,EXPERIMENTAL DETERMINATION OF STRESS INTENSITIES

AND CRACK SHAPES ASSOCIATED WITH THE NOZZLE

CORNER CRACK PROBLEM/

by

Thomas Sherwood Fleischman1 ·':---. I

Thesis submitted to the Graduate Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

in

Engineering Mechanics

APPROVED:

C. W. Smith, Chairman

E. G. Henneke R. A. Heller

August 1978

Blacksburg, Virginia

ACKNOWLEDGEMENTS

My deepest appreciation is extended to my committee chairman,

teacher, employer, guidance counselor, associate and friend --

Professor C. W. Smith, for his time spent in all the above capacities.

Similar regards are due for his constant technical

assistance and friendship. The time and effort contributed by

committee members and is also appreciated,

as is the laboratory assistance of and

Oak Ridge National Laboratories of the Union Carbide Corporation

are noted for their financial support of the problem under

W-7405-ENG-26 subcontract number 7015.

***** The interest and support of my friends, relatives, family and

parents especially, are gratefully acknowledged.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS

LIST OF TABLES .

LIST OF FIGURES

NOMENCLATURE . .

. I. INTRODUCTION

II. ANALYTICAL CONSIDERATIONS

III. EXPERIMENTAL METHOD ..

IV. DISCUSSION OF RESULTS

V. SUMMARY AND CONCLUSIONS

REFERENCES

VITA . . .

iii

Page

ii

iv

v

vi

l

5

11

20

31

33

36

LIST OF TABLES

Table Page

4.1 Crack Geometries . . . . . . . . . . . . . . . . . . . 21

iv

Figure

1.1

2. 1

2.2

3. 1

3.2

3.3

3.4

3.5

LIST OF FIGURES

Initi a 1 crack 1 ocations . . . .

Problem geometry and notation

Photoelastic fringe pattern for mode I

Geometry of the BWR mode 1 . . . . . .

BWR model and calibration rig in stress-freezing oven

Typical crack shape photograph

Typical slice locations ....

Typical plot of computer output

Page

2

6

8

12

14

15

17

18

4.1 Test results--SIF distributions 23

4.2 Test results--crack shapes 24

4.3 Comparison of current results with Besuner's analysis . 28

4.4 Comparison of current results with Peters' results 29

v

a

c ,m

n,t,z

n'

N

p

r,0

R

SIF

t'

T

NOMENCLATURE

Flaw depth at 8 = 45° (mm)

Crack length along nozzle wall (mm)

Crack length along vessel wall (mm)

Characteristic crack length at 8 = 45° (mm)

Material properties associated with crack growth rate

equation (units of c depend on value of m)

Young's modulus of elasticity (kPa)

Material fringe constant (kPa - m/fringe)

Mode one stress intensity factor (kPa - m112)

Apparent mode one stress intensity factor

[Tmax(87rr) 1/2] [ kPa - ml/2}

Local orthogonal coordinate system along the flaw border

(mm)

Stress fringe order

Number of cycles in crack growth rate equation

Internal pressure (kPa)

Polar coordinates with origin at crack tip (mm, radians)

Vessel radius (mm)

Perpendicular distance from the centerline of the nozzle

to the origin of the crack (mm)

Stress intensity factor

Slice thickness (mm)

Nozzle wall thickness at 8 = 45° (mm)

vi

v

crij i,j=n,z

0 a .. i,j=n,z

lJ

'max

Angle measured from nozzle wall toward vessel wall

(degrees)

Angle measured from nozzle wall to point of intersection

of flaw border and vessel wall (degrees)

Poisson's ratio

Stress components in plane normal to flaw surface and

flaw border near crack tip (kPa)

Part of regular stress field near crack tip (kPa)

Maximum shearing stress in n-z plane

vii

I. INTRODUCTION

Failure analysis of heavy section steel structures requires

fracture mechanics treatment due to the tendency of thick sections to

fail in a brittle rather than ductile fashion. For most structures,

fatigue crack propagation of subcritical flaws to critical size is the

primary mechanism of failure. Predictions of service lives of cracked

structures, then, is dependent on the accurate estimation of both

stress intensity factor (SIF) distributions and crack shapes.

One such cracked structure which has received considerable atten-

tion from designers and analysts alike is the nuclear reactor pressure

vessel. In particular, the nozzles of these vessels have exhibited an

abundance of cracks occurring at the juncture of the nozzle wall with

the vessel wall. The nucleation of these cracks is attributed to

thermal shocks experienced by the nozzles in the course of normal

operating procedure. A variety of techniques have been applied to

this nozzle corner crack problem for cracks located in the plane normal

to the maximum principal (hoop) stress in the vessel or at the zero

degree (0°) location depicted in Figure 1.1. Treatments of the

problem include both numerical [l] - [6] and the experimental [7] -

[10] techniques; however, the complex geometry of a nozzle causes the

problem to be intractable to purely analytical solution.

Cracks at locations other than the 0° location have been ob-

served in reactor vessel nozzles but analysis of such cracks has only

recently been attempted. In fact, the only treatment of this problem

1

2

VESSEL AXIS

0°LOCATION

45° LOCATION

---;--- + ---=====---900 LOCATION

INNER BOUNDARY OF NOZZLE

Figure 1.1 Initial crack locations

3

found in the literature to date is a study by Besuner [4] who employs

a hybrid influence function and boundary integral equation method to

calculate the average SIF for various cracks. Since this is a

numerical technique, the crack is assumed to be part circular in

shape to facilitate finite element mesh generation. Although this is

a valid assumption in some instances, naturally occurring cracks

generally exhibit non-circular shapes. Hence, Besuner [4], in his

closing remarks, recommends further investigations into the effects of

non-circular crack shapes and SIF variation around the crack tip.

One method of analysis which is ideally suited for the treatment

of three-dimensional cracked body problems, and which lends itself to

the study of crack shapes and SIF variations, is an experimental stress-

freezing photoelastic method developed by Professor C. W. Smith and

associates at VPI & SU. The object of this thesis, then, is to evaluate

the results obtained by applying the principles of the above experi-

mental method to the nozzle corner crack problem for cracks at loca-

tions other than the 0° location. In particular, cracks at the 45°

and 90° locations, as depicted in Figure 1.1, were chosen for study.

In a later chapter, results from this study are compared with the

results of Besuner's work [4] for the average SIF 1 s of varying depth

cracks. It is also deemed useful to compare the current results with

the crack shapes and stress intensities for cracks at the 0° location

as determined in similar experiments [9].

Additionally, a discussion of the factors which affect crack

shape is included. Advantages and limitations of the method employed

here with resp~ct to reproducing crack shapes in actual structures

4

is also discussed.

II. ANALYTICAL CONSIDERATIONS

The stress intensity factor as proposed by Irwin [11] is the

key quantity required for the prediction of fatigue crack growth [12].

Certain material properties are also needed but the SIF contains the

effects of both the problem geometry and the magnitude of the loading.

The current problem involves a mode I condition characterized by load-

ing normal to· the crack surface. Summarized below, then, are the

analytical foundations for the extraction of the mode I srF (Kr) from

the experimentally measured quantities. A more general and historical

discussion of the basis for the experimental photoelastic method of

determining SIF's is given by Jolles [13].

Based on the early work by Irwin, Kassir and Sih [14] showed that

the singular elastic stress field surrounding the tip of an elliptical-

ly shaped flaw border in a three-dimensional problem can be expressed

in the same form as for the two-dimensional case. This is accomplished

by employing a local orthogonal coordinate system with components

normal and tangent to the flaw border and a third component perpen-

dicular to the crack surface. In this case, with the notation given

in Figure 2.1, the stresses may be written as follows:

Kr 8 (1 . 8 . 38) 0

crnn = cos - sin -sin - - (} (27rr)l/2 2 2 2 nn

Kr 8 [ . 0 . 38 J 0

crzz = cos 2 1 + sin 2 sin z- - (J (2 .1) (2'TTr)l/2 zz

Kr . 0 8 38 0

0 nz = sin - cos - cos - - 0 nz (27rr)l/2 2 2 2

5

VESSEL WALL

Figure 2.1

6

a )'/ T

z r

n

d notation Problem qeometry an

NOZZLE WALL

7

The terms containing Kr represent.the singular part of the stress 0

field due to a factor of r in the denominator while the crij terms

represent the regular part of the stress field in the form of Taylor

series expansions of the regular stress components near the crack tip.

Note that Equations (2.1) are applicable for any flaw border which can

be considered as locally elliptic in shape [15].

The maximum shearing stress, Tmax' in the n-z plane which is ob-

tained via photoelastic fringe patterns may also be expressed in the

following analytical form. 1

1 [ 2 2] 2 Tmax = 2 (crnn - crzz) + 4crnz (2.2)

As observed in Figure 2.2, a typical mode I fringe pattern, iso-

chromatic fringes are most readily discriminated along a line normal

to the crack surface and passing through the crack tip. The equation

of this line is given as e = TI/2. Hence, Equations (2.1) are

evaluated at e = TI/2 and substituted into Equation (2.2) to yield an

expression which when truncated to the same order as Equations (2.1)

appears as

A Tmax = -- + B rl/2 (2. 3)

1/2 ° where A = Kr/[(8TI) ] and B is a constant containing crij. Equation

(2.3) may be normalized with respect to p(Tia) 112 and expressed as

(2.4)

8

Figure 2.2 Photoelastic fringe pattern for mode I

9

or

_K_A..,_p~= KI + B(B)l/2 { .!:.}1/2 p(rra)l/2 p(rra)l/2 p a

(2.5)

where KAp = 'max(8nr) 112 is defined as an apparent stress intensity

factor. When Equation (2.5) is plotted as KAp/p(rra) 112 vs. (r/a) 112,

a straight line extrapolated from the linear zone intersects the

ordinate at the normalized stress intensity factor Kr/p(rra) 112 for

(r/a) 112 = O.

The stress optic law given below as Equation (2.6) relates the

experimental quantities of stress fringe order (n'), material fringe

constant (f), and slice thickness (t') to the maximum shearing stress

('max) in the plane of the slice.

(2.6)

Thus 'max is the key quantity which relates analytical expressions to

experimental measurements and allows the determination of the desired

normalized SIF through computations involving the appropriate equa-

tions.

It should be noted that truncation of Equation (2.2) yielding a

two parameter (A,B) model was originally suggested only for the case

where the remote stress field was uniform and no surfaces other than

the crack surfaces were present. The linearity, then, of Equations

(2.3) through (2.5) may be confined to a region where such additional

effects as front and back surface effects are negligible. Thus if

10

the linear region of Equation (2.5) can be located experimentally,

the two parameter model is still valid, but only data within this

linear region is considered in SIF determination. On the other hand,

additional terms may be required if this linear region cannot be

located leading to an equation of the form

A j B.ri/2 •max = --,---/2 + r r i=O 1

(2.7)

Such an equation would necessarily require some truncation but

suitable criteria for this procedure have not yet been established.

However, for problems in mode I, both current and past, a mathematical

model of more than two parameters has been unnecessary.

III. EXPERIMENTAL METHOD

Investigation of three-dimensional stress and strain conditions

by the method of photoelasticity dates back to a study by Oppel [16].

When loaded, certain materials exhibit both birefringent optical

properties and linear-elastic stress-strain response above some

critical temperature. Such materials are ideally suited to the

process of stress-freezing. Tests required for the current nozzle

corner crack problem were conducted using scaled models of a nuclear

reactor pressure vessel. Details of the geometry for these tests are

given in Figure 3.1. The models, purchased from Photolastic, Inc.,

were cast in four sections using PSM-8, a material which displays the

required stress-freezing qualities mentioned above. All material was

delivered stress-free from the vendor and end cap sections which were

to be used in multiple tests were annealed before being fitted with

new nozzle sections.

The initial step for each test was the initiation of a small

starter crack at the juncture of the vessel and nozzle walls using a

thin blade guided by a jig and impacted by a hammer. The starter

crack was required to extend ahead of the blade since simply wedging

apart the material with the blade did not always provide a suf-

ficiently sharp root radius to allow the desired crack growth. When

each of the two nozzles for a vessel had been pre-cracked, the model

was glued together and fitted with an air inlet nozzle in the bottom

end cap. The completed vessel was then loaded into the stress-freezing

11

13

oven where it was held by a base Which uniformly supported the dead

weight of the model. Figure 3.2 shows this set-up along with the

four-point-bend calibration specimen loading rig required for the

analysis. A regulated and filtered air supply was connected through

flexible tubing to the air inlet nozzle of the vessel and monitored

with a pressure gage and a manometer.

To initiate the stress freezing cycle, the model was heated

slowly to a selected temperature above critical and then soaked for a

few hours to insure thermal equilibrium. At this point, the pressure

was raised inside the vessel until the starter cracks were grown to

some desired size. Generally, the final crack size in only one nozzle

per vessel could be controlled since the starter cracks could not be

made identical and since both nozzles had to be loaded simultaneously.

Next, the pressure was reduced by a factor of about three to preclude

any further crack growth and the vessel was allowed to cool slowly,

usually over a period of two days, to room temperature. The remaining

pressure was then relieved leaving the model in a state with displace-

ments and stresses "frozen-in." A physical explanation of this frozen-

stress phenomenon is presented by Peters [9] and more detailed

temperature vs. time data for the stress freezing cycle is provided by

Jolles [13].

The next step in the process was to cut the nozzles from the

vessel in such a way as to facilitate crack shape documentation,

generally in the form of photographs. A typical crack shape photograph

is presented as Figure 3.3 .. With the aid of a jeweler's saw, nozzle

sections were then cut into l-3mm thick slices containing the n-z

~

. ~;~· ! .'·il«i•~ if~ I~ ''.,:

14

Figure 3.2 BWR model and calibration rig in stress-freezing oven

•

15 •

Figure 3.3 Typical crack shape photograph

16

plane; i.e., a plane mutually perpendicular to the crack front and

crack plane. Figure 3.4 displays typical slice orientations. The

slices were then sanded smooth, coated with an oil of matching re-

fractive index, and analyzed in an optical comparator fitted with a

crossed circular polariscope at lOX magnification with a white light

source. The Tardy Method of compensation was employed to read partial

fringe orders along a line normal to the crack surface and passing

through the crack tip as discussed in Chapter II. A computer program

adapted to a desk top calculator and printer was then employed to con-

vert the raw data of fringe value vs. micrometer reading to the

normalized values of KAp/p(na) 112 vs. (r/a)l/2. Figure 3.5 indicates

how a plot of the data for a single slice would appear. Extraction of

the SIF is accomplished by fitting the data points in the linear region

with a linear regression (least squares) routine. The intersection of

the resulting straight line with the ordinate, then, is the required

normalized SIF, Kr/p(na) 112. ~

Before concluding this chapter, it is deemed important to note

that the modelling material used in the above experiments exhibits two

properties which are rarely seen in structural materials. The first

is the material 1 s linear elastic behavior above critical temperature

which precludes a crack-tip plastic, non-linear zone common to most

structural materials. There are, however, additional effects such as

crack-tip blunting, material stiffness variations, and optical non-

linearities which cause non-linear zones to be established at crack

tips in the mode.lling material [17][18][19]. Hence, the data points

closest to the ordinate in Figure -3.5 deviate somewhat from the

17

rs - MEASURED FROM NOZZLE WALL

Figure 3.4 Typical slice locations

18

LL ~ 12--~~--~~--~~--~~~~~

~ o TEST DATA w o DATA USED IN K1 DETERMINATION ~ ~ 10- a/T =0.590 f3 =75° o_ -<( 0

~ ..__. 0 0.. w ......... N o_ -' <1: <( ~ s __.. e.::... 0:: 0 z

8 ii_.

I 1/2 j K1 /p(7Ta) =8.58

6 ~ ft

0 0.1 0.2 0.3 0.4 SQUARE ROOT OF NORMALIZED

DISTANCE FROM CRACK TIP ( r/a)l/2

Figure 3.5 Typical plot of computer output

19

straight line extrapolated from the linear zone. In any event, the

above method is intended for application to problems which exhibit

slow stable crack growth and small scale yielding at the crack tip.

Further implications of the plastic zone size due to the nature of the

loading will be discussed in the next chapter.

The second property of the modelling material which deserves

note is Poisson's ratio (v). Although v; 0.3 for most structural

materials, the modelling material has the incompressible characteristic

of v = 0.5 above critical temperature. In two-dimensional problems

which may have a state of plane stress remote from the crack tip with

plane strain in the vicinity of the crack tip, the Poisson's ratio

difference may prove to be significant and must be compensated by a

correction factor [19]. On the other hand, based on studies of highly

three-dimensional problems by Smith. [20] it is estimated that the in-

fluence of the Poisson's ratio difference causes an error of at most

6% which is of the same order as experimental scatter. One further

note, the error causes SIF estimations to be slightly raised rather

than lowered and thus causes results to deviate to the conservative

side only.

IV. DISCUSSION OF RESULTS

Five vessels containing a total of ten nozzles were tested and

analyzed by the method described in the previous chapter. The first

piece of information yielded by these tests was qualitative, for it

was questionable whether cracks being tapped in at the go 0 location

(see Figure 1.1) parallel to the maximum hoop stress would turn so

that their faces would be oriented normal to this maximum principal

stress. It was found that provided a sufficient starter crack along

the go 0 axis of symmetry, the cracks would continue in the same plane.

However, noting Table 4-1, some data were lost for cracks which turned

when the starter crack was not exactly aligned with the go 0 axis, or

when the starter crack was not sizeable enough to indicate a preferred

direction. A preliminary test dealing with the problem of cracks

located at the 45° location in Figure 1.1 indicates that such cracks

will immediately turn to become normal to the maximum principal stress.

The go 0 crack, then, seems to be controlled by some sort of stability

consideration in that it will only remain in its plane provided that

the stresses are symmetric on either side and there is a sufficient

starter crack along the axis of symmetry.

It should be noted here, perhaps, that there do exist radial

cracks at the 90° location of reactor vessel nozzles since the

mechanism of crack initiation is due to thermal stresses which act

independently of pressure induced stresses. The mechanism of growth

of these cracks, on the other hand, is believed to be due to fatigue-

20

21

TABLE 4.1

Crack Geometries

Test p(kPa) av(mm) aN(mm) a(mm) a/T

7-1* 13. 7 10. 3 6. 1 7.7 0.506

7-2 13. 7 3.9 3.5 3.7 0.244

8-1 12.7 11.0 6.8 8.3 0.550

8-2* 12.7 3. 1 3.5 2.2 0.148

9-1** 8.5 16. 7 8.7 13. 0 0.860

9-2 8.5 4.4 3. 1 2.7 0. 176

10-1***

10-2 10.3 6.3 4.7 4.3 0.287

11-1 12.9 12.2 8.0 8.9 0.590

11-2 12.9 6.5 4.8 4.5 0.301

T = 15.l mm for all tests * - Yielded no significant results

** - Crack turned 36° out of plane *** - Crack broke thru outer surface

22

type loading caused by fluctuating internal pressure. Hence, this

crack turning effect becomes significant only after the crack has

penetrated the surface layer since thermal stresses govern the crack's

behavior during initiation.

Test results for cracks which remained in their planes are pre-

sented in Figure 4.1 in the form of SIF distributions. The data for

these tests are normalized with respect to a characteristic crack

depth, a*= 6.74 mm, to display the true effect of increasing crack

depth on the SIF. Increasing crack depth is represented by an in-

creasing a/T ratio since Tis a constant equal to 15.l mm for all

tests.

The second feature to be noted in this figure is the increasing

concavity of the distributions with increasing crack depth. To explain

this effect, first examine the crack shapes resulting from the tests

as depicted in Figure 4.2. Note that the growth is nearly self-similar

except for a slightly higher growth rate along the vessel wall; the

more important consideration, however, is that the crack shapes do not

bulge in the central region but instead maintain an approximately

quarter-elliptical shape. Next, as discussed later in this chapter,

it has been shown that crack shapes obtained by the current method

correspond to crack shapes obtained by fatigue tests of cracked

structures of the same geometry, provided there is only small scale

yielding. Although the current problem is, of course, highly three-

dimensional, consider first the two-dimensional crack growth rate

equation in the form [21][22]

23

(a/T)AVG = 0.57 9 AVERAGE OF TWO TESTS

E ~ 8-1'-t6 II

C\.!

' --l;E-0

~ ...._... 0..

' H Y.:

l 5

LI -r

a/T=0.197

3~~~~~~-"'-~~--"'~~~--~~~ 0 0.2 NOZZLE WALL

0.4 0.6 ts I /3MAX

Figure 4.1 Test results--SIF distributions

0.8 1.0 VESSEL

WALL

VESSEL VJ ALL

24

NOZZLE WALL

Figure 4.2 Test results--crack shapes

25

( 4.1)

where c and m represent material constants obtained via two-dimensional

fatigue experiments. Note that by assuming small scale yielding and

applying the relation

da dN a COD (crack opening displacement)

It has been shown that [21]

2 da _ A, [11EKJ dN -

where A' is a constant and E is Young's modulus.

Equation (4.3) is also two-dimensional but applies at least

(4.2)

(4.3)

qualitatively to the current three-dimensional problem. Hence, the

question now arises as to why there is not an increased growth rate in

the central region of a flaw border due to the higher relative SIF

determined to exist there by observing Figure 4.1. After all, Equation

(4.3) clearly states that the crack growth rate is directly proportional

to the square of the SIF. However, Equation (4.3) also exhibits the

inverse proportionality of crack growth rate with the square of Young's

modulus. Hence, the explanation lies in that the effective modulus of

the material, E, varies around the flaw border reaching a maximum in

the center and so compensates the effect of the increasing SIF.

This consideration of material property variation around the

crack front has just recently made its appearance in the literature

from authors such as Hodulak, Sommer, Kordisch [23] - [25], and Peters

[9]. The explanation of this phenomenon is really quite elementary.

A state of plane stress is known to exist at the intersection of the

26

flaw border with the nozzle and vessel surfaces, but as one moves

around toward the center of the flaw border, a transition toward a

state of plane strain takes place. This transition toward a plane

strain condition is accompanied by an increased state of triaxial

tension which stiffens the material and raises its effective modulus.

Of course, the deeper the crack has penetrated into the material the

more highly developed the triaxial tensile stress may become, and the

more concavity may be observed in the SIF distribution.

It was alluded to earlier in this chapter that the stress

freezing method yields crack shapes comparable to those observed in

steel models of the same geometry fatigued under tension-tension

loading. Peters [9] exhibits the comparison utilizing results obtained

by Broekhoven [10]. Additional discussion of the influence of load

level on crack development is being advanced by Hodulak et al. [25],

currently for a surface flaw in a plate. It appears that there are

no significant crack shape changes at moderate depths for different

tension-tension fatigue loading schemes so long as the yield point of

the material is not reached. If the yield point is reached, a plastic

zone occurs first at the intersection of the flaw border and the free

surface since there is little constraint at this point. This newly

formed plastic zone has the effect of pinning down the crack at the

free surface and any subsequent fatiguing causes increased growth

elsewhere. Thus, since the material employed in the photoelastic

experiments is linear elastic above critical temperature and cannot

model a plastic zone, the method should not be used to predict crack I shapes or SIF's for problems where fatigue loads may reach the yield

27

stress of the material. Load levels in reactor vessels are suf-

ficiently conservative, however, to preclude any large scale yielding.

Hence, the photoelastic method should accurately predict crack shapes

and SIF distributions which might occur in actual service vessels.

Referring to Figure 4.3 the current results are compared with

Besuner's analysis [4]. Besuner's hybrid influence function and

boundary integral equation method yields only average SIF values for

the entire flaw border so the current results were similarly averaged

for comparison. Note that the curves converge for moderate depth

cracks but diverge for shallow cracks. Differing geometries of the

nozzles are known to affect the results, and the inner fillet radius

at the juncture of the vessel and nozzle walls has a considerable ef-

fect on the stress intensity factors for shallow cracks. Also,

Besuner models crack shapes with quarter circles following a trend

initiated by the earlier work of Gilman and Rashid [6]. It has since

been observed, however, that the imposition of an unnatural crack

shape has the effect of raising the stress intensity factor above the

value which would be observed for a natural crack of the same average

depth. Reference [26] may be consulted for additional discussion of

these geometric effects. But, for the present discussion, it suffices

to note that the curves in Figure 4.3 diverge due mainly to geometric

effects including crack shape differences.

Consider now the final figure, Figure 4.4, which exhibits the

average SIF's for varying depth cracks. The results due to Peters

[9] for cracks at the 0° location are compared with the current study

of cracks at the 90° location (see Figure 1.1). Stress intensities

N ........... -c

!::: -0.. ...........

H ~

28

a/rn 0 0.2 0.4 0.6 0.8 1.0 12----------------0.6

--- ------ N 8 ........... 0.4-_

c ~ -60;)

4 0.2 ...........

~CURRENT STUDY H ~

---- BESUNER [4] 0 0

0 0.2 0.4 0.6 0.8 a/T

Figure 4.3 Comparison of current results with Besuner's analysis

29

40-----....------------E E

¢ -o- CURRENT STUDY I': -o- PETERS [9] tO 30· II

C\J 2.0r

I

' -~:c .. 0 10 ~

--0..

al ' 1-1 y:

! !

0 0.2 0.4 a/T

0.6 0.8

Figure 4.4 Comparison of current results with Peters' results

30

appear to differ by about a factor of three for any given crack depth.

The implication here is that a crack at the 0° location is three times

more critical than the same size crack at the go 0 location. Moreover,

a shallow crack with a/T = 0.2 at the 0° location exhibits a higher

average SIF than a moderate depth crack with a/T = 0.6 at the go 0

location. Therefore, it is concluded that although there is a possi-

bility of a failure resulting from a crack initiated at the go 0 loca-

tion, there are certainly other locations around the nozzle, the 0°

location in particular, where cracks are more likely to propagate.

Furthermore, since thermal shock causes more or less random crack

initiation around the nozzle, it is conjectured that the ultimate

failure of a vessel caused by a nozzle crack would be due to a crack

nucleated somewhere in the proximity of the 0° location.

V. SUMMARY AND CONCLUSIONS

The method of stress freezing photoelasticity has been applied to

the nozzle corner crack problem with results in the form of stress

intensity factor distributions and crack shapes. Experimental scatter

for the data reported is expected to be less than ± 7%. The important

conclusions which may be drawn from this study include the following:

1. Cracks will turn so that their faces are aligned perpen-

dicular to the maximum principal stress direction unless

the crack lies along an axis of symmetry.

2. Stress intensity factor distributions around the crack

front exhibit increased downward concavity as the crack

depth increases. This result is d~e to the stiffening of

the material caused by a state of triaxial tension at the

center of deep cracks.

3. Nearly self-similar flaw growth is observed for shallow to

moderate depth cracks with slightly higher growth along the

vessel wall. In all cases, the crack shapes are approximate-

ly quarter-elliptical in shape.

4. Average SIF results agree with Besuner's numerical solution

for moderate depth cracks but diverge for shallow cracks.

This divergence is conjectured to be due to geometric

effects including crack shape and inner fillet radius.

5. Nozzle cracks at an angle of 90° to the vessel axis are less

critical by a factor of three than nozzle cracks which are

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parallel to the vessel ·axis.

Application of the technique requires prior consideration of the

effects of elevated Poisson's ratio and linear elasticity of the

modelling material above critical temperature. The photoelastic method

best reproduces crack shapes from tension-tension fatigue tests con-

ducted below the yield stress of the material.

REFERENCES

1. Hellen, T. K. and Dowling, A.H., "Three Dimensional Crack Analysis Applied to an LWR Nozzle-Cylinder Intersection", International Journal of Pressure Vessels and Piping, Vol. 3, pp. 57-74, 1975.

2. Reynen, J., "On the Use of Finite Elements in the Fracture Analysis of Pressure Vessel Components", ASME Paper No. 75-PVP-20, June 1975.

3. Broekhoven, M. J. G., "Computation of Stress Intensity Factors for Nozzle Corner Cracks by Various Finite Element Procedures", Rep. MMPP119 (Paper G 416 Third SMIRT Conference, London 1975) Delft University of Technology Laboratory for Thermal Power and Nuclear Engineering, May 1975.

4. Besuner, P. M., Cohen, L. M., and McLean, J. L., "The Effects of Location, Thermal Stress, and Residual Stress on Corner Cracks in Nozzles with Cladding", Transactions of the Fourth International Conference on Structural Mechanics in Reactor Technology, Vol. G, Structural Analysis of Steel Reactor Pressure Vessels, Paper No. G 4/5, Aug. 1977, 14 pp.

5. Kobayashi, A. S., Polvanich, N., Emery, A. F., Love, W. J., "Stress Intensity Factors of Corner Cracks in Two-Nozzle-Cylinder Intersections", Transactions of Fourth International Conference on Structural Mechanics in Reactor Technology, Vol. G, Structural Analysis of Steel Reactor Pressure Vessels, Paper No. G 4/4, August 1977.

6. Gilman, J. D. and Rashid, Y. R., "Three Dimensional Analysis of Reactor Pressure Vessel Nozzles", Transactions of First Inter-national Conference on Structural Mechanics in Reactor Technology, Vol. G, Paper G 2/6, 1971.

7. Derby, R. W., "Shape Factors for Nozzle-Corner Cracks", Journal of Experimental Mechanics, Vol. 12, No. 12, pp. 580-584, Dec. 1972 .

. 8. Pearson, G. and Ruiz, C., "Stress Intensity Factors for Cracks in Pressure Vessels", International Journal of Fracture, Vol. 13, No. 3, June 1977.

9. Peters, W. H., "An Experimental Study of the Cracked Nozzle Problem Occurring in Pressure Vessels", Ph.D. Dissertation, VP I & SU, 1977.

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34

10. Broekhoven, M. J. G., "Fatigue and Fracture Behavior of Cracks at Nozzle Corners; Comparison of Theoretical Predictions with Experimental Data 11 , Proceedings of Third International Conference on Pressure Vessel Technology--Part !!--Materials and Fabrication, pp. 839-852, April 1977.

11. Irwin, G. R., "Onset of Fast Crack Propagation in High Strength Steel and Aluminum Alloys", Proceedings of the 1955 Sangamore Conference on Ordinance Materials, Vol. II, 1956.

12. Paris, P. C., "The Fracture Mechanics Approach to Fatigue", Fatigue--An Interdisciplinary Approach--Proceedings of the 10th Sa amore Arm Material Research Conference, J. J. Burke, N. L. Reed and V. Weiss, eds. , Syracuse, New York: Syracuse University Press, 1964.

13. Jolles, M., 11 A Photoelastic Technique for the Determination of Stress Intensity Factors", Ph.D. dissertation, VP! & SU, 1976.

14. Kassir, M. and Sih, G. C., "Three Dimensional Stress Distribution Around an Elliptic Crack Under Arbitrary Loading", Journal of Applied Mechanics, Vol. 33, No. 3, pp. 601-611, 1966.

15. Sih, G. C. and Liebowitz, "Mathematical Theories of Brittle Fracture'', Fracture (Vol. II, Mathematical Fundamentals), pp. 68-188, 1968.

16. Oppel, G., 11 Photoelastic Investigation of Three-Dimensional Stress and Strain Conditions", NACA TM 824 (Translation by J. Vanier), 1937.

17. McGowan, J. J. and Smith, C. W., 11A Plane Strain Analysis of the Blunted Crack Tip Using Small Deformation Plasticity Theory", Advances in Engineering Science, Vol. 2, pp. 585, 1976.

18. Smith, C. W., McGowan, J. J. and Peters, W. H., 11 A Study of Crack Tip Non-Linearities in Frozen Stress Fields", VPI-E-77-25, 1977.

19. Smith, C. W., McGowan, J. J. and Jolles, M., "Effects of Arti-ficial Cracks and Poisson's Ratio Upon Photoelastic Stress Intensity Determination", Experimental Mechanics, Vol. 16, No. 5, pp. 188-193, May 1976.

20. Smith, F. w., "Stress Intensity Factor for a Surface Flawed Fracture Specimen", TR-1, Dept. of Mech. Eng., Colorado State Uni-versity (Sept. 1971).

21. Hahn, G. T., Sarrate, M., and Rosenfield, A. R., "Experiments on the Nature of the Fatigue Crack Plastic Zone", Proceedings of the

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Air Force Conference on Fatigue and Fracture of Aircraft Structures and Materials, pp. 425-449, December, 1969.

22. Broek, D., Elementary Engineering Fracture Mechanics, Noordhoff International Publishing, 1974.

23. Sommer, E., Hodulak, L., and Kordisch, H., "Growth Characteristics of Part Through Cracks in Thick Walled Plates and Tubes", Journal of Pressure Vessel Technology, Feb. 1977, pp. 106-111.

24. Hodulak, L., "Development of Part Through Cracks and Implications for the Assessment of the Significance of Flaws", Paper No. C89/78 (in Press), Transactions of Institute of Mechanical Engineering, 1978.

25. Hodulak, L., Kordisch, H., Kunzelmann and Sommer, E., "Influence of Load Level on the Development of Part Through Cracks", Inter-national Journal of Fracture, Vol. 14, 1978, pp. R35-R38.

26. Smith, C. W., Jolles, M., and Peters, W. H., "Geometric In-fluences upon Stress Intensity Distributions Along Reactor Vessel Nozzle Cracks'', Transactions of Fourth International Conference on Structural Mechanics in Reactor Technology, Vol. G, Structural Analysis of Steel Reactor Pressure Vessels, Paper No. G 4/3, 1977.

The vita has been removed from the scanned document

EXPERIMENTAL DETERMINATION OF STRESS INTENSITIES

AND CRACK SHAPES ASSOCIATED WITH THE NOZZLE

CORNER CRACK PROBLEM

by

Thomas Sherwood Fleischman

{ABSTRACT)

The method of stress freezing photoelasticity has been applied

to a phase of the nozzle corner crack problem yielding results in the

form of crack shapes and stress intensity factor distributions.

Results of the current study are compared with alternate methods and

different phases of the same problem. A discussion of the variational

material behavior around the flaw border is also included. The

applicability of the current method to the prediction of fatigue crack

shapes is also discussed.