the t-stress solutions for through-wall circumferential cracks in cylinders subjected to general...

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Engineering Fracture Mechanics 75 (2008) 3206–3225

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The T-stress solutions for through-wall circumferential cracksin cylinders subjected to general loading conditions

Timothy Lewis, Xin Wang *

Department of Mechanical and Aerospace Engineering, Carleton University, Ottawa, Ontario, Canada K1S 5B6

Received 26 April 2007; received in revised form 27 November 2007; accepted 3 December 2007Available online 8 December 2007

Abstract

The elastic T-stress is a parameter used to define the level of constraint at a crack tip. It is important to provide T-stresssolutions for practical geometries to apply the constraint-based fracture mechanics methodology. In the present work, T-stress solutions are provided for circumferential through-wall cracks in thin-walled cylinders. First, cylinders with a cir-cumferential through-wall crack were analyzed using the finite element method. Three cylinder geometries were considered;defined by the mean radius of the cylinder (R) to wall thickness (t) ratios: R/t = 5, 10, and 20. The T-stress was obtained ateight crack lengths (h/p = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, h is the crack half angle).Both crack face loading and remote loading conditions were considered including constant, linear, parabolic and cubiccrack face pressures and remote tension and bending. The results for constant and linear crack face pressure were usedto derive weight functions for T-stress for the corresponding cracked geometries. The weight functions were validatedagainst several linear and non-linear stress distributions. The derived weight functions are suitable for T-stress calculationsfor circumferential cracks in cylinders under complex stress fields.� 2007 Elsevier Ltd. All rights reserved.

Keywords: T-stress; Weight function; Through-wall circumferential crack; Cylinders; General loading

1. Introduction

The elastic T-stress, or the second term of Williams [1] series expansion for linear elastic crack tip fields,represents the stress acting parallel to the crack plane. It has been demonstrated [2,3] that the sign and mag-nitude of T-stress can substantially alter the level of crack tip stress triaxiality, hence influence crack tip con-straint. Positive T-stress strengthens the level of crack tip stress triaxiality and leads to high crack tipconstraint; while negative T-stress reduces the level of crack tip stress triaxiality and leads to the loss of thecrack tip constraint. Therefore, T-stress has been used as a constraint parameter, in addition to the classicalcrack tip one parameter K or J-integral, to provide an effective two-parameter characterization of elastic–plas-tic crack tip fields in a variety of crack configurations and loading conditions [4–6].

0013-7944/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.engfracmech.2007.12.001

* Corresponding author. Tel.: +1 613 520 2600x8308; fax: +1 613 520 5715.E-mail address: [email protected] (X. Wang).

mailto:[email protected]

Nomenclature

a crack depthd a reference distanceD1, D2 weight function parametersE Young’s modulusf point forceI interaction integralJ J-integralK mode I stress intensity factorM external bending momentP external tensile forceqk function defines the virtual crack extensionR mean radius of the cylinderr radius of polar co-ordinatet cylinder wall thicknessT T-stresst(x,a) T-stress weight function for 2D crack problemst(b,h) T-stress weight function for cylindersTir reference T-stress solutionsui; uL

i displacement components and auxiliary displacement componentsV0, V1 normalized T-stress valuesh crack half anglem Poisson’s ratior0 nominal stressrb nominal bending stressrij; rL

ij stress components and auxiliary stress componentseij; eL

ij strain components and auxiliary strain componentsrt nominal tensile stressa angle of polar coordinateb angular coordinate

T. Lewis, X. Wang / Engineering Fracture Mechanics 75 (2008) 3206–3225 3207

Applications of the two-parameter fracture mechanics methodology to deal with the effects of constraint onboth fracture and fatigue problems can be found in the literature. For example, in [7,8], the failure assessmentprocedures including constraint effects were established. The effects of T-stress on fatigue crack propagationwere investigated in [9,10]. To apply these two-parameter fracture mechanics methodologies, it is essential toprovide T-stress solutions for the crack configurations under consideration.

Circumferential cracks in cylinders are encountered in a variety of applications, such as nuclear pressurevessels, pipelines and tubular structures. A circumferential through-wall crack is one of the most severe defectsin these cylindrical structures (Fig. 1). Stress intensity factors, J-integral and limit load solutions have beenpublished extensively in the literature (see [11,12], for example). Knowledge of the constraint level for theseapplications would lead to more realistic fracture assessments and fatigue analyses. Recently work has beendone by Jayadevan et al. [13] and Skallerud et al. [14] to obtain the T-stress for circumferential surface crackson pipes using the line-spring method, but no comprehensive solutions for T-stress is available for through-wall cracks.

The aim of the present work is to provide T-stress solutions for circumferential through-wall cracks in cyl-inders under general loading conditions. A circumferential through-wall crack in a cylinder considered isshown in Fig. 1. Finite element method is used to obtain the T-stress solutions for various crack face andremote loading conditions. The loading conditions considered include: constant, linear, parabolic and cubiccrack face pressures and remote tension and bending. Three cylindrical geometries will be considered, given

Fig. 1. Circumferential through-wall flaw in a cylinder.

3208 T. Lewis, X. Wang / Engineering Fracture Mechanics 75 (2008) 3206–3225

by mean radius to thickness ratio, R/t = 5, 10, and 20. Each will include eight crack lengths given byh/p = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000, where h is the crack half angle.The T-stress results for constant and linear crack face pressures were used to derive weight functions forT-stress. The weight functions were then verified using the results for several linear and non-linear stressdistributions.

2. Finite element analysis for circumferential cracks

2.1. Finite element model

Finite element analyses are conducted for circumferential cracks in cylinders using ABAQUS 6.4 [15], with8-noded thick shell elements and reduced integration (S8R). The material properties were defined as Young’sModulus E = 207 GPa (30E6 lb/in.2), and Poisson’s ratio v = 0.30. Since there are two planes of symmetryonly 1/4 of the cylinder was modelled. A typical model is shown in Fig. 2 for a crack length ofh/p = 0.2500 in a cylinder of R/t = 5.

To facilitate the mesh generation, a very small key-hole was introduced at the crack tip with a radius (1/10,000) � R. This technique allowed the creation of a web-mesh design focused at the crack-tip, as shownin Fig. 2. The mesh is extensively refined around the crack tip area. The nodes in the contour integral areawere defined using a MATLAB program. This allowed easy manipulation of the node positions within thecrack region.

2.2. Extraction of T-stress values

Consider the general case of a three-dimensional crack front with a continuously turning tangent as shownin Fig. 3a. Assume a line-load of magnitude fk = flk(s) to be applied along the crack front as illustrated inFig. 3b. In the figure, lk(s) defines the direction normal to the crack front and in the plane of the crack atpoint s. The solution for this problem is the case of a plane strain semi-infinite crack with a point force f

applied at the crack tip in the direction parallel to the crack. Using superscript ‘L’ to designate the stressand displacement fields, the analytical solution gives [16]

rL11 ¼ �

fpr

cos3 a; rL22 ¼ �

fpr

cos a sin2 a; rL33 ¼ �

fpr

m cos a

rL12 ¼ �

fpr

cos2 a sin a; rL13 ¼ rL

23 ¼ 0ð1aÞ

Fig. 2. Finite element model with crack length h/p = 0.2500, R/t = 5.


s

Crack Frontμk

nj

fk = f μk

1

2

3αs

Crack Frontμk

nj

fk = f μk

13

Fig. 3. A schematic of elements used in the definition of the interaction integral: (a) crack front and contour, (b), line-load applied alongthe crack front.


and

uL1 ¼ �

1� v2

E� fp

lnrd

� �þ sin2 a

2ð1� vÞ

� �

uL2 ¼ �

1þ v2E� fpfð1� 2vÞa� cos a sin ag

uL3 ¼ 0

ð1bÞ

where the local co-ordinate system is used as shown in Fig. 3a; d is a reference distance.Employing the above solution as an auxiliary field, Kfouri [17] extracted the T-stress for 2D crack problems

by introducing an interaction J-integral based on Eshelby’s theorem. Nakamura and Parks [18] extended thismethod to 3D crack problems and provided the domain-integral formulations. Combined with the finite ele-ment method, it was shown to be a simple and effective method for the determination of the T-stress for 2Dand 3D cracked specimens. For shell problems considered in the present work, the interaction integral assumesthe following domain-integral form:

I ¼ 1

Da

ZS

rijouL

i

oxkþ rL

ij

oui

oxk

� �oqk

oxj� rL

ijeijoqk

oxk

� dS ð2Þ

where Da is the virtual crack advance, S is an area which encloses the crack tip, qk defines the virtual extensionof the crack front; rij, eij and ui are the stress, strain and displacement components of the crack problem underconsideration; rL

ij; eLij and uL

i are the corresponding components in the line-load auxiliary solution given byEqs. (1a) and (1b).

The crack tip T-stress is related to the interaction integral, I, by the following [18]

T ¼ E1� m2

Ifþ e33

� �ð3Þ

where e33 is the extensional strain in the direction normal to the shell plane.The computation of the domain-integral, Eq. (2), is readily compatible with finite element formulations. In

the current analysis, ABAQUS [15] was used to solve the finite element model. The T-stress are then calculatedfrom Eq. (3) using domain-integral I of Eq. (2). Twelve contours of integration were used. The outputs fromthe outer six contours were averaged to give the values of T-stress. The mesh was refined until contour inde-pendence was achieved (see Fig. 2).

2.3. Validation of the finite element models

To validate the finite element models, the stress intensity factors were calculated first using the domain-inte-gral method [15] and compared with published values. The stress intensity factor for a circumferentialthrough-wall crack in a cylinder under axial tension and bending moment can be written as

KI ¼ ðF trt þ F brbÞffiffiffiffiffiffiffiffiffipRhp

ð4Þ
where Ft and Fb are the boundary correction factors for tension and bending, respectively. The nominal stres-ses are calculated using


rt ¼P

2pRtð5Þ

rb ¼M

pR2tð6Þ

for tension and bending, respectively. The boundary correction factor for through-wall cracks in cylinders hasbeen thoroughly investigated in the literature. Here the boundary correction factors are compared to early re-sults from Zahoor [19] and more recent results from Takahashi [11]. The empirical formulas provided byTakahashi are given in Appendix A1, and those provided by Zahoor are given in Appendix A2.

The stress intensity factors obtained from the present finite element analysis for 0 < h/p 6 0.5 and R/t = 5,10 and 20, were compared to the empirical formulas given in Appendices A1 and A2. Figs. 4 and 5 provide agraphical comparison for tension and bending, respectively. Excellent agreements for the boundary correctionfactors were achieved. The percentage difference between solutions provided by Takahashi is better than 3%,and the percentage difference with those provided by Zahoor is between 1% and 7% for most results. This con-firms that the finite element models are suitable for the present linear elastic analysis of circumferential cracksin cylinders.

2.4. Finite element results for T-stress

The T-stress results for a variety of crack face pressures and remote loadings are summarized as follows.

2.4.1. Crack face pressures

First, four crack face loading conditions are analyzed. These loading conditions are considered for the pur-pose of derivation of weight functions in Section 3. They are constant, linear, parabolic and cubic crack facepressures, corresponding to n = 0–3 in the following equation:

rðbÞ ¼ r0 1� bh

� �n

n ¼ 0; 1; 2 and 3 ð7Þ

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Takahashi: R/t = 5Takahashi: R/t = 10Takahashi: R/t = 20Zahoor: R/t = 5Zahoor: R/t = 10Zahoor: R/t = 20FEM Data: R/t = 5FEM Data: R/t = 10FEM Data: R/t = 20

Crack Length, θ / π

Boundary Correction Factor: Tension

Bou

ndar

y C

orre

ctio

n Fa

ctor

, Ft

5

10

20

R/t

Fig. 4. Comparison of boundary correction factor for remote tension.

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Takahashi: R/t = 5Takahashi: R/t = 10Takahashi: R/t = 20Zahoor: R/t = 5Zahoor: R/t = 10Zahoor: R/t = 20FEM Data: R/t = 5FEM Data: R/t = 10FEM Data: R/t = 20


Bou

ndar

y C

orre

ctio

n Fa

ctor

, Fb

Boundary Correction Factor: Bending

5

10

20

R/t

Fig. 5. Comparison of boundary correction factor for bending moment.

Fig. 6. Crack face pressure loading: (a) constant and (b) linear varying (symmetric quarter shown).


where r0 is the nominal stress and h is the crack half angle. Fig. 6 demonstrates the loading cases correspond-ing to n = 0 and 1.

The T-stress results for h/p = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000 withR/t = 5, 10, and 20 are determined. The results of normalized T-stress, V = T/r0, are summarized in Tables1–3, and plotted in Figs. 7–9.

2.4.2. Remote tension and bending

Next, the T-stress results are obtained for remote tension and bending. The finite element results for T-stress were normalized using

Table 1Normalized T-stress, V/r0, for crack face pressure loadings – R/t = 5

Crack length h/p Constant n = 0 Linear n = 1 Parabolic n = 2 Cubic n = 3

0.0625 0.0279 0.0136 0.0091 0.00680.1250 0.0690 0.0331 0.0248 0.01890.1875 0.1037 0.0560 0.0395 0.03020.2500 0.1174 0.0608 0.0436 0.03310.3125 0.1065 0.0645 0.0367 0.02720.3750 0.0962 0.0440 0.0255 0.01760.4375 0.1365 0.0499 0.0280 0.01820.5000 0.2605 0.1171 0.0723 0.0513



0.0625 0.0290 0.0171 0.0121 0.00930.1250 0.0891 0.0524 0.0372 0.02860.1875 0.1015 0.0573 0.0395 0.02990.2500 0.0643 0.0269 0.0100 0.00610.3125 �0.0489 �0.0509 �0.0450 �0.04500.3750 �0.1737 �0.1404 �0.1137 �0.09330.4375 �0.2959 �0.2287 �0.1811 �0.14720.5000 �0.3253 �0.2610 �0.2255 �0.1825



0.0625 0.0529 0.0312 0.0221 0.01700.1250 0.1148 0.0664 0.0465 0.03560.1875 0.0774 0.0361 0.0210 0.01450.2500 �0.0154 �0.0364 �0.0428 �0.03720.3125 �0.1674 �0.1443 �0.1199 �0.09900.3750 �0.3045 �0.2419 �0.1931 �0.15680.4375 �0.4306 �0.3274 �0.2547 �0.20430.5000 �0.4717 �0.3583 �0.2966 �0.2352


V t ¼Trt

ð8Þ

and

V b ¼Trb

ð9Þ

for tension and bending, respectively. Here Vt and Vb are normalized T-stresses. The nominal stresses rt andrb are given by Eqs. (5) and (6). The finite element results for normalized T-stress are provided in Tables 4–6,and are plotted in Figs. 10 and 11.

It is important to note that under remote tension and bending loads, all T-stress are less than zero, indicat-ing low constraint conditions. The only exception is the deep cracks in thicker walled cylinders (R/t = 5) underbending, where as the crack increases in lengths to h/p = 0.5, the T-stress rises above zero. All the normalizedT-stress values starts close to �1 when the crack is very small. This result is expected since a very small cir-cumferential crack in a cylinder should be converging to the limit case of a centre crack in an infinite plateunder remote tension (whose normalized T-stress is �1). After h/p = 0.2 the T-stress solutions start to divergefor different R/t ratios, with thicker cylinders predicting higher constraint levels.

Crack Face Pressure, R / t = 5

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.00 0.10 0.20 0.30 0.40 0.50 0.60


Nor

mal

ized

T-s

tres

s, V

FEM Data: ConstantFEM Data: LinearFEM Data: ParabolicFEM Data: CubicWeight Function Predictions

Fig. 7. Normalized T-stress for crack face pressure loading, R/t = 5.

Crack Face Pressure, R / t = 10

-0.35

-0.30

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.00 0.10 0.20 0.30 0.40 0.50 0.60Crack Length, θ / π

Nor

mal

ized

T-s

tres

s, V

FEM Data: Constant

FEM Data: Linear

FEM Data: Parabolic

FEM Data: Cubic

Weight Function Predictions



2.4.3. Empirical equations for tension and bending

For the engineering applications, solutions in the form of equations are more preferable. Since remote ten-sion and bending loads are among the most common loading conditions, here the finite element results forthese two loadings were approximated using least-squares fitting. A 10th order polynomial was used toapproximate the normalized T-stress for all three geometries and both loadings conditions. Subsequentlythe coefficients for tension and bending were approximated using second order polynomials. This providedan approximation for any geometry in the range 5 P R/t P 20. The approximation for normalized T-stressis given by

Crack-Face Pressures, R / t = 20

-0.60

-0.50

-0.40

-0.30

-0.20

-0.10

0.00

0.10

0.20

0.00 0.10 0.20 0.30 0.40 0.50 0.60Crack Length, θ / π

Nor

mal

ized

T-s

tres

s, V

FEM Data: Constant

FEM Data: Linear

FEM Data: Parabolic

FEM Data: Cubic

Weight Function Predictions


Table 4Normalized T-stress for remote tension and bending – R/t = 5

Crack length h/p Normalized T-stress, V, tension Normalized T-stress, V, bending

0.0625 �0.9686 �0.93550.1250 �0.9274 �0.84170.1875 �0.8924 �0.70970.2500 �0.8786 �0.58140.3125 �0.8895 �0.44050.3750 �0.8998 �0.28970.4375 �0.8592 �0.10130.5000 �0.7350 0.1621



0.0625 �0.9730 �0.94330.1250 �0.9126 �0.82480.1875 �0.9003 �0.72360.2500 �0.9379 �0.64180.3125 �1.0512 �0.60200.3750 �1.1763 �0.54790.4375 �1.2982 �0.47060.5000 �1.3275 �0.3095


V ¼ C1 þ C2

hp

� �2

þ C3

hp

� �4

þ C4

hp

� �6

þ C5

hp

� �8

þ C6

hp

� �10

ð10Þ

where V is the normalized T-stress, either Vt or Vb; and C1, . . . ,C6 are coefficients determined using secondorder polynomials applicable for either tension or bending. The coefficients for remote tension, Vt, are



0.0625 �0.9492 �0.91130.1250 �0.8872 �0.80220.1875 �0.9253 �0.76010.2500 �1.0188 �0.73920.3125 �1.1710 �0.73840.3750 �1.3082 �0.69150.4375 �1.4342 �0.60550.5000 �1.4755 �0.4334

-1.60

-1.40

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Curve-fit,Curve-fit,Curve-fit,FEM Data: R/t = 5FEM Data: R/t = 10FEM Data: R/t = 20


Nor

mal

ized

T-s

tres

s, V

Normalized T-stress: Tension

Fig. 10. Comparison of T-stress obtained from empirical formulas to finite element results for remote tension.


C1 ¼ �0:9613� 0:0062ðR=tÞ þ 0:0003ðR=tÞ2

C2 ¼ 1:0457þ 0:8247ðR=tÞ � 0:0278ðR=tÞ2

C3 ¼ 33:671� 22:126ðR=tÞ þ 0:5365ðR=tÞ2

C4 ¼ �210:6þ 120:23ðR=tÞ � 1:5892ðR=tÞ2

C5 ¼ 852:45� 322:72ðR=tÞ þ 0:2417ðR=tÞ2

C6 ¼ �1272þ 351:17ðR=tÞ þ 3:9989ðR=tÞ2

The coefficients for remote bending, Vb, are

C1 ¼ �0:934� 0:01ðR=tÞ þ 0:0005ðR=tÞ2

C2 ¼ 4:3575þ 1:2671ðR=tÞ � 0:0451ðR=tÞ2

C3 ¼ 74:273� 35:312ðR=tÞ þ 0:9889ðR=tÞ2

C4 ¼ �680:39þ 266:98ðR=tÞ � 6:3389ðR=tÞ2

C5 ¼ 2688:6� 957:82ðR=tÞ þ 20:135ðR=tÞ2

C6 ¼ �3852:8þ 1325ðR=tÞ � 25:744ðR=tÞ2

-1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Normalized T-stress: Bending

Nor

mal

ized

T-s

tres

s, V


Curve-fit,Curve-fit,Curve-fit,FEM Data: R/t = 5FEM Data: R/t = 10FEM Data: R/t = 20

Fig. 11. Comparison of T-stress obtained from empirical formulas to finite element results for bending moment.


The percentage difference between the values of the least-squares curve-fitting and the finite element results arebetter than 1%, except for bending with R/t = 20 where the percentage difference is better than 3%. Eq. (10) isgraphed along with the finite element results in Figs. 10 and 11 for tension and bending, respectively. Overall,excellent agreements is achieved.

Although the present empirical equations can be used to predict the T-stress for simple loading conditionssuch as remote tension, bending and internal pressure loadings, they cannot be used to calculate the T-stressfor more complex loading conditions. A weight function is required to obtain T-stress solutions for these load-ing conditions.

3. Weight functions for T-stress

Finite element calculations of T-stress for circumferential cracks in cylinders under various loading condi-tions have been presented in Section 2. However, it is impossible to foresee the variety of loading occurring inengineering applications, especially when complicated transient or residual stresses are commonly involved inthe cylindrical structures such as pressure vessels. The weight function method is one of the most efficientmethods to derive stress intensity factors for complex stress distributions (Bueckner [20] and Rice [21]). Thismethod was also applied to derive T-stress solutions for complex stress distributions for 2D and 3D crack con-figurations [22–25]. In this section, first the method is reviewed and extended for cylindrical geometries, andthen weight functions for T-stress for circumferential cracks in cylinders are derived and validated.

3.1. Principle of superposition

To facilitate the T-stress calculation using weight function method, the principle of superposition is appliedfirst [23]. Consider a two-dimensional cracked specimen loaded by mode I load system Q, as demonstrated inFig. 12a. The stress field of this problem can be divided into two parts: the regular field which appears underthe same loading conditions in the un-cracked specimen (problem Fig. 12b), and a corrective field due to thepresence of the crack (Fig. 12c). Note that the corrective field (Fig. 12c) is generated by the crack face pressure,r (x), induced by the load system Q in the uncracked body. Therefore, the elastic T-stress for the problemshown in Fig. 12a can be calculated from the summation of the T-stresses for these two problems:

a

t

x

y

Q

a

t

y

Q

σx

σy

α

r

a

t

x

y

Q

a

t

y

Q

σx

σy

α

r

x

y

F = 1

a

t

x

y

σ(x)

a

x

y

F = 1

a

t

x

y

σ(x)

a

Fig. 12. Weight function for T-stress for 2D crack problems.


T ¼ T uncrack þ T crack pressure ð11Þ

where Tuncrack is the T-stress generated in the regular field and Tcrack pressure is the T-stress generated by crackpressure. Applying the 2D case of William’s crack tip stress fields, the corresponding T-stress can be found[23]:

T uncrack ¼ ðrx � ryÞjx¼a;y¼0 ð12Þ

Here rx and ry are the stress components from the uncracked body under applied load system Q, and the loca-tion is at the crack tip (x = a, y = 0). Now, substitute Eq. (12) into (11), the T-stress for the problem we areconsidering, Fig. 12a, is then

T ¼ ðrx � ryÞjx¼a;y¼0 þ T crack pressure ð13Þ

Now, let’s consider the circumferential cracked cylinder (Fig. 13). Here a cylindrical co-ordinate system isused. The measurement along the angular b coordinate axis is defined in radians and h is the crack half angle.From superposition as shown in Fig. 13, the Eq. (11) is still applicable. The uncracked T-stress for Fig. 13(b),the counterpart of Eq. (12), can be obtained:

T uncracked ¼ ðrb � rzÞjb¼h ð14Þ

Fig. 13. Weight function for T-stress for circumferential cracks.


where rb and rz are the circumferential direction and axial direction stresses, respectively. The T-stress for theproblem of Fig. 13a is then:

T ¼ ðrb � rzÞjb¼h þ T crack pressure ð15Þ

where Tcrack pressure is the T-stress under crack face pressure load (problem shown in Fig. 13c).

3.2. Weight function method

For the 2D crack problem represented by Fig. 12c, the corresponding T-stress can be calculated by integrat-ing the product of the weight function for T-stress, t(x,a), and the stress distribution, r (x), on the crack plane:

T crack pressure ¼Z a

0

rðxÞtðx; aÞdx ð16Þ

where r (x) is the stress distribution on the crack face and t(x,a) is the weight function for T-stress. The weightfunction for T-stress depends on the crack geometry and is independent of the loading conditions. Mathemat-ically, the weight function, t(x,a), is the Green’s function for the T-stress. It represents the T-stress at the cracktip for a pair of unit point loads acting on the crack face at the location x as shown in Fig. 12d.

Substituting Eq. (16) into Eq. (13), the T-stress for cracked body as shown in Fig. 12a, loaded by a stressfield Q, is obtained:

T ¼ ðrx � ryÞjx¼a;y¼0 þZ a

0

rðxÞtðx; aÞdx ð17Þ

Eq. (17) provides a very efficient way for the calculation of T-stress for 2D crack problems [22–24].Now consider the circumferential crack problem as shown in Fig. 13. From the derivation of last section,

the T-stress can be calculated from Eq. (15). The second term in Eq. (15) represents the T-stress under crackface loading. Based on the analogy to weight functions for T-stress for 2D crack problems, for a general crack


face stress distribution varying with respect to b, r(b), the corresponding T-stress can be calculated by inte-grating the product of the weight function t(b,h), and the stress distribution, r(b), on the crack plane:

T crack pressure ¼Z h

0

rðbÞtðb; hÞdb ð18Þ

Physically, the weight function t(b,h) represents the T-stress under a pair of unit load acting on the crack faceof location of angle b. Substituting Eq. (18) into Eq. (15), the T-stress can be calculated from

T ¼ ðrb � rzÞjb¼h þZ h

0

rðbÞtðb; hÞdb ð19Þ

Eq. (19) is the counterpart of Eq. (17) for circumferential crack problems. It is a very efficient method for cal-culating T-stress for circumferential cracks in pipes under general loading conditions. Once the weight func-tion t(b,h) is obtained for given crack length h, the corresponding T-stress at this point for any general loadingconditions can be calculated. Note that here the r(b), and (rb � rz)jb=h are the outcome of a stress analysis ofthe uncracked body. In the next sections, Eq. (19) is used to calculate the T-stress for circumferential cracks incylinders under general loading conditions.

3.3. Approximate weight functions

There are different ways to obtain the weight function for T-stress for 2D cracks, t(x,a), in Eq. (17). In [22],Fett has proposed closed form approximations for t(x,a), based on which the t(x,a) can be obtained from ref-erence T-stress solutions. More recently, Wang [23,24] has demonstrated that the following general expressioncan be used to accurately approximate the weight functions for T-stress for a variety 2D internal and externalcrack problems:

tðx; aÞ ¼ 2

paD1 1� x

a

� �1=2

þ D2 1� xa

� �3=2�

ð20Þ

where D1 and D2 are parameters which depend only on the geometry of the crack body. The existence of ageneral expression simplifies significantly the determination of weight function t(x,a). For any particulargeometry, it reduced to the determination of two parameters D1 and D2.

In the present work, Eq. (20) is extended for circumferential cracks in cylinders. Eq. (20) can be re-writtenin terms of the b � z coordinate system (see Fig. 13) as follows:

tðb; hÞ ¼ 2

phD1 1� b

h

� �1=2

þ D2 1� bh

� �3=2" #

ð21Þ

where the constants D1 and D2 are determined using two reference T-stress solutions for the geometry underconsideration. This weight function form of Eq. (21) is used here to derive T-stress weight functions for cir-cumferential cracks.

3.4. Derivation and validation of weight functions

For circumferential through-wall cracks, two reference solutions for T-stress are used to decide D1 and D2

in Eq. (21): constant and linear varying crack face pressures corresponding to n = 0 and 1 in Eq. (7), respec-tively (see Fig. 6).

3.4.1. Reference T-stress solutions

The numerical T-stress results presented in Section 2 were approximated by empirical formulae fitted with5% or better.

The results for the constant crack face pressure:

rðbÞ ¼ r0 ð22Þ


are

T 1r ¼ r0 � V 0 ð23Þ

where V0 represents the normalized T-stress. The results for a linearly decreasing crack face pressure:


� �ð24Þ

are

T 2r ¼ r0 � V 1 ð25Þ

where V1 represents the normalized T-stress for linearly varying loading condition. The expressions for V0 andV1 for the three R/t ratios considered in the present work are presented as follows. For R/t = 5,

V 0 ¼ 0:0121þ 4:4589hp

� �2

� 63:468hp

� �4

þ 351:83hp

� �6

� 764:84hp

� �8

þ 604:86hp

� �10

V 1 ¼ 0:0069þ 1:951hp

� �2

� 19:158hp

� �4

þ 39:745hp

� �6

þ 127:02hp

� �8

� 304:43hp

� �10

For R/t = 10,

V 0 ¼ 0:0095þ 6:494hp

� �2

� 133:32hp

� �4

þ 827:97hp

� �6

� 2335:7hp

� �8

þ 2624:1hp

� �10

V 1 ¼ 0:0055þ 3:9115hp

� �2

� 84:303hp

� �4

þ 521:66hp

� �6

� 1467:1hp

� �8

þ 1643:8hp

� �10

For R/t = 20,

V 0 ¼ 0:0376þ 6:7142hp

� �2

� 203:47hp

� �4

þ 1673:6hp

� �6

� 6092:1hp

� �8

þ 8373:7hp

� �10

V 1 ¼ 0:0233þ 3:8695hp

� �2

� 128:58hp

� �4

þ 1043:7hp

� �6

� 3690:9hp

� �8

þ 4913:6hp

� �10

3.4.2. Weight functions for T-stressBy substituting Eqs. (22)–(25) into Eq. (19), two equations with two unknowns were established. Note that

the uncracked T-stresses are zero in Eq. (19) for those crack pressure cases. The parameters in the weight func-tion expression were solved and are

D1 ¼15

16pð5V 0 � 7V 1Þ ð26Þ

D2 ¼5

16pð35V 1 � 21V 0Þ ð27Þ

The weight function for T-stress can then be determined from Eq. (21). Further discussions on the determi-nation of weight functions using reference T-stress solutions can be found in Refs. [23] and [24].

3.4.3. Validation of weight functions

In this section, the weight functions were verified using T-stress solutions for several linear and non-linearstress fields.

3.4.3.1. Two non-linear stress distributions. First, the weight functions derived above were validated using finiteelement results for two non-linear stress fields. Using Eq. (19), T-stress values were calculated for the followingcrack face stress distributions:



� �2

ð28Þ


� �3

ð29Þ

Note that under those crack face pressure, the uncracked T-stress in Eq. (19) is zero. The T-stress results cal-culated from the derived weight functions and from the present finite element calculations for the above stressdistributions are shown in Figs. 7–9. As a comparison, the T-stress results for constant and linear stress dis-tributions Eqs. (22) and (24) are also plotted in Figs. 7–9. Very good agreements have been achieved. The dif-ferences between the normalized T-stress values were generally less than 5% for all the R/t ratios.

3.4.3.2. Remote tension and bending. The derived weight functions were then used to calculate the T-stress val-ues under remote tension and bending cases. Note that the un-cracked T-stresses are no longer zeros in thosecases. For far-field tension,

T uncracked ¼ ðrb � rzÞjb¼h ¼ �r0 ð30Þ

and the crack face pressure is

rðbÞ ¼ r0 ð31Þ

where r0 is the nominal stress for tension.For far-field bending,

T uncracked ¼ ðrb � rzÞjb¼h ¼ �r0 cos h ð32Þ

and crack face pressure

rðbÞ ¼ r0 cos b ð33Þ

where r0 is the nominal stress for bending.Using the derived weight functions, the T-stresses were calculated using Eq. (19) for tension and bending

(after substituting Eqs. (30) and (31) for tension; and Eqs. (32) and (33) for bending). The results are compared

Weight Function: R/t = 5Weight Function: R/t = 10Weight Function: R/t = 20

-1.60

-1.40

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

Nor

mal

ized

T-s

tres

s, V

0.00 0.10 0.20 0.30 0.40 0.50 0.60


FEM Data: R/t = 5FEM Data: R/t = 10FEM Data: R/t = 20

Normalized T-stress, Tension

Fig. 14. Comparisons between weight function predictions and FEM data, remote tension.

-1.20

-1.00

-0.80

-0.60

-0.40

-0.20

0.00

0.20

0.40

0.00 0.10 0.20 0.30 0.40 0.50 0.60

Weight Function: R/t = 5



FEM Data: R/t = 5

FEM Data: R/t = 10

FEM Data: R/t = 20


Nor

mal

ized

T-s

tres

s, V

Normalized T-stress, Bending

Fig. 15. Comparisons between weight function predictions and FEM data, remote bending.


to the finite element data obtained in Section 2. They are shown in Figs. 14 and 15 for tension and bending,respectively. Excellent agreements are achieved.

4. Conclusions

Finite element analyses for circumferential cracks in a cylinder under a variety of loading conditions wereconducted. The analysis procedures were verified extensively. The T-stresses were then determined for cracksin three cylinder geometries: R/t = 5, 10, and 20. For each cylinder geometry, the T-stress was obtained foreight crack lengths, given by h/p = 0.0625, 0.1250, 0.1875, 0.2500, 0.3125, 0.3750, 0.4375, and 0.5000. Loadingconditions considered were: constant, linear, parabolic and cubic crack face pressures and remote tension andbending.

For the convenience of engineering application, empirical formulas of T-stress were obtained using least-squares curve-fitting for remote tension and bending. The formulas are valid for any geometry in the range5 P R/t P 20 and for crack lengths in the range 0 6 h/p 6 0.5.

The weight function method was extended for the determination of T-stress in circumferential cracks in cyl-inders. The T-stress results for constant and linear crack face pressure loadings obtained from FE analyseswere used to derive weight functions for T-stress for the corresponding cracked geometries. The weight func-tions were validated against several linear and non-linear stress distributions. Very good agreements have beenachieved. The derived weight functions can be used to calculate T-stress for any given complex stress field r(b).

It is important to note that the T-stress is the constraint parameter characterizing the stress triaxiality underplane strain conditions. Therefore, the present solutions should be used as the constraint parameter for theinterior portion of the circumferential crack (points inside the cylinder thickness, away from the free surfaces).

When combined with the corresponding K or J solutions, these T-stresses solutions developed here are suitablefor the analysis of constraint effects for circumferential cracks in cylinders under complex loading conditions.

Acknowledgements

The authors gratefully acknowledge the financial supports from the Natural Sciences and Engineering Re-search Council (NSERC) of Canada and Ontario Centers of Excellences (OCE).


Appendix A1

Boundary correction factors for stress intensity factor for a circumferential through-wall flaw in a cylinder,Takahashi [11].

Applicable range: 1.5 < R/t < 80.5, and 0 < h/p < 0.611.Boundary correction factor for axial tension

F t ¼ At þ Bthp

� �þ Ct

hp

� �2

þ Dthp

� �3

þ Ethp

� �4" #

ðA:1Þ

At ¼ 1

Bt ¼ �1:040� 3:1831k� 4:83k2 � 2:369k3

Ct ¼ 16:71þ 23:10kþ 50:82k2 þ 18:02k3

Dt ¼ �25:85� 12:05k� 87:24k2 � 30:39k3

Et ¼ 24:70� 54:18kþ 18:09k2 þ 6:745k3

k ¼ logtR

� �

Boundary correction factor for bending moment

F b ¼ 1þ t2R

� �Ab þ Bb

hp

� �þ Cb

hp

� �2

þ Db

hp

� �3

þ Eb

hp

� �4" #

ðA:2Þ

Ab ¼ 0:65133� 0:5774k� 0:3427k2 � 0:0681k3

Bb ¼ 1:879þ 4:795kþ 2:343k2 � 0:6197k3

Cb ¼ �9:779� 38:14k� 6:611k2 þ 3:972k3

Db ¼ 34:56þ 129:9kþ 50:55k2 þ 3:374k3

Eb ¼ �30:82� 147:6k� 78:38k2 � 15:54k3

k ¼ logtR

� �

Appendix A2

Boundary correction factors for stress intensity factor for a circumferential through-wall flaw in a cylinder,Zahoor [19].

Applicable range: 5 6 R/t 6 20, and 0 < h/p < 0.5.Boundary correction factor for axial tension

F t ¼ 1þ A 5:3303hp

� �1:5

þ 18:773hp

� �4:24" #

ðA:3Þ

Boundary correction factor for bending moment

F b ¼ 1þ A 4:5967hp

� �1:5

þ 2:6422hp

� �4:24" #

ðA:4Þ


For both Ft and Fb

A ¼ 0:125Rt� 0:25

� �0:25

for 5 6Rt6 10

A ¼ 0:4Rt� 3:0

� �0:25

for 10 6Rt6 20

References

[1] Williams ML. On the stress distribution at the base of a stationary crack. ASME J Appl Mech 1957;24:111–4.[2] Rice JR. Limitations to the small scale yielding approximation for crack tip plasticity. J Mech Phys Solids 1974;22:17–26.[3] Larsson SG, Carlsson AJ. Influence of non-singular stress terms and specimen geometry on small-scale yielding at crack-tips in

elastic–plastic materials. J Mech Phys Solids 1973;21:447–73.[4] Betegon C, Hancock JW. Two-parameter characterization of elastic–plastic crack-tip fields. ASME J Appl Mech 1991;58:104–10.[5] Du ZZ, Hancock JW. The effect of non-singular stresses on crack-tip constraint. J Mech Phys Solids 1991;39:555–67.[6] O’Dowd NP, Shih CF. Family of crack-tip fields characterized by a triaxiality parameter: part I – structure of fields. J Mech Phys

Solids 1991;39:989–1015.[7] Ainsworth RA, O’Dowd NP. Constraint in the failure assessment diagram approach for fracture assessment. ASME J Press Vessel

Technol 1995;117:260–7.[8] Ainsworth RA, Sattari-Far I, Sherry AH, Hooten DG, Hadley I. Methods for including constraint effects within SINTAP

procedures. Engng Fract Mech 2000;67:563–71.[9] Roychowdhury S, Dodds RH. Effect of T-stress on fatigue crack closure in 3-D small scale yielding. Int J Solids Struct

2004;41:2581–606.[10] Wang B, Siegmund T. Numerical simulation of constraint effects in fatigue crack growth. Int J Fatigue 2005;27:1328–34.[11] Takahashi Y. Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part I:

stress intensity factor and limit load solutions. Int J Press Vessels Piping 2002;79:385–92.[12] Takahashi Y. Evaluation of leak-before-break assessment methodology for pipes with a circumferential through-wall crack. Part II:

J-integral estimations. Int J Press Vessels Piping 2002;79:393–402.[13] Jayadevan KR, Thaulow C, Ostby E, Berg E, Skallerud B, Holthe K, et al. Structural integrity of pipelines: T-stress by line-spring.

Fatigue Fract Engng Mater Struct 2005;28:467–88.[14] Skallerud B, Berg E, Jayadevan KR. Two-parameter fracture assessment of surface cracked cylindrical shells during collapse. Engng

Fract Mech 2006;73:264–82.[15] ABAQUS User’s Manual, Version 6.4. Hibbitt, Karlsson & Sorensen, Inc.; 2005.[16] Timoshenko SP, Goodier JN. Theory of elasticity. New York: McGraw Hill; 1970.[17] Kfouri AP. Some evaluations of the elastic T-term using Eshelby’s method. Int J Fract 1986;30:301–15.[18] Nakamura T, Parks DM. Determination of elastic T-stress along three-dimensional crack fronts using an interaction integral. Int J

Solids Struct 1992;29:1597–611.[19] Zahoor A. Ductile fracture handbook. Circumferential through-wall cracks. Ductile fracture handbook, vol. 1. Palo Alto, CA,

USA: Electric Power Research Institute; 1989. EPRI Report NP-6301-D.[20] Bueckner HF. A novel principle for the computation of stress intensity factors. Z Angew Math Mech 1970;50:129–46.[21] Rice JR. Some remarks on elastic crack tip field. Int J Solids Struct 1972;8:751–8.[22] Fett T. A Green’s function for T-stress in an edge cracked rectangular plate. Engng Fract Mech 1997;57:365–73.[23] Wang X. Elastic T-stress for cracks in test specimens subjected to non-uniform stress distributions. Engng Fract Mech

2002;69:1339–52.[24] Wang X. Determination of weight functions for elastic T-stress from reference T-stress solutions. Fatigue Fract Engng Mater Struct

2002;25:965–73.[25] Yu X, Wang X. Weight functions for T-stress for semi-elliptical surface cracks in finite-thickness plates. J Strain Anal Engng Des

2005;40:403–19.

the t-stress solutions for through-wall circumferential cracks in cylinders subjected to general...

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