NATL INST. OF STAND & TECH R.I.C.

A 11 10 3 453553

United States Department of Commerce

National Institute of Standards and Technology

PUBLICATIONS

NISTIR 3952

STUDY OF STATIC

AND DYNAMIC FRACTURE

USING STRAIN MEASUREMENTS

John R. Berger

James W. Dally
Research Mormatm Center
NISTIR 3952

STUDY OF STATIC

AND DYNAMIC FRACTURE

USING STRAIN MEASUREMENTS

John R. Berger

James W. Dally

Materials Reliability Division

Materials Science and Engineering Laboratory

National Institute of Standards and Technology

Boulder, Colorado 80303-3328

September 1990

U.S. DEPARTMENT OF COMMERCE, Robert A. Mosbacher, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY, John W. Lyons, Director

CONTENTS

CHAPTER 1 INTRODUCTION 1

1.1 THE PROPAGATION TOUGHNESS-CRACK SPEED CONSTITUTIVE

RELATION AND CRACK ARREST 2

1.2 OVERVIEW OF THE REPORT 6

REFERENCES 8

CHAPTER 2 STATIC ANALYSIS 14

2. 1 STATIC STRAIN FIELD REPRESENTATION 14

2. 1. 1 SINGLE-ELEMENT STRAIN GAGE 18

2. 1.2 STRAIN GAGE ROSETTE 18

2.2 OVERDETERMINED ANALYSIS. 19

2.3 EXPERIMENTAL VERIFICATION 21

REFERENCES 24

CHAPTER 3 DYNAMIC ANALYSIS 36

3.1 DYNAMIC STRAIN FIELD REPRESENTATION 36

3.1.1 SINGLE-ELEMENT STRAIN GAGE 42

3.1.2 STRAIN GAGE ROSETTE 44

3.2 TRIANGULATION AND ITERATION ALGORITHM 46

3.3 SPATIALLY OVERDETERMINED ANALYSIS 49

3.4 EXPERIMENTAL VERIFICATION 52

REFERENCES 57

iii
CHAPTER 4 ACCURACY OF THE PARAMETER DETERMINATIONS 84

4.1 LIMITS ON THE VALIDITY OF THE THREE-PARAMETER

MODEL 85

4.2 DETERMINATION OF CRACK TIP POSITION USING

STRAIN GAGE ROSETTES. . 89

4.3 ACCURACY OF THE DETERMINATIONS 93

REFERENCES 96

CHAPTER 5 CRACK ARREST AND THE PROPAGATION TOUGHNESS - CRACK

SPEED RELATION FOR A REACTOR-GRADE PRESSURE VESSEL

STEEL 115

5. 1 WI DE-PLATE SPECIMENS. 116

5.2 TEST RESULTS AND STRAIN ANALYSIS FOR A LOW-UPPERSHELF STEEL. 117

REFERENCES 128

CHAPTER 6 CONCLUSIONS AND DISCUSSION 153

6. 1 SUGGESTIONS FOR FUTURE WORK 154

6. 2 ACKNOWLEDGEMENTS 155

APPENDIX A THE LINEAR LEAST SQUARES PROBLEM 156

A. 1 THEORY OF LINEAR LEAST SQUARES 157

A. 2 THE QR DECOMPOSITION 159

A. 3 ELEMENTS OF THE MATRICES 164

REFERENCES 165

i v
APPENDIX B THE NONLINEAR LEAST SQUARES PROBLEM 167

B. 1 THEORY OF NONLINEAR LEAST SQUARES 168

B. 2 THE NEWTON-RAPHSON ITERATIVE PROCEDURE 169

B. 3 ELEMENTS OF THE MATRICES 170

REFERENCES 175

v
LIST OF TABLES

2.1 Results for with 29 strain gage readings 28

2.2 Results for with 48 data points. 29

2.3 Comparison of higher order coefficients (48 data points).. 30

3. 1 Results from a dynamic analysis of data from gages 2

and 3 61

3.2 Average results from a dynamic analysis of data for all

adjacent gage pairs 62

3.3 Average results from a static analysis of data for all

adjacent gage pairs 63

4. 1 Non-singular coefficients determined from the local

collocation of the finite element data 98

4.2 Synthetic strain data generated for a = 0° and a = 45°. . . . 99

4.3 Sign of the errors applied to the synthetic data to

study error propagation 100

4.4 Summary of the error propagation study 101

5.1 Summary of the mechanical properties of the WP-2.6

material at room temperature 131

5.2 Gages available for use in the strain analysis of

WP-2.6 with changing crack length 132

5.3 Comparison of the arrest conditions for WP-2.6 133
LIST OF FIGURES

1.1 General relationship between propagation toughness

and crack velocity 13

2.1 Definition of the rotated coordinate system 31

2.2 Single crack tip, infinite body fracture problem 32

2.3 Strip gage locations for the A1 6061-T6 Compact

Specimen 33

2.4 Geometry of the A1 6061-T6 Compact Specimen 34

2.5 Experimentally recorded radial strain distributions

for the A1 6061-T6 Compact Specimen 35

3. 1 Definition of the coordinate systems used for the

development of the dynamic strain field equations 64

3.2 Definition of the rotated coordinate system 65

3.3 Normalized strain response for a single element

strain gage (a = 118.7, y

= 10.5 mm, c = 656 m/s) 66

9

3.4 Normalized strain response for a single element

strain gage (a - 61.3, y

= 10.5 mm, c = 656 m/s) 67

9

3.5 Normalized strain response for a rosette strain

gage (a = 45.0, y =10.5 mm, c = 656 m/s) 68

9

3.6 Normalized strain response for a rosette strain

gage for y =10, 50, and 90 mm (a = 45, c = 656 m/s) 69

9

3.7 Maximum gage height for existence of a zero crossing for

a single element gage 70

3.8 Maximum gage height for existence of a zero crossing for

a rosette 71

3.9 Strain gage positions relative to a propagating crack 72

3. 10 Angle 0 as a function of c/c 73

o R

vii
3.11 Angle 0 as a function of y 74

0 g

3. 12 Angle 0 as a function of A /A 75

0 10

3.13 Crack-tip position relative to a gage pair at c = 0 76

a

3.14 Flowchart for the triangulation and iteration algor ithm. .. 77

3. 15 Spatial distribution of strain gages above a propagating

crack 78

3.16 Geometry of the 4340 steel compact specimen 79

3.17 Strain-time records for the compact specimen 80

3. 18 Relative position of the crack as a function of time

using the triangulation and iteration algorithm 81

3. 19 Propagation toughness as a function of time from the

spatially overdetermined analysis 82

3.20 Crack position as a function of time from the

spatially overdetermined analysis 83

4. 1 Final geometry of the finite element mesh (only the

top half of the specimen is shown) 102

4.2 Finite element stresses along the crack line 103

4.3 Convergence of the local collocation of the finite

element stresses 104

4.4 Comparison of finite element stresses with stresses

computed with a twelve parameter model. The data

points are finite element values, the solid lines

are the twelve parameter series values 105

4.5 Valid zones for the rosette strain gage using a

two percent error 106

4.6 Valid zones for the rosette strain gage using a

five percent error 107

4.7 Valid zones for the rosette strain gage using a

ten percent error 108

vi i i
4.8 Valid zones for the single element strain gage using a

two percent error 109

4.9 Valid zones for the single element strain gage using a

five percent error 110

4.10 Valid zones for the single element strain gage using a

ten percent error Ill

4.11 Gage positions for generating synthetic strain data 112

4. 12 Least squares solution space for the a = 0° rosette.

Contour labels are in percent 113

4. 13 Least squares solution space for the a = 45° rosette.

Contour labels are in percent 114

5. 1 Crack ?rrest toughness as a function of temperature

for a nuclear pressure vessel steel (from [5.3]) 134

5.2 Geometry of the wide plate specimen (from [5.3]) 135

5.3 Schematic representation of the propagation toughness,

crack velocity, temperature surface 136

5.4 Geometry of the WP-2.6 specimen (from [5.4]) 137

5.5 Typical strain-time records from WP-2.6 138

5.6 Temperature gradient for WP-2.6 139

5.7 Crack position as a function of time for WP-2.6 using

the static representation of the strain field 140

5.8 Propagation toughness as a function of time for WP-2.6

using the static representation of the strain field 141

5.9 Propagation toughness as a function of position for

WP-2.6 using the static representation of the

strain field 142

5.10 Crack velocity as a function of time for WP-2.6 143

5.11 Fracture surface of WP-2.6 showing the first arrest

location (a = 45° rosettes are on the lower side of

the specimen 144

ix
5.12 Crack position as a function of time for WP-2.6 using

both static and dynamic representations of the strain

field 145

5. 13 Propagation toughness as a function of time for

WP-2.6 using the dynamic representation of the

strain field 146

5. 14 Propagation toughness as a function of temperature for

WP-2.6 using the dynamic representation of the

strain field 147

5. 15 Propagation toughness as a function of position for

WP-2.6 using the dynamic representation of the

strain field 148

5.16 Propagation toughness computed from both the static

and dynamic strain field representations as a function

of crack velocity 149

5. 17 Crack acceleration as a function of time 150

5. 18 Propagation toughness-crack velocity relation

for WP-2.6 151

5. 19 Propagation toughness-crack velocity-temperature

contour for WP-2.6 152

A. 1 Graphical interpretation of the linear least squares

problem 166

x
ABSTRACT

The analysis of strain fields surrounding both stationary and

propagating cracks is presented. Series expansions of the static

and dynamic strain fields are developed. Gage orientation angles

are then studied to optimize the strain response. The orientation

angles are found to be dependent on gage type and material.

Algorithms are developed which use the temporal or spatial

strain variations to extract fracture parameters . The accuracy of

the parameter determinations is shown to be excellent, and limits

are placed on the validity of the developed methods. The methods

are then applied to the analysis of a large-scale crack arrest

test conducted in a pressure vessel steel. The behavior of the

crack-tip position with time and the propagation toughness with

time, temperature and position are determined. From this

information^ details of the conditions at crack arrest are

extracted. The propagation toughness-crack-velocity relation is

then constructed.

Key Words: crack arrest; dynamic fracture; fracture mechanics;

propagation toughness; strain gages; stress intensity
'

.
CHAPTER 1

INTRODUCTION

Characterizing fracture in engineering materials requires

measuring stress field quantities relevant to the type and mode of

fracture and comparing them with the appropriate material property

that describes the fracture resistance. In terms of driving

force, which is the approach taken here, the characterization can

be accomplished in terms of the stress intensity factor, K, for

fracture that is primarily cleavage (or satisfies small-scale

yielding criteria) or in terms of Rice’s J-integral for ductile

hole joining fractures. These quantities depend not only on the

mode of fracture, that is, opening or forward shear modes for

planar problems, but also on the extent of dynamic effects in the

body.

Previous experimental approaches to the general problem of

determining K or J in opaque materials have emphasized optical

methods including reflection photoelasticity [1. 1-1.3], caustics

[1.4-1. 6], and moire interferometry [1.7-1. 9]. Additionally,

methods using acoustic emission [1.10] and acoustic birefringence

[1.11] have been employed. Most of these methods are full field

measurements and are experimentally complex. The method of strain

gages has started to receive attention only in recent years

[1.3,1.12-1.18], although their use to determine K was suggested

by Irwin more than thirty years ago [1.19].

Strain gage methods are extremely attractive in terms of

experimental simplicity. However, since a measurement is made

only at a material point specified by the gage position and not

over an entire region, the strain field surrounding the crack tip

must be well understood in order to position the gage properly for

optimum response. Therefore, a necessary precursor to studying

fracture behavior using strain gages is to investigate the strain

field surrounding the crack tip.

1
In this report we first establish methodologies for analyzing

the response of both single and multi-element strain gages near

either a stationary or moving crack tip. Proof-of-concept

investigations will be described for each of the developed

analysis procedures. The developed methodologies will then be

applied to two current topics in the mechanics of fracture: (1)

determining crack arrest toughness and (2) the propagation

toughness-crack speed constitutive relation in a nuclear pressure

vessel steel.

1.1 THE PROPAGATION TOUGHNESS - CRACK SPEED CONSTITUTIVE RELATION

AND CRACK ARREST

The general relationship between propagation toughness, K

and crack speed, c, at constant temperature is illustrated in

figure 1.1. Provided that linear elastic conditions prevail, that

is, the one-parameter characterization of the stress intensity

factor holds, the propagation toughness-crack speed relationship

can be considered a constitutive law in the sense that it

completely characterizes the dynamic fracture behavior of a

material

.

Based on extensive experimental data [1.20, 1.21] the K -c

relation is unique for a given material at a given temperature and

is therefore independent of specimen size and geometry. Some

evidence of nonuniqueness in K -c relations has been presented in

the literature [1.22-1.24]. However, in [1.22] extensive

nonelastic deformation occurred during crack growth in the

material being used, which invalidates the one-parameter

characterization of linear elastic fracture mechanics. The

experimental method of caustics employed in [1.23] and [1.24] has

unresolved geometric effects which raises questions concerning the

conclusions of this particular study [1.25].

Referring to figure 1.1, in region I on the K -c curve, the

2
crack is propagating rapidly and large increases in driving force

increase crack speed only a small amount. A terminal crack speed,

c , is indicated in figure 1.1; above c^ the amount of energy

going into the fracture process is so large that the crack

branches in order to consume energy in the creation of multiple

fracture surfaces. Although branching behavior is of interest in

fragmentation and blasting studies, it is not of immediate

interest for structural steels and will not be studied in this

investigation.

In region II there is a transition in which increasing the

driving force brings about a corresponding increase in crack

speed. Finally, in region III small changes in driving force

bring about very large changes in crack speed. The arrest

toughness of the material can be found in region III where c = 0.

It is in this context that crack arrest is viewed here; as a crack

approaches arrest, a point on the K -c curve is tracked from

region I or 1 1 into region III and finally to arrest where c = 0.

The approach taken here toward crack arrest measurement falls

between the fully dynamic approach advanced by Hahn and his

co-workers at Battel le Columbus Laboratories (BCL) [1.26] and the

static approach advanced by Crosley and Ripling at Materials

Research Laboratory (MRL) [1.27]. The BCL approach accounts for

the dynamics of the specimen through the use of reference curves

generated from a finite-difference computer code for standard

specimen geometries. The reference curves enable us to determine

K from the stress intensity at initiation, K , and the crack

length at arrest. However, the crack-speed dependence of K

jd

was

not accounted for in the generation of the reference curves.

The MRL procedure for determining K uses the crack opening

I a displacement (COD) measured approximately 1 ms after arrest and

the final crack arrest length to predict from an elasto-stat ic

equation. This procedure therefore assumes that the value of K

determined 1 ms after arrest is not significantly different from

3
the value of K at arrest. Furthermore, the response of the COD

may lag the events at the crack-tip by times exceeding the 1 ms

interval [1.21]. The current ASTM standard for determining crack

arrest toughness in ferritic materials is based on this static

approach.

In either the BCL or MRL procedure for determining arrest

toughness, measurements of the final crack length are combined

with a "remote" variable (either a relatively low-frequency

measurement of COD or a value of K ) to determine K . However,

Iq la

in the approach developed here, K is determined at the moment of

la

arrest from strains in the immediate vicinity of the arresting

crack tip. This provides a more meaningful insight into the field

surrounding the arresting crack. Although a dynamic (crack speed

dependent) strain field is employed in portions of this study, the

contribution of kinetic energy to the dynamic stress intensity is

not accounted for as it is in a fully dynamic analysis. However,

the manifestation of the energy return in terms of the resulting

perturbation of the local strain field is used.

For brittle materials, crack propagation occurs by cleavage,

and little plasticity develops near the propagating or arresting

crack tip. Under these conditions of small scale plasticity, the

linear elastic approach is entirely valid, and the strain gage

methodologies developed in this dissertation can be applied

directly to develop the K -c relation.

The characterization of propagation toughness as a function

of crack speed for a ductile, high toughness steel can be

accomplished, to some extent, by the same methods used for the

brittle material characterization. However, for a ductile

material the problem is more complex due to the transition from

cleavage to fibrous fracture modes in a run-arrest event, as well

as mode change from fibrous to cleavage in initiation and

post-arrest conditions. No attempt will be made here to

characterize the ductile tearing portion of the fracture event

4
because the need to characterize the cleavage behavior is more

critical

.

As described in [1.28] and [1.29], cleavage fracture occurs

in a ductile steel when the normal stress some distance ahead of

the ductile, blunted crack tip reaches a critical value over a

potential initiation site. For the pressure vessel steel analyzed

in Chapter 5, the initiation site is usually at an inclusion or at

a cluster of inclusions [1.28]. Once cleavage initiates, the

crack runs at high speed with little plasticity, primarily due to

4 -

1

the large strain rates ( 10 s or more [1.30, 1.31]) near the

crack tip and the associated increase in yield stress. For this

portion of the fracture, linear elastic conditions prevail and the

fracture analysis methodologies using strain gages may be directly

applied.

As the crack slows to arrest, the strain rate near the crack

tip decreases, the ability of the material to flow increases, and

the plastic zone increases in size. At some point the plastic

zone and the stresses and strains on its boundary are no longer

fully described by the stress intensity factor—the fracture can

no longer be analyzed by a linear elastic model. The question of

whether the linear theory fails before or after the arrest event

has recently been examined [1.32,1.33]. These studies concluded

that cleavage conditions exist up to and including arrest. This

result justifies the application of the methods developed in this

dissertation to the analysis of cleavage crack run-arrest events

in ductile steels. Once arrest occurs, the plastic zone size

increases and, depending on testing machine compliance and the

kinetic energy available in the specimen, the crack tip blunts and

ductile tearing begins. Cleavage initiation may again occur after

this point depending on the availability of suitable initiation

sites.

5
1.2 OVERVIEW OF THE REPORT

In the chapters to follow we develop the tools necessary to

analyze the strain field measured in the vicinity of a stationary

or moving crack tip in order to extract fracture parameters.

Since the stationary crack problem is considerably simpler, we

concentrate on it first. Equations suitable for modeling the

response of both single element strain gages and two-element

strain gage rosettes are developed. An overdetermined analysis

scheme for analyzing the strains in order to determine values

is then presented. The static methodology has potential use in

the analysis of complex geometries such as structural components,

where suitable expressions for the stress intensity factor are not

available. Although finite element analysis can be employed to

obtain this information, it is usually desirable to obtain some

measure of experimental validation before proceeding with a

fracture mechanics analysis.

The analysis of strains near a moving crack is presented

next. We develop expressions for both single element and rosette

strain gages using the strain field representation for a

propagating crack. The orientation of the gage with respect to

the crack path is studied in order to optimize the response in

terms of the analysis. Two analysis schemes from which both the

strain field parameters and crack-tip position are obtained are

presented. Which algorithm to employ depends on the velocity

gradient of the crack tip.

Due to the increased complexity of the dynamic problem, a

separate chapter is devoted to several issues related to the

strain analysis. The first issue concerns the extent of validity

of the strain representation in terms of the distance of the gage

from the crack tip. Using the results of this study, we specify

limits on which gages to employ in the strain analysis.

The second issue concerns the determination of the crack tip

6
position when strain gage rosettes are used. There are data which

were recorded during dynamic fracture experiments where rosettes

which were not oriented to optimize the strain response were used.

We study the difficulty in extracting crack tip position from

these records as opposed to extracting the information from gages

whose orientation is optimized. Finally, the accuracy of the

crack tip determination is discussed.

The analysis of the dynamic crack problem is undertaken to

determine arrest toughnesses as well as propagation

toughness-crack speed relations. As discussed above, the arrest

toughness determined by such a procedure is accomplished using

information (strains) local to the arresting crack. This approach

is certainly more attractive than inferring an arrest toughness

from a more remote measurement. The methods developed for dynamic

analysis are used to analyze the dynamic fracturing of a

large-scale, wide-plate test of a nuclear pressure vessel steel.

For this material, the propagation toughness-crack speed relation

as well as the arrest toughness is determined.

7
REFERENCES

[1.1] Kobayashi , T. , and Dally, J.W. , "Dynamic Photoelastic

Determination of the c-K Relation for 4340 Alloy Steel," Crack

Arrest Methodology and Applications, ASTM STP 711, G.T. Hahn and

M.F. Kanninen, Eds., American Society of Testing and Materials,

1980, pp. 189-210.

[1.2] Der, V. K.

,

Barker, D. B.

,

and Holloway, D.C. ,"A Split

Birefringent Coating Technique to Determine Dynamic Stress

Intensity Factors," Mechanics Research Communications, Vol. 5,

No. 6, pp. 313-318 (1978).

[1.3] Shukla, A, Agarwal , R. K.

,

and Nigam, H. , "Dynamic Fracture

Studies on 7075-T6 Aluminum and 4340 Steel Using Strain Gages and

Photoelastic Coatings," Engineering Fracture Mechanics, Vol. 31,

No. 3, pp. 501-515 (1989).

[1.4] Shockey, D. , Kalthoff, J.F.

,

Klemm, W. , and Winkler, S. , "Simultaneous Measurements of Stress Intensity for Fast Running

Cracks in Steel," Experimental Mechanics, Vol. 23, No. 2., pp.

140-145 (1983).

[1.5] Ravi-Chandar , K. , and Knauss, W. G.

,

"An Experimental

Investigation into Dynamic Fracture I: Crack Initiation and

Arrest," International Journal of Fracture, Vol. 25, pp. 247-262

(1984).

[1.6] Rosakis, A.J., "Analysis of the Optical Method of Caustics

for Dynamic Crack Propagation," Engineering Fracture Mechanics,

Vol. 13, No. 2, pp. 331-347 (1980).

8
[1.7] Epstein, J.S.

,

Deason, V. A.

,

and Reuter, W. G.

,

"Dynamic

Moire Interferometry Studies of Stress Wave/Crack Tip Diffraction

Events in 1018 Steel," Fracture Mechanics: Nineteenth Symposium,

ASTM STP 969, T.A. Cruse, Ed., American Society for Testing and

Materials, pp. 482-503 (1988).

[1.8] Kang, B. S. and Kobayashi, A. S.

,

"J Resistance Curves in

Aluminum SEN Specimens Using Moire’ Interferometry," Experimental

Mechanics, Vol. 28, No. 2, pp. 154-158 (1988).

[1.9] Barker, D.B.

,

Sanford, R.J. and Chona, R. , "Determining K

and Related Stress-Field Parameters from Displacement Fields,

Experimental Mechanics, Vol. 25, No. 4, pp. 399-407 (1985).

[1.10] Dunegan, H.L. , Harris, D.O.

,

and Tatro, C.A.

,

"Fracture

Analysis by Use of Acoustic Emission, " Engineering Fracture

Mechanics, Vol. 1, No. 1, pp. 105-122 (1968).

[1.11] Clark, A. V. , Mignogna, R.B.

,

and Sanford, R.J.,

"Acousto-elastic Measurement of Stress and Stress Intensity

Factors Around Crack Tips, " Ultrasonics, March 1983, pp. 57-64.

[1.12] Dally, J.W. and Sanford, R. J.

,

"Strain Gage Methods for

Measuring the Opening Mode Stress Intensity Factor, K ,"

Experimental Mechanics, Vol. 27, No. 4, pp. 381-388 (1988).

[1.13] Berger, J.R. and Dally, J.W.

,

"An Overdeterminist ic

Approach for Measuring Using Strain Gages, " Experimental

Mechanics, Vol. 28, No. 2, pp. 142-145 (1988).

[1.14] Dally, J.W. and Berger, J.R.

,

"Determining K^ and K

^ ^ in a

Mixed Mode Stress Field Using Strain Gages, " Proceedings 1986 SEM

Spring Conference on Experimental Mechanics, pp. 603-612, New

9
Orleans, LA.

[1.15] Dally, J.W. and Sanford, R. J. , "On Measuring the

Instantaneous Stress Intensity Factor for Propagating Cracks,

Proceedings, International Conference on Fracture Mechanics VII,

Houston, TX (1989).

[1.16] Dally, J. W. , Sanford, R. J. , and Berger, J.R. , "An Improved

Strain Gage Method for Measuring K(t) for a Propagating Crack,"

Journal of Strain Analysis for Engineering Design, Vol. 25, No.

3 (1990).

[1.17] Berger, J.R. , Daily, J.W. and Sanford, R.J., "Determining

the Dynamic Stress Intensity Factor with Strain Gages Using a

Crack Tip Locating Algorithm, " Engineering Fracture Mechanics,

Vol. 36, No. 1, pp. 145-156 (1990).

[1.18] Read, D.T., "Analysis of Strains Measured During a Wide

Plate Crack Arrest Test, " Proceedings of the 20th Session of the

German Society for Materials Testing, Working Group on Fracture,

Frankfurt, Federal Republic of Germany (1988).

[1.19] Irwin, G. R. , "Analysis of Stresses and Strains Near the

End of a Crack Traversing a Plate, " ASME Journal of Applied

Mechanics, Vol. 24, No. 3, pp. 361-364 (1957).

[1.20] Fourney, W. L. , Chona, R. , and Sanford, R.J., "Dynamic

Crack Growth in Polymers, " Workshop on Dynamic Fracture,

California Institute of Technology, pp. 75-99 (1983).

[1.21] Irwin, G. R. , Kobayashi , T. , Fourney, W. L.

,

Metcalf, J.

T. , and Dally, J. W. , "Photoelastic Studies of Crack Propagation

10
and Arrest in Polymers and 4340 Steel," U.S. NRC Report,

NUREG/CR-0542 , University of Maryland (1978).

[1.22] Dahlberg, L. , Nilsson, F. , and Brickstad, B. , "Influence

of Specimen Geometry on Crack Propagation and Arrest Toughness,

Crack Arrest Methodology and Applications, ASTM STP 711, G.T. Hahn

and M. F. Kanninen, Eds., American Society for Testing and

Materials, pp. 211-227 (1980).

[1.23] Knauss, W. G. and Ravi-Chandar , K. , "Fundamental

Considerations in Dynamic Fracture," Engineering Fracture

Mechanics, Vol. 23, No. 1, pp. 9-20 (1986).

[1.24] Takahashi, K. and Arakawa, K. , "Dependence of Crack

Acceleration on the Dynamic Stress Intensity Factor in Polymers,"

Experimental Mechanics, Vol. 27, No. 2, pp. 195-200 (1987).'

[1.25] Kobayashi, T. , discussion of reference [1.24],

Experimental Mechanics, Vol. 27, No. 2, p. 200 (1987).

[1.26] Hahn, G.T., et. al . , "A Cooperative Program for Evaluating

Crack Arrest Testing Methods, " Crack Arrest Methodology and

Applications, ASTM STP 711, G.T. Hahn and M. F. Kanninen, Eds.,

American Society for Testing and Materials, 248-269 (1980).

[1.27] Crosley, P.B. and Ripling, E.J., "Comparison of Crack

Arrest Methodologies," Crack Arrest Methodology and Applications,

ASTM STP 711, G.T. Hahn and M. F. Kanninen, Eds., American Society

for Testing and Materials, pp. 211-227 (1980).

[1.28] Heerens, J. and Read, D.T., "Fracture Behavior of a

Pressure Vessel Steel in the Duct i le-to-Britt le Transition

Region, " National Institute of Standards and Technology Internal

11
Report, NISTIR 88-3099, Boulder, CO, (1988).

[1.29] Ritchie, R. 0. , Knott, J.F. , and Rice, J.R. , "On the

Relationship Between Critical Tensile Stress and Fracture

Toughness in Mild Steels," Journal of the Mechanics and Physics of

Solids, Vol . 21, pp. 395-410 (1973).

[1.30] Freund, L. B. and Hutchinson, J.W.

,

"High Strain Rate Crack

Growth in Rate Dependent Plastic Solids," Journal of the Mechanics

and Physics of Solids, Vol. 33, No. 2, pp. 169-191 (1985).

[1.31] Freund, L. B.

,

Hutchinson, J.W.

,

and Lam, P. S.

,

"Analysis

of High Strain Rate Elastic Plastic Crack Growth," Engineering

Fracture Mechanics, Vol. 23, No. 1, pp. 119-129 (1986).

[1.32] Naus, D.J., et. al . , "Crack Arrest Behavior in SEN Wide

Plates of Quenched and Tempered A533 Grade B Steel Tested Under

Non-Isothermal Conditions," U.S. NRC Report, NUREG/CR-4930, Oak

Ridge National Laboratory (1987).

[1.33] Fields, R.J. and deWit, R. , "Plastic Zone Formation Around

an Arresting Crack," International Journal of Fracture, Vol. 42,

pp. 231-238 (1990).

12
PROPAGATION TOUGHNESS

Fig. 1.1 General relationship between propagation toughness and

crack velocity.

13
CHAPTER 2

STATIC ANALYSIS

In this section we consider the problem of analyzing the

strain field surrounding a stationary crack subjected to

time- independent loading. The strain field will be modeled using

the generalized Westergaard series stress functions appropriate

for single crack tips subjected to remote loading in opening mode

(Mode I). The problem is formulated as a linear least squares

problem in the unknown series coefficients which yields the stress

intensity factor upon solution.

2.1 STATIC STRAIN FIELD REPRESENTATION

We begin the analysis of the static mode I problem by

formulating the equations describing the strain sensed by a gage

oriented at an angle, a, situated at a material point P as

originally examined in [2.1], figure 2.1. In previous static

analyses [2. 2-2. 4] the Westergaard method of analysis proved to be

quite useful and convenient for representing the static stress

field; it is used here as well.

As first demonstrated by Irwin [2.5], modifications to the

original two-dimensional Westergaard equations are necessary to

completely model two-dimensional, finite body, opening-mode crack

problems. By re-examining the work of Sih [2.6] and Eft is and

Liebowitz [2.7], Sanford [2.8] has shown that the most general

form of the Westergaard equations for the finite body problem

follows from an Airy stress function of the form

<f> = Re Z(z) + y Im Z(z) + y Im Y(z) ( 2 . 1 )

where

d d

2 Z(z) Z(z)

2 Z(z), (2.2)

dz dz

14
d

Y(z)

,

Y(z) = (2.3)

dz

and

z = x + i y. (2.4)

The function Z(z) is one of a family of Westergaard stress

functions that have the required property

Re Z(z) = 0 (2.5)

on the traction-free portions of the crack faces. The choice of

Z(z) depends on the geometry and type of loading under

consideration. Sanford and Berger [2.9] have described a

methodology for choosing appropriate forms for Z(z) for the finite

body problem.

The function Y(z) is a second complex-valued function which

has the required property

Im Y(z) = 0 on y

= 0. (2.6)

This function is representative of the generalization of the

Westergaard equations due to Sanford, since the most general

solution to eq (2.6) is not a real constant as was selected in

previous analyses [2. 6, 2. 7]. Y(z) can be represented generally as

a power series in z,

00 Y(z) = Z B z

m

; Im B = 0 (2.7)

m m

m = 0 since, on y

= 0, Y(z) reduces to a power series in the real

variable, x. For infinite body problems only the constant leading

term B is admissible for Y(z) to remain bounded at infinity.

0

The Cartesian stress components referred to a coordinate

system situated at the crack tip are given in terms of Z(z) and

Y ( z ) as

15
cr = Re Z(z) - y Im Z’ (z) - y Im Y’ ( z ) + 2 Re Y(z) (2.8)

xx

cr = Re Z(z) + y Im Z’ (z) + y Im Y’ (z) (2.9)

yy

cr = - y Re Z’ ( z ) - y Re Y’(z) - Im Y(z), (2.10)

xy

where the primes indicate derivatives with respect to the complex

variable, z. Substituting eqs (2.8)-(2.10) in Hooke’s law yields

the strain field equations,

Eg = (1 - v) Re Z(z) - (1 + i>) y Im Z’ (z) -

XX

(1 + v) y Im Y’(z) + 2 Re Y(z), (2.11)

Eg = (1 - v) Re Z(z) + (1 + r) y Im Z’ (z) +

yy

(1 + v) y Im Y’(z) - 2 v Re Y(z), (2.12)

jly = - y Re Z’ (z) - y Re Y’ (z) - Im Y(z), (2.13)

xy

where E is Young’s modulus and p is the shear modulus.

The strain in the rotated coordinate system (x’,y’) shown in

figure 2. 1 can be found through the complex form of the strain

transformation equation,

g - g + i y

y'y' x'x' x'y'

and the first strain invariant,

G + G = C + G . y/y/ X'X' yy XX

Substituting eqs (2. 11) — (2. 13)

strain field is

e

2l<X (e -c + i? ), (2.14)

yy xx xy

(2. 15)

in eqs (2.14) and (2.15), the

il-v)

{ Re Z(z) + Re Y(z)> - {y Im Z’(z)

(1+r)

+ y Im Y’ (z) - Re Y(z)} cos (2a) - {y Re Z’ (z)

+ y Re Y’ (z) + Im Y(z)> sin (2a), (2.16)

16
n-v)

{ Re Z(z) + Re Y(z)> + {y Im Z’ (z)

(1+v)

+ y Im Y’ (z) - Re Y(z)> cos (2a) + {y Re Z’ (z)

+ y Re Y’ (z) + Im Y(z)> sin (2a), (2. 17)

nr = { y Im Z’ (z) + y Im Y* (z) - Re Y(z) > sin (2a)

x'y'

+ { - y Re Z’ (z) - y Re Y’ (z)

- Im Y(z) > cos (2a)

.

(2. 18)

Before developing the strain equations further a selection

must be made for the form of the Westergaard function, Z(z). For

purposes of this investigation the problem involves a single crack

tip subjected to an arbitrary time- independent remote stress with

no point forces acting on the crack faces or anywhere inside the

body, figure 2.2. A suitable form for Z(z) for the finite body

problem is

The series representations of eqs (2.7) and (2.19) involve an

infinite number of terms to completely solve the finite body

problem. However, if a small error is acceptable, we can truncate

the series at a finite number of terms. In boundary collocation

studies [2.10-2.12], the number of terms is large, since

information is obtained only at the boundary. For experimental

analysis the number of terms that must be retained is relatively

small, since experimental data are obtained in the local region

surrounding the crack tip. The method of obtaining the singular

and nonsingular parameters from experimental near tip data has

been termed "local collocation" [2.13-2.15] due to its

00 Z(z) = Z A z

J

‘ 1/2

- « i (2. 19)

where K = A V2n.

I 0

17
mathematical similarity to the boundary collocation method.

Hybrid approaches incorporating both local and boundary

collocation procedures have recently been developed by Kirk

[2. 16]

.

2.1.1 SINGLE-ELEMENT STRAIN GAGE

For a single-element strain gage coincident with the x’-axis of

figure 2.1, the strain response of the gage, c , using three-term

9 expansions of eqs (2.7) and (2.19) is

- 1/2

2pc = Ar {k cos (0/2) - (1/2) sin0 sin (30/2) cos(2a)

9 0

+ (1/2) sin0 cos(30/2) sin(2a)> + B {k + cos(2a)>

0

+ A^r^2 cos(0/2) {k + sin2 (0/2) cos(2a)

- (1/2) sin0 sin(2a)> + B^r cos0 {(k + cos(2a)) cos0

3/2 - 2 sin0 sin(2a)> + A r {k cos (30/2)

2 - (3/2) sin0 sin(0/2) cos (20) - (3/2) sin0 cos (0/2) sin(2a)>

+ B r

2 {k cos(20) - 2 sin20 cos(2a) - 2 sin0 cos0 sin(2a)

2 - cos(20 - 2a)>, (2.20)

where k = (1 - r)/(l + r). For brevity we introduce the notation

2^g = V, + Vo + Vi + Vi + V2 + V2

’ (2 ' 21)

where the f. and g. are separable functions of r and 0 given

above.

2.1.2 STRAIN GAGE ROSETTE

In some applications it is desired to compensate the strain

gage response for temperature. This can easily be accomplished

using a strain gage rosette with its individual sensors connected

to adjacent arms of the Wheatstone bridge. For this case the gage

response, r e , is the difference in strains c - e : a v'v'

18
(1/2) *r

o

2/LtC c ~c sin9 < sin (30/2) cos(2a)

y/y/ X 'X'

- 1/2

- cos (30/2) sin(2a)> - B cos(2a)

0

+ (1/2) A^r

1/2 sin0 {cos (0/2) sin(2a)

- sin (0/2) cos(2a)>

+ { 2 sin0 sin (2a) - cos0 cos(2a) >

+ (3/2) A r

3/2 sin0 { sin(0/2) cos(2a)

2

+ cos(0/2) sin(2a)>

+ B r

2 {2 sin20 cos(2a) + 2 sin0 cos0 sin(2a)

2 - cos(20 + 2a)>. (2.22)

When eq (2.22) is written in the form of eq (2.21), the f and g

J' j are obtained from eq (2.22).

2.2 OVERDETERMINED ANALYSIS

Dally and Sanford [2.1] previously described a single-gage

technique for the static mode I case where the gage is positioned

at an orientation angle, a, determined by

cos 2a = -(1 - r)/(l + v) (2.22)

to eliminate the B

q contribution and positioned along a radial

line at an angle 0 determined by

tan (0/2) = cos 2a (2.23)

to eliminate the A^ contribution. The stress intensity factor,

K , is then given to three-term accuracy by

K = v^(8/3)7rr E e (2.24)

I 9 9 where r is the radial position to the gage and e is the strain

9 9 sensed by the gage.

19
The present single-gage analysis builds on the results

described above to eliminate two constraints associated with the

original approach. By increasing the order of the series

representation from three terms to six terms, we enlarge the field

of data acquisition, and prior knowledge of the extent of validity

of a three-term model is not required. Second, using a large

number of gages positioned farther from the crack tip eliminates

the need for a plastic zone correction.

The six undetermined coefficients in eq (2.21) can be

determined by analyzing the data taken from gages oriented at

various angles around the crack tip. The orientation angles for

the gages are determined by eqs (2.22) and (2.23). Provided

that a sufficient number of data points are obtained to ensure

adequate redundancy, a system of equations linear in the unknown

coefficients can be formed. The linear problem can then be solved

directly in a least squares sense.

For n strain sensors eq (2.21) can be applied repeatedly to

form the system

2lie =

9

! A f

0 0

B g

cro A f

1 1 B g A f

2 2 . B g

262

(2.25)

2fJE

g = Vo + B

ogo + Vi + B

i gi + Vz + B2

8

;

or equivalently in matrix form

D c = b, (2.26)

where c is the vector containing the unknown coefficients.

Traditionally, linear least squares problems of the type given by

eq (2.26) are solved through the formation of the normal

equations,

20
c = (D

T D)'1 DT b, (2.27)

which can be shown to minimize the Euclidean norm of the residual

vector [2.18], II r II , where

r = b - ID c (2.28)

and c is the least squares estimate of c. Appendix A includes a

formal discussion of the linear least squares problem and a

derivation of the solution given by eq (2.27).

Alternative solutions of systems of equations of the type

given by eq (2.26), based on orthogonal transformations, have been

found to have broader application in linear least squares theory

[2.19, 2.20]. Specifically, the normal equations tend to exhibit

numerical instabilities when either the row-space or column-space

dimension of the coefficient matrix ID becomes large. One such

orthogonal izat ion process is the QR decomposition. The theory of

the QR decomposition is detailed in [2.21] and summarized in

Appendix A. The decomposition was implemented using the Linpack

collection of FORTRAN subroutines [2.22].

2.3 EXPERIMENTAL VERIFICATION

A compact tension specimen with W = 305 mm was fabricated

from a 6.4 mm thick plate of 6061-T6 aluminum for verifying the

analysis procedure described above. A simulated crack of length

153 mm was machined into the specimen, providing an a/W ratio of

0.5. For aluminum, Poisson’s ratio is 0.33, and the orientation

angles are equal: a = 0 = 60°. Since for this case the strain

c is equivalent to the radial strain e , the specimen was

. x'x' rr instrumented with 10-element strip strain gages with active gage

lengths of 1.6 mm on 2.0 mm centers positioned at 0 = 0°, 45°, and

90° to provide 30 data points for the analysis. The strip gages

were located 5.8 mm, 11.1 mm, and 4.6 mm from the crack tip,

respectively. The specimen geometry and gage layout are shown in

21
figure 2.4.

The specimen was loaded in a servo-hydraulic closed- loop

material-test system. Strains were recorded at 1780 N loading

increments to a maximum load of 8900 N. The strain distributions

at maximum load are shown in figure 2.5 for each of the strip

gages. Only 29 data points are shown on the graph, because one

element was damaged during installation or testing and was not

functioning.

The overdetermined analysis was implemented in two ways.

First, only the data corresponding to the actual gage readings

were used to form the system of equations given by eq (2.25). The

order of the model was increased sequentially up to a maximum of

six parameters. The results of each analysis are presented in

table 2.1. The value of K is obtained with relatively low error

regardless of the number of terms retained in the model. The

results of the two- and three-parameter models provide the lowest

error in estimating K , while the largest errors were obtained for

the five- and six-parameter models.

A second overdetermined analysis was performed using the

smooth distributions of strains shown in figure 2.5. Data from

the curves were obtained at 2.5 mm increments in radial position,

r, to increase the number of data points from 29 to 48. This

increased the redundancy in the analysis from five to eight. The

results of the second analysis are presented in table 2.2.

Clearly, the degree of redundancy was adequate in the first

analysis, and no further improvement in calculating was

obtained.

Values of the first five nonsingular coefficients for the

specimen geometry employed here were previously determined by

Chona [2.17] using photoelasticity. For the six-parameter

analyses performed above we would expect to obtain reliable

results for the values of the first three coefficients. The

values of B

q and A obtained through the strain field analysis are

22
compared with Chona’ s values in table 2.3. Examination of the

results indicates that the value of B obtained with a fifth-order

0

model compares favorably with Chona’ s value. However, the value

of does not compare well with the photoelast ical ly determined

value, even though the coefficient value has stabilized.

To examine the reason for failing to accurately obtain the

value of A , Chona developed a plot showing areas in the field

where one-, two-, three- and four-parameter models can be used to

describe the strains to within five percent accuracy. The

comparison is based on the six-parameter solution of Chona.

Presented in [2.24], this plot indicates that the regions in which

the gages were positioned were insensitive to the value of A^

For the gages positioned on 0 = 45° and 90° the strain field is

adequately described by one or two parameters. The data taken

along 0 = 0° do require a three-parameter model for an accurate

description of the strain field; however, the coefficient

multiplying the A term when 0 = 0° is (r'

/2/2) which is too weak

in comparison to the multipliers of A

q

, (r /2), and B

q

, (3r°/2), to strongly influence the results.

The analysis developed in this section is accurate enough to

evaluate fracture parameters obtained from strain gages within

engineering accuracy. The value of the first nonsingular

coefficient, B

q

, was obtained as well. This coefficient is of

interest since it is relates directly to the value of the constant

stress acting parallel to the crack faces. The value of A^ was

not obtained to sufficient accuracy from the strain field

measurements primarily due to the positioning of the gages.

23
REFERENCES

[2.1] Dally, J.W. and Sanford, R. J. , "Strain-Gage Methods for

Measuring the Opening-Mode Stress Intensity Factor, K, "

Experimental Mechanics, Vol. 27, No. 4, pp. 381 - 388 (1987).

[2.2] Chona, R. , Irwin, G. R.

,

and Sanford, R.J., "Influence of

Specimen Size and Shape on the Singularity-Dominated Zone,

Fracture Mechanics: Fourteenth Symposium-Volume I: Theory and

Analysis, ASTM SIP 791, J.C. Lewis and G. Sines, Eds., American

Society for Testing and Materials, pp. 1-3 - 1-23 (1983).

[2.3] Wah, T. , "Stress Intensity Factors Determined by Use of

Westergaard’ s Stress Functions,", Engineering Fracture Mechanics,

Vol. 20, No. 1, pp. 65-73 (1984).

[2.4] Clark, A. V.

,

Mignogna, R. B. and Sanford, R.J.,

"Acousto-elastic Measurement of Stress Intensity Factors Around

Crack Tips,

Ultrasonics, March, 1982, pp. 57 - 64.

[2.5] Irwin, G. R.

,

Discussion of: "The Dynamic Stress Distribution

Surrounding a Running Crack - a Photoelastic Analysis,

Proceedings SESA, XVI, pp. 93-96 (1958).

[2.6] Sih, G.C.

,

"On the Westergaard Method of Crack Analysis,"

International Journal of Fracture Mechanics, Vol. 2, pp. 628-631

(1966).

[2.7] Eftis, J. and Liebowitz, H. , "On the Modified Westergaard

Equations for Certain Crack Problems," International Journal of

Fracture Mechanics, Vol. 8, pp. 383-392 (1972).

[2.8] Sanford, R. J.

,

"A Critical Re-Examination of the Westergaard

24
Method for Solving Opening-Mode Crack Problems, " Mechanics

Research Communications, Vol. 6, pp. 289-294 (1979).

[2.9] Sanford, R. J. and Berger, J.R. , "The Numerical Solution of

Opening Mode Finite Body Fracture Problems Using Generalized

Westergaard Functions," Engineering Fracture Mechanics (In Press).

[2.10] Newman, J.C., "Stress Analysis of the Compact Tension

Specimen Including the Effects of Pin Loading, " Fracture Analysis,

ASTM STP 560, American Society for Testing and Materials, pp.

105-121 (1974).

[2.11] Srawley, J.E. and Gross, B. , "Stress Intensity Factors for

Crackline Loaded Edge Crack Specimens," Materials Research and

Standards, Vol. 7, No. 4, pp. 155-162 (1967).

[2. 12] Pei-Qing, P. , "Stress Intensity Factors for a Rectangular

Plate with a Point -Loaded Edge Crack by a Boundary Collocation

Procedure, and an Investigation into the Convergence of the

Solution," Engineering Fracture Mechanics, Vol. 22, No. 2, pp.

295-305 (1985).

[2.13] Sanford, R. J. and Link, R. E.

,

"Local Collocation - A Hybrid

Technique in Fracture Mechanics," Proceedings SECTAM XII, The

Southeastern Conference on Theoretical and Applied Mechanics,

1984.

[2.14] Sanford, R. J. and Chona, R. , "Photoelastic Calibration of

the Short-Bar Chevron-Notch Specimen, " Chevron-Notched Specimens:

Testing and Stress Analysis, ASTM STP 855, J.H. Underwood, S.W.

Freiman, and F. I. Baratta, Eds., American Society for Testing and

Materials, Philadelphia, pp. 81-97 (1984).

25
[2.15] Sanford, R. J. , "Determining Fracture Parameters with Full

Field Optical Methods," Experimental Mechanics, Vol. 29, No. 3,

pp. 241-247 (1989).

[2.16] Kirk, M.T., A Hybrid Collocation Technique for Solution of

the Finite Body Single Ended Crack Problem, M. S. Thesis,

University of Maryland, 1989.

[2.17] Chona, R. , Non-Singular Stress Effects in Fracture Test

Specimens - A Photoelastic Study, M. S. Thesis, University of

Maryland, 1985.

[2.18] Leon, S.J., Linear Algebra with Applications, Macmillan

Publishing Company, New York, NY, 1980.

[2. 19] Eason, E. , "A Review of Least Squares Methods for Solving

Partial Differential Equations, " International Journal for

Numerical Methods in Engineering, Vol. 10, pp. 1021-1046 (1976).

[2.20] Longley, J.W.

,

Least Squares Computations Using

Orthogonal izat ion Methods, Marcel Dekker, Inc. , New York, NY,

1984.

[2.21] Stewart, G. W.

,

Introduction to Matrix Computations,

Academic Press, Inc. , New York, NY, 1973.

[2.22] Dungarra, J.J., Moler, C.B.

,

Bunch, J.R.

,

and Stewart,

G. W. , Linpack Users Guide, Society for Industrial and Applied

Mathematics, 1979.

[2.23] ASTM E 399-83, "Standard Test Method for Plane-Strain

Fracture Toughness of Metallic Materials," 1989 Annual Book of

ASTM Standards, Vol. 03.01, American Society of Testing and

26
Materials, Philadelphia, Pennsylvania (1989).

[2.24] Berger, J. R. , and Dally, J. W. , "An Overdeterminist ic

Approach for Measuring Using Strain Gages, " Experimental

Mechanics, Vol. 28, No. 2, pp. 142-145 (1988).

27
Table 2.

1

Results for K

I with 29 strain gage readings.

Parameters

A , B , A

0 0 1 A , B , A , B

0 0 1 1 A , B , A , B , A

0 0 1 1 2 A , B , A , B , A , B

0 0 1 1 2 2

K r

I (Mpa Vm) (pe)

25.4 1033.

6

24.4 290. 1

24.8 229. 8

25.2 168.

3

25. 4 139.7

25. 8 121.

1

$

Kj = 24.5 Mpa '/m from [2.23]

* Error

(%)

3. 59

-0.45

1. 35

2.69

3.59

5.38

28
Table 2.2

Results for with 48 data points.

Parameters

A

I (Mpa Vm)

25.3

r

(pe)

1049.8

Error

(50

3. 14

A , B

0 0

24.3 272.9 i o CD O

V Bo’ A

i 24.7 224.0 0.90

A

o’ Bo- V B

, 25. 1 166.7 2.24

A

o' B0

- A

r

B

r

A

2 25.4 126.2 3.59

A

o’ Bo’ Ar

B

r

A , B

2 2 25.7 121.9 4.93

Kj = 24.5 Mpa v'm from [2.23]

29
Table 2.3

Comparison of higher order coefficients (48 data points).

B

n Error A

1 (MPa-m' 1/2

)

Error

of Parameters

u (MPa) (%) (°/o)

2 6.51 33. 1 — —

3 9. 81 0.9 -100.4 -21.

9

4 11. 1 14.2 -156.6 22.0

5 9.49 -2. 1 -157.5 22.6

6 9. 17 -5.7 -161.8 25.9

30
31
Fig. 2.2 Single crack tip, infinite body fracture problem.

32
Fig. 2.3 Strip gage locations for the A1 6061-T6 compact specimen

33
Fig. 2.4 Geometry of the A1 6061-T6 compact specimen.

34
0

x:

4->L

O

w

C

o

•rH

-t-> .Q

• rH

t-w

• rH T3

Cti L

4->0)

•rH • •v c

a 0

l e

•rH T3 O

0 0

T3 a

o

o

0

L

w

a

4->c

0

£

•rHC-0

a

xw

cn

CNJ

ti i —i i—r i—i—r —i—

r

o

(O O) CD h*

• • • o o o o

ob

•rH Lu

( £ _0I x ) JJ5 ‘UJDJ^S PlPDy

35

A1

6061-T6

compact
CHAPTER 3

DYNAMIC ANALYSIS

The analysis of the strain field surrounding a propagating

crack subjected to time- independent loading will be considered in

this section. A series solution to the mode I problem will be

used to examine the strain field in order to optimize the

positioning of the strain gage. Analysis procedures based on the

temporal and spatial variations of strain sensed by a gage or

series of gages will be developed.

Unlike the static problem, the dynamic problem is inherently

non-linear due to the unknown position of the crack tip at a given

time. For a temporal approach to the analysis, an algorithm which

uses both iteration and triangulation to determine the crack tip

position and the series coefficients is developed. For a spatial

analysis approach, a modified version of the nonlinear least

squares technique similar to that used in photoelastic analysis

[3.1] will be used.

3.1 DYNAMIC STRAIN FIELD REPRESENTATION

We begin the analysis by considering which solution to employ

for the dynamic problem. It must be recognized that there are

three separate types of dynamic fracture problems: a stationary

crack subjected to time-dependent loading [3. 2-3. 7], a propagating

crack subjected to time-dependent loading [3.6 -3.9], and a

propagating crack subjected to t i me- independent loading

[3.10-3.15]. Here we concentrate on a propagating crack subjected

to time- independent loading. This analysis has been used

extensively for crack propagation analyses and crack arrest

analyses [3.10, 3.11, 3.22-3.24].

The assumption of constant velocity and t i me- independent

loading allows us to transform the time-dependent portion of the

36
equations of motion to the spatial variable in the direction of

crack propagation. A series solution technique can then be

employed to satisfy the elastic field equations and boundary

conditions. Without assuming constant velocity, the

transformation of variables cannot take place and series solutions

cannot be employed [3.16]. Although several solutions have been

advanced [3.10,3.12-3.15] for the constant velocity crack problem,

the solution of Irwin [3.14] maintains many of the advantages of

the Westergaard method of analysis for static problems and is used

here.

Irwin used a dynamic transformation of the y-coordinate for a

coordinate system attached to the moving crack tip given by

y, = x, y.

y

2 = *2 y.

with the velocity dependent functions /^and defined as

(c/c )

1

(3.1)

(3.2)

(3.3)

(c/c

2

) (3.4)

where c^ and c^ sire the longitudinal and shear wave speeds,

respectively, and c is the crack speed (for plane stress problems,

the plate wave speed is used for c^). The dilatation, A, and

rotation, w, defined by

Su dv

A = + dx

(3.5)

dy

dv <9u

w = - , (3.6)

dx dy

can then be written as harmonic functions in the transformed

37
variables,

V

2 A = 0,

' (3.7)

V

2

0) = 0, (3.8)

where the notation,

2 2 a a v

2 = +

; j = 1, 2, (3.9)

J ax ay^.

has been used. By noting the form of the dilatation and rotation

from the static analysis, the dynamic forms can be expressed as

A = p (1 - X

2

) Re r (z ), (3. 10)

u = <P (1 - X

2

) Re r (z ). (3. 11)

The analytic functions F 112 (z ) and F (z ) are complex-valued stress

2 functions of the velocity transformed variables z and z defined

1 2 by

z = x + i y ; k = 1, 2 (3. 12)

k k as shown in figure 3. 1. The exact form of the stress functions

depends on the geometry of the problem under consideration. The

constants, p and <p, are to be determined after a suitable

selection for the stress functions has been made.

Following an integration procedure and introducing Hooke’s

law, the stresses can be calculated as

<r = fi {p (1 + 2A

2 - A2

) Re r - 2<pX Re T >,

<r - fi <-p (1 + A

2

) Re r <j + 2<p\

2 Re F^}

,

cr - ii {-2pX Im T + (p (1 + A

2

) Im T }.

xy 11 2 2 where p is the shear modulus. Assuming that a

singularity exists at the crack tip as in the static

series stress functions similar to the static form, eq

be considered. For the dynamic problem then,

(3. 13)

(3. 14)

(3. 15)

square-root

situation,

(2. 19) , can

00 r (z ) = ZJzJ = E A. z['

VZ

; k = 1, 2. (3.16)

k k k k j k

J =0

For this choice of stress functions, the crack faces are

38
traction-free. To satisfy the condition of = 0 on y = 0, the

xy

constants p and <p must be related as

2 X,

<f> = — P- (3. 17)

(1 + X‘)

In this derivation we assume that the A appearing in the f

j 1 series are the same as the A. appearing in the r

? series. The

leading coefficient, A

q

, is again related to the opening-mode

stress intensity factor by

KT = A V2n . (3. 18)

I 0

The existence of a square-root singularity is well documented

for the problem being considered here [3.12,3.13,3.15]. This is

distinctly different from the other types of dynamic problems

mentioned previously. Both Freund [3. 17] and Rosakis [3. 18] found

that a square-root singular field does not develop under

time-dependent loading conditions for some time after the crack

tip is loaded. This remains an open question for the analysis of

stress wave- loaded fracture events.

A second choice of stress functions to be used in eqs

(3. 13) — (3. 15) can be made again in direct parallel to the static

Y(z) series as

r (z ) = Yfz) = S B z

m

; k = 1. 2. (3.19)

k k k k m k The leading term of the above stress function is a constant, as in

the static case. This choice for the stress function satisfies

the condition of <r = 0 on y

= 0. To satisfy the boundary

xy

condition of traction-free crack surfaces the constants p and <p must be related as

<P

1 + A

2

2

2 A

2

P- (3.20)

The constant, p, can be specified through the definition of K_

after superposing the stress functions in eqs 3. 14 and 3. 17. The

39
result is

p

= — . (3.21)

4A A - (1 + A

2

) 2

1 2 2 Substituting eqs (3. 16)-(3. 21 ) into eqs (3. 13)— (3. 15) we

arrive at the following general expressions for the dynamic stress

field:

<r = 0

1 <02 Re - 03

Re Z

? + 0? Re Y

1 - 04

Re Y

2

> , (3.22)

= 0

i {-0

4 Re Z i| + Re - 04 Re Yi| + 0

4

Re Y

2

> , (3.23)

<r = 2A 0 {-Im Z + Im Z - Im Y +0 Im Y), (3.24)

xy 1 1 1 2 15 2 where the velocity-dependent functions 0^., j = 1,..., 5, have been

introduced for brevity. The values of the 0. are

4X^

2 - (1 + A*)

2

2 2

1 + 2A, - A*

1 2 4A A

1 2

(3.25)

(3.26)

(3.27)

(3.28)

(1 + X

2

) 2

0

= . (3.29)

5 4A A

1 2 Substituting eqs (3. 22) -(3. 24) into Hooke’s law provides

expressions for the Cartesian strain components,

40
0,

e = - { O + i>0, )Re Z, - 0 A 1 + i>)Re Z, + (0 + K0, )Re Y,

x x „ 2 4 1 3 2 2 4

-0,(1 + i>)Re Y >

,

4 2 (3.30)

e = -

1 {-(8 + vB ) Re Z, + 8(1 + r)Re Zn yy g, 4 2 1 3 2

+8 (1 + v)Re Y>,

4 2

(0 + vp )Re Y

4

k

2 1

(3.31)

2A 0

y = — {-Im Z + Im Z - Im Y +0 Im Y > . (3.32)

xy 12 15 2

An approach similar to that taken in previous studies

[3.19-3.24] will be used to determine the orientation of the

strain gage relative to the crack propagation path. However, the

influence of the r term, which was neglected in [3.24], is

shown to be of paramount importance in this study.

Consider a rotated coordinate system (x’,y’

)

as shown in

figure 3.2. The strains in the rotated coordinate system are

obtained by combining eqs (3. 30 ) — ( 3 . 32)

,

the first strain

invariant and the strain transformation equation,

(e

y'y' - e ) X'X' iy

x'y'

2ia, ^. e (c - e + ly )

,

yy xx xy

(3.33)

where a is the angle of rotation. Using eqs (3. 30)-(3. 33) the

strain field in the rotated coordinate system is

2lie = 0 {[k(A2-A

2

) + (l+A2 )cos2a] Re Z - 0 cos2a Re Z

x'x' 112 1 13 2

+ [k(A2-A

2

) + ( l+A

2 )cos2a] Re Y - 0 cos2a Re Y 12 1 14 2 +2A sin2a [ -Im Z + Im Z - Im Y + 0 Im Yo ]>, (3.34)

1 12 15 2

41
2lie = 0 {[k(A2-A

2

) - ( l+A

2 )cos2a] Re Z + 0 cos2<x Re Z

y'y# 112 1 13 2 +[k(A2 -A

2

) - ( l+A

2 )cos2a] Re Y + 0 cos2a Re Y 12 1 14 2 +2A sin2<x [ Im Z - Im Z + Im Y - 0 Im Y ]}, (3.35)

1 1 2 1 *5 2

pr = 3, [-( 1+A

2

) Re Z, + 8 Re Zo - (1+A

2

) Re Y

x'y' 1 1 132 1 1

+ 0 Re Y ] sin2a + 2A 0 [ - Im Z + Im Z - Im Y

4 2 f 1 12 1

+ 0^ Y

?

] cos2a, (3.36)

where k = ( 1-y )/( 1+r)

.

3.1.1 SINGLE-ELEMENT STRAIN GAGE

The strain, e , on a single-element gage aligned with the

9

x’-axis is given by eq (3.34). Using the results obtained in

Chapter 4 of this report allows suitable locations for the gages

to be specified where a three-term series expansion is valid. The

dynamic strain field described by a three-parameter model along

the gage axis is

2per

= Af + A f + B g , (3.37)

g 0 0 1 1 0

&

0

where

f

0

0 {cos(0 /2) [k(A 2-A

2

) + ( 1+A

2 11 )cos2oc] +2A sin(0 /2)sin2a> 12 1 11 - 1/2

r 0 {- 0 cos (0 /2)cos2a-2A sin(0 /2)sin2a}, (3.38)

2 13 2 12

f 1111 = r

1/2

' 0 <cos(0 /2)[k(A

222

-A )+(l+A )cos2a] - 2A sin(0 /2)sin2a> 12 1 11

+ r

1 ' 2 8 {- 6 cos(0 /2)cos2a+2A sin(0 /2)sin2a>, (3.39)

2 13 2 12

42
g (3.40)

Q = ^(k+cos 2a) ( A^-A^) .

The coefficient of the B term in eq (3.37) can be eliminated by

0

selecting the orientation angle, a, such that

cos2a = -k. (3.41)

Then eq (3.37) becomes:

2fi€ = A f + A f . (3.42)

g 0 0 11

Values of a for various Poisson’s ratios are given in [3,20]. For

the steel used in this investigation, v = 0.30 and a = 61 . 3°or a = 118.7° to satisfy eq (3.41). It is important at this point to

examine which of the two choices for a gives strain data more

suitable for the accurate determination of K (t) using

ID single-element strain gages.

The first issue to address in this selection of a is the

features observed in the e-t trace that are the most useful for

determining either the crack position or velocity. To examine the

strain-time trace it is convenient to rewrite eq (3.42) in a

modified form as

2pe /A = f + ( A /A ) f . (3.43)

g 0 0 10 1

For specified values of crack velocity, position of the gage line

and the ratio A^/A

q

, the modified strain defined by eq (3.43) can

be determined as a function of crack tip position, time or the

angle 0. For the choice of a = 118.7°, as used in [3.24],. the

modified gage response is shown in figure 3.3. These results show

that the maximum value of 2pc^/A

Q

is critically dependent (± 30%)

on the magnitude of A^/Ag. This indicates that an analysis with a

= 118.7° and A /A taken as zero gives unacceptably large errors

in any A or K determination. Furthermore, the peak in 2ue /A

0 ID g 0

will occur at the position of the gage (x = 0) if and only if

A /A„ = 0. For nonzero values of A /A , which are common, the 10 10

peak value of 2/ie^/A^ does not occur until the crack tip has

43
passed beyond the gage. In general, the peak value of 2\ic /A

9 o does not locate the position of the crack tip. Finally, a zero

crossing does not occur as the gage is unloaded by the passage of

the crack.

Selecting the alternative choice of a = 61.3° results in an

improved strain response that is presented in figure 3.4. The

peak modified strain varies only ±3% with large changes in A^/Ag.

In an engineering approach to the analysis, this error is

negligible and the effect of A^ could be ignored in determining

A

q

. Such a procedure, however, would not locate the crack tip

which is needed for a full dynamic fracture characterization of

the material. A more refined analysis would therefore determine

stress intensity and crack tip position by making use of the large

amount of data available in addition to the peak values. Towards

the development of such a procedure, note in figure 3.4 that a

zero crossing occurs after the crack tip has passed the gage. The

time of the zero crossing is dependent on A^/Ag. This

characteristic is exploited in Section 3.2 to locate the crack tip

both spatially and temporally.

3.1.2 STRAIN GAGE ROSETTE

As previously mentioned in Section 2.1.2, it may be desired

to have temperature compensation for some material testing

applications. Again we consider the response of a strain gage

rosette with its individual sensors connected to adjacent arms of

a Wheatstone bridge. The bridge output is then equivalent to the

difference in strain, c -c :

p(e -e ) = = 0 [-( 1+A

2

) Re Z + 0 Re Z - (1+A

2

) Re Y ^ y'y' x'x' *11 1

' 3 2 1 1

+ 0 Re Y 3 cos2a + 2A 0 [ - Im Z +ImZ - Im Y *4 2 11 12 1

+ 0 Y ] sin2a. (3.44)

5 2

44
t

The gage response with a three-parameter model is

2tlc

3 = A

o fo + A

1 fi * Bo V (3 - 45)

where

f = 2r 1/2 {-( 1+A

2 )cos( 0 /2)cos2a - 2A sin(0 /2)sin2a>

0 1111 11

+2r

2

1/2

£

i <:-£3

cos(0

2/2)cos2a + 2A

i sin(0

2/2)sin2a> , (3.46)

f = 2r| /2 l+X

2 )cos(0

i /2)cos2a + 2Ai sin(0i /2)sin2a}

+2r 1/28 {/3 cos(0 /2)cos2a - 2A sin(0 /2)sin2a>, (3.47)

2 13 2 12

g

= - 23

i (A

2 -X2

) cos2a. (3.48)

A factor of two has been introduced on both sides of eq (3.45) to

allow direct comparison with the single-element gage response of

eq (3.42) The coefficient of the B term in eq (3.45) can be

0

eliminated by selecting the orientation angle, a, so that

cos2a =0. (3.49)

Then eq (3.45) becomes:

2ue = A f + A f . (3.50) ^ g 0 0 11

Unlike the analysis for the single-element gage, the orientation

angle, a, specified by eq (3.49) is material independent; a = ±

45° to eliminate the B

q contribution to the gage response for any

given elastic material. It is helpful to examine a strain-time

trace predicted by eq (3.50) by writing the equation in modified

form as

2pe /A = f + (A/A ) f . (3.51)

g 0 0 10 1 For specified values of crack velocity, position of the gage line,

and the ratio the modified strain defined by eq (3.51) can

be determined as a function of crack tip position, time, or the

angle 0. For a = 45°, the predicted gage response is shown in

45
figure 3.5 for a variety of values for A^/Ag. The gage height was

10.2 mm and the ratio of crack speed to Rayleigh wave speed was

c/c = 0.22 for this plot.

R Several features of the rosette response are evident when

comparing the response shown in figure 3.5 with the response of

the single-element gage shown in figure 3.4. First, the positive

peak-to-negat ive peak amplitude is much larger for the rosette

than for the single-element gage. This is potentially useful in

the development of feature-extraction procedures similar to [3.19]

and [3.22]. Second, the position of the peak in the modified

strain for the rosette is insensitive to the value of A /A . 1 o However, the magnitude of the peak is sensitive to A^/A^.

3.2 TRIANGULATION AND ITERATION ALGORITHM

As discussed above, the zero crossing in strain shown in

figures 3.4 and 3.5 can be used to locate the crack tip in both

space and time. Before developing such a procedure, it is

necessary to comment on the conditions under which a zero crossing

will exist. Specifically, as the height of the gage above the

crack path increases, the zero crossing eventually does not occur.

This can be seen in the predicted rosette responses shown in

figure 3.6 for increasing y -values. Results for the maximum

9

possible

y^

as a function of the A^/Ag ratio are presented in

figure 3.7 for the single-element gage and in figure 3.8 for the

rosette.

Consider two single-element gages, a and b, each located

along a gage line located y above the crack propagation line with

9 a selected to satisfy eq (3.41), as shown in figure 3.9. The

crack is propagating in the positive x-direction with a velocity

c, which is considered constant between adjacent gage pairs. This

velocity can be determined from the spatial separation of the

gages, s , and the time between peaks or zero crossings observed

46
on the e-t traces. The values of A and A are also considered to

0 1 be constant between adjacent gage pairs. This is essentially a

discretization of the variation of A

q and A^ with crack length,

where the discretization interval is defined by the gage spacing.

The first step in the analysis is to locate the position of

the crack tip on the crack line with respect to the gage pair. As

previously discussed, the proper selection of the orientation

angle, <x, and gage height, y , results in a clearly defined zero

9

crossing that can be used to locate the crack tip. Referring to

figure 3.9, consider the zero crossing of gage a to occur at time

t = 0 in the coordinate system (x,y,t) translating with the crack

tip. The time at which this event occurs is easily extracted from

the experimental data. Writing eq (3.43) for gage a at the zero

crossing gives

w( 0 ) = f + (A /A ) f = 0, (3.52)

where the f are defined in eqs (3.38) and (3.39) and r =

i y /sine. For specified values of A /A , y and c, eq (3.52) can

g 1 0 g be solved for the angle (0 ) locating the crack tip with respect

0 a to gage a when the gage is sensing zero strain. Note that y and

9 c are known a priori but the value of A /A must be assumed.

1 0

Since eq (3.52) is nonlinear in 0, the solution uses the

Newton-Raphson iterative procedure; that is,

w(0 ) n 0,^0 - — . (3.53)

n+1 n „ _ dw/30

n The convergence is rapid with 0 obtained in one to five

iterations. No problems were encountered with the multiple-valued

trigonometric functions (f ) or their derivatives, since all of

1 the branch cuts were taken into account. The angle (0 ) which

o a locates the crack tip relative to gage a at the time associated

47
with the zero crossing condition is dependent on the crack

velocity, gage height y , and the ratio A^/Ag. Results showing 0

Q

as functions of these variables are presented in figures 3.10 to

3. 12.

For prescribed values of c, y , and an assumed value of A^/A^

the angle (0 ) relative to gage a is now known. Next, the spatial

0 a positions of both gages a and b may be calculated, figure 3.13,

from the relations

x = y /tan (0 ) (3.54)

a § 0 a

x = s + x , (3.55)

b ab a

(0 ) = tan \ y /[s + x ]) , (3.56) Ob g ab a

where x and x are the x-coordi nates of gages a and b relative to

a b

the origin of the (x,y,t) system and (x,y,0) is taken at the time

of the zero crossing of gage a on the e-t trace.

A test is now made to check the value of A /A used in the

1 0

solution of eq (3.52) and to refine the estimation if necessary.

In the verification procedure, use is made of the assumption that

the ratio A^/A^ is constant as the crack propagates between the

gages. Advantage may then be taken of the wealth of data

available from the experimental c-t traces. The most direct

approach is to shift back from the time of the zero crossing by an

increment At and calculate the ratio (c /e ) . . using the

estimated value of A /A . This strain ratio is compared to the

1 0

experimentally recorded ratio at the time At. If the difference,

£, is within some acceptable error bound, £, then the current

value of A ^ /Ais acceptable. If the difference exceeds £ then

the estimate for A /A must be revised by some amount <5 and the

1 0

entire procedure, starting with the (0 ) calculation, is repeated

0 a

48
until £ ^ £ and an accurate estimate of A /A is established. As 10

a final check on the estimate of A /A the individual strain

1 0

components are recalculated and compared to the experimental

values. This simple check is necessary, since the ratio of the

strains is being compared and not their absolute values.

At this point the precise position of the crack tip at

(x,y,0) is known as well as the ratio A /A . The dynamic stress 10

intensity factor, K^, may then be calculated from the strain

sensed by gage b , e , at the time t = 0 when gage a is sensing

b

zero strain:

2n(c

( f n

) 0 b + ( VV (f A

(3.57)

where the (f.) are evaluated using (0 ) as defined in figure

i b Ob

3.13. The entire iterative procedure is illustrated in the

flowchart shown in figure 3. 14.

3.3 SPATIALLY OVERDETERMINED ANALYSIS

Instead of using the temporal variation in strain at a

particular material point, as was done in the preceding section,

consideration is now given to analyzing the spatial variation in

strains recorded by a series of strain gages at one particular

point in time. By analyzing the strains over a sequence of time

steps for crack length, a, and stress intensity, K^, complete

histories for a(t) and K^Ct) may be formed. The method is

essentially the dynamic parallel to the overdetermined static

analysis described earlier. However, the situation is made more

complicated due to the unknown crack-tip position and the

resulting nonlinearities in the system of equations.

49
The analysis applies either eq (3.42) or eq (3.50) to J (J >

3) different strain-time traces which are showing changes in

strain due to the dynamic crack growth at a prescribed time, t =

t^. In general, there are three independent unknowns in eq (3.40)

or eq (3.50): A

q

, A^, and the crack tip position. The series

coefficients, A and A , will be the same for each individual

o 1 strain-time trace; but the angle, 0, locating the position, P,

relative to the crack tip is different for each gage. In order to

make the analysis tractable it is useful to relate all of the

angles, 0, to the angle the crack tip makes with the first gage,

0

(1>

, by using the spatial separations of the gages, s.. :

0* J

' * = tan

1

|

» j— 2,3,..., J (3.58)

x + s

1 1j

where y , x and s are defined in figure 3.15.

9 1 i j Using either eq (3.42) or eq (3.50) and eq (3.58) for J

strain gages at time t = t^, we develop the following system of

equations,

2fi(e ) 9 1 A (f ) + A (f ) 0 0 1 111

2p(e )_

9 2 A (f ) + A (f ) 0 0 2 1 1 2

2/i(e ) 9 3

•

•

A (f ) + A (f ) 0 0 3 1 1 3

•

•

2p(e ) L

9 J

J A (f ) + A (f ) 0 0 J 1 1 J

The overdetermined system given by eq (3.59) is linear in the

( 1

)

unknown coefficients A and A but nonlinear in the angle 0

0 1 locating the crack tip with respect to the first gage. The

50
overdetermined system can be solved by a variety of nonlinear

least squares methodologies. Since the number of unknowns is

small, the Newton-Raphson iterative procedure is used here. The

details of the solution procedure are presented in Appendix B.

Because an iterative technique is needed to solve the

nonlinear least squares problem, a criterion must be specified for

defining convergence. The criterion used here is based on a

normalized residual defined as

r

J i [s (e - e. ) 1 / (3.60)

Lj.i J J J L j=i J J where

e

e = experimentally recorded strain at gage j,

e

c = calculated strain at gage j,

jJ = total number of gages.

The iterative solution procedure is repeated until the value of

the normalized residual, r, is less than 0.5%.

After solving eq (3.59) at t = t , the system of equations is

formed for the next temporal data point taken at t = t .

( 1

)

Proceeding in this way gives A

Q

(t), A^(t), and 0 (t) over the

entire time history. However, care must be taken to avoid

extending the region of data analysis to points P, where the

three-parameter representation given by eqs (3.42) and (3.50) do

not describe the strain field with sufficient accuracy. This

issue is addressed in Chapter 4.

This particular method of analysis appears most attractive

for applications to nonconstant crack velocity problems, since no

assumptions are required concerning the behavior of the crack

velocity or coefficient value between gages. However, the amount

of data available to introduce redundancy into the analysis is

limited to those gages close enough to the crack tip where a

51
three-parameter model is adequate.

3.4 EXPERIMENTAL VERIFICATION

An experiment was performed to demonstrate the methods

described here. The sample used for the fracture experiment was a

compact tension specimen fabricated from 12.7 mm thick 4340 alloy

steel, as illustrated in figure 3.16. The alloy was heat treated

to a hardness of = 51, obtained with a straight quench in oil.

The simulated crack was saw-cut into the specimen, and a blunted

chevron notch was used to control the load required for

initiation. Shallow side grooves on each side equal to 5% of the

specimen thickness were used to control the crack line.

To account for the presence of side grooves a correction

factor, C , was applied to the calculated K values. The

9 id

correction factor is defined by

1/2

C = (B/B ) , (3.61)

g n

where B is the gross specimen thickness and B is the reduced

n thickness through the side-grooved region. For the specimen used

in this experiment, B = 12.7 mm, B - 11.6 mm, and C = 1.048.

n g This correction factor is very small; however, for the sake of

completeness, it will be applied to all calculated values of

stress intensity.

The specimen was instrumented with 6 single-element strain

gages, each with an active grid length of 3.18 mm. Following the

argument presented in [3.20] errors due to strain gradient over

the gage length were negligible. The gages were oriented at a = 61.3° and were placed along a gage line with y

= 10.4 mm on 12.7

9 mm centers. The gages were connected to high-performance bridge

and amplifier units with a frequency response from DC to 120 kHz.

The strain gages were calibrated using shunt calibration values

52
obtained immediately prior to the experiment. Digital

oscilloscopes were then used to record the output signals using a

sampling rate of 200 ns per point. A closed-loop, servo-hydraulic

test system was used to load the specimen. Voltage-time traces

were downloaded from the oscilloscope bubble memories onto a

personal computer. A commercial spreadsheet program was then used

to postprocess the data. The resulting e-t traces for the 6 gages

referred to a common time base are shown in figure 3. 17.

Excellent agreement was obtained between the experimental traces

and the predicted behavior shown in figure 3.4. Figure 3.17 shows

that the predicted zero crossing is well defined, as required for

the triangulation analysis. Also, the peak amplitude for gage 1

is larger than the peak amplitudes of the other 5 gages. We think

that the crack velocity was higher upon initiation and that the

velocity was decreasing as the crack approached the first gage.

Additionally, the stress intensity sensed by the first gage was

higher as the transition from the initiation stress intensity to

the running stress intensity occurred.

Before performing the dynamic analysis, the stress intensity

factor at initiation, K , was determined. Following [3.20], the

iq

strain sensed by a gage at a distance r from the crack tip is

9 related to the stress intensity factor by

2p<] 2nr

K = C ?— e

S

, (3.62)

I 9 9

f S0

where e

s = static strain level at initiation,

9

f

s - kcos(0/2) - ( l/2)sin(0)sin(30/2)cos(2a)

0 + ( l/2)sin(0)cos(30/2)sin(2a)

,

a = 61.3°,

53
r = distance from the crack tip to the strain gage,

9

C = side groove correction factor.

9 -3

For gage 1 the static strain was 1.661 x 10 prior to crack

initiation. Gage 1 is located at 8 - 22.2° and r = 27.4 mm from

9 the static crack tip; therefore, from eq (3.62) K = 160.6 MPa

Iq

-Jm. Adjacent gage pairs were analyzed following the triangulation

and iteration procedure using an error bound £

= 0.001 and an

-

1

iteration increment on A /A of 6 - 0.492 m A /A was 10 10

initially assumed to be zero to start the algorithm. Convergence

to the correct value of A /A was sensitive to the time interval,

1 0

At, used in the strain ratio test. The most consistent results

were obtained when the time interval, At, was in the range -20 ps

< At < -30 ps. This result follows from figure 3.17, because for

times less than -20 ps before the zero crossing the strain is

changing rapidly and any small error in time resolution leads to

large errors in strain. Additionally, as figure 3.17 shows, the

strain response in the region of t < -20 ps (x < 13 mm for c = 650

m/s) is not sensitive to the value of A /A . Therefore, in this

1 0

experiment, times of less than -20 ps are inherently prone to

error. For times greater than -30 ps the gage is nearing the

limit where a three-parameter model fails to adequately describe

the strain field.

Detailed results obtained from the data from gages 2 and 3

at five different times over the interval between -20 ps and -30

ps relative to the zero crossing as the crack propagated between

the two gage stations are presented in table 3.1. The value of

A /A varies slightly with the time; however, K (t) remains 10 ID essentially constant. As a final test, the strain at the two

gages is calculated at each time step using the values of K , A^/Ag, and crack length established from the analysis. Close

agreement between the measured and computed strains (-5.0% to

54
-7.3%) at both gages is noted as shown in table 3.1.

Results for K and A /A from the data for all five gage

pairs are given in table 3.2. All of the results shown represent

mean values taken over five time steps -20 ps < At < -30 (is.

Standard deviations and coefficients of variation are given for

K and A /A . The time-averaged results indicate little

id 1 o variation in K or A /A with respect to the mean values. As

id 1 0 noted in the discussion of table 3. 1, the calculated values of K

id

exhibit the least amount of variance over the analysis interval.

The two assumptions made in the analysis, namely constant

crack tip velocity and constant A^/A^ between adjacent gage pairs,

are examined in light of the results given in table 3.2. Consider

first the assumption made of constant crack speed. Using the

calculated value 8

q with the known time of the zero crossing and

the spatial separation of the gages, a graph can be constructed of

crack tip position versus time. As shown in figure 3. 18, the

slope of this curve is constant, which verifies the constant

velocity assumption (c = 640 m/s).

For determining the effect of velocity on the determination

of K (t), a parallel analysis was conducted using the first three

ID terms of the static expressions given by eq (2.20) to represent

the strain field. The same analysis procedures and the

restrictions on the time steps as in the dynamic case were used to

obtain the results presented in table 3.3. For the crack velocity

c/c^ = 0.22 for this experiment, the difference between the static

and dynamic results are negligible.

The assumption regarding constant A^/A^ between adjacent gage

pairs must be addressed in two parts, since, in general, the ratio

depends on velocity and crack length. The discussion above has

verified the adequacy of a constant velocity assumption for the

particular experiment which was analyzed. Therefore, it is

justifiable to assume that the velocity-dependent portion of the

A /A ratio does not change since the velocity is not changing.

55
In regard to the variance with crack length of A^/A^, it is

useful to recall that the assumption of constant A /A is made

1 o only over discrete intervals which are defined by the gage

spacings. For the particular experiment analyzed the gage spacing

was 12.7 mm. This corresponds to an assumption of constant A^/A^

over an increment of crack growth a/W = 0.05. The effect of such

a discretization interval for the smoothly varying behaviors of A

q and A with crack length is limited.

1A second analysis was performed using the spatially

overdetermined approach of Section 3.3. Results were obtained

which are consistent with the results of the

triangulation-iteration algorithm. Plots of K^Ct) and a(t) from

the overdetermined analysis are presented in figures 3. 19 and

3. 20.

56
REFERENCES

[3.1] Sanford, R. J. , "Application of Least Squares to

Photoelastic Analysis," Experimental Mechanics, Vol. 20, pp.

192-197 (1980).

[3.2] Thau, S. A. and Lu, T. -H.

,

"Transient Stress Intensity

Factors for a Finite Crack in an Elastic Solid Caused by a

Dilatational Wave," International Journal of Solids and

Structures, Vol. 7, pp. 731-750 (1971).

[3.3] Achenbach, J. D. , "Extension of a Crack by a Shear Wave,"

Journal of Applied Mathematics and Physics (ZAMP), Vol. 21, pp.

887-900 (1970).

[3.4] Jahanshahi, A., "A Diffraction Problem and Crack

Propagation," Journal of Applied Mechanics, Vol. 34, pp. 100-103

(1967).

[3.5] Achenbach, J. D. and Nuismer, R. , "Fracture Generated by a

Dilatational Wave," International Journal of Fracture Mechanics,

Vol. 7, No. 1, pp. 77-87 (1971).

[3.6] Freund, L. B. , "Crack Propagation in an Elastic Solid

Subjected to General Loading - III. Stress Wave Loading," Journal

of the Mechanics and Physics of Solids, Vol. 21, pp. 47-61 (1973).

[3.7] Freund, L. B. , "Crack Propagation in an Elastic Solid

Subjected to General Loading - IV. Obliquely Incident Stress

Pulse," Journal of the Mechanics and Physics of Solids, Vol. 22,

pp. 137-146 (1974).

[3.8] Baker, B. R. , "Dynamic Stresses Created by a Moving Crack,"

Journal of Applied Mechanics, Vol. 29, pp. 449-458 (1962).

57
[3.9] Homma, H. , Ushiro, T. , and Nazakawa, H. , "Dynamic Crack

Growth Under Stress Wave Loading, " Journal of the Mechanics and

Physics of Solids, Vol. 27, pp. 151-162 (1979).

[3. 10] Freund, L. B. , "Crack Propagation in an Elastic Solid

Subjected to General Loading - I. Constant Rate of Extension,"

Journal of the Mechanics and Physics of Solids, Vol. 20, pp.

129-140 (1972).

[3.11] Freund, L. B. , "Crack Propagation in an Elastic Solid

Subjected to General Loading - II. Non-Uniform Rate of

Extension," Journal of the Mechanics and Physics of Solids, Vol.

20, pp. 141-152 (1972).

[3. 12] Radok, J. R, M. , "On the Solution of Problems of Dynamic

Plane Elasticity," Quarterly of Applied Mathematics, Vol. 14, pp.

289-298 (1956).

[3.13] Nishioka, T. and Atluri, S. N. , "Path- Independent

Integrals, Energy Release Rates, and General Solutions of Near-Tip

Fields in Mixed-Mode Fracture Mechanics, " Engineering Fracture

Mechanics, Vol. 18, No. 1, pp. 1-22 (1983).

[3.14] Irwin, G. R. , "Constant Speed Semi-Infinite Tensile Crack

Opened by a Line Force, P, at a Distance, b, from the Leading

Edge, " Lehigh University Lecture Notes on Fracture Mechanics

(1968).

[3.15] Nilsson, F. , "Dynamic Stress-Intensity Factors for Finite

Strip Problems," International Journal of Fracture Mechanics, Vol.

8, No. 4, pp. 403-411 (1972).

58
[3. 16] Knauss, W. G. and Ravi-Chandar, K. , Fundamental

Considerations in Dynamic Fracture," Engineering Fracture

Mechanics, Vol. 23, No. 1, pp. 9-20 (1986).

[3.17] Freund, L. B. , "Some Theoretical Results on the Dependence

of Dynamic Stress Intensity Factor on Crack Speed, " Workshop on

Dynamic Fracture, Knauss, W. G. , Ravi-Chandar, K. and Rosakis, A.

,

Eds., California Institute of Technology, pp. 129-136 (1983).

[3.18] Krishnaswamy, S. , and Rosakis, A. J. , On the Extent of

Dominance of Asymptotic Elasto-Dynamic Crack-Tip Fields; Part 1:

An Experimental Study Using Bifocal Caustics, Graduate

Aeronautical Laboratory Report SM 88-21, California Institute of

Technology (1988).

[3.19] Dally, J. W. and Sanford, R. J. , "On Measuring the

Instantaneous Stress Intensity Factor for Propagating Cracks,

Proceedings, International Conference on Fracture Mechanics VII,

Houston TX, (1989).

[3.20] Dally, J.W. and Sanford, R. J. , "Strain-Cage Methods for

Measuring the Opening-Mode Stress Intensity Factor, K^,"

Experimental Mechanics, Vol. 27, No. 4, pp. 381 - 388 (1987).

[3.21] Berger, J. R. and Dally, J. W. , "An Overdeterminist ic

Approach for Measuring Kj. Using Strain Gages, " Experimental

Mechanics, Vol. 28, No. 2, pp. 142-145 (1988).

[3.22] Dally, J. W. , Sanford, R. J. , and Berger, “J. R. , "An

Improved Strain Gage Method for Measuring Kj(t) for a Propagating

Crack," Journal of Strain Analysis for Engineering Design (In

Press)

.

59
[3.23] Berger, J. R. , Dally, J. W. , and Sanford, R. J.

,

"Determining the Dynamic Stress Intensity Factor with Strain Gages

Using a Crack Tip Locating Algorithm", Engineering Fracture

Mechanics (In Press).

[3.24] Shukla, A., Agarwal, R. K. , and Nigam, H. , "Dynamic

Fracture Studies on 7075-T6 Aluminum and 4340 Steel Using Strain

Gages and Photoelastic Coatings," Engineering Fracture Mechanics

Vol . 31, No. 3, pp. 501-515 (1989).

60
Table 3.

1

Results from a dynamic analysis of data from gages 2 and 3.

Experimental Computed

At ( c ) g 2 (c )_

g 3

k

id

A /A

1 0 (e ) 9 2 U )_

9 3

Ids fim/m /im/m MPa \/m

-1 m lim/m fdm/m

20 2039 1171 123.

9

-28.5 1926 1104

22 1998 1087 123.

9

-29. 1 1883 1023

24 1956 1045 123.8 -27. 6 1817 969

26 1872 962 123.8 -28. 1 1734 895

28 1747 920 123.5 -25. 1 1654 874

61
Table 3.2

Average results from a dynamic

adjacent gage

S

(K)

analysis

pairs.

of data for all

S

( A 1 /Ao k

id

A /A

1 0 k

id Al/Ao

Gage

Pair MPa'/m %

-1 m 7.

1 - 2 131.

4

0.27 -26. 2 -10. 4

2-3 123.8 0. 15 -27.7 -5. 6

3-4 122.4 0.33 -27.5 -11.7

4-5 120. 1 0. 19 -27.5 -5.7

5-6 120.9 0.20 -30.9 -6.9

62
Table 3.3

Average results from a static analysis of data for all

adjacent gage pairs.

K

ID

S

(K)

S

( Al/Ao )

1 ° Al/Ao

Pair MPav^m %

-1 m %

1 - 2 138.0 0. 38 -26.9 -11.4

2-3 126.5 0.21 -28.7 0

1 CD 3-4 125. 3 0. 36 -28. 4 -10. 1 4-5 122.7 0. 15 i [V) 00 00 iCD 5-6 124. 1 0.55 -32.2 1 CJ] no

63
64

Fig.

3.1

Definition

of

the

coordinate

systems

used

for

the

development

of

the

dynamic

strain

field

equations.
65
in

r^

i

0

in

1

in

OJ

( uj / /2ri) uie j;s pszi leuuJON

n

co

bb •H

U,

66

Normalized

strain

response

for

a

single

element

strain

gage

(oc=

118.7,

y

=

10.5

mm,

c

=

656

m/s).
c

o

CD

O

CL

CO ID fO CM

( W/ / 111) UIBJ^S paZIIBLUJON

67

Fig.

3.4

Normalized

strain

response

for

a

single

element

strain

gage

(a

=

61.3,

y

=

10.5

mm,

c

=

656

in/s).
o

CO

O

fO

O

ro

X

c

o

-

1—

I

4->

• I—

I

cn

o

Q_

CDCO

QO

•rH

t-u

( / 2//) uiej^s pazn^LUJON

68

Normalized

strain

response

for

a

rosette

strain

gage

(a=45,

y

=10.5

mm,

c=

656

m/s).
ujbjjs pazneiujON

69

Normalized

strain

response

for

a

rosette

strain

gage

for

v

=10,

50,

and

90

mm

(a

=

45,

c

=

656

m/s)
130

h*

&b U«

(tutu) )t|6!9H @6bo iunui|xe|/\|

70

Maximum

gage

height

for

existence

of

a

zero

crossing

for

a

single

element

gage
130

<

<

- RDBB 1e

DC

(lulu) }L)6!3H a6eo luniupceifl

71

Fig.

3.8

Maximum

gage

height

for

existence

of

a

zero

crossing

for

a

rosette
72

Strain

gage

positions

relative

to

a

propagating

crack.
115 Tf-rOCNI — OaiOONUDLD^-rOCN’-OCTOON ^^-r-^-^-OOOOOOOOOOOlO^O)

(S90J63Q) °Q 9I6UV

73

Fig.

3.10

Angle

0

as

a

function

of

c/c
126 (see j6sq) °e 016UV

o> >>

O

C

74

Gage

Line

Position
120

o

(S8SJ60Q) °0 3l6uv

75

Fig.

3.12

Angle

0

as

a

function

of

A

/A

.
o o

it

76

Fig.

3.

13

Crack-tip

position

relative

toa

gage

pair

at
Fig. 3. 14 Flowchart for the triangulation and iteration algorithm

77
78

Fig.

3.15

Spatial

distribution

of

stra
254

cozo

f—I

CDz

LU

79

Fig.

3.

16

Geometry

of

the

4340

steel

compact

specimen.
2.6

80

Fig.

3.

17

Strain-time

records

for

the

compact

specimen.
(UJUJ) UOIJJSOd d|J_ >|ODJQ

81

Fig.

3.18

Relative

position

of

the

crack

as

a

function

of

time

using

the

triangulation

and

iteration

algorithm.
140

o

©

(UJ/BdW)

ai

>l ‘SS3NH0DO1 NOIlVOVdOdd

82

Fig.

3.

19

Propagation

toughness

as

a

function

of

time

from

the

spatially

overdetermined

analysis.
154

o

o

ro

O

CN

O

CM O

in in

oo

m- 10

•<* CM o

M00

ro

CO ^

ro ro CM

ro O

ro

oo

CM

t£> Tj- CM CM CM

CM

LU

(lulu) H10 N31 >IOVya

83

Fig.

3.20

Crack

position

as

a

function

of

time

from

the

spatially

overdetermined

analysis.
CHAPTER 4

ACCURACY OF THE PARAMETER DETERMINATIONS

Before applying the methodologies developed in the previous

two chapters, we examine several important aspects of strain field

analysis. The first, and perhaps most important, concerns the

extent of validity of the three-parameter expansions for the

strain field. Knowledge of these valid zones is essential in

determining which gages can be used in a spatially overdetermined

analysis of the strains. Selecting gages that lie outside the

region of validity of the three-parameter model will introduce

errors into the analysis.

The second issue addressed in this chapter concerns the

determination of crack tip position from the strain analysis when

rosette gages are used. As demonstrated in Chapters 2 and 3,

orienting the rosette at a = 45° eliminates the contribution of B

o (cr ) to the strain response. However, we will show that there

ox are other, more important, reasons for prescribing this gage

orientation in terms of the analysis for crack tip position.

Finally, conclusions will be made concerning the anticipated error

in determining the crack tip position.

84
4.1 LIMITS ON THE VALIDITY OF THE THREE-PARAMETER MODEL

In Section 3.3, an analysis method was presented which made

use of the spatial variations in the recorded strains to determine

both propagation toughness and crack tip position. However,

information from gages remote from the current crack-tip position

may introduce error into the analysis if they lie outside the

region where the three-parameter model adequately describes the

strain field. In this section we study the extent of validity of

the three-parameter model to arrive at criteria for inclusion or

exclusion of gages from the analysis.

To assess the strain field surrounding the crack, the static

representation presented in Chapter 2 is used, since velocity

effects are presumed to be small for this portion of the

dissertation. Through knowledge of the higher order (nonsingular)

terms, we will examine the error over a reasonably sized area

between strain calculated from a three-parameter model and a

strain calculated using a high-order model.

Although detailed studies of the nonsingular terms have been

carried out for a variety of fracture test specimens [4.1, 4.2],

no reliable data are available in the literature for nonsingular

terms in the single-edge-notch (SEN) geometry. This particular

geometry is of interest because the large-scale, crack-arrest test

to be analyzed in Chapter 5 was an SEN specimen. Therefore, we

first determine the nonsingular terms for this geometry.

In previous studies [4.1, 4.2], photoelasticity was used to

determine the nonsingular terms. In [4.3], Sanford and Link

demonstrated that the same information could be obtained by

analyzing data from a finite element analysis using the same

algorithms as were used for photoelastic analysis. Here, we

follow [4.3] and determine the nonsingular coefficients from a

finite element analysis (FEA). The FEA stresses local to the

85
crack tip are collocated in an overdetermined fashion to extract

both the stress intensity factor and the nonsingular terms.

The finite element analysis was performed using the

two-dimensional, elasto-stat ic PC-based version of the ANSYS code

[4.4]. An SEN specimen was modeled using a width of 1.0 m and a

crack length-to-width ratio of 0.5. The specimen was modeled to a

height of only 1.15 m after it was determined that the stresses

were uniform beyond this height. The bottom edge of the specimen

was constrained against motion in the vertical direction, and an

applied pressure of 1000 MPa was imposed on the specimen’s top

edge. Finally, the node at the crack tip was constrained against

motion in the horizontal direction.

Two-dimensional, isoparametric elements were used throughout

the model. No attempt was made to include the singular behavior

of the stresses near the crack tip. Using a sufficiently fine

mesh and taking advantage of the local collocation procedure with

higher order terms, modeling the exact singular behavior is not

necessary. 1200 elements were used with the element mesh graded

to become finer as the crack-tip was approached. The r.m.s.

wavefront for the problem was 55.4 with 2140 degrees of freedom.

Several stages of mesh refinement were used until stress

intensity values determined from the FEA agreed with values

calculated from Tada [4.5]. For this geometry and loading, the

exact value of K = K = 354.0 MPa v'm. The final mesh is shown

I TADA

in figure 4.1. At this stage of the mesh refinement, = K - 350.0 MPa /m and K /K = 1.01. The modeling was therefore

FEA TADA

considered to be adequate. The distribution of the in-plane

stresses determined from the finite elements (<r , <r , and <r ) xx yy xy

along the crack line is shown in figure 4.2.

Stresses were collocated across a 0.50 m x 0.50 m area

surrounding the crack tip. As suggested in [4.3], no data were

used from a small region (0.05 m x 0.05 m) surrounding the crack

86
tip where the finite elements are incapable of modeling the

singularity in stresses. Nodal stresses were used as data for the

collocation procedure. This provided three equations (cr , cr , xx yy

and cr ) for each of the 191 nodes in the collocation region. The

xy

numerical details of collocation methods can be found in [4.6] and

[4.7].

The collocation was performed using the series expansions

from the static representation of stresses presented in Chapter 2.

Initially, one term from each of the Z(z) and Y(z) series was

used. Additional terms were then added in pairs (one from each

series) until the residual error converged. The residual used to

determine convergence was

where

r =

1 r J "i Z (tr

C0LLC

- r

FEV

Jo- Lj.i «J '> J

00

1/2

(4. 1)

J = total number of data points,

cr = remotely applied stress,

00

o' = stress component calculated from the collocation,

cr = stress component from the finite elements,

i J As shown in figure 4.3, the residual decreased with increasing

order of the model until it stabilized at 10 coefficients (5 terms

in each of the Z(z) and Y(z) series). The value of the residual

at 10 coefficients, r = 0.075 %, indicates good modeling of the

stresses across the collocated area. For comparison, in the

boundary collocation study of [4.7], the residual at convergence

calculated in a similar manner was r = 0.20 %.

We will therefore assume that a twelve-parameter model more

than adequately describes the stress field across the collocation

region. The numerical values of the coefficients are given in

table 4.1. Shown in figure 4.4 is a comparison of the finite

87
element stresses with the stresses calculated using the

twelve -parameter model. The best agreement is with the cr stress

yy

component due its large magnitude. The <r and <r components are

xx xy

matched well, considering their smaller magnitudes. The data

plotted in figure 4.4 are across y

= 0.123 m, roughly halfway

between the crack plane and the top of the collocated region.

Similar behavior for the comparison was found on other nodal lines

as well.

Having an "exact" solution for the strain field around the

crack tip now enables us to study the error in using a

three-parameter strain field representation. We study this error

by simply choosing a point in the field, calculating the strain at

this point using both three- and twelve-parameter models, and

comparing the two strain values. As we vary the position of the

point throughout the field, we can begin to see where strain gages

provide information usable in the spatially overdetermined

analysis procedure developed in Chapter 3.

The first comparison is done for the strain gage rosette,

figures 4.5 - 4.7 show areas in the field surrounding the crack

tip where the difference between three- and twelve-parameter

strain calculations are less than 2, 5, and 10%. Note the

dramatic increase in the size of the valid zone when a 10% error

is acceptable. In all cases the region extends at least twice as

far ahead of the crack tip as behind it.

For the analysis of dynamic crack propagation, the 5% error

plot provides an acceptable basis for establishing criterion for

gage selection. The 1.0 m width of the modeled specimen allows

for an easy conversion to a/W ratios for the criterion. We

conclude that for the strain gage rosette, data from gages a

distance x from the crack-tip may be included in an

9

overdetermined analysis, provided -0.15 ^ (x /W) ^ 0.30 with

9

(y /W) ^ 0.1. The value of y is usually constrained by thickness

9 9

88
considerations and is typically taken as (y ) = t/2, where t is

9 MIN

the specimen thickness [4.8]. For larger values of y the usable

9

range for x decreases but may be obtained from figure 4.6.

9

Finally, for the single-element strain gage, error plots for

2, 5, and 10% are shown in figures 4.8 - 4.10. In comparison to

the rosette, the region of validity is markedly smaller for the

single element gage. Again using the 5% error plot for the case

of dynamic crack propagation, we conclude that -0. 175 =£ (x /W) ^

9 0. 1 with (y /W) ^ 0. 13. Again, y must satisfy the thickness

9 9

criterion mentioned in the previous paragraph.

4.2 DETERMINATION OF CRACK TIP POSITION USING STRAIN GAGE ROSETTES

Arguments have been presented for orienting strain gage

rosettes at a = 45° in order to eliminate the contribution of the

Bg-term to the strain response. However, many crack propagation

experiments have been performed using strain gage rosettes

oriented at a = 0°. In the course of analyzing one of these tests

with rosettes at a = 0°, we encountered problems in determining

the crack tip position. These problems persisted even if the

nonsingular coefficients used in the series representation were

prescribed to their exact values. The study conducted to determine

the reasons for the difficulty in determining the crack-tip

position is presented in this section.

To examine the problem of locating the crack tip from the

strain analysis when a = 0°, synthetic data were generated using

the static representation of the strain field presented in Chapter

2. These synthetic data provided input free from experimental

error for a spatially overdetermined analysis. Data were generated

for both a = 0° and a = 45° under identical states of stress for a

series of six strain gage rosettes spaced 40 mm apart and located

65 mm above the crack path, figure 4.11. Using coefficient values

89
for the SEN specimen analyzed in Section 4.1, the series

coefficients were prescribed as A

q = 139.0 Mpa Vm, = -270.7 Mpa

- 1/2

m , and B = -12.7 MPa. The strains for each of the six

0

0 o rosettes with orientation a = 0 and a = 45 are listed in table

4.2.

For the a = 0° rosettes, there are four unknowns in the

analysis: A , A , B , and the angle the crack tip makes with the

0

n

10

)

first gage, 0 . The overdetermined analysis provided meaningful

(1

)

results for these variables only when the value of 0 was

initially estimated at nearly its exact value. Even if the

( 1 ) initial estimate for 0 was in error by as little as 10% from

the exact value, the algorithm would converge to a false solution.

That is, further iterations of the solution algorithm would not

alter the values of the unknown variables. Convergence to a false

solution would occur regardless of the accuracy of the estimates

for the series coefficients. Furthermore, similar problems were

encountered with the estimate for the B term. If the initial

o estimate for B was in error by only 10% and exact values were

o provided for the remaining variables, the algorithm would not

converge to the exact solution.

In contrast with the analysis for a = 0°, the same analysis

for the a = 45° rosettes was numerically well behaved. Initial

estimates for the three variables (A , A , and 8 ) could be

o 1 given any reasonable values and the algorithm quickly converged to

the exact solution. Only when physically unreasonable values were

specified for the initial estimate of O' would the algorithm

diverge. For example, if 0

(1)

was initially estimated to be 120°

Cl) O instead of the exact value of 0 = 45 , the algorithm diverged.

( 1

)

If 0 was initially estimated to be within 40% of its exact

value, the algorithm converged regardless of the estimates for A

q

( 1

)

and A . Since 0 can easily be estimated to within 40%, this

limitation does not present any difficulty. Most important, for

90
data from the rosettes with a = 45°, the algorithm would either

converge to the exact solution or it would completely diverge . In

no instance did the algorithm converge to a false solution.

Since the convergence of the solution for the a = 45°

rosettes is completely controlled by the nonlinear parameter,

0

(1>

, the implementation of the solution from a practical

standpoint is much simpler. After making the initial estimates

for A

q

, A^ and 0

(1 \ the algorithm will either converge to the

exact solution or it will diverge. If it diverges, simply change

the estimate for 0

(1) until the algorithm converges to the exact

solution.

There are two reasons for the behavior of the numerical

results for data taken from the a = 0° rosettes. First, the B

o term is spatially constant; that is, it is independent of the

(r,0) coordinates. Therefore, it contributes no information

pertaining to the location of the crack tip. A parallel can be

made to the general problem in instrumentation studies of

measuring a small signal over a large dc voltage offset. The B

q term acts as the dc level in the strain response and shifts the

response, positively or negatively, depending on the sign of B

q

. This shifting contains no information relating to the position of

the crack tip.

Second, the behavior of the numerical results also suggests

that the solution surface in least squares space for a = 0° is

either relatively flat or contains many local minima when compared

to the surface for a = 45°. Since the number of variables

precludes viewing the entire solution surface (a five-dimensional

( 1 ) "surface"), a three-dimensional projection of the A

q

, 0

solution surface was constructed with A and B prescribed to

1 0

their exact values. A contour plot of the surface for a = 0° is

shown in figure 4. 12. The contours represent constant values for

the normalized least squares residual defined as

91
r (4.2)

r

J _ 1 1/2 r J

II [,?, k-i > ]

/

J E e

S

j-i j J where

e

s = synthetic strain at gage j,

£

C = calculated strain at gage j using current values of A and

J e'

1 ’,

J = total number of gages.

Typically, for analyzing actual strain records, the criterion for

convergence is r ^ 0.5%.

The surface in figure 4. 12 exhibits some rather interesting

characteristics. First, the contours are highly elongated and

most do not close on themselves over the large range of values for

(A

q

, 9

(1)

). Second, local minima occur in the long valley

bracketed by the r = 1.5 contour. We can therefore see that there

are two problems with the a = 0° analysis which are typical of

problems in nonlinear least squares analysis; namely, the

existence of many local minima as well as a long, flat valley.

The Newton-Raphson technique used to solve the nonlinear system

relies on derivatives with respect to the unknown variables to

provide a direction for the solution. In the valley bracketed by

the r = 1.5 contours the derivatives cannot direct the solution to

the correct, absolute minimum.

In [4.9], a thorough discussion is made on the shape of the

contours in the residual space as they relate to "ill

conditioning" or parameter identification. Elongated contours

(banana shaped) indicate that the function of the variables can be

estimated precisely; yet the individual parameters can not be

obtained. Note in figure 4. 12 the elongation of the contours

which indicates exactly the type of problem described in [4.9].

92
For comparison, a contour plot of the three-dimensional

projection for the least squares solution surface with a = 45° is

shown in figure 4. 13. In contrast with the surface of figure

4. 12, this surface contains a distinct minimum near the exact

values of A = 139.0 MPa /m and 0

(1) = 45°. The topology of the

0

surface allows the Newton-Raphson procedure to provide an accurate

direction for the solution to follow. This explains the

difference in the behavior of the numerical results for a = 0° and

a = 45°.

4.3 ACCURACY OF THE DETERMINATIONS

One final issue to discuss in terms of the strain analysis is

the accuracy of the determination of the crack tip position and

series coefficients using strain gage rosettes. The series

coefficients determined in Section 4. 1 provide an opportunity to

address this question. The synthetic strain data which was

generated in Section 4.2 for a series of rosettes oriented at a = 45° was used for this study. The strains (c - e ) at

y/ y

/ x

/ x, these gages were used for input into the spatially overdetermined

analysis algorithm with crack tip position as an unknown. The

calculated value for crack tip position was then compared to the

actual value.

The spatially overdetermined analysis for these strains was

made using inexact values for the initial estimates of the

variables (A = 100.0 MPa v'm, A = -200.0 MPa m

1/2

, and 0

(1> =

0 1 65.0 ). The algorithm converged rapidly, with a resulting value

of 0

(1) = 45.1° with a normalized residual error of 0.01%. This

result for 0

(1)

compares well (0.2%) with the exact value of 0

(1)

= 45.0° obtained from the rosette positions indicated in figure

4.11. In terms of crack length, x = 64.8 mm which compares well

9

with the actual value of x =65 mm. The difference of 0.3% in

9

93
locating the position of the crack tip is certainly acceptable.

The values for and at convergence were 138.6 MPa \6n and

-273.2 MPa m

1/2

, which compare well (0.3% and -0.9%) with the

actual values of A = 139.0 MPa Vm and A = -270.7 MPa m

1/2

. 0 1 To study the propagation of error due to perturbation of the

strain field, the synthetic strain data were modified by errors of

2, 5, and 10%. Instead of uniformly applying the error across all

of the gages, we introduced various combinations of the sign of

the error. As shown in table 4.3, four cases were considered for

each of the error levels. The perturbed strains were then

reanalyzed with the spatially overdetermined algorithm, using the

same initial estimates given above.

Summarized in table 4.4 are the results of the error

propagation analysis. The range of values in the table represent

the maximum and minimum errors over the four cases (table 4.3) at

each error level (2, 5, and 10%). The last column of the table

shows the average residual over all four cases for each error

level. As expected, the residual at convergence increases as the

error was increased. Note especially for the 5% and 10% errors

the residual at convergence exceeds 0.5%, which was the value used

in most portions of this dissertation as a convergence criterion.

The value of A^ was influenced the most by the introduction

of error into the strain field, as can be seen by the large errors

for this term in table 4.4. This is not an unexpected result.

Typically, in any collocation procedure such as the analysis used

here, the highest-order term in the series expansion collects the

"numerical noise" from the analysis. However, this enables us to

determine the lower order parameters with greater precision. As a

( 1 ) result, A and 0 show a lower error in table 4.4.

0

We conclude from this analysis that as the strain field is

( 1

)

perturbed, the errors in the A

q and 0 determinations roughly

scale with the error; that is, if the error in the strain field

94
measurement is on the order of 5%, we can expect a 5% error in A

q and 0

(1 \ Finally, it is difficult to determine A with any

degree of accuracy if the strain field is perturbed. We noticed

similar behavior in the static analysis of the compact specimen in

Chapter 2.

95
REFERENCES

[4.1] Sanford, R. J. , Chona, R. , Fourney, W. L. , and Irwin, G.

R. , A Photoelastic Study of Non-Singular Stresses in Fracture Test

Specimens, NUREG/CR-2179 (ORNL/Sub. -7778/2 ) , University of

Maryland (1981).

[4.2] Chona, R. , Irwin, G. R. , and Sanford, R. J. , "Influence of

Specimen Size and Shape on the Singularity-Dominated Zone,

Fracture Mechanics: Fourteenth Symposium - Volume I: Theory and

Analysis, ASTM STP 791, J. C. Lewis and G. Sines, Eds., American

Society for Testing and Materials, pp. 1-3 - 1-23 (1983).

[4.3] Sanford, R. J. and Link, R. E. , "Local Collocation - A

Hybrid Technique in Fracture Mechanics, " Proceedings - Southeast

Conference on Theoretical an Applied Mechanics, SECTAM XII, Auburn

University (1984).

[4.4] ANSYS-PC/ED/LIN 4.3, Swanson Analysis Systems, Inc.,

Houston, Pennsylvania (1988).

[4.5] Tada, H. , Paris, P. , and Irwin, G. R. , The Stress Analysis

of Cracks Handbook, Del Research Corporation, Hellertown,

Pennsylvania (1973).

[4.6] Hulbert, L. E. , The Numerical Solution of Two Dimensional

Problems of the Theory of Elasticity, Bulletin 198, Engineering

Experiment Station, Ohio State University, Columbus, Ohio (1963).

[4.7] Berger, J. R. , An Improved Method of Boundary Collocation

for the Analysis of Finite Body Opening Mode Fracture Problems, M.

S. Thesis, University of Maryland (1986).

96
[4.8] Rosakis, A. J. , and Ravi-Chandar, K. , "On Crack Tip Stress

States: An Experimental Evaluation of Three-Dimensional Effects,

California Institute of Technology Technical Report FM-84-2

(1984).

[4.9] Seber, G. A. F. and Wild, C. J. , Nonlinear Regression,

Wiley and Sons, New York, NY (1989).

97
Table 4. 1 Non-singular coefficients determined from the local

collocation of the finite element data.

Coefficient Value

A

0

139.0 MPa 1/2m

A

1 -270.7 MPa

-1/2

m

A

2 -240.2 MPa

-3/2

m

A -334.5 MPa

-5/2

m

3

A -388.

2

MPa

-7/2

m

4 A -226.

1

MPa

-9/2

m

5 B -12.7 MPa

0

-1 B 30.3 MPa m

1 -2

B 153.3 MPa m

2 -3

B 178.2 MPa m

3

-4

B 86.8 MPa m

4

-5 B 807.9 MPa m

98
Table 4.2

Synthetic strain data generated for a == 0° and a = 45°.

r>° c , a = 0

9

(tie)

e , a = 45°

9 (tie)

2254 -1155

1252 -1225

770 -1029

534 -846

408

334

-707

-s* -603

99
Table 4.3

Sign of the errors applied to the synthetic data to study error

propagation.

Case

Sign of the applied error

Gage Number

1 2 3 4 5 6

1 + - + - +

2 + + -- + +

3

4

100
Table 4.4

Summary of the error propagation study.

All Values are in Percent.

&U ) 9

€(e) £(A J0

SiA)

1 r

+ 1.6 + 2.2 + 7.2

2

- 1.3 - 2.5 - 5.7

0. 27

+ 3.9 + 5.7 + 20.2

5

- 3.2 - 6. 1 - 13.4

0.65

+ 9.5 + 10. 1 + 54.5

10

- 6. 1 - 12.3 - 22.6 1.32

101
1000 MPa

rrrrm

1.0 m

20 ELEMENTS

TOP HALF OF SPECIMEN

Fig. 4. 1 Final geometry of the finite element mesh (only the top

half of the specimen is shown).

102
(spuDsnom)

(VdW) SS3dlS

103
0.24

B O

cs

4-1C

<D£

<D

0)

<D 4-> •rHC

•rH

e*H

<d x:

4->

O

C

O

»F—

3

4->RjCJO

o

o

RjO

O

CD •C

4J^4o

CDCJC

<D cn DO CD L C/3 CD C/3 > CD C C-. O 4-> CJ C/3 n

DO

•rH

U-.

(%) ivnaisad asznv^aoN

104
500

X>0

4->

O 3 -t-> CX

«-* £ C O -r-t U «M

in0

cn

cn

4)Gp

W1 .c

-p

*rH3

Cfl 0

cn

OT0g4-) cn

<DG

cd

cn

4->C• rHOa

cd -P

cd T30

JCH

-P 0 C "O 0 O E E 0' G 0 0

-P 0 0£

«-• cd C G

.r4 Cd Cm a

G0

-Po£

cdG

cda0>

r-H03P0

.C

-P0G

cd

cn0c

CmOGO

O

cn0

-C

-p

cn • 0 cn 3 03

cn -p

•rHG cd

cda

a £ -p oU 3

cd>

-Pc

cd>

cn 0 0 £ -p 0 G

r—t

D 0 cn

oo

•rH U. ( D dk*0 SS3dlS

105
ERROR

=

TWO

PERCENT

¥x>x>ooeeeee^o<x>eoeeo<>o<xxx>oo<xxxxxxxxx>ooooooooooee*- >o<xxxxxxxxxxxxxxxxxxx>ockxxxxxxx>c<>o<>c>oc>o<xxx>o<xxxxxx

)boc<xx>o<>Dooco<x>CKX>o<xxx>oooo<xxx>ooooooooooo<xxxxxxx:(

) c>c<x><xxkxxx>c<x>o<xxxxxx>o<xxxxxxx>c>cooooooooooooooooc < xxx>oo<xoo<xxx>dooooooo<xxx>ooooooooooooooooooooo<x:c

xx>oo<xx>o<x>o<xxxxxx>c>o<xxxxx>o<>o<x>o<xxxxxxxxxxx>c < XXXXXXX>O<XXXXXXXXXX>0<XXXX>O<XXXXXXXXXXXXXXXXX

XX>C<X>OC<XXXXXXXXXX<XXXXX>O<XXXXXX>0<XXXXXXXX

xxxxxxx>o<>o<x>c<x>c<xxxxxxxxxxxxxxxxxx>c>ooc(

*x >d<xxxxxxxxx>oo<xxx>c>c<><xx>^oooooooc>oooocc

>5<x x>>oc<x>cooc<x><>c>o<xx>c<x>o<^

)po<xx xx>o<xxxkxx><>o<x>ooo<x>o<xxx>ooo<x>co<>cc

XXXXXX XXXXXXXXX<XX>0<XXXXX>SO<XXXXXXXXXXX

)OOCOO<X XX>000<XXX><XXXX>C^5<XXX)0<XXXX>00<X

XXXXXXXX

xx>oooo<xx

)OOOOOOOOOCK

XXXXXX>QOOOO<

5 OOOOOOOCKXXXX

) 000<XX>0000000<

) DOOOOOOOOOOOOOO<

> oooooooooocxxxxxxx

>C<XXXXXXXXXXXXXXXXXX

53

XXXXXXXXXXXXXXX>C<XXX>0<XXX>C>O0<X

>Q<XX>OC<>D<XXXXXXXXXXXXXXXXXXXX

xxx>oo<xxxx>c<xxxxxxxxxxxxx>o<x

xxxx>o<x>c<x><x>^ooooooooooooc<

>o<xxxxxxxxxxx>c<x>ooooooooc ( >o<xxx>o<xxxxxxxxxxxxxxxxx: < XXXXXXXXXXXXXXXXXXXXXX>OC (r xxxxxxxxxxx>ooo<xxxx>ooock

XXXXXXXXXXXXXXXXXXXXXX) C )C<XX><XXX>000000000000000< xxxxxxxxxxxxxxxxxxxx>occ

B<X>0<XXXXXXX>OOOOC<X>0<XXXXXXXX XXXXXXXXXXXXXXXXXXX

)o<xx>oooo<xx>oooc<xxxxx>oooo<xx>< xxxxxxxx>ooo<x

>c<x>o<>c^>oooooooooooooooooooo<xx XXXXX>OOOC(

)oo<>o<xxxxx>o<xx>oo<x>ocooooooooo< xxxxxxxxxxx

>C<XXXXXXX><XXX>5lOOOOOC<XXXXX><X>D<X xxkxxxxxxxxokx: c )b^00<>^<xx>0<xx>00000<x>0000000000< xxx>ckxxxxxxxxxxx

)ee^0OOO<XKX><XXXXXXXKX>0OOO<XXXXX X>X>OOOOOOCKXXX>(>e ^

) 0<XXXX>C*XXXXX><X>O<X><XXXXXXXXXXX

) 0<XXXXXXXXX>OOOO<XXXXX>O<XXXXXX

>o<xxxxxxxxxxxxxxxxxxx>oo<xxx

; o<xxxxxxx>oooc<xxxxxxxxxxxx

) OOOOOOOOOOOOOOOOOOOOOOCK

)booo<xxx>Do6oc<xxx>occ<x

xxxxxxxx>oooo<xx>o<xx^

> OOOCKXXXXXXXXXXXXXX

) ooo<xx>ocooc>oc>oo<x

> ooooooooooooocx

)OOOOOOOOOCOO<

xxxxxxxxxxx

) 0<XXXXXXX

)OOC<XXX

> 0<XX

XX

)<XXXXXXXXXX

) COOOOOOOCKXXX

5 cxxxxxxxxxxxxxx

) ooooooooooocxxxxx

) c>oooo<xx>ooooooooo<

) ooooooooooooooooocx

> ooooooooo<xxxxxxxxxx

; o<xxxxxxxxxxxxxxxxxxxx

5 o<xxxxxxxxxxxxxxxxxxxxx

) oooooc<x>ooooo<xxxxxxxxx

)0<XXX>O<XXXXXXXXXXXXXX>C<X

XXXXXDO<XXXXXXXXXX

xxxxxxx>ooooooooc c xxxxxxxxxxxxxxxxx

xxx>oooooooooooc c XXXXXXXXXXX>OOOC< >xxxxxx>oooooo<x

XXXXXXXXXXXXXX

xxx>ooooooooc c xxxxxxxxxxxx

xxxxxxxxxx*-

xxxxxxxx>c c XXXXXXXXX

XXXXXX

xx

XXXX

XX

XX

XX

xx

XX

XXX

xxxxxx

xxxxxxx

XXXXXXXX

n

d

Q

ql

o

o

o

rOd

cn

00

»rH u,

ind d

rod

CNd

(UJ) QdOOO-A

106

Valid

zones

for

the

rosette

strain

gage

using

a

two

percent
MAX.

ERROR

=

FIVE

PERCENT

>00000000000ckx>ooooo<xxx><xxxxooooooooooooockxx>ooooqi<

:-

> COOOOOOOOOOOOOOOOOOOOOCOOOOOOOOOOOOOOOOOOOOOOOOOOC c >cooooooooooooooooooooooooooooooooooocxxxxxxxxxxxxxx

> OOOOOOOOCOOOOOOOOCOOCOOOOCOOOCOOCOCOOOOOOOOOOOOOOC ( >000000000000000000000000000000000000000000000000 * >0000000000000000000000000000000000000000000000 * >00000000000000000000000000000000000000000000 * >00000000000000000000000000000000000000000 * >000000000000000000000000000000000000000 * XX >oooooooooocooooooooooooooooooooooooooq*

>oo< >000000000000000000000000000000000000*-

>*XXXX >0000000000000000000000000000000000 *

) 00000< >000000000000000000000000000000000 * >*XXXXXX >0000000000000000000000000000000 * >ooooooo< >000000000000000000000000000000*

>OOOOOCOO< >00000000000000000000000000000 * >oooooooooo< X>OOOCOOOOOOOOOOOOOOOOOOOOOOC*

xxxxxxxxxxxx >000000000000000000000000000*

> 0OOO<XXXXXXXX X>OOOOOOOOOOOOOOOOCOOOOOOOC*

>OOCOOOOOOOOOO< >0000000000000000000000000 * xx>oooocooooooo<x xxxxxxxxxxxxxxx>o<xxxxxxx>e

>ooooooooooooooooo< >000000000000000000000000* >*xxxxxxxxxxxxxxxxxxx >00000000000000000000000

(

>*xxxxxxxxxxxxxxxxxxxxxxx >0000000000000000000000 ( >oooooooooooooooooooooooooooo< >000000000000000000 c >ooooooooooooooooooooooooooooo< >000000000000 c >oooooooooooooooooooooooooooooooo< xxxxxxxxxx

> 00000000000000000000000000000000000000000000000000 *

> 00000000000000000000000000000000000000000000000000 (

> 0000000000000000000000000000000000000000000000000c (

> cooooooooooooooooooock>ck>ocooooooc>oooooooooooooooo6 ^ >00000000000000000000000000000000000000000000000000*

> 00000000000000000000000000000000000000000000000000 c

> 00000000000000000000000000000000000000000000000000 *

> 00000000000000000000000000000000000000000000000000 c > 00000000000000000000000000000000000000000000000000 c

> 0000000000000000000000000000000000000000000000000c c >oooooooooooooooooooooo< >00000000000000000000 *

> oooooooooooooooooooo<

> ooooooooooooooooo<

> ooooooooooooooo<

> ooooooooooocxx

>oooooooooo<

>oooooooo<

>oooooo<

>ooo<

>*x

>*XXXXXXXXXX

>oooooooooooo<

>cooocooocoooocx

> oooooooooooooooo<

> OOOOOOOOOOOOOOOOO*

> oooooooooooooooooo<

>*xxxxxxxxxxxxxxxxxxx

> OOOOOOOOOOOOOOOOOOOOCK

> oooooooooooooooooooooo<

> oooooooooooooooooooooo<

> ooooooooooooooooooooooo<

) |»XH»>XO<>0|000^

>000000000000000000

c

>00000000000000000 * >0000000000000000 *- X>OOOOOOOOOOOOOC*

>00000000000000

c

>0000000000000 * >ooooooooooc|<

>oooooooo<

>ooo<

x>c x>e

XX

XX

XX

XX

XX

XX

>oooc*

xxxxxxx

xxxxxxxx

>00000006 *

fO6

fOd

Q

a:o

o

0

1 X

m

o

•M" d

rno

CMd

(w) QdOOO-A

107

Fig.

4.6

Valid

zones

for

the

rosette

strain

gage

using

a

five

percent
MAX.

ERROR

=

TEN

PERCENT

>o<xx>c><xxxkxx>ckxxxx>o<xxxxxxxxx>c>o<x>ooooock>xw

jJoc<j<xj<xo<xxc<xj<><xoo<xo<xoockxo<xxoc<xoco<xckxoo<xoooo<xoo<xo<xocc

) CXOC<J<XCXO<XO<XC<XO<XCXO<XC<XO<XO<XO<XOC<J<XO<XO<XO<XO<XO<XO<XO<XOC<J<XOOC c ^o<xx>g<>o<xxx>o<xxx><xxxxxx>oooo<x>d<xxxxxxx>ooooco<xxx>occ

xxxxxxxxxxxxxxxxxxx>c<xxxxxxxx>ooo<xxx>ooooooooooocc

xx><xxxxxx>o<xxxxxxxxxxx>ooooo<xxxxxxxxxxx>oooooocc

xooooooooooockjoooooooocooooooooooooooooooooocc

xo<xooo<xo<xo<xo<j<joc<j<j<xoco<xc<jo<jo<xo<joo<xo<joooooc<jc

><xxo<xo<xoc<xo<jo<joo<xc<xo<xoc<j<xo<>o<j<jo<jc<xo<joo<joc<

jp< xxxxxxxxxxxxxxx>o<xx>dooooooooooooo<xxxx>cc

jkxx xxx><>cx>c<xxxxxxxxxx>c<xx>o<xxxxxxxxx>ooc>c^

joooc< xxxxxx>o<xxxxx>^o<xx>>o<xxxxxxxxxxxxxx;

jbooooc xxxxx>o<>o<xxxxxxxxxxxxxxxx)oooooooooc

JO<J<XO<J<X J<XOO<JOO<XCXXCKJO<J<XC<JO<XOO<JOOOO<XOC<XO<XOC<

JOOOOOOCX XXXXXXX>0<XXXXX>D<XXXXXXXXXXXXXX>C!:

JO<J<J<XO<XOO< XJOOOOOOOOOOOOOOOOOOOOOOCKJO<J<J<JC<

jpocoooooooc jooooooooocoockjoooooooooooooocc

jPckjooooooock joooocoockjoooooococooccooooocc

jboooooooooooc xcoooooooooooooocoooooooooocc

jooooooooooooo: joooooccoockxooocoooooooooocc jbooooooooooooocK jocoooooooxooooooooooooooce

JOOOOOO<JOC<JKXJOO<J<J<X jockjkjooockxxxjockjckjckjcooocc jpoooooocoooooooooooc joc<xockjo<xjc<joooc<xo<xo<j<j<xc<

jooooockjoooooooooooooockx xjocckjoooooooockjoocooocc

JOOOOOOOOOCKJKJOCJOOOOOOOOOOOCKXX XXO<XX<XO<XXXXXXCOOCCC

: C<XXC<XXO<XXXXXO<XXXXXXC<XC<XXXXO<XXXO< xxxxxxxxoooocc

XXXXXO<XXXXXXOOOO<XXXXXXOOOOOO<XXXXXXXXO<XXXXXO<XXXXXXXC

JO<XOOOOO<XXXO<XCXXXOOOO<XXOO<XO<XO<XO<XXXO<XCXO<XOC<XXO<XXOOO<XXC

JOO<XXO<XXXXOOO<XXXXO<XXOO<XO<XO<XO<XXXX<XO<XXO<XXO<XO<XO<XXO<XXC

JOOOO<XO<XOOOOOO<XOOOO<XOOOOOOOO<XOO<XOOO<XOO<XOOOO<XOOOOO<XC

j ockxxxxx>oo<xxxooo<xxxxkxkxho<xkxxnoo<xko<>oo<xx>oo<xxx>o ^ JCXO<XO<XO<J<XXOOO<XXXXO<XXXOO<XO<XXXO<XO<XXOO<XXO<XXOOO<XO<XXO<XOCC

JCXO<XXXOO<XO<XOO<XXXO<XOOOOOOOOOO<XXO<XO<XXOC<XOOOO<XOO<XXXXOCC

JOOOOOO<XXXOO<XO<XOOOOOOOOO<XXXXXXXXO<XO<XXXXXXXXXXO<XO<XOCC

JCXOO<XXXXO<XO<XJ<XOOOOOOOOOOOOOO<XXXXO<XO<XXOOOOOOOO<XOC<XOCC

JO<XXXO<XXXXXOO<XXXXXXXO<XXOOOOO<XO<XXOOO<XXXO<XXXXXOOOOOOCC

JOC<OC<J<XO<XOC<XXXXO<XO<XXJ<XXXXXOOOOC<XXXOOC<XXXO<XO<XOO<XOOOCC

JO<XO<XOO<XO<XO<XO<XXOOOO<XO<XOOOOOOOOC<XO<XOO<XXOOOOOOO<J<XO<XOCC

JC<XOO<XO<XOOO<XOC<XO<XOO<J<XOOO<J<J<XO<J<XOOOOOOOO<XOOO<XO<J<XO<XXOCC

JC<XXO<XXOOOC<XO<J<XXXO<XJ<X>OCOOOC<XXXOOOOO<XO<XOOOO<XJ<XOOOOCC

JO<XO<XXXOOOCOO<XOOO<X XXOC<XO<XO<XJ<J<>OOOC<XOC<XO<XO<J^-

JOOOOOC<XJ<XOOO<XO<

JCXXCXO<XXO<XXO<X

JC<XO<XXO<XXX

JOOOOOOC

JO<XXX

JCX

JO<XO<XO<XO<XXO<

JOO<J<XXOOC<XOOO<

J cxoocxxxxxxxxxxx

JC<XO<XO<XXXJ<XO<XO<XXX

J oooooooooooocxxxxx

JOOO<XXXOOOOOOO<XXO<XX

JP<XOOOCKXJOOOO<XOOOOO<X

J OOOOOOOOOOOCKXXXXXXXXX

joooooooooooooooooooockx

JO<XXO<XXOOOOOO<XO<XO<J<XO<XXX

JC<XC<XC<XO<XO<XOOOOOO<XOO<XO<J<X

j oo<xxoo<xo<xyo^^

m

6 d

tod

XOOCOC<XXXXO<XXO<XXO<XOOOCC

XO<XO<XOC<XXOO<XOO<XOOOOCC

XXO<XXOOO<XOOOC<XXOOOCC

XOOO<XOO<XXOC<XO<XXOOCC

xc<xoo<>coooc<xo<xooqc

XXJOOOOCOOOOCX

XXOOOC<J<XOO<XO<

XXXO<XJ<XXOO<

J<XO<XXO<X XX

XXXX XJC- XX

>cc JCC

JCC

XJC xxooocc

xxoooocc

xxooooocc

xxxxxooocc

XOOOOOOOOC f

CNo

roo

Q

o

:

o

o

0

1 X

rod

bb

rrH u.

(>-u) aaooo-A

108

Valid

zones

for

the

rosette

strain

gage

using

a

ten

percent
ERROR

=

TWO

PERCENT

(w) 3ivNiaaooo-A

109

Valid

zones

for

the

single

element

strain

gage

using

a

two

percent

error.
ERROR

=

FIVE

PERCENT

110

Valid

zones

for

the

single

element

strain

gage

using

a

five

percent

error.
ERROR

=

TEN

PERCENT

roO

roo

E

LU<

Q

OZO

o

o

X

(«0 3ivNiaaooo-A

111

Fig.

4.10

Valid

zones

for

the

single

element

strain

gage

using

a

ten

percent

error.
112
0

(DEGREES)

ALPHA 0

A

q

(MPa— 0.5)

Fig. 4.12 Least squares solution space for the a - 0

rosette. Contour labels are in percent.
0

(DEGREES)

ALPHA = 45

A 0 (MPa-m~ 0.5)

Fig. 4. 13 Least squares solution space for the a

rosette. Contour labels are in percent.

45

114
CHAPTER 5

CRACK ARREST AND THE PROPAGATION TOUGHNESS - CRACK SPEED

RELATION FOR A REACTOR-GRADE PRESSURE VESSEL STEEL

The Nuclear Regulatory Commission has conducted a test

program at Oak Ridge National Laboratory (ORNL) to assess the

crack-arrest behavior of reactor pressure vessel steels at

relatively high temperatures. Using large-scale, wide-plate

specimens in a single-edge-notch (SEN) geometry, the objective was

to simulate propagation of a crack from a low- temperature,

low-toughness region into an increasing temperature-toughness

field. This test effectively simulates a pressurized

thermal-shock scenario in a reactor pressure vessel.

The American Society of Mechanical Engineers’ (ASME) Boiler

and Pressure Vessel Code (BPVC) [5.1] contains provisions limiting

arrest toughness, K , to 220 MPa v^m for light water reactor

I a pressure-vessel steels as shown in figure 5. 1. Although K

la

increases with temperature for these steels, it was unknown

whether or not an upper shelf occurred in similar to that

observed for energy absorbed in failures of Charpy specimens. The

uncertainty pertaining to the existence of an upper shelf formed

the basis for ASME’s limiting value of K^. The wide-plate test

program was intended to provide arrest toughness data at

temperatures, exceeding the start of ASME’s upper limit. The

results were expected to verify or refute the validity of such a

limit on K . Ia In this chapter the strains measured during the first crack

run-arrest event in a wide-plate test of a low-upper-shelf

reactor-grade steel will be analyzed using the methods developed

in this dissertation. Results for the behavior of the propagation

toughness with time, temperature, crack tip position and crack tip

velocity will be presented. Finally, results of the strain

115
analysis will be compared with results from previous analyses to

demonstrate the validity of the methods used here.

5.1 WIPE-PLATE SPECIMENS

The specimens used in the wide-plate test program were

designed to minimize the possibility of reflected elastic stress

waves from interacting with the dynamic crack propagation [5.3].

The single-edge-notch (SEN) geometry was selected to provide an

increasing stress intensity with crack length. The dimensions of

the wide-plate specimens are illustrated in figure 5.2.

To establish the increasing toughness field in the specimen,

the left (crack opening) side of the specimen was cooled using

liquid nitrogen, and the right side was heated using

electrical-resistance strip heaters. The specimen was

instrumented with strain gages and thermocouples using procedures

developed over the duration of the testing program [5.2].

Two-element rectangular strain gage rosettes were placed on the

front and back of the specimen along a line located 0.65B above

the anticipated crack path, where B is the specimen thickness.

Each element in the rosette was connected to an adjacent arm in a

Wheatstone bridge circuit to cancel apparent strains induced by

temperature changes during the loading period. The bridge

amplifiers and recording instrumentation were estimated to have

time resolution between 2 and 4 ps [5.5].

The specimens were loaded in tension using the 26 MN capacity

testing machine at the National Institute of Standards and

Technology (NIST) in Gaithersburg, Maryland. The desired

temperature gradient was obtained by -'adjusting the power to the

heaters and the flow of LN^ to the specimen’s sides. After the

desired gradient was obtained, final shunt calibration of the

strain gages was made, and the specimen was then loaded to failure

116
at a loading rate of 11 to 14 kN/s.

5.2 TEST RESULTS AND STRAIN ANALYSIS FOR A LOW-UPPER-SHELF STEEL

The second series of wide-plate tests (WP-2 series) examined

a 2.25 Cr-1 Mo steel (ASTM type A387 grade 22) heat treated to

produce a low-upper-shelf toughness and a high transition

temperature. ORNL evaluated the mechanical properties of the

material [5.4] and reported drop-weight nil-ductility transition

(NDT) temperature, Charpy V-notch energy, tensile properties,

initiation toughness, J-R behavior and arrest toughness. The

averaged mechanical properties are given in table 5.1.

Small-specimen crack-arrest tests performed at ORNL followed

the procedure in ASTM E 1221-88. The arrest data were then fit

with regression analysis to an equation relating K to

I a temperature,

K =55 + 5 e°*

0287 T MPa v'm, (5. 1)

la

where T is the temperature in °C. Equation (5.1) can also be

written as

K = 34 + 36 e

°- 02413(T ' NDT) MPa v'm, (5.2)

la

where NDT = 60 °C for this material.

The propagation toughness-crack speed relation was estimated

to be [5.4]

K = K (T) + A(T) c

2

, (5.3)

ID la

where c is the crack speed and, for T - RT > -13.9 K,

A(T) = (329.7 + 16.25 (T - NDT)) x

10' 6

, (5.4)

or, for T - NDT < -13.9 K,

A(T) = (121.7 + 1.296 (T - NDT)) x

10‘ 6

. (5.5)

Note especially the temperature dependence in eq (5.3). This

changes the appearance of the K^-c curve from that shown in

117
figure 1.1 to the three-dimensional illustration of the K -c-T

ID surface that is shown in figure 5.3.

The geometry of the sixth specimen in the WP-2 series,

WP-2.6, is shown in figure 5.4. The back of the specimen was

instrumented with strain gage rosettes oriented at a = 45° to the

crack plane to eliminate the contribution of B

q to the gage

response as demonstrated in previous chapters and in [5.63. The

specimen was loaded until initiation occurred at 19.33 MN. The

crack propagated in a series of eight run-arrest events followed

by a final tearing fracture. Typical strain gage traces from the

45° rosettes recorded during the run-arrest events are shown in

figure 5.5. Of the eight rosettes which were oriented at a = 45°

(gage numbers 13 to 20), only seven gages were useful for the

analysis presented here, since one arm of gage 15 failed before

providing meaningful output. Finally, the temperature gradient

established from thermocouple data immediately before fracture of

the specimen is shown in figure 5.6.

The strain records for this test were analyzed using the

spatially overdetermined method presented in Section 3.3.

Attention was focused on the period when the crack initiated in

rapid fracture to the first arrest. The behavior of the crack

following the first arrest was not studied.

Before beginning the analysis of the strain records, gages

were identified which satisfied the requirements given in Chapter

4 for the extent of validity of the three-parameter model used.

As the crack propagated into the specimen the "window" of gages

usable in the overdetermined analysis changed. Table 5.2 lists

which gages could be used as the crack extended.

The strain data were analyzed from 0 ^ t ^ 0.60 ms in 0.01 ms

increments. Since the crack velocity was not known a priori, as

required to employ the 'dynamic strain field equations, the first

analysis was performed with the static rosette expressions for

c —c given in Section 2.1.2. Results obtained from the

y'y' x'x'

118
static analysis yield the crack tip position as a function of time

as shown in figure 5.7. The continuous curve through the data

points was obtained using a combination of smoothing techniques

and least squares analysis.

The smoothed data for a(t) were used as input for an

overdetermined analysis with the crack position prescribed.

Although this did not drastically change the values for A

q and A

determined in the initial analysis (with crack position unknown),

it did smooth out the behavior of A and A as functions of time.

o 1 Results from this static strain field analysis for the variation

of propagation toughness, K , with time and position are shown in

figures 5.8 and 5.9. The toughness values appearing in these

figures and those reported later have been modified to account for

the presence of side grooves. The correction factor for the side

grooves is defined as

C = y/ B/B , (5.6)

9 n where B is the gross section thickness and B is the reduced

n section thickness at the side-grooved region. For the WP-2.6

plate, B = 152.4 mm, B - 113.9 mm, and C = 1.16.

n g The results of the overdetermined analysis at t = 0 with the

static representation of the strain field provide an opportunity

to evaluate the initiation toughness, . The initiation

toughness determined from the strain analysis with the side groove

correction was K = 202.5 MPa /m. It is interesting to compare

this value with K determined from the load at initiation (19.33

Jq

MN). Following [5.7],

K = <r i/ 7i a F(a /w) + K , (5.7)

eff eff B where <r is the remote applied stress, a is the

eff

plasticity-corrected crack length, F(a/w) is a geometric shape

function, and K is the stress intensity generated by the

temperature-induced bending. For the WP-2.6 plate geometry, the

initiation toughness value is then K •- 196.6 MPa Vm. This

iq

119
result agrees to within three percent of the K determined from

iq

the strain analysis.

Using the smoothed data for a(t), we calculated the

velocities, c(t), shown in figure 5.10. Although these results

can be considered preliminary until the analysis using the dynamic

equations is presented, there are several features worth noting on

figure 5.10. The maximum velocity of 729 m/s is attained at 0.11

ms. It is likely that the crack propagated at an even greater

velocity immediately following initiation. However, we have no

reliable velocity data for this period for the reasons discussed

below. The crack speed then drops as the arrest is approached.

After 0.37 ms, the velocity maintains a near constant value of

approximately 50 m/s. The final interpretation of the time of

arrest and a discussion of the apparent crack velocity after

arrest are deferred until after the presentation of -the analysis

using the dynamic representation of the strain field.

Reviewing the curves for a(t), figure 5.7, and c(t), figure

5.10, the behavior of the crack before t < 0.1 ms deserves

attention. First, there is an apparent discrepancy for the crack

length at initiation determined from figure 5.7 and the fracture

surface shown in figure 5. 11. Second, the motion of the crack

after initiation is not rapid; that is, rapid propagation is not

occurring until approximately 0. 1 ms after initiation, when this

behavior is expected almost immediately. Each of these apparent

discrepancies are addressed in turn.

To examine the crack length discrepancy, it is useful to

recall the Irwin approximation to the plastic zone and its effect

on the apparent position of the crack tip [5.8]. The radius of

the plastic zone in plane strain is

1 2

r = (K / <r ) , (5.8)

y 6 *

1 ys

where cr is the yield stress. Under the influence of such a

ys

plastic zone the effective position of the crack tip is shifted by

120
r for the elastic stress distributions of linear elastic fracture

y

mechanics,

a = a + r , (5. 10)

eff phys y

where a is the effective crack tip position and a is the

eff phys

physical crack tip position.

Now, in the strain analysis we are determining crack tip

position through the use of linear elastic fracture mechanics

expressions for the strain field. Therefore, if a plastic zone is

present in the specimen the crack tip position determined from the

analysis is actually the effective crack tip position and not the

physical crack tip position. Therefore, crack tip positions

determined at initiation where the plastic zone was large will

show significant differences with the physical crack tip position

until the crack is propagating in cleavage with the associated

shrinkage of the plastic zone.

To quantify the above discussion, we can calculate a plastic

zone diameter at initiation using K = 202.5 MPa V'm and an

iq

appropriate value of the yield stress. From the NIST thermocouple

data, the crack tip temperature at initiation was 65 °C. The ORNL

data on tensile properties gives cr = 270 MPa at this

ys

temperature. The value of r is then 29.8 mm. The effective

y

crack tip position determined from the strain analysis was 243.2

mm. Therefore, the physical crack length is a = 213.4 mm,

phys

which is within five percent of the measured crack length of 224

mm before initiation.

The second issue concerning the "lag" in when rapid

propagation occurs can be examined by considering two physical

phenomena that occur simultaneously: (1) propagation of the crack

across the plastic zone generated before initiation and (2) the

generation of elastic waves at initiation and their transit times

to the gages. Calculating the time for the crack to propagate

across the plastic zone requires velocity values immediately

121
following initiation. Although precise values for the velocity

are not available from figure 5.10 for this period, a reasonable

estimate is from 100 to 500 m/s. Using the plastic zone radius at

initiation calculated above, 29.8 mm, the time required to

propagate across the plastic zone diameter ranges from 0.30 ms for

c = 100 m/s to 0.06 ms for c = 500 m/s. Until the crack has

propagated through the plastic zone, reliable calculations of the

physical crack tip position cannot be made.

To evaluate the delay due to elastic wave propagation,

consider the distance to the most remote gage used in the analysis

for t < 0.10 ms, gage 18, as 218.0 mm. Since the shear wave

generated at initiation carries the most significant information

about the dynamic crack growth, we calculate the transit time to

gage 18 of this wave. For steels, c = 3220 m/s, and the transit

s time is 0,068 ms. Since at least two transit times are probably

required before the gage is truly responding to the dynamic crack

growth, the data for t < 0.10 ms are not representative of the

dynamic crack propagation. Based on both of these considerations,

the data analysis using the dynamic representation of the strain

field will only be performed for t £ 0. 10 ms. Using the velocity data shown in figure 5.10 as input, the

strain analysis in 0. 10 ^ t ^ 0.60 ms was repeated using the

dynamic equations presented in Section 3.1.2. It was anticipated

that the crack tip position determined using the dynamic

representation would not differ significantly from the position

determined using the static representation, since the spatial

distribution of strain predicted by the two equations is similar.

However, a change in the value of the propagation toughness (the

scalar multiplier for the strain field) was anticipated. As

indicated on the a(t) curve in figure 5.12, the data points are

essentially identical when using either of the two formulations.

Consequently the velocity data did not need to be readjusted.

Data for t > 0.40 ms are not shown on figure 5.12 because of

122
the low crack velocity associated with crack growth during this

time interval (c < 50 m/s). The dynamic strain field

representation does approach the corresponding static

representation in the limit as c -» 0 is approached. However,

examining the dynamic field equations for strain, the limit must

be taken carefully, since both the 8. functions and the

J velocity-transformed coordinates r , 0 are functions of the crack

j j speed. As a result, these equations produce numerical

instabilities at low crack speeds when using finite precision

arithmetic. To avoid these instabi 1 it ies, for velocities below 50

m/s the static representation should be used. The absolute value

of this "cutoff" velocity is not too critical, since, as discussed

below, the difference in K values computed using either

formulation at low velocities is small.

The smoothed curve for a(t) was used to prescribe the crack

tip position for the next analysis, following the same procedure

developed previously with the static strain field representation.

Results for propagation toughness with time, temperature and

position are shown in figures 5. 13-5. 15. A comparison of these

figures with figures 5.8 and 5.9 shows that predictions based on

the static representation are the same as those determined with

the dynamic representation. However, the magnitude of changes

depending on the instantaneous value of the crack velocity. As

shown in figure 5. 16, for crack speeds less than 300 m/s, the

difference in calculating K

id

by the dynamic or static equations

is negligible (less than 1% difference). The difference then

becomes larger as velocity increases. For this analysis of

WP-2.6, the maximum difference was 5.5% at c = 729 m/s.

In [5.3], an approximate correction factor is used to

facilitate scaling K values obtained from a static analysis to K

values for running cracks. The correction factor is

123
K = K 1 - (c/c ) , (5. 11)

DYNAMIC STATIC R where c is the Rayleigh wave speed (c = 2980 m/s in steels).

R R The ratio, K /K , is presented in figure 5. 16 as a

DYNAMIC STATIC

function of c/c

r and is compared with the results from the strain

analysis of WP-2.6. Figure 5.16 indicates some difference in

actual values predicted by eq (5.11) when compared to the strain

analysis. However, the overall trend with increasing velocity is

similar.

To ascertain when arrest occurred, it is useful to calculate

crack tip accelerations from the c(t) data in figure 5.10. As

shown in figure 5. 17, the accelerations are largest after

initiation and then gradually decrease as the crack arrests. Since

a necessary condition for crack arrest is for the acceleration to

be zero, figure 5.17 indicates arrest occurring near t =0.38 ms. However, the behavior of a(t) and c(t) near t = 0.38 ms must be

reviewed to determine if arrest did indeed occur.

Examining figure 5.10 at t = 0.38 ms, we note that the

velocity is constant at c = 50 m/s. This behavior is consistent

with an argument that the physical crack has arrested and that the

"effective" crack is extending due to the expansion of the plastic

zone at arrest. We can estimate this growth rate through a simple

differentiation of eq (5.8). For a first approximation we assume

that the time rate of change of the yield stress and shear modulus

is small during the time of interest at arrest (distinctly

different than if we were looking at cleavage-crack propagation

where such approximations would be dubious). After making these

assumptions and substituting K = A^V^tt, we obtain

2 p

Using values for gage 16 obtained at the time of arrest, t = 0.38

e c . (5. 12)

9 9

124
ms, r = 47.8 m/s. This is in good agreement with the results of

y

the strain analysis and is equivalent to an extension of the

effective crack tip at the same velocity. Slightly higher values

for the growth of the elastic-plastic boundary at arrest were

found in [5.5]. However, our result is within the estimated error

reported in [5.5].

Examination of the velocity data suggests that the crack

arrested at t = 0.38 ms followed by growth of the plastic zone.

It is unlikely that the plastic zone ever reaches its full static

equilibrium size, since a second initiation was estimated to occur

at t = 1.0 ms [5.4], and in [5.5] it was estimated that 4 to 5 ms

are required to fully develop the equilibrium plastic zone in a

transition from a running to an arrested crack.

One further comment should be made here concerning plasticity

effects for a running crack. In [5.5], the authors found that the

velocity of the elastic-plastic boundary was weakly dependent on

the applied stress intensity. Furthermore, an upper limit for the

plastic zone velocity was found to be approximately 250 m/s. This

fact implies that there is insufficient time for the plastic zone

to develop for a propagating cleavage crack running in excess of

250 m/s. Similar conclusions were obtained in [5.10] following a

dislocation model of fracture.

As a final check on the hypothesis that arrest occurred at t = 0.38 ms, we note in figure 5.12 at t = 0.38 ms that the crack

length is a = 360.4 mm. An examination of the fracture surface

characteristics presented in figure 5.11 indicates that the crack

front at the first arrest is curved significantly. The crack

front intersects the front surface at 320 mm, the back surface

(location of the a = 45° gages) at 300 mm, and has a maximum

extension of 390 mm. The result for the arrest position a = 360

a mm obtained from the strain analysis represents an apparent crack

tip position for a two-dimensional analysis. Although significant

125
progress has been made in formulating an approach to the

three-dimensional problem [5.11, 5.12], a full three-dimensional

treatment will not be considered here.

Based on the above observations and discussion, we conclude

that arrest occurred at t = 0.38 ms with a crack length at arrest

a of a = 360.4 mm. From the temperature-position data given in

a figure 5.6, the temperature at arrest was determined as T = 103.6

a °C. From the results of K (t) shown in figure 5.14, the arrest

ID toughness was K = K^(0. 38 ms) = 230.0 MPa Vm. For purposes of

comparison, in [5.4] the arrest was determined to occur at t =

a 0.31 ms, a = 340.0 mm, K = 253.0 MPa v'm, and T = 97.6 °C.

a la a These results are summarized in table 5.3.

The arrest toughness determined from the strain analysis can

also be compared with the ASME curve and 0RNL’ s small specimen

data. The ASME limits arrest toughness at T = 103. 6°C to 85 MPa

a Vm, clearly an extremely conservative value based on the arrest

toughness obtained here. From eq (5.1), an estimate of the arrest

toughness from ORNL’s small specimen data is = 184 MPa /m.

This lower value of the arrest toughness can be attributed to both

the difficulty in translating small-scale test data to large

structures as well as the inherent problems with the ASTM

crack-arrest standard discussed in the introduction to this

report

.

Having evaluated the arrest conditions (toughness, time,

temperature, and position), we can now focus on the relationship

between propagation toughness and crack velocity. Shown in figure

5. 18, the characteristic shape of this curve is not of the gamma

(D shape described in Chapter 1. However, by plotting the

propagation toughness-crack velocity relation in this manner we

have ignored the temperature dependence in the constitutive

relation. We know that arrest toughnesses increase with

increasing temperature from the data in the wide-plate test

series. This behavior is indicated in the schematic K -c-T

ID

126
surface shown in figure 5.3. Since we are following a contour on

this surface for this test, the effect is to produce additional

curvature for the lower segment of the - c curve when we

ignore the temperature dependence. The imposition of a

temperature gradient forces us to use the three-dimensional

constitutive surface shown in figure 5.19 to properly interpret

the crack propagation.

127
REFERENCES

[5.1] Boiler and Pressure Vessel Code, Section XI, American

Society of Mechanical Engineers.

[5.2] deWit, R. , Low, S. R. Ill, and Fields, R. J. , "Wide Plate

Crack Arrest Testing: Evolution of Experimental Procedures,"

Fracture Mechanics: Nineteenth Symposium, ASTM STP 969, T. A.

Cruse, Ed. , American Society for Testing and Materials,

Philadelphia, pp. 679-690 (1988).

[5.3] Naus, D. J. , et al., Crack-Arrest Behavior in SEN Wide

Plates of Quenched and Tempered a 533 Grade B Steel Tested Under

Noni so thermal Conditions, NUREG/CR-4930 (ORNL 6388), Martin

Marietta Energy Systems, Inc.

,

Oak Ridge National Laboratory,

August, 1987.

[5.4] Naus, D. J. , et al., Crack-Arrest Behavior in SEN Wide

Plates of Low-Uppei—Shelf Base Metal Tested Under Nonisothermal

Conditions: WP-2 Series, NUREG/CR-5451 , Martin Marietta Energy

Systems, Inc., Oak Ridge National Laboratory (Preliminary Draft).

[5.5] Fields, R. J. , and deWit, R. , "Plastic Zone Formation

Around an Arresting Crack," International Journal of Fracture,

Vol . 42, pp. 231-238 (1990).

[5.6] Dally, J. W. , and Sanford, R. J. , "Strain Gage Methods for

Measuring the Opening Mode Stress Intensity Factor, K ,

Experimental Mechanics, Vol. 27, No. 4, pp. 381-388 (1987).

[5.7] deWit, R. , et al .

,

Wide Plate Crack Arrest Testing: A

Description and Discussion of the First Two Wide Plate Tests and

the Results of Six, Full Thickness, Bend Bar Tests, NBSIR 87-3629,

128
National Institute of Standards and Technology, Gaithersburg,

Maryland (1987).

[5.8] Irwin, G. R. , "Plastic Zone Near a Crack and Fracture

Toughness, " Proceedings of the Seventh Sagamore Ordinance Material

Research Conference, Report No. MeTE 661-61 1/F, Syracuse

University Research Institute, pp. IV-63 (1960).

[5.9] Freund, L. B. , "Crack Propagation in an Elastic Solid

Subjected to General Loading - III. Stress Wave Loading," Journal

of the Mechanics and Physics of Solids, Vol. 21, pp. 47-61 (1973).

[5.10] Lin, I.-H.

,

and Thompson, R. M. , "Relatavist ic BCS-OHR

Model," International Conference on Fracture Mechanics VII,

Houston, Texas (1988).

[5.11] Solecki, J. S. , and Swedlow, J. L. , "On the

Three-Dimensional Implications of LEFM: Finite Element Analysis of

Straight and Curved Through-Cracks in a Plate,

Fracture

Mechanics: Sixteenth Symposium, ASTM STP 868, M. F. Kanninen and

A. T. Hopper, Eds.

,

American Society for Testing and Materials,

Philadelphia, Pennsylvania, pp. 535-553 (1985).

[5. 12] Swedlow, J. L. , "Three-Dimensions Versus Two-Dimensions in

Fracture Mechanics, " Fracture Mechanics: Nineteenth Symposium,

ASTM STP 969, T. A. Cruse, Ed. , American Society for Testing and

Materials, Philadelphia, Pennsylvania, pp. 88-98 (1988).

[5.13] Irwin, G. R. , Kobayashi , T. , Fourney, W. L.

,

Metcalf, J.

T. , and Dally, J. W. , "Photoelastic Studies of Crack Propagation

and Arrest in Polymers and 4340 Steel," U.S. NRC Report,

NUREG/CR-0542, University of Maryland (1978).

129
[5. 14] Schwartz, C. W. and Bass, B. R. , "Crack Speed Relations

Inferred from Large SEN Specimens of A533B Steel," Engineering

Fracture Mechanics, Vol. 34, No. 5/6, pp. 1209-1223 (1989).

130
Table 5.

1

Summary of the mechanical properties of the WP-2 material

at room temperature.

Charpy V-Notch Energy at the

Nil-Ductility Temperature 18.3 J

Ultimate Tensile Strength 596 MPa

Yield Strength 315 MPa

Reduction in Area 50 %

Elongation 25 %

131
Table 5.2

Gages available for use in the strain analysis of WP-2.6 with

changing crack length.

Crack Length, a

(mm) Usable Gages

a

0 < a < 249 13 - 17

250 < a < 289 13 - 18

290 < a < 329 13 - 19

330 < a < a

a 13 - 20

Gage 15 not used

132
Table 5.3

Comparison of the arrest conditions for WP-2.6.

This Analysis Ref. [5.4]

K

la

(Mpa ’/m) 230.0 253.0

a

a

( mm) 360. 4 340.0

T

a (°C) 103.6 97.6

t

a (ms) 0.38 0.31

133
Fig. 5. 1 Crack arrest toughness as a function of temperature for a

nuclear pressure vessel steel (from [5.3]).

134
CRACK PLANE (SIDE GROOVED)-;

&

0.1 J

_

T

• 9 . 6

- 7 ^ « 10 ?

9

0 * !

i

. - 7

&

INITIAL FLAW

(•1-0*1

0.102 w )

ELEVATION DIMENSIONS IN met«rs

Fig. 5.2 Geometry of the wide plate specimen (from [5.3i),

135
c

Fig. 5.3 Schematic representation of the propagation toughness,

crack velocity, temperature surface.

136
Fig. 5.4 Geometry of the WP-2.6 specimen (from [5.4]).

137
WP-2.

6.

Arrest.

Strain

Gage

113

WP-2.

6.

Arrest.

Strain

Gage

#14

138

Typical

strain-time

records

from

WP-2.

6.
300

(o 0) aaruvysdwsi

139

Temperature

gradient

for

WP-2.6.
370

UDd

o O o o O o o o o o o o o o

CO in to CO o a

t

GO (O in 't ro 0 to fO to to to to CO CO CO CO CO CO CO

ooc

• rHCOo

CD00

I

(1)

Q

G

o c

Cm • rH $ cd <f 0) G $ s -P

•pH CO -P 0

+ Cm •C O -p C Cm O o

•rH

H-> c

,—

\

o o

n c •i—4 E 3 -p Cm cd

LlI cd -p 2 G

0

CO hCO

cd 0

G

c a

< o 0

(— •pH L

< -p a •pH o

CO • rH a O -p

UJT a cd ± -p o CO o o

2 cd 0

cn G •G u -p

0~

Lf)

00

•pH Uu

NOIllSOd d!±-><OVdO

140
250

e

o O o O O O

ro CN O O)

CM CM CM CM CM %

(S*0^uJ-Dd^) SSBNHOnOl ‘dOdd

141

Propagation

toughness

as

a

function

of

time

for

WP-2.6

using

the

static

representation

of

the

strain

field.
250

(S'O^W-DdW) SS3NHOnO! *dOdd

Propagation

toughness

as

a

function

of

position

for

WP-2.6

using

the

static

representation

of

the

strain

field.
800

m

E

LlJ 2

(s/uj) A1I0013A

143

Fig.

5.10

Crack

velocity

as

a

function

of

time

for

WP-2.6.
144

Fig.

5.11

Fracture

surface

of

WP-2.6

showing

the

first

arrest

location

(a

=

45

rosettes

are

on

the

lower

side

of

the

specimen).
x

4->0

X!

T3

DO r-H C <D o • rH • rH <2: >- wD

CD

<XC• rH

Q • CX) c_ 1 4-> Oh CO O 0) L X O 4-> 4h Ch 0 0 e• rH cn.

4-> c0

<X • rH 0 4-3

O KJ c 4-3 < 0 c 1—

c/)

/• s • rH

4-> 0W

m

O 0 E C L . D a

UJ + 2

Ch

Cti 0

L, 1- U

Cfl • rHe

ctf c c 0 >5

•rH T3

4-> • rH TS W C 0 cti a 0• rH 0 4-> < ctf aJ

c_ 4-> S u> in Q

Ul eg X r-H I-O CD 2

in ob • rH 1 Uh

NOIllSOd dll-:HOVidO

145
250

(S‘0^LU-Ddw) SS3NH0H01 *dOdd

146

Fig.

5.13

Propagation

toughness

as

a

function

of

time

for

WP-2.6

using

the

dynamic

representation

of

the

strain

field.
250

(s’Q^w-Pdw) ssaNHonoi *dOdd

147

Fig.

5.14

Propagation

toughness

as

a

function

of

temperature

for

WP-2.6

using

the

dynamic

representation

of

the

strain

field.
250 o O O O O O o

•'t ro CN T— O O) 00

CN CN CN CN CN <—

(S*0^w-DdW) SS3NH0D01 ‘dOdd

148

Fig.

5.15

Propagation

toughness

as

a

function

of

position

for

WP-2.6

using

the

dynamic

representation

of

the

strain

field.
105

o

o

GO

i

j

i

I

t

l

i

I

rj- ro O O

% ‘0I1V1S ^/OIWVNAQ >1

149

VELOCITY,

M/S

DATA

+

EQ.

(5.11)

Fig.

5.16

Propagation

toughness

computed

from

both

the

static

and

dynamic

strain

field

representat

ions

as

a

function

of

crack

velocity.
(SUOjIUW)

( 3 vs/uj) NOIlVa31300V

150

Fig.

5.17

Crack

acceleration

as

a

function

of

time.
800

m

ro

CM CD

o o o o o O

o o o o o O

h* co m m CM

(s/uj) A1I0013A *ovao

151
700

in

152

Fig.

5.19

Propagation

toughness-crack

velocity-temperature
CHAPTER 6

CONCLUSIONS AND DISCUSSION

A variety of analytic techniques suitable for studying the

strain field surrounding either a stationary or propagating crack

have been presented in this report. The use of strain

measurements in fracture analysis is attractive due to the

relative simplicity of the experiments and the measurement of

quantities in immediate vicinity of the crack tip. By employing

the methodologies developed here, fracture parameters can be

extracted from the strain records.

In reviewing the work presented in this dissertation, several

important conclusions can be made concerning fracture analysis

based on strain measurements:

(1) The feasibility of using strain measurements to obtain

fracture parameters has been demonstrated By using the series

representations of the strain field for stationary or propagating

cracks, several analysis schemes were developed.

(2) The importance of including the nonsingular term

became evident in the development of the algorithms for crack

propagation analysis. Even though this particular parameter could

not be determined accurately, it was necessary to include it in

order to determine with sufficient accuracy.

(3) For dynamic crack propagation, two algorithms were

developed to take advantage of either the spatial or temporal

variation in the strain field. The viability of these algorithms

was proved on a demonstration experiment conducted in 4340 steel.

(4) Orientation of the strain gage with respect to the crack

propagation path is of paramount importance to optimize the strain

signal for analysis and to locate the crack tip. A significant

number of experiments in which the orientation suggested here was

not employed have been conducted, and crack tip location was shown

to be a significant problem in attempting to analyze these

153
records.

(5) The algorithms developed here were successfully applied

to analyze a large-scale, crack-arrest test conducted in a nuclear

pressure vessel steel. Both the arrest toughness and the

propagation toughness-crack speed relation along a contour

determined by the imposed temperature gradient were determined

through the strain analysis.

6.1 SUGGESTIONS FOR FUTURE WORK

The work reported here has generated many areas worthy of

investigation. These can be broadly divided into numerical

analysis and fracture mechanics issues:

1. Numerical analysis issues. The large amount of data in

existence for dynamic crack propagation experiments where the

gages were not oriented as suggested here deem the crack tip

location problem worthy of further investigation. For developing

improved numerical schemes to analyze of crack tip position with a

= 0° rosettes we offer the following suggestions:

(i) The algorithms available in the optimization literature

may offer improvements over the Newton-Raphson method used here.

Although many of these algorithms are simply modifications to this

method, other techniques which use direct-search methods,

restricted iteration steps, conjugate gradients, or robust loss

functions are available. In certain optimization case histories a

simple scaling of a parameter, say scaling x to ax, changes the

contours in the residual space from a highly elongated to a more

desirable circular shape. By employing one or a combination of

these techniques, the analysis of a = 0° rosettes for crack-tip

position may be possible.

(ii) The difference between a true convergence criterion

versus a termination criterion for the solution algorithm should

be examined. No distinction is necessary for the a = 45° rosette

154
analysis, but a clear distinction is evident for the a = 0°

analysis. Perhaps a true measure of convergence based on

concurrently evaluating several factors (such as the match to the

applied strain field, expected coefficient values, relative

changes in the parameters) would improve the a = 0° results.

2. Fracture mechanics issues.

(i) The three-dimensional nature of fracture is an extremely

difficult problem, especially as it relates to curvature of the

crack front. However, three-dimensional elasto-plast ic finite

element analysis may yield useful results for relating crack tip

positions determined from surface measurements to the curved crack

front

.

(ii) The development in Chapter 4 of regions of validity for

the three-parameter model were performed at only one crack

length-to-specimen width ratio and with a stationary crack.

Validity zones taking crack velocity and crack extension into

account would be useful for more accurately defining criteria for

inclusion or exclusion of strain gages in the analysis.

(iii) The gage orientation developed here was based on

eliminating nonsingular terms, primarily cr , from the gage

ox response. However, further studies on gage orientation may

provide useful results for determining only individual parameters

such as crack tip position, propagation toughness, or crack

velocity.

6.2 ACKNOWLEDGEMENTS

The work presented in this report represents the dissertation

submitted by the first author to the Department of Mechanical

Engineering at the University of Maryland, College Park, Maryland.

The authors thank Dr. H. I. McHenry, Chief of the Materials

Reliability Division at the National Institute of Standards and

Technology in Boulder, Colorado, for providing both time and

facilities to complete this work.

155
APPENDIX A

THE LINEAR LEAST SQUARES PROBLEM

156
In Chapter 2, the formulation of the strain analysis for the

stationary crack problem required the solution of a linear least

squares problem. In this Appendix, we detail the theory of the

linear least squares problem, present the orthogonal decomposition

used to solve the system of equations, and provide the elements to

the system matrices of Section 2.2.

A.

1

THEORY OF LINEAR LEAST SQUARES

The general, linear least squares problem,

Ax = b, (A. 1)

involves finding a vector x which minimizes the residual

r

2 = II r II 2 = II b - Ax II 2

, (A. 2)

2 2 where A is a fixed m x n , m > n, matrix of constant multipliers,

b is a known m-vector, and x is the n-vector of unknown

coefficients. The norm indicated by II •

II ^ represents the Euclidean

2-norm of a vector,

II r II =

2 (A. 3)

Since the system given by eq (A. 1) is overdetermined, a consistent

solution usually does not exist. However, we can obtain a least

squares solution to eq (A. 1) by finding the best-fit vector x

which minimizes the residual in eq (A. 2).

Before establishing the procedure by which a least squares

solution to eq (A. 1) can be found, it is necessary to review some

basic definitions and theorems from linear algebra. The theorems

(offered without proof) and the definitions are standard and can

be found in [A. 1, A. 2] or other books dealing with linear algebra.

First, the column space of an m x n matrix, A, written R(A),

is the subspace which is spanned by the columns of A. Second, two

complementary vector subspaces are said to be orthogonal

157
complements if and only if they are orthogonal to one another.

That is, if if and % are orthogonal complements, any vector, x, can

be written as x = s + t where s € if, t e J

,

and s is orthogonal to

t. Finally, if a vector, r, is the projection of any vector onto

the orthogonal complement of ?i(A), then r is orthogonal to K(A)

and A

Tr = 0 (where A

r is the transpose of A).

Using these concepts, we can now formally present the

solution to the linear least squares problem and outline its proof

following [A. 1]. If x is a solution to the least squares problem

of minimizing eq (A. 2), then the residual vector satisfies

A

T r = 0. (A. 4)

By substituting r = b - Ax into eq (A. 4) we have the equivalent

statement

,

A

1 A x = A

T b, (A. 5)

which are commonly called the normal equations of eq (A. 1). From

eq (A. 4) we can see that the residual vector must therefore be

orthogonal to iR{ A).

To understand the proof that eq (A. 4) solves the linear least

squares problem, it is easiest to consider the problem from a

graphical viewpoint and then offer the formal proof. To construct

the graphical presentation, note that as the vector x varies the

vector y

= A x varies over the column space of A, ?((A).

Therefore, minimizing eq (A. 2) is equivalent to finding the vector

2 y which minimizes II b - y II . This residual is obtained when b

min 2 - y is orthogonal to ft(A) as can be seen in figure A. 1. In the

figure, the minimum length of the vector, r, is attained when it

is orthogonal to y (keep in mind that b is of "fixed" length and

we are varying y). This is a simple two-dimensional view of the

multi-dimensional least squares problem, but it is quite helpful

in understanding the role of orthogonal projections in linear

least squares.

With reference to figure A. 1, let b^ lie in Hi A) and b^ lie

158
in the orthogonal complement of ft(A). Then b = the

vector y = b and the residual vector r = b . The linear least

min 1 2 squares problem can thus be considered a multi-dimensional

orthogonal projection problem. On obtaining y , we have

MIN

minimized eq (A. 2).

Now, returning to the proof that eq (A. 4) represents the

least squares solution to eq (A. 1), if x minimizes eq (A. 2) then

Ax = b and r -- b = b - b . However, as discussed in the

1 2 1 previous paragraph, b^ is in the orthogonal complement of ft(A).

Then, by definition, A

T b

2 = 0. So, A

Tr = 0 which is eq (A. 4).

Therefore, the solution of eq (A. 1) by the normal equations has

been proved.

The implementation of the normal equations has been found to

generate numerical instabilities in certain cases [A. 3, A. 4], not

all of which involved models with large numbers of terms. Even

Stewart [A. 1] states that the normal equations are "tricky to

use,

even if double precision is used on the computer, since the

condition of the normal equations may be even worse than the

condition of the original least squares problem. A numerically

stable method of calculating the solution to the linear least

squares problem can be obtained using a method of

orthogonal izat ion known as the QR decomposition.

A. 2 THE OR DECOMPOSITION

To present the details of the QR decomposition, it is again

necessary to review several concepts from linear algebra [A. 1,

A. 2]. Most of these deal with the properties of orthogonal

vectors, matrices, and transformations which play a central role

in the theory of the decomposition described here. Orthogonal

transformations are inherently stable and have low storage

requirements and are therefore of great interest in numerical

159
linear algebra.

Constructing the QR decomposition requires the use of an

orthogonal transformation which is called a variety of names

including an elementary reflector, an elementary orthogonal

matrix, or an elementary Hermit ian matrix. These transformations

are matrices of the form,

!H = 0 - 2 u u\ (A. 6)

where u is an n-vector which satisfies

II u II = u

T u = 1 , ( A. 7

)

2 and fl is the identity matrix. In general, a matrix (D is

orthogonal if

Q' 1 = Q

T (A. 8)

and

II Q y II = II <D T y II = II y II (A. 9)

2 2 2 for any vector y. The matrix IH is called an elementary orthogonal

matrix because it satisfies

IH*

1 = |H T = IH, (A. 10)

which can be proved through the use of eq (A. 7). Finally, IH is

completely determined by the vector, u. This is helpful in

computations since only u need be stored.

Recall from basic linear algebra that the method of Gauss

elimination can be performed by applying a series of

transformations which transform a matrix into one that is in upper

triangular form; that is, the transformations introduce zeros into

the matrix below the main diagonal. Similarly, we can choose the

form of u in eq (A. 6) so that when a series of these

transformations are applied to a matrix it is reduced to upper

triangular form. The particular form of IH which allows us to

perform this operation is known as a Householder transformation.

160
Without proof, we define the Householder transformation as

1

[H = 0 - - v v\ (A. 11)

0

Details regarding the proof of orthogonality, etc.

,

can be found

in [A. 2]. The terms in eq (A. 11) are defined as

0 = a (a - x ) (A. 12)

1 a = II x II (A. 13)

2 v = (x - a, x . x, x )

,

(A. 14)

1 2 3 n where x is the first column of the matrix A which is to be

reduced. Finally, the vectors u and v are related by

1 u = v, (A. 15)

Itvll 2 which is simply a normalization of v. Iterative formulas similar

to eqs (A. 12) - (A. 14) are given in [A. 2] for constructing the

Householder transformation for the remaining columns of A.

Now, if A is an m x n matrix, we can apply a series of

Householder transformations to reduce A to triangular form. We

apply the first transformation, IH , to introduce zeros in the

first column of A below (A) . A second transformation, IH , is

11 2 then applied to IH A, which introduces zeros in the second column

of IH^A below (IH^A)^. Continuing this sequence n

- 1 times, we

arrive at the matrix IR, which is upper triangular with a nonzero

diagonal

,

IH

• • • IH IH IH A = IR. (A. 16)

n-1 321

Now we take advantage of the property of elementary orthogonal

transformations given in eq (A. 10) to write

A = IH'

1 IH'1 IH'1

• •

• IH'

1 123 IR n-1

= IH IH IH • •• IH IR. 123 n-1

(A. 17)

161
Now, let us define

Q = EH EH EH

• • • IH , 1 2 3 n-1

where Q is orthogonal based on the properties of the !H

j then have the orthogonal decomposition

(A. 18)

s. We

A = (D IR. (A. 19)

Since IR is an m x n upper triangular matrix, we can partition it

as

IR =

IR

1 0

(A. 20)

We now return to the linear least squares problem where we

wish to minimize the residual

r = b - A x.

Multiplying eq (A. 21) by Q

T yields

Q

T r = 0T b - QT A x.

(A. 21)

(A. 22)

If we define the partitioned product

j

Q b =

then eq (A. 22) becomes

c

d

<D r =

c

d j - q

t A

Now, since A = Q ER and Q is orthogonal (Q

T = Q1 ),

°

T r =

[ d ]

' R xSubstituting the partitioning of eq (A. 20) we obtain

(A. 23)

(A. 24)

(A. 25)

Q

T p

IR.

o

c - ER x

1 (A. 26)

162
Now, let us form the expression for the norm of eq (A. 26) where we

can take advantage of the orthogonality of Q defined in eq (A. 9),

II Q

T r II 2 = II r II 2 = II c - [R x II 2 + II d II 2

. (A. 27)

2 2 12 2

Now, since d is "fixed" we can minimize the residual by finding x

2 such that II c - IR x II = 0. This is a straightforward task since

IR is n x n\ i.e., we simply solve the determined system

1 c - IR x = 0. (A. 28)

Note that we can also directly calculate the residual since, if we

solve eq (A. 28),

II r II 2 = II d II 2

. (A. 29)

2 2 Also, from eq (A. 26) with c - IR x = 0, we can form r from

0

T r = r ° 1. (A. 30)

Since Q is orthogonal we can directly calculate r as

r = Q [ ° 1. ( A. 31

)

The description of the QR decomposition given above provides

the outline for an efficient algorithm. Well developed, stable

algorithms for the decomposition have been included in the Linpack

package of linear algebra subroutines [A. 5]. The Linpack package

was used throughout this dissertation for solving both the linear

least squares problem and the linear portion of the nonlinear

least squares problem.

163
A. 3 ELEMENTS OF THE MATRICES

In Section 2.2, the strain gage analysis was formulated as a

linear least squares problem,

ID c = b. (A. 32)

In this section we define the elements of the matrix and vectors

in eq (A. 32). First, we define the multiplier matrix ID using the

definitions of the f and g from Section 2.2. The lowermost

J j subscript indicates the gage number:

(A. 33)

g. g. g.

The vector c contains the unknown series coefficients,

and the vector b contains the data from the n strain gages,

(A. 34)

(A. 35)

164
REFERENCES

[A. 1] Stewart, G. W. , Introduction to Matrix Computations,

Academic Press, New York, New York (1973).

[A. 2] Leon, S. J. , Linear Algebra with Applications, Macmillan

Publishing Co., New York, New York (1980).

[A. 3] Eason, E. D. , "A Review of Least-Squares Methods for

Solving Partial Differential Equations, " International Journal for

Numerical Methods in Engineering, Vol. 10, pp. 1021-1046 (1976).

[A. 4] Longley, J. W. , Least Squares Computations Using

Orthogonal izat ion Methods, Marcel Dekker, Inc. , New York, New York

(1984).

[A. 5] Dungarra, J.J., Moler, C.B.

,

Bunch, J.R.

,

and Stewart,

G. W. , Linpack Users Guide, Society for Industrial and Applied

Mathematics,

1979.

165
166

Fig.

A.1

Graphical

interpretat

ion

of

the

linear

least

squares

problem.
APPENDIX B

THE NONLINEAR LEAST SQUARES PROBLEM

167
In Chapter 3, an algorithm was presented which took advantage

of the spatial variations in strain to determine the fracture

parameters. The resulting system of equations was overdetermined

and nonlinear in one of the three variables. In this appendix, we

discuss the theory of nonlinear least squares problems, present

the Newton-Rapheson (or Newton) method for iteratively solving the

system of equations, and provide the elements of the system

matrices in Section 3.3.

B. 1 THEORY OF NONLINEAR LEAST SQUARES

The general problem of determining the least squares solution

to a system of equations

y = f(t) (B. 1)

requires us to minimize the 2-norm of the least squares residual

r = II r II 2 = II y

- f(t) II 2

, (B.2)

2 2 where t are unknown parameters, some or all of which are related

nonlinearly to y; and the defintion of the 2-norm is given in

Appendix A. Although the general problem presented here holds

many parallels to the linear least squares problem presented in

Appendix A, one important difference is the fact that we may find

many local minima to eq (B.2). In linear least squares analysis

we are guarnteed to only have one minimum. This fact, together

with the inability to analytically solve the normal equations for

the system, renders nonlinear least squares analysis considerably

more difficult than its linear counterpart.

At a minimum, say at t = t , the derivative of r must satisfy

Sr

St t=t

*

0 . ( B. 3

)

From eqs (B.2) and

which were shown

(B.3), one can arrive at the normal equations

in Appendix A. However, here the normal

168
equations can not be solved analytically, and we are forced to use

iterative techniques.

B. 1 THE NEWTON-RAPHSON ITERATIVE PROCEDURE

A variety of techniques are available for the solution of

systems of nonlinear equations. Here we will show the details of

a solution using the Newton-Raphson, or Newton, method. Details

of other methods can be found in [B.l - B.3]. Although the

Newton-Raphson method is perhaps the most elementary of the

methods, the algorithm has been found to work well in a variety of

applications

.

Restating the problem, we wish to find a zero of the residual

function

r = y

- f(t) = 0. (B. 4)

Expanding eq (B.4) into a Taylor series and retaining only the

first two terms we obtain

r

i+1

r +

i

df

At,

at

(B.5)

where i is the

parameters t.

linearization,

iteration step and At is the

Since the desired result is r

i+l

increment in the

= 0, we obtain the

af

-r . = At. (B. 6)

1 at

Note that eq (B.6) is simply a multivariable form of the well

known Newton’s method of finding roots of equations. With the

linearization of eq (B.6), we can solve for the increments in the

parameters, At, using the methods presented in Appendix A. The

solution is then iterated until a reasonable measure of

convergence is attained.

169
B. 3 ELEMENTS OF THE MATRICES

In this section we will present the details of implementing

the Newton-Raphson method for the strain field equations presented

in Chapter 3. Referring to eq (B.5), we recast the strain

response equation, eq (3.50), as

(g.). = (A f ). + (A f ). - 2pc , (B.7)

j

i 0 0 i 111 g

J J j where i is the iteration counter and j is the gage number. From

eq (B.6) we then obtain

dg,- r ; (gj. =

3 A

CH i AA +

0 3 A

i AA +

1

3g ;

36

A0 ( B. 8

)

for the i th iteration. From eq (B.7), we can evaluate the partial

derivatives in eq (B.8), —3g

= f (B. 9

)

« . 0

dg

= f (B. 10)

1 a a

i

a g af af

= a — + a — . (B. 11)

36 ° 36 36

Equations (B.9) and (B. 10) can be evaluated by inspection from eqs

(3.38) and (3.39) for the single-element -gage or eqs (3.46) and

(3.47) for the rosette. Equation (B. 11) requires performing a

partial derivative with respect to 0. For the case of the

single-element gage, the required expressions are:

170
3f dr —0 = - 0.5 r‘

3/2

1 30 30

cos(0 /2) A + 2A sin(0 /2)sin2a

1 1 1

- 1/2 _ + r

l 0

30

1 -0.5 sin(0 /2)A + A cos(0 /2)sin(2a) i 1 M L

1 11 J

dr

p -0.5 2 0^

^ -0^ cos( 0

2/2) cos(2a)

- 2A sin(0 /2) sin(2a)

1 2

[»

30

+ r"

1/2

0

2

I 0.5 j3 sin(0 /2) cos(2a)

2 1 30 1 3 2

- A^ cos(0

2/2)sin(2a)

J

(B. 12)

3f _ dr —1 = 0.5 r"

/2

1 30 30

•*.[ cos(0 /2) A - 2A sin(0 /2)sin2a>

1 1 1

30 p + r

1/2

g

1 -0.5 sin(6 /2) A

1 1 89 L1

- A^ 003(0^2) sin(2a)

]

dr j- + 0.5 r

1/2 2 0 - 0 cos (0 /2) cos(2a)

2

se

1 L3 2

+ 2 A sin(0/2) sin(2a)

1 2

171
50

1/2 0

2 + r 8 —

2 1 ae

0.5 ,8^ sin(0^/2) cos(2a)

+ A cos(0 /2) sin(2a)

1 2 (B. 13)

For the case of the rosette, the required expressions are:

5f dr —o = _ r

-J/2 _1 p

30

1 30

1 - (1+A ) cos(0 /2) cos2a

1 1

2A sin(0 /2) sin2a

1 1

50 r 2r

1/2

13

1 0.5 (1+A

2

) sin(0 /2) cos2a

1 1 50 L

1 1

- A cos(0 /2) sin2a

1 1

5r

+ r'

3/2 —2 p

2 30 1 l

-(3^ cos (0^/2) cos2a

+ 2A sin(0 /2) sin2a

1 2

50

+ 2r'

1/2

13 2

2 1 30

-0.5 /3 sin(0 /2) cos2a

3 2

+ A cos(0 /2) sin2a

1 2 (B. 14)

172
3f 3r

1 - 1/2 _ r

1 30 dQ

1 ^i[

- (i+x

i

) 003(0^/2) cos2a

+ 2A sin(0 /2) sin2<x

1 1 J

30 r 2r

1/2

/3

1 0.5 (1+A ) sin(0 /2) cos2a

1 1 30 L

+ A

i 005(0^2) sin2aj

dr

p + r

1/2 2

13 (3 cos(0 /2) cos2a

2 30

1 L3 2

- 2A sin(0 /2) sin2a

1 2

30

+ 2r 1/2

(3 2

2 1 30 b 5 13 sin(0 /2) cos2a3 2

- A cos(0 /2) sin2a

1 2 ]• (B. 15)

where the (3 . and k have been defined in Chapter 3 and

A = k (A

2-A2

) + (1+A

2

) cos2a, 12 1 (B. 16)

30 A sec 0

j _ j 30 1 + (A tan0)

j

; J = 1, 2 (B. 17)

3r. cot0 esc 0 _ =

’ 7cot 2 e + 1 ,/2

; J = 1, 2 . 30

(B. 18)

Using eqs (B.6) - (B. 11) we can form the following system of

173
equations linear in the increments AA , AA , and A0 for the i 0 1 increment

:

. th

' ~8 1

r (f n ),

0 1 (f )1 1 (A f

9 +A f ’ ) 0 0 111

"g

2

( f ) 0 2 (f )

1 2 (A f’+A f’ ) 0 0 112

.

~ 8

" . (f ) 0 n (f )

1 n (A f’+A f’ ) 0 0 1 1 n

AA

AA

A0

(B. 19)

where for brevity we have dropped the iteration counter i and used

primes to denote the derivatives with respect to theta. Writing

eq (B. 19) in matrix form we obtain

b = M x. ( B. 20

)

Equation (B.20) is a linear least squares problem which is solved

using the methods of Appendix A. Upon solution of the system we

revise the values of A , A , and 0 and continue on to the next

0 1 iteration step. We then repeat the procedure until the changes in

A

q

, A^, and 0 are small and the convergence criteria of eq (3.60)

is satisfied.

174
REFERENCES

[B.l] Seber, G. A. F. and Wild, C. J. , Nonlinear Regression,

Wiley and Sons, New York, NY (1989).

[B.2] Rheinboldt, W. C. , Methods for Solving Systems of Nonlinear

Equations, Society for Industrial and Applied Mathematics,

Philadelphia, Pennsylvania (1974).

[B.3] Vanderplaats, G. N. , Numerical Optimization Techniques for

Engineering Design, McGraw-Hill, New York, New York (1984).

175
4. TITLE AND SUBTITLE

Study of Static and Dynamic Fracture Using Strain Measurements

5. AUTHOR(S) John R. Berger and James W. Dally

6. PERFORMING ORGANIZATION (IF JOINT OR OTHER THAN NIST, SEE INSTRUCTIONS)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BOULDER, COLORADO 80303-3328

7. CONTRACT/GRANT NUMBER

8. TYPE OF REPORT AND PERIOD COVERED

BL-114A

(5-90)

U.S. DEPARTMENT OF COMMERCE

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

BIBLIOGRAPHIC DATA SHEET

1. PUBUCATION OR REPORT NUMBER

NISTIR 3952

2. PERFORMING ORGANIZATION REPORT NUMBER

3. PUBUCATION DATE

September 1990

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (STREET, CITY, STATE, ZIP)

10.

SUPPLEMENTARY NOTES

11.

ABSTRACT (A 200-WORD OR LESS FACTUAL SUMMARY OF MOST SIGNIFICANT INFORMATION. IF DOCUMENT INCLUDES A SIGNIFICANT BIBUOGRAPHY OR

UTERATURE SURVEY, MENTION IT HERE.)

The analysis of strain fields surrounding both stationary and propagating cracks is presented,

Series expansions of the static and dynamic strain fields are developed. Gage orientation

angles are then studied to optimize the strain response. The orientation angles are found

to be dependent on gage type and material.

Algorithms are developed which use the temporal or spatial strain variations to extract

fracture parameters. The accuracy of the parameter determinations is shown to be excellent,

and limits are placed on the validity of the developed methods. The methods are then

applied to the analysis of a large scale crack arrest test conducted in a pressure vessel

steel. The behavior of the crack-tip position with time and the propagation toughness with

time, temperature and position are determined. From this information, details of the

conditions at crack arrest are extracted. The propagation toughness-crack-velocity

relation is then constructed.

12.

KEY WORDS (6 TO 12 ENTRIES; ALPHABETICAL ORDER; CAPITAUZE ONLY PROPER NAMES; AND SEPARATE KEY WORDS BY SEMICOLONS)

13. AVAILABIUTY

X

X

UNUMITED

FOR OFFICIAL DISTRIBUTION. DO NOT RELEASE TO NATIONAL TECHNICAL INFORMATION SERVICE (NTIS).

ORDER FROM SUPERINTENDENT OF DOCUMENTS, U.S. GOVERNMENT PRINTING OFFICE,

WASHINGTON, DC 20402. ORDER FROM NATIONAL TECHNICAL INFORMATION SERVICE (NTIS), SPRINGFIELD, VA 22161.

14. NUMBER OF PRINTED PAGES

188

15. PRICE

ELECTRONIC FORM