicholson, J.W., Simmonds, J.G., 1980. Sanders’ energy-release rate integral for arbitrarily loaded shallow shells and its asymptotic evaluation for a cracked cylinder. J. Appl.

Mech. 47, 363–369.Nicholson, J.W., et al., 1983. Sanders’ energy-release rate integral for a circumferentially cracked cylindrical shell. J. Appl. Mech. 50, 373–378.Paris, P.C., Tada, H., 1983. The Application of Fracture Proof Design Methodology using Tearing Instability Theory to Nuclear Piping Postulating Circumferential throughwall

Crack, NUREG/CR-3464, Nuclear Regulatory Commission.Ramberg, W., Osgood, W.R., 1943. Description of stress-strain curves by three parameters. NACA TN-902.Rooke, D.P., Cartwright, D.J., 1976. Compendium of Stress Intensity Factors. Her Majesty’s Stationaries Office, London.Sanders Jr., J.L., 1960. On the Griffith-Erwin fracture theory. J. Appl. Mech. 27, 352–353.Sanders Jr., J.L., 1982. Circumferential through-wall cracks in cylindrical shells under tension. J. Appl. Mech. 49, 103–107.Sanders Jr., J.L., 1983. Circumferential through-wall cracks in a cylindrical shell under combined bending and tension. J. Appl. Mech. 50, 221.Takahashi, Y., 2002.Evaluation of leak-before-break assessment methodology for pipes witha circumferential through-wall crack. Part I. Stress intensityfactor and limit load

solutions. Int. J. PVP 79, 385–392.Wuthrich, C., 1983. Crack opening areas in pressure vessels and pipes. Eng. Frac. Mech. 18, 1049–1057.Zahoor, A., 1985. Closed form expression for fracture mechanics analysis of cracked pipes. Trans. ASME J. Pres. Ves. Technol. 107, 203–205.Zahoor, A., 1986. Fracture of Circumferentially Cracked Pipes. Trans. ASME J. Pres. Ves. Technol. 108, 529–531.Zahoor, A., 1989. Ductile Fracture Handbook, EPRI NP-6301.