Astm stp 995 v2 1999

Contents Overview ANALYSIS Experimental and Numerical Validation of a Ductile Fracture Local Criterion Based on a Simulation of Cavity Growth—JEAN-CLAUDE DEVAUX, FRANCOIS MUDRY, ANDR

Nonlinear Fracture Mechanics:

Nonlinear fracture mechanics

(STP; 995)

Papers presented at the Third International Symposium on Nonlinear Fracture

Mechanics, held 6-8 Oct 1986 in Knoxville, Tenn., and sponsored by ASTM Committee

E-24 on Fracture Testing

Vol 2 edited by J D Landes, A Saxena, and J G Merkle

"ASTM publication code number (PCN) 04-995002-30."

Includes bibliographies and indexes

Contents: v 1 Time-dependent fracture—v 2 Elastic-plastic fracture

1 Fracture mechanics—Congresses I Landes, J D (John D.) II Saxena, A

(Ashok) III Merkle, J G IV ASTM Committee E-24 on Fracture Testing

V International Symposium on Nonlinear Fracture Mechanics (3rd; 1986: Knoxville,

Tenn.) VI Series: ASTM special technical publication; 995

TA409.N664 1988 620.1126 88-38147 ISBN 0-8031-1174-6 (v 1)

ISBN 0-8031-1257-2 (v 2)

Copyright © by American Society for Testing and Materials 1988

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

Peer Review Policy

Each paper pubhshed in this volume was evaluated by three peer reviewers The authors

addressed all of the reviewers' comments to the satisfaction of both the technical editor(s)

and the ASTM Committee on PubUcations

The quality of the papers in this publication reflects not only the obvious efforts of the

authors and the technical editor(s), but also the work of these peer reviewers The ASTM

Committee on Publications acknowledges with appreciation their dedication and contribution

of time and effort on behalf of ASTM

Printed in Ann Arbor, MI May 1989

It was with great sorrow that we learned of the death of William "Gomer" Pryle on July

6, 1987 Although Gomer seldom sought or received much public recognition for his work,

he was a vital part of a team which advanced fracture mechanics from the earliest days We

have lost a great friend, one who enriched the lives of his fellow workers and made working

in fracture mechanics a constant pleasure

Gomer grew up near Pittsburgh, Pennsylvania, and served in the U.S Air Force from

1947 to 1951 He began his technical career at Westinghouse R & D Center in February

1952, where he continued working until his death During most of his career at Westinghouse

he was part of a widely recognized team, headed by Ed Wessel, which made numerous

contributions to the advancement of testing, analysis, and applications of fracture mechanics

technology Although his work is reflected in many places in the fracture mechanics

liter-ature, his contributions are not always readily apparent The work was presented

anony-mously and can be recognized only by those associates of his who remember his contributions

This is most notable in fracture toughness test standards where, beginning with the

devel-opment of the compact specimen and the ASTM E 399 Kx^ test standard, his work on

specimen design, machining, and precracking technique played a vital role in making this

standard a model for those which would follow He played a similar role in some of the

newer fracture mechanics test standards, contributing to ASTM Standards E 647, E 813,

cording, analysis, and reporting

• Development of modern precracking techniques, taking the process from the earliest

approach of thermal-mechanical induced cracking to the modern computer-controlled

fatigue precracking techniques

• Development of precracking techniques for difficult materials, including beryllium alloys

and ceramics

• Development of systems for identifying specimen size and orientation

• Development of modern specimen inventory control methods

• Author or coauthor of 36 fracture mechanics papers and reports, most notably ones

relating to the development of the compact specimen and the testing of large (12T)

compact specimens

Besides his technical career, Gomer was dedicated to his wife Barbara, his three children

Lynn, John, and Barbie, and his granddaughter Debbie He also showed his concern for

people through his association with the fracture mechanics family at Westinghouse He was

a continual source of encouragement, bringing hope with his familiar, "Hang in there

Tiger."

Now that he is gone, the world of fracture mechanics has lost a colleague whose

contri-butions have advanced the technology in more ways than can be counted Those of us who

knew him have lost a great friend; we will miss him

National Laboratory Both men, along with A Saxena, Georgia Institute of Technology,

served as editors of this publication

Contents

Overview

ANALYSIS

Experimental and Numerical Validation of a Ductile Fracture Local Criterion

Based on a Simulation of Cavity Growth—JEAN-CLAUDE DEVAUX,

FRANCOIS MUDRY, ANDRE PINEAU, AND GILLES ROUSSELIER 7

Numerical Comparison of Global and Local Fracture Criteria in Compact

Tension and Center-Crack Panel Specimens—FRANCOIS MUDRY,

FRANCOISE DI RIENZO, AND ANDRE PINEAU 24

Evaluation of Crack Growth Based on an Engineering Approach and

Dimensional Analysis—JEAN BERNARD 40

Comparison Between Experimental and Analytical (Including Empirical) Results

of Crack Growth Initiation Studies on Surface Cracks—

WALTER G REUTER 59

Defect, Constitutive Behavior, and Continuum Toughness Considerations for

Weld Integrity Analysis—PETER MATIC AND MITCHELL I JOLLES 82

Plasticity Near a Blunt Flaw Under Remote Tension—DENNIS M TRACEY AND

COLIN E F R E E S E 93

Nonlinear Work-Hardening Crack-Tip Fields by Dislocation Modeling—

FERNAND ELLYIN AND OMOTAYO A FAKINLEDE 107

FRACTURE TOUGHNESS

Geometry Effects on the /?-Curve—JOHN D LANDES, DONALD E MCCABE, AND

HUGO A ERNST 123

Evaluating Steel Toughness Using Various Elastic-Plastic Fracture Toughness

Parameters—ALEXANDER D WILSON AND J KEITH DONALD 144

Gas Welds—MICHIHIKO NAJCAGAKI, CHARLES W MARSCHALL, AND

FREDERICK W BRUST 214

Effect of Prestrain on the /-Resistance Curve of HY-100 Steel—ISA BAR-ON,

FLOYD R TULER, AND WILLIAM M HOWERTON 244

APPLICATIONS

A Viewpoint on the Failure Assessment Diagram—DONALD E MCCABE 261

Simplified Procedures for Handling Self-Equilibrating Secondary Stresses in the

Deformation Plasticity Failure Assessment Diagram Approach—

JOSEPH M BLOOM 280

Further Developments on the Modified /-Integral—HUGO A ERNST 306

Stable Crack Growth and Fracture Instability Predictions for Type 304 Stainless

Steel Pipes with Girth Weld Cracks—JOSEPH W CARDINAL AND

MELVIN F KANNINEN 320

A Methodology for Ductile Fracture Analysis Based on Damage Mechanics: An

Illustration of a Local Approach of Fracture—GiLLES ROUSSELIER,

JEAN-CLAUDE DEVAUX, GERARD MOTTET, AND GEORGES DEVESA 332

A Closer Look at Tearing Instability and Arrest—JAMES A JOYCE 355

Crack Growth Instability in Piping Systems with Complex Loading—

JAMES E NESTELL AND ROBERT N COWARD 371

Critical Depth of an Internal or External Flaw in an Internally Pressurized

Tube—BRIAN w LEITCH 390

Use of a Ductile Tearing Instability Procedure in Establishing

Pressure-Temperature Limit Curves—KENNETH K YOON, JOSEPH M BLOOM,

AND W ALAN VAN DER SLUYS 404

Strength Since Occurrence of a Postulated Underclad Crack During

Manufacturing—JEAN-CLAUDE DEVAUX, PATRICK SAILLARD, AND

ANDRE PELLISSIER-TANON 454

An Analytical and Experimental Comparison of Rectangular and Square

Crack-Tip Opening Displacement Fracture Specimens of an A36 Steel—

WILLIAM A SOREM, ROBERT H DODDS, AND STANLEY T ROLFE 470

M O D E L S AND MECHANISMS

Metallurgical Aspects of Plastic Fracture and Crack Arrest in Two

High-Strength Steels—JOHN p GUDAS, ROBERT B, POND, AND GEORGE R IRWIN 497

Effect of Fracture Micromechanisms on Crack Growth Resistance Curves of

Irradiated Zirconium/2.5 Weight Percent Niobium Alloy—

C K CHOW AND LEONARD A SIMPSON 537

A Combined Statistical and Constraint Model for the Ductile-Brittle Transition

Region—TED L ANDERSON 563

Kinetics of Fracture in Fe-3Si Steel Under Mode I Loading—

MICHAEL H BESSENDORF 584

Separation of Energies in Elastic-Plastic Fracture—MARION F MECKLENBURG,

JAMES A JOYCE, AND PEDRO ALBRECHT 594

INDEXES

Author Index 615

Subject Index 617

Elastic-plastic fracture mechanics (EPFM) had its birth in the late 1960s and early 1970s

After nearly two decades of steadily growing effort, the field has seen a maturing as well

as a change in emphasis EPFM developed in response to a real technology need The parent

technology, linear elastic fracture mechanics (LEFM), did not apply to many of the

engi-neering materials used in modern structures New and better materials were developed to

attain more ductiUty and higher fracture toughness, and where LEFM could no longer be

used for analyzing failures in these materials, EPFM provided the solution

To organize and document the results of the growing research effort in the field, ASTM

Committee E-24 on Fracture Testing sponsored the First International Elastic-Plastic

Frac-ture Symposium in Atlanta, Georgia, in 1977 The bulk of this symposium, as peer-reviewed

papers, is pubhshed in ASTM STP 668, Elastic-Plastic Fracture Subsequently, a second

international symposium on this subject was held in 1981, resulting in the two-volume ASTM

STP 803, Elastic-Plastic Fracture: Second Symposium

The 1980s saw a rise in more general interest in nonlinear fracture mechanics topics,

particularly time-dependent fracture mechanics It became apparent that the title for the

next symposium would have to be modified to include this emerging field As a result, that

symposium was called the Third International Symposium on Nonlinear Fracture Mechanics

and it was held in Knoxville, Tennessee, in 1986 This symposium, sponsored by ASTM

Committee E-24 and its Subcommittee E24.08 on Elastic-Plastic Fracture and Fully Plastic

Fracture Mechanics Terminology, featured both time-dependent and elastic-plastic topics

in fracture mechanics The time-dependent fracture mechanics papers (as peer-reviewed

papers) are pubhshed in Volume I of this Special Technical Publication (ASTM STP 995);

this book Volume II of ASTM STP 995, features elastic-plastic contributions to the

sym-posium

In the early years of the field, EPFM activities centered on the power generation industry,

particularly the nuclear power industry, where the needs for safety and reliabihty were at

an all-time high and a new level of technology was required to satisfy those needs The

earliest work concentrated on the development of characterizing parameters and the

de-velopment of test methods Debate was often centered on two or more candidate parameters

or test methods, among them the /-integral, the crack-tip opening displacement (CTOD),

and various energy approaches After more than a decade of this debate, it was recognized

that the leading candidate parameters were all related and the various test methods produced

complementary results Therefore, what was needed was not more work on basic approaches

but rather work on standardizing methods of testing and seeking new and better methods

of applying the technology

Test standardization for EPFM began with the development of the ASTM Test for 7i„ a

Measure of Fracture Toughness (E 813-81) in 1981, the year of the second international

symposium The ASTM Test for Determining J-R Curves (E 1152-87) was developed in

1987, and a standard on CTOD testing is presently in ballot The goal of applying the

technology was slower in developing However, with the ever-expanding capabilities of

modern computers, numerical solutions to nonlinear fracture problems become easier to

attain Most recently, interest in EPFM has expanded to the study of models and mechanisms

of fracture at the microstructural level

The elastic-plastic portion of the symposium included all of these topics of current interest

This volume is divided into four major sections, covering the topics of analysis, fracture

toughness, appUcations, and models and mechanisms

Analysis

The section on analysis contains a variety of new topics that make use of the capabilities

of modern computers One of the new areas included is that of the local criterion for fracture,

a topic that has gained in importance in recent years A feature of many recent studies is

the comparison between experimental and analytical techniques and results The

improve-ment in analytical capabilities has helped EPFM to grow and has been particularly helpful

in the development of application techniques

Fracture Toughness

The section on fracture toughness features results from experimental studies Some areas

of study include size and geometry effects, the effect of material quality, the effect of

prestrain, and the study of weldments Experimental results on the various fracture behaviors

continues to provide one of the cornerstones of the methodology, that of determining the

material behavior

Applications

The section on appUcations represents the largest section of papers, which suggests that,

at present, this is the most important aspect of EPFM development The list of topics for

application is still dominated by the interests of the power generation industry Components

include pressure vessels, pipes, and tubes, with an interest in welded components prevailing

Approaches to appUcation remain varied, ranging from well-documented ones—such as the

failure assessment diagram, tearing instability, and leak-before-break applications—to newly

developed methods being presented for the first time in this volume Many of the application

approaches are accompanied by experimental results to illustrate their success with the

particular problem addressed

Models and Mechanisms

The final section, on models and mechanisms, represents the newest area of interest in

EPFM It features the study of both metallurgical and microstructural features, as well as

models, based on macroscopic continuum aspects Although this section is the smallest one

in this volume, it is nevertheless an important one Use of a technology and its characteristic

parameters to formulate models and study mechanisms of behavior indicates a level of

confidence in that technology After nearly two decades of EPFM, this level of confidence

is evident from this volume and is one of its important results

nated by the interests of the power generation industry, new areas of interest are emerging, especially in critical structures for the defense industry Elastic-plastic fracture is still pro- gressing, and a fourth symposium will probably be needed in the not-too-distant future

Oak Ridge National Laboratory, Oak Ridge,

TN 37831; symposium cochairman and editor

Analysis

Simulation of Cavity Growth

REFERENCE: Devaux, J.-C, Mudry, R, Pineau, A., and Rousselier, G., "Experimental

and Numerical Validation of a Ductile Fracture Local Criterion Based on a Simulation of

Cavity Growth," Nonlinear Fracture Mechanics: Volume 11—Elastic-Plastic Fracture, ASTM

STP 995, J D Landes, A Saxena, and J G Merkle, Eds., American Society for Testing

and Materials, Philadelphia, 1989, pp 7-23

ABSTRACT: A local criterion based on the simulation of hole growth by plastic deformation

has been evaluated Fracture of a material volume is reached for an assumed critical value of

cavity growth This critical value is determined from notched tensile tests When deahng with

cracked geometries, a process zone is introduced at the crack tip This zone is modeled as the

first mesh element (Aa)c at the crack tip in a finite-element code The size of this element,

which is a material constant, is measured from a conventional compact tension (CT) test

Different tests with cracked geometries were carried out on side-grooved CT specimens of

different sizes (25 and 50-mm width) and on axisymmetrically cracked tensile bars (TA) with

15,30, and 50-mm outer diameters In all cases the fracture was flat with no shear lips Keeping

the parameters of the fracture local criterion constant, crack initiation and crack propagation

were modeled using the node release technique The numerical procedure and results are

described in detail The model results are shown to be in good agreement with the experimental

results

KEY WORDS: fracture local criteria, cracked round bars, ductile fracture, crack initiation,

stable crack growth, /-integral, A508 steel, fracture mechanics, nonhnear fracture mechanics

In a previous numerical work d'Escatha and Devaux [7] proposed a very simple criterion

to predict ductile failure in low-alloyed steels The growth of an assumed cavity in a mesh

element of a finite-element method (FEM) program was simulated using the formula

pro-posed by Rice and Tracey [2]

where

R= the radius of the cavity,

(T„= the hydrostatic stress,

defq = the incremental Von Mises equivalent plastic strain, and

(Tf,= the yield stress in a perfectly plastic material

Integrating this equation along the strain path yields an assessment of the cavity growth

ratio

L n ( | ) = / 0 2 8 3 e x p [ i f - ] e f , (2)

where Ro is the initial radius of the cavity

The failure criterion assumes a critical volume fraction {R/l)c where (lis the distance

between the cavities Here, £is also changed during straining

I = min, (o exp (e,) (3)

where e; is a principal strain, and fj, is the initial distance between the cavities

Since (Rli)o is a material constant, this criterion is equivalent to a critical cavity density

growth ratio [(i?/«„)/(K/Q],

D'Escatha and Devaux [1] showed that this very simple criterion can be used in a numerical

simulation of crack initiation and stable growth The cavity growth ratio is calculated in a

square mesh element at the crack tip This element, which is a representation of the process

zone, is a material constant During loading of the crack, the cavity growth computed at

the crack tip eventually reaches the critical value The node at the tip of the crack is released

This simulates a stable crack advance, the length of which is equal to the process zone size

Using this procedure, the stable crack growth in three-point-bend specimens with various

widths was simulated The results were found to be in qualitative agreement with known

experimental results, such as the specimen size effect Moreover, it was shown that an even

simpler criterion that does not include the distance between cavities was equivalent The

critical cavity growth ratio {R/Ro)c could be used instead of the critical cavity density growth

ratio

Starting from this work, more recently, the Beremin group investigated ductile fracture

of A508 steel, using quantitative metallography [3] This group showed that cavity nucleation

is negUgible, and that strain hardening must be taken into account in the formula for hole

growth [4] The simplest way to do so is to change from the yield stress to the actual flow

stress (Teq- However, a more sophisticated formulation was proposed to take strain hardening

more precisely into account It was based on the problem of a cavity in a spherical stress

field, the solution of which is given by Hill [5] Details are given in the Appendix It is

shown that both formulations are almost equivalent Quantitative measurements of hole

growth [3,6] are in good agreement with the prediction, except for the multiplicative factor

0.283, which seems to underestimate the experimental results Therefore, Eqs 1 and 2 can

be used to predict hole growth, provided that do is changed to o-jq- The results of the following

formula are proportional to the true cavity enlargement at failure

Ln(|-J = p^'0.283 exp 1.5a„ ^ < (4)

of the experimental results has already been given elsewhere [7] Moreover these specimens,

as well as side-grooved compact tension (CT) type specimens, were numerically calculated

The numerical results are compared with the experimental results Furthermore, we discuss

the benefits and drawbacks of this approach when compared with the more commonly used

techniques such as the /-integral

Experimental Program

All experimental procedures and results are given in detail in Ref 7 Here we concentrate

on the main results Three kinds of specimens were investigated:

1 A notched tensile bar with a 10-mm minimum diameter, an 18-mm outer diameter,

and a 2-mm notch radius The details of this specimen geometry, which was the one primarily

investigated, are given elsewhere [4,7]

2 Cracked round bars with an outer circular crack The crack was introduced using

rotative fatigue with an imposed deflection Figure 1 shows the final fracture surface aspect

Three different dimensions were used The outer diameters were 15, 30, and 50 mm,

des-ignated TA 15, TA 30, and TA 50, respectively The ratio of the diameter of the initial

uncracked ligament to the outer diameter was 0.555

3 Conventional CT specimens with 25% side grooves The specimens were tested using

the ASTM Test for /,„ a Measure of Fracture Toughness (E 813-81) Two specimen sizes

were used: 25 and 50 mm They were designated CTJ 25 and CTJ 50, respectively All

specimens were loaded at 100°C The material was a A508 forged steel taken from a nozzle

shell of a pressurized water reactor (PWR) nuclear vessel The chemical composition, heat

treatment, and inclusion content are given in Ref 7 Here, it is enough to say that the sulfur

content was 50 ppm The yield stress at 100°C was 450 MPa, the ultimate tensile strength

555 MPa, and the elongation 25%

Notched specimens were loaded to final fracture The minimum diameter was continuously

recorded Using this procedure, it is possible to define unambiguously the diameter <))« for

which initiation of final failure took place in the center of the specimen An average strain

at failure is defined as

e« = 2 L n f (5)

where <|)o is the initial minimum diameter For this steel, the experimental results are

FIG 1—Macroscopic aspects of the fracture surface of two cracked round bars: (a) specimen

loaded at 373K, unloaded and broken at 77K; (b) specimen broken at 77K to check circularity

of the fatigue crack

including the stretched zone width (Fig 1) Around crack initiation, the specimens were

unloaded, sectioned, and polished in order to measure the crack-tip opening displacement

(CTOD) from polished sections (Fig 2) Stable crack growth, Aa, was taken as the sum of

one half of the measured CTOD and crack advance due to ductile tearing With this

defi-nition, stable crack growth before crack initiation corresponds to the blunting line, which

was experimentally determined This procedure facilitates the measurement of crack

initi-ation

In CT specimens, / was determined using ASTM Test E 813-81 In cracked round bars,

/ was evaluated using the following formula

where K is the stress-intensity factor, v is Poisson's ratio, E is Young's modulus, P and dp

are the load and plastic displacement for a given point on the loading curve, and 4> is the

initial radius of the uncracked ligament This formula was found to be in reasonable

agree-lOOfim

FIG 2—Experimental determination and numerical simulation of crack-tip opening

dis-placement: (a) optical micrograph of a specimen loaded and section polished, (b) finite-element meshes and definition of crack-tip opening displacement (CTOD)

ment with numerical results using the M-integral, an equivalent to the /-integral for

axi-symmetric problems [9]

Crack growth resistance curves for all the cracked geometries tested are given in Fig 3

Table 1 gives all relevant results It is observed that the effect of specimen size is small

Moreover, both kinds of specimens yield similar results, which is rather surprising since the

round bars were heavily deformed under large-scale yielding This could be related to the

fact that the CT specimens contained side grooves It is worth recalling that the load is very

different for the different geometries This load is proportional to the square of the diameter

Figure 4« gives the loading curves for the CT specimens in nondimensional coordinates by

FIG 3—Experimental results of the crack growth resistance curves (J — Aa) TA stands

for cracked round bars and CTJ for compact tension specimens

on cracked

CTOD

at Initiation, H.m

fracture with no shear Ups In particular, no slant-type shear mode of fracture was noticed

It is also interesting to notice that for the two larger round cracked specimens (outer

diame-ter of 30 and 50 mm), final fracture occurred by unstable cleavage at 100°C, while the

nil-ductility temperature (NDT) of this material was found to be - 15°C Discussion of this

behavior is out of the scope of the present study; however, it clearly indicates that the

transition between ductile rupture and cleavage fracture is specimen dependent

Nnnerical Procedure

The numerical calculations used the FEM code TITUS of Framatome, which solves

elas-toplastic problems This code uses an incremental Von Mises rule with kinematical

harden-ing Eight four-sided nodes and six triangular isoparametric element nodes were used with

reduced Gauss integration Two different large-strain procedures were used The first one

is the simple updating of the Lagrangian technique, while the second one is a large-strain,

large-displacement scheme This last procedure uses the definition of stresses and strains

used by Mandel [10] Very little difference between the procedures was noticed, probably

because the loading is mainly tensile No special attention was paid to incompressibility

since the calculation is elastoplastic with elastic compressibility As already noted in the

literature, parabohc isoparametric elements with reduced Gauss integration are well adapted

for solving almost incompressible problems All calculations involved two-dimensional FEM

simulation in axisymmetry, except for the CT specimens, which were assumed to be in plane

strain

As explained in the introduction, meshing at the crack tip uses square elements in regular

arrays in order to simulate crack growth (Fig 2b) The size of these meshes is a parameter

of the fracture local criterion since it is a model of the process zone

The fracture criterion has two independent parameters The first one is the critical cavity

growth ratio; the second one is the process zone size According to the local criterion theory,

the critical cavity growth ratio can be assessed from the notched tensile test Since stress

and strain gradients are low in these specimens, the size of the mesh is unimportant The

second parameter can be fitted from a given cracked geometry (for example, TA 50

spec-imens) in order to simulate crack initiation Once these two parameters are determined,

they are used to predict crack initiation and stable crack growth behavior for all the specimen

geometries used in the present study

Measuring the Critical Cavity Growth Ratio

The numerical simulation of the notched tensile bar is straightforward, taking advantage

of the axisymmetry A piecewise linear stress-strain curve, deduced from the tensile test is

used This curve is extrapolated beyond necking, using a power law The loading curve of

the notched specimen is computed up to failure The comparison with experimental results

is excellent An accuracy better than 2% is noticed on the load for a given displacement

The cavity growth ratio and the cavity density ratios are computed in the center of the

specimen using both expressions, taking into account strain hardening (respectively, Eq 4

and the Appendix) Figure 5 gives the cavity growth ratio, RIRo, as a function of the average

strain, e, appUed to the specimen The ductility at failure, €«, was used to calculate {RIR^^,

which was found to be 1.83 ± 0.03 and 1.81 ± 0.03, respectively, depending on the equation

used Note that both results are similar The variation of cavity distance il% (Eq 3) was

calculated to be 0.857 ± 0.02 The critical cavity growth is in good agreement v^dth the

correlations with the inclusion volume fraction [10,11], which predict a value of 1.77 It is

1.50

1.25

FIG 5—Cavity growth ratio in the center of the notched tensile specimen as a function of

the average strain determined from the minimum section reduction: (1) from Eq 4, (2) from

the formula in the Appendix

worth emphasizing the fact that these experiments and these calculations are very simple

for this geometry Moreover, only a small amount of material is required

Measuring the Process Zone Size

A rough estimate of this process zone size, (Aa)c, can be made from the knowledge of

inclusion distribution [12]

where (Aa)^ is the size of the process zone, and N„ is the number of inclusions per unit

volume Details to measure this last value can be found elsewhere [77-75] Using this

procedure, (Aa)^ was found to be 215 jtm Another rough estimation of (Aa), can be made

from/,c measurements, using the following approximate formula, valid for small-scale

yield-ing [77]

where a is a numerical constant which depends on the exact meshing of the crack tip Here,

for square meshing and reduced Gauss integration, a = 4 Choosing / t = 200 MPa • mm

FIG 6—Comparison of experimental and computed loading curves The hatched area

de-notes experimental scatter The discontinuous line is the computed curve Net thickness is used

in the CT specimens, (a) Axisymmetrically cracked specimens, 50-mm outer diameter; (b) CT

specimen with 50-mm thickness

Simulation of Stable Crack Growth

All geometries were simulated using exactly the same meshes at the crack tip, that is, a

grid with 0.2-mm-square elements For the cracked round bars, the calculated loading curves

are in good agreement with the experimental ones (Fig 6) The simulation was slightly too

stiff, but the load predicted is always within ±2% of the measured one However, the

computed curve for the smaller 15-mm diameter specimens was found to be noticeably too

stiff This could be due to the fact that for this specimen, which was largely deformed at

failure, the extrapolated stress-strain curve was not strictly adequate For this reason, this

specimen was not numerically simulated

In CT specimens, very accurate comparisons are difficult to make because the exact

thickness of the specimen is difficult to define If B is the total thickness and B^, is the net

thickness, an effective thickness, B^,, must be defined In the elastic domain, B^f is found

to be roughly equal to the geometrical average of the two others After large plasticity, B^t

is almost equal to B„et, but not exactly, so that a precise verification of the loading curve

was difficult (Fig 6b)

Another way of verifying the quality of the solution was a prediction of the crack-tip

opening displacement In a procedure similar to that used in the experiments, the specimens

were unloaded and the displacements of the crack faces were compared with the CTOD

measured in the experiments (see Fig 2) The results, given in Table 2, show that the

comparison is actually very good This table also includes the values of the CTOD calculated

before releasing the load appUed to the specimens

Stable crack propagation was simulated using a node release technique When the critical

cavity growth rate is reached, the node representative of the crack tip and the following

middle node are released in six to ten steps, keeping the outer displacement constant It

was shown in Ref 1 that the procedure yields similar results when the nodes are released

one after another and when the load is kept constant during the node release process (except

after maximum load, of course) During release of the previous crack-tip element, the actual

crack-tip element is strained, and the cavities that had already grown a little are further

TABLE 2—Comparison of numerical and experimental results relative to the crack-tip opening

displacement {all lengths are in micrometres)

Specimen

TASO"

Experimental Results

210 ± 20

250 ± 20

Computed Loaded CTOD

265

317

Computed Unloaded CTOD

220

265

enlarged In all cases, this growth was insufficient to reach the critical value again This

means that crack growth was always stable Therefore, an increase of the outer displacement

was necessary to increase the damage of the material at the crack tip It eventually reaches

the critical value until the process is repeated

A process zone square element on the future path of the crack tip is first damaged during

the loading and breaking events before the crack tip reaches it We call this part accumulated

damage Hole growth continues during the release of the element just before, and then the

critical cavity growth ratio is reached during reloading The accumulated damage is

re-sponsible for the lowering of the slope of the crack growth resistance curve

Results of the simulations are shown in Fig 7, which gives the predicted crack advance

as a function of the displacement Crack growth, computed using either Eq 4 or the

Ap-pendix, is given The difference is small and is impossible to detect from inspection of the

1 1

Aa (mm)

FIG 7—Comparison of stable crack growth calculations and experiments; Load-line

dis-placement, d, as a function of crack growth: (a) CT specimens, (b) Axisymmetrically cracked

specimens The curves, calad«ted with the formula given in the Appendix and Eq 4, are drawn

in fuH lines and dotted lines, respectively

FIG 8—Comparison of stable crack growth experiment and calculations Load-line

dis-placement, d, as a function of crack growth (Aa) The experimental results are compared with the curves computed using the cavity growth formula derived in the Appendix

loading curves At first glance, it seems that the formula derived in the Appendix gives

better predictions Figure 8 gives the predictions using this expression However, in view

of the uncertainties of the parameters and experimental results, a more precise fit of the

parameters in Eq 4 would also have given good results

Discussion

We must stress the fact that the local fracture criterion used in this study involves only

two parameters:

(a) the critical cavity growth ratio {RIRo)c, and

(b) the process zone size

{Aa\-With these two parameters, reasonable predictions could be made for the following:

(a) the failure of the notched specimens,

(b) ductile crack initiation of cracked round bars with outer diameters equal to 30 and

50 mm,

(c) stable crack growth of cracked round bars, and

(d) stable crack growth of CT specimens

fraction and from the steel strain-hardening behavior, while the process zone size is related

to the number of inclusions per unit volume [12,13] Moreover, the process zone size can

be assumed to be independent of temperature, strain rate, and irradiation effect since it is

related to inclusion distribution This allows an easy estimate of the variation of the

param-eters from very simple experiments on notched tensile tests

In the examples given here, rather simple geometries loaded by simple tension were used,

and the rather heavy calculations presented here might seem out of proportion for such

simple problems Moreover, from Fig 3, it is apparent that the /-integral approach provides

good results This numerical and experimental program was undertaken in order to vaUdate

the local criterion approach Obviously, for these simple geometries, the more usual methods

are much easier to apply However, local criteria were primarily derived for much more

complex situations in which the stress-strain field at a crack tip cannot be described using

a single parameter such as / , for example:

1 Large-scale yielding The field is very different in a CT specimen and in a center-crack

panel, for example A paper is devoted to this subject in this publication [14]

2 Nonsymmetric, mixed Modes I, II, and III loadings

3 Complex thermomechanical loadings

4 Loading in the transition region in which two kinds of failure mechanisms are

com-peting In that case, another local criterion for cleavage fracture is necessary [75]

5 Situations with different materials such as welds and claddings

Since there is no theoretical limitations to the application of local criteria, they can be

used in much more complex situations The results given here show that the predictions are

precise

However, the node release technique is somewhat tedious and cannot be used for large

crack advances Moreover, for real nonsymmetric three-dimensional apphcations, the

mesh-ing procedure is very complex That is why another technique has been developed in which

the yield criterion is modified to take into account hole growth This induces a reduction

of the element stiffness This procedure, which is very similar to the one presented here, is

also presented in this pubUcation [16]

For typical problems involving small cracks in real structures subjected to complex

inci-dental loading, the local criterion developed here allows a quick and realistic modeling,

which gives an accurate estimate of the safety margins An example of such a problem is

given in this publication [77] In that case, involving an underclad defect, this local criterion

is the only possible realistic method to apply This local criterion has also been used

suc-cessfully to predict ductile tearing of stainless steel welds [18]

Conclusions

A numerical and experimental program was carried out in order to predict crack initiation

and stable crack growth, using local fracture criteria Several specimens were tested and

numerically simulated This included:

(a) a notched tensile round bar which gives a measure of the steel ductility,

(b) cracked tensile round bars with different dimensions, and

(c) cracked compact tension specimens with different dimensions

In the ductile local criterion adopted, hole growth by plastic flow is simulated using Rice

and Tracey's formula Cavity growth is limited by a critical value inferred from notched

In view of the small number of parameters used in the criterion, the results of the

numeri-cal simulations are in good agreement with the experimental results This suggests that lonumeri-cal

criteria can be used in more complex situations in which the stress and strain field at the

crack tip cannot be described using a single parameter such as J

Acknowledgments

The authors greatly acknowledge Electricite de France (Septen), Framatome, and the

Delegation de la Recherche Scientifique et Technique (DGRST) for financial support

APPENDIX

Hill [5] computes the limit load of a spherical hole of a radius, R, submitted to an internal

pressure, P, for a perfectly plastic material Then, he gives an implicit solution for the same

problem in a strain-hardening material, the behavior of which is given by

a^ = a, + H(e.,) (8)

The material is incompressible Therefore, the incremental equivalent Von Mises strain

at a distance C, from the center is given by

FIG 9—Comparison of the different cavity growth expressions: (1) the Rice and Tracey

formula with a constant yield stress {Eq 1), (2) the Rice and Tracey formula with a variable

flow stress (Eq 4), and (3) the formula deduced from the calculations given in the Appendix

Results for (a) aja,^ = 1.50 and (b) (Tm/a,, = 2

which is almost exactly the results of Rice and Tracey [2] for a perfect spherical hole

Eliminating CIR between Eqs 9 and 11 must be done numerically

These two formulas were compared for a special remote field so that (T„ = ka^^ as a

remote field Here CTQ is replaced by Weq in the equation

Results are given in Fig 9 for ^ = 1.5 and A: = 2 It is apparent that the differences are small The formula of Rice and Tracey with a constant yield stress is also given for com-

parison

Fracture, D Francois, Ed., Pergamon Press, Cannes, France, 1981, pp 809-816

Beremin, F M., Metallurgical Transactions, Vol 12, 1981, pp 723-731

Hill, R in The Mathematical Theory of Plasticity, Clarendon Press, Oxford, England, 1985, p

Beremin, F M., "Calculation and Experiment on Axisymmetrically Cracked Tensile Bars:

Pre-diction of Initiation, Stable Crack Growth, and Instability, Paper L/G No 2/3, SMIRT6

confer-ence, Paris, France, 1981

Haigh, J R and Richards, C E., "Yield Points and Compliance Functions of Fracture Mechanics

Specimens," Internal Report No RD/L/M 461, Central Electricity Research Laboratories,

Leath-erhead, England, May 1974

Mandel, J., International Journal of Solids and Structures, Vol 17, 1981, pp 873-878

Mudry, F in Elastic-Plastic Fracture Mechanics, M H Larsson Ed., Fourth ISPRA Conference,

Joint Research Centra, Ispro, Italy, 1983, pp 263-284

Mudry, F in Plastic Behaviour of Anisotropic Solids, i P Boehler, Ed., Edition du CNRS,

Proceedings, CNRS International Colloquium 379, Grenoble, France, 1981, pp 521-546

Lautridou, J C and Pineau, A., Engineering Fracture Mechanics, Vol 15, 1981, pp 55-71

Mudry, F , di Rienzo, E, and Pineau, A., "Numerical Comparison of Global and Local Fracture

Criteria in CT and CCP Specimens," this publication, pp 24-39

Beremin, F M., Metallurgical Transactions, Vol 14A, 1983, pp 2277-2287

Rousselier, G., Devaux, J C , Mottet, G., and Devesa, G., "A Methodology for Ductile Fracture

Analysis Based on Damage Mechanics," this publication, pp 332-354

Devaux, J C , Saillard, P., and Pellisier-Tanon, A., "Elastic-Plastic Assessment of a Cladded

PWR Vessel Strength Since Occurrence of a Postulated Underclad Crack During Manufacturing,"

this publication, pp 454-469

Devaux, J C , Mottet, G., Balladon, P., and Pellisier-Tanon, A., "Determination des parametres

d'endommagement et de rupture ductile d'un joint sonde austeno-ferritique," Proceedings,

In-ternational Seminar on Local Approach of Fracture, Moret sur Loing, France, June 1986,

pp 321-334

Numerical Comparison of Global and Local

Fracture Criteria in Compact Tension and

Center-Crack Panel Specimens

REFERENCE: Mudry, E, di Rienzo, R, and Pineau, A., "Numerical Comparison of Global

and Local Fracture Criteria in Compact Tension and Center-Crack Panel Specimens,"

Non-linear Fracture Mechanics: Volume II—Elastic-Plastic Fracture, ASTM STP 995, J D Landes,

A Saxena, and J G Merkle, Eds., American Society for Testing and Materials, Pliiladelphia,

1989, pp 24-39

ABSTRACT: Numerical computation of the stress-strain fields ahead of the crack tip of

compact tension (CT) and center-crack panel (CCP) specimens were performed Various crack

length to width ratio and strain hardening exponents were investigated The results of these

calculations were used to simulate numerically either brittle fracture or ductile rupture For

this purpose, the comparison between global fracture criteria, such as /^ or the critical

crack-tip opening displacement [(CTOD)^], and local fracture criteria is made It is shown that both

types of criteria provide similar results when the specimens are loaded under small-scale

yielding On the other hand, under large-scale yielding significant differences are found Local

fracture criteria are used to specify the vaUdity range for the global parameter, Ju- It is shown

that this validity range depends both on the specimen shape and on fracture micromechanisms

KEY WORDS: ferritic steels, ductile rupture, cleavage fracture, transition behavior, fracture

mechanics, /-integral, large-scale yielding, nonUnear fracture mechanics

The main object of the present study is to provide an answer to the well-known question:

What are the size requirements which must be fulfilled in a fracture mechanics test in order

to measure a valid value of /j^ It is clear that the answer to this question has important

practical implications as far as materials testing is concerned In previous studies, a number

of elements were obtained using both experimental results and the analysis of the

stress-strain field ahead of a crack tip determined either numerically or analytically In these

studies, in particular those reported by McMeeking and Parks [1], Hutchinson [2], and Shih

[3], different commonly used specimens were modeled using the finite-element method

(FEM) The stress distribution ahead of the crack tip was analyzed in detail and compared

with small-scale yielding solutions given, for example, by McMeeking [4]

Figure 1, taken from the work by McMeeking and Parks [7], illustrates this kind of

comparison The stress-strain distribution ahead of the crack tip is plotted as a function of

the distance normalized by a factor proportional to the plastic zone size, //CTQ, where / is

the contour integral while Vo is the yield strength In this figure, e,,^ is the maximum principal

stress ahead of the crack tip, e^, is the von Mises equivalent strain, while CQ is the yield

strain This specific case corresponds to a deeply edge-cracked bend specimen loaded in

such a way that bl{JI(Ja) = 16 where b is the ligament size The small-scale yielding solution

' Vice president for research, scientist, and professor, respectively Centre des Materiaux, Ecole des

Mines de Paris, 91003 Evry Cedex, France

100

R/U/Oo)

Small-scale yielding solution by McMeeking [4]

Solution by McMeeking and Parks [1] for edge-cracked bend specimen:

f>/(//ao) = 16, a/w = 0.9, and « = 0

This study, for CT specimen: a/w = 0.6, n = 0.005, and b/(J/(Jo) = 16

FIG 1—Stress and strain distribution ahead of a crack tip

by McMeeking [4] is also included For this geometry, it was found that the stress-strain

field was largely different from the small-scale yielding solution when the nondimensional

ratio b/{J/(To) was smaller than 25 The same kind of calculations led to a much larger value (=200) for the b I {J I a a) ratio required in a center-crack panel (CCP) Experiments were

then carried out to validate these size requirements, which are only indicative From this work several questions can be raised:

1 What is the minimum value of the ratio bl{Jluo) for a cracked body of any given shape

in order for the small-scale yielding solution to be adequate to represent the stress-strain distribution?

2 Is this minimum size requirement independent of the failure micromechanisms tigated?

inves-3 Under general yielding are the values of J^^ and dJIda determined, for instance, on

CT specimens necessarily conservative?

Following the original work by McClintock [9], these studies started from a modelization

of the physical events leading to final fracture in this specific steel Based on numerous

experimental data and microscopic observations, a damage function was estabUshed for each

fracture micromechanism Implemented in a FEM code, these functions are used to simulate

crack initiation and, eventually, stable crack growth

In this study, an attempt is made to show how these local criteria can be used to investigate

the effect of general yielding on fracture predictions In other words, our aim is a comparison

of global fracture criteria like the critical crack-tip opening displacement [(CTOD),], /]„ or

X'lc with local criteria previously proposed, that is, the cleavage Weibull statistic and critical

ductile void fraction In the first section, local fracture criteria are briefly reviewed Then

the results of FEM numerical calculations are presented Compact tension (CT) and CCP

specimens with different ligament sizes and two different strain hardening rates were modeled

The results of the calculations are compared with already published solutions In the last

section, the results obtained from different fracture criteria are compared in order to discuss

the validity of specimen size requirements

Brief Description of Local Fracture Criteria

Local Cleavage Criterion

The associated local criterion is very simple It originates from the well-known concept

of the existence of a critical fracture stress However, because of the large scatter noticed

in cleavage experiments, the critical stress is given a statistical meaning using Weibull's

theory In this theory, the probability of failure Pf of a specimen of volume V submitted

to a homogeneous stress state cr is given by

where V^ is an arbitrary unit volume, cr„ is the average cleavage strength of that unit volume,

and m is the Weibull exponent In three dimensions, with smooth stress gradients, this

where a^, which has the dimension of a stress, is referred to as the "Weibull stress," though

its definition depends on m and y„ This equation shows that the Weibull stress, directly

the Ritchie, Knott, and Rice model [11] For very large stress gradients, the variation of

stress on the unit volume is not negligible In that case, an averaged value on the volume

is used The dimension of this unit volume is usually of the order of one to two grain sizes

Local Ductile Criterion

The physical basis of this criterion is also very simple Cavities are assumed to initiate

with the onset of plastic deformation It was shown previously that this condition is rapidly

fulfilled because of the large stress triaxiality prevailing ahead of the crack tip [5,10] Hole

growth is described using a formulation originally proposed by Rice and Tracey [12], slightly

modified to take into account the strain hardening effect

dR

—- = 0.283rfe?, exp

where R is the actual cavity radius, rfe?, is the incremental von Mises equivalent strain, u„,

is the hydrostatic stress, and cr,, is the equivalent von Mises stress Metallographical

obser-vations showed that Eq 4 accounted reasonably well for the experimental results [6,10,13]

Integration along the strain path yields the average cavity growth ratio

As already proposed by other authors, fracture is assumed to occur when a critical cavity

volume fraction is reached This failure criterion was already used in FEM calculations by

D'Escatha and Devaux [14]

Thus, the cavity volume fraction at failure/ = fc can be written as

where /o is the initial volume fraction

Therefore, the local ductile rupture criterion is simply expressed as

[''" 0.283 exp 1.5— dt^ = Ln 111 (7)

the material properties and not to the conditions for a precise numerical solution A precise

solution would require very fine meshes and special crack-tip elements However, since the

material is heterogeneous at such small distances, an averaging technique would be necessary

for calculating the metallurgical damage For example, Eq 7 is meaningless when applied

at crack-tip distances that are less than the inclusion interspacing

D'Escatha and Devaux [14] showed that a more direct way of modeling these local criteria

is an experimental fit of the characteristic distance \ This distance is used as the size of the

first element at the crack tip More precisely, the local criteria (<j^,m for cleavage; {R/Ro)c

for ductile fracture) are fitted by using experiments performed on homogeneously stressed

volumes, such as tensile or mildly notched tensile bars Then, further experiments are

performed on cracked specimens These experiments are modeled using FEM with a given

choice for the numerical solution (for example, four Gauss points integration, square

ele-ments, and so forth) Then, the size of the first element is fitted in order to reach a critical

value of the criterion in the first mesh element for the experimentally determined initiation

conditions This mesh size and mesh pattern are then used without change for other cracked

specimens Details can be found in another paper in this volume [75] Here, we need, for

purpose of comparison, a rather fine description of the stress-strain distribution ahead of

the crack tip This is the reason we used triangular elements at the crack tip instead of the

square elements used in a previous study [15]

Theoretical Relationships Between J,c and Local Criteria Under Small-Scale

Yielding Conditions

Using known analytical relationships and FEM plane-strain small-scale yielding results,

it is possible to derive theoretical relationships between J^^ (or Ki^ and the local criteria

For cleavage fracture, it was shown earlier that the Weibull stress could be expressed as [7]

In this equation, B is the specimen thickness, (j„ is the yield stress, and C„ is a numerical

factor Here, C„ = 1.5 x lO*" for a material almost perfectly plastic

Equations 2, 3, and 8 are used to describe the theoretical scatter in /jc for cleavage fracture

1 - exp JlE^Bu,"'-'C„

which means that /i^ obeys a Weibull statistical distribution with a Weibull exponent equal

to 2 As shown earlier [7,76] this prediction is in good agreement with the experimental

results

For ductile rupture, it was shown that /jc at initiation of stable crack growth was related

to critical void growth by the following expression [10]

7„ = a(T„X Ln ij] (10)

where a is a numerical factor dependent on the exact mesh used at the crack tip, for example,

the geometry used to model the process zone

From Eqs 9 and 10 the toughness transition curve (/,„ versus temperature) can be calculated

Numerical Simulations

Specimens and Material

Both CT and CCP specimens were modeled The specimen geometries are shown in Fig

2 The numerical calculations were made with the same kind of FEM code used in Ref 15

More precisely, most of the calculations dealing with local criteria were performed using the TITUS code A small part of this code was rewritten by the authors for their own purpose It allows two-dimensional simulation of elasto plastic materials The numerical calculations involved eight-node or six-node isoparametric elements with reduced Gauss integration An updated Lagrangian procedure was used with an implicit scheme to solve

the incremental Levy-Mises equations of plasticity As explained in detail elsewhere [15],

the comparison with the experimental results is excellent (see for example, Fig 6 in Ref 75) Of course, the detailed solution very near the crack tip is approximate because of the

square elements used at the crack tip Their size is of the order of 200 ixm This size is usual

for the mean inclusion spacing in A508 steel These calculations were performed in

plane-strain condition A detailed comparison with the experiments is impossible since

three-dimensional effects are neglected This means that, as explained in detail in Refs 1 and 2,

the lack of plastic constraint arises from the finite in-plane dimensions of the specimens Of

course, on true specimens, the thickness effects related to three-dimensional aspects are

also very important The present two-dimensional modeling does not take this problem into

account

The material is assumed to be elasto plastic with a Young's modulus, E, equal to 200 000

MPa and a Poisson's ratio, v, equal to 0.3 The stress-strain relationship is given in the

following form

= - Q " (11)

where ao is the initial yield stress, and eo is the total longitudinal strain in a tensile test

Here, (Xo was chosen so that E/vo = 300 Two values for the strain hardening exponent, n,

were used, that is, 0.1 and 0.005 The effect of initial crack length was also investigated

(Table 1) In all cases, the loading of the specimens was simulated by imposing an increasing

displacement

Computation of the Fracture Parameters

The value of / was derived from the calculated load-displacement curves, as it is usually

made experimentally, and not from the definition of the contour integral This was done in

order to be as close as possible to an experimental procedure The formulas used are the

following:

For CT specimens

where U is the area under the load-line displacement curve, b is the ligament size, and B

is the specimen thickness In this study, 5 = 1 mm For alw = 0.45 and 0.60, aj was taken

as equal to 2.2896 and 2.2126, respectively These values were determined from the ASTM

Test for / ] „ a measure of Fracture Toughness (E813-81)

TABLE 1—Specimen geometries and material characteristics

Specimen No Geometry alw Strain Hardening Exponent

0.45 0.60 0.45 0.60 0.75 0.45 0.75 0.45

258 CCP 0.75 0.1

259 CCP 0.45 0.1

268 CCP 0.75 0.005

269 CCP 0.45 0.005

excluding the elastic part, 8p is the plastic displacement, and P is the load For a/w = 0.75, tti = 1.020 and a^ = -0.512, while for a/w = 0.45, a, = 0.695 and az = -0.195 These

values were adjusted to be consistent with those taken from Ref 17

In a similar manner, the crack-tip opening displacement (CTOD), is estimated from the displacement of the crack mouth in a portion behind the crack tip where the crack faces are approximately parallel Here it should be noticed that we are essentially interested in modeling relatively large values of CTOD (^0.10 mm), which are usually measured in low-strength materials, such as A508 steel Under these conditions the definition used to calculate the CTOD is similar to the experimental procedures when poUshed sections normal to the crack tip are observed [5]

The local criteria are evaluated according to the procedure already described However,

a difficulty arises because the characteristic distances may be different for the cleavage criterion (usually about 50 jjim) and for the ductile criterion (usually about 200 (xm) In the transition region, the size of the first element is not very important for cleavage because the highest stresses are found some distance ahead of the crack tip On the contrary, it is

of prime importance for ductile fracture Therefore, the first mesh element size used in the

present study was of the order of 200 [x.m

Comparisons with Former Calculations

Figures 1 and 3 compare the computed stress distribution ahead of the crack tip of CT

and CCP specimens to other solutions obtained by McMeeking and Parks [1] and Hutchinson

[2] Strains are more difficult to compare because only a few results are published Figure

1 gives our results for a CT specimen Strain gradients are so large that detailed comparisons

are very difficult to make In both figures, the exact specimen geometries vary slightly with

the various problems However, the results are very similar It can therefore be concluded

that our results compare well with others as far as stresses are concerned The comparison

with strain distributions is more difficult to make

Results and Discussion

Comparison of J and CTOD

Figure 4 compares the opening of the crack tip for similar values of the / parameter

obtained for both kinds of specimens It is apparent that for small-scale yielding the results

are very similar On the other hand, when fully plastic, CCP specimens give rise to much

larger values of CTOD than CT specimens In both cases, beyond general yielding the

variation of CTOD as a function of J is roughly linear The results are given in Table 2,

where the calculated proportionality factor between AJ and A(CTOD) is compared with

theoretical values determined from Ref 4 In this table the values of m and a are those used

in the folowing expressions

CT specimen: alw = 0.45, n = 0.005

CCP specimen: a/w = 0.45, n = 0.005

FIG 4—Comparison of crack-opening displacement, U,, in CCP and CT specimens for

I J "r~ o 1 ' — y

It is apparent that the above relationship for flow stress accurately takes into account the

effect of strain hardening It is also worth noting that if the CTOD was a relevant parameter

for ductile fracture initiation, then Ji^ should be lower in a CCP specimen than in a CT

specimen, which seems a rather strange prediction