Astm stp 803 v2 1983

ERNST Specimen Geometry a n d Extended Crack Growth Effects on J r R Curve Characteristics for HY-130 and ASTM A533B Steels-- An Elastic-Plastic Fracture Mechanics Study of Crack Init

FRACTURE:

SECON SYMPOSIUM,

VOLUM~ II FRACTURE RESISTANCE CURVES

AND ENGINEERING

APPLICATIONS

A symposium sponsored by ASTM Committee E-24 on Fracture Testing Philadelphia, Pa 6-9 Oct 1981

ASTM SPECIAL TECHNICAL PUBLICATION 803

C F Shih, Brown University, and

J P Gudas, David Taylor Naval Ship R&D Center, editors

ASTM Publication Code Number (PCN) 04-803002-30

1916 Race Street Philadelphia Pa 19103

Copyright 9 by AMERICAN SOCIETY FOR TESTING AND MATERIALS 1983

Library of Congress Catalog Card Number: 82-83520

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

Printed in Baltimore, Md (b) November 1983

The Second International Symposium on Elastic-Plastic Fracture Mechan-

ics was held in Philadelphia, Pennsylvania, 6-9 Oct 1981 This symposium

was sponsored by ASTM Committee E-24 on Fracture Testing C F Shih,

Brown University, and J P Gudas, David Taylor Naval Ship Research and

Development Center, presided as symposium chairmen They are also editors

of this publication

Related ASTM Publications

Fracture Mechanics (13th Conference), STP 743 (1981), 04-743000-30

Fractography and Materials Science, STP 733 (1981), 04-733000-30

Crack Arrest Methodology and Applications, STP 711 (1980), 04-711000-30 Fracture Mechanics (12th Conference), STP 700 (1980), 04-700000-30

Elastic-Plastic Fracture, STP 688 (1979), 04-688000-30

to Reviewers

The quality of the papers that appear in this publication reflects not only

the obvious efforts of the authors but also the unheralded, though essential,

work of the reviewers On behalf of ASTM we acknowledge with appreciation

their dedication to high professional standards and their sacrifice of time and

effort

A S T M Committee on Publications

ASTM Editorial Staff

Janet R Schroeder Kathleen A Greene Rosemary Horstman Helen M Hoersch Helen P Mahy Allan S Kleinberg Virginia M Barishek

The editors would like to acknowledge the assistance of Professor G R

Irwin, Dr J D Landes, Professor P C Paris, and Mr E T Wessel in plan-

ning and organizing the symposium We are grateful for the support provided

by the ASTM staff, particularly Ms Kathy Greene and Ms Helen M

Hoersch The timely submission of papers by the authors is greatly appreci-

ated Finally, this publication would not have been possible without the tre-

mendous effort and dedication that was put forth by the many reviewers

Their high degree of professionalism ensured the quality of this publication

The editors also wish to acknowledge the diligent assistance of Ms Susan

Beigquist, Ms Ann Degnan, Mr Steven Kopf, and Mr Mark Kirk in

preparing the index

J P Gudas

C F Shih

Contents

Introduction

E N G I N E E R I N G A P P L I C A T I O N S

A Method of Application of Elastic-Plastic Fracture Mechanics to

Nuclear Vessel Analysis P c P ~ x s AND R E JOHNSON

Evaluation of the Elastic-Plastic Fracture Mechanics Methodology on

L ISSLER

Studies of Different Criteria for Crack Growth Instability in Ductile

M a t e r l a ] s - - s KAISER AND A J CARLSSON

Further Developments of a J-Based Design Curve and Its

Application of Two Approximate Methods for Ductile Failure

Development of a Plastic Fracture Methodology for Nuclear

AND D F MOWBRAY

Some Salient Features of the Tearing Instability

T h e o r y - - H A ERNST

Verification of Tearing Modulus Methodology for Application to

Reactor Pressure Vessels with Low Upper-Shelf Fracture

T o u g h n ~ - - s s TANG, P C RICCARDELLA, AND R HUET

Ductile Tearing Instability Analysis: A Comparison of Available

T e c h n i q u e s - - G G C H E L L AND I M I L N E

Validation of a Deformation Plasticity Failure Assessment Diagram

Approach to Flaw Evaiuation j M BLOOM

C F SHIH, V KUMAR, AND M D GERMAN

Lower-Bound Solutions and Their Application to the Collapse Load

of a Cracked Member Under Axial Force and Bending

M o m e n t - - H OKAMURA, K KAGEYA_MA, AND Y TAKAHATA

Ductile Crack Growth Analysis Within the Ductile-Brittle

Transition Regime: Predicting the Permissible Extent of

Ductile Crack Gmwth L MILN~ AND D A CURRy

Ductile Fracture of Clrcumferentially Cracked Pipes Subjected to

Bending Loads A ZAHOOS AND M F KANNINEN

Engineering Methods for the Assessment of Ductile Fracture Margin

in Nuclear Power Plant Piping s RANGANATH AND

H S MEHTA

Fracture of Circnmferentlally Cracked Type 304 Stainless Steel Pipes

Under Dynamic Loading G M WILKOWSKI, J AHMAD,

A ZAHOOR, C W MARSCHALL, D BROEK, I S ABOU'SAYED,

On the Unloading Compliance Method of Deriving Single-Specimen

R-Curves in Three-Point Bending A A WXLLOUGHBY ANt)

Evaluation of Several Jlc Testing Procedures Recommended in

J a p a n - - E o m i , A OTSUKA, AND r~ KOBAYASHI I I - 3 9 8

Evaluation of Blunting Line and Elastic-Plastic Fracture

Toughness H KOBAYASHI, H NAKAMURA,

Instability Testing of Compact and Pipe SpeCnnens Utilizing a Test

System Made Compliant by Computer Control~j A JOYCE II-439

Computer-Controlled Single-Specimen J - T e s t - - w A VAN DER SLUYS

AND R J FUTATO

Quantitative Fractographlc Definition and Detection of l~acture

AND J F KNOTI"

Combined Elastlc-Plastic and Acoustic Emission Methods for the

Evaluation of Tearing and Cleavage Crack Extension

An Analysis of Elastic-Plastic Fracture Toughness Behavior for J~c

Measurement in the Transition Region T IWADATE,

Y TANAKA, S - I ONO, AND J WATANABE

Characterlzation D E McCABE, I D LANDE$,

AND H A ERNST

Specimen Geometry a n d Extended Crack Growth Effects on J r R

Curve Characteristics for HY-130 and ASTM A533B Steels

An Elastic-Plastic Fracture Mechanics Study of Crack Initiation in

316 Stainless Steel v H DAVIES

Thlclmess Effects on the Choice of Fracture Crlteda a.-w LIU,

W.-L HU, AND A S KUO

Experimental Validation of Resistance Curve Allfl]y$[li I MILNE

CYCLIC PLASTICITY E F F E C T S AND M A T E R I A L CHARACTERIZATION

Elastic-Plastic Fracture Mechanics Analysis of Fatigue Crack

G r o w t h - - ~ H EL HADDAD AND B MUKHERYEE 11-689

EIastle-Plastlc Crack Propagation Under High Cyclic Stresses

Load History Effects on the JR-Curve J Do LAN'DES AND

Ductile F r a c t u r e with Serrations i n A I S I 310S Stainless Steel a t

J - R Curve C h a r a c t e r i z a t i o n of I r r a d i a t e d Low-Shelf N u c l e a r Vessel

S t e e l s - - F J LOSS, B H MENKE, A L HISER,

STP803-EB/Nov 1983

Introduction

In October 1981, ASTM Committee E24 sponsored the Second Interna- tional Symposium on Elastic-Plastic Fracture Mechanics which was held in Philadelphia, Pennsylvania The objective of this meeting was to provide a forum for review of recent progress and introduction of new concepts in this field The impetus for this symposium was the historical development of elas- tic-plastic fracture technology Concepts such as the J-Integral, COD, and HRR field generated tremendous interest which led to the First International Symposium held in 1977 The presentation and publication of works on such topics as J-controlled crack growth, tearing instability, and numerical de- scription of crack tip fields, among others, led to major, broad-based re- search activities and application-oriented developments This sustained growth provided the motivation for another meeting devoted solely to elastic- plastic fracture

The call for papers for the Second International Symposium generated an overwhelming response This was reflected by the number of papers pre- sented, and the attendance which exceeded 300 participants The papers sub- mitted to this symposium underwent rigorous review The works contained in these two volumes reflect the high degree of interest in this subject and the quality of the efforts of the individual authors

In the first ASTM publication devoted to elastic-plastic fracture (ASTM STP 668), there were three major groupings including elastic-plastic fracture criterion and analysis, experimental test techniques and fracture toughness data, and applications of elastic-plastic methodology The present collection

of papers shows substantial growth in theoretical and analytical areas which now include topics ranging from fundamental analysis of crack growth under static and dynamic conditions, finite strain effects at the crack tip, elevated temperature effects, visco-plastic crack analysis, and tractable treatments of fully plastic crack problems and surface flaws These theoretical and analyti- cal developments, combined with progress in test method development and ductile fracture toughness characterization led to substantial growth in engi- neering application of elastic-plastic fracture methodologies as evidenced by the large selection of papers on this topic

The papers in these two volumes have been grouped into six topic areas including elastic-plastic crack analysis, fully plastic crack and surface flaw analysis, visco-plastic crack analysis and correlation, engineering applica- tions, test methods and geometry effects, and cyclic plasticity effects and ma-

Copyright* 1983 by ASTM lntcrnational

I1-1 www.astm.org

terial characterization The first grouping contains papers on crack propaga- tion under static and dynamic conditions, crack growth and fracture criteria, finite strain effects on crack tip fields, and plasticity solutions for important crack geometries and structural configurations Fully plastic crack solutions, elastic-plastic line-spring models, approximate treatment of surface flaws, and surface flaw crack growth correlations are the focus of the second group- ing of papers An area not addressed in the first symposium and which has since attracted significant attention is crack growth at elevated temperature, and time dependent effects Papers on this topic address theoretical aspects

of creeping cracks, computational procedures, microstructural modelling, creep crack growth correlations, and materials characterization

The second volume of this publication begins with the major section on engineering applications This includes several papers on tearing instability, J-based design curves, evaluations of several fracture criteria, further devel- opments of fracture analysis diagrams, and flawed pipe analyses This is fol- lowed by a series of papers on test methods and geometry relationships A majority of the papers focus on J1e and JI-R- curve test procedures and compu- tations, and several papers address the test specimen geometry dependence of these parameters In the last section several papers are devoted to prior load history effects, crack growth in the elastic-plastic regime under cyclic loading, and micromechanism studies of the fracture process

The collections of papers from this and the previous symposium contain many of the major works in the rapidly evolving subject of elastic-plastic frac- ture It is hoped that these two volumes will serve to stimulate further prog- ress in this field

J P Gudas

David Taylor Naval Ship Research and Devel- opment Center, Annapolis, Md 21401; symposium chairman and editor

C 1; Shih

Division of Engineering, Brown University, Providence, R.I 02912; symposium chair- man and editor

Engineering Applications

A Method of Application of

Elastic-Plastic Fracture Mechanics to Nuclear Vessel Analysis

REFERENCE: Paris, P C and Johnson, R E., "A Method of Application of Elastic-

Plastic Fracture Mechanics to Nuclear Vessel Analysis," Elastic-Plastic Fracture: Second Symposium, Volume II Fracture Resistance Curves and Engineering Applications,

A S T M STP 803, C F Shih and J P Gud.as, Eds., American Society for Testing and

Materials, 1983, pp II-5-II-40

ABSTRACT: The primary purpose of this work was to develop analytical relationships which could be used to assess the safety of irradiated nuclear reactor pressure vessels against unstable fracture The need for such a calculation occurs when the Charpy upper- shelf energy of the vessel steel is predicted to fall below the required 50 ft- lb (67.8 J) level from accumulated neutron radiation damage The method used was based on "tearing instability" concepts under "J-controlled growth" conditions for the crack stability criterion The aforementioned purpose was served by developing fracture mechanics methods of wider applicability than previously available and applying them in analyses at upper-shelf conditions (above the transition temperature) Elastic-plastic fracture mechanics concepts were used to extend recognized linear elastic fracture mechanics flaw analysis equations for through-the-thickness flaws and surface flaws into the plastic range The approach also made use of J-R curve characterization of the material fracture resistance

A crack stability diagram in the form of J as a function of T plot was shown to be useful

in demonstrating safe levels of loading (applied J ) by comparison with the material J-R curve, reduced onto the same diagram Consequently, a safe level of applied load, Js0 [for

J / T = 50 in.-lb/in, 2 (8.756 kJ/m2)], was suggested and the possibility of its correlation

with upper-shelf Charpy energy values discussed

KEY WORDS: fracture mechanics, elastic-plastic fracture, analysis, J-integral, J-R

curve, tearing modulus, pressure vessel, surface flaw, through-wall flaw, crack growth in- stability, yielding, Charpy (energy), upper shelf (energy)

N o m e n c l a t u r e

a C r a c k l e n g t h ( f o r t h r o u g h - t h i c k n e s s c r a c k s ) o r c r a c k d e p t h ( f o r

s u r f a c e c r a c k s ) 1Professor of mechanics, Washington University, St Louis, Mo 63105; Fracture Proof De- sign Corporation, St Louis, Mo., 63108

2Task manager, U.S Nuclear Regulatory Commission, Washington, D.C 20555

11-5

11-6 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

ae~ Effective crack size with a plastic zone correction added

a0 Initial crack size prior to growth

Aa Crack length change

Uncracked ligament size Half the surface length of a surface crack Modulus of elasticity

A surface flaw geometry correction

A coefficient in a hardening stress bracket

A coefficient in a hardening stress bracket

A coefficient in a hardening stress bracket

A coefficient in a hardening stress bracket

A coefficient in a hardening stress bracket Rice's J-integral

Intensity of the crack-tip field, J, applied Material's resistance value of J for an observed crack length change, A~

Intensity of an elastic crack-tip field

A hardening exponent (for describing material properties) Applied load

Radial distance from a crack front

A plastic zone correction to be added to crack length Radius of a pressure vessel

Arc length on a contour around a crack tip Wall thickness of a pressure vessel

Tearing modulus Applied tearing modulus Material's resistance to tearing modulus for an observed crack length change, ~a

Ti Applied traction (stress)

ui Displacement corresponding to an applied traction

U Pseudo-elastic system energy stored (corresponding to deformation

plasticity theory) Pseudo-strain-energy density Rectangular coordinates measured perpendicular and parallel to a crack surface

A geometrical correction for crack-tip field intensity in shells (a function of ~ = a/x/fit)

Coefficients in hardening laws for stress-strain curves Stress state (plane stress versus plane strain) coefficient in a plastic zone correction

3' A coefficient adjusted for stress state in relating t5 to J

I' A contour around a crack tip

Crack opening stretch (displacement)

&p Load-point displacement

A shell parameter, a / ~ / - ~ -

Applied (tension) stress Components of stress Flow stress (in tension) Net ligament nominal (effective) stress

A complete elliptic integral (of the second kind [19])

A coefficient in a hardening stress bracket Hutchinson's J-controlled growth validity assurance parameter Prime derivative with respect to the argument

Stress brackets or factors in equations for J and T applied Geometry brackets or factors in equations for J and T applied The American Society of Mechanical Engineers (ASME) Boiler and Pres-

sure Vessel Code for nuclear reactor pressure vessels has for some time per-

mitted the use of linear elastic fracture mechanics (LEFM), specifically in

Appendix A of Section XI This has allowed clear and conservative evalua-

tions of any potential danger due to flaws found in inspections of reactor

vessels However, LEFM, as incorporated in the Code, has a limited range

of direct applicability without large and perhaps undue conservatism

Moreover, the Code version of LEFM makes use of the Kit - - K i d concept of

impending failure (little or no crack growth), instead of more advanced con-

cepts of flaw or crack stability permitting limited stable flaw growth Under

the use of LEFM, the Code itself acknowledges ranges of inapplicability such

as well above the transition temperature where LEFM cannot produce ap-

plicable quantitative results Appendix A provides no specific criteria for

upper-shelf toughness; the situation was discussed earlier [1] 3 In Title 10 of

the Code of Federal Regulations, Part 50 (10 CFR 50), a lower limit is imposed

on Charpy upper-shelf energy (USE), namely, 50 ft Ib (67.8 J); for materials

of less USE, unspecified methods must be used to assure safety On the other

hand, the current ASME Code provisions using fracture mechanics have

served very well in cases of appropriate quantitative applicability

In recent years, a great deal of progress has been made in J-integral based

elastic-plastic fracture mechanics (EPFM) In particular, a more advanced

crack stability criterion has been developed [2,3] and widely accepted [4,5]

which depends on the whole J-integral R-curve for material characterization

(rather than a single value such as Kic, which is more limited) These and

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

11-8 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

other advances in EPFM make possible the suggestion of new methods for application to nuclear vessels

The new methodology presented in this paper is proposed on its own merit but is phrased with the existing Code in mind in order to supplement it with alternative methods in areas such as upper-shelf conditions where the ex- isting Code seems lacking Indeed, the most realistic postulated vessel failure conditions are usually well within the elastic range for gross section stresses but may include occasional cases of large-scale yielding Therefore, only modest modifications of current methods of vessel flaw stress analysis will be suggested On the other hand, more ductile, perhaps fully plastic, failures are characterized by significant amounts of stable flaw growth Therefore a more advanced (R-curve) stability concept will be suggested, especially for material property evaluation purposes The new methodology can be con- sidered as" an extension of the existing Code methods written in terms of J-integral EPFM, for which LEFM is simply a special case

Indeed, the only really new embellishment to be presented herein is the use

of a J versus T diagram to assess crack instability It is simply a new diagramatic representation of J-R curve material representation and applied J-T curves from established methods It is proposed to clarify situations which will lead to crack instability, to simply delineated regions of rigorous applica- bility of the analytic concepts, to clearly demonstrate safety margins for approaching instability, etc However, the use of J versus T diagrams involves

no new assumption, it is just a new representation method which clarifies many matters One further result, which will be demonstrated, is that the limiting allowable J-values suggested herein to avoid crack instability on the Jversus T diagram have, so far, shown good correlation with Charpy upper-shelf energies This can be of great practical significance where only Charpy data are available

The Plane-Strain J-Integral R-Curve

According to developments by Hutchinson [6] and Rice and Rosengren [7] (HRR), the value of the J-integral (or./applied) c a n be seen to be a parameter characterizing the intensity of the plastic stress-strain field surrounding the crack tip Their results lead to the following form for the stress-strain field, the HRR field

plus higher-order terms (negligible near a crack tip) The coordinates r and 0 are the usual cylindrical coordinates measured from the crack tip The anal- ysis was based on adopting a deformation theory of plasticity for a stress- strain curve whose latter portion (well beyond the elastic range) can be represented by a power law or

1 that conditions in the material's crack-tip fracture process zone are plane-strain,

2 that conditions which disrupt the HRR field are avoided, such as avoiding concentrated slips direct from the crack tip to nearby boundaries or cross-slip (slip at 45 deg through the thickness),

3 that crack growth does not disrupt the HRR fields, and

4 that cleavage does not intercede on the J-R curve

Indeed, J-R curves produced by the types of test conditions proposed by ASTM Committee E-24 for standards at least attempt to be sufficient to avoid Conditions 1 and 2 as problems Indeed, Condition 4 is thought not to

be a problem at temperatures exceeding 100~ (212~ above the transition temperature (beginning of upper shelf); but more data on this point may be needed Finally, Condition 3 is not a problem under conditions proposed by Hutchinson [2], which are

11-10 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

Hutchinson [3] showed by differentiating Eq 1, obtaining the increments of

the strain, &,> that these increments deo are sufficiently proportional % to

assure appropriate use of deformation theory The use of J itself here is also

based on having conditions sufficiently appropriate for deformation theory

Hence Eq 3 also assures sufficient conditions for the definitions of J in its in-

tegral forms to follow [It should be noted that sufficient conditions are

distinct from necessary conditions and therefore Eqs 3 may not always be re-

quired for appropriate use of J.]

Therefore under the given conditions the applicability of "strict deforma-

tion theory" is appropriate, the conditions for so called "J-controlled crack

growth" are met, and J may be defined with equal validity either by its con-

tour integral or compliance counterparts, which are [8] (see also [Ref 9] for

Consequently, the "plane strain J-R curve" as shall be adopted here is

assumed to be produced under appropriate conditions as discussed under the

preceding four conditions J should be measured by a method consistent with

applying Eqs 4, including crack length change, ~a, corrections The J-R

curve is then a plot of J versus Aa points as loading progresses on a cracked

specimen of the material at a given temperature

Further, the J-R curves available may not always have been produced

under ideal conditions (often undersized test specimens) This will not rule

out their use if they can be shown to be conservative For example, slightly

subsized specimens or the use of side grooves or both with appropriate data

reduction methods have been shown to give conservative J-R curves for

bending-type tests As used here, conservatism is taken with respect to safety

when using the test results to evaluate applications by the methods developed

later in this paper

The Tearing Instability Criterion

In the previous section in Eqs 1 it was noted that J is the intensity of the

crack-tip stress and strain field Moreover, with proportional straining as

guaranteed by meeting the conditions of Eqs 3 it can be argued that ap-

propriate use of "strict deformation theory" and "J-controlled crack growth"

will result Therefore, at least under these conditions, the second definition

of J in Eqs 4 implies that

d U

where U is pseudo-elastic energy per unit thickness stored (that is, for the

nonlinear elastic analog to an elastic-plastic material) by applying loads or

deformation to the cracked body of interest Regarding crack length change,

da, as a displacement, Japplied takes on the connotation of a generalized force

and Jmaterial may be regarded as the material's resistance to that force Conse-

quently, a statement of equilibrium with respect to crack extension is

The stability of the equilibrium expressed by Eq 6 can be found by examining

the second derivative of system energy Using Eq 5 the stability criterion can

where E is elastic modulus and o0 the flow stress Then the stability criterion,

Eq 7 may be expressed in nondimensional terms by

(stable) Tapplied < Tmaterial (indifferent) (9)

(unstable) Now, Japplied may be found from the stress analysis solution for the cracked

body, applying Eqs 4 to make the determination Consequently Japplied will

depend on applied loads, P, or deformations, A, and crack size, a, hence

Japplied : Japplied (P, a ) OrJapplie d (A, a) (lO)

On the other hand, Jmaterial depends on the materials' resistance or its J-R

curve, which is a plot of J versus Aa characterizing the material's resistance

to crack extension Consequently

Therefore, when derivatives d / d a are taken of Eqs 10 and 11 to form Tapplie d

and Tmateriai as indicated in Eq 8, it should be noted that Trnateria I may be

11-12 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

formed from the slope of the J-R curve, dJraaterial/da, taken at a given level of

J That is to say

On the other hand

_ O~JappliedOA / O A / ~ _+_ o~Japp|iedoa

where the partial derivatives of ']applied on the right side of Eqs 13 are found

from Jappiiea solutions in the form of Eqs 10 The other ( ) partial derivatives

in Eqs 13 depend on the load application system compliance and must be

evaluated accordingly Furthermore, assuming that the quantities in Eqs 13

are properly evaluated, it is observed that

Tapplie d = Tapplie d (e, a)

: Tapplie d (A, a)

Regarding Eqs 10 and 14 as parametric equations for Japplied and Tappiied,

the loading parameter P or A may be eliminated between them Making use

of the statement of equilibrium from Eq 6

Japplied ~ - Jmaterial ~ J

Tapplie d -~ Tapplie d (J, a) The result is that both Tapplie d and Tmater~ in Eqs 12 and 15 may be thought

of as functions of J, where increasing J is viewed as the variable indicating in-

creasing load or deformation applied to the body Moreover, the crack size,

a, in Eq 15 may be regarded as the initial crack size, a0, plus the change in

crack size, Aa, from the increase in J as determined from Eq 11 or the

material's J-R curve That is to say that

a : a 0 + A a

Aa = Aa (J)

is determined by the J-R curve (for Aa may be negligible compared with a0 in some cases) Therefore as loading progresses and J increases, Ta0plied may be computed by Eq 15 with Eq 16 and Tmaterial by Eq 12 then compared, accord- ing to Eq 9, to determine the first value of J or the loading which causes instability

This approach to determining instability will be exploited graphically in the next section where J versus T diagrams will be used as a method of ex- ploring crack instability problems

T h e J V e r s u s T S t a b i l i t y D i a g r a m

Consider a schematic representation of Jmaterial and Tmaterial on a J versus T diagram, using a side-by-side plot of the material's J-R curve, Fig 1 Given the material's J-R curve, the left-hand diagram of Fig 1, at any J such as in- dicated by the arrow, the slope, dJmaterial/da, may be determined As defined

by Eq 8

dJmaterial E

(17) Tmaterial da Go 2

which establishes a point on the J versus T diagram on the right in Fig 1 Repeating the procedure at various J-values will result in the J versus T material curve Note that below JIr no crack extension takes place, so that Zmaterial is very large (that is, off scale) In this way J-R curves can be transformed directly into the J versus T-mat curves

In a typical J-R test, the remaining uncracked ligament, b, is the proper dimension to determine w as defined by Eq 3 Therefore, dividing Jmate,a~ by Zmaterial

11-14 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

Consider the conditions of assured validity, Eqs 3 As shown in Fig 1, a crack extension limit (Aa << b) may be placed on the R-curve with a cor- responding mark at the same J-level on the J versus T-mat curve Another limit (from Eq 3) can be represented as in Fig 2 by a line of slope oo2b/Eo~

through the origin representing Eq 18 The actual material properties (o0 and E), specimen size (b), and smallest acceptable o~ (perhaps five or smaller) determine the slope and, therefore, the intersection with the ~-limit of, the materials curve Therefore the J versus T-mat curve may be doubtful above the lower of these two limits (It is presumed that all other J-R curve test re- quirements and practices are met satisfactorily.)

All J versus T material curves which have been plotted to date have shown concave upward behavior Physical reasons why this should be observed will be omitted here Accepting this empirically observed behavior, the material curve from below the limit marks at least could be extrapolated upward as a straight- line extension of the valid curve to determine a safe J versus T loading region as shown in Fig 3 That is to say that if a cracked specimen of the same material

is loaded to a certain J-level, and the applied (Japplied Tapplioa) point is in the

"safe region" as shown in Fig 3, then for that J-level, Tapplie d < Tmate~i~ and the crack is stable according to Eq 9

It remains to determine the trace of the (Japol~d, TappJieO) points for Tappr~a (J)- curve, as loading or J increases starting with no load However, it is sufficient

to observe that for the applications to be considered here, 4 the Zapplie d c u r v e s always increase monotonically with J whereas the Zmaterial c u r v e s decrease monotonically with J, so the intersection of the two curves uniquely indicates the onset of instability; that is, no prior instabilities (intersections) can occur This is illustrated in Fig 4

Analysis of typical Tapplied curves for the applications of interest will follow to demonstrate the monotonic increasingJ versus Tappiied behavior

On the other hand, there are other applications such as testing for which

J

LIMIT

sLoPE

T

FIG 2 Assured validity limits noted on a ]-T diagram

4Cracks in pressure vessel walls primarily loaded with internal pressure are considered here

<T,,A~

9 ( S T A B L E )

T

FIG 4 A schematic T-applied curve extending to instability

always stable conditions are sought (in bending where TappJiea = negative)

These are treated in earlier studies [2] sufficiently for the objectives of this

current work Nevertheless, it is noted and the reader is warned that other

relevant considerations must be made where widely different loading condi-

tions and crack configurations exist, such as plastic bending of nuclear pip-

ing with through cracks However, for the normal conditions and postulated

flaws for pressure vessels, the J versus Taopl~ behavior will follow a consistent

pattern, as will be shown

Analysts of Y Versus T Applied Curves for Through Cracks In Pressure

Vessel Walls

Under the actual pressures expected in nuclear pressure vessels, the shell

stresses remain linear elastic and LEFM conditions apply At a temperature

high enough to be well into the Charpy upper-shelf region and for flaw sizes of

11-16 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

interest, it may take stresses approaching yield or higher to cause actual crack instabilities Moreover, in assessing measured crack instabilities in model or full-scale vessel tests, the necessary pressures resulted in stresses near or ex- ceeding the yield of the material Therefore, along with the previously developed J versus T diagram, stability analysis, and material characteriza- tion, it is necessary to develop analytical equations for Japplie d and Tapplied which are accurate when applied in the LEFM range and also can be applied in the range where stresses exceed the yield strength Thus factors of safety or results

of vessel tests or both may be assessed at least approximately

where )~ = a/',,/-R-[ and Y is a geometrical correction factor for the effect of shell curvature and bending Substituting Eq 20 into Eq 19 and rearranging leads to a convenient form

f"Tra2 "~

which contains the same stress bracket as Eq 21 but a new geometry bracket

To identify the implied Japplied versUS Tapplie d curve on a J versus T diagram by

eliminating load or e, simply divide Eq 21 by Eq 23, to obtain

Japplied : ~ [ 1 1

Tapplie d " E " 1 + 2~Y'/Y (24) For constant crack size, a, and for a given material, the ratio of Japplied

to Tapplie d is a constant according to Eq 24, which can be represented as a

straight line through the origin on a J versus T diagram as in Fig 5

As loading occurs, that is, as stress o is applied, from Eq 21, J applied

starts from zero (the origin of Fig 3) and proceeds to increase with the

square of the applied stress If J exceeds JIc crack extension, Aa (actual)

begins to occur so the trace of Japplied versus Tapplie d would depart slightly

from a straight line But the crack length changes prior to the onset of in-

stability are likely to be small in heavy sections so this slight departure will be

neglected for the moment, s

It remains to show how the Japplied VCrsUS Tapplie d curve behaves as stresses

exceed the range of applicability of LEFM But first it is relevant to establish

values for the geometry brackets as given in Eqs 21, 23, and 24

Shell Correction Factors or Geometry Brackets for Through Cracks in

Cylindrical Shells with Internal Pressure

The shell correction factors for longitudinal through cracks in shells as

developed first by Folias [10] and modified by Erdogan and Kibler [11] and

FIG 5 A typical J versus T applied curve (almost straight)

5Even if they are not small, their effects can easily be incorporated into the analysis, as a

perturbation

11-18 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

verified by Krenk [12] are perhaps most conveniently shown in Rooke and

Cartwright's work [13] For the longitudinal crack they can be empirically

expressed over the range of interest by the approximations ( _ 1 percent)

Y = (1 + 1.25)`2) 1/2 for(0 _< h -< 1)

= (0.6 + 0.9),)for (1 _< k ~ 5) where as before

a

Similar expressions may be developed for circumferential cracks again see

Ref 13 but are of lesser interest since the applied longitudinal stresses are a

factor of 2 less than the hoop stresses, and longitudinal cracking is favored

Using expressions such as Eq 25 or curves from [Ref 13], the geometry

brackets required in Eqs 21, 23, and 24 have been computed and are given

here graphically in Fig 6-9 for both longitudinal and circumferential

through cracks

In the following discussion it will be of special interest to note that the

geometry bracket associated with Eq 24 (dashed curves in Figs 6-9) is always

a number smaller than 1 and greater than 1/3 Indeed for most vessels, R / t _~

10 and the usual leak-before-break assumption of a = t gives )` -_ 0.31 and

the [ ] is between 1 and 0.8, that is, always nearly 1 in Eq 24

Plastic Zone Corrected L E F M Conditions

Historically the first attempts to extend LEFM toward the elastic-plastic

range included correcting the crack length for the plastic zone at the crack

tip to obtain an effective crack size, aef f, that is

aeli = a -4- ry

where

~lrao 2 where

fl -=- 2 (for plane stress)

-= 6 (for.plane strain)

(26)

In applying the plastic zone correction to Eq 21, for example, the crack

size, a, might be replaced by aeff, both where it appears explicitly and in Y

However, its use here shall be restricted to relatively low nominal stress

I.i.1 1- (/) 0.4 ~

FIG 6 Shell correction factors for longitudinal cracks in cylinders (for low Kt

levels, for example, (a/o o) < ~3, so ry << a, so that its effect on the value of

the geometry bracket [ y2] and others will be small and can be neglected

Correcting only the explicit appearance of "a" in Eq 21 and rearranging

gives

ao2a I r(a/ao)2 1

Japplied = E 1 ( y 2 / ~ ) (o/ao)2 [ y2] (27) For through cracks in nuclear vessels, where instability is approached at

stress levels of 2/3 yield or less, the crack-tip plastic zone stress state will be

closer to plane strain than plane stress Hence, to simplify the stress bracket

in Eq 27 taking y2 = 1 (but using the actual values from curves) for the

geometry bracket and ~ = 2 (plane stress thus conservative), a conservative

estimate of ']applied is achieved

1620 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

Japplied = E 1 - 1/2 (0/00) 2 [ y2] (28)

Indeed, most often the 1/2 in the stress b r a c k e t might be too conservative b u t

it can be no less t h a n 1/6 For the range of interest Fig 10 shows a plot of these extremes for the stress bracket Using the conservative value t/2 also compensates for the slight underestimate of the geometry bracket, [ y2], by neglecting the plastic zone correction in it

D 0.6 0

a ILl

"T" 0")

0 4 t'~

D

LO

r r 0.2

FIG 8 Shell correction factors for circumferential cracks in cylinders (for low X)

Finally, it is noted that the simplifying assumptions leading to Eq 28 not only result in a good (perhaps slightly conservative) approximation for./applied, but most importantly result in an especially convenient format The stress bracket and geometry brackets in Eq 28 completely separate the stress and geometry effects on Japplied into independent factors Because of the separa- tion and following the analysis represented by the sequence Eq 21 to Eqs 23 and 24, operating on Eq 28, the results are

11-22 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

- 0 8 t ) t:)

to

"l- or)

- 0 6 < a

,< LL.I

t h r o u g h the origin of t h e J versus T d i a g r a m of a slope given by

FIG lO Stress correction factors f o r J and T f o r low stress, a/a o <- 0.67

A Note on Further Extrapolation of the Stress Bracket

The analysis of actual nuclear vessels at nominal stress levels above 2/3 yield

is not realistically associated with any known operating or even faulted condi-

tions However, for the purpose of comparison of analytical methods with test

results from model vessel tests pressurized to crack instability, extrapolation

of the foregoing methods to obtain fair approximations is relevant

Moreover, at stress levels higher than 2/3 yield, interest becomes centered

on rather short through cracks, a << t, so that X << 1 and the geometry cor-

rection effects become small Under such conditions, the separation as in Eq

11-24 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

28 to independent stress brackets and geometry brackets is no less justified; thus it need not be discussed further here For the stress bracket functions derived in the following, it must be noted that they should be applied only for low X (X < 1) so Figs 6 and 8 will be relevant but Figs 7 and 9 should be excluded

The Strip Yield Model Stress Bracket

Using the so-called Dugdale strip yield model to develop the stress bracket, the development of the function follows Eqs 21, 23, and 24 or equally well Eqs 28-30, repeated here for emphasis

a92 a Japplied : E { }[ y2]

where { } is the stress bracket

From the solution for the strip yield model for a center through-cracked plate for example, see Ref 14 and comparing results with the first of Eqs

32, the stress bracket for strip yielding is

where 0.7 (for plane strain) _< 3' -< 1 (for plane stress)

This stress bracket might be used for stress levels from 2/3 yield up to (but not including) the yield strength (it assumes elastic-perfectly plastic nonhard- ening material) It is appropriate to go on to hardening solutions for ex- trapolation of the stress bracket for stresses at or above the yield strength

The Power-Hardening Stress Bracket

For a power-hardening approximation of a material's stress strain curve by

EO

the numerical solutions for center-cracked plates under both plane stress and plane strain have been presented by Hutchinson and co-workers [15] Their results were compiled and applied to develop tearing instability parameters

by Zahoor [16] and tabulated by Tada [17] Taking their plane stress results

in the same form as the first of Eqs 32, the stress bracket becomes

Above yield the stress is always near the yield stress a = % (or Eq 34 can be adjusted) Hence in the above-yield range the stress bracket is almost propor- tional to the strain or more properly the stress times the strain Substituting

Eq 36 into the first of Eqs 32

Noting that in this relationship "/applied varies approximately linearly with nominal stress, o, with nominal strain, e, and with crack size, a, is of con- siderable assistance in intuitively understanding the role in loading deforma- tion and size as variables affecting Japplied"

However, the simple power-hardening model of a material's stress-strain curve Eq 34, is inadequate to represent the detailed behavior of both the elastic range and the hardening range A better representation is found through the Ramberg-Osgood approximation

The Ramberg-Osgood Stress Bracket

The Ramberg-Osgood representation of a material's stress-strain behavior

is

(38)

\ a o /

~o ao

11-26 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

Again, from Hutchinson's results [15] as compiled by others [16,17], com- paring terms in the same form as the first of Eqs 32, the stress bracket may

be written

k \ o 0 / o 0 / ) The parameters q,* and G * vary in a complex way with ~x and n, which can

be determined from analysis in Refs 16 and 17 The limiting case for elastic material, ~x = 0, is ~I,* = 7r (G* ;~ 0o n # Do) Thus Eq 39 is seen to reduce

to a form proper for insertion in Eq 21 At the other limit with the stress above yield, o > o0, the xI,*-term is negligible and then ~x G* =- c~f*, which produces agreement with Eq 35

It would be cumbersome to present stress-strain curve-fitting considera- tions using Eq 38, as well as corresponding determinations of q * and G*, for all materials here More to the point is to consider a typical material, AS33B,

at 93~ (200~ for which Shih [18] obtained the following curve-fiRing results

o o = 60ksi

E = 2 9 • 103ksi

= 1.115

n = 9,7 Following Refs 16 and 17 for plane stress and using these results, one obtains

q'* 4.3

~' G * = 11.8 which, when substituted in Eq 39, gives

Summary on Through-Crack Analysis

In summary, a method has been developed to analyze through cracks in nuclear pressure vessels to determine '/applied, rapplied, and Japplied/Tappiied

-4 C- II

s 73

S8 3A I:IV:I7OI'IN

NO NO

s C]NV 811::IVd

Copyright by ASTM Int'l (all rights reserved); Wed Dec 23 18:02:42 EST 2015

11-28 ELASTIC-PLASTIC FRACTURE: SECOND SYMPOSIUM

Neglecting both a plastic zone correction to the geometry factor and geometry correction to the stress factor forced a separation of effects which was compensated by developing stress factors for plane stress (conservative) For application at stress levels below 2/3 yield or low values of X (h << 1), the method is accurate and slightly conservative At stresses above 2/3 yield or with high X(X > 1), but not both, the method will give good approximations This permits comparison of analytical predictions with many test results [The method is not intended to treat long through cracks (;~ > 1) concurrent with high nominal stresses (approaching or above yield), but this combina- tion is never encountered in nuclear vessel analysis.]

The resulting equations for all cases were reduced to Eqs 32 The geometry brackets were given in Figs 6-9 and the stress brackets in Fig 11 (and Fig 10) Finally, the loading line on a Jversus T diagram for the trace of Japplie d versus

times a factor which ranges from 0.5 to 1 This result is independent of the stress bracket model employed

For a surface flaw of depth, a, and length, 2c, in a vessel wall of thickness,

t, the form of the elastic solution for K is often given as

Japplied- E - T 71" F(a/c)G(a/t) (42)

where

dG G(a/t)=[g(a/t)] 2 and G ' ( a / t ) - -

d(a/t)

for subsequent use

Differentiating under constant

Tapplie d gives

pressure stress, o, as before to obtain

where the derivatives of F are neglected since they are slightly negative for in- creasing "a" compared with "c." This gives a conservative result for Tapplied Proceeding as before, dividing Eq 42 by Eq 43

which for 0 _< (a/t) <_ 1/2 takes on values which range from 1 to 0.57 Hence,

as before, the geometry bracket in Eq 44 is slightly less than but nearly equal to

1 for cases of interest This result is also independent of adjustments to the stress bracket, does not enter Eq 44, and is independent of the crack shape aspect ratio; that is, it does not include the function F (a/c) This elastic analysis should be tentatively restricted to avoid yielding of the uncracked re- maining ligament, t a, behind the crack It is certainly acceptable if

cr/tr 0 < (1 a/t)

For additional reasons associated with corrections of the form of the forego- ing elastic analysis, it is prudent to restrict its use to a/t-values equal to or less than 1/2