Mechanical Engineering Series

J. Angeles, Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms

P. Basu, C. Kefa, and L. Jestin, Boilers and Burners: Design and Theory

J.M. Berthelot, Composite Materials: Mechanical Behavior and Structural Analysis

U. Busch-Vishniac, Electromechanical Sensors and Actuators

J. Chakrabarty, Applied Plasticity

G. Chryssolouris, Laser Machining: Theory and Practice

V.N. Constantinescu, Laminar Viscous Flow

G.A. Costello, Theory of Wire Rope, 2nd ed.

K. Czolczynski, Rotordynamics of Gas-Lubricated Journal Bearing Systems

M.S. Darlow, Balancing of High-Speed Machinery

J.F. Doyle, Nonlinear Analysis of Thin-Walled Structures: Statics, Dynamics, and Stability

J.F. Doyle, Wave Propagation in Structures: Spectral Analysis Using Fast Discrete Fourier Transforms, 2nd ed.

P.A. Engel, Structural Analysis of Printed Circuit Board Systems

A.C. Fischer-Cripps, Introduction to Contact Mechanics

J. Garcia de Jalon and E. Bayo, Kinematic and Dynamic Simulation of Multibody Systems: The Real-Time Challenge

W.K. Gawronski, Dynamics and Control of Structures: A Modal Approach

K.C. Gupta, Mechanics and Control of Robots

J. Ida and J.P.A. Bastos, Electromagnetics and Calculations of Fields

M. Kaviany, Principles of Convective Heat Transfer, 2nd ed.

M. Kaviany, Principles of Heat Transfer in Porous Media, 2nd ed.

E.N. Kuznetsov, Underconstrained Structural Systems

P. Ladeveze, Nonlinear Computational Structural Mechanics: New Approaches and Non-Incremental Methods of Calculation

A. Lawrence, Modern Inertial Technology: Navigation, Guidance, and Control, 2nd ed.

(continued after index)

Dominique P. Miannay

Time-Dependent Fracture Mechanics

With 252 Figures

Springer

Dominique P. Miannay Institut de Protection et de Surete Nuc\eaire Departement d'Evaluation de Surete BP6 Fontenay aux Roses F-92265, France dominique.miannay@ipsn.fr

Series Editor Frederick F. Ling Emest F. Gloyna Regents Chair in Engineering Department of Mechanical Engineering The University of Texas al Austin Austin, TX 78712-1063, USA

and William Howard Hart Professor Emeritus Department of Mechanical Engineering,

Aeronautical Engineering and Mechanics Rensselaer Polytechnic Institute Troy, NY 12180-3590, USA

Library of Congress Cataloging-in-Publication Data Miannay, Dominique P.

Time-dependent fracture mechanics / Dominique P. Miannay p. cm.- (Mechanical engineering series)

Inc1udes bibliographical references and index. ISBN 978-1-4612-6537-5 ISBN 978-1-4613-0155-4 (eBook) DOI 10.1007/978-1-4613-0155-4 1. Fracture mechanics. 1. Title. II. Mechanical engineering series (Berlin, Germany)

TA409.M525 2001 620.1 '126-dc21 20001018393

Printed on acid-free paper.

© 2001 Springer Science+Business Media New York Originally published by Springer-Verlag New York, Inc. in 2001 AlI rights reserved. This work may not be translated or copied in whole or in part without the written permission ofthe publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval , electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.

Production managed by Steven Pisano; manufacturing supervised by Joe Quatela. Camera-ready pages prepared from the author's Microsoft Word files.

9 8 7 6 543 2 1

ISBN 978-1-4612-6537-5 SPIN 10791580

To myfamity

Mechanical Engineering Series

Frederick F. Ling Series Editor

Advisory Board

Applied Mechanics

Biomechanics

Computational Mechanics

Dynamical Systems and Control

Energetics

Mechanics of Materials

Processing

Production Systems

Thermal Science

Tribology

F.A. Leckie University of California, Santa Barbara

V.C. Mow Columbia University

H.T. Yang University of California, Santa Barbara

K.M. Marshek University of Texas, Austin

J.R. Welty University of Oregon, Eugene

I. Finnie University of California, Berkeley

K.K. Wang Cornell University

G.-A. Klutke Texas A&M University

A.E. Bergles Rensselaer Polytechnic Institute

W.O. Winer Georgia Institute of Technology

Series Preface

Mechanical engineering, an engineering discipline borne of the needs of the industrial revolution, is once again asked to do its substantial share in the call for industrial renewal. The general call is urgent as we face profound issues of productivity and competitiveness that require engineering solutions, among others. The Mechanical Engineering Series features graduate texts and research monographs intended to address the need for information in contemporary areas of mechanical engineering.

The series is conceived as a comprehensive one that covers a broad range of concentrations important to mechanical engineering graduate education and research, Weare fortunate to have a distinguished roster of consulting editors on the advisory board, each an expert in one of the areas of concentration. The names of the consulting editors are listed on the facing page of this volume. The areas of concentration are: applied mechanics; biomechanics; computational mechanics; dynamic systems and control; energetics; mechanics of materials; processing; thermal science; and tribology.

Austin, Texas Frederick F. Ling

Preface

This book is the natural continuation of the book entitled Fracture Mechanics, published fIrst in French in 1995 by Les Editions de Physique, and then in Eng­lish in a revised, updated, and expanded form in 1997 by Springer-Verlag, New York.

The first book reviewed the failure of cracked bodies under static loading and with possible quasi-static or stable crack propagation in isotropic and homoge­neous continua. The present work is divided into three main parts. This con­tinuation begins with a general treatment of the practical use of fracture me­chanics in terms of relevant material properties and loading. Then the basic knowledge of dynamic-fracture mechanics and of creep-fracture mechanics in isotropic and homogeneous continua is successively presented. The other im­portant issue in fracture mechanics, fatigue-fracture mechanics, is excluded, but some elements can be found here. It is assumed that readers will be familiar with the matter contained in the fIrst book and acquainted with the elementary definitions.

The book aims to be a reference book for postgraduate students, scientists and engineers working in the fIeld of materials, structural design, non­destructive evaluation and safety assessment. The book is more particularly devoted to metallic materials.

The first two chapters feature details of the most up-to-date review of the development of fracture analysis, and of the practical uses of fracture mechan­ics. Basic ideas, as well as current estimation approaches, for the structural analysis of industrial components are given. These chapters can also be consid­ered a practical illustration of volume I, and of parts on dynamic- and creep­fracture mechanics of this volume.

In the first chapter, emphasis is placed on the ductile-to-brittle transition of steels. In chronological order, after measuring toughness by the ductility from the tension test of smooth specimens, the noxiousness of notch and loading was recognized and Charpy testing was developed to screen metals with considera­tion of fracture initiation. With the appearance of large monolithic structures such as the ship hull, fully dynamic aspects of crack propagation and arrest were treated by Pellini and colleagues. Subsequently, thick structures such as nuclear pressure vessels appeared and the effect of thickness was taken into account by Wessel and colleagues at Westinghouse, and then codifIed by the American

x PREFACE

Society of Mechanical Engineers. However, the need to infer toughness charac­teristics of materials, and not from geometry from small specimens, particularly when the available amount of metal is small, was always present and the master­curve concept for the transition toughness is proposed. Other methods, such as the small-punch test, the indentation test, and the miniaturized-specimen tests, are developed to study flow properties, fracture-impact energy and toughness transition. The end of the chapter is an extension of the fIrst volume and is de­voted to the constraint effect, and to the warm-pre-stressing effect.

The second chapter is devoted to the historical development of engineering methods, also called simplifIed methods, for assessing the risk of fracture of components under design or in service, and giving a suffIcient basis for the comprehension of the procedures. Different methods under development in different countries are described, and more particularly the English R6 rules in the case of brittle and ductile fracture at relatively low temperatures, and the English R5 and R6 rules and the French Appendix A 16 procedure of the RCCM-MR code in the case of creep fracture at high temperature. The French work on simplifIed methods in the case of thermal shocks is also given and analyzed. The classifIcation into primary and secondary stresses is explained, and the calculus of safety margins based on probabilistic and deterministic mod­els for measuring the reliability of structures is given. These bases allow for the comprehension of the French RSEM code.

The next three chapters describe the dynamic aspect of fracture mechanics. The third chapter presents an approach to high-loading-rate characteristics

with inertia effect and an elementary background on the knowledge of the con­stitutive-flow rules under dynamic conditions in terms of stress, strain harden­ing, strain rate and temperature. Thermally activated and phonon viscous-drag controlled deformation rates are introduced. The more usual elastic and visco­plastic laws are described and the experimental procedures to identify them are given. Next, the micromechanisms of dynamic fracture by cleavage, identical to that of static loading but with different criteria, ductile tearing identical to that of static loading but with different criteria, and adiabatic shear specifIc to dynamic loading are reviewed, the experimental investigation procedures are described. Finally, equations of continuum-damage mechanics for ductile rupture are given.

The fourth chapter presents the behavior of a stationary crack in an elastic continuum under stress-wave loading. This behavior is given by an analytical time-dependent stress-intensity factor that describes the stress-and-strain singu­larity fIelds around the crack tip with a spatial distribution identical to that of a crack under static loading. Optical experimental methods to measure the stress­intensity factor are described. After describing the testing procedures allowing a very high loading rate, the main types of variation of dynamic brittle- or ductile­initiation and stable-propagation toughness of polymers and metals with loading rate are provided, and their links to fracture criteria, such as they are now more or less empirically understood, are given. Some insight is gained by analyzing the build-up of the visco-plastic zone in the small-scale yielding regime.

In chapter fIve, the dynamic propagation of a crack in an elastic medium subjected to loading is studied. The elastic singularity fIeld is described by a

PREFACE xi

stress-intensity factor depending on the crack velocity, and by spatial functions depending also on velocity. To adequately describe the strong transient propa­gation, the higher-order terms of the asymptotic development are to be consid­ered. The energy concept is presented either under its analytical form as a J­contour integral, or as an energy balance to define a crack driving force. No direct equivalence between stress-intensity factor and crack driving force exists, and these two quantities are used to describe the dynamic crack resistance. Crack arrest toughness is defined and the experimental procedures to determine it are given. When the visco-plastic response of the material is taken into ac­count under the small-scale yielding-regime hypothesis, a description of visco­plastic zone is available and dynamic-crack resistance is explained with the help of microscopic fracture criteria identical to those of the static case, or by contin­uum-damage mechanics for ductile tearing. However, understanding is made difficult by the actual heterogeneous and nonsimilar nature of propagation.

The theory for analyzing creep-crack initiation and propagation is the topic of the last two chapters.

The sixth chapter presents, firstly, an elementary background on the knowl­edge of constitutive-flow rules under creep conditions in terms of stress, strain rate and temperature. The dislocation, and the point-defect motions responsible for the deformation, are described. The deformation-mechanism maps with lines of equal strain rate are defined. The more customary elastic and visco-plastic laws are described. The concepts of skeletal point and reference stress are intro­duced. Thereafter, the micromechanisms of creep fracture are reviewed. Dam­age develops by void initiation and diffusion growth enhanced or hindered by dislocation creep constraints. The results are conveniently displayed as void­growth maps. The map is constructed with lines of equal damage. Final rupture occurs by void coalescence or facet micro crack coalescence. Phenomenological and micromechanism continuum-damage mechanics are given.

In chapter seven, the creep initiation and propagation of a crack in a creeping elasto-plastic medium is studied. The time-dependent singularity fields of the stationary crack in an elasto-plastic material, creeping according to the three fundamental usual laws first studied by Riedel and Rice, are described by spe­cific load parameters, such as the elastic-stress intensity factor, the plastic J­contour integral and the creep-contour integrals C( t), C*, C\, and so forth, depending on the extemalloading level, and on temporal and spatial distribution similar to that of a purely plastic material. This leads to the drawing of load­parameter maps with lines of transient time. For a growing crack, the stress­singularity field is described for steady propagation according to the Hui and Riedel model by means of an amplitude factor uniquely determined by the crack velocity, and thus independent of the applied load. For K-controlled growth, the singularity field is composed by this field, by the Riedel and Rice field, and by the singular-elastic field with domain sizes depending on time and load. For the extensive creeping regime, the singularity field is described by the Riedel and Rice model. The Ct parameter, proposed by Saxena and an extension of the stress-power dissipation-rate interpretation of C* into the transient regime, is introduced. The crack-growth data are discussed according to the these singular­loading parameters.

xii PREFACE

The morphology of crack initiation and growth, and incubation times, are discussed in terms of rnicromechanism and local damage according to the brit­tleness or the ductility of the material. Growth is analyzed according to a local approach with a stress-and-strain rate field for a damage-free material or a dam­aged obeying the model of Hutchinson on one hand, and with a fracture-strain criterion on the other hand. Such a treatment explains the transient regimes. Continuum-damage mechanics are also used to explain crack branching. Thus, the practical use of the different load parameters and their limitation is outlined.

In chapter eight, fatigue in the whole temperature range, and its interaction with creep at high temperature, are reviewed. Damage as it occurs in smooth specimens and structure is presented. Micromechanisms are described and the damage laws in terms of stress, strain and energy are given. Modifications commonly used when creep occurs are discussed. For fatigue-crack growth, the stress-ratio effect, the closure phenomenon for explaining the threshold, and overload, leading to retardation are presented according to their phenomenol­ogical aspect, and the local approach in terms of damage at the crack tip. When creep occurs at high temperature, the role of frequency with a transition value and hold time under constant load are presented. Limitation of the summation rule is emphasized, and the history effect linked to the loading phase and the transient nature of creep as it is currently envisaged is given.

There are exercises at the end of each chapter. Some of these include exten­sions of the text material. Answers to approximately one-half of the exercises are given at the end of the book.

This book results from a course taught at the Institut Superieur des Materiaux et de la Construction Mecanique. It also represents the fulfillment of thirty years of experience in the field of fracture mechanics at the Commissariat a 1 'Energie Atomique.

The author should like to thank the Institut de Protection et de Surete Nucle­aire and the Commissariat a l'Energie Atomique for their encouragement, and for the opportunity to publish this work.

June 2001 Dominique P. Miannay

Contents

Preface ................................................................................................................ ix

Symbols ........................................................................................................... xvii

1. STRUCTURAL-INTEGRITY ASSESSMENT: THE RELEVANT FRACTURE-TOUGHNESS EVALUATION

1.1 The Energy Approach: The Ductile-to-Brittle Transition (DBT) and the Fracture Analysis Diagram (FAD) ............................................... 3 1.1.1 The Notched-Bar Impact Tests ....................................................... 3 1.1.2 The Fracture Analysis Diagram .................................................... 10

1.2 Low-Toughness Materials ....................................................................... 20

1.3 Determination of Dynamic Fracture Toughness by Instrumented Pre-Cracked Charpy Impact Testing ....................................................... 22 1.3.1 Initiation and Crack Resistance Toughness .................................. 22 1.3.2 Crack-Arrest Toughness and the Master-Curve Approach .......... 23

1.4 The Ductile to Brittle Transition and Fracture Toughness ...................... 24 1.4.1 Elastic Toughness and the Correlations with the Charpy

V-Notch Impact Energy for Common Structural Steels ............... 24 1.4.2 Toughness Scatter and Constraint Effects in the

Ductile-Brittle Transition of Ferritic (Bainitic) Low-Alloy Pressure-Vessel Steels .................................................................. 27

1.5 Measurement of Flow and Fracture Behaviors Using Miniature-Scale and Compound Specimens ............................................................ 42 1.5.1 Flow Behavior .............................................................................. 42 1.5.2 The Energy Approach .................................................................. 45 1.5.3 The Toughness Approach ............................................................. 51

Appendix I: The Constraint Issue ...................................................................... 63 Appendix II: Warm Pre-Stressing Effect ........................................................... 69 Appenidix III: Choice of Steel Qualities with Regard to the Risk of Brittle Fracture ................................................................................................... 76

xiv CONTENTS

2. STRUCTURAL INTEGRITY ASSESSMENT: THE RELEVANT LOADING EVALUATION

2.1 Procedures for Pure Mechanical Loading at Low Temperature .............. 82 2.1.1 Basic Approaches ......................................................................... 82 2.1.2 Assessment Procedures ................................................................ 86 2.1.3 The Constraint Effect in the R-6 Defect Assessment Procedure ............................................................................. 95

2.2 Procedures for Combined Loading ......................................................... 96 2.2.1 Stress Classification ...................................................................... 96 2.2.2 Assessment Methods Based on Linear Elastic

Fracture Mechanics ...................................................................... 98 2.2.3 Assessment Methods Based on Elasto-Plastic

Fracture Mechanics ...................................................................... 99

2.3 Safety Margins ...................................................................................... 116 2.3.1 The R-6 Rules ............................................................................. 116 2.3.2 The RSEM Code ........................................................................ 117

2.4 Assessment Methods at High Temperature ........................................... 121 2.4.1 The R-5 Procedure ..................................................................... 121 2.4.2 The Modified R-6 Procedure ...................................................... 125 2.4.3 The Annex A16 Draft Document Procedure .............................. 128

Appendix I: Probabilistic Integrity Analysis of Mechanical Components ...... 136

3. DYNAMIC FRACTURE: ELEMENTARY DYNAMICS AND MICROSCOPIC FRACTURE

3.1 Dynamic Behavior of a Material... ........................................................ 142 3.1.1 Elementary Notions on Wave Theory ........................................ 142 3.1.2 Constitutive Equations ............................................................... 146

3.2 Fracture Micromechanisms ................................................................... 162 3.2.1 Brittle Fracture ........................................................................... 162 3.2.2 Ductile Fracture .......................................................................... 163

4. DYNAMIC FRACTU RE: TH E STATIONARY CRACK

4.1 Linear Elastic Analysis of the Two-Dimensional Crack ....................... 181 4.1.1 Description of the Stress and Strain Fields ................................. 181 4.1.2 Analytical Expressions of the Stress Intensity Factor ................ 182 4.1.3 Methods of Direct Measurement of the Stress Intensity ............ 189

4.2 Crack Speed and Thermal-Field Measurements ................................... 193 4.2.1 Methods of Measurement of Crack Speed ................................. 193 4.2.2 Methods of Measurement of the Thermal Field ......................... 194

4.3 Testing Methods and Fracture Criteria in Mode 1.. ............................... 195

CONTENTS xv

4.3.1 Preliminary Comment on Toughness Determination and Value ................................................................................... 197

4.3.2 Loading Procedures .................................................................... 199

4.4 Main Results on Toughness .................................................................. 215

4.5 Local Approach to Fracture and Continuum Damage Mechanics ........ 217 4.5.1 Microscopic Observation of Fracture ......................................... 217 4.5.2 Cleavage Fracture ....................................................................... 217 4.5.3 Ductile Tearing ........................................................................... 218

4.6 Testing Methods and Fracture Criteria in Mode II ............................... 219

4. 7 Mixed-Mode Dynamic Crack Initiation ................................................ 221

5. DYNAMIC FRACTURE: THE MOVING CRACK

5.1 Linear Elastic Modeling of Crack Growth ............................................ 227 5.1.1 Mott's Approach ........................................................................ 227 5.1.2 Analytical Description of the Elastic Asymptotic

Stress-and-Strain Fields .............................................................. 229 5.1.3 The Dynamic Stress Intensity Factor and the Dynamic

Strain Energy-Release Rate ........................................................ 236 5.1.4 Energy Concept for Any MateriaL ............................................ 239 5.1.5 Fracture Criteria ......................................................................... 246 5.1.6 The Arrest Phenomenon in an Elastic Continuum ..................... 249

5.2 Elastic-Plastic Modeling of Crack Growth ........................................... 256 5.2.1 Modeling with Strain Rate-Independent Plastic

Flow Behavior ............................................................................ 256 5.2.2 Modeling with Strain Rate-Dependent Plastic

Flow Behavior ............................................................................ 262

5.3 Mechanistic Models of Crack Growth .................................................. 264 5.3.1 Experimental Observations ........................................................ 264 5.3.2 Continuum Damage Mechanics ................................................. 266

6. CREEP FRACTURE: CREEP LAWS AND ELEMENTARY MICROSCOPIC FRACTURE MODELS

6.1 Creep Flow Behavior ............................................................................ 275 6.1.1 Creep Deformation Mechanisms ................................................ 276 6.1.2 Constitutive Equations ............................................................... 283 6.1.3 The Behavior of the Plain Bar and the Reference Stress ............ 285

6.2 Mechanistic-Creep Failure Analysis ..................................................... 287 6.2.1 Macroscopic Fracture ................................................................. 288 6.2.2 Fracture Mechanisms ................................................................. 290

6. 3 Fracture Parameters ............................................................................... 312 6.3.1 Creep-Damage Tolerance ........................................................... 313

xvi CONTENTS

6.3.2 The Time-Temperature Creep Failure Parameters ..................... 313

6.4 Continuum-Damage Mechanics ............................................................ 319 6.4.1 Introduction to the Damage-Constitutive Equations .................. 320 6.4.2 Phenomenological Continuum-Damage Mechanics .................. 321 6.4.3 Mechanistic Continuum-Damage Mechanics ............................. 323

Appendix I ....................................................................................................... 340 Appendix II ...................................................................................................... 341 Appendix III ...................................................................................... , ............. 342

7. CREEP FRACTURE MECHANICS

7.l The Stationary Crack ............................................................................ 343 7.1.1 Crack Tip Stress-and-Strain Fields Under the Small-

Deformation Theory ................................................................... 343 7.1.2 Crack Tip Stress-and-Strain Fields in Large

Deformation Theory ................................................................... 362 7.1.3 Estimation Scheme of the C(T) Integral by the

CcParameter ............................................................................... 362

7.2 The Moving Crack ................................................ " ............................... 366 7.2.1 Crack Tip Stress-and-Strain Fields Under Small

Deformation Theory ................................................................... 367 7.2.2 Experimental-Creep Crack-Growth-Rate Correlation ................ 370 7.2.3 Initiation of Crack Growth and Transient

Non-Planar Propagation ............................................................. 373

7.3. Relationship Between Micromechanisms and Macromechanics .......... 375 7.3.1 Micromechanisms Local Approach to Creep-

Crack Growth ............................................................................. 376 7.3.2 Continuum-Damage-Mechanics Approach to

Crack Growth ............................................................................. 385

7.4. Recommended Practice for Determining Creep-Crack Growth in a Material .......................................................................................... 388

7.5. The Stationary Crack Under Mixed-Mode Loading ............................. 390 7.5.1 Global Approach ........................................................................ 390 7.5.2 Local Approach .......................................................................... 394

8. FATIGUE AND CREEP-FATIGUE

8.1 Fatigue, Creep, and Creep-Fatigue Damage ......................................... 401 8.1.1 Fatigue ........................................................................................ 401 8.1.2 Creep Damage ............................................................................ 408 8.1.3 Creep-Fatigue ............................................................................. 408

8.2 Fatigue, Creep, and Creep-Fatigue Crack Propagation ......................... 410

CONTENTS xvii

8.2.1 Fatigue Crack Propagation ......................................................... 410 8.2.2 Creep Crack Propagation ............................................................ 427 8.2.3 Creep-Fatigue Crack Growth (CFCG) Law ............................... 428

Guide for Further Reading and Bibliography .................................................. 441

Answers to Selected Exercises ......................................................................... 443

Index ................................................................................................................ 455

Symbols

SYMBOL

F

H

J

Je

JBB

lE,l,

lF8

Jp

ls

JS RSE_M

JSR6

JsAI6

JSSY

Ko

Ko

KY

KCY

K-EE

Kmat

DESCRIPTION

Force (see also Pl.

Constraint parameter in terms of stress triaxiality ratio.

Rice's path integral.

Cleavage elastic-plastic fracture toughness.

Bounded body solution of J.

Elastic component of 1.

Finite-body solution of J.

Plastic component of J.

l-value estimated according to a simplified method.

l-value estimated according to the simplified method of the RSE-M code.

l-value estimated according to the simplified method of the R6 rule.

J-value estimated according to the simplified method of the A 16 appendix.

Small-scale yield solution of J.

Stress-intensity factor at crack initiation, or the value of K at the onset of rapid-fracturing ASTM E 122!.

Stress-intensity factor level corresponding to a 63.2% cumulative failure probability.

Absorbed energy by a broken standard Charpy Y -notch impact-test specimen.

Absorbed energy per unit ligament area by a broken standard Charpy Y­notch impact-test specimen.

Equivalent-energy fracture toughness or fracture toughness of steel determined according to the equivalent-energy methodology.

An elastic-plastic equivalent stress-intensity factor derived from the J­integral.

An elastic-plastic equivalent stress-intensity factor derived from the J-integral at the point of onset of cleavage fracture, Je.

Stress-intensity-factor level corresponding to a 50% cumulative failure probability.

Material toughness according to the R6 method.

Constraint-corrected material toughness according to the R6 method.

Master-curve Weibull fitting parameter, or master-curve reference-fracture toughness, corresponding to a 63.2% failure probability.

xx

Ks

Kr

KIR

Lr

Q

Sr

T

To

TIO(

Tt

USE

p oS y

p

cr

SYMBOLS

Stress-intensity factor corresponding to primary stresses.

Non-dimensional stress-intensity factor parameter according to the R6 method.

Stress-intensity factor corresponding to secondary stresses.

Sum of Kp and Ks.

Reference-fracture toughness or lower-bound initiation toughness.

Master-curve x% lower-bound fracture toughness.

Non-dimensional load parameter according to the R6 method.

Ratio of applied load to yield-collapse load.

Lower shelf energy.

Load at an arrest event.

Cumulative failure probability.

Fast-brittle fracture load.

Maximum load on an instrumented impact-test record.

Unstable crack-propagation load in an instrumented impact-test record.

General yield load.

Maximum load.

Q-stress, amplitude of the second term of an asymptotic solution for the stress field around a crack tip in a power-law hardening material.

Constraint parameter in terms of opening stress.

Constraint parameter in terms of hydrostatic stress.

Nil-ductility-temperature reference temperature.

Standard deviation.

Non-dimensional load parameter according to the R6 method.

Ratio of the applied load to flow-strength collapse load.

T-stress, constant stress parallel to the crack in the linear elastic crack-tip solution.

Master-curve material-specific reference transition temperature correspond­ing to median fracture toughness of 50%.

Transition temperature indexed at an X J energy level when using standard Charpy V-notch impact-test specimen.

Transition temperature corresponding to 50% shear fracture.

Upper shelf energy.

Reliability index.

CTOD spanning the original fatigue crack tip over a gauge length of 5 mm.

Partial safety factor.

Shift parameter according to the R6 method.

Standard deviation.

Plastic flow stress of a metal.

Static yield stress.

SYMBOLS xxi

• Dynamic Behavior

B

D

GID

K1 (v,t)

KU I

Ka

KA

m

m

v

v

V

H(t)

p

1 +a/ ' a function of crack-tip speed. B = ----"-----:::-::-

4 a l a s - (1+a/)2

D = 4 a a _ (1 +a 2)2 , a function of crack-tip speed. 1 s s

Elastic longitudinal wave speed in a rod.

Elastic dilatational wave speed.

= Cd, elastic longitudinal wave speed.

Heat capacity.

Elastic Rayleigh surface-wave speed.

Elastic shear-wave speed.

Dynamic elastic toughness.

Dynamic elastic-plastic initiation toughness.

Thermal conductivity.

Dynamic stress-intensity factor for a stationary crack.

Dynamic stress-intensity factor for a growing crack.

Dynamic stress-intensity factor for a growing crack.

Crack-arrest fracture toughness, or the value of the stress-intensity factor shortly after crack arrest.

Plane-strain crack-arrest toughness, or the value of the crack-arrest fracture toughness, Ka, for a crack that arrests under conditions of crack-front plane­strain.

Dynamic crack-arrest fracture toughness.

Dynamic initiation toughness.

Dynamic elastic propagating fracture toughness, or dynamic crack propaga­tion resistance.

Strain-rate sensitivity exponent.

Normalized crack-tip growth rate, vlcs or v/cR'

Crack growth rate or crack speed or crack velocity.

Particle velocity.

Impact velocity.

Unit step function, or Heaviside function = 0 for t < 0 and = I for t 2: O.

= ~l - v2 / c/

=~I_v2/C/. Taylor-Quinney exponent, fraction of plastic work-rate density that is converted into heat.

Density.

Dynamic yield stress.

xxii SYMBOLS

• Creep Behavior

it A

A,m,M,M

C(t)

C*(t)

C(t)/scc

C,

d

Ogb

Os

f

k

L

m

m

n,nl,n2

n

p

= da / dt, crack growth rate.

Cavity area on a grain boundary.

Constants in creep-constitutive equations.

Constant in power-law creep expression or secondary-creep coefficient.

Primary-creep coefficient.

Secondary-creep coefficient.

Tertiary-creep coefficient.

Constant in Larson-Miller parameter.

Creep compliance.

Steady state or secondary-creep characterizing parameter for a power-law creep-constitutive law.

Long-time creep-characterizing parameter for an elastic power-law secon­dary-creep constitutive law.

The parameter is a line-integral or the amplitude of the stress field, stress singularity in the creep zone.

Path-independent extensive secondary-creep integral.

Transient creep characterizing parameter for a power-law creep-constitutive law.

Line-integral for a material with a secondary (power) creep-constitutive law.

Amplitude of the stress field, stress distribution.

Time-varying value of C*.

Small-scale creep component of C(t).

Primary-creep characterizing parameter.

Primary-creep path independent integral.

Short-time creep characterizing parameter or time-dependent transient-creep fracture-mechanics parameter.

Creep component of the power-release rate.

Value of C* for power-law creep.

Value of C* estimated from reference stress.

Value of C* estimated from experimental data.

Facet diameter.

Atomic grain-boundary diffusivity.

Atomic surface diffusivity.

Stress-enhancement factor of creep assisted by grain-boundary sliding.

Boltzmann's constant, = 1.38 10.23 J K-1.

Cavity half spacing.

= (Ogb8gbnlkT)crj E ,)113, characteristic diffusion length scale parameter.

Time exponent in creep law.

Material constant in creep-strain law.

Stress exponents in creep laws.

Stress exponent in Hollomon and Ramberg-Osgood laws.

Larson-Miller parameter.

R

R

S

Sr

St

tr

tr

tru

trm

T

U

V

V

y,

I)

SYMBOLS xxiii

Activation energy.

Dimensionless-constant, second-order stress term in the series expansion of the stress field for a material with a secondary (power) creep-constitutive law.

Gas constant, = 8.315 J mol- l K- l

Cavity radius.

Creep process zone size.

Externally applied (remote) axial stress.

Mean stress to rupture.

Time-dependent allowable stress.

Time.

Small-scale primary to extensive-primary creep transition time.

Extensive-primary to extensive-secondary creep transition time.

Time to fracture.

Hold time under constant load or displacement.

Initiation time.

Normalized time given by tN = (aN/E)1 [E 0 I (a N I a 0) n 1 ,which is the time

required for the creep strain to equal the elastic strain under a uni-axial ten­

sile stress aN'

Time to rupture.

Rise time to maximum load during cycling.

Uni-axial creep rupture life.

Multi-axial creep rupture life.

Redistribution time prior to steady-state creep.

Reference time for steady-state crack growth under small-scale creep conditions.

Transition time between small-scale and large-scale, or extensive, creep conditions (behavior, regimes) for a material with a secondary-creep consti­tutive law; transition time prior to global steady-state creep.

Transition time between small-scale and large-scale, or extensive, primary­creep conditions for a material with primary and secondary-creep constitu­tive laws.

Transition time between small-scale and large-scale secondary-creep condi­tions for a material with primary and secondary-creep constitutive laws.

Remote transverse stress.

Absolute melting temperature.

Creep usage factor.

Creep-rupture usage factor.

Cavity volume.

Constants in damage-constitutive equations.

Grain -boundary free energy.

Surface free energy.

Grain-boundary separation.

Grain-boundary thickness.

xxiv

y

. c Es

. c E· 1

v

p

a,

a* ,

crnet

(0

(()

SYMBOLS

Multi-axial creep-rupture exponent in the Code of Practice MSRC.

Creep strain.

Creep-strain rate.

Elastic-strain rate.

Effective creep-strain rate.

Uni-axial creep ductility.

Creep ductility under multi-axial stress.

Equivalent creep-strain rate.

Initial-creep-strain rate.

Primary-creep-strain rate.

Secondary-creep-strain rate .

Initial-creep-strain rate.

Reference-strain rate, defined as the strain rate at the reference stress aref'

Factor relating C(t) to area under load-displacement-rate curve in creep regime.

Uni-axial creep-rupture exponent.

Density of cavitating facets.

Maximum principal stress.

Maximum principal stress at skeletal point.

Stress at distance d from crack tip.

Effective, equivalent or Von Mises stress.

Effective, equivalent or Von Mises stress at skeletal point.

Mean or hydrostatic stress.

Mean or hydrostatic stress at skeletal point.

Average normal facet stress at grain boundary.

Net-section stress defined as applied load divided by the uncracked ligament area.

Reference stress defined by aref = P crolP 0, where P is the load and PO is the limit load for a rigid plastic material with yield stress ao.

Sintering stress

Non-dimensional time C*/Jo'

Cavitation damage parameter.

Cavitation damage-rate parameter.

Atomic volume.

SYMBOLS XXV

• Fatigue Behavior

daldN

I'>a

C

C,m

f

I'>J

KM,,, KM

Kop

K,

K'

I'>K

t,p

R

R( C)

R(O)

R(P)

Fatigue crack growth per cycle.

Crack extension.

Normalized K-gradient, C = (l/K)dKJda.

Constants in crack growth law.

Frequency.

Cyclic J-integral.

Stress-intensity factor at closure.

Fatigue notch factor.

Minimum value of the stress-intensity factor during a cycle.

Maximum value of the stress-intensity factor during a cycle.

Stress-intensity factor at opening.

Elastic-stress concentration factor.

Cyclic-strength coefficient in the relation I'>EPI2=(l'>cr/2K') lin'.

Stress-intensity range.

Effective stress-intensity range, =KM-Kop­

Threshold stress-intensity range.

Conservative lower-bound to long-crack threshold values.

Number of cycles.

Number of cycles.

Number of cycles to failure.

Number of cycles to failure.

Number of reversals to failure.

Cyclic strain-hardening exponent.

Maximum applied load in fatigue.

Minimum applied load in fatigue.

Range of applied load in fatigue.

Ratio of minimum load (or stress-intensity factor) to maximum load (or stress-intensity factor), =PM/P m=KM/Km.

Cyclic or reversed plastic zone size.

Overload plastic zone size.

Monotonic plastic zone size.

Holding period at constant load during cycling.

Plastic work per unit area to advance crack.

Elastic-strain energy during tensile loading during load cycling.

Plastic-strain energy in the hysteresis loop.

Strain amplitude, E, =I'>E/2.

Minimum strain during cycling.

Maximum strain during cycling.

Mean strain during cycling.

Strain range.

Strain amplitude, £..E/2 = (EM - Em)/2.

xxvi SYMBOLS

6.0"

6.0"/2

Elastic-strain amplitude.

Plastic-strain amplitude.

Stress amplitude, 0". =6.0"/2.

Fatigue (endurance) strength limit.

Minimum stress during cycling.

Maximum stress during cycling.

Mean stress during cycling.

Stress range.

Stress amplitude, 6.0"/2=( O"M - O"m)/2.

ABBREVIATIONS

Acronyms CAT

CVN DBTT

DT

DWTT

FAD

FAD

FATT

FIE FIP NOT

RTndt

LSE

USE

EC

HRR

HR

MSRC

RR

SSC

TC

I mil = 2.54 10.5 m

Description Crack arrest temperature.

Charpy V-notch.

Ductile-to-brittle transition temperature.

Dynamic tear test.

Drop-weight tear test.

Fracture analysis diagram.

Failure assessment diagram.

Fracture appearance transition temperature.

Fracture transition elastic.

Fracture transition plastic.

Nil-ductility transition.

Reference temperature ofthe ASME.

Lower shelf energy.

Upper shelf energy.

Extensive creep state or regime.

Hutchinson/Rice and Rosengren stress-and-strain field near the crack tip of a stationary crack for a power-law plastic material.

Hui and Riedel stress-and-strain field near the crack tip of a steady-state moving crack for a power-law creeping material.

Multi-axial stress rupture criterion.

Rice and Riedel stress-and-strain field near the crack tip of a stationary crack for a power-law creeping material.

Small-scale creep state or regime.

Transient creep state or regime.

I ksi Ji; = 1.095 MPa..r;;;

I ftlb = 1.36 J 1 J = 0.74 ft.lb