Elsiever Crack

7/30/2019 Elsiever Crack

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Elasticplastic Jand COD estimates for axial through-wall cracked pipes

Yun-Jae Kim, Nam-Su Huh, Young-Jae Park, Young-Jin Kim*

SAFE Research Centre, School of Mechanical Engineering, Sungkyunkwan University, 300 Chunchun-dong, Jangan-gu, Kyonggi-do,

Suwon 440-746, South Korea

Received 5 January 2002; revised 19 March 2002; accepted 19 March 2002

Abstract

This paper proposes engineering estimation equations of elasticplastic Jand crack opening displacement (COD) for axial through-wall

cracked pipes under internal pressure. On the basis of detailed 3D nite element (FE) results using deformation plasticity, the plastic

inuence functions for fully plastic J and COD solutions are tabulated as a function of the mean radius-to-thickness ratio, the normalised

crack length, and the strain hardening. On the basis of these results, the GE/EPRI-type J and COD estimation equations are proposed and

validated against 3D FE results based on deformation plasticity. For more general application to general stressstrain laws or to complex

loading, the developed GE/EPRI-type solutions are re-formulated based on the reference stress (RS) concept. Such a re-formulation provides

simpler equations for Jand COD, which are then further extended to combined internal pressure and bending. The proposed RS based Jand

COD estimation equations are compared with elasticplastic 3D FE results using actual stressstrain data for Type 316 stainless steels. The

FE results for both internal pressure cases and combined internal pressure and bending cases compare very well with the proposed Jand COD

estimates. q 2002 Published by Elsevier Science Ltd.

Keywords: Axial through-wall crack; Crack opening displacement; J-integral; Reference stress approach; Finite element; Plastic inuence functions

1. Introduction

Leak-before-break (LBB) analysis is an important frac-

ture mechanics concept for design and integrity evaluation

of nuclear pressurised piping. In this respect, signicant

efforts have been made over the last two decades on elastic

plastic fracture mechanics methods for LBB analysis [13].

However, a majority of research activities have been

focused on analyses of circumferential cracked pipes, but

reports on axial cracked pipes are rare. For instance, noting

that application of LBB analysis requires estimates of the J-

integral and the crack opening displacement (COD), there

are currently a number of engineering methods available to

estimate elastic plastic J and COD for circumferential

through-wall cracked (TWC) pipes [411], whereas few

methods are yet available for axial TWC pipes. In the GE/

EPRI handbook [12], the Dugdale model is given for esti-

mating elastic plastic Jof axial TWC pipes under pressure,

and a small scale yielding model for estimating COD.

Although this may be due to the fact that axial cracks in

pipes would be less signicant than circumferential cracks,a reliable non-linear fracture mechanics method for the LBB

analysis of axial cracked pipes is still desirable.

The goal of this paper is to develop an elasticplastic

fracture mechanics method to estimate J and COD for

axial TWC pipes under internal and combined pressure

and bending. To achieve this goal, 3D nite element (FE)

analyses based on deformation plasticity are carried out to

determine fully plastic components of J and COD for axial

TWC pipes under internal pressure. These results are re-

formulated in the form of the reference stress (RS)

approach, which is then validated against further elastic

plastic 3D FE analyses using realistic stressstrain data.

Finally, the extension of the proposed RS based Jestimation

method to combined pressure and bending and to estimate

other non-linear fracture mechanics parameters, such as the

Cp-integral, is discussed.

2. Fully plastic J and COD solutions

2.1. FE analysis based on deformation plasticity

Fig. 1 depicts an axial TWC pipe under internal pressure

p, with relevant dimensions, considered in the present work.

International Journal of Pressure Vessels and Piping 79 (2002) 451464

0308-0161/02/$ - see front matter q 2002 Published by Elsevier Science Ltd.

PII: S0308-0161(02) 00030-3

www.elsevier.com/locate/ijpvp

* Corresponding author. Tel.:182-31-290-5274; fax:182-31-290-5276.

E-mail address: [email protected] (Y.-J. Kim).

Abbreviations: COD, crack opening displacement; ERS, enhanced refer-

ence stress; FE, nite element; GE/EPRI, general electric/electric power

research institute; LBB, leak-before-break; RO, RambergOsgood;

TWC, through-wall cracked


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Some important dimensions for the pipe should be noted.

The mean radius and the thickness of the pipe are denoted as

Rm and t, respectively, and the half crack length is denotedby c. The plastic limit load solution (see e.g. Miller [13])

suggests that important non-dimensional variables are the

ratio of mean radius to the thickness, Rm=t; and the normal-

ised crack length parameter r, dened by

r cRmt

p 1

Elastic plastic analyses of the FE model for the axial TWC

pipe (Fig. 1) were performed using the general-purpose FE

program, ABAQUS [14]. The tensile properties for the FE

analysis are assumed to follow the RambergOsgood

(RO) relation:

1

10 s

sy1 a

s

sy

2 3n

2

where 10, sy, a and n are constants, with E10 sy whereE and sy are Young's modulus and the yield strength,

respectively. The deformation plasticity option with a

small geometry change continuum model was invoked. In

the present FE calculations, specic values of the variables

a , E and sy were used; a 1; E 190 GPa and sy 400 MPa: It should be noted, however, that such specic

values do not affect fully plastic J and COD solutions

based on deformation plasticity, which will be proposed in

the present work, as plastic inuence functions do not

depend on these variables (see Section 2.2 for details). For

the strain hardening exponent n, on the other hand, three

values were selected, n 1; 3 and 7. Note that the case ofn 1 corresponds to the elastic case with Poisson's ratio ofn 0:3:1 Regarding other variables, two values of Rm=twere considered, Rm=t 5 and 20, and four values of rwere considered, r

0:5; 1.0, 2.0 and 3.0. Thus a total of

24 calculations were performed in the present work.The number of elements and nodes in a typical FE

mesh are 1440 elements/8485 nodes. Two elements

were used through the thickness, which has been

shown to provide the most reliable results for COD

calculation [15,16]. Although the aspect ratio of the

near-tip elements is quite high, it does not affect the

present FE computations of J and COD, as the stress

gradient through the wall is low for the present

problem. For problems where the stress gradient through

the wall is high, for instance when through-wall bend-

ing is applied or when welding residual stress is con-

sidered, more elements should be used through thethickness. Considering symmetry conditions, only one

quarter of the pipe was modelled. Fig. 2 shows the

FE mesh for r 1 and Rm=t 5: To avoid problemsassociated with incompressibility, reduced integration

20 node elements (element type C3D20R in ABAQUS)

were used. Internal pressure was applied as a distributed

load to the inner surface of the FE model, together with

an axial tension equivalent to the internal pressure

Y.-J. Kim et al. / International Journal of Pressure Vessels and Piping 79 (2002) 451464452

Nomenclature

c half crack length

E Young's modulus, E0 E=12 n2 for planestrain; E for plane stress

h1, h2 fully plastic inuence functions for the GE/

EPRI methodJ J-integral

K linear elastic stress intensity factor

n strain hardening index 1 # n , 1 forRambergOsgood model, Eq. (2)

nI strain hardening indices for the ERS-based

COD estimation equation, Eq. (32)

p internal pressure

pL plastic limit pressure assuming the limiting

stress ofsypoR optimised reference pressure

Rm mean radius of pipe

t pipe wall thickness

a coefcient of RambergOsgood modeld crack opening displacement at centre of crack

1 strain, general

n Poisson's ratio

r normalised crack length, c=Rmtps stress, general

sref reference stress

sy yield strength

Fig. 1. Schematic illustration of axial TWC pipes under internal pressure p.

1 The effect of n on J was found to be minor for the present problem.

For instance, the value ofJusing n 0:5 differs within 3% from that usingn 0:3 for all cases considered.


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applied at the end of the pipe to simulate the closed

end. More importantly, 50% of the internal pressure was

applied to the crack face to consider the effect of the

crack face pressure.

The J-integral values were extracted from the FE resultsusing a domain integral, as a function of the applied internal

pressure. The J values are averaged, whereas the COD

values, were determined from the FE displacement results

in the mean thickness of the centre of the crack.

2.2. FE based plastic inuence functions

Elastic FE calculations (with n 1) gave the elastic J, Je,from which the shape factor Ffor the elastic stress intensity

factor K was found:2

Je

K2

E 1

Es1 pcp F 2; s1

pRi

2t 3

Note that the plane stress condition was assumed to calcu-

late the values of F.3 Resulting values of F are given in

Table 1, and shown in Fig. 3. Fig. 3 also compares the

present results with published results [12], showing good

agreement. Similarly, the shape factor V, associated with

the elastic COD, de, can be found from

de 4

Es1cV 4

Note again that the plane stress condition was assumed to

calculate the values ofV. Resulting values ofVare given in

Table 2, and shown in Fig. 3. Fig. 3 also compares thepresent results with those in the GE/EPRI handbook [12]

and in Refs. [17,18]. Noting that the solutions in Refs.

[17,18] were obtained from detailed 3D FE analysis, excel-

lent agreement between the present solutions and those in

Refs. [17,18] gives condence in the present FE calcula-

tions. On the other hand, approximate solutions given in

the GE/EPRI handbook slightly underestimate the COD.

For RO materials, the plastic components ofJand COD,Jp and dp, can be expressed as

Jp asy10ch1np

pL

!n11

5

dp a10ch2np

pL

!n

6

where pL denotes the plastic limit pressure for axial TWC

pipes, of which the expression used in the present work is

the solution based on detailed FE limit analyses [19]:

pL 23

p syt

Rm

1

11 0:34r1 1:34r2p 7

where r is dened in Eq. (1). Fig. 4 compares this solution

with the limit pressure resulting from detailed 3D FE limit

analyses [19], together with two published solutions. The

rst one is the limit pressure solution due to Folias [20],

which is given by

pL syt

Rm

111 1:05r2

p 8

Y.-J. Kim et al. / International Journal of Pressure Vessels and Piping 79 (2002) 451464 453

Fig. 2. Typical nite element meshes for axial TWC pipe with Rm=t 5 andr 1:0:

Table 1

Values of the shape factor F for the stress intensity factor and the plastic

inuence h1-functions for the plastic J-integral

Rm=t r F h1n 1 h1n 3 h1n 7

5 0.5 2.743 3.859 5.656 6.710

1 3.604 3.740 4.730 4.367

2 5.576 3.409 3.578 2.8663 7.465 3.055 2.851 2.270

20 0.5 2.545 3.897 5.806 6.901

1 3.344 3.779 4.927 4.648

2 5.240 3.533 4.012 3.606

3 7.113 3.255 3.430 3.324

Table 2

Values of the shape factor V for the elastic COD and the plastic inuence

h2-functions for the plastic COD

Rm=t r V h2n 1 h2n 3 h2n 7

5 0.5 2.632 4.460 5.617 6.388

1 3.922 4.980 5.824 5.460

2 8.582 6.723 7.223 6.334

3 15.417 8.540 8.606 7.656

20 0.5 2.452 4.500 5.695 6.407

1 3.627 4.989 5.946 5.611

2 8.093 6.868 7.915 7.649

3 14.913 8.949 10.100 10.744

2 The stress on the end of the pipe, s1, in Eq. (3), is the thin-shell

approximation. The thick-shell averaged stress is slightly different.

However, for Rm=t$ 5 considered, there is not much difference. When

the correct expression for s1 is used, the corresponding value of F can

easily be found from Eq. (3). Thus the use of the correct expression ofs1 is

not so important, and for clarity, the thin-shell approximation is used in the

present work.3 This plane stress assumption may not be valid for thick-walled pipes.

However, the plane stress assumption does not affect the present solution,

as the fully plastic solutions do not depend on elastic solutions.


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The other solution is one due to Erdogan [21]

pL syt

Rm

1

0:6141 0:87542r1 0:386 exp22:275r

!9

Fig. 4 shows that Eq. (7) agrees very well with the FE results

for all ranges of r, whereas agreement between the FEsolutions and the above published solutions is excellent

for r. 0:5; but not so good for 0 , r, 0:5: This is

because in the limiting case of an uncracked cylinder r!0; the above two solutions converge to the Tresca solutionnot to the Mises solution, and thus the factor 2=

3

pis

missing.

Note that in Eqs. (5) and (6), the plastic inuence func-

tions, h1 and h2, are functions ofRm=t; the normalised crack

length r and the strain hardening exponent n. Values of h1and h2 were calibrated from the present FE analysis as

follows. Firstly, the plastic components of the FE J and d

values were calculated by subtracting their elastic com-

ponent from the total FE J and d values:

JFEp JFE 2

1

E

pRi

2t

2pcF

2 10

dFEp

dFE 24

E

pRi

2t cV 11

Then the values of h1 and h2 were calibrated from Eqs. (5)

and (6), respectively. Note that the calculated values of h1and h2 depend on the load magnitude, as shown in Fig. 5. In

the present work, the value was chosen as the (almost)

asymptotic value at large loads. Resulting values of h1 and

h2 are tabulated in Tables 1 and 2, respectively.

3. J and COD estimations based on GE/EPRI method

The plastic inuence functions, reported in Section 2.2,


Fig. 3. Variations of the shape factors, Fand V, for the stress intensity factor and the elastic COD with r: the F-solutions for (a) Rm=t 5 and (b) Rm=t 20;the V-solutions for (c) Rm=t 5 and (d) Rm=t 20: The present solutions are compared with Ref. [12] and Refs. [17,18].


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can be used to estimate J and COD for axial TWC pipes,

based on the GE/EPRI approach (see for instance Refs.

[4,12]). For instance, the J-integral can be estimated from

J 1E

pRi

2t

pce

pFce

!21asy10ch1n

p

pL

n11

12

where the effective crack length ce is estimated from

ce c1 wry;

w 111 p=pL2

; ry 12p

n2 1n1 1

Ksy

2 32 13On the other hand, the COD can be estimated from

d 4E

pRi

2t

ceVce1 a10ch2n

p

pL

!n

14

where the values ofh1(n) and h2(n) can be determined using

the data given in Tables 1 and 2 with appropriate interpola-

tion/extrapolation. Fig. 6 compares estimated J, according

to Eq. (12), with the FE results for four cases of a and n

(Note that for all cases, sy is xed tosy 400 MPa). Fig. 7,on the other hand, compares the estimated COD, according

to Eq. (14), with the FE results. They show that the proposed

GE/EPRI-type J and COD estimations are quite good. It is

worth noting, however, that the FE results shown in Figs. 6

and 7 are based on the idealised stressstrain data according

to the RO relation, see Eq. (2).The GE/EPRI-type J and

COD estimation equations, given above, have some inher-

ent problems. First of all, this method requires the RO

idealisation of the tensile data, and there can be inaccuracy

associated with this process. The RO idealisation is known

to be a poor approximation to tensile data for typical

materials, which consequently can produce inaccuracy in

the estimated J and COD. Readers can refer to other

published papers (e.g. Refs. [5,6,9,10,22,23]). The second

problem is that it is difcult to generalise this method tomore complex problems, such as to combined loading cases.

To provide relevant solutions for combined loading, in prin-

ciple more extensive FE calculations have to be performed.

To overcome these problems, the GE/EPRI-type Jand COD

estimation results, given in this section, are re-formulated in

the form of the RS approach [24] in Section 4.

4. J and COD estimations based on reference stress

concept

4.1. Reference stress formulation

For the elastic case n 1; the elastic component of Jand COD, Je and de, in Eqs. (3) and (4) can be re-written as

Je asy10ch1n 1p

pL

!215


Fig. 5. Variation of the FE results for h1 and h2 with the load magnitude for Rm=t 5 and r 0:5:

Fig. 4. Comparison of the FE limit pressure solutions for axial TWC pipes

under internal pressure with known solutions. The FE result for r 0corresponds to that for uncracked pipes.


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de a10ch2n 1p

pL

!16

where h1n 1 and h2n 1 denote the values ofh1 andh2 for elastic materials, respectively. Comparing Eq. (15)

with Eq. (3) gives the values of h1n 1; which are tabu-lated in Table 1. Normalising Eq. (5) with respect to Eq. (15)

gives

Jp

Je a

h1nh1n 1

p

pL !

n21

17

Variations ofh1n=h1n 1 with n are shown in Fig. 8 for

Rm=t 5 and 20. Similarly, comparing Eq. (16) with Eq. (4)gives the values of h2n 1; which are tabulated in Table2. Normalising Eq. (6) with respect to Eq. (16) gives

dp

de a h2n

h2n 1p

pL

!n21

18

Variations ofh2n=h2n 1 with n are also shown in Fig. 8for Rm=t 5 and 20. The results in Fig. 8 show that thevalues of h1n=h1n 1 and h2n=h2n 1 are rather

sensitive to strain hardening n, that is ranges from ,0.7 to

,1.8 for n ranging from 1 to 7.

Introducing another normalising (reference) pressure pref,

and re-phrasing Eqs. (17) and (18) gives

Jp

Je a h1n

h1n 1pref

pL

!n21

& 'p

pref

!n21

19

dp

de a h2n

h2

n

1

pref

pL

!n21& ' p

pref

!n21

20

Noting that h1n=h1n 1; h2n=h2n 1 and pref=pL arenon-dimensional variables, Eqs. (19) and (20) can be written

as

Jp

Je aH1

p

pref

!n21

21

dp

de aH2

p

pref

!n21

22

where non-dimensional functions, H1 and H2, presumably

depend on Rm=t; r and n. An important underlying idea of


Fig. 6. (ad): Comparison of FE J results for axial TWC pipes under internal pressure with the GE/EPRI estimates. Note that the FE results are based on

RambergOsgood materials with deformation plasticity.


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the RS based J and COD estimation approach is that a

proper denition of pref can minimise the dependence of

H1 and H2 on Rm=t; r and n in Eqs. (21) and (22)

[8,10,24]. Suppose such a load has been found, which will

be termed `optimised reference pressure', poR. On the basis

of the present FE results, the following expressions are

proposed for poR:

poR crpL 23

cr 20:06r2 1 0:21r1 0:82 for r, 1:5

1 for r$ 1:5@ 24

where the expression for pL is found from Eq. (7). Note that

for r! 0; cr ! 0:82; whereas for r$ 1:5;cr 1:Introducing these expressions for pref poR into Eqs. (21)and (22) gives the values of H1 and H2. Variations of the

resulting H1 and H2 values with n are shown in Fig. 9. The

results in Fig. 9 rstly show that the sensitivity in

h1n=h1n 1 and h2n=h2n 1 is reduced in H1 andH2. For instance, for the range of 1 # n # 7; the values of

h1n=h1n 1 and h2n=h2n 1 range from ,0.7 to,1.8, whereas those for H1 and H2 from ,0.8 to ,1.2.

Noting that the values of both H1 and H2 are now closer to

unity, Eqs. (21) and (22) can be approximated as

Jp

Je< a

p

poR

!n21

25

dp

de< a

p

poR

!n21

26

Noting that for the RO materials, the plastic strain is

related to the stress as

1p a sE

s

sy

2 3n2

1 27

Eqs. (25) and (26) can be written explicitly in terms of the

RS, sref, and the reference strain, 1ref, as

Jp

Je

E1ref

sref; sref

p

poRsy 28

dp

de

E1ref

sref; sref

p

poRsy 29

In Eqs. (28) and (29), sy denotes the 0.2% proof stress, and


Fig. 7. (ad): Comparison of FE COD results for axial TWC pipes under internal pressure with the GE/EPRI estimates. Note that the FE results are based on

RambergOsgood materials with deformation plasticity.


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1ref is the true strain at s sref; determined from the truestressstrain data.

4.2. Proposed reference stress based J and COD estimation

Eq. (28) gives the estimate of the plastic J-integral, and

the total J-integral can be estimated by adding the elastic

component with a plasticity correction [25]:

J

Je

E1ref

sref

11

2

sref

sy2 3

2sref

E1ref

; sref

p

poR

sy

30

where poR is given in Eq. (23). The COD can be estimated

from Refs. [811]

d

de

E1ref

sref1

1

2

sref

sy

2 32sref

E1reffor 0 # sref, sy

d

de

1

sref

sy

2 3n121

for sy # sref

VbbbbbbbbbbX

31In Eq. (31), (d/de)1 denotes the value of (d/de) at sref=sy 1;

calculated from the rst equation in Eq. (31), so that Eq. (31)

is continuous at sref sy: The strain hardening index n1 inEq. (31) should be estimated from

n1 ln1u;t 2 su;t=E=0:002

lnsu;t=sy32

where su,t and 1u,t denote the true ultimate tensile stress and

percentage uniform elongation at s su; respectively.These are obtained from the corresponding engineering

values using

su;t 11 1usu; 1u;t ln11 1u 33

4.3. FE validation

To validate the proposed RS based Jand COD estimation

equations for axial TWC pipes under internal pressure, addi-

tional elasticplastic 3D FE analyses were performed. The

main difference between these calculations and the previous

ones in Sections 2 and 3 is the material properties. The

previous cases considered idealised RO materials with

deformation plasticity, whereas the present cases used


Fig. 8. Variations of h1

n

=h1

n

1

for the J-integral with n for (a) Rm=t

5 and (b) Rm=t

20; variations of h2

n

=h2

n

1

for the COD with n for (c)

Rm=t 5 and (d) Rm=t 20:


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actual experimental uni-axial stressstrain data of Type 316

stainless steel at the temperature, T 288 8C; withincremental plasticity option. Stressstrain curves for the

material are shown in Fig. 10, and the relevant data are

summarised in Table 3. Two values of Rm=t and r were

considered, Rm=t 5 and 20, and r 0:5 and 2.0, givinga total of four cases.

Elasticplastic analyses of this FE model were performed

using ABAQUS [14]. The experimental true stressplastic

strain data were directly given in the FE analysis. Materials

were modelled as isotropic elastic plastic materials that

obey the incremental plasticity theory, and a small geometrychange continuum FE model was employed. Detailed

information on the FE model is in Section 2.1.

Fig. 11 compares the FE Jand COD results for Rm=t 5with the predictions based on the proposed RS method,

denoted as the `enhanced reference stress (ERS)' method.

In Fig. 11, the Jvalues are normalised with respect to sy and

c, while the COD (d) values with respect to c. The load,

internal pressure, is normalised with respect to the opti-

mised reference pressure, poR. (see Eq. (23)). The results

are also compared with two other methods: the GE/EPRI

method and the RS method. Noting that the GE/EPRI Jand

COD estimations are developed in the present work (see

Section 3), the resulting J and COD are also compared

with the FE results. Application of the GE/EPRI method

rstly requires that the material's tensile data should be


Fig. 9. Variations ofH1 for the J-integral with n for (a) Rm=t 5 and (b) Rm=t 20; variations ofH2 for the COD with n for (c) Rm=t 5 and (d) Rm=t 20:

Fig. 10. Stressstrain curve for SA312 Type 316 (288 8C) and the resulting

RambergOsgood t.


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tted using the RO relation, see Eq. (2). In the presentwork, the entire true stressstrain data up to the ultimate

tensile strength were tted4 using the ROFIT program [26],

developed by Battelle. The resulting RO parameters, a

and n, are listed in Table 3, and the resulting RO ts are

compared with the experimental tensile data in Fig. 10.

Once the RO parameters, a and n, are determined, then

Jand COD can be estimated using Eqs. (12)(14) in Section

3, with the values ofh1(n) and h2(n) obtained by interpolat-ing the present FE results (tabulated in Tables 1 and 2). The

resulting values of J and COD are denoted as `Present GE/

EPRI' in Fig. 11. The RS method is similar to the ERS

method, except that the RS is dened using the limit pres-

sure, Eq. (7), instead of the optimised reference pressure,

Eq. (23). For r 2; the optimised reference pressure is thesame as the limit pressure, and thus the ERS-based predic-

tions are same as those based on the RS method. The

comparisons in Fig. 11 show that the proposed ERS-based

J and COD estimates are in overall good agreements with

the FE results. On the other hand, the GE/EPRI J and COD


Table 3

Summary of tensile properties for SA312 Type 316 stainless steel at 288 8C, used in the present FE analysis

Material E (GPa) sy (MPa) su (MPa) RO Parameters ERS Parameters

a n 1u n1

SA312 Type 316 (288 8C) 190 165 455 8.42 2.92 0.3 3.82

Fig. 11. (a d) Comparison of FE Jand COD results for axial TWC pipes with Rm=t 5 under internal pressure with the engineering estimates: (i) the proposedenhanced reference stress (ERS) method, (ii) the GE/EPRI solutions, developed in the present work (Present GE/EPRI), and (iii) the reference stress (RS)

method.

4 There are other ways to t the tensile data using the RO relation.

Typical ways include to t the data only up 5% strain and to t the data

from 0.1% strain to 0.8 1u,t, where 1u,t denotes the true ultimate strain.


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estimates are not so accurate, compared to the FE results.

Such results are consistent with our earlier nding [10,22]

and such inaccuracy is associated with the RO t. In fact, if

different ways of tting the RO equation are performed,

accuracy can be improved, but no guidance on the best RO

t can be given since it depends on material [22]. The Jand

COD estimates based on the RS method are good but less

accurate than those based on the ERS method. Fig. 12

repeats the results for Rm=t

20: It can be seen that the

effect of Rm=t on estimated J and COD is minimal, andthus the same conclusions as those for Rm=t 5 can bedrawn.

5. Discussion

In this paper, engineering Jand COD estimation equations

for axial TWC pipes under internal pressure are developed.

On the basis of detailed 3D FE results with deformation

plasticity, fully plastic components ofJand COD estimation

equations for RO materials are given, which lead to the GE/

EPRI-type estimation equations. The developed solutions are

re-formulated based on the RS concept, to overcome

problems associated with the RO tting. Comparison with

elasticplastic 3D FE results using actual stressstrain data

for Type 316 stainless steels with the proposed J and COD

estimates shows excellent agreement.

The present work considers internal pressure only.

However, typical pressurised piping components are subject

to combined internal pressure and global bending. It has

been found that a bending loading has only a slight effecton plastic limit load for axial TWC pipes [13]. Noting that

the denition of the RS in the proposed enhanced RS

approach is related to the plastic limit load, it can be argued

that the proposed Jand COD estimation equations for inter-

nal pressure can be equally applied to combined pressure

and global bending loading.5 To verify our proposal, the

proposed J and COD estimates for internal pressure,

Eqs. (30)(33), are compared with the FE results for axial


Fig. 12. (ad) Comparison of FE J and COD results for axial TWC pipes with Rm=t 20 under internal pressure with the engineering estimates: (i) theproposed enhanced reference stress (ERS) method, (ii) the GE/EPRI solutions, developed in the present work (Present GE/EPRI), and (iii) the reference stress(RS) method.

5 This statement is true not only for the proposed enhanced RS based J

and COD estimations but also for the GE/EPRI and RS based estimations.


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TWC pipes under combined pressure and global bending.

The load proportionality factor l for combined loading is

dened as

l MpR2i pRm34

In the FE analysis, internal pressure and bending moment

are increased in a proportional manner. For the proportion-

ality factor, only one value of l was considered, l 0:5:The axial crack was located at the position of the maximum

tensile stress due to global bending. The resulting FE J andCOD results are compared with the proposed ERS method,

the GE/EPRI method and the RS method in Fig. 13 for

Rm=t 5 and in Fig. 14 for Rm=t 20: See Section 4.3 fordetailed descriptions on presentation of results. It can be

seen that the bending moment in fact has a minimal effect

on estimated J and COD and thus those proposed for inter-

nal pressure can be used for combined pressure and global

bending. Although the results for one value ofl were given

here, it would be sufcient to show that the proposed J and

COD estimates can be used for combined pressure and

global bending.

Whenthe cracked pipeis operatedat elevatedtemperatures,

assessment should be carried out against creep crack growth,

which in turn requires estimation of the Cp-integral and the

COD due to creep [27]. On the basis of the analogy between

plasticity and creep, the present estimation equations can be

used to estimate the Cp-integral and the COD rate, _dc; due to

creep [28]

Cp EE0

K

2_1c

sref35

_dc

de _1csref=E

with dct 0 0 36

where _1c isthecreepstrainrateattheRS s sref; determinedfrom the actual creep-deformation data. Validation of these

estimation equations will be given in a separate paper [29].

6. Conclusions

This paper proposes engineering estimation equations of

elasticplastic Jand COD for axial TWC pipes under inter-

nal pressure. On the basis of detailed 3D FE results using


Fig. 13. (ad) Comparison of FE Jand COD results for axial TWC pipes with Rm=t 5 under combined internal pressure and bending with the engineeringestimates: (i) the proposed enhanced reference stress (ERS) method, (ii) the GE/EPRI solutions, developed in the present work (Present GE/EPRI), and (iii) thereference stress (RS) method.


13/14

deformation plasticity, the plastic inuence functions for

fully plastic Jand COD solutions are tabulated as a function

of the mean radius-to-thickness ratio, the normalised crack

length and the strain hardening index. On the basis of these

results, GE/EPRI-type J and COD estimation equations are

proposed and validated against the 3D FE results based on

deformationplasticity. Formore general applicationto general

stressstrain laws or to complex loading, the developed GE/

EPRI-type solutions are re-formulated based on the RS

concept. Such a re-formulation provides simpler equations

for Jand COD, which are then further extended to combinedinternal pressure and bending. The proposed RS based Jand

COD estimation equations are compared with elasticplastic

3D FE results using actual stressstrain data for Type 316

stainless steels. The FE results for both internal pressure

cases and combined internal pressure and bending cases

compare very well with the proposedJand COD estimations.

Acknowledgements

The authors are grateful for the support provided by a

grant from Safety and Structural Integrity Research Centre

at Sungkyunkwan University.

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