Progress Report No - · PDF file6 stress distribution (x) in the prospective crack plane and integrating the product along the crack length „a‟. The success of the weight function

1

Progress Report No.1

DEVELOPMENT OF WEIGHT FUNCTIONS AND

COMPUTER INTEGRATION PROCEDURES FOR

CALCULATING STRESS INTENSITY FACTORS AROUND

CRACKS SUBJECTED TO COMPLEX STRESS FIELDS

Prepared for

Analytical Services& Materials, Inc.

107 Research Drive,

Hampton, VA 23666, USA

by

G. Glinka, Ph. D.

(on behalf of)

SaFFD, Inc.

Stress and Fatigue-Fracture Design

RR #2, Petersburg

Ontario, Canada N0B 2H0

March 1996

2

Summary

The report consist the first part of the project described in the proposal associated with

the Contract No. 554. The Progress Report No.1 consists of the theoretical background,

the integration methods and the collection of weight functions as described in Tasks 2.1,

2.2 and 2.3. The development of the coded numerical routines is being carried out and it

will be submitted in the next Report.

3

1. INTRODUCTION 5

2. STRESS INTENSITY FACTORS AND WEIGHT FUNCTIONS - TECHNICAL

BACKGROUND 5

3. UNIVERSAL WEIGHT FUNCTIONS FOR ONE-DIMENSIONAL CRACKS 9

4. UNIVERSAL WEIGHT FUNCTIONS FOR TWO-DIMENSIONAL PART-

THROUGH SURFACE AND CORNER CRACKS 11

5. SEQUENCE OF STEPS FOR CALCULATING STRESS INTENSITY

FACTORS USING WEIGHT FUNCTIONS 13

6. DETERMINATION OF WEIGHT FUNCTIONS 13

7. NUMERICAL INTEGRATION OF THE WEIGHT FUNCTION AND

CALCULATION OF STRESS INTENSITY FACTORS 17

7.1 Integration by using the centroids of areas under the weight function curve 17

7.2 Analytical integration of the linearized piecewise stress distribution and the

weight function 23

8. WEIGHT FUNCTIONS FOR CRACKS IN PLATES AND DISKS 26

8.1 Through crack in an infinite plate subjected to symmetrical loading 26

8.2 Central through crack in a finite width plate subjected to symmetric loading 28

8.3 Edge crack in a semi-infinite plate 29

8.4 Edge crack in a finite width plate 31

8.5 Two symmetrical edge cracks in a finite width plate subjected to symmetrical

loading 32

8.6 Embedded penny-shape crack in an infinite body subjected to symmetric

loading 33

8.7 Shallow semi-elliptical surface crack in a finite thickness plate (a/c < 1) 36

8.8 Deep semi-elliptical surface crack in a finite thickness plate (a/c > 1) 41

4

8.9 Quarter-elliptical corner crack in a finite thickness plate 46

8.10 Radial edge crack in a circular disk 51

8.11 Edge crack in a finite thickness plate with the right angle corner 53

8.12 Edge crack in a finite thickness plate with an angular corner 55

9. WEIGHT FUNCTIONS FOR CRACKS IN CYLINDERS 57

9.1 Internal circumferential edge crack in a thick cylinder subjected to

axisymmetric loading (Ro/Ri=2.0) 58

9.2 Two internal symmetrical axial edge cracks in a thick cylinder subjected to

symmetrical loading (Ro/Ri=2.0) 61

9.3 External axial edge crack in a thick cylinder (Ro/Ri=2.0) 64

9.4 Internal circumferential semi-elliptical surface crack in a thick cylinder

(Ro/Ri=1.1) 67

9.5 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0) 71




9.9 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0) 88




10. NOTATION 106

11. LITERATURE 108

5

1. INTRODUCTION

Durability and strength evaluation of notched and cracked structural elements requires

calculation of stress intensity factors for cracks located in regions characterized by

complex stress fields. This is particularly true for cracks emanating form notches and

other stress concentration regions that might be frequently found in mechanical and

structural components. In the case of engine components complex stress distributions are

often due to temperature effects. The existing stress intensity factor handbook solutions

are not sufficient in such cases due to the fact that most of them have been derived for

simple geometry and loading configurations. The variety of notch and crack

configurations and the complexity of stress fields occurring in engineering practice

require more efficient tools for calculating stress intensity factors than the large but

nevertheless limited number of currently available ready made solutions, obtained for a

few specific geometry and load combinations.

Therefore, it is proposed to develop a methodology for efficient calculation of stress

intensity factors for cracks in complex stress fields by using the weight function

approach.

2. STRESS INTENSITY FACTORS AND WEIGHT FUNCTIONS -

TECHNICAL BACKGROUND

Most of the existing methods of calculating stress intensity factors require separate

analysis for each load and geometry configuration. The weight function method

developed by Bueckner [1] and Rice [2] simplifies considerably the determination of

stress intensity factors. If the weight function is known for a given cracked body, the

stress intensity factor due to any load system applied to the body can be determined by

using the same weight function. Therefore there is no need to derive ready made stress

intensity factor for each load system and resulting internal stress distribution. The stress

intensity factor can be obtained by multiplying the weight function m(x,a) and the internal

6

stress distribution (x) in the prospective crack plane and integrating the product along

the crack length „a‟.

The success of the weight function technique for calculating stress intensity factors lies in

the possibility of using superposition. It can be shown [3] that the stress intensity factor

for a crack body (Fig. 1) subjected to the external loading S is the same as stress intensity

factor in geometrically identical body with the local stress field (x) applied to the crack

faces. The local stress field (x) in the prospective crack plane is due to the external load

S and it is determined for uncracked body by ignoring the presence of the crack.

The unique feature of the weight function method is that once the weight function for a

particular cracked body is determined, the stress intensity factor for any loading system

applied to the body can be calculated by a simple integration of the product of the stress

field, (x), and the weight function, m(x,a).

The idea of using weight functions is illustrated in Fig.2, where the through thickness

crack is used as an example. The weight function m(x,a) can be interpreted as the stress

intensity factor that results from a pair of splitting forces applied to the crack face at

position x. Since the stress intensity factors are linearly dependent on the applied loads,

the contributions from multiple splitting forces applied along the crack surface can be

superposed and the resultant stress intensity factor can be calculated as the sum of all

individual load contributions. This results in the integral (1) of the product of the weight

function m(x,a) and the stress field (x) for the case of continuously distributed stress

field.

(1) dxa,xmxKa

0

7

Fig. 2. a) Weight function and stress intensity factors for a through crack in an infinite

plate.

8

Fig.2. b) Weight function and stress intensity factors for a through crack in an infinite

plate subjected to multiple loads or/and continuously distributed stress.

9

3. UNIVERSAL WEIGHT FUNCTIONS FOR ONE-DIMENSIONAL

CRACKS

The weight function is dependent on the geometry only and in principle should be derived

individually for each geometrical configuration. However, Glinka and Shen 4 have

found that one general weight function expression can be used to approximate weight

functions for a variety of geometrical configurations of cracked bodies with one

dimensional cracks of Mode I.

The system of coordinates and the notation for internal through and edge cracks are given

in Fig. 3.

In order to determine the weight function m(x,a) of eq(2) for a particular cracked body it

is sufficient to determine [5] the three parameters M1, M2, and M3. Because the weight

function of form (2) is the same for all cracks then the same integration routine can be

used for calculating stress intensity factors from eq.(1). Moreover, only limited number

of generic weight functions is needed to enable the determination of stress intensity

factors for a large number of load and geometry configurations.

(2) a

x1M

a

x1M

a

x1M1

xa2

2a,xm

2

3

3

1

2

2

1

1

10

Figure 3. Single Edge Crack and Central Through Crack Under Symmetrical Loading

11

4. UNIVERSAL WEIGHT FUNCTIONS FOR TWO-DIMENSIONAL

PART-THROUGH SURFACE AND CORNER CRACKS

In the case of 2-D cracks such as semi-elliptical and corner surface cracks in plates and

cylinders the stress intensity factor changes along the crack front. However, in most

practical cases the deepest point A (Fig. 4) and the surface point B are associated with the

highest and the lowest value of the stress intensity factor along the crack front. Therefore,

the weight functions for the points A and B have been derived [6] analogously to the

universal weight function (2).

For point A (Fig. 4)

For point B (Fig. 4)

The weight functions mA(x,a) and mB(x,a) given above corresponding to the deepest and

the surface point A and B respectively have been derived for one-dimensional stress

fields (Fig. 4), dependent on one variable , x , only.

(4) a

xM

a

xM

a

xM1

x

2ta,ca,a,xm

2

3

B3

1

B2

2

1

B1B

(3) a

x1M

a

x1M

a

x1M1

xa2

2ta,ca,a,xm

2

3

A3

1

A2

2

1

A1A

12

Fig. 4. Semi-elliptical surface crack in an infinite plate of finite thickness, t.

13

5. SEQUENCE OF STEPS FOR CALCULATING STRESS

INTENSITY FACTORS USING WEIGHT FUNCTIONS

In order to calculate stress intensity factors using the weight function technique the

following tasks need to be carried out:

Determine stress distribution (x) in the prospective crack plane using linear elastic

analysis of uncracked body (Fig. 1a), i.e. perform the stress analysis ignoring the

crack and determine the stress distribution (x) = 0 f(S,x);

where: 0 - nominal stress, x - coordinate, S - external load

Apply the “uncracked” stress distribution (x) to the crack surfaces (Fig. 1b) as

tractions

Choose appropriate generic weight function (i.e. choose appropriate M1, M2, and M3

parameters).

Integrate the product of the stress function (x) and the weight function m(x,a) over

the entire crack length or crack surface, eq.(1).

6. DETERMINATION OF WEIGHT FUNCTIONS

The derivation of the weight function of eq.(2) for a one-dimensional crack is essentially

reduced to the determination of parameters M1, M2, and M3. There are several methods

available depending on the amount of information available as described in reference [5].

The most frequently used method based on two reference stress intensity factors is

described below.

Supposing that two reference stress intensity solutions Kr1 and Kr2 for two different stress

fields r1(x) and r2(x) respectively are known, a set of two equations can be written by

using eqs.(1) and (2). The third equation necessary to determine parameters M1, M2, and

14

M3 can be formulated based on the knowledge of the crack surface slope [5,6] at the crack

center (central through cracks) or from the crack surface curvature at the crack mouth

(edge cracks). The three pieces of information result in three independent equations

which can be used for the determination of unknown parameters M1, M2, and M3.

Central through cracks under symmetric loading (Fig. 3a)

Edge cracks

The two reference cases can be obtained from the literature or they can be derived using

finite element method. Most often available are the solutions for the uniform and linear

(7) 0

a

x1M

a

x1M

a

x1M1

xa2

2

x

(6) dxa

x1M

a

x1M

a

x1M1

xa2

2xK

(5) dxa

x1M

a

x1M

a

x1M1

xa2

2xK

0x

2

3

3

1

2

2

1

1

2

3

3

1

2

2

1

1

a

02r2r

2

3

3

1

2

2

1

1

a

01r1r

(10) 0

a

x1M

a

x1M

a

x1M1

xa2

2

x

(9) dxa

x1M

a

x1M

a

x1M1

xa2

2xK

(8) dxa

x1M

a

x1M

a

x1M1

xa2

2xK

0x

2

3

3

1

2

2

1

1

2

3

3

1

2

2

1

1

a

02r2r

2

3

3

1

2

2

1

1

a

01r1r

15

stress distributions induced by simple tension or bending load and those cases were

predominantly used to derive the weight functions listed below.

Semi-elliptical surface and corner cracks

The set of equations necessary for deriving MiA parameters for the weight function

corresponding to the deepest point A on the crack front (Fig. 3) was found to be identical

to the set of equations derived for the edge crack.

In the case of the surface point B the first two equations were the same as previously and

the additional equation was derived [6] by satisfying the condition that the weight

function must vanish for x=0.

(13) 0

a

x1M

a

x1M

a

x1M1

xa2

2

x

(12) dxa

x1M

a

x1M

a

x1M1

xa2

2xK

(11) dxa

x1M

a

x1M

a

x1M1

xa2

2xK

0x

2

3

A3

1

A2

2

1

A1

2

3

A3

1

A2

2

1

A1

a

02r

A2r

2

3

A3

1

A2

2

1

A1

a

01r

A1r

(16) MMM10

(15) dxa

xM

a

xM

a

xM1

x

2xK

(14) dxa

xM

a

xM

a

xM1

x

2xK

B3B21B

2

3

B3

1

B2

2

1

B1

a

02r

B

2r

2

3

B3

1

B2

2

1

B1

a

01r

B

1r

16

The three unknown parameters Mi can be determined by simultaneously solving one of

the set of equations presented above.

17

7. NUMERICAL INTEGRATION OF THE WEIGHT FUNCTION

AND CALCULATION OF STRESS INTENSITY FACTORS

The calculation of stress intensity factor from the weight function requires integration of

of the product “s(x)m(x,a)” along the crack length according to eq.(1). The weight

function can always be written in the general form of eq.(3) or eq.(4). However the stress

distribution “s(x)” can take any form depending on the problem of interest. If the stress

distribution is given in the form of a mathematical expression analytical integration can

be performed and closed form integrals of eq.(1) are sometimes feasible. However, very

often the stress distribution “s(x)” is obtained from finite element calculations and the

results are given as a series of stress values corresponding to a range of point of

coordinate “x”. Therefore, a numerical integration technique is needed for the integration

of equation (1) and the calculation of stress intensity factors. Two methods of efficient

integration of eq.(1) are described below.

7.1 Integration by using the centroids of areas under the weight function curve

The integration method using the area centroids is based on the following theorem:

If m(x,a) and (x) are monotonic and linear function respectively and both depend on

variable x only, (Fig. 5), then the integral (1) can be calculated from expression (14),

representing the product of the area S under the curve m(x,a) and the value of the function

(X) corresponding to the coordinate x=X of the centroid C.

The weight functions, m(x,a), are monotonic and non-linear. The stress functions,

(x), are usually nonlinear as well. Therefore, in order to apply the theorem above to the

integral (1), the integration interval is divided into “n” sub-intervals in such a way that,

the stress function (x) is approximated by the secant line drawn between the end points

of each sub-interval Fig. (6). Thus, the stress function (x) over the sub-interval “i”, may

be written in the form of Eq. (15),

(14) XSK

18

After substitution of expression (15) into eq. (1) and summation over all sub-intervals the

following expression for the stress intensity factor can be derived.

Each integral in eq.(16) can be computed by using the simplified (Fig. 5) integration

method given in the form of Eq.(14). Thus, the stress intensity factor, K, can be finally

written in the form of expression (17).

In order to calculate the stress intensity factor given in the form of Eq. (17), it is necessary

to calculate the areas Si under the weight function curve m(x,a), and the co-ordinates of

their centroids, Xi. Both the areas Si and the centroid co-ordinates Xi for each sub-interval

can be calculated once in a general form based on the generalised weight function (2).

However, they appear to be too lengthy for an easy hand calculation. Fortunately, further

simplification of the integration routine is possible due to the fact that the weight

functions are smooth within their ranges of integration. Therefore the procedure can be

reversed (Fig. 7) by calculating first the areas Si* under the stress function (x) and the

coordinates Xi* of their centroid. Then the appropriate values of the weight function

m(X*,a) can be calculated from expression (2). It is worth noting that in the case of the

piece-wise approximation of the stress function, (x), the areas Si* and the coordinates of

their centroids, Xi*, can be easily calculated from relations (18) and (19) respectively.

(15a) xAxB and

xx

xxA

:where

(15) xx1xfor BxAx

iiii

1ii

1iii

iiiii

(16) dxa,xmBxAKn

1i

x

x

ii

i

1i

(17) n 1,2,i here w,XSK i

n

i

i

19

Finally the stress intensity factor, K, is calculated from expression (20).

Thus the numerical procedure for calculating the stress intensity factor using the

integration method described above requires the calculation of appropriate parameters

using equations (2) and (18-20). The method described above is recommended for quick

approximate calculations with the help of a hand calculator.

(19) xx3

xx2xxxX

(18) xxxx2

1S

1ii

i1i1iii

*i

1ii1ii*i

(20) a,XmSK *i

n

1i

*i

20

Fig. 5.Graphical representation of simplified integration of the product of two one-

dimensional functions, a) monotonic non-linear weight function m(x,a), b) linear stress

function s(x).

21

Fig. 6. Application of the simplified integration method to the case of non-linear stress

distribution; a) weight function m(x,a), b) linearized stress function.

22

Fig. 7. Integration method based on the sub-areas and centroids of the linearized

piecewise stress function, a) weight function m(x,a), b) linearized piecewise stress

function.

23

7.2 Analytical integration of the linearized piecewise stress distribution and the weight

function

The integration technique described in Section 7.1 is convenient for hand calculator

calculation when the stress distribution can be approximated by a few linear segments.

However, the segments adjacent to the crack tip can not be large because the weight

function tends to infinity near the crack tip and if the stress is the highest near the crack

tip the method described above might be sometimes inaccurate. Moreover, in the case of

stress distributions characterized by high gradients accurate approximation of the stress

distribution requires relatively large number of linear pieces and the integration has to be

carried out with the help of a computer. However, the computer integration routine can

also be significantly simplified because closed form solution to the integral (16) can be

derived analytically for each linear piece (Fig. 7) of the stress function s(x).

The stress function s(x) over the linear segment “i” can be given in the form of the linear

equation (15). Thus the contribution to the stress intensity factor associated with the stress

segment “i” can be calculated from eq.(1) after substituting appropriate expressions for

the stress and the weight function. The solutions given below have been derived for the

deepest and for the surface point of semi-elliptical crack using the weight function (3) and

(4) respectively. The solution to one-dimensional cracks is the same as for the deepest

point of semi-elliptical cracks.

Deepest point A (Fig. 4)

Surface point B (Fig.4)

(21) dxa

x1M

a

x1M

a

x1M1

a

x1a2

2BxAK

2

3

A3

1

A2

2

1

A1

x

xii

Ai

i

1i

(22) dxa

xM

a

xM

a

xM1

x2

2BxAK

2

3

B3

1

B2

2

1

B1

x

xii

B

i

i

1i

24

The closed form expressions resulting from the integration of eq.(21) and eq.(22) are

given below.

Deepest point A (Fig. 4)

Surface point B (Fig.4)

3

i

3

1i16

2

5

i2

5

1i5i

2

i

2

1i14

2

3

i2

3

1i3i

1

i

1

1i12

2

1

i2

1

1i1i

iiiii

6iA35iA24iA13ii

4iA33iA22iA11iiAi

a

x1

a

x1

3

aC

a

x1

a

x1

5

a2C

a

x1

a

x1

2

aC

a

x1

a

x1

3

a2C

a

x1

a

x1aC

a

x1

a

x1a2C

aA and aAB

:where

(23) CMCMCMC

CMCMCMC

a

2K

25

Equations (23) and (24) can be used for calculating stress intensity contributions due to

each linear piece of the stress distribution function by substituting appropriate values for

a, xi-1, xi, Ai and Bi. The stress intensity factor K can be finally calculated as the sum of

allcontributions Ki associated with the linear pieces within the range of 0 = x = a.

Thus the integration can be reduced to the substitution of appropriate parameters into

equations (23 -24) and summation according to eq(25). This makes it possible to develop

very efficient numerical integration routine which is important in the case of lengthy

fatigue crack growth analyses.

(25) KKn

i

i

3

1i

3

i16

2

5

1i2

5

i5i

2

1i

2

i14

2

3

1i2

3

i3i

1

1i

1

i12

2

1

1i2

1

i1i

iiiii

6iB35iB24iB13ii

4iB33iB22iB11iiiB

i

a

x

a

x

3

a D

a

x

a

x

5

a2D

a

x

a

x

2

a D

a

x

a

x

3

a2D

a

x

a

xa D

a

x

a

xa2D

aA and aAB

:where

(24) DMDMDMD

DMDMDMD

a

2K

26

8. WEIGHT FUNCTIONS FOR CRACKS IN PLATES and DISKS

The weight functions are given in the form of expressions describing the Mi parameters as

functions of crack dimensions and geometry of the cracked body. Given are the range of

application, the accuracy and the source of the reference stress intensity factors for each

set of parameters Mi including the generic geometry of the crack body.

8.1 Through crack in an infinite plate subjected to symmetrical loading

WEIGHT FUNCTION (Fig. 8a)

PARAMETERS

M1= 0.0698747

M2= -0.0904839

M3= 0.427203

ACCURACY: Max. error less than 1% when compared to exact solution

REFERENCES:

weight function:

Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I

“Engng. Fract. Mech”, Vol. 40, No. 6, pp. 1135-1146.

reference data:

Tada H., Paris P. C., Irwin G.R., 1985, The Stress Analysis of Cracks Handbook, 2nd ed.,

Paris Productions Inc., St. Louis, Missouri, pp. 5.11-5.11a.

2

3

3

1

2

2

1

1a

x1M

a

x1M

a

x1M1

xa2

2a,xm

27

Figure 8. Central through-crack in an infinite plate (a) and finite-width plate (b).

28

8.2 Central through crack in a finite width plate subjected to symmetric loading

WEIGHT FUNCTION (Fig. 8b )

PARAMETERS

RANGE OF APPLICATION: 0 < a/w < 0.9.

ACCURACY: Better than 1% when compared to the reference solution.

REFERENCES:

- weight function:

Moftakhar A., Glinka G., 1992, “Calculation of Stress Intensity Factors by Efficient

Integration of Weight Functions,“ Engng. Fract. Mech., Vol. 43, No. 5, pp. 749-756.

Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I,

“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.

- reference data:


Paris Productions Inc., St. Louis, Missouri, pp. 2.1-2.2, 2.34.

2

3

3

1

2

2

1

1a

x1M

a

x1M

a

x1M1

xa2

2a,xm

765

432

3

65

432

2

765

432

1

w

a1575.68

w

a882.295

w

a128.574

w

a255.347

w

a40.138

w

a6349.29

w

a56001.2427216.0M

w

a128.111

w

a445.392

w

a599.210

w

a6553.89

w

a5325.22

w

a14886.209049.0M

w

a218.140

w

a939.377

w

a395.552

w

a699.200

w

a0886.50

w

a5407.5

w

a40117.006987.0M

29

Sih G., 1973, Handbook of Stress Intensity Factors, Institute of Fracture and Solid

Mechanics, Lehigh University, Bethlehem, P.A.

8.3 Edge crack in a semi-infinite plate

WEIGHT FUNCTION (Fig. 9a)

PARAMETERS

M1= 0.0719768

M2= 0.246984

M3= 0.514465

ACCURACY: Max. error less than 1% when compared to the reference solution

REFERENCES:

- weight function:

Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode

I,“ Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.

- reference data:


Paris Productions Inc., St. Louis, Missouri, pp. 8.3-8.3a.

Sih G., 1973, Handbook of Stress Intensity Factors, Institute of Fracture and Solid

Mechanics, Lehigh University, Bethlehem, P.A.

2

3

3

1

2

2

1

1a

x1M

a

x1M

a

x1M1

xa2

2a,xm

30

Fig 9: Edge crack solutions in semi-infinite and finite-width plates.

31

8.4 Edge crack in a finite width plate

WEIGHT FUNCTION (Fig.9b)

PARAMETERS



REFERENCES:

- weight function:



Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode

I,“ Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.

- reference data:


Paris Productions Inc., St. Louis, Missouri, p. 2.27.

2

3

3

1

2

2

1

1a

x1M

a

x1M

a

x1M1

xa2

2a,xm

8765

432

3

109876

5432

2

109876

5432

1

w

a7.35780

w

a106246

w

a128746

w

a6.81388

w

a2.28592

w

a53.5479

w

a074.532

w

a3235.22529659.0M

w

a127291

w

a531940

w

a936863

w

a904370

w

a520551

w

a181755

w

a8.37303

w

a76.4058

w

a456.176

w

a47583.6246984.0M

w

a49544

w

a208233

w

a368270

w

a356469

w

a205384

w

a7.15907

w

a8.14583

w

a95.1554

w

a1001.61

w

a51346.10719768.0M

32

Kaya A. C.., Erdogan F., 1980, “Stress Intensity Factors and COD in an Orthotropic Strip,

“Int. J. Fracture, Vol. 16, pp. 171-190.

8.5 Two symmetrical edge cracks in a finite width plate subjected to symmetrical

loading

WEIGHT FUNCTION (Fig.9c)

PARAMETERS



REFERENCES:

- weight function:





2

3

3

1

2

2

1

1a

x1M

a

x1M

a

x1M1

xa2

2a,xm

65

432

3

65

432

2

8765

433

1

w

a8893.59

w

a976.131

w

a933.115

w

a9251.47

w

a5597.10

w

a7512.04983.0M

w

a4731.86

w

a634.182

w

a937.151

w

a5326.56

w

a1687.11

w

a6146.02234.0M

w

a009.127

w

a591.336

w

a837.336

w

a266.481

w

a6924.19

w

a64559.4

w

a41028.1

w

a02230.008502.0M

33

- reference data:


Paris Productions Inc., St. Louis, Missouri, p. 2.31.

8.6 Embedded penny-shape crack in an infinite body subjected to symmetric loading

WEIGHT FUNCTION (Fig. 10) - Point A

m x a

a xM

x

aM

x

aM

x

a

M

M

M

A A A A

A

A

A

( , )

.

.

.

/ /

2

21 1 1 1

0 646714

0 303783

0527654

1

1 2

2 3

3 2

1

2

3

PARAMETERS

WEIGHT FUNCTION (Fig. 10) - Point B

m x ax

Mx

aM

x

aM

x

a

M

M

M

B B B B

B

B

B

( , )

/ /

21

1

0

0

1

1 2

2 3

3 2

1

2

3

PARAMETERS


REFERENCES:

- weight function:

Glinka G., Shen G., 1991, “Universal Features of Weight Functions for Cracks in Mode I

,“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146.

- reference data:

Gallin L. A., 1953, Contact Problems in the Theory of Elasticity, (in Russian),

translation: North Carolina State College Publications (1961).

34


Paris Productions Inc., St. Louis, Missouri, pp. 24.2-24.3.

35

Fig. 10. Embedded penny-shape crack in an infinite body subjected to

symmetrical loading

36

8.7 Shallow semi-elliptical surface crack in a finite thickness plate (a/c < 1)

WEIGHT FUNCTION (Fig.11) - Deepest Point A

PARAMETERS

where:

and

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8YY2

Q2

6M

3M

5

24Y3Y2

Q2

2M

01A3

A2

10A1

071.8

3

953.9

2

688.0

2

1

32

0

6

3

4

2

2

100

65.1

c

a1211.24

c

a014.0

1459.1A

c

a10.22

c

a027.0

1995.0A

c

a147.0

1

c

a007.2

c

a045.3456.0A

c

a4394.0

c

a7703.0

c

a2581.00929.1A

t

aA

t

aA

t

aAAY

c

a464.11Q

37

WEIGHT FUNCTION (Fig.11) - Surface Point B

PARAMETERS

where:

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

xa2

2a,xm

203.9

3

286.9

2

846.0

2

1

32

0

6

3

4

2

2

101

c

a1924.21

c

a020.0

1

c

a643.3228.4B

c

a1259.17

c

a041.0

1

c

a126.3148.3B

c

a198.0

1

c

a534.0

c

a665.1652.1B

c

a460.0

c

a7412.0

c

a1231.04537.0B

t

aB

t

aB

t

aBBY

8F7F10Q

3M

15F3F2Q

15M

8F3F5Q

3M

01B3

10B2

01B1

38

and

RANGE OF APPLICATION: 0 = a/t =0.8 and 0 = a/c = 1.0

ACCURACY: Better than 3% when compared to the FEM data.

REFERENCES:

- weight function:

Wang X., Lambert S. B., 1995, “Stress Intensity Factors for Low Aspect Ratio Semi-

Elliptical Surface Cracks in Finite-Thickness Plates Subjected to Nonuniform Stresses,“

Engng. Fract. Mech., Vol. 51, No. 4, pp. 517-532.

Shen G., Glinka G., 1991, “Weight Functions for a Surface Semi-Elliptical Crack in a

Finite Thickness Plate,“ Theor. Appl. Fract. Mech., Vol. 15, No. 2, pp. 247-255 (this

paper contains the older version of the weight function).

c

a157.0

879.0101.1C

c

a5128.0

c

a3219.15083.1C

c

a0185.0

c

a1548.029782.1C

c

a

t

aC

t

aCCF

c

a464.11Q

2

2

1

2

0

4

2

2

100

65.1

c

a035.0

199.0

c

a608.0190.0D

c

a5876.0

c

a2289.11207.1D

c

a7250.0

c

a4646.1

c

a0642.12687.1D

c

a

t

aD

t

aDDF

2

2

1

32

0

4

2

2

101

39

Shen G., Plumtree A., Glinka G., 1991, “Weight Function for the Surface Point of Semi-

Elliptical Surface Crack in a Finite Thickness Plate,“ Engng. Fract. Mech., Vol. 40, No.

1, pp. 167-176 (this paper contains the older version of the weight function).

- reference data:

Shiratori M., Miyoshi T., Tanikawa K., 1986, “Analysis of Stress Intensity Factors for

Surface Cracks Subjected to Arbitrarily Distributed Surface Stress (2nd Report, Analysis

and Application of Influence Coefficients for Flat Plates with a Semielliptical Surface

Crack),“ Trans. JSME, Vol. 52, pp. 390-398.

Gross B., Srawley J. E., 1965, “Stress Intensity Factor for Single-Edge-Notch Specimens

in Bending or Combined Bending and Tension by Boundary Collocation of a Stress

Function,“ NASA TN, D-2603.

Gross B., Srawley J. E., 1966, Plane Strain Crack Toughness Testing of High Strength

Metallic Materials, ASTM STP 410.

Raju I. S., Newman J. C., 1979, “Stress Intensity Factors for a Wide Range of Semi-

Elliptical Surface Cracks in Finite Thickness Plate, “Engng. Fracture Mech., Vol. 11, pp.

817-829.

Newman J. C., Raju I. S., 1981, “An Empirical Stress Intensity Factor Equation for the

Surface Crack, “Engng. Fracture Mech., Vol. 15, pp. 185-192.

40

Fig. 11. Shallow semi-elliptical surface crack in a finite thickness plate.

41

8.8 Deep semi-elliptical surface crack in a finite thickness plate (a/c > 1)

WEIGHT FUNCTION (Fig. 12) - Deepest Point A

PARAMETERS

where:

and

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8YY2

Q2

6M

3M

5

24Y3Y2

Q2

2M

01A3

A2

10A1

2

2

2

1

2

0

4

2

2

100

265.1

65.1

c

a0934.0

c

a5517.07855.0A

c

a2454.0

c

a01106.108461.1A

c

a03526.0

c

a12945.013047.1A

t

aA

t

aAAY

0.1c

afor

c

a

a

c464.11

0.1c

a0for

c

a464.11

Q

42

WEIGHT FUNCTION (Fig. 12) - Surface Point B

PARAMETERS

where:

2

2

2

1

2

0

4

2

2

101

c

a0731.0

c

a4177.06459.0B

c

a1492.0

c

a6352.07259.0B

c

a0529.0

c

a2609.05044.0B

t

aB

t

aBBY

2

3

B3

1

B2

2

1

B1Ba

x1M

a

x1M

a

x1M1

xa2

2a,xm

8F7F10Q

3M

15F3F2Q

15M

8F3F5Q

3M

01B3

10B2

01B1

43

and

RANGE OF APPLICATION: 0 = a/t = 0.8 and 0.6 = a\c = 2.0

ACCURACY: Better than 2% when compared to the reference FEM data.

REFERENCES:

- weight function:

Wang X., 1995, “Weight Functions for High Aspect Ratio Semi-Elliptical Surface Cracks

in Finite-Thickness Plates”, University of Waterloo, to be published.

- reference data:

Shiratori M., Miyoshi T., Tanikawa K., 1986, “Analysis of Stress Intensity Factors for

Surface Cracks Subjected to Arbitrarily Distributed Surface Stress (2nd Report, Analysis

and Application of Influence Coefficients for Flat Plates with a Semielliptical Surface

Crack),“ Trans. JSME, Vol. 52, pp. 390-398.

2

2

2

1

2

0

4

2

2

100

265.1

65.1

c

a1894.0

c

a4423.008429.0C

c

a37185.0

c

a5275.1757673.1C

c

a08125.0

c

a29091.033469.1C

c

a

t

aC

t

aCCF

0.1c

afor

c

a

a

c464.11

0.1c

a0for

c

a464.11

Q

2

2

2

1

2

0

4

2

2

101

c

a1864.0

c

a4784.02246.0D

c

a23315.0

c

a98743.015312.1D

c

a0781.0

c

a2065.011855.1D

c

a

t

aD

t

aDDF

44

Murakami Y., et al., 1992, Stress Intensity Factors Handbook, Vol. 3, Pergamon Press,

pp. 588-590.

45

Fig. 12. Deep semi-elliptical surface crack in a finite thickness plate

46

8.9 Quarter-elliptical corner crack in a finite thickness plate


PARAMETERS

where:

and

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8YY2

Q2

6M

3M

5

24Y3Y2

Q2

2M

01A3

A2

10A1

32

4

32

3

32

2

32

1

32

0

4

4

3

3

2

2100

65.1

c

a393.8

c

a797.10

c

a073.3572.0A

c

a441.13

c

a906.12

c

a857.1293.1A

c

a918.1

c

a747.6

c

a216.13972.4A

c

a028.0

c

a310.1

c

a953.1599.0A

c

a111.0

c

a186.0

c

a016.0041.1A

t

aA

t

aA

t

aA

t

aAAY

c

a464.11Q

47


PARAMETERS

where:

32

4

32

3

32

2

32

1

32

0

4

4

3

3

2

2101

c

a804.5

c

a250.8

c

a977.2162.0B

c

a370.6

c

a357.5

c

a731.1359.1B

c

a135.0

c

a349.6

c

a028.9468.3B

c

a184.0

c

a740.0

c

a373.1507.0B

c

a052.0

c

a213.0

c

a323.0500.0B

t

aB

t

aB

t

aB

t

aBBY

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

8F7F10Q

3M

15F3F2Q

15M

8F3F5Q

3M

01B3

10B2

01B1

48

and

RANGE OF APPLICATION: 0 = a\t =0.8 and 0.2= a\c = 1.0

ACCURACY: Better than 1.5% when compared to the FEM data.

REFERENCES:

32

4

32

3

32

2

32

1

32

0

4

4

3

3

2

2100

65.1

c

a92.99

c

a6.212

c

a9.14280.30C

c

a9.145

c

a0.302

c

a1.19416.39C

c

a92.72

c

a2.152

c

a9.10095.22C

c

a504.1

c

a658.1

c

a2261.02318.0C

c

a7278.0

c

a016.3

c

a495.4340.3C

c

a

t

aC

t

aC

t

aC

t

aCCF

c

a464.11Q

32

4

32

3

32

2

32

1

32

0

4

4

3

3

2

2101

c

a195.19

c

a228.30

c

a176.9141.1D

c

a910.13

c

a848.8

c

a932.15682.9D

c

a677.2

c

a624.12

c

a057.15019.4D

c

a226.13

c

a001.29

c

a498.20600.4D

c

a511.0

c

a477.2

c

a840.3831.2D

c

a

t

aD

t

aD

t

aD

t

aDDF

49

- weight function:

Zheng X. J., Glinka G., Dubey R. N., 1994, “Stress Intensity Factors and Weight

Functions for a Corner Crack in a Finite Thickness Plate,“ Engng. Fracture Mech.,(to be

published, 1996)

- reference data:

Shiratori M., Miyoshi T., 1985, “Analysis of Stress Intensity Factors for Surface Cracks

Subjected to Arbitrarily Distributed Surface Stress (Analysis and Application of Data

Base of Influence Coefficients Kij),“ Proc. Third Conf. Fract. Mech., Soc. Materials

Science Japan, pp 82-86.

Murakami Y., et al., 1992, Stress Intensity Factors Handbook, Vol. 3, Pergamon Press,

pp. 591-597.

Raju I. S., Newman J. C., 1988, “Stress Intensity Factors for Corner Cracks in

Rectangular Bars,“ in Fracture Mechanics: Nineteenth Symposium, ASTM STP 969,

T.A.Cruse, ed., pp. 43-55.

50

Fig. 13. Quarter-elliptical corner crack in a finite thickness plate

51

8.10 Radial edge crack in a circular disk

WEIGHT FUNCTION (Fig. 14)

PARAMETERS

RANGE OF APPLICATION: 0 < a/D< 0.9.

ACCURACY: Better than 1.5% compared to the reference solution.

REFERENCES:

- weight function:

Kiciak A., 1995, University of Waterloo, to be published.


“Engng. Fract. Mech., Vol. 40, No. 6, pp. 1135-1146 - this paper contains the original

version of the weight function with coefficients Mi given in tabular form.

- reference data:

Wu X. R., Carlsson J. A., 1991, Weight Functions and Stress Intensity Factor Solutions,

Pergamon Press.

2

3

3

1

2

2

1

1a

x1M

a

x1M

a

x1M1

xa2

2a,xm

7654

32

3

7654

32

2

654

32

1

D

a1270.36

D

a35817.61

D

a6.91042

w

a61627.54

D

a14897.48

D

a85709.16

D

a06605.309836.1

expM

D

a34267.125

D

a66155.356

D

a48183.428

D

a25303.280

D

a21153.112

D

a90993.30

D

a48276.858602.0

expM

D

a87969.9exp

D

a74080.0

D

a83026.2

D

a56109.4

D

a96175.3

D

a94141.1

D

a49586.004732.0

M

52

Fig. 14. Radial edge crack in a circular disk

53

8.11 Edge crack in a finite thickness plate with the right angle corner


m x aa x

Mx

aM

x

a,

2

21 1 11 2

2

PARAMETERS

Ma

t

a

t

a

t

a

t

a

t

a

t

Ma

t

a

t

a

t

a

t

a

t

a

t

1

1 5 3 4 5 6

7 5 9

2

1 5 3 4 5 6

7 5

0 6643 12 7438 397 8081 3285.181 14162 587

30127 158 258119 535

0 1117 3 857 30127 158 285.4393 647 6118

934 4538 596 8319

. . . .

. .

. . . .

. .

. .

.

. .

.

9

RANGE OF APPLICATION: 0 = a\t = 0.5, a = 90o.

ACCURACY: Better than 3% compared to the reference data.

REFERENCES:

- weight function:

Niu X., Glinka G., 1990, “Weight Functions for Edge and Surface Semi-Elliptical Crack

in Flat Plates and Plates with Corners,“ Engng. Fract. Mech., Vol. 36, No. 3, pp. 459-

475.

- reference data:

Hasebe N., Ueda M., 1981, “A Crack Originating from an Angular Corner of a Semi-

Infinite Plate with a Step,“ Bull. JSME, Vol. 24, pp. 483-488.

Hasebe N., Matsuura S., 1984, “Stress Analysis of a Strip with a Step and a Crack,“

Engng. Fract. Mech., Vol. 20, pp. 447-462.

54

Fig. 15. Edge crack in a finite thickness plate with the right angle corner

55

8.12 Edge crack in a finite thickness plate with an angular corner


m x a

a xM

x

aM

x

aF( , )

2

21 1 11 2

2

PARAMETERS

Ma

t

a

t

a

t

a

t

a

t

a

t

Ma

t

a

t

a

t

a

t

a

t

a

t

1

1 5 3 4 5 6

7 5 9

2

1 5 3 4 5 6

7 5

0 6643 12 7438 397 8081 3285.181 14162 587

30127 158 258119 535

0 1117 3 857 30127 158 285.4393 647 6118

934 4538 596 8319

. . . .

. .

. . . .

. .

. .

.

. .

.

9

1 2 3 2 2

5 2 3

16

2 0 0355 3 3324 21 5999 58 8513 81 6246

56 9396 15.8784

Fa

t

a

t

a

t

a

t

a

t

a

t

. . . . .

.

/ /

/

RANGE OF APPLICATION: 0 = a/t = 0.5 and /6 = a = /3 for a in radians.

ACCURACY: unknown.

REFERENCES:

- weight function:

Niu X., Glinka G., 1990, “Weight Functions for Edge and Surface Semi-Elliptical Crack

in Flat Plates and Plates with Corners,“ Engng. Fract. Mech., Vol. 36, No. 3, pp. 459-

475.

Niu X., Glinka G., 1987, “The Weld Profile Effect on Stress Intensity Factors in

Weldments,” Int. J. Fract., Vol. 35, pp. 3-20.

- reference data:

Hasebe N., Ueda M., 1981, “A Cracks Originating from an Angular Corner of a Semi-

Infinite Plate with a Step,“ Bull. JSME, Vol. 24, pp. 483-488.

Hasebe N., Matsuura S., 1984, “StressAnalysis of a Strip with a Step and a Crack,“

Engng. Fract. Mech., Vol. 20, pp. 447-462.

56

Fig. 16. Edge crack in a finite thickness plate with an angular corner

57

9. WEIGHT FUNCTIONS FOR CRACKS IN CYLINDERS

The weight functions are given in the form of expressions describing the Mi parameters as

functions of crack dimensions and geometry of the cracked body. Given are the range of

application, the accuracy and the source of the reference stress intensity factors for each

set of parameters Mi including the generic geometry of the crack body. The geometry of a

cylinder is characterized by the ratio of the external to internal radius, Ro/Ri.

58

9.1 Internal circumferential edge crack in a thick cylinder subjected to axisymmetric

loading (Ro/Ri=2.0)


m x aa x

Mx

aM

x

aM

x

a,

2

21 1 1 11

1

2

2

1

3

3

2

PARAMETERS

MY

at

Y

M

M YY

at

1

1

0

2

3 0

1

2

2

3 24

5

3

6

2

2 296

15

where:

t = R0 - Ri (cylinder wall thickness)

Y A Aa

tA

a

tA

a

tA

a

tA

a

t

A

A

A

A

A

A

0 0 1 2

2

3

3

4

4

5

5

0

1

2

3

4

5

1 033505

2 41397

15 34436

46 42214

22 49592

31 75198

26 94214

exp

.

.

.

.

.

.

.

RANGE OF APPLICATION: R0/Ri = 2 and 0 = a/t = 0.8


REFERENCES:

- weight function:

59

Liebster T. D., Glinka G., Burns D. J., Mettu S. R., 1994, “Calculating Stress Intensity

Factors for Internal Circumferential Cracks by Using Weight Functions,“ ASME PVP-

Vol. 281, High Pressure Technology, J. A. Kapp, ed., pp.1-6.

- reference data:

Mettu S. R., Forman R. G., 1993, “Analysis of Circumferential Cracks in Cylinders Using

the Weight Function Method,“ Fracture Mechanics: 23rd Symposium,ASTM STP 1189,

R. Chona, ed., American Society for Testing and Mateials, pp. 417-440.

60

Figure 17. Internal circumferential edge crack in a thick cylinder subjected to

axisymmetric loading.

61

9.2 Two internal symmetrical axial edge cracks in a thick cylinder subjected to

symmetrical loading (Ro/Ri=2.0)


PARAMETERS

MY

s

Y

sY

MY

s

Y

s

Y

MY

s

Y

sY

1

1 2

2 0

2

1 2

2

0

3

1 2

2 0

6

2

13 142

48

5

105

2

3 3

221

12

2

22 214

64

5

where:

s = a/( R0 - Ri), and

for 0 s 0.1

Y s s s s s

s s s s s

0

2 3 5 4 7 5

9 6 10 7 11 8 12 9 12 10

111198 0 978998 360 49 42949 2 261945 10 9 5658 10

219014 10 31481 10 2 74822 10 132888 10 2 72711 10

. . . . . .

. . . . .

Y s s1

20 000291 0 66207 0 224015 . . .

Y s s2

6 4 28 01761 10 7 74379 10 0534838 . . .

for 0.1 s 0.8

Y s s s s s0

2 3 4 51071 0 424314 120826 511629 9 74362 6 08975 . . . . . .

Y s s s s s1

2 3 4 50 009535 0866583 135905 589469 7 68059 4 25993 . . . . . .

Y s s s s s2

2 3 4 50 007826 0161374 0 63532 375368 4 85997 2 85235 . . . . . .

RANGE OF APPLICATION: R0/Ri = 2.0 and 0.1 = s = 0.8

2

3

3

1

2

2

1

1a

x1M

a

x1M

a

x1M1

xa2

2a,xm

62


REFERENCES:

- weight function:

Shen. G., Glinka G. ,1993, “Stress Intensity Factors for Internal Edge and Semi-Elliptical

Cracks in Hollow Cylinders,“ ASME PVP-Vol. 263, High Pressure - Codes, Analysis, and

Applications, A. Khare, ed., pp.73-79.



- reference data:

Andrasic C. P., Parker A. P., 1984, “Dimensionless Stress Intensity Factors for Cracked

Thick Cylinder Under Polynomial Crack Face Loading,“ Engng. Fract. Mech., Vol. 19,

1984, pp. 187-193.

Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press,

pp. 309-317.

63

Fig. 18. Two internal symmetrical axial edge cracks in a thick cylinder subjected

to symmetrical loading

64

9.3 External axial edge crack in a thick cylinder (Ro/Ri=2.0)


m x aa x

Mx

aM

x

aM

x

a,

2

21 1 1 11

1

2

2

1

3

3

2

PARAMETERS

MY

s

Y

sY

MY

s

Y

s

Y

MY

s

Y

sY

1

1 2

2 0

2

1 2

2

0

3

1 2

2 0

6

2

13 142

48

5

105

2

3 3

221

12

2

22 214

64

5

where:

s a R Ro i / ( ) , and

for 0 = s = 0.1

Y s s s s s

s s s

0

2 3 5 4 5 5

6 6 8 7 8 8

111201 167679 7 38413 354383 10534 10 7 33 10

101865 10 168865 10 652672 10

. . . . . .

. . .

Y s s1

4 20 907986 10 0 676241 0 310302 . . .

Y s s2

6 4 28 69582 10 9 8064 10 054827 . . .

for 0.1 = s = 0.8

Y s s s s s

s s

0

2 3 4 5

6 7

164 12 2614 121488 574 313 1542 3 2349 04

189584 628 971

. . . . . .

. .

Y s s s s s1

2 3 4 50 0243751 0 200288 342782 887021 12 4676 589425 . . . . . .

Y s s s s2

2 3 40 00172167 0 0261316 0 42169 0 230129 0 342948 . . . . .

RANGE OF APPLICATION: R0/Ri = 2.0 and 0.1 = s = 0.8

65


REFERENCES:

- weight function:

Shen. G., Liebster T. D., Glinka G. ,1993, “Calculation of Stress Intensity Factors for

Cracks in Pipes,” Proc. of the 12th Int. Conf. on Offshore Mech. and Arctic Engng..,

ASME,M. Salama et al., ed., Vol. III-B, Materials Engineering, pp.847-854.



- reference data:



1984, pp. 187-193.

Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon Press, pp.

309-317.

66

Fig. 19. External axial edge crack in a thick cylinder

67

9.4 Internal circumferential semi-elliptical surface crack in a thick cylinder

(Ro/Ri=1.1)


m x a

a xM

x

aM

x

aM

x

aA A A A( , )

/ /

2

21 1 1 11

1 2

2 3

3 2

PARAMETERS

MQ

Y Y

M

MQ

Y Y

A

A

A

1 0 1

2

3 1 0

2

22 3

24

5

3

6

22

8

5

where:

Qa

c

1 1464

1 65

.

.

, and t R Ro i

Y A Aa

tA

a

t

Aa

c

a

c

Aa

c

a

c

Aa

c

a

c

a

c

0 0 1 2

2

0

2

1

2

2

2 3

11378 0 4259 01299

01223 1502 08013

2 0572 9 4342 138825 65333

. . .

. . .

. . . .

Y B Ba

tB

a

t

Ba

c

a

c

Ba

c

a

c

a

c

Ba

c

a

c

a

c

1 0 1 2

2

0

2

1

2 3

2

2 3

05116 0 373 01129

0 3724 2 3922 33694 15589

13148 5771 7 9406 36193

. . .

. . . .

. . . .


68

m x ax

Mx

aM

x

aM

x

aB B B B( , )

/ /

21 1

1 2

2 3

3 2

PARAMETERS

MQ

F F

MQ

F F

MQ

F F

B

B

B

1 1 0

2 0 1

3 1 0

35 3 8

152 3 15

310 7 8

where:

Qa

c

1 1464

1 65

.

.

and t R Ro i

F Ca

c

Ca

t

a

t

Ca

t

a

t

a

t

a

t

C

0 0

0

2

1

2 3 4

1

0 9242 0 6172 01379

05437 21302 8 0279 118896 58708

. . .

. . . . .

and

F Da

c

Da

t

a

t

Da

t

a

t

a

t

a

t

D

1 0

0

2

1

2 3 4

1

0 7631 0 4891 01164

0 6287 388279 155542 241589 12 6441

. . .

. . . . .

RANGE OF APPLICATION: R0/Ri = 1.1, 0.1 = a/t = 0.8, and 0.2 = a/c = 1.0

ACCURACY: unknown.

REFERENCES:

- weight function:

69

Shen. G., Glinka G. ,1993, “Stress Intensity Factors for Internal Edge and Semi-Elliptical

Cracks in Hollow Cylinders,“ ASME PVP-Vol. 263, High Pressure - Codes, Analysis, and

Applications, A. Khare, ed., pp.73-79.

- reference data:

Shiratori M., 1989, “Analysis of Stress Intensity Factors for Surface Cracks in Pipes by an

Influence Function Method,“ ASME PVP-Vol. 167, pp. 45-50.

70

Figure 20. Internal circumferential semi-elliptical surface crack in a thick cylinder

71

9.5 Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0)


PARAMETERS

where:

and

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8Y2Y

Q2

6M

3M

5

24Y3Y

Q2

2M

10A3

A2

10A1

432

3

5432

2

5432

1

5432

0

4

3

2

2100

io

65.1

c

a084.52

c

a941.118

c

a243.85

c

a828.18140.0A

c

a532.131

c

a920.343

c

a535.323

c

a935.132

c

a265.20498.1A

c

a219.35

c

a89741

c

a649.83

c

a455.36

c

a205.7111.0A

c

a933.0

c

a007.2

c

a305.1

c

a153.0

c

a207.012.1A

t

aA

t

aA

t

aAAY

R-R tand c

a464.11Q

72


PARAMETERS

where:

432

3

5432

2

5432

1

5432

0

3

3

2

2101

c

a841.16

c

a425.33

c

a172.20

c

a742.3087.0B

c

a196.15

c

a961.44

c

a993.54

c

a313.33

c

a481.7821.0B

c

a554.23

c

a282.56

c

a963.41

c

a725.7

c

a496.1163.0B

c

a015.5

c

a482.12

c

a671.10

c

a617.3

c

a377.0687.0B

t

aB

t

aB

t

aBBY

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

8F10F3Q

3M

15FF3Q

15M

8F5F2Q

3M

10B3

01B2

10B1

73

and

RANGE OF APPLICATION: R0/Ri = 2.0, 0 = a/t = 0.8, a/c = 0 and 0.2 = a/c = 1.0


REFERENCES:

- weight function:

Zheng X. J., Glinka G., 1995, “Weight Functions and Stress Intensity Factors

for Longitudinal Semi-Elliptical Cracks in Thick-Wall Cylinders,“ J. Press.

Vess.Technol., ASME, vol. 117, 1995, pp 383 - 389

432

3

432

2

432

1

432

0

4

3

2

2100

io

65.1

c

a124.48

c

a822.144

c

a449.151

c

a389.64586.9C

c

a149.28

c

a904.86

c

a643.112

c

a902.7214.19C

c

a774.3

c

a935.9

c

a138.14

c

a686.11607.3C

c

a37.10

c

a634.31

c

a937.36

c

a55.20923.5C

c

a

t

aC

t

aC

t

aCCF

R-R tand c

a464.11Q

432

3

432

2

432

1

432

0

4

3

2

2101

c

a236.11

c

a036.27

c

a603.21

c

a54.6666.0D

c

a055.29

c

a735.82

c

a828.87

c

a608.42504.8D

c

a244.6

c

a881.17

c

a195.19

c

a399.9797.1D

c

a567.0

c

a981.1

c

a718.2

c

a821.1687.0D

c

a

t

aD

t

aD

t

aDDF

74

- reference data:

Mettu S. R., Raju I. S., Forman R. G., 1988, “Stress Intensity Factors for Part-Through

Surface Cracks in Hollow Cylinders,“ NASA Technical Report, No. JSC 25685 LESC

30124.



1984, pp. 187-193.


pp. 309-317.

75

Figure 21. Internal axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0).

76



PARAMETERS

where:

and

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8Y2Y

Q2

6M

3M

5

24Y3Y

Q2

2M

10A3

A2

10A1

c

a165.4exp46.1140.0A

c

a386.3exp161.1498.1A

c

a393.3exp665.0111.0A

c

a051.5exp07.0044.1A

t

aA

t

aA

t

aAAY

R-R tand c

a464.11Q

3

2

1

0

4

3

2

2100

io

65.1

77


PARAMETERS

where:

c

a079.0exp284.1398.1B

c

a025.4exp753.0307.0B

c

a574.5exp265.0225.0B

c

a035.0exp16.2825.2B

t

aB

t

aB

t

aBBY

3

2

1

0

3

3

2

2101

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

8F10F3Q

3M

15FF3Q

15M

8F5F2Q

3M

10B3

01B2

10B1

78

and



REFERENCES:

- weight function:

Zheng X. J., Kiciak A., Glinka G., 1996, “Determination of Weight Functions and Stress

Intensity Factors for Semi-Elliptical Cracks in Thick-wall Cylinders,“ Nucl. Engng.

Design, to be published.

- reference data:

2

3

2

2

2

1

2

0

4

3

2

2100

io

65.1

c

a817.9

c

a959.9exp33.28104.0C

c

a092.1

c

a081.4exp784.8119.0C

c

a009.4

c

a053.46exp239.10199.0C

c

a568.1

c

a061.5exp163.5972.0C

c

a

t

aC

t

aC

t

aCCF

R-R tand c

a464.11Q

432

3

432

2

432

1

432

0

4

3

2

2101

c

a433.30

c

a555.76

c

a546.65

c

a677.21243.2D

c

a636.3

c

a05.25

c

a644.45

c

a622.30535.6D

c

a099.13

c

a498.42

c

a221.50

c

a231.24448.3D

c

a690.2

c

a397.8

c

a708.9

c

a842.4033.1D

c

a

t

aD

t

aD

t

aDDF

79



30124.



1984, pp. 187-193.


pp. 309-317.

80



PARAMETERS

where:

and

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8Y2Y

Q2

6M

3M

5

24Y3Y

Q2

2M

10A3

A2

10A1

c

a354.1exp061.0149.0A

c

a859.3exp269.3057.0A

c

a17.31exp366.0055.0A

c

a15.13exp0998.0010.1A

t

aA

t

aA

t

aAAY

R-R tand c

a464.11Q

3

2

1

0

4

3

2

2100

io

65.1

81


PARAMETERS

where:

c

a0434.0exp538.1552.1B

c

a562.4exp787.0269.0B

c

a663.8exp436.0136.0B

c

a012.0exp944.5594.6B

t

aB

t

aB

t

aBBY

3

2

1

0

3

3

2

2101

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

8F10F3Q

3M

15FF3Q

15M

8F5F2Q

3M

10B3

01B2

10B1

82

and

RANGE OF APPLICATION: R0/Ri = 1.25, 0 = a/t = 0.8, and a/c = 0 and 0.2 = a/c = 1.0


REFERENCES:

- weight function:

Zheng X. J., Glinka G., Dubey R. N., 1995, “Calculation of Stress Intensity Factors for

Semielliptical Cracks in a Thick-Wall Cylinder,“ Int. J.Pres.Ves. & Piping, Vol. 62, pp.

249-258.

- reference data:

432

3

432

2

432

1

432

0

4

3

2

2100

io

65.1

c

a507.35

c

a289.93

c

a789.83

c

a49.29468.3C

c

a55.9

c

a216.45

c

a59.72

c

a067.49497.12C

c

a616.1

c

a44.10

c

a618.16

c

a514.975.1C

c

a507.11

c

a705.33

c

a335.37

c

a583.19566.5C

c

a

t

aC

t

aC

t

aCCF

R-R tand c

a464.11Q

432

3

432

2

432

1

432

0

4

3

2

2101

c

a853.10

c

a37.26

c

a548.21

c

a605.6569.0D

c

a826.1

c

a235.12

c

a358.22

c

a985.15116.4D

c

a333.0

c

a999.0

c

a412.3

c

a626.2533.0D

c

a212.0

c

a793.0

c

a161.1

c

a879.0486.0D

c

a

t

aD

t

aD

t

aDDF

83



30124.



1984, pp. 187-193.


pp. 309-317.

84



PARAMETERS

where:

and

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8YY2

Q2

6M

3M

5

24Y3Y2

Q2

2M

01A3

A2

10A1

983.0

2

2

2

1

32

0

4

2

2

100

io

65.1

c

a061.0

1

c

a027.13

c

a456.20519.8A

c

a931.6

c

a531.1084.3A

c

a5202.0

c

a0464.1

c

a6699.01449.1A

t

aA

t

aAAY

R-R tand c

a464.11Q

85


PARAMETERS

where:

92.0

2

2

32

1

32

0

4

2

2

101

c

a090.0

1

c

a097.8

c

a282.13251.6B

c

a291.1

c

a928.5

c

a901.6415.2B

c

a4417.0

c

a7576.0

c

a4967.04732.0B

t

aB

t

aBBY

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

8F7F10Q

3M

15F3F2Q

15M

8F3F5Q

3M

01B3

10B2

01B1

86

and

RANGE OF APPLICATION: R0/Ri = 1.1, 0 = a/t = 0.8, and 0 = a/c = 1.0


REFERENCES:

- weight function:

Wang X. J., Lambert S. B., 1996, “Stress Intensity Factors and Weight Functions for

Longitudinal Semi-Elliptical Surface Cracks in Thin Pipes,“ Int. J.Pres.Ves. & Piping,

Vol. 65, pp 75 -87.

Shen. G., Liebster T. D., Glinka G. ,1993, “Calculation of Stress Intensity Factors for

Cracks in Pipes,”Proc. of the 12th Int. Conf. on Offshore Mech. and Arctic Engng..,

05.12

5432

1

2

0

4

2

2

100

io

65.1

c

a2.0

1

c

a15.1065.2C

c

a502.92

c

a492.270

c

a879.293

c

a547.143

c

a96.271256.0C

c

a1203.0

c

a2935.02959.1C

c

a

t

aC

t

aCCF

R-R tand c

a464.11Q

05.12

5432

1

2

0

4

2

2

101

c

a299.0

1

c

a087.1879.1D

c

a309.52

c

a789.153

c

a357.167

c

a361.81

c

a433.153311.0D

c

a4901.0

c

a8104.02959.1D

c

a

t

aD

t

aDDF

87

ASME,M. Salama et al., ed., Vol. III-B, Materials Engineering, pp.847-854. (this paper

contains the older version of the weight function, limited to 0 = a/t = 0.8 and 0.2 =

a/c = 1.0)

- reference data:

Raju I. S., Newman J. C., 1982, “Stress Intensity Factors for Internal and External Surface

Cracks in Cylindrical Vessels,“ J. Press. Ves. Technol., Vol. 104, pp. 293-298.



1984, pp. 187-193.

88

9.9 External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0)


PARAMETERS

where:

and

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8Y2Y

Q2

6M

3M

5

24Y3Y

Q2

2M

10A3

A2

10A1

432

3

432

2

432

1

432

0

4

3

2

2100

io

65.1

c

a64254.63

c

a12933.118

c

a77065.71

c

a84462.23exp78423.1707482.0A

c

a88622.55

c

a30990.118

c

a48907.87

c

a34798.22exp41917.041663.0A

c

a1922.85

c

a87345.185

c

a51103.138

c

a69496.35exp04469.002208.0A

c

a48712.21

c

a45540.53

c

a74122.44

c

a93568.11exp10409.098463.0A

t

aA

t

aA

t

aAAY

R-R tand c

a464.11Q

89


PARAMETERS

where:

6

5432

3

432

2

432

1

432

0

4

3

2

2101

c

a23.75335

c

a66299.84

c

a96622.118

c

a96909.85

c

a59049.35

c

a68986.800553.1B

c

a96618.58

c

a18837.103

c

a68409.66

c

a79057.14exp46420.012875.0B

c

a15246.87

c

a27997.158

c

a67649.103

c

a45619.24exp08062.002433.0B

c

a09569.16

c

a63312.39

c

a30859.82

c

a42681.25exp00525.071344.0B

t

aB

t

aB

t

aBBY

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

8F10F3Q

3M

15FF3Q

15M

8F5F2Q

3M

10B3

01B2

10B1

90

and



REFERENCES:

- weight function:

Kiciak A., Glinka G., Burns D. J., 1996, “Weight Functions and Stress Intensity Factors

for Longitudinal Semi-Elliptical Cracks in Thick-wall Cylinders,“ J. Press. Vess.

Technol., ASME (to be published)

432

4

432

3

432

2

432

1

5432

0

5

4

4

3

2

2100

c

a35426.14

c

a66380.26

c

a30584.19

c

a87203.9exp

c

a88925.01

c

a95222.16C

c

a66400.2

c

a52675.7

c

a20568.8

c

a21823.10exp

c

a17520.11

c

a90204.55C

c

a51091.7

c

a10918.17

c

a42291.9

c

a60488.0exp111589.1C

c

a78266.6

c

a55459.13

c

a78270.5

c

a64659.3exp

c

a17520.11

c

a12461.15C

c

a80402.8

c

a68609.28

c

a14294.36

c

a48633.21

c

a93070.6exp140290.1C

t

aC

t

aC

t

aC

t

aCCF

32

2

432

1

432

0

2

2101

c

a33890.1

c

a71158.1

c

a27930.5exp

c

a90366.01

c

a67302.14D

c

a84826.4

c

a82767.10

c

a99914.4

c

a22628.4exp

c

a09108.11

c

a73349.9D

c

a86369.9

c

a62707.30

c

a07640.28

c

a38873.11exp120478.0D

t

aD

t

aDDF

91

- reference data:



30124.



1984, pp. 187-193.


pp. 309-317.

92

Figure 22. External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=2.0).

93

External axial semi-elliptical surface crack in a thick cylinder (Ro/Ri=1.5)


PARAMETERS

where:

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8Y2Y

Q2

6M

3M

5

24Y3Y

Q2

2M

10A3

A2

10A1

94

5

c

a43104.150

4

c

a88103.419

3

c

a31551.400

2

c

a59759.148

c

a66325.1218341.0

exp024225.03

A

c

a50662.23

c

a40776.28

c

a41241.3

c

a46411.338467.0

exp25833.0A

c

a02799.32

c

a84164.109

c

a22714.142

c

a13521.84

c

a36041.2189775.1

exp83400.0A

c

a92014.0

c

a38695.2

c

a88533.1

c

a14072.085208.0

exp69716.0A

t

aA

t

aA

t

aAAY

R-R tand c

a464.11Q

4

32

2

54

32

1

4

32

0

4

3

2

2100

io

65.1

95

and


PARAMETERS

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

4

c

a67589.24

3

c

a51119.25

2

c

a30479.53

c

a17795.044818.0

exp02704.03

B

c

a080007.37

c

a92487.74

c

a49509.55

c

a46249.1311085.1

exp14247.0B

c

a43167.43

c

a35299.132

c

a08510.154

c

a19018.84

c

a51173.2027787.2

exp43259.0B

c

a25050.0

c

a04996.0

c

a19984.0

c

a07367.078477.0

exp22856.0B

t

aB

t

aB

t

aBBY

4

32

2

54

32

1

4

32

0

4

3

2

2101

96

where:

and

8F10F3Q

3M

15FF3Q

15M

8F5F2Q

3M

10B3

01B2

10B1

322

322

2

432

1

322

0

4

3

2

2100

io

65.1

c

a53042.7

c

a22175.21

c

a21310.17exp

c

a89888.12

c

a91620.1452544.27

c

a

3C

c

a62398.144

c

a84981.84

c

a29526.23exp

c

a72855.0

c

a50789.1170539.2C

c

a25396.27

c

a27126.55

c

a49788.38

c

a47111.16exp

c

a89759.68208999.55

c

aC

c

a37581.382

c

a14830.139

c

a87363.29exp

c

a34206.0

c

a31674.0169519.0C

t

aC

t

aC

t

aCCF

R-R tand c

a464.11Q

97



REFERENCES:

- weight function:

Kiciak A., Glinka G., Burns D. J., 1996, University of Waterloo, to be published.

- reference data:



30124.



1984, pp. 187-193.

Murakami Y., et al., 1986, Stress Intensity Factor Handbook, Vol. 2, Pergamon

Press, pp. 309-317.

25.032

2

2

32

1

32

0

4

3

2

2101

c

a19501.52exp1

c

a861145.2

c

a64517.4

c

a02421.223183.0

3D

c

a38520.1

c

a31235.21

c

a57311.3exp107021.2D

c

a58371.0

c

a66328.0

c

a30088.21

c

a75502.11exp1

c

a66304.0D

c

a55715.27exp

c

a79576.0

c

a27723.1

c

a51333.1109646.0D

t

aD

t

aD

t

aDDF

98



PARAMETERS

where:

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

5

8Y2Y

Q2

6M

3M

5

24Y3Y

Q2

2M

10A3

A2

10A1

54

32

3

432

2

432

1

432

0

4

3

2

2100

io

65.1

c

a71664.98

c

a74017.354

c

a99358.497

c

a20961.340

c

a58315.11294324.15

exp198498.0A

c

a95069.5

c

a48057.11

c

a515307.4

c

a16636.358221.1exp57088.0A

c

a23850.4

c

a96283.11

c

a06481.12

c

a66494.482502.0exp38656.0A

c

a62456.10

c

a59266.26

c

a08395.23

c

a56013.868499.1exp94235.0A

t

aA

t

aA

t

aAAY

R-R tand c

a464.11Q

99

and


PARAMETERS

4

c

a64059.4

3

c

a85419.25

2

c

a70843.33

c

a85207.710419.2

exp01828.04

B

4

c

a50684.8

3

c

a45236.2

2

c

a45912.13

c

a88270.055548.0

exp02569.03

B

c

a32259.1

c

a84029.304497.0exp04652.0B

c

a37007.1

c

a08396.3

c

a21570.2

c

a59120.031448.0

exp53122.1B

c

a71914.4

c

a00652.10

c

a58393.7

c

a14909.271519.1

exp050249B

t

aB

t

aB

t

aB

t

aBBY

2

2

4

32

1

4

32

0

4

3

3

2

2101

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

100

where:

and

8F10F3Q

3M

15FF3Q

15M

8F5F2Q

3M

10B3

01B2

10B1

22

322

2

2

1

2

0

4

3

2

2100

io

65.1

c

a79275.3

c

a18934.3exp

c

a01701.1

c

a08060.2124024.0

3C

c

a88988.78

c

a86421.34

c

a39742.11exp

c

a55332.0

c

a27898.1180687.1C

c

a11736.14exp

c

a47068.1

c

a79131.1222958.0C

c

a02505.14exp

c

a19842.0

c

a58454.1147251.0C

t

aC

t

aC

t

aCCF

R-R tand c

a464.11Q

101



REFERENCES:

- weight function:

Kiciak A., Glinka G., Burns D. J., 1996, University of Waterloo, to be published.

- reference data:



30124.



1984, pp. 187-193.


pp. 309-317.

54

32

2

2

22

1

0

4

3

2

2101

c

a09600.18

c

a62440.60

c

a95158.77

c

a80652.46

c

a47981.1208880.1

c

a

3D

c

a96829.0

c

a89058.11

c

a21697.1exp117158.2D

c

a35842.2

c

a24795.5exp

c

a03870.1

c

a30358.2115562.0D

c

a19737.3exp

c

a54116.0112603.0D

t

aD

t

aD

t

aDDF

102



PARAMETERS

where:

and

5

8YY2

Q2

6M

3M

5

24Y3Y2

Q2

2M

01A3

A2

10A1

2

3

A3

1

A2

2

1

A1Aa

x1M

a

x1M

a

x1M1

xa2

2a,xm

094.1

2

2

2

1

2

0

4

2

2

100

io

65.1

c

a066.0

1

c

a305.8

c

a559.1444.7A

c

a29.5

c

a98.800.4A

c

a2984.0

c

a4322.01492.1A

t

aA

t

aAAY

R-R tand c

a464.11Q

103


PARAMETERS

where:

2

3

B3

1

B2

2

1

B1Ba

xM

a

xM

a

xM1

x

2a,xm

8F7F10Q

3M

15F3F2Q

15M

8F3F5Q

3M

01B3

10B2

01B1

006.1

2

2

2

1

32

0

4

2

2

101

c

a097.0

1

c

a062.5

c

a653.969.5B

c

a850.2

c

a0937.54478.2B

c

a453.0

c

a788.0

c

a5211.04840.0B

t

aB

t

aBBY

104

and

RANGE OF APPLICATION: R0/Ri = 1.1, 0 = a/t = 1.0, and 0 = a/c = 1.0


REFERENCES:

- weight function:

Wang X. J., Lambert S. B., 1996, “Stress Intensity Factors and Weight Functions for

Longitudinal Semi-Elliptical Surface Cracks in Thin Pipes,“ Int. J.Pres.Ves. & Piping,

Vol. 65, pp 75 -87.

- reference data:

Raju I. S., Newman J. C., 1982, “Stress Intensity Factors for Internal and External Surface

Cracks in Cylindrical Vessels,“ J. Press. Ves. Technol., Vol. 104, pp. 293-298.

2

2

5432

1

2

0

4

2

2

100

io

65.1

c

a262.2

c

a746.3405.1C

c

a8339.56

c

a848.177

c

a832.210

c

a107.115

c

a953.251110.0C

c

a0895.0

c

a2532.02964.1C

c

a

t

aC

t

aCCF

R-R tand c

a464.11Q

2

2

5432

1

2

0

4

2

2

101

c

a58529.0

c

a3817.17704.0D

c

a6732.38

c

a252.115

c

a93.129

c

a574.67

c

a4065.142128.0D

c

a4668.0

c

a7814.02531.1D

c

a

t

aD

t

aDDF

105



1984, pp. 187-193.

106

10. NOTATION

a - crack length for one-dimensional cracks or crack depth (minor semi-

axis) for semi-elliptical cracks

Ai, Bi - coefficients of the linearized stress function in the sub-interval “i”

c - crack length (major semi-axis) for semi-elliptical crack

Cij - coefficients of the integrated weight function mA(x,a) in association with

the linearised stress field s(x)

Dij - coefficients of the integrated weight function mB(x,a)in association with the

linearised stress field s(x)

K - stress intensity factor

KI - mode I stress intensity factor (general)

K0A - mode I reference stress intensity factor for the deepest point A of a crack

under the uniform stress field

K0B

- mode I reference stress intensity factor for the surface point B of a crack

under the uniform stress field

KA - mode I stress intensity factor for the deepest point A on the crack front

KB

- mode I reference stress intensity factor for the surface point B on the crack

front

Mi - parameters of the weight function (i = 1, 2, 3)

MiA - coefficients of the weight functions for the deepest point A (i= 1,2,3)

MiB - coefficients of the weight functions for the surface point B (i= 1,2,3)

m(x,a) - weight function

mA(x,a) - weight function for the deepest point A of a crack

mB(x,a) - weight function for the surface point B of a crack

S - area under a monotonic curve m(x,a)

Si*

- area under the linearized stress function s(x) corresponding to the

sub-interval “i”

Si - area under the weight function curve m(x,a) corresponding to the sub-

107

interval “i”

Q - the semi-elliptical crack shape factor: Q = 1 +1.464(a/c)1.65

for a/c < 1

t - wall thickness of a cylinder or plate thickness

x - the local, through the thickness co-ordinate along the crack depth

X - co-ordinate x of the centroid of the area S

Xi - co-ordinate x of the centroid of the area Si corresponding the sub-

interval “i”

Xi*

- co-ordinate x of the centroid of the area Si* corresponding to the sub-

interval “i”

(x) - stress distribution

(xi) -value of stress function at x = xi

(X) - value of stress function at x = X

(x) - a reference stress distribution

YnA - the geometric stress intensity correction factor for the deepest point A

YnB - the geometric stress intensity correction factor for the surface point B

108

11. LITERATURE

1. H. F. Bueckner, A novel principle for the computation of stress intensity

factors. Zeitschrift fur Angewandte Mathematik und Mechanik, 50, 529-546,

1970

2. J. R. Rice, Some remarks on elastic crack-tip stress field. International Journal of

Solids and Structures, 8, 751-758, 1972

3. D. Broek, The Practical Use of Fracture Mechanics, Kluwer, 1988.

4. G.Glinka and G.Shen, Universal features of weight functions for cracks in mode I.

Engineering Fracture Mechanics, 40, 1135-1146, 1991

5. G. Shen and G.Glinka, Determination of weight functions from reference stress

intensity factors. Theoretical and Applied Fracture Mechanics, 15, 237-245, 1991

6. Shen G., Glinka G., 1991, “Weight Functions for a Surface Semi-Elliptical Crack in a

Finite Thickness Plate,“ Theor. Appl. Fract. Mech., Vol. 15, No. 2, pp. 247-255

7. T. Fett, C. Mattheck and D. Munz,”On the Evaluation of Crack opening Displacement

from the Stress Intensity Factor, Engineering Fracture Mechanics, vol. 27, No. 6,

1987, pp-697-715

8. A. Moftakhar and G. Glinka, “Calculation of Stress Intensity Factors by Efficient

Integration of Weight Functions”, Engineering Fracture Mechanics, vol. 43. No. 5,

1992, pp. 749 - 756.

Documents

Progress Report No - · PDF file6 stress distribution (x) in the prospective crack plane and integrating the product along the crack length „a‟. The success of the weight function