Astm stp 536 1973

Wnuk 64 Quasi-Static Extension of the Crack 65 Subcritical Growth 68 Subcritical Crack Growth Threshold for Fatigue Crack Propagation and the Effects of Load Ratio and Frequency-/?.. Bar

PROGRESS IN FLAW GROWTH

AND FRACTURE TOUGHNESS

TESTING STP 536

AMERICAN SOCIETY FOR TESTING AND MATERIALS

PROGRESS IN FLAW GROWTH

AND FRACTURE TOUGHNESS

TESTING

Proceedings of the 1972 National Symposium on Fracture Mechanics

A symposium presented by Committee E-24

on Fracture Testing of Metals, AMERICAN SOCIETY FOR TESTING AND MATERIALS Philadelphia, Pa., 28-30 Aug 1972

ASTM SPECIAL TECHNICAL PUBLICATION 536

J G Kaufman, symposium chairman

List price $33.25 04-536000-30

AMERICAN SOCIETY FOR TESTING AND MATERIALS

1916 Race Street, Philadelphia, Pa 19103

© by American Society for Testing and Materials 1973 Library of Congress Catalog Card Number: 73-76198

NOTE

The Society is not responsible, as a body, for the statements and opinions advanced in this publication

Printed in Baltimore, Md

July 1973

Foreword

The Symposium on Progress in Flaw Growth and Fracture Toughness Testing was presented 28-30 August 1972, in Philadelphia, Pa and was sponsored by Committee E-24 on Fracture Testing of Metals of the American Society for Testing and Materials J G Kaufman, Aluminum Company of America, presided

as the symposium chairman, and the six sessions were presided over by J L Swedlow, H T Corten, J E Srawley, R H Heyer, E T Wessel, and G R Irwin

Related ASTM Publications

Fracture Toughness Testing at Cryogenic Temperatures, STP 496

(1971), $5.00, 04-496000-30

Probabilistic Aspects of Fatigue, STP 511 (1972), $19.75,

04-511000-30 Fracture Toughness, STP 514 (1972), $18.75, 04-514000-30

Some Observations on Fracture Under Combined Loading—G H Lindsey 22 Correlation of Fracture Criteria 23 Example Plexiglass 26 Summary 31

Interaction of Cracks with Rigid Inclusions in Longitudinal Shear

Deformation II Further Results—G P Sendeckyj 32 Crack Between Two Rigid Inclusions 33 Pull-Out of Partially Bonded Fiber 39 Debonding of a Rigid Fiber 40 Discussion 42

Local Stresses Near Deep Surface Flaws Under Cylindrical Bending Fields—

M A Schroedl and C W Smith 45 Analytical Considerations 46 The Experiments 51 Results and Discussion 53 Summary and Conclusions 60 Prior to Failure Extension of Flaws in a Rate Sensitive Tresca Solid—

M P Wnuk 64 Quasi-Static Extension of the Crack 65 Subcritical Growth 68

Subcritical Crack Growth

Threshold for Fatigue Crack Propagation and the Effects of Load Ratio

and Frequency-/? A Schmidt and P C Paris 79 Results on Frequency Effects 80

vi CONTENTS

The Effect of Load Ratio 80 Load Ratio and Crack Closure 81

A Crack Closure Explanation of Data Trends for Load Ratio Effects 83

Conclusions 90

Overload Effects on Subcritical Crack Growth in Austenitic Manganese

Steel-/? C Rice and R I Stephens 95 Nomenclature 95 Material and Test Procedures 97 Test Results 101 Discussion and Results 108 Conclusions 111 Discussion 113

Effect of Multiple Overloads on Fatigue Crack Propagation in 2024-T3

Aluminum Alloy— V W Trebules, Jr., R Roberts, and

R W.Hertzberg 115 Nomenclature 115 Experimental Procedures 118 Testing Procedure 119 Test Results 120 Summary and Interpretation of the Multiple Overload Curve

Using Closure Concepts 139

Fatigue-Crack Growth Under Variable-Amplitude Loading in ASTM

A514-B Steel-/ M Barsom 147 Material and Experimental Work 149 Results and Discussion 155 General Discussion 161 Summary 162

Temperature and Environment

Effect of a Loading Sequence on Threshold Stress Intensity

Determination-W C Harrigan, Jr., D L Dull, and L Raymond 171 Experimental Procedure 172 Results 174 Discussion 178 Conclusions 180

CONTENTS vii

Fatigue and Corrosion-Fatigue Crack Growth of 4340 Steel at Various

Yield Strengths-^1 / Itnhof and J M Barsom 182 Materials and Experimental Work 183 Results and Discussion 192 Summary 204

Fatigue Crack Propagation and Fracture Toughness of 5Ni and 9Ni Steels

at Cryogenic Temperatures—/? / Bucci, B N Greene and

P C Paris 206 Materials 208 Specimens 211 Test Apparatus and Experimental Procedures 212 Experimental Results and Discussion 216 Summary 226

Methods

Some Further Results on J-Integral Analysis and Estimates—/ R Rice,

P C Paris, andJ G Merkle 231 The Double Edge Notched Plate in Tension 233 The Internally Notched Plate in Tension 234 The Notched Round Bar in Tension 235 The Remaining Uncracked Ligament Subject to Bending 235 Charpy and "Equivalent Energy" Toughness Measures 237 Estimates of/ From Single Points on Load Displacement Records 238 Summary 244

A Comparison of the J-Integral Fracture Criterion with the Equivalent

Energy Concept—/ A Begley and J D Landes 246 J-Integral 247 The Equivalent Energy Concept 248 /Ic and the Equivalent Energy Procedure in the Linear Elastic

Range 250 The Lower Bond Equivalent Energy Procedure and Approximate

J-Solutions 251 /Ic and Equivalent Energy for a General Load Versus Load Point

Displacement Curve 253 Examination of the Condition for Agreement of/jc and the

Equivalent Energy Procedure 255

A Graphical Interpretation of the Constant J/A Condition 258 Summary and Conclusions 259

viii CONTENTS

Analytical Applications of the J-Integral—/ G Merkle 264 Nomenclature 264 Current Approaches to the Development of Elastic-Plastic

Fracture Analysis 267 Discussion and Conclusions 279

Experimental Verification of Lower Bond K\c Values Utilizing the

Equivalent Energy Concept-C Buchalet and T R Mager 281 Equivalent Energy Method 282 Materials, Specimens, and Test Procedure 285 Method of Analyzing the Test Data 285 Experimental Results 286 Discussion 291

A Method for Measuring Kic at Very High Strain Rates—D A Shockey

and D R Curran 297

A Method for Achieving Very High Crack-Tip Loading Rates 298 Experimental Procedure 300 Results 303 Discussion 306 Summary 309

Influence of Stress Intensity Level During Fatigue Precracking on

Results of Plane-Strain Fracture Toughness Tests—/ G Kaufman

and P E Schilling 312 Material 313 Test Procedure 314 Results and Discussion 315 Conclusions 319

Materials

Influence of Sheet Thickness upon the Fracture Resistance of Structural

Aluminum Alloys-A M Sullivan, J Stoop, and C N Freed 323 Experimental Parameters 324

Applicability of a Model for the Sheet Thickness Dependency of Kc 329 Thickness Reduction and Crack-Tip Opening 330 Critical Crack Length at Various Levels of Operating Stress 331 Summary 332

CONTENTS ix

Plane-Stress Fracture Toughness and Fatigue-Crack Propagation of

Aluminum Alloy Wide Panels-Z) Y Wang 334 Test Program 336 Discussion 340 Conclusions 349 Fracture Toughness of Plain and Welded 3-In.-Thick Aluminum Alloy

Plate-F G Nelson andJ G Kaufman 350 Material 351 Procedure 353 Discussion of Results 357 Summary and Conclusions 374

Dynamic Tear Tests in 3-In.-Thick Aluminum Alloys—R W Judy, Jr.,

and R J Goode 377 Materials and Procedures 378 Discussion of Results 380 Conclusions 389

Structure of Polymers and Fatigue Crack Propagation-/? W Hertzberg,

J A Manson, and W C Wu 391 Experimental Procedure 393 Experimental Results and Discussion 394 Conclusions 401

Effects of Strain Gradients on the Gross Strain Crack Tolerance of

A533-B Steel-P N Randall and J G Merkle 404 Procedure 405 Experimental Results 411 Discussion 415 Conclusion 419

Applications

Applications of the Compliance Concept in Fracture Mechanics—

H Okamura, K Watanabe, and T Takano 423 Deformation of a Cracked Member 424 Analysis of Statically Indeterminate Structure Containing a

Cracked Member 427 Extension to the Multiple Loads 430

X CONTENTS

Analogy by Equivalent Electric Circuit 432 Application to the Vibration of Cracked Member 432 Deformation and Strength of Cracked Column Under Eccentric Load 433 Fatigue Crack Propagation and Final Fracture Under Various

Constraint Conditions at the Ends 435 Application to the Arrest of Brittle Fracture 436 Conclusion 438

Fracture Mechanics Technology for Optimum Pressure Vessel Design—

J G Bjeletich and T M Morton 439 Review of Fracture Mechanics 441 Discussion: Vessel Design Optimization 442

Failure Stress Levels of Flaws in Pressurized Cylinders—

/ F Kiefner, W A Maxey, R J Eiber, and A R Duffy 461 Description of the Experiments 462 Analysis of the Experiments 463 Summary 480

Experimentally Determined Shape Factors for Deep Part-Through

Cracks in a Thick-Walled Pressure Vessel—^? W Derby 482 Procedure 483 Results 484 Discussion 488 Summary and Conclusions 490

Introduction

This publication Progress in Flaw Growth and Fracture Toughness Testing contains the papers presented at the Sixth National Symposium on Fracture Mechanics at American Society for Testing and Materials Headquarters, in Philadelphia, on 28-30 August 1972 It was the first Fracture Mechanics Symposium conducted under the sponsorship of ASTM Committee E-24 on Fracture Testing of Metals, and so it was appropriate that one major thrust of the papers presented was the progress in testing reflected in the book's title

In essence, the volume provides the 1972 state-of-the-art in the analysis and measurement of fracture toughness and flaw growth resistance, and so it should prove useful to theoreticians and experimentalists in keeping abreast of developments in these fields The coverage includes items on the theory, analysis, and understanding of fracture behavior, including comment by Trebules and others on the crack closure phenomenon In the regime of new methods development, the attention to the J-integral should prove particularly interesting

to expermentalists, as considerable progress has been made by Rice, Paris, Landes, Begley, and Merkle in simplifying and presenting in a meaningful manner this new approach, first described at the Fifth Symposium The new papers on subcritical crack growth are also of considerable importance, as they add to our understanding of threshold behavior and of the usefulness of crack closure in interpreting stress ratio and overload effects

The use of the fracture mechanics approach to the evaluation of a number of relatively tough materials is given considerable attention in this volume; the real engineering problems in dealing in fracture mechanics terms with the kinds of materials of which most structures are made are faced Specifically, the problems

in interpretation of plane-strain fracture toughness test results for tough aluminum alloys and steels are described, and the useful relationships between such data to other fracture related indices are presented While no final answers are prescribed, the experiences of individuals who have had to deal with the problem today are discussed in the papers by Sullivan, Wang, Nelson, Judy, and Hertzberg, and their respective coauthors Similarly, the papers by Okamura, Bjeletich, Kiefner, and Derby provide some current thoughts on the application

of fracture mechanics concepts to design problems

The success of the Sixth National Symposium on Fracture Mechanics, as evidenced by this volume, could not have been achieved without the strong support of many individuals far too numerous to mention The encouragement

1

STP536-EB/Jul 1973

Copyright © 1973 by ASTM International www.astm.org

2 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

and advice of P C Paris, Chairman of ASTM Committee E-24 Symposium Test Group was a primary factor, but without the very hard work of the ASTM Staff

it could not have happened I believe the Sixth National Symposium, through its initiation of sponsorship by and cooperation with ASTM, was a hallmark in the advancement of knowledge on fracture mechanics

J G Kaufman

Alcoa Research Laboratories New Kensington, Pa

Theory and Stress Analysis

C L Ho,1 O Buck,2 and H L Marcus'

Application of Strip Model to Crack Tip

Resistance and Crack Closure Phenomena

REFERENCE: Ho, C L., Buck, O., and Marcus, H L., "Application of Strip Model to Crack Tip Resistance and Crack Closure Phenomena," Progress in Flaw Growth and Fracture Toughness Testing, ASTM STP 536, American Society for Testing and Materials, 1973, pp 5-21

ABSTRACT: This paper systematically investigates plasticity effects in fatigue crack propagation in structural materials These effects lead to the crack closure phenomena and the crack tip resistance to propagation due to residual compressive stresses A simple application of the strip-model, elastic-plastic analysis leads to the exact distribution of the residual stresses at the crack tip The compressive stresses are explicitly demonstrated and computed with their significance discussed Experimental techniques based on ultrasonic surface waves are used to directly measure the effects of the residual stresses Structural materials of aluminum, steel, and titanium are used for these studies The experimental data clearly reveal the extents of crack closure and crack tip resistance to applied loadings It is found that the amount of crack closure increases with the actual crack length before gross scale plastic flow occurs

From this investigation, it is particularly significant that the concepts of crack closures and crack tip resistive forces to propagation are not only demonstrated by direct experimental means, but also their extents can be quantitatively measured Such measurements will facilitate modifications in analytic models for predicting crack growth rates

KEY WORDS: fracture properties, mechanical properties, crack propagation, residual stress

The present paper is concerned with continuum plasticity effects which have long been recognized to play an important role in fatigue crack propagations in common structural materials [1 ] 3 These effects arise due to stress concen-

1 Formerly, member of Technical Staff, Science Center, Rockwell International, Thousand Oaks, Calif 91360; now, medical student, Pritzker School of Medicine, The University of Chicago, Chicago, 111 60637

2 Member of Technical Staff and group leader, respectively, Science Center, Rockwell International, Thousand Oaks, Calif 91360

3 The italic numbers in brackets refer to the list of references appended to this paper

STP536-EB/Jul 1973

Copyright © 1973 by ASTM International www.astm.org

6 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

trations at crack tips [2] coupled with the material ductility As the high level of stresses induced there by the applied load reaches the yield strength of the material, local plastic flow occurs even under loads within the elastic range More precisely, for fatigue cracks under cyclic tensile loads, the plastic zone formed by the increasing part of the load cycle will be called the "forward yield zone" and that caused by the unloading part the "reversed yield zone" embedded in the former[3,4] In the forward yield zone, permanent sets of plastic tensile straining occur which will subsequently experience compressive actions by the surrounding elastic matrix during unloading Thus, after a complete cycle of loading and unloading, the material in the immediate neighborhood of the crack tip is in a state of plastic tensile strains and compressive stresses This state will be called the residual deformation whose effects on fatigue crack propagation based on presently obtained experimental data are of interest here

Each complete load cycle will set up a residual deformation As the crack grows during subsequent load cycles, it extends into regions of such deformation due to previous load cycles, associated with which there are two important mechanical quantities to be considered They are the distribution of residual stresses and that of residual strains The former will be shown in the next section

to vary from compression to tension while the latter is tensile in nature due to plastic permanent sets in the forward yield zone Hence, during the course of crack propagation, there will be accumulation of residual tensile strains along the crack surfaces, and the compressive stress of each load cycle will resist the applied loads of the next cycle Their combined effects are to render the loads less effective for crack extension and to cause crack closure behaviors [5-8] The influence of residual strains on fatigue crack was discussed in a previous paper [8]; that of residual stresses will be the subject of this work

In the present case, the experimental evidence for crack closure is given by acoustic measurements[7,8] in part-through crack specimens The crack interferes with the acoustic wave and crack closure is indicated by a received signal which is higher with the crack partially closed than with the crack fully open

Crack Tip Resistance and Closure

The nature of residual stresses can be seen by a simple application of the Dugdale crack hypothesis (also known as the strip model [4,9]) to the residual deformation of the first complete loading cycle applied to a virgin material containing a crack This model is chosen because its predictions agree quite well with test data in certain situations[/0] and its simplified elastic-plastic analysis leads to an exact mathematical solution for our problem under consideration Dugdale's model treats a plane problem by confining the plastic deformation

HO ET AL ON CRACK CLOSURE PHENOMENA

11 II I'M 111

iisiii-

y

» a-

FIG 1—Dugdale crack model

to a thin strip extending out of the crack tip As shown in Fig 1, this is done by subjecting a traction free crack of length 2a in an infinite medium to a remotely applied stress, a«., and a distributed load of the material yield strength, ay, over the plastic strip, fy, which will be taken here as the forward yield zone size The reversed yield zone for unloading will be denoted by ry as in Fig 2 which schematically represents the problem of residual deformation after a complete load cycle: a tensile load cycle is given in Fig 2a The situation corresponding to the increasing part of the load cycle up to the maximum value of a„ is described

by Fig 2b, while that for unloading by Fig 2c Because the yield criterion of plastic deformation in the strip model is expressed in terms of the boundary condition of tractions along the plastic zone and because the field equations of elasticity are used, the elastic-plastic analysis is, in fact, linear in nature Hence, superposition of the solutions for loading and unloading so as to produce the

PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

I TvM

State of Unloading (c)

state of residual deformation of zero applied load (Fig 2d) is allowed The details of the derivation are given in the Appendix

We are chiefly interested in the residual stress component r which is normal

to the y plane containing the crack Taking into account the symmetry of the problem, we can express the stress distribution along the x axis[4]:

T = —<Jy, for 0 <JC < r y

HO ET AL ON CRACK CLOSURE PHENOMENA

A semilog plot of the normalized residual stress distribution Eq 1 for various load levels is shown in Fig 3 This indicates that the range (along which compressive stresses occur ahead of the crack tip) extends itself beyond the reversed yield zone somewhere into the forward plastic zone when the stress becomes tension which attains a maximum value at the end of the forward yield zone beyond which it rapidly decays off Furthermore, the range of existence of the compressive stress increases with, and the peak tensile stress decreases with, the load ratio, a^/ays

The previous discussion is based on the residual deformation of the very first load cycle The strip model, however, is too simple to treat the accumulated result of residual deformations due to many load cycles, because it predicts that the solution for subsequent cycles is identical to that of the initial one [4] Such

a shortcoming of the present theory could be removed by taking into account (1) the incremental crack growth for each load cycle, and (2) the strain

10 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

The residual compressive stresses in the neighborhood of the crack tip tends

to resist the applied tensile load upon reloading, because part of this load must

be consumed to overcome the resistance before becoming effective for further crack growth

In fact, as the crack propagates into the region of residual deformation during subsequent load cycles, the residual compressive stresses (together with the tensile strains) close portions of the opposite crack surfaces, leading to the crack closure concept[5,6] As will be discussed in the section on experimental procedure, experimental data show that at low levels of the applied load, the two crack faces near the tip would appear partially closed As the load increases, it gradually overcomes the resistance until a crack opening load, P0 (Fig 4), is reached when the crack becomes fully open and stays open for the rest of the rising load cycle The situation then reverses itself during unloading Hence, the apparent crack length of fatigue cracks would vary with the applied load, P; and the true, geometric crack length, a0, is the one when the crack is fully open foxP

> P0 This is schematically shown in Fig 4 where the apparent crack length, a, versus the applied load, P, for any single load cycle is given This has been

HO ET AL ON CRACK CLOSURE PHENOMENA 11

S a min

min o max

APPLIED LOAD, P

FIG 4-Schematic description of apparent crack variation with applied load

verified for crack propagations in aluminum alloys by direct experimental evidence presented in the previous work[7,8], using ultrasonic techniques Similar results from other materials in addition to aluminum alloys will be given later in this work to study the residual stresses and the associated crack closure concept The magnitudes of the amount of crack closure Aa and the extent of resistance AP due to crack tip plasticity effects can be quantitatively measured by our experimental technique

Experimental Procedure

The experimental technique of the present work has its origin in Ref 7 which describes a simple application of ultrasonic surface waves for continuously monitoring crack propagations The test arrangement is schematically shown in Fig 5 where a part-through crack (PTC) specimen with a semielliptical pre-crack

of depth, a, and length, 2c, (formed by electro-discharges (Elox)) is used A lucite holder which contains ceramic transmitter and receiver transducers (PZT5)

of about 1 cm diameter is attached to either the front or the back face of the specimen, depending on the depth of the propagating crack so as to maximize the attenuation of the surface wave by the crack In our tests, this was done by mounting the transducers on the front face for shallow flaws, namely, when the crack depth is less than half the thickness of the specimen and on the back face for deep flaws The wedge angle OR is chosen always to maximize the wave energy along the surface of the test specimen For aluminum, steel, and titanium alloys tested here, the angle 6R equals 27 ± 2 deg

12 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

FIG 5-Schematic description of experimental arrangement

The typical specimen in use has dimensions: 1.25 cm thick (r), 10 cm wide (W), and a gauge length of about 20 cm long (L), while the pre-crack is 1.2 mm

in depth (a) and 7 mm in length (2c) According to programmed loading spectra and frequencies, the specimen is cyclically loaded in an MTS electrohydraulic system perpendicular to the crack plane which is, in turn, parallel to the wave front of the surface wave The crack, therefore, partially scatters the incoming wave with the attenuation of the received signal depending on the apparent flaw depth, a, which is measured from a calibration curve of crack depth versus received signal strength of the surface wave; two calibration curves are used for

HO ET AL ON CRACK CLOSURE PHENOMENA 13

this work, one associated with the front-face mounting of the lucite bridge and the other associated with the back-face mounting These curves are, however, always calibrated by using the true crack depth, a0

In view of the above, by continuously recording the received wave signal and its corresponding load level and with the aid of the calibration curves, we can detect the amount of crack tip resistance and the variation of the apparent crack depth for a complete load cycle at any stages of crack propagation (Fig 4) The amount of crack closure Aa and the extent of resistance AP associated with such

a load cycle (Fig 4) estimate the strength of crack tip compressive stresses The present experimental system has been successfully applied for this purpose at a transducer frequency of 500 kHz

References 7 and 8 should be consulted for more details of the experimental arrangement Allied test techniques can also be found in Refs 14 through 18 Experimental Results and Conclusions

As the crack moves through the displacement field of the surface wave in part-through crack specimens, the attenuation of this wave by the crack is used

as a direct measure of crack depth Looking at the amplitude of the received wave as a function of the applied load during any load cycle exhibits the crack closure effect: the received amplitude is larger at small loads than at high loads This observation is interpreted as an indication that at low loads the crack is tightly closed over at least part of the fracture surface and is completely open only at high load levels as shown schematically in Fig 4 Such an effect is readily explainable in terms of (1) Elber's crack closure concept [5,6,8] with residual tensile deformations left behind the moving crack tip and, (2) by the state of compressive stresses in the reversed plastic zone, as discussed in the present paper

Experiments on the phenomena of crack closure using surface waves were carried out in steel, titanium, and aluminum alloys The results on the amounts

of crack closure, Aa, and resistance, AP, for several values of the actual crack depth, a0, are given in Figs 6 through 8

The magnitudes of Aa and AP are determined by taking a0 and P0 as the initial tangent point to the experimental a versus P curves in the increasing direction of P These measurements indicate that Aa is close to ry along which residual compressive stresses predominate (Fig 3) Because of this observation and Eq 3, Aa is expected to increase with the true crack deptha0 This is indeed observed as shown in Figs 9 through 11 for steel, titanium, and aluminum specimens being considered here The effect that Aa is an increasing function of

a0 is lost when the remaining ligament between the crack and the back face of the PTC specimen undergoes gross plastic flow, because the restraints from the surrounding elastic matrix disappear in those circumstances

The retardation phenomena at the end of multiple peak overloads in fatigue

14 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

0.4 0.3 0.2 0.1 3

crack propagations, as reported in Refs 8, 11-13 can also be explained by the crack tip residual stresses[12] In Ref 8 it was pointed out that rather than responding immediately to new lower loading conditions, the material retained the plasticity character of the previous loading, for many new loading cycles Hence, immediately after the overloads, the new load has to work against the residual deformation of the peak overloads Associated with this residual deformation, the range of the compressive stress distribution in front of the crack tip is larger while the peak tensile stress is smaller, according to Fig 3 and

HO ET AL ON CRACK CLOSURE PHENOMENA 15

FIG 8-Crack closure behavior in aluminum 2024-T851

Eqs 1 and 3 Therefore, immediately after lowering the load level the new applied load is, in fact, being more effectively impeded by a residual deformation in front of the crack tip caused by the previously higher applied load, as the crack grows across the plastic zone This leads to excessive retardation as observed in Ref 8 It took many load cycles before the material

TRUE CRACK DEPTH a Q (inch) 0.09 0.10 0.11 0.12 0.13 0.14 0.15 0.16 0.17

0.125 r^ , , 1 , -T- 1 , , 1 ,0.05 0.100

TRUE CRACK DEPTH a Q (cm)

FIG 9—Crack closure dependence on crack depth in steel 4340

-0.04 '

0.03 *

UJ

0.02 § 0.01 §

16 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

TRUE CRACK DEPTH a Q (inch) 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.08

0.06; -0.04S 0.02 :

5 0.20 0.25 0.30 0.35 0.40 0.45 0.50

TRUE CRACK DEPTH a Q (cm)

FIG 10—Crack closure dependence on crack depth in titanium-6-4

responded to the new loading conditions This indicates that the crack closure is important but that the actual governing of the retardation is due to the residual stress field in front of the crack

In conclusion, the hypothetical effects of the residual deformation due to crack tip plasticity have been supported by experimental observations here and

in Refs 7 and 8 according to the interpretations given above Their quantitative

\ 0 1 1 1 1 1 1 i i

0.10

- 0.08 0.06

- 0.04 0.02

g

0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50

TRUE CRACK DEPTH a 0 (cm) FIG 11-Crack closure dependence on crack depth in aluminum 2024-T851

HO ET AL ON CRACK CLOSURE PHENOMENA 17

measurements in terms of Aa and AP, we believe, will facilitate modifications in analytic models for predicting crack growth rates and improve our ability to analyze fracture mechanics data Such efforts are currently under investigation Acknowledgments

This work was sponsored in part by the Interdivisional Technology Program

of Rockwell International The authors gratefully acknowledge the experimental help of R V Inman and J D Frandsen

APPENDIX

The following derivation for the residual stress distribution given in Eq 4 will

be based on results of Rice[4] after corrections in the representation of stresses

by stress functions are made To make the present work self contained, we apply the Dugdale crack problem of Fig 1 to the configuration shown in Fig 12a where the coordinate position and notations for the stresses are introduced Since the yield criterion of plastic deformation in the strip model is chosen as the boundary condition of tractions along the plastic zone, there is actually no nonlinearity involved, as indicated before Hence, the problem of Fig 12a can be equivalently decomposed into two problems of Fig 12b and Fig 12c where the stress tensors are related by

18 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

symmetrically loaded about the x axis can be expressed in terms of Westergaard's stress function F(z)[19,20]:

j — a«,/2 for remote loading, perpendicular to the crack

C = \ (8)

0 for wedge loading normal to the crack while Re[F] and Im[F] stand for the real and imaginary parts of F, respectively; F\z) denotes the derivative df/dz In view of Eqs 4 through 8, we can write

The stress functions on the right of Eq 9 correspond to the problems of Figs

the inclusion of C in Eqs 5 and 8, these stress functions found in Ref 4 are herein modified to be

—aoo which was needed in the less general representation[4]

HO ET AL ON CRACK CLOSURE PHENOMENA 19

F(z) = ov tan l I (a+ry)

2-a2

(z+a)2 -(a+rv)2 (17) where ry is given by Eq 3 As pointed out before, the plastic deformation in Dugdale's crack model is expressed in terms of the boundary distributed forces prescribed along the plastic strip, the elastic-plastic analysis is, in fact, linear in nature and, hence, superposition of Eq 16 for the state of loading and Eq 17 for the state of unloading is allowed to render the state of residual deformation This is why —2y[3,4] is used in Eq 17 so that the reversed yielding strength will

be — ay after superposition, as shown in the diagram at the bottom of Fig 2 We also note that ry can be obtained from fy, if we replace in Eq 2 a„ and ay by

—Ooo and — 2ay for the state of unloading

Substitution of Eqs 2 and 3 into Eq 16 and Eq 17, respectively, and simplification lead to

20 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING and

Since the stress component normal to the plane of the crack is of importance and to simplify the mathematics, we shall only consider the distribution of this

loading, unloading, and residual deformation, respectively; we have from the boundary conditions (Fig 2), Eqs 6, 18, and 19, and the symmetry of the problem about the j> axis:

By superposition,

T = Ty + Ty (22) hence Eq 1 is derived

HO ET AL ON CRACK CLOSURE PHENOMENA 21

References

[1] Ronay, M in Fracture, Vol 3, H Liebowitz, Ed., Academic Press, New York, 1971,

p 431

[2] Irwin, G R., Handbuch derPhysik, Vol 6, 1958, p 163

[3] Rice, J R in Fatigue Crack Propagation, ASTM STP 415, American Society for Testing and Materials, 1966, p 247

[4] Rice, J R., Proceedings, First International Conference on Fracture, Sendai, Japan, Vol 1,1965, p 283

[5] Elber, W., Engineering Fracture Mechanics, Vol 2, 1970, p 37

[6] Elber, W in Damage Tolerance in Aircraft Structures, ASTM STP 486, American Society for Testing and Materials, 1971, p 230

[7] Buck, O., Ho., C L., Marcus, H L., and Thompson, R B in Stress Analysis and Growth of Cracks, ASTM STP 513, American Society for Testing and Materials, 1972,

[14] Bradfield, G in The Physical Examination of Metals, B Chalmers and A G Quarrel, Eds., Edward Arnold Publishers, London, 1960

[15] Rasmussen, J G.,Nondestructive Testing, Vol 20, 1962, p 103

[16] Klima, S J., Lesco, D J., and Freche, J C, NASA Technical Note D-3007, National Aeronautics and Space Administration, Washington, D C, Sept 1965

[17] Viktorov, I A., Rayleigh and Lamb Waves, Plenum Press, New York, 1967

[18] Lynworth, L C, Materials Evaluation, Vol 25, 1967, p 265

[19] Westergaard, H M., Journal of Applied Mechanics, Vol 6, No 2, 1939

[20] Sih, G C, International Journal of Fracture Mechanics, Vol 2, No 4, 1966, p 628

pp 22-31

ABSTRACT: Observations are made concerning comparative fracture behavior

of cracked and uncracked specimens under combined loading conditions A theory is proposed which provides a general fracture criterion for mixed mode crack extension and which unifies the treatment of elastic fracture problems in initially cracked and uncracked geometries All observations are supported by experimental fracture results taken from the literature on polymethyl- methacrylate and glass

KEY WORDS: cracks, failure, fractures, crack propagation, brittle fracturing polymethyl methacrylate, criteria, fracture properties, mechanical properties

Much of the traditional theory of fracture has been cast in terms of failure surfaces in principal stress space, which have been given physically descriptive names such as maximum principal stress, distortional energy, etc., and these theories have been applied to bodies that are assumed to contain no dominant flaws On the other hand, in more recent times an equally strong theory in the form of "fracture mechanics" has been developed upon the premise that a crack already exists within the body As a result of these two developments, fracture analysis is compartmentalized into problems of one type or the other

The two approaches have been further dissimiliar in the types of stress fields typically treated While the use of combined stress states is very common in the instance of failure surface theories, it is not at all common to apply fracture mechanics to combined loading conditions, and, in fact, a general criterion is not available where more than one mode of fracture is present In this paper an attempt is made to bridge between the two for elastic materials, linking them more closely together, and providing a more unified perspective By bringing

22

STP536-EB/Jul 1973

Copyright © 1973 by ASTM International www.astm.org

LINDSEY ON FRACTURE UNDER COMBINED LOADING 23

both of these theories together, a general fracture criterion has emerged for combined loading and mixed mode crack propagation

Work on the problem of crack propagation in sheets under combined loads was undertaken by Erdogan and Sih [1 ] ,2 who proposed an explanation of crack trajectories in plane two-dimensional stress fields based upon experimental results obtained with polymethylmethacrylate (PMM) Later Panasyuk et al[2] independently proposed similar explanations based upon their work with glass Cottrell[5] has examined the energy released from a system when a crack extends radially along an arbitrary trajectory He found that the direction of propagation hypothesized by the above investigators corresponds to the direction of maximum energy release, which is consistent with the general fracture theory proposed by Griffith

In the experimental studies that were conducted relative to the aforemen- tioned investigations, the stress conditions which precipitated crack instability were only of secondary interest; consequently, observations are offered herein to complete the picture by describing the local conditions of fracture at a crack tip

in an arbitrary two-dimensional stress field by relating them to fracture conditions in unflawed geometries This leads quite naturally to a fracture criterion representable in Ki-K2 space Verification of the theory has been demonstrated on PMM for two-dimensional plane geometries

Correlation of Fracture Criteria

For the unflawed case, that is, for specimens without cracks, fracture in multiaxial stress states is describable in terms of a surface in principal stress space Over the years, it has been found by experience that the continuous surfaces are quadric surfaces and are representable by second order equations in stress A very general representation of such surfaces can be made in terms of the first two stress invariants V\ and I2; however, it is physically more meaningful to use Ix and J2, the second deviatoric stress invariant The equation of the surface

in functional form is

where 1\2 is selected for dimensional homogeniety

Mathematical expressions such as Eq 1 represent the manner in which a material reacts fracturewise to applied loads and theoretically will predict fracture regardless of the geometry or character of the stress state A crack in a body introduces a geometrical change, but it should not influence the fundamental fracture behavior Therefore, a similar function should exist, which

is related to this one, and which describes fracture of cracked bodies However,

24 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

due to the singularity at the crack tip, there is always a region, usually very small, which has exceeded the fracture criterion of Eq 1 without producing specimen fracture When the loadingKlt K2 is such that fracture is imminent, there is a boundary r0 = r(0), which depends upon Kx and K2, where Eq 1 is satisfied At the same time there is a function of Kt and K2 that arises from fracture mechanics which must be satisfied at the crack tip when fracture is imminent It is this function of Ki andK2 that can be found via Eq 1

For geometries containing cracks, Barenblatt[4] has proposed that the local distribution of cohesive forces over the crack surface at fracture is a material property, independent of external loading and body geometry; thus, for any body constructed of a given material there exists a function of stress intensity factors which describes elastic fracture conditions For two-dimensional stress states limited to the opening and sliding mode, his criterion reduces to

where Kia and K2a are the portion of the stress intensity factors which are due

to the cohesive forces at the crack tip The total stress intensity factor is obtained by adding the cohesive force contribution to K± and K2, which result from the external forces

where /,/' =1,2

From this equation, one can solve for Kt and K2 in terms of the two in-plane principle stresses or in terms of the two stress invariants, both of which can be used to relate fracture of cracked geometries to fracture of uncracked

LINDSEY ON FRACTURE UNDER COMBINED LOADING 25

geometries Continuing in terms of invariants, Eq 6 can be used to formulate Ix2

1 and 8 describe a boundary between fractured and unfractured states, the function $ must coincide with / for r = r0 in order for the two fracture criteria

to be consistent along that contour Expressing this in functional form,

<*> (r„/i2, r0J2) = r0 \f{h 2, J2)] r=ro (9)

In other words, if the fracture surface /is known, the fracture condition, $, for bodies containing cracks may be obtained from / at r = r0 From this a general fracture criterion can be generated in terms of Kx and^T2 •

To illustrate, let / = J2 - C\ ; then at r = r0, 4> = r0(J2 - C\) Since / = O at fracture, $ will also be zero, which requires that r0J2 be equal to a constant From Eq 6

Evaluating this at a convenient point, we let K2 = 0 from which d0 = 0 and

Kl =Klc Equation 10fl then becomes

which is the desired fracture criterion for combined loading

26 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

Example Plexiglass

Figure 1 shows the uncracked multiaxial fracture behavior of PMM obtained

by Thorkildsen and 01szewski[6], where it has been plotted in two-dimensional principal stress space A more precise reduction of this data in terms of invariants makes possible the correlation with results obtained by Erdogan and Sih [1 ] on cracked specimens made from the same material

Stress invariants, /j and I2, are defined as:

Ii = Oi + a2 + o3

h = - (oi a2 + a2 a3 + a, a3)

A plot was made in Fig 2 of l2 versus I\2 for the experimental data, which appears to yield a clearly defined linear relationship The data was fitted by least squares to a linear equation, which resulted in

where Of is the failure stress in uniaxial tension This expression in terms of invariants is postulated to be the description of a failure surface for all possible tensile stress states Instead of leaving the final expression in terms of l2, which

HUBER-MISES- HENCKY THEORY

O LONGITUDINAL FRACTURE SURFACE

D CIRCUMFERENTIAL FRACTURE SURFACE

A NO PREFERENCE

4 6 8 10 12 14 16 NOMINAL AXIAL STRESS, <TX (KSI) FIG 1 -Fracture stress for PMM[6]

18 20

LINDSEY ON FRACTURE UNDER COMBINED LOADING 27

°~f

Hz)

crf2 = 2-93 crf2 0.96

0.2 0.4 0.6 0.8 1.0

FIG 2-Least squares fit to fracture data for PMM

has no particular physical interpretation, it is preferred to use J2, which is related to It and I2 by

28 PROGRESS IN FLAW GROWTH AND FRACTURE TOUGHNESS TESTING

in the same type of material In other words, for bodies containing a crack, fracture will occur by an incremental extension of the crack along the path normal to the greatest tension, and when it occurs, the stresses on that trajectory combine to satisfy the fracture criterion found for uncracked specimens at r =

r0 This requires that the direction of propagation, 0O, be known a priori; however, this information can be calculated from the work of Erdogan and Sih

By Eqs 8,9, and 13 propagation of a crack is given by

[r0J2 + 008/-0/i2 - Const] =0 (14)

To apply this criterion, the crack tip stress field is used The coordinate description is standard, and the equations are classical as given by Sih and Liebowitz[7]

a, = —l— Cos j |(3 - Cos 6)Ki + (3 Cos 6 - \)K2 Tan y] (15a)

oe= —^7 Cos 4 10+ Cos 6)Kl - 3K2 Sin 6] (156) 2(2r)/2 ^

T'0 = ^TW Cos T I*' Sin ° +K^3 Cos 6 ~ ])] (15c)

J2 in polar coordinates is given by,

J2 = j [a2

r - arae + a\ + 3T% ] (16) which in terms of stress intensity factors from Eqs 15a, b,c give

roJ* = "Z )| [7 + 4 Cos 0 - 3 Cos 26] K\ + [3/2 Sin 26 - Sin 6] KtK2

+ y [7 - Cos 0 - | Sin2 6]K2

2\ (17) From Eqs 15a,b,c, fj can also be readily computed,

[«!

rQl\ = 2 \K\ Cos2 y - KXK2 Sin 6 +K\ Sin2 y 1 (18)

Combining Eqs 17 and 18 for a surface of the general form that was exhibited

by PMM in Eq 14, we have

LINDSEY ON FRACTURE UNDER COMBINED LOADING 29

r0(J2 +0!])= T\\Y (7 + 4 Cos 0 - 3 Cos 20) + 6a (1 + Cos 0)1 K\

+ »y Sin 20 - Sin 0] - 12a Sin 0 A,A:2

+ |y (7 - Cos 0 - ! Sin20j + 6a(l - Cos 0)1 K\\ (19)

The constant in Eq 14 is found by examining the fracture condition of symmetric loading, for which A^ ->KXC, K2 =0 and 0O = 0 From Eq 19

[r0(J2 + a/2)] = Const =-7(1 + l2a)Klc2 (20)

6

Thus, the criterion of fracture for arbitrary combined loads, assuming a meanr0

obtained from opening mode failure, is given by

(1 + \2a)K\c = \(j + 6a) + P- + 6a) Cos 0 - | Cos 20 1 K\

+ |-| Sin 20 - (1 + 12a) Sin 0I AjAz + \jX +6OL\-(J + 6a\ Cos 0 - | Sin20| K\ (21)

where for PMM, a = 0.008

To limit check the skew-symmetric fracture condition, Kx = 0,K2 -* K2c and Erdogan and Sih's theory and measurements give 0O = —70.5° Equation 21 gives the critical K2 value in terms of the critical A\ value,

_|-3(l + 12a)f

*2C-[.4(l+3a)J KlC (22)

Substantiation of this prediction from the theory is provided by the experi- mental results of Erdogan and Sih, who found for PMM, Kxc = 472 lb in"3/2 and

K2C = 422.1 lb in"3/2 Using this value of A", c in Eq 22, and a = 0.008 from Eq

14, the theory predicts K 2 Q = 421.5 lb in~3/2 This very close correlation provides meaningful substantiation to the theory A check on r0 made from a comparison of Eq 11 and Eq 14 reveals it to be of the order of 10"3 in

Further evaluation of the theory can be obtained by plotting Eq 21 for a