Astm stp 559 1974

FRACTURE TOUGHNESS AND SLOW-STABLE CRACKING Proceedings of the 1973 National Symposium on Fracture Mechanics, Part I A symposium sponsored by Committee E-24 on Fracture Testing of Metals, AMERICAN SOCIETY FOR TESTING AND MATERIALS University of Maryland, College Park, Md., 27-29 Aug 1973 ASTM SPECIAL TECHNICAL PUBLICATION 559 P C Paris, chairman of symposium committee G R Irwin, general chairman of symposium List price $25.25 04-559000-30 ~(~l~ AMERICAN SOCIETY FOR TESTING AND MATERIALS 1916 Race Street, Philadelphia, Pa 19103 1974 Library of Congress Catalog Card Number: - 1 (~) AMERICAN SOCIETY FOR TESTING AND MATERIALS NOTE Thc Society is not responsible, as a body, for the statements and opinions advanced in this publication Printed in Baltimore, Md August 1974 Foreword The 1973 National Symposium on Fracture Mechanics was held at the University of Maryland Conference Center, College Park, Md., 27-29 Aug 1973 The symposium was sponsored by the American Society for Testing and Materials through Committee E-24 on Fracture Testing of Metals Members of the Symposium Subcommittee of Committee E-24 selected papers for the program Organizational assistance from Don Wisdom and Jane Wheeler at ASTM Headquarters was most helpful G R Irwin, Dept of Mechanical Engineering, University of Maryland, served as general chairman Those who served as session chairmen were H T Corten, Dept of Theoretical and Applied Mechanics, University of Illinois; C M Carman, Frankford Arsenal; J R Rice, Div of Engineering, Brown University; D E McCabe, Research Dept., ARMCO Steel; J E Srawley, Fracture Section, Lewis Research Center, NASA; E T Wessel, Research and Development Center, Westinghouse Electric Corp.; and E K Walker, Lockheed-California Co The Proceedings have been divided into two volumes: Part Fracture Toughness and Slow-Stable Cracking and Part II Fracture Analysis Related ASTM Publications Stress Analysis and Growth of Cracks, STP 513 (1972), $27.50 04-513000-30 Fracture Toughness, STP 514 (1972), $18.75 04-514000-30 Fracture Toughness Evaluation by R-Curve Methods, STP 527 (1973), $9.75 04-527000-30 Progress in Flaw Growth and Fracture Toughness Testing, STP 536 (1973), $33.25 04-536000-30 Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized Contents Introduction Fracture Toughness Test Methods for Abrasion-Resistant White Cast Irons Using Compact Specimens D E DIESBUR6 Experimental Procedure Results and Discussion Conclusions 13 Development of Fast Fracture in a Low Alloy Steel J c RADON AND A A POLLOCK Material, Specimen Geometry, and Fracture Mechanics Results Acoustic Emission Fractography Discussion Conclusions 15 16 19 25 27 29 Acoustic Emission from 4340 Steel During Stress Corrosion Cracking-H H CHASKELIS, W H CULLEN, AND J M KRAFFT Experimental Procedures Data Reduction Comparison of Acoustic Emission to Crack Growth Comparison of Acoustic Emission to the Fracturing Process Discussion Conclusions 31 32 35 36 36 42 43 Effects of Shot-Peening Residual Stresses on the Fracture and CrackGrowth Properties of D6AC Steel WOLF ELBER Nomenclature Residual Stress Model Experiments Results and Discussion Conclusion 45 45 47 50 53 57 Fracture Properties of a Cold-Worked Mild Steel E, J RIFLING Materials and Procedure Results and Discussion Conclusions 59 60 62 72 More on Specimen Size Effects in Fracture Toughness Testing~J KAUFMANAND F G NELSON Material Procedure Results Discussion Implications in Size Requirements Value of Kmax Summary 74 75 75 79 79 83 85 85 Dynamic Compact Tension Testing for Fracture Toughness -P c pARIS, R 3" BUCCI~ AND L L LOUSltIN Alloys Experimentation Equipment and Instrumentation Test Records Analysis of Test Records Summary 86 88 89 91 93 96 98 Further Aspects of Fracture Resistance Measurement on Thin Sheet Material: Yield Stress and Crack LengthwA M SULLIVAN ANt) STOOP Experimental Parameters Degradation of K e with Increased Yield Strength Relationship Between Initial and Final Crack Length Screening Specimen Conclusions 99 101 103 104 109 109 Further Studies of Crack Propagation Using the Controlled Crack Propagation Approach R H WEITZMANN AND I FINNIE Results and Discussion General Test Aspects Conclusions 111 117 124 125 Double Torsion Technique as a Universal Fracture Toughness Test Method J o OUTWATER, M C MURPHY, R G KUMBLE, AND J T BERRY Fundamental Concepts of Fracture Mechanics Theoretical and Experimental Limitations Theoretical Basis for the Double Torsion Technique Experimental Procedure and Results Discussion Conclusions Author's Note Added in Proof 127 128 129 130 133 135 136 137 Measurement of KIe on Small Specimens Using Critical Crack Tip Opening Displacement J N ROBINSONAND A S TETELMAN Experimental Procedure Results and Discussion Summary 139 141 146 156 Correlation Between Fatigue Crack Propagation and Low Cycle Fatigue Properties -SAURINDRANATH MAJUMDAR AND JODEAN MORROW Nomenclature Description of the Fatigue Crack Propagation Model Mechanics and Fatigue Analysis of the Crack Tip Region Influence of Material Properties on the Coefficient of Eq 13 Effect of Microstructure Size Comparison of Eq 14 with Barsom's Data on Steel Conclusions Effect of Stress Concentration on Fatigue-Crack Initiation in HY-130 Steel J M BARSOM AND R C M e NICOL Materials and Experimental Work Results and Discussion Summary 159 159 163 164 168 169 172 173 183 184 187 199 Evaluation of the Fatigue Crack Initiation Properties of Type 403 Stainless Steel in Air and Steam Environments -w G CLARK, JR Material Experimental Procedure Experimental Results Crack Initiation Data Discussion Summary Conclusions 205 206 207 211 215 218 222 223 Subcritical Crack Growth Under Single and Multiple Periodic Overloads in Cold-Rolled S t e e l ~ F H GARDNER AND R I STEPHENS Material and Experimental Procedures Test Results Discussion of Results Conclusions 225 227 229 239 242 Effect of Mean Stress Intensity on Fatigue Crack Growth in a 5456H l Aluminum AlloymH P CHU Experimental Procedure Results and Discussion Conclusions Discussion 245 246 246 259 261 Effects of R-Factor and Crack Closure on Fatigue Crack Growth for Aluminum and Titanium Alloys M KATCHER AND M KAPLAN Nomenclature Experimental Procedure Results Discussion Conclusions 264 264 266 270 277 280 Proposed Fracture Mechanics Criteria to Select Mechanical Fasteners for Long Service Lives -A F GRANDT, J R AND 3" P GALLAGHER Approach Stress Intensity Factor Analysis Application of the Linear Superposition Method to the Fastener Hole Problem Interference Fit Fasteners Achieving a Fatigue Stress Intensity Threshold Implications to Fastener Design Summary 283 284 285 287 289 290 294 295 Rapid Calculation of Fatigue Crack Growth by lntegration T R ~RUSSAT 298 The Existing Summation Procedure 299 The Proposed Integration Procedure 300 Application and Comparison 307 Conclusions 310 STP559-EB/Aug 1974 Introduction Readers of this volume will not be disappointed with regard to novelties of current and practical interest in fracture toughness and slow-stable cracking These range from unusual test methods to a puzzling effect of lateral specimen dimensions on Kic values for an aluminum alloy Observational techniques include acoustic emission, both in relation to onset of rapid fracture and stress corrosion cracking, tape recordings as an assist for rapid load testing, and use of rubber castings to verify measurements of crack opening stretch Toughness measurements are reported for white cast irons and cold-rolled steel The papers dealing with fatigue cracking include a low cycle fatigue viewpoint on fatigue crack growth, effects of shot peening, initiation of fatigue cracking as a function of notch root radius, as well as effects of overloads, mean K, and mechanical fastener pressure The development of technology in this field has prospered over the years so that often novel approaches soon become routine techniques to solving problems This volume is another contribution to the engineer and metallurgist faced with fracture problems With two exceptions, all of the papers in this volume were presented at the 1973 National Symposium on Fracture Mechanics held at the College Park campus of the University of Maryland, - Aug 1973 The two exceptions were a paper offered for this symposium but not presented and a late submission of a paper from the 1972 symposium The companion volume, STP 560, covers fracture analysis G R Irwin Department of Mechanical Engineering, University of Maryland, College Park, Md Copyright*1974 by ASTMInternational www.astm.org D E Diesburg Fracture Toughness Test Methods for Abrasion-Resistant White Cast Irons Using Compact Specimens REFERENCE: Diesburg, D E., "Fracture Toughness Test Methods for Abrasion-Resistant White Cast Irons Using Compact Specimens," Fracture Toughness and Slow-Stable Cracking, A S T M STP 559, American Society for Testing and Materials, 1974, pp 3-14 ABSTRACT- The fracture toughness of abrasion-resistant white cast irons has been measured, using precracked compact specimens Some procedures used for precracking the brittle cast irons were outside the ASTM Test for Plane-Strain Fracture Toughness of Metallic Materials (E 399-72) requirements but still gave valid results The excellent reproducibility, combined with a range in toughness values of 17.5 to 28.5 ksiVTff (19.2 to 31.4 M N / m 8/~) for abrasion-resistant white cast irons, provided the sensitivity necessary to distinguish differences in the toughness of white cast irons resulting from variations in composition or microstructure The fracture toughness of three commonly used irons, 27Cr, 9Cr-6Ni, and 20Cr-2Mo1Cu, was compared in the as-cast (and stress-relieved) condition Heat treating the 20Cr-2Mo-lCu iron substantially increased the hardness and reduced the fracture toughness slightly KEY WORDS: abrasion-resistant iron, white cast iron, fracture properties, toughness, evaluation, mechanical tests, fatigue (materials), mechanical properties T h e mining i n d u s t r y uses a b r a s i o n - r e s i s t a n t m a t e r i a l s in large q u a n t i t y in various stages of o r e beneficiation T h e m a t e r i a l s u s e d for jaw crusher plates, c r u s h e r liners, mill liners, grinding balls, c o n v e y i n g systems, a n d slurry p u m p s all m u s t be able to w i t h s t a n d a b r a s i v e wear T h e w e a r rate in these pieces of e q u i p m e n t contributes significantly to the e c o n o m i c s of m i n i n g o p e r a t i o n s U n f o r t u n a t e l y , the m o s t a b r a s i o n - r e s i s t a n t m a t e r i a l s Senior research associate, Ann Arbor Research Laboratory of Climax Molybdenum Company of Michigan, a subsidiary of American Metal Climax, Inc Copyright9 1974 by ASTM lntcrnational www.astm.org T R B r u s s a t ~ Rapid Calculation of Fatigue Crack Growth by Integration REFERENCE: Brussat, T R., "Rapid Calculation of Fatigue Crack Growth by Integration," Fracture Toughness and Stow-Stable Cracking, ASTM STP 559, American Society for Testing and Materials, 1974, pp 298-311 ABSTRACT: A procedure is described for drastically reducing the computation time in calculating crack growth for variable-amplitude fatigue loading when the loading sequence is periodic By the proposed procedure, the crack growth, r, per loading period is approximated as a smooth function and its reciprocal is integrated, rather than summing crack growth cycle by cycle The savings in computation time result since only a few pointwise values of r must be computed to generate an accurate interpolation function for numerical integration Further time savings can be achieved by selecting the stress intensity coeff• (stress intensity divided by load) as the argument of r Once r has been obtained as a function of stress intensity coefficient for a given material, environment, and loading sequence, it applies to any configuration of cracked structure Any of a broad range of prediction models (such as "retardation models") may be used in conjunction with the proposed procedure Agreement with results obtained using cycle by cycle summation is demonstrated, for two retardation models KEY WORDS: fatigue (materials), crack propagation, cyclic loads, mathematical prediction In the past, calculation of crack growth for variable-amplitude fatigue loading has been carried out by essentially a cycle-by-cycle summation Even for one-time-only calculations on today's high-speed computers, this brute-force procedure already requires excessive computation time If fatigue crack growth analyses are to be included in the iterative computations of automated design, more efficient procedures are crucially needed A procedure is described for streamlining the calculation of crack length, a, as a function of time, t, under a periodic fatigue loading function, P ( t ) Research specialist, Fatigue and Fracture Mechanics, Science and Engineering, Lockheed-California Company, Burbank, Calif 91520 298 Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Copyright* 1974 by by ASTM International www.astm.org Downloaded/printed University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized BRUSSAT ON RAPID CALCULATION BY INTEGRATION 299 Typically, one to three orders of magnitude reduction in computer time can be achieved, compared to cycle-by-cycle summation Although not required for the proposed procedure, P(t) usually consists of an ordered sequence P1, P2 Pr of discrete loading cycles Let (~a)~ denote the crack growth due to the jth loading cycle in this sequence In the proposed procedure, the crack growth per period, r, defined by T r= E (Aa)j (1) j=l is calculated as a smooth function of crack length, and its reciprocal is integrated numerically to obtain the crack growth lifetime L in number of periods: L-= f0 L dt= fa "'~~ r(a) da (2) Initial This integration replaces the cycle-by-cycle procedure of calculation, which involves a summation over perhaps millions of intervals Since many numerical methods of integrating Eq are well-known, attention here is limited to estimation of the crack growth per period, r, as a function of crack length The concept of using numerical integration in calculating crack growth for variable-amplitude fatigue loading was used in an earlier paper ([1],2 p 135) More recently, the integration concept proved useful in simulated design and analysis of damage tolerant aircraft structure [2, 3] In each of these references the effects of previous loadings were neglected in the crack growth prediction model used In contrast, this paper addresses the case in which crack growth per period r is calculated by prediction models such as the "retardation models" of Wheeler[4] and of Willenborg et al[5], which include effects of previous loadings The Existing Summation Procedure Suppose that a structure containing an initial crack of length ai.i.a~ is to be subjected to an infinite sequence P1, P2 of fatigue loading cycles Figure diagrams the existing summation procedure for calculating the growth of the crack Starting at the initial crack length, an increment of crack growth (~a)l (caused by P~) is calculated in accord with some crack growth prediction model such as that of Wheeler[4] or Willenborg et al[5] and added to *"The italic numbers in brackets refer to the list of references appended to this paper Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions auth 300 FRACTURE TOUGHNESS A N D SLOW-STABLE CRACKING (START) Let a = ainitial; i = !Aa)j = f~aj,Pj, ) [ ~ Crack I Growth = L=j/T Life (END) FIG Existing summation procedure ainttial From this new crack length, a2, and the next loading cycle, P2, the next crack-growth increment (aa)e is calculated This iterative process is repeated until the final or "critical" crack size is exceeded, which marks the end of the crack growth life In the past, modifications have been developed to improve the efficiency of this procedure For example, when several identical load cycles of equal magnitude occur, the increment of crack growth due to all of them combined may be computed simultaneously and added to the crack length as a single entity Even with such a modification, the essence of the procedure is a brute-force summation of small increments of crack growth taken over the entire lifetime The longer the life, the more time-consuming is the computation The Proposed Integration Procedure The Basic Concept Drastic reductions in computation time are possible if the calculation centers around a numerical integration rather than a summation However, in order to use integration in the manner described later it is necessary to represent the loading sequence as a periodic function of time P(t), with period, T, as illustrated schematically in Fig 2, where the period T is small compared to the crack-growth lifetime In aircraft design, for example, T Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized BRUSSAT ON RAPID CALCULATION BY INTEGRATION h T 301 LOAD P(t) TIME FIG Periodic loading sequence could be one simulated flight or, if necessary, a sequence of 10 or so representative flights from the anticipated loading history The basic concept behind the proposed integration procedure for calculation of crack growth under a periodic loading sequence P(t) is depicted in Fig The procedure is as follows A small number n of distinct crack lengths al a~ are selected such that the smallest is no larger than the initial crack length, ainiual, and the largest is no smaller than the final crack length, a~i,~l At each selected crack length, the increments of crack growth ( a a ) j due to each cycle in P(t) are computed This is done using the same prediction models as used in the summation procedure However, the crack FIG Proposed integration procedure Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 302 FRACTURETOUGHNESS AND SLOW-STABLE CRACKING length is not increased, but is held constant throughout one entire period T At the end of one period the increments of crack growth per cycle are summed to compute the crack growth per period, r, for the selected crack length A curve-fitting technique is used to connect the computed points and obtain 1/r (the inverse of crack growth per period) as a function of crack length The crack growth lifetime L in number of periods is calculated by numerical integration of this function between the desired initial and final crack lengths The rapidity of this approach is apparent from the fact that only a few periods of the periodic sequence need to be considered by direct calculation For example, the life might consist of thousands of periods, whereas in Fig (which is schematic only, but typical nonetheless) only five periods are directly considered in the integration procedure Rapid numerical techniques for integration are well known The number n of point values of crack length required for a good fit of a versus 1/r can depend on the configuration of the cracked structure For example, the crack growth rate in reinforced structure does not change monotonically Instead, as shown schematically in Fig 3, the crack slows down as it approaches a reinforcement, and then accelerates as the reinforcing member is bypassed Therefore, at least the five points shown in Fig 3, and perhaps more, are required for an accurate curve fit It is explained in the following how the growth per period r can be made always monotonic, so that fewer points are required for curve fitting, and how the same function r can be applicable to various configurations, without recomputing its point values The Use of Stress Intensity Coefficient The integration procedure for crack growth calculation as just introduced is general and can be used even when the crack growth rate does not depend exclusively on the current and previous values of K, the crack tip stress intensity However, significant further improvements in the computation efficiency of the integration procedure are possible when used in conjunction with "linear elastic fracture mechanics prediction models" (such as the prediction models of Refs and 5) in which exclusive dependence on K is assumed The term "exclusive dependence" here means that the fatigue crack growth rate depends on the loading history P(t) and the macroscopic configuration of the cracked structure only through the past and present values of K This assumption is valid[6] as long as the crack tip plastic zone Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions aut BRUSSAT ON RAPID CALCULATION BY INTEGRATION 303 size Ry remains small compared to all macroscopic dimensions such as crack size, a condition commonly referred to as "small-scale yielding." In the small-scale yielding case the stress intensity can be written in the form K(t, a) = P ( t ) ~z(a) (3) where 0t(a) is called the stress intensity coefficient This factorization of K is always possible since K, when not adjusted for crack-tip plasticity, is always proportional to the applied load P Since the stress intensity coefficient a is simply stress intensity K normalized by load, it is often a known function of crack size Otherwise, it can be determined using classical elasticity The schematic in the upper right-hand corner of Fig depicts stress intensity coefficient as a function of crack size for a crack in an unreinforced panel and in a reinforced panel Because values of stress intensity coefficient are monotonically related to crack tip severity, the crack growth per period, r, is always a smooth, monotonically-increasing function of ot Thus, an adequate curve fit of r(0t) can often be obtained with just two or three computed points, as the schematic in the upper left-hand corner of Fig illustrates The improved integration procedure, using the stress intensity coefficient, is depicted in Fig The crack growth per period r is computed at selected values of stress intensity coefficient, and curve fitting is employed to connect the computed points and express r as a continuous function of ~ This function is transformed by means of the known relationship between stress intensity coefficient and crack length for the configuration of interest This transformation produces i/r as a function of crack size, and numerical integration between the initial and final crack lengths results in the estimate of the crack-growth life The curve at the lower left in Fig is the same as the curve in Fig 3, but this curve can be accurately obtained with fewer computed points using the approach depicted in Fig 4, because crack growth per period varies smoothly and monotonically with stress intensity coefficient There is another advantage of the approach depicted in Fig Once the relationship between crack growth per period r and stress intensity coefficient ~ has been computed, it applies for any configuration of crack and structure provided the same loading sequence and environment is applied, the material is the same, and the same prediction model is used Figure illustrates this applicability of a single r(~) function to crack growth in an unreinforced panel as well as a reinforced panel As shown, the changes in configuration affect the crack growth life only because the relationship between stress intensity coefficient and crack length has changed Thus, the pointwise calculation of crack growth per period at specific values of ~ (the Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions 304 FRACTURETOUGHNESS AND SLOW-STABLE CRACKING FIG Use of stress intensity coefficient in the integration procedure most laborious part of the computation) is done only once for all configurations Beyond examining various crack configurations, a common exercise in design is to proportionally change the loading sequence by a factor, searching for improved sizing of the structure Such proportional changes in P(t) can also be handled without recomputing the pointwise values of r ( a ) The factor by which P(t) changes can be absorbed in the definition of stress intensity coefficient As a result, the a(a) functions shown in the upper righthand comer of Fig are merely shifted vertically by this factor prior to the transformation and iteration steps It is worth noting that instead of using a prediction model, the function r ( a ) can be experimentally determined for the given loading spectrum, environment, and material, using any simple specimen configuration Then this function, together with the applicable relationships between stress intensity and crack length (determined from analysis), can be used in pre- Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized BRUSSAT ON RAPID CALCULATION BY INTEGRATION 305 diction for configurations that would be more expensive and difficult to test This semi-empirical approach may prove to be extremely useful in design substantiation However, the emphasis of the present paper is on the case in which pointwise values of r are calculated, rather than experimentally determined Pointwise Calculations of Crack Growth per Period The calculation of crack growth per period r at selected values of stress intensity coefficient a is described in the following This procedure, diagrammed in Fig 5, is similar to the existing summation procedure (see Fig 1) and therefore is straightforward except for an initialization step that precedes each pointwise computation The initialization is needed because the crack growth life does not, in general, begin at the times when the values of stress intensity coefficient selected for pointwise computation occur Rather, each of these a-values will occur at some intermediate time in the life The growth of the crack at that time can depend strongly on prior values of cyclic stress intensity, as is recognized, for example, in "retardation"-type prediction models One way to consider effects of prior stress intensity values is in terms of material state At any moment during crack growth, past values of K have determined the present material state at the crack tip The present applied (START) INITIALIZE OTHER MATERIAL STATE VARIABLES [ SELECT 'Otk Rep I AND LET j - I v j= 11 (,~ a)j = fe(Pi Otk, Repj, ) I' I RePi + ==fl (Pj ak' Repj'"') NO ~ %= ~(zxa)j lEND) FIG Crack growth during one period Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions authorized 306 FRACTURETOUGHNESS AND SLOW-STABLE CRACKING K carries the material from its present state to a new material state Material state is described by material state variables, one of which is the stress intensity coefficient Another, as in the models of Refs and 5, may be Rep, the distance from the crack tip to the elastic-plastic interface (the point ahead of the crack tip separating material that has been plastically deformed by present and prior loadings from material not yet plastically deformed) It is the material state variables other than stress intensity coefficient that must be initialized at each selected value of a prior to computing crack growth per period Initialization of Rep in the models of Wheeler [4] and Willenborg et al [5] is straightforward Let/'max be the loading cycle in the sequence P(t) with the highest peak Immediately after Pmax has been applied, Rev will equal Rv~max~, the plastic zone size for that highest peak Therefore by starting the periodic loading at Pmax, Rep is equal to Rv~max),and the proper initial conditions are established for the next loading cycle in the sequence For other prediction models, other material state variable values (such as the minimum stress intensity required for the crack tip to open, for example) may be operative that are not so easily initialized at the selected values of stress intensity coefficient For each of these material state variables it should be possible to assume an initial value and correct it by iteration through a few periods During the iteration the stress intensity coefficient would be held constant As a result, the stress intensity K would be periodic with period T during the iteration (because P(t) is periodic) Since the material state variables are functions of K only, their correct values would also be periodic with period T, as long as the stress intensity coefficient tz were held constant Thus, the iteration to correct the assumed values of material state variables other than tz should converge to a set of periodic functions (Convergence properties of this initialization procedure would have to be verified for the particular prediction model in question.) The computation of crack growth per period r at selected values of stress intensity coefficient 0~ is carried out as shown in Fig After selecting the value of ~ to be considered, the other material state variables are initialized as just discussed Then for the first cycle in the loading sequence, an increment of crack growth (Aa)l is calculated according to the prediction model being used The material state variables other than ~ are adjusted in preparation for the next loading cycle For the next cycle, the same procedure is repeated to compute the growth increment (Aa)2 due to that cycle When every loading cycle in the period has been considered, the growth increments are summed to arrive at r, the crack growth per period, for the selected value of ~ Then a new value of tx is selected and the entire procedure is repeated Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproductions au BRUSSAT ON RAPID CALCULATION BY INTEGRATION 307 This provides the points leading to the r(a) function plotted in the upper left-hand comer of Fig The rest of the integration procedure for crack growth calculation has already been discussed Application and Comparison Two Existing Prediction Models It should be possible to use the integration procedure in conjunction with any crack growth prediction model The choice of prediction model affects only the equations used to compute (aa)j, the crack growth per cycle, for use in Eq As examples of how (Aa)j is calculated, the prediction models of Wheeler[4] and Willenborg et all5] are summarized here This provides background for a comparison, described in the section on comparison, and at the same time emphasizes the distinction between the calculation procedure itself and the prediction model selected for use within the procedure Let Kmin~ and Kmaxj denote, respectively, the stress intensities caused by the minimum and maximum values of load for the jth cycle in the ordered sequence P(t) at the appropriate value of stress intensity coefficient a If there were no effects of prior loadings on material state variables other than crack length, then the crack growth could be determined directly from the experimentally-determined crack growth rate function for constant-amplitude fatigue loading da dN const-ampl ~-~f (Kmin' Kma x ) (4) In the prediction models of both Refs and 5, the extent of crack-tip plasticity is used to characterize the effects of prior loadings on material state Let ~ys denote the tensile yield strength of the material For smallscale yielding the plastic zone size due to the ]th cycle in P(t) is [Kmaxi ~ z ) (5) where c~ is a selected constant Let Repj denote the distance from the crack tip to the elastic-plastic interface before the ]th cycle, and let (aa)~ denote the increase in crack length during the ]th cycle Then Repj M A X ( Rep,j_ O - - (Aa)(j'-I ), Ru~) (6) where the function MAX (x,y) takes the value of the larger of its two arguments Wheeler[4] defines a retardation parameter c~j by Copyright by ASTM Int'l (all rights reserved); Mon Dec 13:10:55 EST 2015 Downloaded/printed by University of Washington (University of Washington) pursuant to License Agreement No further reproducti 308 FRACTURETOUGHNESS AND SLOW-STABLE CRACKING where m is an empirical constant Crack growth is calculated from (Aa)j:cp i f(Kminj, K j) (8) The function f is crack growth rate for constant-amplitude loadng, Eq In the model of Willenborg et al[5], Kmll~and Kmax are replaced by "effective" stress intensity values K,nin