CLOSURE OF FATIGUE CRACKS AT HIGH STRAINS

by

Nagaraja S. Iyyer

Thesis submitted to the Faculty of the

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

APPROVED:

J. N. Reddy

MASTER OF SCIENCE

in

Engineering ~echanics

N. E. Dowling, Chairman

September, 1985 Blacksburg, Virginia

C. W. Sm~ th

CLOSURE OF FATIGUE CRACKS AT HIGH STRAINS

by

Nagaraja S. Iyyer

{ABSTRACT)

Experiments were conducted on smooth specimens to study the

closure behavior of short cracks at high cyclic strains under

completely reversed cycling. Testing procedures and methodology, and

closure measurement techniques, are described in detail. The strain

~cvels chosen for the study cover from predominantly elastic to

grossly plastic strains. Crack closure measurements were made at

different crack lengths. The study reveals that, at high strains,

cracks close only as the lowest stress level in the cycle is

approached. The crack opening was observed to occur in the

compressive part of the loading cycle. The applied stress needed to

open a short crack, under high strain was found to be less than for

cracks under small scale yielding. For increased plastic

deformations, the value of a0 p/amax is observed to decrease and

approaches the value of R. Comparison of the experimental results

with existing analysis has been made and indicates the limitations of

the small scale yielding aporoach where gross plastic deformation

behavior occurs.

ACKNOWLEDGEMENTS

The author would like to thank his committee chairman, Or. N. E.

Dowling, for imparting knowledge and valuable guidance during the

course of this study. In particular, his patience, helpful

criticisms, suggestions and constant encouragement throughout this

study is deeply appreciated. The author would also like to

acknowledge the guidance and suggestions of Professor J. N. Reddy and

Professor C. W. Smith in the preparation of this thesis.

The author thanks Mr. Robert F. Davis, Mr. Kenneth McCauley and

Mr. George C. Lough for their constant help and assistance while

conducting experiments. A warm and sincere thanks is also extended

to Ms. Paula Lee and the secretari~~ staff of Engineering Science and

Mechanics Department.

A very special thanks to Mr. Tom W. Orange of NASA Lewis

Research Center who was the technical monitor of this study.

Financial support of NASA Lewis Research Center uner grant no. NAG-3-

438 is acknowledged.

( i i i )

Chapter

Abstract

TABLE OF CONTENTS

Page

( i i )

List of Figures......................................... (vi)

List of Tables •..•..............•...••.................. (viii)

1 Introduction and Literature Review

1.1 Short Cracks and Their Importance............. 1

1.2 Fatigue Crack Propagation..................... 3

1.3 Crack Closure................................. 6

1.3.1 Plasticity Induced Closure............. 6

1.3.2 Roughness Induced Closure.............. 11

1.3.3 Oxide Induced Closure.................. 12

1.3.4 Discussion of Crack Closure Effects.... 12

2 Experimental Program

2.1 Material...................................... 15

2.2 Specimen Design and Geometry.................. 16

2.3 Test Equipment................................ 16

2.4 Mounting the Specimen and Alignment........... 18

2.5 Test Procedures and Methods................... 19

3 Experimental Results and Discussion

3.1 Experimental Results.......................... 25

3.2 Jiscussicn of the Results..................... 26

3.2.1 An Estimate and the Model of Budiansky

and Hutchinson......................... 27

( i ")

3.2.2 Models of Newman and of Nakai.......... 31

3.2.3 Comparison of Experimental Results

with Existing Models................... 32

4 General Discussion................................... 36

5 Conclusions and Scope for Further Study.............. 42

6 References........................................... 46

(v)

Fig. 1:

Fig. 2:

Fig. 3:

Fig. 4:

Fig. 5:

Fig. 6:

Fig. 7:

Fig. 8:

Fig. 9:

Fig. 10:

Fig. 11:

Fig. 12:

Fig. 13:

Fig. 14:

Fig. 15:

Fig. 16:

Fig. 17:

Fig. 18:

Fig. 19:

LIST OF FI GU RES

Schematic diagram of plasticity induced closure

Schematic diagram showing the wake effects

Crack tip parameters

Crack surface displacements and stress distributions (26)

Schematic diagram of roughness induced closure

Schematic diagram of oxide induced closure

Monotonic and cyclic stress-strain curve for AISI 4340 steel [411

Strain life curve for AISI 4340 steel (40)

Specimen geometry

Hydraulic actuator with the stiffener

Assembled view of the grip

Details of the grip

Deflection control testing with t~o clip gauges

(a) Schematic diagram of strain measurements (b) Correlation between plastic strains

Strain control testing with the clip gauge mounted across the grip ends.

A typical measurement of cack length vs cycles

da/dn vs ~K for AISI 4340 steel (40)

Schema:ic diagram of closure measJrement

Typical crack at varic~s stress levels in one ccmp1ete cycle

( v i )

Fig. 20-38: Crack opening displacements measured in one complete cycle

Fig. 39:

Fig. 40:

Fig. 41:

Fig. 42:

Fig. 43:

Fig. 44:

Fig. 45:

(a) Load displacement loop as obtained from clip gauge mounted across the grip ends and the points (corresponding stress and strain levels shown in table) where closure observations were made.

(b) Crack opening displacement during increasing (loading) and decreasing strain (unloading) at different points along the crack length. Different stress (strain) levels correspond to the points shown in load displacement loop.

(c) Crack opening displacement as a function of stress at different points along the crack length during increasing and decreasing strain.

(d) Crack opening displacement as a function of strain at different points along the crack length during increasing and decreasing strain.

Loading and unloading paths in a typical coo vs stress, and COD vs. strain plots.

Typical measurement of the crack opening stress level (cop).

Crack opening stress level as a function of crack length

Crack opening strain level as a function of crack length

Crack opening stress level as a function of t'.J

Crack opening stress level as a function of R

Schematic diagram of the real crack behavior at high strains and corresponding ideal crack behavior.

('I i i )

LIST OF TABLES

TABLE 1: Chemical Composition

TABLE 2: Mechanical Properties

(viii)

CHAPTER 1

INTRODUCTION ANO LITERATURE REVIEW

The present day concern of fracture mechanics is the study of

critical crack sizes which have a significant effect on the life of a

component. The failure of a structure or a component is often due to

the presence of a crack of critical size. Fatigue, which causes

failure of materials by the incipient growth of flaws, is the most

important cause. Thus, understanding the behavior of microcracking

and growth of small cracks in fatigue leads to the development of

improved methods of predicting lives of components.

Failure of materials under fatigue involves Ill the following:

1. Initial cyclic damage (cyclic hardening or softening)

2. Formation of initial microscopic flaws (microcrack initiation)

3. Microcrack coalescence to form a propagating flaw (microcrack

growth)

4. Macroscopic propagation of this flaw (macrocrack growth)

5. Failure instability

Often steps 1 and 2 described above are referred to as crack

initiation, 3 and 4 as crack propagation.

1.1 SHORT CRACKS ANO THEIR IMPORTANCE

The definition of a short crack depends on the nature of tne

problem being considerec. Reference 2 lists various considerations

for defining short cracks, such as the following:

2

1. Relative size of the crack with respect to the microsructure

(grain size etc., 0.4 x 10-6 in to 2 x 10-3 in.)

2. Relative size of the crack with respect to the plastic zone

(typically 0.004 in. in high strength materials, or 0.04 in. to

0.4 in. in low strength materials, and varying with stress level)

3. Size of the crack with respect to thickness (constraint)

4. Size of the crack with respect to the applicability of linear

elastic fracture mechanics, LEFM.

5. Crack detecting capability i.e., cracks that are so small that

they are difficult to find (0.004 in. to 0.04 in.)

Thus, an exact definition of a short crack cannot be made. The

size of a crack to be considered as short (small) depends on the

perspective of the problem that one is faced with.

A reasonable and improved estimate of the life of a comoone~t

can be made by the study of short crack initiation and growth. Since

most service failures are caused by cyclically varying stresses which

cause progressive failure of a component, the short crack problem in

fatigue is of major concern. Advances in the understanding of short

crack growth have enabled increasing~y quantitative studies to be

pursued into the soecific mechanisms that affect initiation and

growth. Manufacturing related problems associated with small cracks

that affect the lives of structural components have been identified

[31. References [4,5,61 discuss the ;mportance of the short cracK

problem.

3

Since the similitude relative to the metallurgical structure

breaks down for short cracks, the local effects will be dominant in

the materials response. Material inhomogenieties, such as crack

front irregularities, second phase particles or inclusions, and grain

boundaries play a vital part in affecting the local stress field and

hence the materials response. In the case of long cracks, all these

effects are integrated and averaged over many grains. But in the

short crack case, the following are important: applied stress, yield

stress and yield properties, crystallographic anisotropy,

homogeneity, and environment.

The behavior of short cracks as to their propagation is

different from that of long cracks, which can be generally handled by

LEFM. The literature indicates that study of short cracks should

consider the following aspects:

1. Fracture mechanics characteristics involving elastic-plastic

fracture mechanics, and

2. Physics of crack propagation involving microstructure,

environment, crack closure, crack extension, crack size, and

crack shape.

1.2 FATIGUE CRACK PROPAGATION

Rice [71 and Hult and ~cClintock [BJ used plastic superpos~tion

methods which are valid in cases where plastic strains are in

constant proportion to one another and where total strain theories

can be applied. These are ideal cases, and in reality, there are

4

deviations from proportional flow. The ideal case assumed by Rice

[7] permits a general treatment of the reponse to unloading,

reloading and cyclic loading.

Small scale yielding solutions for cyclic loading are obtained

directly by replacing the stress intensity factor by its variation

and doubling the yield stress, J , and yield strain, E • One of the y y

important results from analyzing the elastic-plastic models for small

scale yielding is that plastic deformation is entirely determined by

the history of variation of the stress intensity factor, K. Thus,

two different cracked bodies will exhibit identical fatigue crack

extensions if each is subjected to the same variations of K.

But in large scale yielding, especially when the crack itself is

small, no single parameter is known that plays the role of the

elastic stress intensity factor in determining crack tip

plasticity. Large scale yielding analyses are not available for all

cases, since with perfect plasticity models, unrestricted flow

occurs. Thus, crack propagation under repeated overall plastic

straining has not been analyzed mathematically because of the

complexities involved.

LEFM is based on the result that the strength of the elastic

stress field singularity at the crack tip is expressed by K, which is

a function of the applied load and geometry [9,lOJ. The res4stance

of metals to fracture under static and cyc:ic loading can be

described by this stress intensity factor 'n a geometry independent

5

manner [11]. LEFM based fatigue crack growth models also do not

really include all complexities of the long crack behavior. Hence,

for short crack problems, this method of using LEFM for predicting

crack growth is seriously limited. Kanninen [121 observes that crack

growth is associated with crack tip plasticity, but the result ·is not

unique because of complexities involving similitude. Thus, for short

crack problems, where similitude does not usually exist, LEFM based

relationships cannot be used.

In the case of large scale yielding, crack growth rate is

described by parameters such as the J-integral [13,14,15], crack

opening displacement, COD [16,17,18], and the size of the plastic

zone [19]. Approaches based on small scale yielding generally

underestimate actual growth rates, especially when the crack length

is small.

The unusual behavior of short cracks can thus be atrributed to

the following:

1. Violation of the continuum mechanics assumption used with LEFM,

2. Violation of the linear elasticity assumption.

When crack lengths are small comoared to the plastic zone, cracks are

observed to grow faster than expected from the long cracK LEFM

analysis. This is attributed to inappropriate use of LE~M analysis

as discussed by Dowling i 131. One approach is to remove the

discrepancy between short and long cracks in the use of the J-

integra l.

6

Ritchie [2] lists the following factors which have significant

impact on short crack growth behavior:

1. Plasticity at stress raisers (notches)

2. Microplasticity

3. Grain boundary blocking of slip bands

4. Cessation of growth and crystallographic reorientation of growth

at grain boundaries.

5. Crack closure

1.3 CRACK CLOSURE

Since Elber [20] showed that fatigue cracks can be partially

closed even under tensile loading, crack closure has been widely

investigated and recognized as an important factor affecting fatigue

crack propagation behavior. Briefly, crack closure can be induced by

plasticity, crack surface roughness, or oxide wedging.

1.3.l PLASTICITY INDUCED CLOSURE

A schematic illustration of the mechanism of plasticity induced

closure is shown in Fig. 1.

Plasticity induced closure is due to the contained plasticity

and due to the residual tensile strains left behind the crack tip.

As the load is applied, the material ahead of the crack tip yields

due to the stress concentration, even if the applied stress on the

specimen is below the yield stress. The size of the p1 astic zone is

related to tne crack length and aoplied stress [9J. The ~aterial

7

surrounding the plastic zone remains elastic, and as the load is

decreased, compressive stresses build up in the region of the crack

tip. This compressive stress must be overcome before the crack tip

can open on reloading.

The other mechanism which causes crack closure behavior is due

to the residual strains that exist in the wake of the moving crack

tip. There exists a region of residual tensile strains

(deformations) which are left in the material behind the crack tip.

These are illustrated schematically in Fig. 2. These residual

strains existing inside of the envelope of all previous plastic zones

are also responsible for crack closure [20,21]. These cause the

crack surfaces to come into contact before the minimum stress level

in the cyle is reached. Upon loading, the crack will open only when

the applied stresses overcome the residual compressive stresses

between the crack surfaces. Since crack growth can occur only when

the crack tip is open, an effective stress intensity is defined to

correlate crack growth rates. Elber [211 defines an effective stress

range as

, c - ~ , L ·eff - 'max - ·op ( 1)

where~ is the maximum stress level and 7~o is the opening level. max v

Defining

u 7 1lax 'oo (2)

8

Elber obtained the following crack propagation equation for an

aluminum alloy:

da = C(uK )n = C(U~K)n dN eff

where da/dn is the crack growth rate and uKeff is the.effective

stress intensity range. By his results, he fitted an emperical

relation for U as

u 2 0.5 + O.lR + 0.4R

where R is the ratio of the minimum stress level to the maximum

stress level.

Some other relations developed for U by various workers are

given as

U 0.68 + 0.91 R (Ref. 22)

u (Ref. 23)

K U = l~gx [880R + 6.0J + l.30R + 0.2 (Ref. 24)

Equations (4) - (7) can be expressed in the general form

U f(R, K , material) max

Crack closure is ~hus a complicated process influenced by

cracking mode (I, II, or III) environment, and microstructure.

( 3)

(4)

( 5)

(6)

( 7)

(8)

9

From the various reported work, the following, not fully

consistent, observations have been made:

1. U = f(R) for a material is independent of other parameters

[22,23]

2. U increases with increase in R in all cases, whereas the

relationship of U with Kmax is not usually the same.

3. No crack closure is observed at higher values of R, i.e., U

becomes more than unity [24].

4. Crack closure measurement technique is found to influence the

value of U.

The role of compressive stress in the plastic zone envelope in

the wake of the crack would be limited for a small crack of length

comparable to the plastic zone size ahead of the crack tip. This may

be one of the reasons why short cracks can grow at a level below the

threshold stress-intensity range.

Analytical work [25,26] has been done to predict the crack

opening load. These results may give closure loads Nhich differ from

the opening load [27]. Budiansky and Hutchinson \25] have determined

the residual stresses and crack opening and closure loads for R > O

loading in plane stress situations. Recently, Nakai, et.al. [281

have published results based on Budiansky and Hutc~inson's analysis

which can be extended to R < 0, also for small scale yielding

situations.

10

In their analytical work, Budiansky and Hutchinson [25] solved a

boundary value problem at Kmin with a residual displacement 6r along

the contact region (-oo,o), a compressive yield stress, -J over the y

reversed plastic zone, 6w, in (o,a), and with a displacement 6=6M'

between reversed and maximum plastic zone size '(a,w). The crack tip

parameters that were used in their analysis are shown in Fig. 3. As

is indicated in their work, the Dugadale model which has been used is

most appropriate for plane stress problems, whereas plane strain

conditions are more relevant to fatigue crack growth. It is also

noticed that cyclic hardening produces increased closure effects.

Dill et.al. [29,30] obtained an integral equation formulation to

determine the contact stresses and effective stress intensity by a

different approach.

Thus it is observed that most of the work on crack closure has

been focused on long cracks and small scale yielding conditions.

Newman [26,31,32] has developed a model based on the Dugdale model,

leaving the plastically deformed material in the wake of an advancing

crack tip. The advantage of using this model is that plastic zone

size and crack surface displacements are obtained by superposition of

two elastic problems. Ohji et.al [33,34] also have used finite

element tecniques based on a Dugdale type model to study closure

behavior. Ne·Nman has used elements, which behave as perfectly

plastic material for any applied load. These e1emen~s can be either

11

intact ahead of the crack tip, or broken behind it, to represent

residual plasticity effects. The elements which are not in contact

are used to calculate a 0 p Using the effective stress-intensity,

Newman [26] has used plasticity corrected K values in his

calculations. Figure 4 describes the crack ·surface displacements and

stress distributions along a crack line with the elements as used by

Newman [26].

It is important to observe from Newman's analysis that at equal

K values, the applied stress needed to open a small crack is less

than that required to open a large crack. Consequently, laeff is

greater for small cracks. This correlates ~ell with the high crack

growth rates for short cracks. Thus, the short crack effect may be

at least partly a result of the differences in the crack closure

effect between long and short cracks [26]. The effects of the stress

ratio, R, peak stress, a , and the degree of constraint at the max crack tip can all be included in Newman's analysis.

1.3.2 ROUGHNESS INDUCED CLOSURE

In shear mode (II or III) extension of the crack, the rough

irregular fracture surfaces [35,361 induce roughness induced

closure. In these cases, crack closure ~ill be strongly dependent

upon crack size [36,37]. In the case of short cracks, rougnness

induced closure is less significant because of the near zero crack

lengths. ~ schematic diagram of ~ougnness induced c1osure is snowr

in Fig. 5. The amourt of crack closure has been observed :o

12

correlate with increasing fracture surface roughness and the degree

of crack path deviation from a straight line [46]. Thus, for long

cracks, a zig-zag path of the crack should be considered along the

crack front because of the pronounced crack deflection. The extent

and angles of these deflections and concurrent stresses are thought

to be related to texture and grain size.

1.3.3 OXIDE INDUCED CLOSURE

There are situations in which crack closure occurs because of a

wedging action from oxidation or corrosion products [38J. A

schematic diagram of the oxide induced closure is shown in Fig. 6. A

number of workers have studied oxide induced closure and have offered

explanati0ns for near threshold corrosion fatigue crack growth

behavior.

1.3.4 DISCUSSION OF CRACK CLOSURE EFFECTS

Since short cracks possess a limited wake, it is to be expected

that in general such cracks will be subjected to less closure. There

are difficulties in experimental techniques to observe closure as

such. The experimental techniques reported in the literature on

crack opening/closure measurements vary from one investigator to

another. The location of the crack opening displacement has also

been shown to influence the results significantly [47J. As obser~ed

in Newman's analysis, residual stresses at ~ . have been estimated, ~in

and crack opening loads to overcome them have been calculated. The

13

other analytical work on crack closure has mostly been done for R > 0

loading.

Sehitoglu [271 has observed a difference between crack opening

and closure levels, while in most experimental studies crack opening

and closure levels were assumed to be.equal. The level of crack

closure is lower than the opening level, so that use of the closure

level results in conservative values of ~Keff"

Because of the inadequate characterization of the crack tip

stress and deformation fields and surface interaction effects, crack

closure studies are not complete. Reference [391 reports a crack

that was open throughout the entire cycle under R=-1 loading.

Similar observations have been made under large scale yielding. No

solution yet exists to analyze closure effects in the case of general

yielding.

In most previous work, a Dugdale type of model is used, where

the plastic strain gradient perpendicular to the crack axis is

considered by assuming all the sample to be elastic except a small

strip in the weakest cross section. This severely strained region is

considered to model the redistribution process due to the elastic

material response on its boundary. From this viewpoint, the shape of

the plastic zone is less significant compared to its extent along the

crack axis. Thus, tnis tyoe of model can De applied only to a

soecial case of elastic-plastic fracture mecnanics problems and

cannot be generalized to completely describe real fatigue crack

14

growth behavior. Added to these complexities is the three-

dimensional nature of the crack, where plane strain conditions exist

in the interior and plane stress conditions at the surface. Since

most experimental techniques measure closure by observing surface

cracks, they may not give good ins1ght into the actual crack closure

phenomenon.

From the various reported work, it has been observed that study

of crack closure effects has been limited to mostly elastic analysis,

i.e., small scale yielding. Crack closure observations in completely

reversed cycling at high strains have not been reported. The effect

of plastic strain on the crack closure behavior is thus poorly

understood. For a better understanding of the crack growth behavior,

experiments are to be conducted at different strain levels, at

different R ratios, and on different grain sizes. This in turn will

result in a more general elastic-plastic analysis describing the

crack growth behavior in all cases. This study concentrates on the

closure behavior of cracks at high strains. The experimental results

of this study is hoped to reveal the effects of the residual crack

tip plasticity and its manifestations, i.e., crack closure in a

broader perspective.

CHAPTER 2

EXPERIMENTAL PROGRAM

Controlled strain tests were conducted on uniaxial test

specimens, and detailed observations of crack opening and closing

were made. The details and scope of the testing procedures and

methods are described here.

2.1 MATERIAL

The experiments are conducted on strengthened metal alloy AISI

4340 steel. The chemical composition and the mechanical properties

are tabulated in tables 1 and 2, respectively. The material was

obtained in thick section to obtain the most isotropic and

homogeneous state possible, and was heat treated in slabs

sufficiently thin to obtain through hardening, specifically 2

inches. Also, the material was relatively free of any crack

arresters such as large nonmetallic inclusions. Reference [40]

illustrates typical inclusions and their sizes. The presence of such

crack arresters would invalidate the results, since measurements are

made on surface cracks, which propagate through the thickness also.

The AISI 4340 steel chosen for the study has a mean prior austenite

grain size of 0.00063 in. Cyclic stress-strain and low cycle fatigue

data for this material are shown in Figs. 7 and 8 [401. The data

obtained from the present study are also shown on

15

16

these figures, and these correlate well with the previous work of

Dowling [401 on the same material.

2.2 SPECIMEN DESIGN AND GEOMETRY

Smooth unnotched axial _specimens are used in the present

study. The specimen geometry is shown in Fig. 9. Material for these

specimens was obtained from a 7.5 in. diameter bar, the axis of the

specimens being parallel to the axis of the circular bar. In the low

cycle fatigue region, surface finish in the reduced section is not

that important. A good surface finish of 4 x 10-6 in.) was

nevertheless used in the reduced section to help in differentiating

the crack from polishing or grinding marks. Since longitudinal

strains are measured witn a 0.5 in. gauge length, straight gage

sections were employed. To minimize buckling problems, we have

employed a length to diameter ratio of 2.0. We have been able to

reach strains of 0.02 in/in without buckling. The secimen with

straight gage section helped us in the surface topographical studies

and provided an ample amount of equally strained, bulk material. All

marks from the final polishing were required to be longitudinal,

i.e., parallel to the axis of the specimen, since cracks grow on

planes generally perpendicular to the axis of the specimen.

2.3 TEST EQUIPMENT

All the tests ~ere ccnducted on a closed loop, servo controlled

hydraulic MTS testing system of 20 kios capacity. To arres: tne

17

lateral motion of the hydraulic actuator (ram) during its travel, a

fixture was designed to stiffen the hydraulic actuator against

lateral motion. This fixture is shown in Fig. 10. This ensures

alignment and also minimizes the problem of specimen buckling. This

fixture essentially consfsts of a sleeve shrunk fit on the actuator,

which slides inside a bronze plated demountable bushing, which is in

turn fixed in position by a bottom plate secured to the bottom platen

of the MTS machine.

To avoid extraneous bending and to obtain high quality test

results meeting ASTM standards for alignment, special grips were

designed. An assembled view of the grip and the detailed drawings

are shown in Figs. 11 and 12, respectively. The grips were made out

of Carpenter Custom 450 stainless steel of hardness Rc42. The grips

are hydraulically operated, and the piston is designed for a maximum

of 3000 psi of hydraulic pressure. An essential feature of the grips

is that, once the grips themselves are centered and aligned, each

test specimen is then automatically centered and aligned when

gripped. Operation of these grips involves the movement of the

piston upward due to hydraulic pressure applied at the bottom oil

port, forcing the collet to squeeze on the grip ends of the

specimen. This ensures that the specimen end surfaces are held

evenly by the collet. The collet ends were smoothed and given a

small radius to avoid any fretting problems. Releasing the grips is

also by hydraulic pressure, the piston Jeing made to move downward by

18

oil pressure applied from the top port. Proper sealant (a-rings)

were used in the grips to preserve oil and to avoid any leakage.

These grips are found to be effective in testing smooth

specimens with circular ends. Although these grips are intended for

0.5 in. diameter specimen ends, compatible collets with different

inside diameters are available for testing different specimens.

Several advantages of this gripping arrangement are as follows:

1. specimen alignment and centering are automatic

2. there is no backlash in the grip,

3. total mounting time is less, and

4. it can be used for other tests, such as tension testing, etc.

2.4 MOUNTING THE SPECIMEN ANO ALIGNMENT

Alignment and centering of the grip is checked as follows: With

the head of the hydraulic ram retracted, a specimen is gripped in the

top grip. A dial gage indicator of high resolution (0.0001 in.) is

mounted on the bottom grip, such that the dial gage stylus ~s in

contact with the test section of the specimen. By moving the ram

upward and downward, the parallelism of the test section with the

grip axis is checked, specifically by noting any deflection on the

dial gage. If the gage deflection is greater than 0.0005 in. over

the test section, then the parallelism is not satisfactory. To

obtain good para11elism, circular bevelled shims which give proper

tilt, are to be provided bereath the bearing area of the top grip.

The concentricit; of the spec'men axis wit~ respect to the grip

19

axis is checked by rotating the ram 360° and noting the maximum and

minimum of the readings on the dial gage. It should be noticed that

the maximum and minimum occur at opposite ends of the specimen

diameter. The offset, which is half the difference between the

readings, is then adjusted by loosening the locking nut on top of the

crosshead and moving gently in the direction where correction is

required.

This initial alignment procedure is absolutely necessary to

avoid bending strains and specimen buckling. A further check is made

using a specimen with 6 strain gages mounted on it, 3 gages 120'

apart on top of the test section, and 3 on bottom of the test

section. First, this specimen is held in the top grip and the strain

readings adjusted to zero. After gripping the lower end, the strain

readings of all six gages are noted. The difference between he

readings before and after gripping for the same strain gauge should

not exceed 40xlo- 6 in/in. Nhich corresponds to the maximum bending

strain. Otherwise, the initial alignment procedure is to be repeated

and checked again.

Next, the specimen is cycled at a low stress level, and plots of

strain versus load for each of the gages are then Jbtained. If the

slopes on the plots do not differ more than 2%, then alignment is

considered satisfactory, and the system is ready for testing.

Alignment checks are not considered necessary for individual

specimens, further checks being done only at ~nfrequent intervals.

20

2.5 TEST PROCEDURES AND METHODS

Constant amplitude controlled strain tests were carried out in

the present study. Although the closed-loop hydraulic testing

machine is capable of frequencies upto 100 Hz, the recording

equipment, the dynamic characteristics of the clip gage, and heat

generation in the sample limits the actual testing frequency to the

range 0.01 to 5 Hz. Our strain controlled tests were carried out at

a cyclic frequency given by an ampirical relation [41]

where f is the frequency and _ is the stable plastic strain pa amplitude estimated prior to the test.

(9)

Note that use of the above implies a constant average plastic

strain rate in a cycle. Thus, at higher strain levels, where the

life of the specimen is less than 103 cycles, a frequency of 0.01 to

0.5 Hz is employed, and up to 5 Hz is used for greater lives

corresponding to low strain levels. At low strain amplitudes where

the life is expected to exceed 105 cycles, the tests are usually

carried out in stress control, which allows a higher cyclic frequency

up to about 20 Hz. This modest change in frequency above 5 Hz is not

expected to affect the benavior. Since approximately isothermal

conditions are maintained in the normal room temoerature testing, and

since no other signi•ican: effect of frequency on life is kncwr to

exist in this mater~al at room temperature, the ~ffect of freq~ency

21

on the life is not included in the study.

Since it is desired to follow crack growth with a large degree

of plastic strain, two pairs of identical tests are conducted. The

first of these involves controlling the grip deflection using a

longer clip gage mounted across the grip ends al~ng with a smaller

clip gage mounted on the test section. This is shown in Fig. 13.

Use of this pair of gages provides a correlation between the test

section strain nd grip deflection. Such a correlation is shown in

Fig. 14 along with a schematic diagram of strain measurements. In

particular, there is a correlation between the test section plastic

strain range, 6£P 2, and the plastic strain range on the grip

ends, 6£pl· It is observed from the plot of ~£pl versus ~£P 2 that

the relation between the two strain ranges is almost linear (on a

log-log plot) at higher levels and nonlinear at lower levels. The

plastic strain in the test section is then estimated from the

correlation of Fig. 14, and the elastic strain, known from the

measured stress, is then added to obtain the total strain [131.

The second of these tests has the longer clip gauge mounted

across the grip ends, with the smaller clip gage not present, as

shown in Fig. 15. This is done so as to have access for surface

crack measurements. This type of test allows crack growth data to be

obtained under conditions of known large p1ast'c strains.

Cracks are either naturally initiated or are initiated frcm

artificial defects. The growtn behavior is then monitored by surface

22

crack measurements using cellulose acetate replicas. In the earliest

experiments, defects in the specimen were made by drilling a

hemispherical hole of diameter 0.004 to 0.012 in. In one case, the

crack was initiated by creating a region of residual tensile stresses

by piercing the surface of the specimen with ·the tip of a sharp

needle. Although this is not highly recommended, we tried this as an

expedient. However, the best approach was found to be a small pit

made by electro-discharge machining, EDM. The smallest crack length

that could be observed had to be limited to 0.01 inches, which

includes the size of the artificial flaw, so that the results were

not affected by the proximity of the crack tip to the artificial

flaw.

A dense array of cracks was observed at higher levels of strain

greater than 0.02 in/in. Since the cracks were close to one another,

affecting the stress/strain field and complicating the interpretation

of the data, the tests were mostly limited to a maximum of 0.015

strain amplitude.

Cellulose acetate replica tapes were used to monitor the crack

growth. The replicating tape thickness used was 0.005 in. and

acetone was employed to soften the tape. The tape was wrapped firmly

around the specimen, tak1ng care not to allow excessive air bubbles

inside the area oetween the specimen and the tape. The ~aoe was

removed after it dried, so as to obtain the impression of the

surface. Two or three replicas were taken to cover the entire test

23

section of the specimen.

A low power microscope (up to 280X) was fitted to the system as

shown in Fig. 15. This aided in observing the cracks. Since the

crack could start anywhere in the test section, the microscope

mounted from the top grip could be swivel1ed 360° around the

specimen, enabling detection of cracks on the test section in any

region.

The replicas thus obtained provided a permanent record of crack

growth and its measurements, and cracks of smaller lengths could be

traced back. Crack lengths and closure measurements were then made

by examining and measuring the replicas under a microscope. Typical

crack length versus cycles data obtained from surface replicas are

shown in Fig. 16. Crack growth rate, da/dn, versus stress intensity

range, ~K. as obtained by Dowling [40] on the same materiai is shown

in Fig. 17. Note that the earlier work of Dowling on this material

did not include closure measurements.

Crack closure measurements are made by measuring the offset of

an inclined scribe line intersecting the crack [42[. This is

illustrated in Fig. 18. A scribe line at an angle to the crack axis

is drawn across the crack at the minumum stress level. As the load

level is increased in the cycle, the crack opens. Thus, there

results an offset bet~een the lines. By measuring this offse:, the

crack opening displacement, COO, was computed. Lines drawn a:

several points along the crack give rise to COO measurements at

24

various points along the crack. In a few cases, when the crack was

very nearly parallel to the scribe line, it sometimes became

necessary to compute COD from enlarged pictures, by direct

measurement of the width of the crack surfaces.

CHAPTER 3

EXPERIMENTAL RESULTS AND DISCUSSION

3.1 EXPERIMENTAL RESULTS

The experiments were conducted· to observe the closure behavior

of small cracks. The term small crack in our present study is used

to indicate a crack which is physically small, but which can be

easily observed in detail under a microscope at a magnification of

lOOX. Thus, the short cracks in this context fall in the region of

0.01 to 0.1 inches of crack length.

Constant amplitude controlled deflection was used with a

completely reversed (R=-1.0) sinusoidal wave form. Tests were

conducted at four different values of test section strain, by

employing four different deflection amplitudes on the grip enas. The

four different strain amplitude levels, £ , chosen for the st~dy are a 0.0125, 0.0066, 0.0042, and 0.0024 in/in. These cover conditions

from predominantiy elastic to grossly plstic strain. At each of the

strain levels, crack closure measurements were made at 3 or more

different crack lengths. Crack closure measurements are made by the

offset technique as described in chapter 2. The COD, crack opening

displacement, was obtained at various points along the crack

length. A typica1 observation of the crack at various levels of

stress (strain) 'n one cycle is shown in Fig. 19.

Figures 20-38 illustrate the variation during a :ycle cf the

25

26

crack opening displacement with the stress and strain at various

points along the crack length. Along with this also is shown the

load versus displacement loop obtained from the longer clip gauge

mounted across the grip ends. Data for both loading and unloading

are shown in each case. It is n6teworthy that here we are measuring

crack opening displacement, at the indicated positions, since it is

difficult to measure the crack tip opening displacement, CTOD. Fig.

39 indicates typical loading and unloading paths in COD vs stress,

and COD vs. strain plots.

The determination of the crack opening stress level is shown in

Fig. 40. A scatter of !0.05(CODmax) is attached to each of the

points on the loading part of the cycle, where CODmax is the largest

value in the cycle.

shown in Fig. 40.

A corresponding scatterband is then drawn as

The crack opening stress level, J , is then op defined as the stress level where the center of the scatterband

crosses a COO value of zero. Similarly, the crack opening strain

level, s , is defined as the strain which occurs at the same time op as J op

A plot of J /a versus crack length is shown in Fig. 41 for op max the different strain amplitudes chosen for the study. Figure 42

illustrates a similar plot of " /s versus crack length. oo max

3.2 DISCUSSICN OF THE RESULTS

The work cf Elber 1201 suggested that the p"astic zone left in

the wake of the advancing cracK :io causes the crack :o be closed

27

after unloading even under tension-tension loading. But as observed

from the present study at R=-1.0, i.e., at completely reversed

cycling, no closure c9uld be observed at zero load. The crack closed

only as the lowest stress level, ~ . , was approached. However, the min crack opening was delayed, b~t still occurred in the compressve part

of the loading cycle. Thus, there is a significant difference

observed between the closure and opening levels of the crack, while

some of the other work in the literature assumes the crack ooening

and closure levels to be the same. For the contained plasticity

problem, not involving wake effects, Newman [261 calculates that

cracks are ooen for approximately 1/4 of the total cycle, i.e., half

of the tensile loading part of the cycle. A similar result was

obtained by Budiansky and Hutchinson [25], who included the olastic

wake effect. In the present study, it is observed that the crack is

open more than 1/2 of the loading portion of the cycle. This drastic

difference in behavior is almost certainly cue to the large scale

plasticity involved in the present tests.

3.2.1 AN ESTIMATE AND THE MODEL OF BUDIANSKY AND HUTCHINSON

In the case of small scale yielding, a first order estimate for

crack closure levels can be made. The follcwing is based on

Budiansky and Hutchinson's [251 analysis. Recalling Fig. 3, the

crack tip ooening displacement, ) , Hhich is calculated using the J

Dugdale strip yield model, corresoonding to an applied maximum

load, 'max' is

28

( 10)

where cry is the ideally plstic tensile yield stress, and E is the

elastic modulus.

The corresponding plastic'zone size, w, is given by

Thus, crack tip opening

K 11 ( max)2

IJ.)::: -8 '"y

displacement K2

max 60 "' Ea1 ==

is given by

8 oy w

rrE

( 11)

(12)

The plastic stretch variation, or the crack opening displacement from

the weight function analysis, is then given by

0"'009(11) (13)

where

11 :::: X/w

and I-, ( l - 1;. l + 11 - n g n == /l-11 - n/2 1n I 1 - /y-::-;;-

(14)

It should be noted that the above equation is valid on1y for small

scale yielding. The crack tip displacement variations, upon

unloading to a level K from Kmax' then is

- K)2 (15)

This is based on Rice's [7] work where the plastic flow is

proportional, and the plastic strains are in proportion to one

29

another.

Similarly, the variation of the plastic zone size~ ~w1 is

(K - K) 2 Tl max

6w = ! ( 2o ) y

(16)

Defining, ilw h 1;; "" - , t. en w

i; "" ilw = l ( l _ _K_) 2 w 4 K max

(17)

The function g(x/ilw) similarly takes the form

g(x/6w) = g(~~Jw) = g (n/1;;),

g(.!l) = 1I - n/1;; - l ~n 11+11 - n/1;;1 1;; ' 1 - ll - n/1;;

(18)

If the crack is assumed to have a residual displacement

of 5r appended to its surfaces, then the crack closure can be assumed

to occur at the tip to get a lower bound on (KcloslKmax)• when

o - M - 5R = 0

Normalizing with respect to crack tip opening displacement, 50

a M 6 R -----=O 60 00 60

Observing the crack tip displacement, at n2 0 i.e., when o=a , 0

1 M aR 0 - - - - =

60 00

fK - K ) 2 , ' max , cl os - l 2Eay g(x/tlw)J

(19)

(20)

(21)

(22)

30

Since g(x/6w)lx=O =1 we get

66 = 1 (l _ Kclos)2 60 2 Kmax

(23)

Equation (13) becomes

1 1 - 2 ( 1 0 (24)

or K l 6R

C OS = 1 - 12(1 - ~) Kmax 60

(25)

This closure level is for the crack tip, but first contact of cracks

may occur behind the crack tip, as is the case in Budiansky and

Hutchinson's [251 analysis. A similar form of the equation for the

first contacL closure level has been shown [25,271 to be

K clos Km ax

(26)

During the reloading process, the crack starts opening, and the value

if Kopen when the crack has been fully opened up has been calculated

as for R=O loading as [251

K ~ 0.557 K max

(27)

The above equations have been derived based on the assumptions of

small scale yielding, ideally plastic materials, and plane stress

situations.

From equations (25) and (26) it is observed that the K leve~ at

the contact in the crack ~iD regicn and at first contact anywhere are

31

different. In their analysis, Budiansky and Hutchinson showed that

first contact occurs behind the crack tip. Contrary to this, it has

been observed that in our study, within the resolution experimentally

possible, that contact of cracks occurs first at the tip, in

agreement with the analysis of Newman [ 26]. It was a 1 so observed in

one case only that the crack front irregularities enabled the crack

to close behind the crack tip in a manner consistent with roughness

induced closure.

3.2.2 MODELS OF NEWMAN AND OF NAKAI

An analytical fatigue crack closure model was develooed by

Newman [26] "'hi ch is based on the Dugdale model, but modified to

leave plastically deformed material in the waKe of the advancing

crack tip. A fatigue crack growth analysis program (FASTRAN)

developed by Newman calculates the crack opening stresses under

simulated plane stress and plane strain conditions. The model

developed cannot handle general yielding conditions but is quite

representative of small scale yielding conditions. A simulated plane

strain situation is chosen and the results of Newman's analysis are

shown in Fig. 43.

Recently Nakai et.al. [28] extended Budiansky and Hutchinson's

analysis to short cracks growing from notches under small scale

yielding conditions, and they ar~ived at an equation for ooening the

stress as

32

where ~ = ~w/w is the reversed plastic zone ratio at Kmin'

h=i/w, 1, being the crack length, and R is the ratio of the minimum

stress level to the maximum stress level. The term I{/___!i__h} is a ~-

first order elliptical integral, which is read from mathematical

tables. The reversed plastic zone ratio, ~ at Kmin has been obtained

[27] from Budiansky-Hutchinson's analysis. Such as

1 ~ n-h 112 R = - 2 ~ ((~-n)n) dn

1 l + - J

"!T ~

1/2 1/2 ( 1-h ) l rn 11 + {1-1) I dn (n-~)n 2 1 - (l-1)1/2 (29)

Solving this equation numerically for~ for representative cases

similar to the crack size and plastic zone sizes in our study, the

results are depicted in the crack opening map of Fig. 43. It is to

be noted that the above equations are obtained for small scale

yielding and for cracks growing from nothces where closure cannot

occur over the notch.

Along with these results, values obtained from Elber's estimated

empirical relation (equation 2) is also shown in Fig. 43. The

results from the present study are also indicated in the same fig.

43.

33

3.2.3 COMPARISON OF EXPERIMENTAL RESULTS WITH EXISTING MODELS

The effective stress intensity opening ratio, U, has been found

to increase as the crack length becomes shorter and approaches

unity. When the crack is open throughout the unloading part of the

cycle, closing only at the minumum stress level, the the effective

stress intensity range, defined by

U oK (30)

becomes equal to oK, the overall stress-intensity range. From our

present experimental results, conducted at R=-1 at different strain

levels, it is observed that the crack closes first at the lowest

stress in the cycle, c . , and remains closed for a part of the min loading cycle till it opens at a value J (c > c . ) . This can op op min be seen from the Figures 20-38. If we apply these results to the

equation for crack closure obtained earlier in equation (26) Ne

observe that

K clos K max

. 2 1 - ,, 1 - (~)

~o ( 31)

since sR/5 0 ·1 as R·l, and ~R/S 0 ·0 as R·-n, a reasonable choice for

the estimate of sR/c 0 coLl1d not be made from our present study, since

closure of the cracks was first observed at : . , corresponding to min Kmin· (For R=O loaaings, Budiansky and Hutchinson [25l have

estimated the sR value as 0.856 0 ).

34

Crack closure and opening stress level analysis for cracks

subjected to stress beyond the yield stress does not exist in the

literature. Following the elastic, small scale yielding analysis

gives inconsistent results. This is observed in fig. 43 where the

present data is pictured along with the analytical results.

Our observation that crack closure occurs only at ~ . , suggests mm

that 6R/6 0 =-1, (from eqn. 25) which is not meaningful. If cR taken

to be zero then,

since u Kclos

1 -K-max (---,,.-------=--1 - R 1 (32)

Thus, for short cracks ~hich are subjected to stresses beyond yield,

the crack closure level which occurs at 0 . , and the crack opening mm

level, ~ , may both have significant effects on the growth behavior op of the crack. Since the crack tip advances only in the loading part

of the cycle, the crack opening level, :op' may ~e relatively more

important. If we define the effective stress intensity as

(33)

where, U' is defined as

u-1 - K /K op max 1

1 - R (34)

then, we observe that :he i~creasea plastic deformations, the value

of K0 p/Kmax decreases and approaches the value of R making the va'ue

of J' to approach uni:;. •rom this aefinition of effective stress

35

intensity we observe that for shorter cracks, under large scale

deformations, the value of the effective stress intensity, 6Keff' is

significantly different from the earlier definition of the effective

stress intensity.

From our present results it is observed in most of the cases

that the crack opened in the compressive part of the loading cycle.

This revealed the significant difference in the crack closure and

opening levels, and these do not occur at the same stress level as

assumed in some studies.

From the crack opening displacement versus strain plots in

Figures 20-38, it is also noted that the cracks do not open and close

at the same strain levels. Hence, at high stress-strain levels where

significant plstic strain is involved, the available analysis on

closure/opening of fatigue cracks is insufficient.

CHAPTER 4

GENERAL DISCUSSION

Figure 41 illustrates the crack opening stresses normalized with

respect to the maximum stress, as a function of the crack length at

different strain levels. Since at each strain level, 3 or more crack

length measurements were made, lines are fitted to the data points

representing each strain level. It is observed that at higher strain

levels, as the crack length increases, this relative opening level

increases only slightly with the crack length. At lower strain

levels, the relative opening level is higher for any given crack

length and found to increase more with the crack length.

The corresponding strain level for crack opening, s , normal-op ized with respect to the maximum strain level, E , as a function of max the crack length is shown in Fig. 42 for different levels of strain

amplitude. Here also it is observed that, at higher cyclic strain

levels, the crack opens very early in its loading path, and the crack

opening is delayed more for the lower cyclic strain levels. It is to

be noticed that at low strain amolitude cycling, the crack ooening

level increases ~ith crack lengtn.

It is observed that the stress or the strain opening level is

dependent upon the point along tne crack length where the observa-

tions are made. The difference can De attributec to the irregular

crack front, the ~easurement ~ecnnique, and other microstructural

36

37

features. Stress and strain opening levels in this study are made

considering points which are relatively near the crack tip, typically

0.004 in. Though it would be desirable to measure the crack opening

displacements right at the crack tip, this is not feasible because of

the limitations of the measuring techniques.

Figure 43 describes the variation of crack opening stress level

as a function of 6J. 6J has been calculated without considering

closure effects, but considering plastic strain effects, using the

formula obtained by Dowling [40)

2 (0.714) 2a[• J.EJ + 8.59 Lc6EPJ (35)

where 2a is the crack length, E is the elastic modulus, u~ is the

stress range and 6E is the plastic strain range. p

The data shown in Figure 43 do not indicate a clear correlation

of the trends of the behavior with ~J. Thus no significant

interpretations could be made from this figure. A larger numoer of

tests covering the entire range would be helpful in describing the

behavior with modifications to ~J accounting for closure effec:s.

It is expected that, for an ideal rigid-plastic material, Nhen

the crack opens during the loading part of the cycle, the cracK never

closes unless a compressive strain is imoosed which exceeds the ten-

sile strain reached. This sets one limit and is shown by the solid

line in Fig. 43. For cases where oredominately elastic loading is

38

applied to the specimen, as is done in the usual fracture specimens

under R > 0 loadings, there exists a crack opening level, obtained

from different analyses [26,28] as illustrated in Fig. 43. All the

earlier analyses on crack closure/opening have been done for the case

of small s~ale yielding situations.

Extending the same small scale yielding analyses to R = -1 load-

ing for smooth specimens, as done in our tests, where there are large

scale deformations, the results seems to be not meaningful enough to

describe the phenomenon observed.

Thus we observe that the analysis existing in the literature is

limited to only special cases. We believe that cracks under large

scale yielding conditions behave more similarly to an iaeal crack

with no wake effects and with no contact. Figure 45 illustrates the

behavior of real (fatigue) crack behavior under large scale yielding

conditions. Also shown in the figure is the behavior of an ideal

linear elastic crack with no wake effect. As can be observed from

Fig. 45, the closure level is lower than the opening level. Also it

is to be noticed that the ideal crack closes and opens at ,, = 0.

This behavior of a ideal elastic crack sets one bound, while the

other bound for large scale yielding situations is still to be

analytically investigated.

It is also illus:rated in the Fig. 45, in a manner qualitatively

consistent with the present eperimental results, hew the opening

behavior of a real cracK varies from large scale yielding to smal 1

39

scale yielding conditions. This needs to be investigated and mathe-

matical analysis and modelling done. Such an effort would aid in

bridging the gap between the growth behavior of small microscopic

flaws and that of long cracks.

It ·is observed that in the analysis [25,281 existing in the

literature that the residual displacement, 6R' is assumed to be con-

stant. But this may not be so, since the residual displacements near

the tip of the crack may be different from that of the residual dis-

placements behind the crack tip. This stems from the argument that,

because the contact stress may exceed negative yield, the residual

displacements are changed considerably as the crack front grows far

beyond a given point.

Note that under compressive loading, the crack tip starts

closing and the apparent crack tip recedes. This makes the crack tip

singularity, such as of the type ;r in elastic analysis, to become

weaker and vanish when the crack is fully closed. Considering the

ideal rigid, perfectly plastic behavior of the material, it may be

assumed that the apparent crack tip at any point during. the compres-

sive loading starts receding only when the contact stresses ahead,

between the original crack surfaces, exceed the yield stress. Thus,

a residual compressive deformation of the material exists along the

original crack surface. Exte~ding the same analogy to cyclic loading

situations, it is believed that these contact stresses between the

crack surfaces resulting in compressive residual deformation are

40

responsible for the observed 'no closure' effect in large scale

deformations in R = -1 loading situations, as in our study. Since

these residual deformations must be overcome during reloading, the

crack opening is delayed and as observed takes place at a stress

level a0 p > a .• min It is thus observed that linear residual displacements, as done

in earlier analysis [25,26!, holds if the maximum state fulfils small

scale yielding conditions. This is the case if the monotonic plastic

zone is small compared to the crack length. In the case of a real

fatigue crack under large strains, the above analyses fails to

predict the crack growth behavior accompanied by crack

closure/opening.

Another aspect that is to be noted is the three dimensional

nature of fatigue cracks. The 2-D analyses are only ideal and are

only appropriate for thin sheets. Since the plastic zone ahead of

the crack in a plane strain region is small compared to the plane

stress zone, closure of cracks is less significant in plane strain

situations. The crack closure measured by electric potential [381,

ultrasonic [43j, or compliance [441 methods reveal only an average

obtained at the specimen surface and interior, with stress interac-

tion effects being less significant in the interior. There exists in

the literature [461 data from tne measurement of closure behavior of

cracks under pure plane strain conditions. But in that study also,

the remote load level was gradually decreased as the crack propagated

41

to maintain the small scale yielding situations.

Other factors to be considered while studying crack

closure/opening analysis are the effects of loading condition and

specimen geometry. Dowling and Begley [14] applied the J-integral to

e1astic-plastic and general yield conditions and obtained a good cor-

relation between the crack growth rate and the range of the J-

integral, (uJ). The application of this J-integral to fatigue crack

growth studies at high cyclic stresses and strains at different R

ratios to observe the crack closure/opening phenomenon is planned to

be investigated.

CHAPTER 5

CONCLUSIONS AND SCOPE FOR FURTHER STUDY

The observed closure/opening behavior of cracks reveals the

limitations of the existing elastic-plastic fracture mechanics

approach to the study of crack growth behavior. There exist no

closed form solutions for the redistribution of stresses and

displacements in a cracked body under any general elastic-plastic

conditions. Assumptions made in several analyses just simplify the

problem to a very special case of elastic-plastic fracture

mechanics. The use of the Dugdale model is one such approximation.

In this model, the size of the plastic zone ahead of the crack tip is

completely ignored, and so is the elastic field surrounding it. The

model can be applied to thin sheets under plane stress, where only

the entire strip along the crack axis in front of the crack undergoes

plastic deformation. In real crack situations, the Dugdale model

fails to explain the observed behavior in totality, and in general

cannot be extended to all cases.

An energy balance investigation (model) may be a suitable

approach, thereby the plastic dissipative work within the plastic

zone can be fully considered with the bounded elastic zone in a

cyclic hardening or softening material. The early approach by Rice

[7] to problems of fatigue cracks is to use deformation theory of

plasticity, which is difficult to use for nonlinear cases such as

42

43

crack closure and an extending crack, even under small scale yielding

situations.

Thus, the discrepancies observed between the experimental

observations and the analytical models are because of both mechanics

related factors and material related factors.

The mechanics related factors are:

(1) Re-distribution of stress and displacements in cracked

bodies

(2) Material deformation behavior

(3)

(4)

(5)

Local closure and contact stress effects

Macroscopic closure due to residual stress and deformation

Anisotropic effects and homogeneity

(6) Three dimensional nature of crack

Some of the material related factors are:

(1) Differences in cracking process

(2) Characteristic length comparisons, such as crack length

versus the microstructural dimensions of the material

(3) Transient effects due to grain boundaries, inclusions, grain

to grain non-orientations, etc.

Because of the complexities involved in fatigue crack growth, it

is impossible to single out any one of the factors as a major

controlling parameter. Thus, further research is needed in :his area

to critically analyze the most important parameter (mater~a 1 and/or

geometric) related to plastic strains for the closure behavior of

44

cracks in fatigue.

Further research in this area includes conducting tests at

different R ratios such as -0.5, 0 and 0.7, and examining the

validity of existing analyses of the closure behavior of cracks in

fatigue. This includes various parameters such as different strain

levels, crack lengths, and different specimen geometries. Tests are

being conducted, and the results are expected to provide a good

understanding of this subject. Also, it is planned to carry out

tests on different grain sizes, to expose the effects of grain size,

thus checking the limitations of the continuum mechanics approach

also.

We are at present conducting tests of 2-0 cases on flat

specimens to examine the effects of various parameters on the closure

behavior of cracks. This is expected to bring results leading to

differentiating among the actual mechanism of crack growth and its

closure in fatigue in 2-0 and 3-0 situations. The variables that are

being included in this study are the load ratio, maximum load,

plastic strain level, and crack growth rate. The data will be

analyzed based on J-integral, ard new analysis of the closure of

short cracks will be attempted.

Further work is also aimed at developing a model ~hich describes

the crack tip stresses and displacements, and hence redistributions

of the stresses and displacements, under general elastic-plastic

conditions, which are in :urn expected to help in a better

45

understanding of the crack growth behavior over all strain ranges

from gross plastic to elastic deformation.

REFERENCES

1. Hoeppner, D. W., "Short Crack Fatigue Design Considerations: Modelling Characterization, Interpretation, Detection, Prediction of Behavior," Proc. of the 55th meeting of the AGARD Structures and Materials Panel, Toronto, Canada, Sept. 1982.

2. Ritchie, R. 0., and Suresh, S., "Mechanics and Physics of the Growth of Small Cracks," AGARD report no. CP-328, 1983.

3. Potter, J.M., "Advances in Fastener Hole Quality through the Application of Solid Mechanics," Proc. of the Army Symposium on Solid Mechanics, Sept. 1978.

4. Hudak, S. J., "Small Crack Behavior and Prediction of Fatigue Life," J. of Engg. Matls and Technology, Transactions of ASME, vol. 103, 1981.

5. Schijve, J., "Differences Between the Growth of Small and Large Fatigue Cracks: The Relation to Threshold K Values," in Fatigue Thresholds eds., Backland, J., Blom, A., and Beevers, C. J., EMAS Ltd., Warley, U. K., vol. 2, 1981.

6. Lankford, J., "The Effect of Environment on the Growth of Small Fatigue Cracks," Fatigue of Engg. Matls. Structures, Vol. 5, 1982.

7. Rice, J. R., "Mechanics of Crack Tip Deformation and Extension by Fatigue," Fatigue Crack Propagation, ASTM, STP, 415, 1967.

8. Hult, J. A., and McClintock, F. ii.., "Elastic Plastic Stress Strain Distribution Around Sharp Notches under Repeated Shear," Proc. of the 9th International Congress of Applied Mechanics, Vol. 8, Brussels, 1956.

9. Irwin, G. R., and Wells, A. A., "A Continuum Mechanics View of Crack Propagation," Metallurgical Reviews, Vol. 10, no. 38, 1965.

10. Paris, P. C., and Sih, G. C., "Stress Analysis of Cracks," Symp. on Fracture Toughness Testing and its Application, ASTM, STP, 381, 1965.

11. Paris, P. C., "The Fracture \1ecnanics Approach to Fatigue," Fatigue- An Interdisciplinary Approach, Syracuse University Press, 1976.

12. Kanninen, ~. F., Pope'lar, C.H., arid Broek, D., ''A Critical

46

47

Plastic Fracture to Nuclear Pressure Vessels and Piping," Nuclear Engg. and Design, vol. 67, 1981.

13. Dowling, N. E., "Crack Growth During Low Cycle Fatigue of Smooth Axial Specimens," Cyclic Stress-Strain and Plastic Deformation Aspects of Fatigue Crack Growth, ASTM, STP, 637, 1977.

14. Dowling, N. E., and Begley, J. A., "Fatigue Crack Growth During Gross Plasticity and the J-integral," Mechanics of Crack Growth, ASTM, STP, 590, 1976.

15. El Haddad, M. H., Smith, K. N., and Topper, T. H., "Fatigue Crack Propagation of Short Cracks," J. of Engg. Matls. and Technology, Transactions of ASME, vol. 101, no. 1, 1979.

16. Fuhring, H., "Calculation of Elastic Plastic Loading in Dugdale Cracks with Crack Closure on the Basis of Non-Linear Fracture Mechanics," Report of the Institut fur Statick and Stahlbam, Darmstadt, 1977.

17. Leis. B. N., ''Fatigue Crack Propagation Through Inelastic Gradient Fields," Int. J. of Pressure Vessels and Piping, vol. 10, 1982.

18. Seeger, T., and Heuler, P., "Generalized Application of Neuber's Rule," J. of Testing and evaluation, ASTM, vol. 8, no. 4, 1980.

19. Ohuchida, H., Usami, S., and Nishioka, A. "Fatigue Limit of Steel with Cracks' bulletin of JSME, vol. 18, no. 125, 1975.

20. Elber, W., "Fatigue Crack Closure under Cyclic Tension,'' Engg. Fracture Mechanics, vol. 2, no. 1, 1970.

21. Elber, W., "The Significance of Fatigue Crack Closure," Damage Tolerance in Aircraft Structures, ASTM, STP, 590, i976.

22. Katcher, M., and Kalpan, M., in Fracture Toughness and Slow Stable Cracking ASTM, STP, 559, 1974.

23. Bell, P. D., and Creager, M., in Crack Growth Analysis for Arbitrary Spectrum Loading, AFEDL-TR-74-129, 1974.

24. Satish Chand, and Garg, S.B.L., 'Crack Closure Studies under Constant Amplitude Loading,' Engg. Fracture Mechanics, ·101. 18, no. 2, 1983.

25. Budiansk;, B., and Hutchinson, J. W., ''Prnalysis of Closure in Fatigue :rack Growth," J. of App. Mechanics, vol. 45, 1978.

48

26. Newman, J. C., Jr. "A Nonlinear Fracture Mechanics Approach to the Growth of Small Cracks," Proc. of the 55th meeting of the AGARD Structural and Materials Panel on Behavior of Short Cracks in Airframe Components, Toronto, Canada, Sept. 1982.

27. Sehitoglu, H., "Crack Opening and Closure in Fatigue," Report of the University of Illinois-Urbana, 1984.

28. Nakai, Y., Tanaka, and Yamashita, M., "Analysis of Closure Behavior of Small Fatigue Cracks," J. of Society of Materials Science, Japan, Jan. 1983.

29. Dill, H. 0., and Saff, C.R., "Spectrum Crack Growth Prediction Method based on Crack Surface Displacements and Contact Analysis," Fatigue Crack Growth Under Spectrum Loads, ASTM, STP, 595, 1976.

30. Dill, H. 0., and Saff, C.R., "Analysis of Crack Growth Following Compressive High Loads Based on Crack Surface Displacements and Contact Analysis," MCAIR-76-006, Report of McDonnel Aircraft Company, 1976.

31. Newmann, J. C. Jr., "Prediction of Fatigue Crack Growth under Variable Amplitude and Spectrum Loading Using a Closure Model," NASA Technical Memorandum, Jan. 1981.

32. Newman, J. C. Jr., "A Crack Closure Model Predicting Fatigue Crack Growth under Aircraft Spectrum Loading," NASA Technical Memorandum, Jan. 1981.

33. Ohji, K., Ogura, K., and Ohkubo, Y., "Cyclic Analysis of a Propagating Crack and its Corre 1 at ion ·..;ith Fatigue Crack Growtn, '' Engg. Fracture Mechanics, vol. 7, 1975.

34. Ogura, K., and Kohji, K., "FEM Analysis of Crack Closure and Delay Effects in Fatigue Crack Growth Under Variable Amplitude Loading," Engg. Fracture Mechanics, vol. 9, 1977.

35. Suresh, S., and Ritchie, R. 0, "A Geometric Model for Fatigue Crack C 1 osure by Fracture Surf ace Roughness," Met. Trans. vo 1. 13A, 1982.

36. Mccarver, J. F., and Ritchie, R. 0., ''Fatigue Crack Propagation Thresholds for Long and Short Cracks in Rene 95 Nickel Base Superalloy," Mater. Sci. Eng., '101. 55, 1982.

37. Morris, \~. L., James, ~. R., and Buck, 0., "A Simple Model of

49

Stress Intensity Range Threshold and Crack Closure Stress," Engg. Fracture Mechanics, vol. 14, 1982.

38. Gangloff, R. P., ''Electric Potential Monitoring of the Formation and Growth of Small fatigue Cracks in Embrittling Environments".

39. Murakami, Y., Harada, S.S., Endo, T., Taniishi, H., and Fukushima, Y., "Correlations Among Growth Law and Applicability of Miners Rule," Engg. Frcture Mechanics, vol. 18, no. 5, 1983.

40. Dowling, N. E., "Growth of Short Fatigue Cracks in an Alloy Steel," Paper for the ASME, 4th National Congress on Pressure Vessels and Piping Technology, June, 1983, Portland, Oregon, Also Scientific Paper no. 82-107-STINE-P2, Westinghouse R&O Center, Pittsburg, Pa., Dec. 1982.

41. Dowling, N. E., Private Communication

42. Di l lner, C. W., "A Crack Closure Study of High Carbon Steel," Report of Dept. of Theoretical and Applied Mechanics, University of Illinois, Urbana, 1984.

43. Shih, T. T., and Wei, R. P., in Engineering fracture Mechanics, vol. 6, 1974.

44. Heubaum, F., and Fine, M. E., "Short Fatigue Crack Growth Behavior in a High Strength Low Alloy Steel,'' Scripta Metallurgica, vol. 18, 1984.

45. Kikukawa, M., Jana, ,\1., and Hora, H., "Fatigue Crack Propagation and Closure Behavior Under Plane Strain Condition," Report of the Dept. of Mechanical Engg., Osaka UniversitJ, Japan.

46. Gray, G. T., III, and Luetjering, G., ''The Influence of Crack Closure on the Fatigue Crack Propagation of Ti-6Al-4V and Ti-8-6Al", Fatigue, 1984.

47. Macha, D. E., Corbly, D. M., and Jones, J. '..J., "On the Variation of Fatigue crack 1Jpening Load with Measurement Location," Experimenta 1 Meehan i cs, vol. 19, 1979.

' i

I c Ni

i

1.89 I I 0.38 !

Heat Treatment:

s

TABLE 1

CHEMICAL COMPOSITION

AISI 4340 Steel

I p Si Mn

0.052 0.012 0.29 o.77

I Mo I Cr I

0.21 0.83

1. Austenitize at 1560°F, 5 hours to temperature; hold 3 hours; oil quench.

2. Temper at 1225°F, 4 hours to temperature; hold 8 hours; air cool.

50

TABLE 2

MECHANICAL PROPERTIES

i : Heat No: Ultimate 0.2% yield % Red.Area True Fr.

(Ksi) (Ksi) Strength

61738 114 94 68 225 I I

51

Cf \

' \ \ I \

\· .... :_: .::.\: ··.:· .... · ............. ,,

}·.-:R::rj . . ' " ' . . ' . . . ~

crack fully open

rr

\

' I \

{

" \ .. . . ' \ .. ' .·. '- ..... . .... . : {'

"'.... .. ' .. . .·· .. ' . " "' ·,. \ \ 'J .. ' . . . ·' ' . .

I

~ l

·:" '': .. -~ .. ·. ~ . .

. . . ' :

I

-~r

I

I I.

: : :_.: l: : , ':· . ' . ->11 i, ·-~;~ '-~- ·.: ~~ ).

· ... ·::~v r·_·,1'· .. · .... t

cr~f.k starts closing ' crack fully closed from the tip

crack reopens

/-iy. 1: )clle111dtic l)ia~ra111 of Plasticity induced closure

(J1 N

53

symbolic plastic zone .,, •, .

~ .... ... .. ... . . ....,

... ~ . '

·.

I .. ·. . '

·,~.

' ,, '

..

' ' ':- .

·. \ .

. .. ' ' . '

" . . ...: ~ ' ·, .. ' · .... ~. ' .. ' . ' .

·. ' · .. ,.

\ . .. . . , .

·~ '

' '

--·~/'·'' /.· . . -.::.:.. './\ .. . . / ... ' . . . . . . ,,. . :".... . . ' /\ ~ /

" ""'- / l I / • .,·

\\ \ . -~: ... /, ' \. . ' ~ \ . . ,'. '._" .. ~ . .· ':.. \ . .

"'. ' . \\. . \ ' .

enevelope of all plastic zones

\. I -.

• I I' " I I

I.

Schematic dia'.:lralll sl1owin~ the waKe et'f~cts

54

OR

K=K-

~~222ttr 0 <Ty : <Ty)

w

(a)

K = K,.. • O

Is: w

Fi ':I. 3: CracK tiµ µa rC11;1eters

55

y y

-x -x

a

Ca> Maximum stress Cb> Minimum stress

Fi :.; • If : Crack surtace ois 1Jlace111ents ancJ stress distributions (~l>)

r ,,/~<

'/:

/ "

"'

:~

J I /

~

b

56

= u /)

57

' ....

" l)

L ~

V'I

'.:::l

u

" 'll '.J ~

~ ~

(1)

"'=' x '.:::>

..... J .-o L ;') ro

~

u ')(

< 'cf

~ ....., ro

:ii ..:: u '/

)

_.. j

...... ......

? ..

.,,, .0

.,,, .J'l

/ .......

) <

b

100

-::;; 80 -GJ. ~ = -c. e 60 < "' "' Q) ... VI

58

• - __ -:: ,,. .. I"'\ -- --- -:: :-- - -- ----

AISI 43d0 Yield 94 ksi

f:;t'O 4fr Cycli~ constant Amplitude

- -o-- Cyclic, Incremental Step · --- Monotonic Tension

A Present D.:ita

0 ' '--~·-a-.a·~-----o.-o~os~~~-.0·12--~o-.~01-6~~0-.o~~~~o-.0~2~d--~O.~o~

Fig. 7:

Ea, Strain Amplitude

Monotonic and cyclic stress-strain curve for AISI 4340 steel [411

0.. E .. c: .. ... ...

_, 10

.. -J lO

59

• data points from present study

plastic strain amplitude -

10-• L-----1-:----.....!...7, ..;._ __ _J~.----~.----10-:0, ----:-:: , 10' 10' 10 10 10 10

cycles to failure

Fig. 8: Strain life curve for Arsr 4340 steel (40)

I m I

0 0 .. Q

IO

l'l 0

C\i 0

lL . \

0 l'l IO

t

60

I u.

', LtJ 0: U1 CD

ct IO

0

0 0

I() 0 0

0 0 0 d

(I) _..

I

0 ct

0 ~ + ~

U1 o

O

U1 0

oo

O

c)

...----q 0 (I)

.ff I

0 ct

0 ~·

>,

1-...., (lJ E 0 (lJ O

l

c r• E u (lJ 0

. V

l

.. (]"I . Ol

61

;; I~ I~ I 5 , 1.

0 Ii: " I'' '·

. • a -~ ii ,, I ~ ~I ·• 1 ~

= •I

~ .

~I 0

SPAC£R '\

LOCICN> RINGS

@:GREASE FTTTlNGS TO BE AOOED.

SLEEVE

.._~~~~~~~~~~~~~1ooocu.-~~~~~~~~~~~~~~

Fi'j. lU: Hy<Jraulic dCtuator wit11 the stittner

62

0 I

4•

5

-Part No. Part Name No. Reqd. Material Hardness

1 Universal AF-123 2 (supplied) Co 11 et 2 Cover 2 cc 450SS RC 42

3 Collet Holder .

2 cc 450SS ilr 42

4 3/8-16 Sock Hd Cap Screws 16 (std. bolting) SAE Grade 8, 1.25 in. lonq 5 Housing 2 cc 450SS RC 42

Fig. ll: Assembled view of the grip

0.90001A I el •-a I 0.0005 I REF. DIMENSION GRIND AT FINAl. ASSY 10 FIT COL.LET, SO ™AT ASSEMBLED GRIP WILL ACCEPT O.!lOO DIA ROD

1.35

2.00D

G

Fig. 12 (a) Details of the grip - Cover

All dimensions in inches

5.~ •0001 lei •-a I 0.0005 I

8 HOLES EQUALLY SPM:ED 45° APART 04060IA, C BORE 05940 0325 DEEP AS SHOWN CLEARENCE FOR ll•-16 SOC. HO. CAP SCREWS WITHIN 0.~ OF TRUE POSITION ON 4 5 OIA

075

2.00

0.91

I. 2 5 l r '·---..,;:--,,...,_

~

64

• THO

4 SLOTS 90• APllRT

ORILL a TAP 0.375-16 0.75 OEEP Tl40 8 HOLES 45• APART EOUAU.Y SPACED ON 4.~ DIA. WITMH 0.005 OF TRUE POSITION DRl.L 0.8~ DEEP

Fig. 12 (b) Details of the grtp - Housing

2.995 • 0.001 I e!A-e!o 0005!

3.~

0 0° RING GROOVE 0.22 WIDE

65

/

I 864 & 0.001 lelA-810 00051

TO FIT PRP-33• TO HOLD PR 3000 PSI

0 10•0.00t

i= 0.2 ...1._._

r 0.910 •O 001 --1

I 1Et!.f.-8!ooo<nl 14)/_i_ '0.125. 0 001 -r-

45° CHAMFER

Fig. 12 (c) Details of the grip - Collet Holder

66

Fig. 13: Deflection control testing with 2 clip gauges

,... a.. UJ

N I

w

(T)

<' r-~ _J_ UJ Q

'q' I

w

(a)

I .E-4

Fig. 14:

67

f--

I I I I I I I I I I I I

I ~ I ~I

~ 1f I

I ~

,iu

<:~ v I

I/ I I

I I I I I I / I I

I I I 11 v I I I 111

~} I.I J I

I .Li J9 11 I ~

I 11 I I 11

I 1. E-3 1 .E-2 l . E- I

DELTA EP2

(a) Schematic diagram of strain measurements (b) Correlation between plastic strains

Fig. 15:

68

Strain control testing with the clip gauge mounted across the grip ends.

c ...... -..r::; -0) c Cl)

~ u .,, I.. u

0 -

OJ 0 . 0

c.o 0 . 0

~ 0 0

N 0 . 0

0 0 . 9J.oo

69

~e_eciinen No: 61?-1 Strain amµlitude. € =U.UU~4 a

d' /

/ 20.00 40.00

cycles to failure

1 .i

I

(!)

(!)

I

60.00 z 80.00 100.00 *10

Fig. 16: a typical measurement of cack length vs cycles

70

Hr' AISI 4340 .P++ ++-;* S :700 MPa u

+ ~~ i o·' R =-1 ... : ~ Iii .~~~ u >- :.:~C) v u

i o·· -e e 6 C) ~~ 1i1· : ffiPo + -

......... f!J ~_p IO e::: .i::. i a·' ~ 0 ..... ~

.!!J * ->' u ;. + IO '- 1 a·• 0 u 00 Short Cracks. & 0 z ~ Long Cracks 0 0. 0047 -IO

CJ 0. 0066 . ~

t c·' ):{ Ctr. Hole ~ 0.010 x DEN ¢ 0.018 + Compact + 0. 029

i a·• -~-ut l O' ut tJ<. Stress Intensity Range. MPa rm

Fig. 17: da/dn vs ~K for AISI 4340 steel (40)

Fig. 18:

71

,

C~CK (oPC:U) ( E > £"WI\.)

( C:C1), ~) s'" e

CRAC.IC (CLOSED)

( e'-,..,.i11.)

Schematic Diagram of closure measurement

66. 63 ksi A

-54. 60 ksi Y

72

·f.:.

<!...,.,,,.,, = 74 .8 ksi specimen no . 617-5 strain amplitude= 0. 0066 Nl200R 2a=0 . 0171 in

44. 13 ksi A

Fig. 19: Typical crack at various stress levels in one complete cycle

(XlOO)

73 ·-·t····1···-~····1·-, .. : : __ J._:. ·-:.:::;-:: :.:::.=...

. : ~: . I : ::.j : : : : : · : . :~ -· : I;~.:~ .:.:·:.:=-t--=.::.~::-' {,, ~ ~. :...;. . .. ::1:J:....._ ( ..

_,.;_-:::.:.:·=i=-..:-!.:;. ~~=- - ::_-t+E·-·-£~= 0. 0125 -=-=--=.: :~ :-:~: -~ =-v j .• ..:=:·i.::-:-.:. ~ ·r..:...::=::t::::J_

617-3. NSOOR _:_-:.-=.:...~~~j~:;§"~JJf._:-;~ 2a- Q 035 • -~=-47-:. :::=:-..:.::E::~-f "'::-/,,_ I . ~ - '" _____ .,__ _____ . .._._......,__, . .

. --=z ----·--r----J-t=-·---=-

Fig. 20:

------·- -===:.:=:::.~;::~:--=::i.:t- ~ - ----- -------· -'·-~-:/-· -~ --·- --·- t=:: ·--~---? t.:.=: • - -: :-=-r::===.==-.

7 - --·-·----·,-1·-;-·-·----· -·-" ------~-- -,-----·~ ---T--· ---·--,., ..... -----.---fc- --· ... -·-· ·,....----..-- ' 1 ..:;)-- -===7-'--'--

.~-=-t==-1--

~--- , -----

2000 Ibo

Stl"ess Stl"ain Cksi)

1 -59.13 -0.0115

2 - 7.07 -0.0095

3 44.98 -0.0055

4 75.97 0.00435

5 42.44 0.0105

6 -21. 22 0.007

7 -59.27 0.0013

8 -81. 34 -0.0089

Crack opening displacements measured in one complete cycle (a) Load disolacernent loop as obtained from clip gauge mounted across the grip ends and the points (corresponding stress and strain levels shown in table) where closure observations were made.

74

c:: ·~ ( a) Increasin:, Strain .., c:: 9::! '/

Qi \ u ·'ti

:i vi

"':l

:;> c:: c:: Cl) :i 0

~

u 'ti '-u

=i '::> :..;

~

~ I I

o~ ~

0.GO 0.25 a.so 0.75 I .CO

0 x/2a, Relative Position

0 61 c::

r ·~

j ( b) Decreasiny strain .... I c:

Cl)

I = Qj 5 u J "

I '° \ ..-1 VI

"O g; :n ~t c: c: Cl) "'.l. rr 0

'I ' ....:: [ ,, " u f ' '° 'X '-w IX-

:::i 8u ::> 8 :.J

g .::· r- : n~

0.CJO o.zs n c:.n o. 75 I . ::i

x/C.a, .'<elative Position (b) CracK OiJenin~ disjJlace:nent aur111:,i rncreasin:, ( loadiny) and aecreasin~ Strdin (unloaain::i) at dinert:?nt 1-1oints alon::1 the cracK ·1en~th. Different stress (strain) levels corresµond to the ~oints shmvn in load ais11lacement IOOjJ.

= ·~

....,

~ Cl) u ~

1

"' "O

"J> c: c: Cl) :l. 0

.:.(.

u ~ L

'....)

'.:::l "::l u

= ...., c: J)

1 VI

0

8 0

0 .,, 8 0 0

§ l 0 0 -100.JC

0 0 0 0

0

:g r 8

_;") a

·~

c :lJ 1

0

~ -:i "....)

0

8 8 ~ h• 0 -100.C

75

(a) Increasing Strain

,.... " '

+ x b

J.C

stress, ksi

(D) Uecreasing strain

x $

+ 0

0 0

O"", stress, ksi

x

C)

x + 0

C)

x + 0

:cc .0

C)

~

-t

x

~

x( in)

0.0039

0.0076

0.0133

0.0284

0.0315

(c) Crack openin'.J displacement as a function of stress at aifterent points alon; the crack len~tn auring increasiny ana decreasiny strdih.

x/2a

0.01114

0.02171

0.380

0.8114

0.9

c: ·~

....., c: ~ a; u ·U

1 VI

"'O

::-i ;::

c: a; 1 0

~ u tU '-u

......, s :._.)

0 V1

8 0

76

(a) Increasing Strain

x <5 6.

x

+ 0

0

~

+ x

<;)

§8.l~~~t='+-~~~~---~~~~~~~ - C6 ~ ~ ,...,

0 -0.0ISJO

~ 8 0

0 V1 0 0 0

0

0

§ l •I

0 --0 .0 I 500

a.cocoa £,strain

(D) uecreasing Strain

v ......

l, strain

x + 0

x .J_

I

x C) 0 ~

6.

C)

o.c:::::;o 8.JtS'J

x(in) x/2a

0.0039 0.01114

0.0076 0.02171

0.0133 0.380

0.0284 0.8114

0.0315 0.9

(d) Crack openin::J disiJlacement as a function ot strain at Llitferent µoints alon':1 tne crao~ len':1tn aurrn'j increasin~ and aecreasiny strJin.

77

~ = 0.0125 617-3. N400R 2a= 0.025 in

··-·~·~;j~~=::..1._..:·1.:-. ::: .:1~-~ .:::L :i·· -·~· ·----. -·-· ...;..! .... -4~... '.:-1- ·- • --- -- • A• --

• .. ,,....-• 1-. ::i-·· ·I ···---·--·- .. ·----=:==:-:-:t:-/.. _-:_-:1=:::1..:...:: :...:.::..-:::=..==!~.:- . : ~-::.:::==. - /__,;. __ :·1----···--·~------~ - -- '··---·

gµ

-1

Stress strain Cksil

1 -63.66 -0.01105

2 -14.14 -0.00925 ,...._ CL -~~

~ 8 - 0

Fig. 21:

3 44.09 -0.0045

4 56.59 -0.00225

5 74.67 0.0057 s · 1 6 52.77 0.01215

-=::::::: : =->-- 7 0.00 0.0096

~ '· -8 -42.44 I -0.00615

9 -70.73 - 0.0025

10 -82.76 - 0.0104

Crack opening displacements measured in one ccmp 1 ete cycle (a) Load dis~lace~ent loop as obtained frcm c1'~ gauge mo~nted across the grip ends and :ne aoints (correspondiig stress ara strain levels shewn in tab'e) where closure ooservaticns were made.

c:

...... c: ~ (jj u ~

1 ~

"'O

::n c: c: ll 1 0

~ u ~

'-;.,.)

78

i5 8 0

la) Increasing Strain

0 .,, § 0

5

4 ~ . __ ___:_--~· ~~~----~3~--~__.._~

~~r=-~~~1~1~s2...,,._......,__,=====-==:=.;•~~::;:..;.:~---t o.co 0.25 0.50 0.75 I.GO

- 0 x/2a, Relative Position

c: ·~

Q)

0

8 0

c \D) Decreasing strain 6·

9

' 0

~I 10 a~·~-+:o....--==;..t==-=""'=~==~~~~~~;:;--'.~ 0o. uo 0 . 25 0 . so 0. "5 l . ~;J

x/2a, Kelative Position (b) Crack o~enin~ displacement durin~ increasiny (loaainy) ana aecreasin~ strain tunloadins) at aifrerent points alony the crack len:;ith. lJifterent stress (strain) leve Is correspona to tne points snown in load aiSjJlacement loo..,.

= ..... = -:J u ., ~ /1

-::;

:"'> .::

:J

0

.:...:. u

"' "-"._)

_, _;

:::

..... -. :; :_) "'.)

::.. /1

-::;

71 = :J

~

_;,,{,

J

"' '--_;

8 0 ::i

a :

l l i . I I

"' 5 I ::i

a

I f ! I I

I

i '2 c g a

79

(a) lncreasin'j Strain

+ L

0

-100.GO -SO.GO a.co so.c:

"' § ,, J, ::>tress, i<s1

a !

I (D) Decreasinlj stra1~ -~

I f '

I ' -,

;-' x ~

L '-'

' t I (T'\

! x '-'

' l '

I 6 C) I t x '

C)

~ ,5 ..... ? - ... .J ........

'.J, s::r~s:>, <S1

~r:lc' J""'e''. ..... .: .:~ S...1 _.::'"~cr~'l'C ..;v ..... : ;._,! ...

: ~ ·.: ; ~ :; 5 ... ~ ,.. ) : .~ : .

-'-.

0

6

+

x

x (in)

0.0028

0.00575

0.0204

0.0225

x/2a

0.01143

0.2347

0.8327

0.9183

c:

..... c:: ~ ii u tO

::i. Vl ...... "O

:n c:: c: <l) ::i. 0

~ u tO '-

:..,)

-=:I ~ ~

c:

+-'

C1J E C1J u tO

1 Vl

"O

.:Tl c: c:: C1J a. 0

...:.: u tO '-u

~ '::l u

80 g' 8 0

ta) Increasin:;1 Strain x( in)

(!) 0.0028

A 0.00575

+ 0.0204 0

"' 8 ±; 0 x 0.0225

0

+ (!) 6 x C)

~ 8 C) 0

......,~ 8 0 -0.0ISGO o.cc:::oo 0.01500

t , strain "' 0 0 0 0

t D) Uecreasin':;j -+-strain .6

+ x 6

"' ~l + x C)

6 0

C) x

ZS C) x C)

0 0 • gl .~

0 . ...,....

-0.0ISCO o.c:coo o.o:s £_ , strain

( d) CracK open i rig di sp I a cement dS a funct i un of stress ar. different points a1on 9 the crdCK length during increasing and decreasing strain.

x/2a

0.01143

0.2347

0.8327

0.9183

Fig. 22:

81

E ~=:=-:-_; :;: :..~---=-.---·--=- :.=:=i j ==::--=:..:: ~=::. ~·-:· - -·-===?.3---· i:::::::-:.-=-ro~- ·----.--: . ·--·-

----- - I· -.4 ·----- .. -- .... --.-.. ,t-• ·-----· -r-----+--::-=,.-_::;.::. -

----·-- ----:-i--~ :=:::::._~~--· --~------' ?! o.'.. -----=;:;::..:_-:.=:-__ :::c: __ ,._ ,......,...--:_ ------------------,,.-~ ---·-· --·-·--··---·---. -:.&.::::=-- ·-··--- .--.----... ·:·-=-----:=.:::.:.-:::-----~-·- -- ~--· ----· - ___. ·-·-------· r-::=.--· --·-·-----·-~~ - ---~--=--..:=.!..--=..= _. --, - . --==-· ~---..--·-·- c I ::;:Jr~':"-'··-·-·•- - l --·•---~-·!I

--··--··--... ,...,, .=:::..=::: ;::-: ~-~ :.. :.:. 8 ____ .__. --~- 0

·------ I • 2alO '" ::===:i::--=::: ________ .

~,,_

1

2

3

4

5

6

7

8

9

Stl"ess Cksil

-42.44

-14.14

63.37

81.06

49.57

-11. 88

-44.56

-71. 58

-81. 34 I

St I" a in

-0.0105

-0.0071

-0.0046

0.0081

O.Oll25

0.0085

0.00545

-0.00490

-0.0115

Crack opening displacements measured in one ccmolete cycie (a) Load disalacement :ccp as obtaine1 fr8m clip gauge mounted acrJss the grip enas and :he Joints (ccrresoonding stress arc strain levels src~n ~n tab~e) where c~osure Jbservaticrs ~ere ~ade.

82

~ o'

i:::: I

i \ a) Increasin~ Strain + ...., \ i::::

~ li u ~

.::::...

~./ 3

~I Vl

'O

::;') §1 .::: .-o' i:::: Q) ..., 0

2 ~~ ~ 3 o I '

:::

...., i::::

~ lJ u ~

.::::... •/1

";:J

:;\ ::: i:::: Q) g-

c • ~ ' ', c:i 0.00

0 .,, 8 0 0

1 o.zs a.so o. cs

x/2a, Relative Position

I .CO

I I

(o} Decredsin'::l strain 1

/s .,, N 0 8t 0 , _ _,__~6~--------~:

7

8

x/Zd, Relative Position (::i) Crack opening displacement durin.g increasing (loading) and decreasing strain (unloading) at different points along the craci< length. Oifferent stress (strain) levels correspona to tne poi'ltS snown 1f1 loaa a1spldce111ent loop.

i= ·-...,, ~ Cl) u tO

1 Vl

"O

~

i=

i= Cl) 1 0

.::.!. u tO ~

'....)

=i ":> '....)

Vl

i= :lJ

83 c "' § 0

la) lncreasin':J Strain

"' "' 0 0 0 0

0 0 0 0 a 0

-ICO.CCCCO -sa.cc.:co o.ccc::1 50.CCuCO

fJ, stress, ksi "' 0 0 0

0

(D) u~creasin::i strain-

...:.... b

0 ><

© x

+ 6

x c:;

+

x

x

100.0

~ L~-=---......._ ____ ......._ ________ __; 0 .,,.

-100.CO -SO.CO a.co so.:a 1:-'!n.o

<f, stress, i<si

x(in) x/2a

0 0.00093 0.05123

6 0.0032 0.1778

+ 0.0153 0.85

x 0.01746 0.97

(c) Crack opening displacement dS a function of stress ar. r. different points alon 9 the crdCK length during increasing and decreasing strain.

84

0 "' § 0

= x (in) x/2a (a) Increasing )t:rain

;...) e> 0.00093 0.0512j = ~

ii 6 0.0032 0 .1778 u 'U

.::i... + 0.0153 0.85

Vl

"' ··-"' . -:: 8

=' ~~ 0 ! c:

x x 0.01746 0.97

~

·-= 11 x .::i... 0

~

u I 'U r...

'._.)

t ....., ~ 0 t '._.) a ;

§ ' a -0.01500 o.:cc:o

"' 0 t strain 0 Q

c::i I = tD) uecreasing strain ._; r = I }:! I lJ t u t 'U

.::i... I Vl

"' I ' -:: ~ . :;-, cs_

0 i • c: I ~

6 ' , ,, ~

6 )< ~ u

,..._, v

~ A r... '-. ....) x --J ;::::: •

-~ ........ ;-----c':~.~~~~~~~~~~~~~~~~~

-J . :; : sea :J .::::::.a : ."": : c:::

(c) Crack opening displacement dS a functiun of stress at different points a1on 9 the crdCK length during increasing and decreasing strain.

85

[(\, = 0.0125 617-3. NSOOF 2a= 0.06 in

--~~-- --

Fig. 23:

---···------ -

----·- Stress Strain ----- Cksi>

1 -59.27 -0.0115

2 - 7.07 -0.0095

3 44.98 -0.0055

4 75.97 0.00435

5 42.44 0.0105

6 -21.22 0.007

7 -59.27 0.0013

8 -81.34 -0.0089

Crack opening displacements measured in one complete cycle (a) Load displacement loop as obtained from clip gauge mounted across the grip ends and the points (corresponding stress and strain levels shown in table) ~here closure observations were made.

86

c:: ..... .. ~ c:: • ~ I

i (jj

J u tO ,... a. VI

~t .,.. \j

::n c:: c:: QJ a. 0 t ~

t u tO t.. t u .. •

:::::i ~1 ':;) u .1 0 0.00000 o.2soc:t1 C.SCCO:J 0. 7SOCrC I.CO

OPg~· x/2a, Kelative Position

N 8

c:: al

..... (o) Uecredsin~ strain ~ 51 c: 8 ~ (jj 0

u '° a. VI 8 \j

~ ::'I 0 c:: c:: QJ Q. 0 0

.~ ~ u 0 tO t.. u .. 0 ':;) 0 :..J ~

x 8 0 0.00 o.zs a.so o. 75. I.CO

x/~a, Kelative Position (b) Crack openin~ aisplacement duriny increasiny (loaainyJ ana aecreasin~ strain lunloaain~) at aitferent ~oints alun~ the cracK len~tt1. Uirterent stress (strain) levels corresµona tv tn~ points st10M1 in I udCl ai StJ I ace111ent I ooµ.

c: ..... ..,, c: QJ c: Qj u "' .:::i.. V'l

'O 8 .;'\ 8 = 0

c: QJ :::i.. 0

~ u

"' '-u

c: ..... ..,, c: ~ l'.i u

"' ::l. V'l

'O

:n ;::: .,.... c: QJ '.l.. 0

-"" u "' '-:...,)

'::::l

8 § ':.100.CG

0 N 8 0

~ ~ 0

la) lncreasin~

~ ~

C9

o.o (), stress, ksi

87

!>train

6 I

©

to) Oecreasiny strain

+ x

3 ~ E ':.100.00 l.C

r;, stress, ks i

I x(1n} x/2a

(!) 0.00506 0.0844 I

6 0.01133 0.1888

+ 0.01546 0.2577 +

x 0.0433 0.7217 x <> 0.04946 0.8243

0 + 0.05667 0.9445 m I

I

.00.0JO

'.C'O.O

(c) Crack Ot-1enin:i aist-1lac~1!1ent d::i a function ut stress at airterent ~oint~·alon~ tne cracK len~tn Jurin~ increasin~ and aecreasin~ strain.

..

c

...., ~ a:; u IU

:l. VI

0

::'l

·-= 'lJ :l. 0

.:,,:. u

"' t.. ._, :::::l ~ ._,

c ·-...., c ~ a:; u "' .:l. VI ·-0

':;> c ·-'lJ ~

::>

-""' u 'O t.. ._, =i ::i '....)

88

8' "' 8 0 x (in) x/2a

\ a) Increasin':l )train 0 0.00506 J.0844

5! 8 0

~ 0.01133 0.1888 ....__ __ + 0.01546 0.2577

+ 8 8 x 0

x 0.0433 0.7217 '1

0.04946 0.8243 0 ' c

0

~ § I + 0.05667 0.9445

0 ~ ~ ~

t m C) I

0 -0.0ISCO o.cccco o.01s:::i

&. ' strain 0

"' 0 0 0

\ b) lJecreasin':l strain

0 ~

8 ... 0

+ 8 x 8 0

,,, ~

~ 0

~ ~ El

~I ~ L)

C) 't-4-

8' ~ o I 0 -o. o 1 sea o.cccc::i c.01s:;

[, strain

(a) CraCi< oµenin::i uistJlace;11ent as a tunction ot strain at ditferent f)uints alun':l the cracK len~tn uurin':J increasin:;, and oecreasin~ strain.

·-

·-·-

89

I-· -. . . .. . .. . . . -· .. . . ~. •••I .. '•

I ·••••·••I·-•· . ..: .. ~ I -:-:: : . : - ~ : - . -. - :1-=~- '. ~:-::·: 1-:-_-:-: :~. :::. i-···i-::...::-:-:.•: -~-· ·-,__.:, -~·;c~__:_-._---.~-s{:-=~

........ -~ ~ .. _!.:: ~-~- :.! :=.:...: =J. i~:= _ .. _ ·-·-·-•-· ·-~'~--' -- •. .4..,

0.0125

9 =:=======~~--=--~------·----·-. --

-~----.~-==-=== ~ ·---:...- ~ --=~

_,__ ----3 10·

,.. __________ _ ·------ 0 .__ __

Str'ess Str'ain ·--- Cksil

1 -63.66 -0.01105

2 -14.14 -0.00925

3 44.09 -0.0045

4 56.59 -0.00225

5 74.67 0.0057

6 52.77 I 0.01215

7 0.00 0.0096

8 I -42.44 -0.00615 ~,,_

F'g. 24:

9 -70.73 I - 0.0025

10 -82.76 - 0.0104

Crack ope~ing disJlace~erts measured in one c:mo1ete cycle (a) Load disJlaceme1t lccp as obtained ~ram :lip gauge mounted across :~e 3rio erds and the ooints (ccrrescondjng s:~ess ard stra'n levels shewn in tan"e) N~ere c'osure J~se~~atians ~ere mace.

! I

90 8 8

::::: 0 I (a) Increas i n':t ~train

µ c:: ~ lJ u "O

1 Vl

"O ~ i :n §I ~ 0 c:: <lJ 1 0

;;,,: u "O '-

'....)

.:i x '=> :....> 8 § I .~ . .,.._ 1 0 v

0.00 0.2S a.so 0. 7S I .:'J

CJ

"' x/La, Relative Position

§ 0

c:: (b) lJecreasin~ strain

µ

~ ii u 'O

1 Vl

'::::J .,., ~

lJ 1 :i

.:.: u "O '-

'....)

::::i ':l 0 ....) §1

0 0 a.co a .~s a.SJ a.~s I .CJ

x/La, Relative Position

(D) Crack openin':J dis,ilace•nent Jurin':! increasin':i ( loauiny) and uecreasin':J strain l~nioadin':j) at different points alon':i tt1e cracK len':itri. Uitrerent stress (strain) levels corres;Jonu to tne ;JOlnts shown in loaa ais;Jlctce111ent loop.

c: .... ...., c: ~ a'.i u ,,, ..... ::i.. Ill

"'O

:n c: c: a> i 0

-~ u ,,, L.. :J

c: .... ...., c: ~ a:; u ,,, 1 VI

-::I. ,., c:

-11 ..., 0

.:>It. u ,,, L..

:..J

:::i :::> '..J

8 8 0

0

~ 0

Ill

§ 0

Ill

"' ~ 0

~ ::-.t,.:<

0 ..,,~

91

(a) Increasiny Strain

)~ ._.

' 0

o.c . i:r, stress, ks i

(o) Uecreasin~ strain

+ 6

+ x

+

!C9 9x . -

I x(in) x/2a

I 0 0.0025 0.0694 i i 6 0.0062 0.1889

+ 0.0246 0.6833

+ x 0.0288 0.8 6

(6

:oo.oo

·ICO.O ").C ICO.O

:f, stress, ksi

(c) Crack oµenin':J diSiJldce111ent as a tunction ot stress at dirferent ~oints alun~ the crack len~tn aurin~ incredsin':J an.a decreasin~ strain.

:.-

= ...., ~ 1i u 'i:l

1 <JI

0

";'> = ·~ = JJ -i. 0

.;,,'. u ~

'-'....)

'::l ~

·.:l

c

..... c ~ CJ u ~

1

"" 0

::"I c

JJ 1 0

.;,,'. u ~

'-'...)

92

3 8~~~~~~~~~~~~~~~~~~~-0

x(in) x/2a (aJ Increasin':J ~train

0 0.0025 0.0694

6 0.0062 0.1889

+ 0.0246 0.6833 0

"' 8 0

x 0.0288 0.8 0 '

I g g §~ 0 -0.0ISGO o.c:cco O.CISCO

"' E, ' § strain 0

( b) uecreasin'::l strain

' '

6

~ I + 6 ci + x

Li.\ 6 c x 0 ~ c

[, strain

(a) Crack Of.!en i n:i ai SiJ I acc::nent as a tunct ion or stra i ri at aitterent ,.,oints alun:i tne crack len:it11 aurin:i iricreasin':J dnd uecrcasin~ strain.

. -·-

Fig. 25:

--.... i.:.=r.::-::~:...o :a:=::_ =..:::-r· _ ___,, ____ ,"""_.:.._, ~--·

3.! _;;'.~~-: ~ L -= -~- :.--t=.::::t..: ~ -

-~~=~~-~t~ § ----~=-----t- 0

·=-=..:::-__ t::-:-:J; ·--- ·-·- :100D Ibo

stress strain Cksi>

1 -42.44 -0.0105

2 -14.14 -0.0071

3 63.37 -0.0046

4 81.06 0.0081

5 49.57 0.01125

6 -11. 88 0.0085

7 -44.56 0.00545

8 -71. 58 -0.00490

9 -81. 34 -0.0115

Crack opening displacements measured in one complete cycle (a) Load displacement loop as obtained from clip gauge mounted across ~he grip ends and the points (corresponding stress and strain levels shown in table) where closure observations were made.

c:

...., c: ~

:ii u 'U

:i VI

':>

:n c: c: lJ :i 0

o.! u 'U t...

:...>

94 0 ./) 0 8 0

"' N 0 8 0

'

(a) Increasin~ Strain 4 I

3

~

I I I

3 § I ~~"'~~=2:::=======<=>.,;,,,~~~~~~E;-~-=:;--. 6~.:_:i::;::::;:

2 ~ 1 =

O.GO 0.25 o.:o 0. -:OS I .CO

0 x/);i_ KPliltive Position 0

c: 0

( b) uecreasin~ strain ...., c: 5 ~ <1J u

"' :i VI

-0 "' N c ' ::;> 81 c:

·~ 0 = 'lJ

:i 0

.:.t! u

"' t... :...>

--":> :...> 0

0 r 0

8 ! "" -· . / 0 a.co 0.25 a.so 0.75 I .C:J

x/~a, Kelative ~osition

(0) Crack openin~ uisiJiacernent durin~ increasin~ ( loadinlj) and decreasin~ strain (unluaain~) at different iJOints alony tne cracK len~th. Oifferent stress (strain) levels corresµond to the iJOi nts sno1 ... n in I Odd i..l i SiJ I a cement l OOf.!.

= ..... .., = ~ a:; u <'O

1

"' "O "' N

~ ~

= ..... ..... -~ lj u <'O

1

"' ..... "O

~ = :lJ ""' 0

....: u <'O ~

:....>

::::i ~

c

~ c -100.00

s c

"' N 0 0 0 0

( b)

+ ~

95

I x(in) x/2a (a) Increasin~ Strain +

x ~ 0.00133 0.0523 -· 6 0.0036€ 0.1444

+ 0.00606 0.2386 6 x 0.02133 0.8398

x ~ 0.024061 0.9472

+ C)

+ ~

0 t:;

,,-:,,,. ~ ·-· ..

-so.cc. o.o:. so.cc. 100.co

(j' stress, ksi

Decreasin~ strain + x

+ + x 6

x 6 6 ~

@) C)

~

·~ 0

§~r-E~_....~~--~~~--~----4 0 --=-· -ICC.CO -so.co a.co so.cc

(). stress. 1<.si

(c) Crack uµenin-:J aist-1lacc:~nent as a tunctiun u!' stress at dit'terent ~oints alun':i me crac:<. l~n':itn <Jurin'j incr2asin~ and aecreasin':i strdin.

.

(d) Crack Of-lenin::i aisf)lace111ent as a tunct1on of strain at dirferent µoints alun:; the craci< len:;t11 aur1n~ increasin~ and decreasin~ strain.

c .. ;. . .;::: .... ..!

~'\, = 0. 0066 6. l7.=?_._ N l SOOF 2a= 0.05867

2000 lb•

s ·1

-=-==- ; ~

~,,_

97

.....

1

2

3

4

5

6

7

8

........

. . . . . .

. . . . .

. . . .... . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . .. . . . . . -. . . l.

. . . . . . . .

stress strain Cksil

-50.36 -0.006

0.00 -0.0043

-42.58 0.0014

68.33 0.0038

42.44 0.006

-17.40 0.0032

-53.90 -0.0009

-71.02 -0.0050

Fig. 26: Crack cpening displace~ents measured in one complete cycle (a) Lead displace~ent loop as obtained from clip gauge mountec acrcss the grip ends and the points (corresponding stress and strain levels shown in table) where closure observations were made.

98

0

c:: la) lncreas in~ '.:ltra in

~ 4 c::

~ ' ~ u ~

a. "' " 71

= <lJ ::i.. 0

=

::::> c:: c:: lJ

G

! . 16--= 8, g ~;...._____~

2 .I

1 0 a.co o . zs o. so o . 75

"' x/~a, Kelative Position

0 0 0

0 i l (b) Decreasin:J strain

0 g '.

I. :a

g . / 8 ·---'----------------0 O.CJ 0.2S 0. SJ

x/~a. Kelative ~osi:ion I.::

(b) Craci<. CJµenin~ uis~lacernent aurin'::J increasin<:, ( loadin':J) ana decreas1n~ stroin \unluaJ1n~) at a1rrerent ~oints alon'::J trJ e c r au 1 e 'l ~ tri • 0 1 ~ re re r 1 t st res s ~ st r a i n ) I eve l s corre::; 1;unu to tri~ ,Joi11ts sno· .. 1n in load d1Sf.11Jce:nerit loo,.;.

c:

.µ c: ~ li u ~

1 <J)

"'O

~ c: ·~

ClJ

0

-"" u ~ L

'._)

ClJ 1 0

99

§ a

la) Increasin~ ::itrain tS '

I x (in) x/2a ~

C) 0.0016 0.0272' ~ (2)

'. 0.00746 0.1271 6

<) + 0.01546 0.2635 I

~ -"' N a 8

L. x: x 0.02106 0.3590 a

0 '-' () 0.0273 0.6357 ·-X: .,. 0.04413 0.7521

~ 0.05093 0.8680 x ...... \.:..,

,,.,. --50.~0 o.:o so.cu

'J, stress, ksi

(;:)) Oecreasiny strain

a 8 t

g~I~~~.-;;:~·~~~~~~~~~~~~~~~~~ 0 -100 co -50.:0 o.:o s:.:a :co.~

(), stress, r<. s 1

(c) Crdei<. OJJenin~ uis,.,lacerr;ent as a 1'unct1on or stress at <..11rrerent ;.;01nts atun"' rne cracr<. leri"'Ch aur1n" 1ncreas1n" and uecr~dsin~ str~1n.

= .., c::: ~ .l) u

"' '.::l..

"' "":)

:::> c::: ·~

;:: '.l.J 1 ~

.:..: u

"' ~ '...)

:::i -=-:....;

c: ·~

:J U1

8 0 0

U1

"' 0 8 0

8 CJ I 8 0 -0.01

U1

"'' 0.

81 0

ol

~~

100

(a) lncreasin::1 Strain

,. c .. ,

a.co strain

2S

(b) lJecreasin::1 strain

9'. ..J_ ,::,.

<'.> C:.

x (5 4' ? C)

x (in) x/2a

C) 0.0016 0.0272. . 6 0.00746 0.1271

+ 0.01546 I 0.2635

x 0.02106 0.3590

¢ 0.0273 0.6357 ·-+ 0.04413 0.7521

i;c:: 0.05093 0.8680

~

+ <::;> A

C)

8 t 0 ___ :,;,.·---------------~ 0 -0.01 a. :o a.01

('., strain

( d) Cr a.Cr< o,.;eri in~ l1 is,..; I dce1;:ent as a runct i 0ri or st ra 1 n at dltterent f)uints alonJ tt1e craCi<. len::1tn aurin;j 1ncrea.s1n;i ana aecreasin'::J strain.

101

.:: 0 ('- = 0. 0066 •• I I I I; I

. ~ ! . I I ' l : 1 111" 1 ' ; · 11 . I ; : : ~ I 1 .;Jj;;: ; 6.1.Z~S. N l 200F 2a= 0. 0325 ; n

. . . . . . . . •••••I I

t '. ! : ; i 1 ! : • • • • • • • I . '

• • t t ••I

I 1' ! : 'I I .. ' · 1 • . '•' t I ,.J' . t. I I I'. t

; : I:? l?:: ii ·~~··· ~ I 1 l '1: : ; 1

·1::::::::-: . . ....... . . . . . . ....... .

o .. I I t t 4 t e • 1. ~·: '~ •• ......... • . • • t •••• . . . . . . . . . ........... • o • • • t • I 41

. . . . . . . . . . ....... . . . . . . . . . . . ....... . • I·.',,'·.',,'·,.· • • • t • • • ......... :::;/;: .......... 7;::. : : : : : : s. : : : : : ; : : :

' I ~ • • • I

~ j ii qt' l; . • 1 ··I I' I, "If . ;::::t.::

...... ' . . . . t • • • • • ! • • : • ~ Io o o o t • o • o •Io Io

• • t''. '•. • i •I• r-f ....... l"'l:~ .. ::··: .. ~2 ~:;,~:::1 . .. ~

.••••••••• t

s ·1

. . . . -. ' .. -~· ..... ~ .. t • o o o • o I o

........... . . . ~ ..... : : : ~ ; : ~ ! : .... ' .... o •!•·•I• o o

1

2

3

4

.. . .

. ~ : ' . ' ' :1~·1n-:1: ' .•. 1 . . .... , ... ·- ~ .. ' .. . .. . . ~ . . .. . ~ 4: ~l : : :

• t I 1 I • o e

I :j'Elili~I· uf: ... 1 ~ t I i ~ ~ 0

r-' .•• t . . . ·-· .

. L 2000 Iba

stress Strain Cksi)

-41.59 -0.0058

- 0.28 -0.004

44.13 -0.0011

66.33 0.0037

54.06 0.0061

-27.30 0.00246

5

--= f ~ 6

~" Fig. 27:

7 -54.60 -0.00052

8 -74.27 -0.005

Crack opening displacements measured in one complete cycle (a) Load displacement loop as obtained from clip gauge mounted across t1e grip ends and the points (corresponding stress and strain levels sho~n in table) where closure observations were made.

U1 0

102

8.,..--~~~~~~~~~~~~~~~~~~~~

c <1J 1 0

c

0

::;'\ 0 c

= lJ 1 0

la) Increasin~ ~tra1n

a .zs a . so n • ;5 I .CO

x/~a. Kelative Position

l D) uecreasing strain

0. ZS 0. SJ 0 · ''3

x/2a, ~elative Jos1tion

:i) Crack OiJen1n::1 ais,.ilacement Gurin:; increas1'1'J "i·.:..ac11::• ana decreasin':i sr;:.i1ri \'Jnloaain:;1) at Jl~~er !lC ;.io1nts aion::1 :~e cracK len~::n. J1~ferent s::ress (strain 1eve;s :Jrres,.ior.a ::o :~e ;,G i ~ns sr:own ~ n 1 J.la Ji s.., ace:ne'l: , oo;.;.

c:::

.... c::: ~ :v u IQ

1 <II

"'O

::n c::: c::: <1J 1 0

.:.c u IQ '-:....J

~ ':> w

c::: ..... .... c::: ~ Cii u ~

·1 <II

-0

::n c::: c::: <1J .:i.. 0

.:.c u IQ '-:..> ~

~ :::> u

103

~ 0

( a) Increasin~ Strain x (in} x/2a

C) 0.0045 0.1384 -· 6 0.0087 0.2673

+ 0.0234 0.72 "' "' § 0

>'< x 0.0262 0.806

* 6

¢ ¢ ~ 0.0293!-i 0.9015 C)

C)

::(- 8 0 0 e ¢

§ (T"'\ ~

0 -100.CO -SO.CO a.co so.co ICO.C.0

'.If

8 <r, stress, ks i

0 0

( D) Decreasin~ strain

"' "' + § x 0 e

6 C)

Cs + x 6 ~ + ¢ ® ¢

8 0 0 0 ~

0 -100.00 -SO.CO a.co so.co ICJ.OO

<J, stress, KSi

(c) Crack oµenin~ dis~lacement as a tunction ot stress at ditferent points alon',j tt1e crack lenljth durin::i increasiri~ and decreasiny strain.

'

c::

..... c:: ilJ

Q) u ~

1 "' 'O

.:-1 c:: c:: Q) 1 0

""" u IU t..

:...>

:::i ".:) :...>

c::

..... c:: 9:! li u IU

~

VI

"O

:n c

c:: Q) 1 0

-"" u IU t.. u

-=i '::> u

104

"' § 0

(a) Increasin~ Strain x (in) x/2a

0 0.0045 0.1384 ~-

I 6 0.0087 0.2573

+ 0.0234 0.72 "' N 0 0 0 0

...J.... x I

..i... x 0.0252 0.805 ~ 6

<1> ¢ 0.0293!) 0.9015 <1> C)

~ 8 C)

0 8 e <1>

g (T'\

0 -0.CC750 o.::::o O.C07SO

"' £, , strain 0

8 0

( b) Uecreasin'j strain

"' + N 0 0 x 0

!" B 6 C) ...J....

+ c x 6 ~ + 0 ® <1>

0 g 0 0 0 -0 .007SO a.o:;::o 0.CUiS

£' strain

(d) Crack oµenin~ displacement as a tunctiun of strain at aitterent µ01nts aton'j tne crac~ ten'jtn aur1n~ increasinJ anu aecreasin'j strain.

105

Lt\.-: 0. 0066 61 7-5 • N 1 1 OOF 2a= 0.027 in

stress Strain Cksi>

1 -29.42 -0.0052

2 26.59 -0.0027

3 58.71 0.0009

~,,_ 4 69.03 0.0036

5 55.17 0.006

6 17.83 0.0032

Fig. 28:

7 -72.85 -0.0047

Crack opening displacements measured in one complete cycle (a) Load displacement loop as obtained from clip gauge mounted across the grip ends and the points (corresponding stress and strain levels sho~n in table) where closure observations were made.

=

=

c:

......, c: ~ l) u ~

::i... <ll

"";::)

:n = c: l) ~

'::)

.;,,t. u ~ ....

:._)

~ -s :._)

106

la) Increasin:; Strain

/ g 1~ ,;k:--' -----=-1 -~ 8 !' <"' o.oo

0 I 0 CJ ! 0 0 0 0 co

0.25 a.so 0.75

x/2a, Relative Position

(DJ uecreasin:; scra1r1

7 o.~s 0. SJ o. ~s

x;ia. ~e1ative ~usition

I .CO

I .:C'

(o) Craci<. oµeninj uisrilace!1;ent aurin'::J increasin:; ( ludain'::J) ana aecreasrn'::J strain (unluadin:1) at dltferent ,JOlntS alon:i the crac~ len~tn. uitrerent stress lstrainj levels curreSf!Ona to the iJO l ns snown in I uao a i sri I ac.:1ile'1t I OOiJ.

= ·~

...., = ~ .ii u ru

=-<J'I

-a ~

c lJ -0

~ ·...; '1J ~

'....)

,....,

-~

=

107 O•

8 0 0

(a) Increasin~ ~train x (in) x/2a

C) 0.002 0.0278

.!. 0.0051 0.1834

+ 0.0072 0.2589 ~I ' g t of . - x 0.0213 0.7661 0 '

t ¢ 0.0238 0.8581

' I

f i ... 0.0261 0.9382 i I

t i

g l c gr 0 -100.::a -SO.CO o.co so.:o ::c. ;:

~ T, stress, Ksi

0 0

(D) L.Jecrectsin':j strairi

6

6 x ~

ti tT\ '--' ~

~

§1 g~(. .. 0 :-~~~~~~~~~~~~~~~~-

- .~:.:o -so.:o o.:o so.:J ,:;:.:a

~. s -c re s s , i< s i

1 c) Craci< Jue~in= ~is~iacc~en: as d f~nc:ion of s'C~ess at dif~c:rent ,.;oin:s atun'::J :ne crac:< 1en:i'Cr1 uuriri:; inc~c::sin~ ~no Jec~eas1n:i s:ra1n.

:::

..., ::: ~ ·li u ,,, -:i.. /JI

-0

°' ::: ::: 'lJ 1 0

.;,,: u ,,, L.

:....>

'O

::n ·~

::: <lJ 1 0

:::i

108

"' 8 0 0

l a) Increasin'::l )train

0

¥ + ~ x ~

+ ~ <> §I 0 C) ~

'T'

x

¢

....

0 a. ~ § ~l ___ -o;;· 0- ...... ~~~~~~--+~~~~~~~~~~

-o.oo?so o.ccc~o o.:~1su

a, 0' 81 0

£, , strain

(D) uecreasin'::I strain

+

x(in) x/2a

0.002 0.0278

0.0051 0.1834

0.0072 0.2589 j 0.0213 0.7661

0.0238 0.8581 1

0.0261 0.9382 i

".:) 0

:....> § '. o...-'---"!:".-o ~- ~~~~~~~~~~~~~~-----<

-0. C0750 0. c:::o O :;::i;~

E, Strain

(a) ·rr·acK OiJenin"' aisi-1lace•nerit as a tunction of st:ra1n at aitrerent r1u1nts alon'::I t:ie cracK len'::lt11 auriri"::I increasin'::I ana aecreas1n'::I strain.

&~ = 0. 0066 6 1 7- 5 . N I 500 ;:; 2a= 0.0248

·-· ... · 1 · - -·-.. , ....... c

:~::;~ ~~~<1·~ ~:-: ~_j . _____ .: l . . ., ·--... ·- ... ,... ····---··------···--· 8 ·---!··--·-----..--·--. ·····--·-·---······-0

£

·1 T

....c::::::::: -=----=-

~"

109

2000 lb•

..... - .... . . . . . . . . . . ... . . ............... ·]00······· . . . . . . . . . . .... -.. . . ... ' . . . . . ...... . . . . . . . . . . . . . . . . . . . . . . :::::!:::'.::::. ::::::::

1

2

3

4

5

6

7

9

Stress Cksi>

-50.36

0.00

-t42.58

68.33

42.44

-17.40

-53.90

-71. C2

: : : : : : : i

Strain

-0.006

-0.0043

0.0014

0.0038

0.006

0.0032

-0.0009

-0.0050

F' g. 29: Crack opening di:claceme~ts measured in one comple~~ cycle (a) Lead displacement loop as obtained •ram clip g~~se mcunteo across the grip ends and the ooints (corresponc~ng stress a~d strain 'eve1s sno~n i~ t~c'.e) M~ere closure ooser1aticns Nere mace.

110

Uo 0 8

c 0

(a) lncreasin'j )train ...., c ~ iii u

"' 1 Vl

"'O ~ 1 ~ §i c . -0 c aJ

0

.;,,:. u

"' I.. :...>

:::::i

~ I '=:> :...>

2

a· (") a.co 0.25 a.so 0. 7S I .CO

Lil

8 x/Za, ~elative Position

a 0

c (b) uecreasin'::! strctin

...., c ~ Cii u "' 1 Vl

Lil "::) § i ...,., c 0

c li :i.. ~

:.L u 'l:l I.. '_)

~ ::i g. '_) s:

0 O.C8

I . ' .

x/ ~d, .-<t: I at l ve J-'o:, lt l •Jrl

(b) CracK Of-!enin-:J ais,.ilaceinent ourin"' incrertsin'::i ( loaain~) and aecreasin'::i strain (unloaa1n~) at aitterent ~oints alon'::i tne cracK len~tn. Difterent stress (strain) levels currespona to the ,.,oints shown in loaa aisplacement looµ.

,, 0 8 0

c:

...., c: ~ lJ u 11:)

'1 VI

"' "'O N

8 ~

0

c: 0

c: lJ 1 0

.:.<'. u "O L

'....)

:::l '.:) 0 '....) 0

0 0 0 0 -100.00

"' 0 8 0

c:

...., c: 9:! CJ u 11:)

::i. V'I "' N

"-:J 0 0 0

-:> 0 c: c: <1J 1 :::> ~ u "O L

'....)

~ -:i 0

0 ...) 0 I 8 I

0 -100.00

~;

111

(a) Increasin'.:f )train

+

-SO. 00 a.ca so.ca

~. stress, ksi

(D) Uecreasin~ strain

I z ?

+ 6

~

6

+

+ C)

!

I ' I I

JOG .~J

-SO. DO a.~n 50.:J ICU.:..O

(), stress, ksi

x( in) x/2a -

0 0.0037 0.1491

6 0.0062 0.2742

+ 0.0183 0.7379

x 0.0206 0.8306

( c) CracK Of!en in~ •Ji s~ I acet11ent as a function or str~s s at different µoints alon'.:f me cracK len.,_,t11 aurin'.:f increasin::i and decreasin~ str3in.

= ...., c: ~ ii u 'O

~ <JI

"::l

~ = c: '1J ~ c ~ u ,, '-

'....)

~

~ ._,

= ._, c: ;,! ii u <1:l

~ ../>

~

.::-> = c: ~

c

°"" :...J 'O '-~

~

~

112

"' 8 0 0

( a) lncreasinlj Strain x(in) x/2a ' i 0 0.0037 0.1491 '

6

i 6. 0.0062 0.2742 I

+ 0.0183 I 0.7379 ' "' N

§ x 0.0206 0.8306 0 C)

+ 6 x 0

& 6 x 8· 0 • gi

0 -0.01 a.co 0.01

"' E , strain g 0 0 I

t l l D) uecreasin'j strain I • t

"'' "'' 0 • c' ~~ 0 6

' -r-

6 9

! ~ m '-' x ol

<;? ~ c ; "7 0 22;

--J .Cl a.co J.JI

[, Strain

(a: ~rac~ J~en1n~ J1S~iaceme": as a r~~c:~~~ 0f str31n a: a1r7erent ~01nts alun~ tne crac~ len~tn c~r~nj increas1n~ ana Jec~~as1n~ str~1n.

-

I

£0.. = 0. 0066 617-5. N1200R 2a= 0.0171 in

. I : : : :: : : : : . . .. -.... -.. :: :::1::::::::: ... -. . . . . ....... . I. . . . . . . . . . ...... t • . ···-····· ········-: ::~:~::~: :::;::£::

. ···-·-··· ......... . . -----·--· ·-·'· . -- ·/ ... . - -· ... . : : ==~=·= :i .. -- .... . ::::::/: . -- ..... .

s ., I

.. ; I : . . : : : I ; :

....... ' .. . . . . . . . . . • • • • - '! - ••

:1~!::::~ ............ . . --.•... ; -. -...... • • • • • • • 4 •

113

.... ;:: :,u1: 1~~:1 1l!Hlj~~~ : : '. '. i i ! ; i : ! ! j j j; : d ~ I~~ ~. :;;:·:::i ·;r•11··· 1l·11··j ; •.• 1· 1 • I : I • ' i f I • ' • 1 • • . • • : • • I • I • • • . I I • , I

- . ..

Stress strain Cksi>

1 -41.02 -0.0058

2 - 0.28 -0.004

3 44.13 -0.0011

4 66.33 0.0037

5 52.48 0.0061

6 14.71 0.00464

7 -54.60 -0.00052

8 -74.27 -0.005

Fig. 30: Crack opening displacements measured in one complete cycle (a) Load displacement loop as obtained from clip gauge mounted across the grip ends and the points (corresponding stress and strain levels shown in table) where closure observations were maae.

c

...... c ~

~ u ~

1 V'I

"'O

:n = = 1J :i ::>

.:.!. u '" t... :.....>

~ ~ :.....>

c

......

~ 1J u 'O

1 •./)

'::;J

~ c = lJ i ::>

"" u 'O t...

:.....>

=i ~ ...)

114

Ul

8 0 0

l a) Increasi n'::i ::itrain

Ul

"' § 0

4

I

i I ----.:,

' ~,

' 2 '~! 8 ''\ I + 0 1 ""'-· ' 0 0 0 a.co

X/ .:'.a, t<elative Position "' 8 0 0

lb) Uecreasin1:; strain

5 :"\ ?

~I ~

~l 0

. I I

~ I I

\ I I

7 I 0 -~ o, 8~ 8 '

"'-.~ ! o· 0 '= o.:o a .~s 0.50 a. 75 I .:'.l

x/C.a, r<elative Position

(D) Crack ofJenin:i aisul.:icement aurin'::i increasin':; ( loaain':J) ana aecreas1n:J strain \unloaa1n'::i) at a1rrerent iJOlnts alon':i tne crac( len:Jtn. Different stress (strain) levels corresiJona to the µoints shown in loaa uisiJlace:11ent loop.

= ·-..... = 9d li u 'O

':l.. "./l

":l

~ = ·-lJ :::::.. 0

.:.(.

u ·'O ~

:_)

::::i ~ ~

'...)

= ·-..... = ;! ::i u 'O

•fl

-:l

:;"\

= = :J ~

0

..:><: u 'O ~

'...)

"...)

115

"' c 8 0

(a) Increasiri~ ~train

U1

"' . 0 ' c a_ 0

0 0 0 I 0 0 c c --------.·?r-. ---

-iOO.C.O

,, 0 ~ 0 0

I l l I [

0 ! 8 I -0 I

. I .

'2. §~

-SO.(:u :J.J\) '7, stress, ksi

(b) Uecreasins strain

~

' -x 0

0

6

: +

x

e x

so.::o \CO.:O

~

0

x

5~~s~------------------;.:a ~o -SO . .:::: sa.::i

fJ', s:r::ss, <s1

x(in) x/2a

0.0017 0.0955

0.0032 0 .179 7

0.0152 0.8539

0.0171 0.955

:) enc:< :Jue'l~'l'.:' :~sJiacerr.eri: 35 3 ~Jn::>:;:1 .Ji s::::ss at c1f~erent ~01nts alan~ :ne :rac~ len~:n JJr1~~ 1ri::e3s1n~ aria Jecreas~ns s:ra1n.

::::

.u c ~

Ci u "' :::i.. Vl

"'O

:"\

c a; ::i.. 0

""' u ,, L.. u

::::

..., ~ ii u 'U

::i.. Vl

"Cl ,, c c a;

0 _,,,,_ u 'U L..

'....)

'::l -:::. '....)

J\ § 0

Lil

"' § 0

8 g c_ 0

;.-:-~ ~

-0, C07SJ

U'I 0 0 0 0

U'I

"' c r o, ~ ... 0

0 0

f 0 8 0 -0. CG750

116

( a) Increasin~ Strain

6 C) '/ ' '

6

e x

o.~coco

Et, strain

6

+ C) A

(D) uecreasing strain

C)

0,CCCGO

£, strain

* e x

! 0

6

+ ' r x ' '

O,:JISO

x(in} x/2a

0.0017 0.0955

0.0032 0.1797

0.0152 0.8539

0.0171 0.955

(d) Crack oµenin'::I ais,;lcice1rrent as a function ot strain at aifferent µoints alon'::I tne cracK len~tn aur1n~ increasinj ana aecreasin'::I strain.

Fig. 31:

117

Stress strain Cksi)

1 -29.42 -0.0052

2 25.74 -0.0027

3 58.71 0.0009

4 69.03 0.0036

5 55.17 0.006

6 17.83 0.0032

7 -22.35 0.0027

8 -72.99 -0.0047

Crack opening displacements measured in one c0mplete cycle (a) Load displacement loop as obtained from clip gauge mounted across the grip erds and the points (corresponaing stress and strain ~evels shown in table) where closure observations were made.

118 "' § 0

I ::::: ( a) Increasiny '.:>train I I .....,

I ::::: CJ 4 E I } l.J +-"''

I u tO

1 3 l/l '/ I

" / i I ::;') c ::::: CJ 1 0

.:¥. u tO r-. u

~ 0 2 ,..,..., u 0 -0 ... ,.,.:', g, -r.-- -·

C)

o.co 0.2S O.SU 0.75 I .~O

x/ 2a, f<elative Position

c:: ( D) lJecreasin':J strain

....., ~ c::

~ > ';-<._ l.J u tO

::'.l. l/l

I

~

::;')

c:: c:: :1.' -, 0 7

.:¥. -~-u / \ :o r-. ~ L u

.::i J "::> ;...) ol

01 8 c· O' 0 a.co O.<S o.sa 0.7S I .~0

x/ (a, Kelative rlosition

(D) Crau Of.lenin 0 diScJlace111ent durin':J incre:isin::i ( loaainy) ana aecreas1n~ strain \Jnloaain~) at aitterent rloints alon~ the cracK len~t~. Oitterent stress (strain) levels corres~om1 to tne cJO i nts snown i 11 I oaa a i si-1 I aceli1ent I oo~.

c: ·~

c: lJ

5

c:

<J

"' 0

119

8~-----------------~ 0

(a) Increasin::i ~train

b x I I I I I

Li: C)X

0 g 0

0 .;..· -----+--·•'---.-..--0 .. ,.,,

C)

~

+

x

-1co.oo -so.c::i a.co so.co IGC .C.J

\J, stress, ksi

§~------------------~ 01 ( b) Decreas i ny strain

+

x

({, stress, KSl

x 3:

x( in) ! x/2a '

0.0012 0.0784

0.0027 0.1764

0.0138 0.9019

0.0146 0.9477

( c) Crack OJJt:ni n:J ai S,J 1 ace:nent as a function ot stress at dltterent ;Joints alon'j the cracK len'::ith durin" increasin'j and decreasiri'j strain.

...... ~ 'lJ u ·'U

1 Vl

" ~ c c lJ 1 0

;,,(_ u 'U t...

'.._)

.:::i =i '.._)

~ 0 8 ! 0

0 0 :::l D G I

'· 0 ......

120

(a) Increas i n':J ~train

x

~ x

S, strain

(b) Oecreasiny strain

+

6 12)

6X

x z

C)

~

+

x

x(in) ! 0.0012

0.0027

0.0138

0.0146

x/2a

0.0784

0.1764

0.9019

o. 9477

-0. CG7SO

r, strain

(d) CraCK OfJenin:i uis..,Jace;;1ent as a function or strctin at llirterent. ;.;01nts alon:i me craCK len':Jt.n durin:i increct:>Hl:i and uecrectsin:J strdin.

121

t.~ = 0. 0042 S17-1. NSSOOR 2a= 0. 029 in

Fig. 32: Crack opening displacements measured in one complete cycle (a) Load displacement loop as obtained from clip gauge mounted across the grip ends and the points (corresponding stress and strain levels shown in table) where closure observations were made.

c ·~

~

u

"' '-u

c

..., c::: ~ ~ u

"' ::::i.. VI

"O

".;>

c ~ ::::i.. 0

~ u "' '-._, :::i -::i "...)

122

\a) Increasin::i Strain

5

0 0.00 a. 25 a.so 0.75

"' 8 x/2a, Relative Position 0

0 t (b) Uecreasin::i strain

6

~ 0 0 0 0 ' 7 : '. \ r-8 \\ / ' -\ \~ '/ /,-Y-==----_ 9 x

A // --------~ \ ' // .__. ""'\._ 0 0 '-' () 0

,// 10 ,., "' z ~ -·""' <-;,

,_/ ~· n C.CO 0. ZS 0. ::; a. 75

x/(.a, .-<ela:ive r'osition

1 .:o

l ::

(D) Crao. oµenin':J ais,..;lace:nent aurin·::i incredsin':J ( loaain::i) ano aecreas1n':i strain (eJnloao1n::i) at aitterent ,..ioints alon'j tne crac~ lenjtn. 01rrerent str~ss (strain) levels corres~ono to tne ,;uints snown in loaa oisi-Jlace1nent looµ.

:::

....., c ~

li u "' 1

"' ~

:n c: c <ll 1 0

'°" u 'O ~

:...>

::i "":J •_;

c

.....,

~ (ij u "' 1

"' "O

:;'>

c ~

0 ~

u "' ~ u

~ -=i ·_;

123

"' 8 0 0

la) lncreasin~ )train

0 0 I 8 6 0 6

' ....... ' .......

6 0 0 + x 0 x e

0 0 c 0 0 el

x a

I i I

I I

I

0 I

6 i

+

x

x( in) x/2a

0.0013 0.0448

0.00667 0.2306

0.0236 0.8138

0.0266? 0.9196

-100.CO .so .ca a.co s:.C·'J ICJ.::J

"' 0 0 .;:)

0

0 a 81 0

0 0 0

" I 0 a

:r' stress, ksi

lo) uecreasin::1 strain

6

+

L5 A \..J

v

e

x 0

+ 6

~

6 +

- lC:J. :J -SO.CO o.::i so.:~ :c:J. :::J

cr' ' stress, kSi

(c) CracK o~enin::1 aisplacement as a function of stress at different ~oints alon~ the cracK len~tn aur1n~ increasinj and aecreas1n~ strdin.

.

124

" 8 0

c: a

(a} Increasin~ Strain x( in) x/2a

..., c: ~ <:J u ~

0 0.0013 0.0448 I

I 6 0.00667 0.2306 ~

' :::i.. VI : + 0.0236 0.8138 "O !:: :;'I 01 c: 8. c: 0 1 6

6

I

x 0.0266? 0.9196

aJ ::i ++ @ 0

~ u ~ t.. u

~ "=> ~

6 c:; 0 + x

0 l 6 c

~ g 0

C) x

x 0 -0.JCSCO I) OGGiJO

"' 0 0 0 0

c:

·~ 0

""=' 8 r ::;") ~ -c: 0

c: aJ

5-

0

t ,

( b)

strain

Decreasin~

6

+ x C)

strain

6 +

-

I

0 •

§~l~~~~~y·~~~~~-'-~~~~~~~~~ 0-o.CCSCJ o.:~:~8 o.:~:

E, strain

(a) CraCi< oµenin~ a1sf-llace!nent as a function or strain at a1trerent ;Joints alun:J tne craci< len:Jtn aurin':1 increasin:J ana uecreasins strain.

IOCD 11M

~

£,(\. = 0.0042 617- 1 • 1NSSC8F 2a= 0.0429 in

: :I: .

s ·1 '

125

.• : •. ) ... 1 . . .. 1:· ..

Stress Strain Cksi)

1 -49.37 -0.00364

2 -20.23 -0.0027

3 -13.58 -0.0013

4 34.23 -0.0018

5 64.08 0.00264 =====-

~2• 6 44.84 0.0032

7 11.17 0.0020

8 -23.77 0.00042

= ~ j • 33 :

9 -46.82 -0.0010

10 -63.80 -0.00266

CncK Oµenins; aiSf.iiacerne"t_j :neasurea jn :Jne :01::,..,\et:: C.J-. -:: ',a) Loaa a isµ 1 aCe!ne:it : oo..i as 0ota rneu ; r·J1;1 : I 1 u :;jau ~e ;nountea across :ne -::;r1u enJs ana me ~01·n.s .corres...,ona~·':

stress ana strarn ;eve1s s1own 1ri taD1e) .-;nere :1-Jsure JDse:·1at 1 uns ·"ere ~.;a..:e.

=

1 •..fl

c lJ ::i.. 0

-:i "_)

126

:: 8-..-~~~~~~~~~~~~~~~~~~~~ 0

..., o.~o

:: 0 0 • I

0 <JI.

§I 0

(a) lncreasin':J ::>train

0.25 a.so o. 75 I .CO

x/2a, Kelative Position

( b) Decreasiny strain

0.25 C. SJ 0. -:s I :o

x/!.a, .ielative t>usic10n

(D) Crack oi-ienin;; cJisi-ilac.:1i1ent durin':J increasin;; ( loadin;;) ancJ decreasin':l strain \unloauin;;J at uliterent ,JUlrlts alon;; tne crau. len;;t11. Llirterent str.::ss (strain) levels correSiJOni.l t0 tne ,.,oints snown in luacJ aisi-11ace111ent IOOiJ.

c::

.., c::

127

0

8.,.--~~~~~~~~~~~~~~~~~ 0

(a) Increasin~ Strain

0

6 ~ a; u •'O

x i~

1 V'I 0

" ;g I 8_ ::n 0 c:: c:: a.i 1 0 "r°"

/, .-.

I I + I

: x

I ¢

...

6

z

I ~ 0 A .. -~· ,~

.><'. u ·'O '--. :_)

0 -IC0.00

0

~ 6 L ,·/. "-- e ~ ...... e

-SO.CO 0.00

cr', stress, KSl

(j

so.co t c.~. ::

8~~~~~~~~~~~-=~..::._~~~~~~ c:i'

( D) Uecreasin':i strain

<?-'

~ ~ 2 ;:.

i

0 8 c. 8~!~~~+:.:C.--+-~~~~-+-~~~~-..-~~~--.; 0

"'"" ~ 0 ;;-..:; ~

,A

- i00.00 -SO. :o a.co so.co 1-.;o. :~ '.) -

' stress, KSl

z

x(in) x/2a

0.0024 0.053

0.004 0.0889

0.0056 0 .1244

0.00746 0.1657

0.0117 0.26

0.03 0.6667

0.0362 0.8044

0.0424 0.9424

( c) Crack OiJen i n'j ui s,; I ace111ent as a function of stress at ditferent ~oints alon'j tne crack len'jtn aurin':J increasin'j ana decreasin~ str~in.

I

c ·~

...... c ~ :jj u ~

..,._

"' ~

"::'I ::: -~ c di 0.. 0

~ u ~ L..

:.....>

c

...... c ~ Qj u ~

'1

"' "O ,, c c l) '1 ·:)

~ u ~ L..

:....>

:::i '.:) '..)

128 e. 8 ci

0

"' §

( a) Increasin'::I Strain

I

x (in) x/2a ¢

0 0.0024 0.053

x 6 0.004 0.0889

4- i~ 0.0056 0. 1244 "" A '

I x 0.00746 0.1657 0 6 ¢ 0.0117 0.26

~ c... '

4- ... 0.03 0.6667 l<: 0 I ·~ 0.0362 0.8044 s .. ·.

A .-., 1-~ ~ F' :;,, I Z 0.0424 0.9424

"- -~ I

~ ...... e I ~ I ,.:...

0 -0.00500 (I .IJi)C,::0

l, strain o.c:::::o

8 "

(o) Uecreasiny strain

0 ~ ,. ~ 0

* 4-~ * 4- l<: 6

§ 8, 0

~ + ~ ~

°;!;"

$ 8 (j ~ ~ _.

6 = IT\

-0.00500 o.occco 0.0050

&, strain (d) CracK o~enin':i aistJlace111e11t dS a function ot strain at ditrerent points alon'::I tne cracK len:;tn aurin":i increasin'::I and decreasin'::I strain.

I

129

~

~ = 0.0042 ........ I .•. I.; ..... I. ...

... ,::::1::: .. : 1:.:. I ................ .

•• ·••• ••• • I

~L IXD Moe

stress strain Cksi)

1 -47.67 -0.00364

2 -15.56 -0.00242

3 14.99 I -0.0012

4 39.61 0.00021

5 - 2.82 0.00013 I 6 -29.42 0.00018

7 -63.09 -0.0025 I

Craci< oµe•11n'j •JlS1.1lac2111e,,::s :11~:1sur~·J :n ·Jne ·:J111Jle::e :_.: ~ ,a) ~L)c1G OlS,.;ldCc!l1e'11:. iQL).., ·.JS JOi:..Jlrl~·.J ~rJ;11 ::~.., ::CU::~ .nuun(coJ j(r'.JSS :~e :,r1,.; en'..:s .'ln~ :::e .Jui-::s ~Jr"-:::Su1)''~~--~

S(ress ana :;;:rain :c:veiS snv..11 :·: :a::-:: .... r!-::"~ :'.;SJ"': Joser-·1a::~Jns ... er:: ;:ace.

1 (/)

=

1 (/)

130

§~~~~~~~~~--~~~~~~~~~---; 0

0 a.co

(a) lncreasin~ ~train

0. 2S a.so a. 1s

x/La, Kelative Position

(b) uecreasin':i strain .....____

5

--'\ ' ~1

I ~ ~i

~ o 7 ~I :::> () :..; 8~:_.;.:c__~:::._...:::..~::::.....~~~~~~~~~~~~~I

Cl a.co 0.25 o.so o. ""5 I .CJ

x/La, ~elative Posi~iun

(D) CracK O,Jenin':J dis~lacernent uuriny increasin:J \ loaain'.:j) ana decreasin'J strain (unloauin':i) at aifferent ,;oin::s alon;, tne crack len~tn. Uifferent stress (strain) levels corresµona to tne ~oi nts snown 1 n I oaa 1.11 s~ I acernent I ooµ.

c: ..... ...., c: (1) = ai u ~

'1 Vl

-c ::n c: ..... = (JJ :i 0

~ u ~ I... u

".::) '::l u

=

c: (1) :::>.. 0

N

8 a a

~ a ~ ... a

§ I I a

-IGO.OQ

131

(a) lncreasi il'::J Strrii n

+

x ' + x 6 x C)

/\ 0 + (,i

§ c.

-SO. CO o .:J

<r, st res s , kc: i

x

+

x 6 C)

+ (b) uecreasin~ strain

0

.0.

+

x

1~: .:.s

'._) a I 01 g: 0 _, -----,.'.:,------....,.....----------0 -IC8.~0

c

-SO.C:J O.C:J SJ .CO

'1"', stress. i<.si

x(in) x/2a

0.00173 0.0715

0.00406 0.1678

0.00586 0.2421

0.0233 0.9628

(jf) CraCK oµenin~ aisf)lacen:ent as a function ot strain at oittere~t points alon'.J tne crac1< len~th aurin~ increasin-;; anu aecreasinj s~ra1n.

c:

...., c: ~ a:; u 'U

1

"" -:l

:::" c: c: ;JJ ~

0

.:»{. u 'U I... u

~

·::::S

.l /l

132

N

§ 0

( d) lncreasin':! :::it: rain I

' " '

' 6 i

+ I

+ 0 c; • a: 01 x 0

x

+ + x 6 x 0

6 0 + 8 * o' - '

~ . c :5 . 0-v .CGSCO o.cc::::::o 0.SCSC8

E, ' strain

(b) Decreasin~ strain

' L; T

x 6

x 0 ~i 3~~~~__.;;.:..:_~~~~~~~~~~~~~~-0 -V.~CSCO

t, strain o.:c;.::

x (in) x/2a

0.00173 0.0715

0.00406 0 .1678

0.00586 0.2421

0.0233 0.9628

(d..) '.:nc< 8L.Je'1i.-:"' ·:is;:;ia.:e!:ient as a t· .. rnc:~on or s:ress at: a1n::rent . .Joints :il0n'j :ne crac:< l~n'jrn uur:'l'J incre:JS1n'j ana Jecreas1n~ s:rain.

I

Fig. 35:

133

stress strain <ksi>

1 -46.54 -0.00356 -

2000 lbs 2 -14.28 -0.00239

3 16.97 -0.00106

4 41.16 0.00032

5 60.97 0.00264

6 40. 31 0.00296

7 4.10 0.00126

8 -27.72 0.00016

9 -48.24 -0.0013

10 -61. 82 -0.00410

Crack opening displacements measured in one ccmplete cycle (a) Load displacement loop as obtained from clip gauge mounted across the grip ends and the points (corresponding stress and strain levels shown in table) where closure observations were made.

c:: <lJ 1 0

~

u ·'O "-~ i ~ I "3..;...

3 ~~. ~?I"_ .... ~ 8 -J. ,""!l'l.'00

CD I I

134

(a) Increasing Strain

x/2a, Relative Position

(b) Decreasin~ strain

6/tb ! /. 7

//(a :n c:: c:: 11 1 0

~ u 'O "-~

.zji I I / ti ~

' I Y. i 91 i I

§ y 10 ~ ,..._ .

x/~a, ~elative Position

(::>) CracK uf'enin~ uist-'lace:11e~n: Jurin'::> increasin':::> ( loaain::;) ana aecreasin~ strain \u11luaai;1;,) at airierent 1-'oints alon':::> tne crac~ len':::>tn. Uifferent stress (strain) levels corr'=Sf10nu to tne 1-'uints sn0v111 in loaa aist-'1ace1;1ent loot-'.

c::

=

c::

...... c:: g,i

ii u

"' ::i..

"' "':J

:n c:: c:: QJ 1 0

~ u "' t.. • .....>

:::::i -:i :...i

g 0 0 0 0 -100.00

"' g 0 0 0

\ a) Increasin~ )train

6

~ C)

"'""" '=' '-' -SO.OD a.co

cr-, stre!:>s, ks i

135

. ~

I ! x (in) x/2a

0 0.00013 0.0083 6 C)

6 0.00133 0.0838

C)

I I

so.co ICC.:JO

( b) Oecreasin'.:! strain

0 0 0 0 0 0 -iC0.20

6

C)

6

C)

t;;;: -SO .:'J o.:o SO.C.J tCO. :J

'J, !:>tress, ksi (c) CracK Of!en1n':J dis,.ilacement dS a tunctiun ot stress at ditterent ~01nts dlun~ tne crack lenjtn uurinj increas1n~ ana decreasin~ strain.

c lJ 1 0

= ....., c lJ

li u -.:I ..., "' "::l

:n c ·~

= <lJ

0

.><: u -.:I (....

'.....)

~ -=i '.....)

~ 0

( a)

8 ~

.---~ § 0

-0.CGSOO

a> 0 0

8 0

136

lncreasin':J :::itra in

6

C)

6

A G \..../

a .oscoo

£.., strain

( b) Deere as i ny stra; n

G

0 0 c '°I ,... 6~ -o .~:::sea o.~:::.::co

[, strain

. x (in} x/2a

0 0.00013 0.0083 0

6 0.00133 0.0838

I

I ! - -

a. :::s.~~

(d) Craci< openin':J dis,.;lace111ent as a function ot strain at d1rferent ,.;oints aion':J cne cracK len':Jtn dur1n':J increasin':J and uecr2asin':J strdin.

f;~ = 0. 0024 617-23.N42234 2a= 0.029 in

----2a---

1 ')., ·~'

1

2

3

4

5

6

7

8

9

stress Strain Cksi>

-43.28 I -0.001925

-26.44 -0.00132

7.64 -0.0002

33.94 0.00083

49.22 0.00165

47.24 I 0.002

- 7.07 0.00018

-26.87 -0.00062

-59.54 I -0.002-::::

Crack opening displace~ents ~easured in one complete cycle (a) Load displace~ent loop as obtained from clip gauge mounted across the grio ends and the points (corresponding s:ress and strain levels sho~n in :~ble) where closure observations Nere ~aae.

I

I

lJ 1 0

0 a.co

138

x

(b) 8ecreasiny strain

6

0.25 C.~D

x/~a. Kelative Position

0. 7S I .CD

(D) Crctci< OiJenin::J disi->lace1nent ~urin':J increasin::J ( loadin':J) ano aecreasin':J strain (unloaoin'j) at oitrerent ;Joints alun"' tt1e crctci< len'.Jtn. Different stress (strain) levels correSiJOnu to tne ;.ioi nts snovin in I uaa a is~ 1 ace111ent l OOiJ.

c::

..... c:: ~ iii u

"' '.1 Ill

"'O

:;"I c:: c:: a.I :::i. 0

~ u

"' ~ :.....J

:::::i :::i :.....J

c::

..... c:: ~ a.I u

"' :::i. :/)

"O

=' c:: c:: l! .... 0 .;,,:. u "' ~

:.....J

~

:'.) :.....J

"' 0 0 8 0

8 c 0 ~ 0

- 75.CO

"' 0 0 0 0 0

~ 0 I 0 0 '=' - 75.CO

139

x (in} x/2a (a) Increas i n1:1 Strain

0 0.0028 0.0965

6 0.0204 0.7034

a.co 75. ::J '1, stress, ksi

(D) uecreasin'j strain

C)

o.:~

o-, stress, Ksi 75.~c

(c) CracK oµenin~ dis~lacement as a function of stress at ditterent ~oints alon~ the cracK lenytn aurin~ increasin0 and decreasin~ strain.

140

x(in) x/2a

0 0.0028 0.0965

6 0.0204 0.7034

O.C:JZSO

(d) CraCK ol-'enin::J cis,..,leicc'llent as a function ut strain at ditterent iJOints alon'.J me craci< len'::itn llurin::1 increasin'::i arHl c:ecreasin·J strctin.

• l I •

Fi'::J. 37:

~.-:.. = 0. 0024 617-23.N32S20 2a= 0. 02i in

-..

141

1

2

3

4

5

6

7

8

9

Stress strain <ksi>

-44.28 -0.00185

-21. 64 -0.00125

0.00 -0.0005

24.05 0.0004

42.72 0.00128

39.60 0.0018

5.65 0.00065

-28.28 0.00066

-55.16 -0.00192

CracK OiJenin':J aisiJlacements rneasurea in one cornf-llde cycle \a) LOdO a1sf-llace1;1ent loot-! as ootainetl trur;, ClliJ ::iau::ie :nountea across tne 'fi,., entls ana me ,.;oints \corresf-lonain':J stress ana strain levels snuwn in taDle) ~.nere closure ooserva-t-i uns were :naue.

c:

c:

142

la) lncreasin~ Strain

r/'~ 5 0 3~

0 +---~ '-I' ....__,

o . co o. 2s a . so r . 75 1 . :o

"' 8 x/'C.ct, r<elat1ve P-osi-t1on ~ ,------------- -------

to) uecreasin~ strain

6 C"i '-'

7

8 I 9 I

, " I ~,~

Q "C . ' I .GJ

x/La, Ke1ative Position

'.,D) CrciCk o,.;enin'j uis,.ilace1lient Jurin'.;! increasin:; ( loadin~) anu decreasin':l strarn lunloauin'j) at airterent ,.ioints alon.3

tne cracK ler1':Jtn. Uirrerent stress (strain) levels correStJOnu tu tne ,.;oints snown iri loaa oisrilace1;1ent loori.

c:

....., c: ~ di u IQ

'1

"' "O

::n c: c: ~ 1 0

.:.t. u IQ '-

'...)

=i -::> '...)

c: ·-....., c: Q) E Q) u IQ

:1

"'

143

~ 0

( a) x (in) x/2a Increasin':::I Strain

0 0.0024 0.1091

6 0.0054 0.3545 -

+ 0.0203 0.9227

6 6 6

+ + +

+ g §

0 0 ~ C)

a ,..._ '-'

g c:/ O· 00 , stress, l<si 75.GJ

8.-~~~~~~~~~~~~~~~~~ 0

(D) Decreasin':::I strain

+

<r, stress. ksi

6

+

\c) Crack oµenin9 dis~lacement as a function of stress at different ~oints alon':::I tne cracK len':::ltn durin~ increasin':::I anu decreas1n~ strJin.

-

i= ·-.µ i=

~ Qi u tO

:1 Ill

'O

::n i=

i= (IJ -:i.. 0

~ u ~ L.

:,,,)

:::::l -:i w

i= ·-.µ i= ~ <= iii u tO

:1 Ill

'O

-n i= ·-::: .l) 1 0

~ u tO L. ._, ::i -:i ._,

144

§ 0 ci

(a) Increasi n':J Strain

+ + +

+

~ I;_;; ci

~ ~ ~ ~ ,..... '--'

-0.00250 o.oocco

"' a § c ' strain a

( D) Oecreasin':J strain

+ 0 8 8.i.--e a -0 .00250 0 .CCGCO

f, strain

~

+

C)

~ -+

O.O:JZ50

x(in) x/2a

0.0024 0 .1091

0.0054 0.3545

0.0203 0.9227

(d) Crack OfleninJ dis~laceinent as a fu:iction of strain at different f)Oints alon~ tne crack len~th durin~ increasin~ 3nd decreasiny strain.

Fig. 38:

En. = 0.0024 617-23.N29760 2a= 0.0181 in

L --s

·1

-= i :::::::-

t=,.

145

1

2

3

4

5

6

7

8

:r ., I

T

Stress strain Cksi)

-45.26 -0.001925

-19.80 -0.00117

21. 21 0.0003

49.50 0.00172

32.53 0.00150

4. 24 0.0005

-18.38 0.00025

-53.74 -0.00195

Crack opening displacements measured in one comolete cycle (a) Load displacement looo as obtained from clip gauge mounted across the grip ends and the points (corresponding stress and strain levels shown in table) where closure observations were made.

c:

...... c: ~ Qj u "=' a.. <l'l

-0

:n c: c: ll 1 0

~ u "=' '--

:...>

~ :J :._)

c:

...., c: ~ ii u "=' 1 <l'l

"O

:n c: c: a; '.l.

':>

~ u "=' '--:...>

~ "'.:) :...>

"' § ~ I

0 o.~o

"' ':J 8 0 01

0 ' ~ .L 0 a.co

146

(a) Increasin~ Strain 4

2

1 n. ~s 0.50 0. 75

x/ca, f<elative fJ_gsition

l D) Uecrea::. in':J strain

8

0 .:s O .. ,.5

x/ca, Ke/ative ~ositinn

/ .. I I

I .CO

(D) Crack oµenin·::i aisi-ilacement aurin':J increasin::i ( loaain'j) ana aecreasin::i strain l~nloaoin':J) at aitterent 1-'oints alonj tile crack len':ltn. uirterent stress (strain) levels corresi-iona to tne ,.io1nts snown in loao ais,.ilace1~ent looµ.

c

..... c ~ 1:i u ~

1 V'I

"C

.:Jl c c <lJ 1 0

~ u ~ L.. u

::i "=> :...J

c

':;'> c c ll 1 0

......

"' 0

§ 0

0 0 0 0 0· 0

- 75.00

"' § u

01

3 8

147

' x (in) x/2a '

( a) Increasin~ 6

Strain + I

0 0.0024 0.1326

+ 6 0.0056 0.3093

6 I + 0.0178 0.9834 + I

6 C)

~ '""' '-../

0.00 75.~J

~. ::;tress. K 5 i

( b) uecreasin':J strain

6

6 + +

G o.::::J

0 ~+-~-":::'1"-~~~~~~..:-~~~~~~~~--1 75.C:J -75.C:J

1r, stress, ksi ( c) Crdck open in-:; ai sri I aceH1ent as a runct ion or stress at airterent f!Olnts alun':J tne crao. len::ith d'Jrin-:; increasin'::l ~nu aecreasin-:; strctin.

-

c:: <li ~

0

:::;> c:: c:: <li :i.. 0

148

~....-----___ _ 0

la) lncreasin~ Strain 0

A

+

+

~ ,.~ C) ci~E:S~~~~~~_,_~~~~~~~~~--'

-0.00250 0.CCCC8 o.c:~5C

0

c, strain

(b) Uecreasin~ strain

6 + +

6 T

0 §I ~ o..._~~'='-'--~~~~~-+-.;._~-'-~~~~~~~__J

-0. 00250 0. :::cJ o. :~25J

c, strain

x (in) x/2a

0.0024 0.1326

0.0056 0.3093

0.0178 0.9834

(a) CrdcK Of)enin':i aisf.llacement as a function ot strain at airterent f)Oints alon~ the crack len~tn aurin~ increasin~ ana decreasin~ strctin.

.. g c -i.. '-c:

...., c: ~ ii u ·~

:i.. II')

'O

:n = c: lJ .:i.. 0

.:..! u ~ ~

:..>

r-.. ,-,.... v c ')(. '-

= ...., c: ~ 1i u "' :i.. IJ')

'O

::;') c: c lJ ~

:)

.:..! :.J 'Q ~

:..>

::i '::l :..>

Fig.

~

:1 0

rt! ~

0 .N 0

0

0

8 0 -100.00

::i Jl ~

0 ~ 0

r· 0 CTl

0

0 N

0

0

0

8 0 -0.15

39:

,'

STRAIN

149

o.o

I

/ /

I

oo

/'

.10·1

')(.,distance as measured from crack tip, in

, I

€9 0 .00133

0.00606

100.00

0.15

Loading and unloading paths in a typical and E plots.

vs J,

.., I =>

>(

c:::

...., c::: ~ :::; <.)

" :i.. .,., ·~

~

~ c::: ·~

= '1J :i.. 0

~ <.)

" '-u

~ => '._)

150

0 lf) . 0

COD max

0 "'1"

0

0 en 0

0 N . 0

0

0

--+ I I

' 'I '' ii ll i : I ! I ;

/i ,.·i/

/;~/ ,~ i //

. __ ----1-,..._-f_:_ f- _,.- o.oscoo...,.. ....

I ~~--, '11

o 0.05 C~\ I_~ ~_rl 00. O~~o- 0 _?_-, _____ .,_o....:.."....:..···....:..,. _____ -;l 00. 00

Fig. 40:

er , Stress, i<Sl

4b,jli kSl

~0.05 COO max scatter

at each point (assumed)

Typical measurement of the crack opening stress level ( (J ) • op

0 0 .

151

O..._~~~~..,-~~~~.--~~~--.~~~~---r-~~~~..., I

0 N . 0

I

0 "l:t'

0 I

0 Q)

0 I

0 0

(!) ER=0.0024

t.... :: o . oo1A A ER=0.0042

~ + ER=0.0066

x ER=0.0125

~ = 0.0066

+x

--+-~~~~+-~~~~i--~~~---r~~~~-t-~~~~~

10.00 0. 10 0.20 0.30 0.40 0.50

a, crack len':Jtll, in x iu-1

Fig. 41: Crack opening stress level as a function of crack length

x '.g

w-..._ ~

0 w

0 0 .

152

9-r-~~~~.-~~~~-,-~~~~-r-~~~~--~~~~~

0 (\J . 0

I

. 0

I

0 U) . 0

I

0 (l) . 0

I

0 0 .

1d.oo

Fig. 42:

.!a. +

+

x

ER=0.0024

ER=0.0042

ER=0.0066

ER=0.0125

A c: : 0.004'2. --- '-o .......__ ~-=---_.,...;..!.=- + ~ = 0.0066 ~+ -

xx .!a.+

x

x E... :: 0. 0 7 25

I

0. 10 0.20 0.30 0.40 a, cracK len"'tt1, in x lu- 1

0.50

Crack opening strain level as a function of crack length

0 0 . 0

I

0 N . 0

I

0 """ . 0

I

0 CD 0

I

0 0 .

I

x x

I .E-2

Fig. 43:

+ + + ~

+

153

I . E-1

+

x

(!) A

ER=0.0125

ER=0.0066

ER=0.0042

ER=0.0024

Crack opening stress level as a function of ~J

I .E-0

154

0·8

0·6

0·4

0·2

-0·2

-0·4

-0·6

-0·8

-1·0 ~-.i___.i___.i___.1.-_.1.-_.1.-_.1.-_....___..._____, -1·0 -0·8 -0·6 -0·4 -0·2 O·O 0·2 0·4 0·6 0·8 l·O

R

L=0·04 in L=0·.004 .in- NAKAI (Kmax/0-y =0·375./lnl

-·-· -·-·-NEWMAN (plane st rain)

-------ELSER

Fig. 44: Crack opening stress level as a funct~on of R

o; stress

0-min O-c1os3 =(-0-max)

Fig. 45:

155

---ideal crack (I)

- - - - real crack, lo Ep (2)

-·-real crack, hi Ep ( 3)

Schematic diagram of the real crack behavior at high strains and corresponding ideal crack behavior.

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