Advances in fatigue lifetime perdictive

Use of Mechanistic Life Prediction Methods for the Design of Damage-Tolerant Contribution of Individual Spectrum Load Cycles to Damage in Notch Root Crack Calculation of Spectrum Load No

S'IP 1211

Advances in Fatigue Lifetime Predictive Techniques: Second Vo/ume

ISBN: 0-8031-1874-0

ISSN: 1070-1079

ASTM Publication Code Number (PCN): 04-012110-30

Copyright ©1993 AMERICAN SOCIETY FOR TESTING AND MATERIALS, Philadelphia, PA All rights reserved This material may not be reproduced or copied, in whole or in part, in any printed, mechanical, electronic, film, or other distribution and storage media, without the written consent of the publisher.

Photocopy Rights

Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by the AMERICAN SOCIETY FOR TESTING AND MATERIALS for users registered with the Copyright Clearance Center (CCC) Transactional Reporting Service, provided that the base fee of $2.50 per copy, plus $0.50 per page is paid directly to CCC, 27 Congress St., Salem, MA 01970; (508) 744-3350 For those organizations that have been granted a photocopy license by CCC, a separate system of payment has been arranged The fee code for users of the Transactional Reporting Service is 0-8031-1874-0/93 $2.50 + .50.

Peer Review Policy

Each paper published in this volume was evaluated by three peer reviewers The authors addressed all of the reviewers' comments to the satisfaction of both the technical editor(s) and the ASTM Committee on Publications.

The quality of the papers in this publication reflects not only the obvious efforts of the authors and the technical editor(s), but also the work of these peer reviewers The ASTM Committee on Publications acknowledges with appreciation their dedication and contribution to time and effort on behalf of ASTM.

Printed in Ann Arbor, MI August 1993

This publication, Advances in Fatigue Lifetime Predictive Techniques: Second Volume,

contains papers presented at the Second Symposium on Advances in Fatigue Lifetime dictive Techniques, held in Pittsburgh, PA on 4-5 May 1992 The symposium was sponsored

Pre-by ASTM Committee E-9 on Fatigue and its Subcommittee E09.08 on Fatigue of Materials.Michael R Mitchell, Rockwell International Science Center, and Ronald W Landgraf,Virginia Polytechnic Institute and State University, presided as symposium co-chairmen andare co-editors of the resulting publication

Use of Mechanistic Life Prediction Methods for the Design of Damage-Tolerant

Contribution of Individual Spectrum Load Cycles to Damage in Notch Root Crack

Calculation of Spectrum Load Notch Root Crack Growth Rate Under Elastic and

Inelastic Conditions-R SUNDER,R V PRAKASH,ANDE I MITCHENKO 30Fatigue Damage Due to Sub-Threshold Load Cycles Between Periodic Overloads-

Exact Determination of l1K.rr and Crack Propagation Prediction for Selected

Loading Sequences-so 1. ZHANG,H DOKER,H NOWACK,K SCHULTE,AND

FEM Analysis of Cyclic Deformation Around the Fatigue Crack Tip After a Single

A Creep Cavity Growth Model for Creep-Fatigue Life Prediction of a

Unidirectional W/Cu Composite-y.-s KIM, M.J. VERRILLI,AND

Thermal-Mechanical Fatigue Lifetime Prediction of an Austenitic Stainless

The Cumulative Fatigue Damage Behavior of MAR-M 247 in Air and

High-Pressure Hydrogen-M A MCGAW,S KALLURI,D MOORE,AND1. HEINE 117Crack Density and Fatigue Lifetime of Metals Under Variable Amplitude

Determination of Fatigue Limit Between 10S and 109 Cycles Using an Ultrasonic

Estimation of Fatigue Propagation Life in Resistance Spot Welds-s D SHEPPARD 169

A Reverse Plasticity Criterion for Specifying Optimal Proof Load

Fatigue Lifetime Prediction of Angle-Plied Fiber-Reinforced Elastomer Composites

as Pneumatic Tire Materials-B L LEE, J P MEDZORIAN,P K HIPPO,

Problems with Current Methodology in Using the Arrhenius Equation to Predict

the Long-Term Behavior of Polymeric Materials in Geotechnical

Based on the success of the first symposium on this topic in 1990 (published asASTM

STP 1122), this follow-up symposium was again intended to review recent progress in our

understanding of fatigue phenomena and in the development and application of methodsfor predicting the fatigue performance of materials and structures in service environments.Topical content was purposely kept broad in an effort to represent the efforts and viewpoints

of a range of researchers and practitioners who often participate in more specialized forums.This strategy, it is felt, provides an excellent opportunity for cross-fertilization and estab-lishment of common ground between the various disciplines and interest groups

A cursory scan of the contents clearly reveals the breadth of coverage The 15 papersincluded in the volume cover:

• fundamental issues in damage development and crack growth,

• behavior in both low- and high-cycle regimes, under constant amplitude and spectrumloading conditions,

• the performance of advanced materials in hostile environments, including creep-fatigueand thermomechanical fatigue, and

• predictive techniques for the real-world environment

It is clear that fatigue problems show no sign of disappearing from our increasingly complexand technologically driven society Thus the need to have in hand more powerful and effectivetechniques for assuring the mechanical integrity of our structures, machines, and deviceswould seem more urgent than ever The continuing high level of activity in fatigue relatedtechnologies serves to substantiate the criticality of this failure mode in engineering practice.Because of these ongoing efforts, we continue to see notable improvements in both ourexperimental and computational capabilities and, more frequently, in their productive in-terplay Indeed, well-conceived and executed experiments establish the behavioral databasefor intelligent model development and also provide the requisite validation tool for fine-tuning newly developed analytical tools

General approaches to damage accumulation and life prediction are the subject of thefirst three papers The complexities of composite systems are highlighted in the first paperand a critical element model is presented that provides predictions of effective strength thataccount for the operative failure mode in a given cyclic environment Of note is the estab-lishment of guidelines to help tailor a composite for a desired performance objective Thenext two papers provide a comprehensive damage assessment method that, in one diagram,combines initiation, short crack, and long crack growth responses This approach is shown

to be particularly relevant for the analysis of notch root cracks under spectrum loads.Three papers focus on the important problem of overload effects on fatigue crack growth.Here, advanced experimental techniques (e.g., acoustic emission, DC potential drop, stria-tion spacing) are helping to provide new insights into crack closure behavior as it affectsretardation or acceleration, or both Particularly encouraging is the development of me-chanics models detailing the crack tip deformation responses as an effective means forpredicting fatigue performance

Material performance in nonambient environments is the subject of three papers fatigue responses in a metal matrix composite is interpreted using a microstructural model

Creep-1

2 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

based on cavity growth Next, a nonlinear kinematic hardening stress-strain model is used

to relate damage to strain energy density for thermo mechanical fatigue of an austeniticstainless steel under both in-phase and out-of-phase loading modes Finally, a damage curveapproach is applied to the cumulative damage analysis of a nickel-base alloy under two-level and multiblock loading sequences A productive combination of careful experimen-tation and innovative model development is apparent in these efforts

Valuable insights into sequence effects on damage accumulation is provided by a paperdetailing studies of crack density development in three alloys under a variety of loadingprofiles Rational for the departure from linear damage concepts for these loading patterns

is clearly demonstrated Long-life fatigue behavior at ultrasonic frequencies (20 kHz) is nextstudied using a unique dynamic strain gage and SEM observations as a means to evaluatethe potential of such test techniques for accelerated testing programs Detailed documen-tation of results for three alloys will be of interest to the experimentalist

The final four papers in the volume provide evidence of the successful application ofadvanced predictive techniques in engineering practice A design approach for the difficultproblem of spot weld fatigue is developed using finite element methods in conjunction withlinear elastic fracture mechanics Encouraging results for a variety of practical weld config-urations are presented Using a local plasticity analysis for notched components, the nextpaper provides guidelines for developing optimal proof load levels for engineering structures.Cost-effective test procedures for aircraft tires provide the focus of an investigation of realtime monitoring techniques, (dynamic creep, temperature rise, and acoustic emission) asindicators of damage development in reinforced polymeric materials The use of rate theory

to project the long-term performance (100 years) of polymeric materials used in earthstructures and waste containment is critically examined in the last paper The many problemsassociated with such methods are clearly articulated

In summarizing the symposium content, a number of interesting trends can be identified.Out of necessity, interdisciplinary approaches are increasingly being employed to developmore realistic damage models for use in design applications The growing success record ofmodern analytical tools in engineering practice has served to further establish the credibility

of predictive methods, thereby providing an impetus for further developments In this vein,one can sense improved rapport between researchers and practitioners as they combineefforts to deal more effectively with fatigue on an applied level A conscious attempt hasbeen made to perpetuate such alliances through forums of this type Perceived benefitsinclude identification, by consensus, of key problem areas for research planning and, throughcollaborative efforts, the formulation of improved strategies for technology transfer.Finally, the co-editors are pleased to report that the ASTM Committee E-9 Award forBest Symposium Paper for 1992 was presented to Sheri Sheppard and Michael Strange fortheir paper "Fatigue Lifetime Estimation in Resistance Spot Welds: Propagation Phase."

We congratulate the authors for their fine contribution

Kenneth L Reifsniderl

Use of Mechanistic Life Prediction Methods for the Design of Damage-Tolerant

Composite Material Systems

REFERENCE: Reifsnider, K L., "Use of Mechanistic Life Prediction Methods for the Design

of Damage-Tolerant Composite Material Systems," Advances in Fatigue Lifetime Predictive

Techniques: Second Volume, ASTM STP 1211, M R Mitchell and R W Landgraf, Eds.,

American Society for Testing and Materials, Philadelphia, 1993, pp 3-18

ABSTRACT: An increasing number of engineering applications depend on the use of material

systems such as fiber-reinforced composites For the most part, the manner in which thesesystems are "designed" is presently heuristic Although much analysis and understanding of

"how such materials are made" is available, there is comparatively less systematic rigor that

addresses "how such materials should be made" This is a serious inhibition to the exploitation

of these materials and material systems

During the last few years, a variety of approaches has been developed for the analysis ofcomposite materials, especially fiber-reinforced systems The body of literature is especiallyreplete in the technical area of "effective stiffness" models, many of which are sophisticatedand well founded-and reasonably well validated A comparable body of work which addresses

"effective strength" is not available However, the author and his colleagues have developed

a mechanistic approach of this type, that is generally referred to as the "critical elementconcept," whereby careful laboratory work is used to define representative volumes of materialthat enclose a "typical" failure mode This representative volume is divided into a "criticalelement" that controls the final failure event, and "subcritical elements" that alter the localstress state around the critical element

The present paper extends this concept to the fiber/matrix level by introducing chanical strength models to be used in the critical elements The result of this advance is thatmechanistic models that include explicit representations of the parameters that describe themanner in which the material systems are made can be used to estimate remaining strengthwhen those parameters change during the lifetime of the material Moreover, the model canthen be used to "design" or tailor a material system for specific long-term performance Thislast topic is the focus of the present paper The approach will be demonstrated, and theinfluence of several parameters will be discussed This discussion will then be used to advanceseveral concepts for the rigorous design of material systems for damage tolerance

microme-KEYWORDS: composites, durability, life prediction, damage tolerance, material systems

IAlexander Giacco Professor of Engineering Science and Mechanics, Virginia Tech Blacksburg VA24061-0219

3

4 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

more specific meaning In point of fact, damage tolerance is defined by remaining strength,i.e., by the residual strength of a material, component, or structure at a specific point inthe lifetime of the item in question In these terms, then, damage tolerance is defined asthe "state of the material" which determines the remaining strength Therefore, if we are

to use mechanistic life prediction methods for the design of damage-tolerant compositematerial systems (our present objective), we must devise a mechanistic approach to thedetermination of the state of stress and state of material at any given instance during theloading history of a component, and construct a philosophy for the interpretation of therelationship between those states of stress and states of material in terms of remaining (orresidual) strength

Damage Development, and Property and Performance Evolution

We are concerned with damage-induced changes in the properties and performance ofcontinuous-fiber-reinforced laminated composite material systems, during the application

of sustained and cyclic mechanical, chemical, or thermal loading Extensive documentation

of the details of the events associated with those property and performance changes appears

in the literature [1-4] Figure 1 illustrates the manner in which such damage develops duringlong-term performance Initial properties may be altered by damage development of ageometric nature (such as micro-cracking) which may change the local stress state as well

as constitutive behavior; by chemical activity such as compound formation, molecular linking,

or other stoichiometric activity; or by thermodynamic events such as diffusion, phasechanges, or morphological variations as a function of time We take the position that thesechanges can be analyzed and represented as changes in the stress state or the material state(or both states) in association with the processes that drive those changes Since these changesmay be caused by nonuniform (often localized) processes, we will generally speak in terms

of local stress states and material states, and will provide a more precise definition of the

region in which our mechanistic representations are to be written, in the following section

on mechanistic modeling

The damage processes alluded to in Fig 1 combine in myriad ways to create phenomenathat may be cycle-dependent, time-dependent, rate-dependent, or dependent on combi-nations of those extensive variables Figure 2 enumerates a few of those phenomena thatare commonly discussed and described Of particular importance is the fact that many ofthese phenomena depend directly or indirectly on time Phenomena such as viscoelasticcreep, creep rupture, aging, and environmental degradation are usually explicitly time-dependent Such phenomena as crack growth are generally described as cycle dependent,although, of course, time is involved indirectly In several instances, these phenomena listed,and others as well, are not independent of each other Time-dependent behavior such asviscoelastic creep or diffusion-related environmental degradation may be significantly altered

by the presence of cyclic mechanical loading, especially if attended by micro mechanicalcracking

This brings us to a conclusion of paramount importance, namely, that mechanistic ing of damage development for the purpose of estimating damage tolerance must be con-cerned not only with the determination of stress states and material states, but also withthe correct modeling of the rate of the processes that create changes in those states Sincedamage tolerance is defined byremaining strength after some history of mechanical, thermal,

model-or chemical loading, time enters the problem directly as a parameter Since time also definesthe total life of a material or component, the issue of rate is inextricably tied to the question

of damage tolerance

We will argue that the rate equations that control property and performance evolutioncan be written, ultimately, as constitutive equations which define changes in stiffness,strength, or local geometry in terms of material parameters However, these rate equationsfor the damage processes of concern to us are quite difficult to determine, and sometimesvery difficult to isolate in the laboratory

Mechanistic Modeling

A rather special approach is taken to the mechanistic modeling of damage development.The details of this approach, called the representative volume concept and critical element

method, have been reported earlier [5-6].That approach, illustrated in Fig 3, assumes thatthe damage associated with property and performance degradation is widely and uniformlydistributed It is further assumed that a "representative volume" of material can be chosenfrom that uniform distribution such that the state of material and state of stress in thatvolume are typical of all other such volumes in the material, and that those details aresufficient to describe the final failure event for a specific failure mode Hence, a repre-sentative volume is chosen for each distinct failure mode.

This representative volume is divided into "critical elements" that are thought to definethe failure of the representative volume, and therefore the failure of the component, in thesense that the critical element remains intact until the final failure event The remainder ofthe material in the representative volume is regarded as "subcritical" in the sense that thefailure of the subcritical elements (such as cracking, debonding, delamination, etc.) doesnot cause failure of the representative volume or of the component

Hence, the division of the representative volume into critical and subcritical elementestablishes several essential parts of the approach Critical elements remain as "continua"

in the classical sense, and are described with constitutive theories throughout the life of thecomponent This continuum assumption does not exclude discrete events from occurringwithin the critical elements but does require that these discrete events be unimportant on

an individual basis for the final determination of property or performance that defines thelife of the component Hence, events that do not directly control the final remaining strengthand life of the critical element (i.e., the damage tolerance) are grouped into elements thatcan be described in terms of continuum concepts This sets a "lower limit" on the scale atwhich mechanics must be done and simplifies the inherent complexity of a mechanisticapproach It should also be noted that the scale of the representative volume in the criticaland subcritical elements may vary greatly, typically from the size of a few fiber failures to

a ligament that controls global buckling

Internal stress redistribution and relaxation brought about by changes in the sub criticalelements during damage development may significantly influence the environment in whichcritical elements operate For example, the stress state created by the degradation of sub-critical elements may ultimately exceed the allowable stress states in the critical elementsand cause the initiation of the final failure event Damage modes in subcritical elementsmay be represented, as necessary, by micro mechanics if the representative volume and

REIFSNIDER ON DAMAGE-TOLERANT COMPOSITE MATERIALS 7

critical element are defined at the micro level, by ply-level mechanics if that is appropriate,

or by mechanics formulation at the global level if the final failure mode is global in nature.Failure modes define the mechanics problem that must be solved to determine strength anddamage tolerance At a given time in the life of the critical element, a criterion is used tocompare the state of stress and state of material to predict remaining strength and life, andthe consequent damage tolerance

We can demonstrate the thesis of the present paper by considering a mechanistic sentation of damage tolerance in a laminate, using ply-level modeling We will end ourpaper by indicating recent advances in the use of micro mechanics to bring the descriptiondown to the fiber-matrix level

repre-Mechanistic modeling begins with the determination of the stress state that controls thebehavior of the critical element For unnotched laminates, we can use laminate analysis totranslate the results of global structural analysis (closed form or numerical) to the ply level

If notches are present, the mechanism of damage and failure is directly related to the notchgeometry and specimen thickness, so that the details of the local geometry must be included

in any attempt to consider changes associated with damage development For example,microcracking alone will cause stress relaxation that must be modeled correctly, since itoften results in increases in notched tensile strength (early in the life of the specimen) andmay result in early increases in compressive strength [7] Modeling all of the details of thelocalized matrix cracking is probably not necessary, and almost certainly not feasible sincethe complexity of damage is extreme, and the statistical variation of the details of damagefrom specimen to specimen is remarkably large [7,8] However, it is possible to capture theessence of the physics in a representation of the damage that creates a "typical" relaxation

in the region of the hole, and provides the correct local stress state for ply-level considerations

by linear viscoelastic behavior by the "Ratio" given by

8 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

controlled stiffness of plies by the ratio given in Eq 1, with time shifted by the factor given

in Eq 2 as a function of temperature The data used to generate these curves are from anepoxy matrix Other relationships have been established for other materials, including ce-ramic matrix composites of various types

Ply level strength may also be reduced For the present case, this reduction was empiricallyfit to strength reduction and creep rupture data using a fourth order polynomial Thisreduction, for a given test temperature, was then scaled using a scaling factor in the generalfashion

where Ai and Ti are material constants, and T is the absolute temperature. The scale rameter is adjusted to have the value 0 at the reference temperature TI(for which no time-dependent strength degradation is observed in the laboratory), to have unit value at thetemperature at which the test data were taken, and to seek very large values for temperaturesnear the disintegration temperature of the material (melting point, flash point, etc., asappropriate)

pa-We may also introduce a rate equation that represents the life of the critical elementunder "baseline" conditions, usually ambient or "normal" test conditions These may be

kinetic equations or simple S-N representations at the ply level (for this example) such as

where A, B, and C are material constants, taken to be 0.953, - 0.05, and 1.2, respectively;

in the following examples, Sa is the applied stress amplitude (or stress function, as explained

later) in the ply, and SUit is the ultimate strength of the ply under that stress state (this mayalso be an effective strength in terms of a failure criterion, as discussed subsequently).Many other time- and cycle-dependent reduction rates may be needed If corrosion driveschanges in stiffness or strength, rate equations for those phenomena must be invoked Ifdiffusion rate controls an environmental degradation, that rate must be introduced into thecalculation of changes In this latter context, moisture effects in some polymer matricesoften fall into this category, and may be highly nonlinear (i.e., non-Fickian) [13] Aging ofvarious types is also pertinent to many materials In the case of polymer matrix composites,aging may be related to the action of atomic oxygen and may be driven by diffusion, surfacechemistry, or other thermodynamic processes And finally, there may be nonequilibriumchemistry at the micro-level This may include such things as compound formation at thefiber/matrix boundary, oxidation, reduction, or dozens of other reactions, especially at hightemperatures and in active environments

Life Prediction and Damage Tolerance Models

Perhaps the most challenging aspect of the task of constructing a mechanistic model ofthe remaining strength (damage tolerance) of a composite material system is the question

of how to bring together the representations of changes in stress state and material state toassess the combined effect of all processes in a manner that is correct from the standpoint

of mechanics and materials behavior We have discussed an approach to this problem inearlier publications [5,6,14-16] Figure 4 illustrates the general concepts involved

We begin in the laboratory, with a careful and precise determination of the failure mode

of the composite material under the conditions of interest Although composite materialsystems demonstrate different failure modes under different conditions, the number ofdistinct failure modes is generally found to be relatively few, and well-described in theliterature; such things as critical numbers of fiber failures in tension, micro buckling of fibers

in compression, crack propagation, and debonding or delamination are examples The failuremode defines the critical element, i.e., an element of material that controls the final strength

of the composite in the sense that the composite fails when that element fails Also in thelaboratory we identify the damage modes that accompany the failure mode of interest Thesemay include matrix cracking, delamination, and many of the processes previously discussedthat change the state of stress and, in some cases, the state of material Together, the

"subcritical elements" and "critical element" define the representative volume, the region

of material that we model with our mechanics formulation The boundary value problem

we solve is defined by that representative volume, including the geometric details of suchthings as matrix cracking, and the constitutive equations of the critical element as suggested

in Fig 4

We seek, then, a model of the critical element, since the damage tolerance of the composite

is defined by the remaining strength of the critical element as defined by the state of stressand state of material that results from a given history of loading, interpreted in terms of agiven failure mode To make such an evaluation, we introduce an "evolution equation" that

we use to calculate remaining strength of the critical element, as follows

in which S.(O"it» is a generalized local stress function in the critical element, defined by the criterion for failure that is appropriate for the failure mode in question S'j(Xit» is the

strength function in the critical element in terms of the same criterion, t is sidereal time,

and, is the life of the critical element in terms of the current state of stress and state of

10 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

It should be noted that the total life of the critical element under current conditions, resented byN in Eq 6 or byT in Eq 5, is also a function of the number of cycles, since the

rep-life will be altered by changes in both the local stress state and local material state Fa in

Eq 6 is the "failure function" or failure criterion previously mentioned For tensile failure

of strongly fiber-controlled materials, for example, we have found that an appropriatefunction may be the ratio of stress in the direction of the fibers to the fiber-direction strength;

in the case of compression, a micro buckling criterion may be used, etc

For a ply-level formulation, then, Eq 6 or 7 provides the requisite method of constructing

a mechanistic model of the damage tolerance of a composite material system by providing

a method of bringing together the representations of stress states (and their changes),material states (and their changes), and constitutive rate information to calculate remainingstrength Changes in elastic stiffness (as in Eq 1) and the growth of subcritical damage cause

changes in stress states which change, in turn, Sa, Fa, and Nor T;changes in material strength

as in Eq 3 change S;j' Fa> and N or T; and changes in temperature affect the rate at whichthose changes occur and may have a direct effect on the "current life estimate," Nor T.Although we have listed a "minimum set" of change equations (for lack of space) to dem-onstrate the approach, other processes could be modeled in the same fashion Chemicalaction (such as oxidation) may alter both the stress state (or residual stress state) or thematerial state as a function of time (or cycles), and the rate of that influence can berepresented by the appropriate chemical rate equations (that are generally well-determined)

or by diffusion rates if they control the rate of the chemical action Microdamage growthrates influence stress state change rates Morphological changes (such as crystallographicgrowth, rearrangement of grain boundaries, free volume variations, precipitates, etc.) maycause changes in stiffness and strength, and will be driven by the thermodynamic ratesassociated with those phenomena All of these changes can be entered directly into Eqs 6

or 7, based on the laboratory data that characterize them

A frontier in this process (one of many), is the method of combining effects, even at thislevel Stiffness changes that cause stress state changes due to microdamage growth are clearlyindependent of, say, changes due to morphological variations, and those effects can simply

be summed or multiplied (as the expressions warrant) But some effects are known to benonlinear Diffusion rates may be influenced by the stress state, so that stress state changescaused by, say, damage growth will change the rates that enter the equations because ofdiffusion processes These kinds of interactions appear to be of secondary importance atthe present, but this type of situation must be the object of careful future study It should

be noted that interactions of failure modes are included in the integral Eqs 6 and 7 in thesense that any stress state alteration or material state alteration associated with some damage

REIFSNIDER ON DAMAGE-TOLERANT COMPOSITE MATERIALS 11

event or other failure mode would be included in proper representations of the stress stateand material state in the critical element, since our approach is a mechanistic one In fact,the only difference in the calculations for different failure modes is the difference in theinterpretations, i.e., the difference in the failure function, Fa> and possibly the function used

to determine N or Tfor a given type of critical element

Results and Discussion

To demonstrate the premise of our paper, we consider a (hypothetical) polymer-basedcomposite material, such as a carbon fiber reinforced epoxy We use laminate analysis todetermine ply-level stresses, and solve Eq 6 for the remaining strength using a code calledMRLife6, one of a code series that we have constructed over the last ten years or so forperformance simulation Figure 5 shows the results of a "baseline" calculation of remaining

strength and the value of the normalized failure function, Fa for a loading that is (initially)

12 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

about 50% of the strength of the laminate in tension (R = 0.1) The laminate is symmetricwith a stacking sequence of [0,45,0, - 45,0], The baseline case assumes no temperature,moisture, damage, creep, creep rupture, or other effects-obviously not a "real" situation,but one that provides an analytical starting point The estimated life of the laminate (thepoint at which the remaining strength and the failure function cross-not shown on thefigure) is about 1.5 million cycles

Figure 6 shows the same calculation with reductions in the stiffness of the matrix materialcaused, for example, by matrix cracking or other damage For this demonstration, the matrixdominated ply stiffness quantities (transverse Young's modulus, shear modulus, and Pois-son's ratio) were reduced to nearly zero over one million cycles, with a linear dependence

on the number of applied cycles Figure 6 shows that the failure function, Fa> that is a

REIFSNIDER ON DAMAGE-TOLERANT COMPOSITE MATERIALS 13

measure of the stress state (compared to strength on the basis of stress in the fiber direction,

in this case) rises as the matrix "unloads" and increases the fiber stress, an expected result.The effect on damage tolerance is large and not easily anticipated without the simulationcalculation The strength reduction is distinctly more nonlinear than the earlier prediction,and the life of the laminate is reduced to about 570 thousand cycles, about a third of theprevious value Hence, we can see that a change of about 15% in the local stress state inthe fibers (due to matrix failure over the 570k cycles) is estimated to reduce the life toroughly 38% of the original value

Figure 7 shows the results for the baseline altered, this time, by only a change in strength

of the fibers, again, to a very small value over about one million cycles Again, the effect

is distinct and clearly represented The failure function increases by about 26% over the

14 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

life of the specimen, which is reduced from 1.5 million to about 210 thousand cycles, about14% of the baseline life Also, the basic nature of the damage tolerance is different Theremaining strength curve is flatter at the beginning and drops off more quickly at the end,

a tendency for "sudden death," in the parlance of the fatigue community Hence, if thing like oxidation reduces the strength of the fibers in a composite (which would certainlynot be linear, as in this example), the life is reduced quite severely, and there is a tendencyfor the degradation to be "hidden" by the nonlinearity of the remaining strength curve.Figure 8 shows the results of the baseline calculation altered only by viscoelastic creep

some-by introducing only an increase in temperature from ambient to 100°C The life is reduced

to 42% of its original value by a 9% change in the local stress state over the 630k cycles,

as illustrated by the change in the failure function On the face of it, this behavior should

be compared to the results of the damage-induced stiffness change shown in Fig 6 However,

REIFSNIDER ON DAMAGE-TOLERANT COMPOSITE MATERIALS 15

that stiffness change produced a 15% change in local stress (much more than that shown inFig 8) but caused a similar loss of life The reason is shown by a closer examination of Fig

8 The rise in temperature causes an initial (early) reduction in the stiffness, before thetime-dependent reduction begins, indicated by the early rise in the local stress (and failurefunction) This "early degradation" creates a greater long-term effect than the gradualuniform reduction assumed in the example of Fig 6 This also serves to highlight the factthat the simulation model is highly history-dependent, as one can see from Eqs 6 and 7 [15].Finally, it can be seen that the residual strength curve in Fig 8 is more nearly uniform inslope, i.e., there is less of a "sudden death" effect associated with viscoelastic creep thanwith, for example, damage induced strength reduction Finally, if one looks in the literaturefor comparison data from the laboratory, the present predictions appear to be reasonable

For a T300/SP313 graphite epoxy system, for example, having about the same reduction in

stiffness as a function of temperature as the modeled material (but a lower glass transitiontemperature), the reduction in long life for a temperature of about 100° centigrade wasfound to be about 40% for a unidirectional laminate, compared to the 60% predicted forthe [0,45,0, -45,0], laminate, for our case Unfortunately, we did not have enough creepdata on an existing system for which elevated temperature fatigue data were available tomake a direct comparison

Figure 9 shows the combined effect of elevated temperature (100°C) and damage-inducedmatrix stiffness reduction (the same amount as the example shown in Fig 6) If this were

a linear phenomenological model, one would expect simply the linear sum of the reductionsshown in Figs 6 and 8, i.e., a resulting life of only a few cycles Instead, a life of about420k cycles is predicted, a result that is lower than the damage case alone, but not by alarge amount This is a fairly realistic case, since the matrix would likely sustain considerabledamage in a long-life test, and the elevated temperature (which is about one third of the

"disintegration" temperature in the present model) would cause additional degradation.The local stress state change of 15% shown reduces the life by more than (roughly) thesame change caused by the damage alone, shown in Fig 6, a reflection of the "earlydegradation" effect mentioned in our previous discussion

The predictions in Figs 5 through 9 would require over 100 tests in the laboratory, at acost of more than $100,000 at this writing Moreover, it is often not possible to isolate theeffect of temperature alone from damage, or to separate the effect of stress state fromstrength degradation, etc., in the laboratory, as we have done here Although the predictionsmust be regarded as rough estimates, we can have reasonable confidence in the physicalbehavior they predict since most of the relationships used in the model are based on mech-anistic representations of the processes involved in driving the changes modeled Also, ithas been shown over the last ten years or so that very good agreement with remainingstrength and life data can be obtained when the simulation is done with fairly complete datafor the material strength, stiffness, and basic rate equations [5,6,14-16]

The point of our present discussion is that this type of mechanistic simulation can be usedfor the design of damage tolerance composite systems At the general level, the materialsdesigner can see that stiffness degradation caused by damage such as matrix cracking (Fig.6) has about the same effect on life as viscoelastic creep at a fairly high use temperature(Fig 8), but that the reduction in remaining strength is earlier for the high-temperaturecase, and more "sudden" for the case of damage Moreover, damage to the intrinsic strength

of the material (Fig 7) as one might obtain from damage to the fibers in a polymer-basedcomposite reduces the life and remaining strength much more greatly than the stiffnesschanges noted above, or even more than the combination of elevated temperature anddamage related (matrix) stiffness reduction-perhaps due to matrix cracking (Fig 9) It isalso clear from Fig 9 that matrix damage and viscoelastic creep do not simply "superpose"

in some linear fashion; their combination produces a life of 420k cycles which is not greatlyless than the life of the specimen without viscoelastic effects (570k cycles) or the life of theviscoelastic case alone (630k cycles) On a more specific level, it is possible with carefulcalibration of the model to be precise in this type of "trade study" in exactly the samefashion as trade studies are done in structural design One could be very specific in theperformance requirements, and could "test" candidate materials for acceptable performancebefore the money is spent for laboratory tests or material acquisition.

Continuing Directions

Our discussion has been based on ply-level properties and performance to this point The

"design" of composite materials occurs at several other levels, especially at the constituent

REIFSNIDER ON DAMAGE-TOLERANT COMPOSITE MATERIALS 17

level wherein the constituents, their geometries, and their arrangements are specified Thecurrent approach can be extended to this level by representing ply properties in terms ofmicromechanical representations of the processes that control those fundamental charac-teristics This effort is well under way in our laboratories One can construct micromechanicalmodels of the tensile strength of continuous fiber-reinforced composites in the direction ofthe fibers in terms of the strength of the fibers and the statistics of that strength, strength

of the matrix, interfacial strength, elastic properties of the constituents, and the local ometry Although these models tend to be nonlinear, and generally require iterative solu-tions, they can be used directly in the simulation models, to bring us to the point of specifyingthe constituents and their arrangement for specific long-term performance goals, and inparticular, to make damage tolerant materials Figure 10 shows an example of the preliminaryresults of such calculation The remaining tensile strength (damage tolerance) for an identicalcomposite material for which only the matrix yield strength has been altered (by one order

ge-of magnitude) is shown in that figure (The characteristic strength is referred to as the "sigmacoeff" in that figure.) It is no surprise that a change in the matrix strength alters the fatigueperformance, but the manner in which that alteration acts through the mechanics problemand statistics involved, and the resulting magnitude of the change in behavior, are not easilyestimated without a mechanistic model such as the present one The change is certainly not

a simple linear shift of some sort

Closure

Fatigue is a phenomenon, but it is also a process Mechanistic models provide a viableapproach to the representation of the consequences of the phenomenon (remaining strength/damage tolerance and life) in terms of the details of the process In this manner, one cancreate a tool for the evaluation of the relative effect of those details on the results of the

18 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

phenomenon, and use that tool to "design" composite material systems for specific-andeconomic-long-term performance The present paper provides a few rudimentary examples

of such models and identifies some of the opportunities and limitations of our approach.Future needs are more numerous than current successes and include the need to obtaincharacterizations at the micro-level, especially such things as interfacial (or interphase region)definition, strength, pointwise stiffness; constituent strength, stiffness, fundamental dura-bility, placement, statistical distribution; rate equations for the dominant process such aschemical degradation, creep, and thermodynamic behavior; and transfer coefficients forenergy at the micro-level However, the door of opportunity to design material systems thathave high engineering performance and competitive economic cost is open to us

Acknowledgments

The author gratefully acknowledges the support of the National Science Foundation ence and Technology Center for High Performance Polymeric Adhesives and Compositesunder Grant DMR 9120004 and the support of the Virginia Institute for Material Systems.References

Sci-(1) Composite Materials: Testing and Design, ASTM STP 715, K L Reifsnider, Ed., American Societyfor Testing and Materials, Philadelphia, 1982

[2] Composite Materials: Testing and Design, ASTM STP 787, I M Daniel, Ed., American Society

for Testing and Materials, Philadelphia, 1982

[3] Hahn, H T., "Fatigue Behavior and Life Prediction of Composite Laminates," Composite terials: Testing and Design, ASTM STP 674, American Society for Testing and Materials, Phila-

Ma-delphia, 1979

[4] Evans, A G., Williams, S., and Beaumont, P W R., "On the Toughness of Particulate-filled

Polymers," Ceramic Containing Systems, A G Evans, Ed., 1986, pp 286-305.

[5] Weitsman, Y., "Moisture in Composites: Sorption and Damage," Fatigue of Composite Materials,

K L Reifsnider, Ed., Elsevier Science Publications, 1990

[6] Fatigue of Composite Materials, K L Reifsnider, Ed., Elsevier Science Publications, 1990.

(7) Bakis, C E., "Fatigue Behavior of Notched Carbon Epoxy Laminates During Reversed CyclicLoads," Dissertation, Doctor of Philosophy in Engineering Mechanics, College of Engineering,VPI&SU, Blacksburg, VA, 1988

[8] Stinchcomb, W and Bakis, C E., "Fatigue Behavior of Composite Laminates," Fatigue of posite Materials, K L Reifsnider, Ed., Elsevier Science Publications, 1990, pp 105-180.

Com-[9] Reifsnider, K L., "Performance Simulation of Polymer Based Composite Systems," Proceedings

of the International Symposium on Durability of Polymer Based Composite Systems, for Structural Applications, Elsevier Applied Science, 1991, pp 3-26.

[10] Schapery, R A., "Mechanical Characterization and Analysis of Inelastic Composite Laminates

with Growing Damage," Mechanics of Composite Materials and Structures, ASME AMD Vol 100,

J N Reddy and J L Teply, Eds., 1991, pp 1-9

[11] Dillard, D., "Viscoelastic Behavior of Laminated Composite Materials," Fatigue of Composite Materials, K L Reifsnider, Ed., Elsevier Science Publications, 1990, pp 339-384.

[12] Yeow, Y T., Morris, D H., and Brinson, H F., Composite Materials: Testing and Design (Fifth Conference), ASTM STP 674, 1979.

[13] Weitsman, Y., "Moisture in Composites: Sorption and Damage," Fatigue of Composite Materials,

K L Reifsnider, Ed., Elsevier Science Publications, 1990, pp 385-430

[14] Reifsnider, K L., "Performance Simulation of Polymer Based Composite Systems," Proceedings

of the International Symposium on Durability of Polymer Based Composite Systems for Structural Applications, Brussels, Belgium, 27-31 August 1990, A H Cardon and G Verchery, Eds., Elsevier

Applied Science, New York, 1991, pp 3-26

[15] "Development of Engineering Data on Advanced Composite Materials," AFML-TR-77-151, AirForce Wright Aeronautical Laboratories, OH, 1977

[16] Reifsnider, K and Gao, Z., "Micromechanical Concepts for the Estimation of Property Evolution

and Remaining Life," Proceedings of the International Conference on Spacecraft Structures and

Mechanical Testing, Noordwijk, The Netherlands, 24-26 April 1991, ESA SP-321, October 1991,

pp 653-658

R Sunder1

Contribution of Individual Spectrum Load

Cycles to Damage in Notch Root Crack

Initiation, Short and Long Crack Growth

REFERENCE: Sunder, R., "Contribution of Individual Spectrum Load Cycles to Damage

in Notch Root Crack Initiation, Short and Long Crack Growth," Advances in Fatigue Lifetime

Predictive Techniques: Second Volume, ASTM STP 1211, M R Mitchell and R W Landgraf,

Eds., American Society for Testing and Materials, Philadelphia, 1993, pp 19-29

ABSTRACT: A unified cumulative damage picture of spectrum load notch fatigue is presented,

covering all its three stages: crack initiation, short crack growth, and long crack growth Theanalysis uses rainflow cycle count, low-cycle fatigue, and crack growth constants as inputs.While cumulative damage in crack initiation is handled using conventional methods, recentobservations of a substantial difference between notch root crack closure and crack openingstress serve as the inputs for modeling sequence sensitive damage accumulation in short andlong crack growth Range/damage-exceedance (RDE) diagrams showing contribution of in-dividual load cycles to spectrum load cumulative damage are presented for an AI-Cu alloy(2014-T651l) subject to FALSTAFF and TWIST load spectra These diagrams include infor-mation on load sequence sensitivity of damage in crack initiation and growth RDE diagramscan be used to assess effects of load sequence reconstitution from rainflow cycle count and ofload spectrum editing on cumulative damage

KEYWORDS: cumulative fatigue damage, range/damage-exceedance (RDE) diagram, ceedance curve, crack initiation, short and long crack growth, crack opening and closure,upper and lower bound damage

ex-Complex service load sequences covering thousands of hours of service and hundreds ofthousands of load excursions may appear at the outset to be incomprehensible from theviewpoint of damage potential of individual loads A turning point in transforming suchsequences into a more meaningful format was the discovery of the rainflow technique [1},that now appears as part of ASTM E 1049, Practices for Cycle Counting in Fatigue Analysis.The rainflow cycle count is to fatigue analysis what power spectral density function is tovibration analysis Its appearance made it possible to perform cumulative damage analysisavoiding cycle-by-cycle calculations of damage in crack initiation through growth to failure.Cumulative damage analyses are essential to determine damage potential of individualload cycles They can be used in editing load sequences to accelerate test duration [2] andalso to assess the consequences of resequencing loads Material stress-strain hysteresis canmake notch crack initiation life sensitive to load sequence By positioning all smaller loadcycles either on the falling or rising half of the major load cycle, one can estimate the upperbound or lower bound respectively of crack initiation life This is of particular significance

in fatigue of rotary components, where large numbers of smaller (vibration) loads aresuperposed on a major cycle [3] Reference 3 describes a procedure for determining upperand lower bounds of crack initiation life directly from the rainflow count It uses low-cycle

1National Aeronautical Laboratory, Bangalore 560 017, India

19

20 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

fatigue (LCF) constants and is based on the local stress-strain (LSS) approach Reference

4 describes cumulative damage analysis for long fatigue cracks The salient feature of thestudy was the representation of loading as well as cumulative damage data on a single range/damage-exceedance (RDE) diagram Such a diagram can be used to assess contribution ofindividual cycles in the spectrum to fatigue damage As a continuation of this work, Ref 5included cumulative damage in crack initiation into the RDE diagram by incorporating theaccelerated computational procedure from Ref 3 Cumulative damage in crack initiation hasalso been assessed in other works [2,3,6]

One would expect notch root small cracks to also exhibit trends of upper and lower boundgrowth rates However, the model used in Ref 4 assumes crack closure stress to be constantunder spectrum loading, thereby rendering growth rates insensitive to load resequencing

A recent study demonstrated, that unlike long cracks subject to elastic loading (considered

in Ref 4), notch root short cracks can experience substantial cycle-by-cycle variations incrack opening stress under spectrum loading [7] A Hysteretic Closure Zero Threshold(HCZT) model based on these observations indicates trends in damage sensitivity to loadsequencing that are similar to what one would expect from LCF and LSS considerations.The HCZT model permits damage analysis directly from the rainflow cycle count It relies

on LCF and fracture mechanics (FM) constants along with crack closure and opening stressvalues for the major cycle (largest cycle in the spectrum) to determine crack growth rate as

a function of applied reference stress and crack geometry The HCZT model can handlethe growth of both short as well as long cracks

This study was motivated by the possibility of unifying cumulative fatigue damage analysescovering all three stages of fatigue: crack initiation, short crack growth, and long crackgrowth The next section briefly describes the procedure for cumulative damage analysis.Results of analyses appear as RDE diagrams for L168 (2014-T6511) AI-Cu extruded barstock under FALSTAFF [8] and TWIST [9] load spectra

Assessment of Cumulative Fatigue under Spectrum Loading

Figure I provides a schematic summary of spectrum load interaction effects simulated in

this study Consider the applied stress sequence in Fig la It consists of a number of identical

small load cycles that are either preceded or followed by an overload or underload

The first case was Seer = 275 MPa and K, = 3.7 at crack depth of 0.1 mm, which involvesnotch root cyclic inelasticity The second case was S,ef = 225 MPa, K, = 3.7 and crack depth0.5 mm This case involves notch root monotonic yield The third case was that of a longcrack under elastic loading (100 MPa) Similar reference stresses were used for the TWISTload spectrum (However, tests have not been conducted under TWIST.)

The presence of and extent of inelasticity affected crack damage analysis in two ways.Crack opening and closure stress depend on the extent of inelasticity The three cases ofloading pertain to the three sets of opening and closure stress listed in Table 1 Also, theextent of cyclic inelasticity in individual rainflow counted cycles determines effective stressintensity range as indicated by Eq 3

Results of Analysis and Discussion

The results of analysis appear as RDE diagrams Figs 2 through 4 The RDE diagramconsists of a family of curves that represent both the spectrum itself as well as its damagecontent The rainflow cycle count is arranged in receding order of ranges All results arerepresented in nondimensionalized format as percentage of total or maximum Reference

4 provides a detailed description of the RDE diagram and of the procedure to obtain thecurves that form part of it The damage curves for all three stages of fatigue were computed

in this study using similar procedures In contrast to the procedure in Ref 4, the damageexceedence curve in this study was computed for upper bound damage To avoid clutteringthe diagram, lower bound damage is represented by its ratio to the upper bound

Figures 2a and 2b show RDE diagrams for FALSTAFF (a) and TWIST (b) load spectra

in crack initiation at three stress levels Curve 1 is the range exceedance curve It showspercentage ratio of applied stress range to maximum stress range versus percentage cu-mulative frequency of occurrence of ranges equal to and above that range This is a materialindependent curve that is constant for a given rainflow cycle count (load spectrum) We find,for example, that for the FALSTAFF load spectrum, cycles with range greater than 30%

of highest range account for about 10% of all the cycles In case of the TWIST spectrum

(Fig 2b), they account for under 2% of all load cycles It may be noted that in case of theTWIST spectrum, the highest range seen on the RDE diagram is just under 80% Largerpercentage ranges do not appear because their frequency of occurrence is less than 0.01%,and hence falls beyond the range covered by the diagram.

With the exception of Curve 1, all other curves are material and loading condition pendent Curves 2, 3, and 4 are the upper bound cumulative damage curves for the threeincreasing applied stress levels listed above We find from these curves, for example, that

de-in case of the FALSTAFF load spectrum, de-in all three cases the larger load cycle amountde-ing

to under 2% of all the cycles in the spectrum accounts for all the damage in crack initiation.

Omitting the remaining 98% would not have made a difference to crack initiation life We

also find that with increase in applied stress level, the cumulative damage curve shifts to

the right, indicating that smaller cycles account for a greater fraction of damage A tatively similar picture is observed for the TWIST spectrum

quali-Curves 5, 6, and 7 represent the percentage ratio of cumulative lower bound damage to

cumulative upper bound damage for the three increasing stress levels With increase in

applied stress level, the notch root stress-strain hysteresis loop will widen, thereby increasing

the difference between upper and lower bound local mean stress in the smaller cycles As

shown by Eq 4, this will reduce the ratio of lower bound to upper bound damage This isreflected in Curves 5, 6, and 7 occupying progressively lower positions Finally, one alsoobserves that all three curves progressively fall with exceedance This is attributed to the

increasing difference between upper and lower bound mean stress with decrease in stress

range of the counted cycle with respect to the largest (major) cycle As cycle range approachesthat of the major cycle, difference between upper and lower bound mean stress will tend

to unity For this reason, all three curves originate at the top left point of the RDE diagram.They flatten out at the right end as upper bound cumulative damage saturates

Figures 3a, 3b, and 3c show RDE diagrams in different stages of crack growth under the

FALSTAFF load spectrum Curve 1 represents range exceedance Curves 2 and 3 representpercentage cumulative effective stress range related to maximum applied range Curve 2 isobtained assuming upper bound damage (lower crack opening stress), while Curve 3 applies

to lower bound damage Curve 4 is upper bound percentage cumulative damage, whileCurve 5 is the ratio of lower to upper bound damage

As indicated by Eq 3, effective range is sensitive to crack opening stress as well as k, In

the absence of cyclic inelasticity, left extreme points on Curves 2 and 3 are at the same level

(60%) in Figs 3b and 3c Larger notch strain ranges due to cyclic inelasticity push this point

to as high as 85% in the case shown by Fig 3a If k, were to be even higher, percentage

effective range could in principle exceed 100% of maximum applied stress range.

The shaded area between Curves 2 and 3 reflects on the difference between upper and

lower bound crack opening stress We see that under cyclic inelasticity, Sop drops from about

26 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

0.24 to -0.04 (Table 1) As a consequence, upper bound effective range (Curve 2) coincides

with applied ranges (Curve 1) in Fig 3a This is indicative of a fully open crack during 80%

of the (smallest) load cycles Interestingly, Curve 3 shows that about 10% of the smallestload cycles see a fully closed fatigue crack (zero effective range) We have a situation wherebydepending on whether these smaller cycles are on the rising half or falling of the majorcycle, the crack will be either fully closed or fully open This point is only of academicinterest, considering that under FALSTAFF, the bulk of damage (over 85%) is caused bythe larger load cycles (Curve 4)

Irrespective of applied stress level, Curve 5 indicates lower to upper bound damage ratio

in crack growth between about 65% to 75% under FALSTAFF load spectrum In crackinitiation, the ratio was even higher, particularly at lower stress levels It follows that onewill not see substantial load sequence sensitivity under this load spectrum A similar ob-servation can be made from mean test results on notch root crack initiation and growth in2024-T3 coupons, subject to programmed load spectra, reconstituted from the FALSTAFFcycle count to follow distinctly different (Hi-Lo, Hi-Lo, etc.) sequences [14]

The FALSTAFF load spectrum is relatively insensitive to load sequence because the bulk

of damage is caused by the largest load cycles It was noted earlier that the ratio of upper

to lower bound damage will tend to unity as relative stress range of a given load cycleapproaches that of the largest load cycle in the spectrum This applies to both crack initiation

as well as crack growth

Figures 4a, 4b, and 4c show RDE diagrams for crack growth under the TWIST load

spectrum Loading cases studied and numbering of curves are the same as in Fig 2 forFALSTAFF A number of observations can be made about cumulative damage in crackgrowth under the TWIST spectrum As pointed out in earlier work [4], irrespective ofloading conditions, the crack growth cumulative damage curve (4) does not saturate as inthe case of FALSTAFF The bulk of damage is caused by the large number of load cycles

As evident from Curves 2 and 3, irrespective of loading conditions, the fatigue crack is atleast partly open under even the smallest load cycles This is attributed to the high appliedmean stress which distinguishes transport aircraft load spectra from fighter aircraft spectra

In case of the notch root crack under high applied stress (Fig 4a), we find that the upper

bound effective range curve coincides with applied ranges at stress range under 25%, dicative of a fully open fatigue crack Under such conditions, linear cumulative damageestimates could be un conservative by comparison Curve 5 (ratio of lower to upper bounddamage) shows noticeable variations under the TWIST spectrum This is possible because,

in-in contrast to FALSTAFF, the smaller cycles contribute the bulk of damage Curve 5 touches

a low of under 15% in Fig 4b (notch root crack under monotonic yield) This implies that

by suitably manipulating the TWIST load sequence, one can see differences in growth rate

by a factor of six Ratio of lower to upper bound damage is 60% in the case shown in Fig

4a and just over 45% in Fig 4c (long crack, elastic) The substantially lower ratio in the

case of Fig 4b (notch root, monotonic yield) can be traced to the denominator in Eq 5.The difference between lower and upper bound growth rates is accentuated not only by anincrease between upper and lower bound crack opening stress, but also by increasing lowerbound Sop" As shown by the constants in Table 1, notch root monotonic yield conditionsaccentuate this effect most

These observations suggest that TWIST spectrum will show dramatic effects of spectrumreconstitution from the rainflow cycle count Other transport load spectra where the effect

of Hi-Lo versus Lo-Hi sequencing was studied [15] have shown substantial and systematicvariations in crack growth rates

Figures 5a and 5b are an attempt to consolidate RDE diagrams covering all three stages

of fatigue into a single format for the FALSTAFF (a) and TWIST (b) load spectra Such a

consolidated diagram can serve as a data sheet in determining consequences of spectrum reconstitution and editing for testing purposes in situations where all three stages of fatigue

are of concern to the test designer Curves 2b and 2c are upper bound effective ranges undernotch root cyclic inelasticity and for a long crack under elastic conditions Curves 3a, 3b,and 3c are upper bound cumulative damage in crack initiation, short and long crack growthrespectively Curves 4a, 4b, and 4c are the lower to upper bound ratios for the three stages

of fatigue

From Fig 5a, it would follow that irrespective of whether it is crack initiation, short, orlong crack growth, 20% of the most severe cycles under FALSTAFF contribute over 90%

of fatigue damage Further, irrespective of how the FALSTAFF spectrum is sequenced, one

is unlikely to see more than a 30% variation in fatigue damage rate (Curves 4a, 4b, and4c) These observations can assist the test designer in justifying accelerated testing underFALSTAFF load spectrum of coupons as well as joints and subassemblies Built -up structuresfrom different materials can also be considered, provided RDE diagrams are available forindividual material It must be noted, however, that even though we have considered allstages of fatigue, the crack initiation analysis only simulates crack incubation at a smoothand free notch surface The question of crack initiation due to mechanisms other than localreversed slip still remains to be addressed As a consequence, the RDE diagrams as presented

in this study may not be valid for cases like fretting, which in fact may show greater sensitivity

to smaller load excursions

As indicated by Fig 5b, the TWIST spectrum does not offer simple test design optionslike FALSTAFF While crack initiation life is relatively insensitive to the bulk of smaller

load cycles, the reverse is true in crack growth As pointed out in earlier work [4], omitting

smaller cycles will result in systematic reduction in damage rate More significantly, crackgrowth under the TWIST spectrum will be extremely sequence sensitive Both load omissioneffects and sequence effects can be accounted for by suitably correcting test results obtained

28 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES

under controlled conditions However, if damage analysis is being carried out based onrainflow cycle count without accompanying data on load sequence, large errors in crackgrowth rate estimates can occur

This discussion implies that in Individual Aircraft Tracking (IA T) schemes for combataircraft, it is adequate if flight data recorders record the rainflow cycle count of serviceloads (resequencing effects will be negligible) This may not be true for transport aircraft

It should be noted that this study considered crack initiation using LCF /LSS concepts.However, many structural joints are likely to see crack initiation due to fretting It has beenshown that fretting sensitivity to load resequencing can be the opposite of what one observes

in fatigue [14] Overlooking this detail may result in unexpected test results in fatigue testing

of joints and subassemblies

Conclusions

(1) A unified cumulative damage picture of spectrum load notch fatigue is presented,covering all its three stages: crack initiation, short crack growth, and long crack growth.(2) Cumulative fatigue analysis was carried out using rainflow cycle count, low-cyclefatigue, and crack growth constants as inputs

(3) Observations of a substantial difference between notch root crack closure and crackopening stress serve as the inputs for modeling sequence sensitive damage accumu-lation in short and long crack growth

(4) Range/damage-exceedance diagrams showing contribution of individual load cycles

to spectrum load cumulative damage are presented for an AI-Cu alloy subject toFALST AFF and TWIST load spectra

(5) Load sequence sensitivity in fatigue under spectrum loading is substantial when thebulk of damage is caused by the smaller load cycles

References

[1] Endo, T., Mitsunaga, K., and Nakagawa, H., "Fatigue of Metals Subjected to Varying

Stress-Prediction of Fatigue Lives," Preliminary Proceedings of the Chugoku-Shikoku District Meeting,

The Japan Society of Mechanical Engineers, November 1967, pp 41-44

[2] Socie, D F and Artwohl, P J., "Effect of Spectrum Editing on Fatigue Crack Initiation andPropagation in a Notched Member," in Effect of Load Spectrum Variables on Fatigue Crack

Initiation and Propagation, ASTM STP 714, D F Bryan and J M Potter, Eds., American Society

for Testing and Materials, Philadelphia, 1980, pp 3-23

[3] Dowling, N E and Khosrovaneh, A K., "Simplified Analysis of Helicopter Fatigue Loading

Spectra," Development of Fatigue Loading Spectra, ASTM STP 1006, J Potter and R T Watanabe,

Eds., American Society for Testing and Materials, Philadelphia, 1989, pp 150-171

[4] Sunder, R., "Contribution of Individual Load Cycles to Crack Growth Under Aircraft Spectrum

Loading," Advances in Fatigue Lifetime Predictive Techniques, ASTM STP 1122, M R Mitchell

and R W Landgraf, Eds., American Society for Testing and Materials, Philadelphia, 1992, pp.176-190

[5] Sunder, R., "Rainflow Applications in Accelerated Cumulative Fatigue Analysis," The Rainflow

Method in Fatigue, Y Murakami, Ed., Butterworth Heineman, 1992, pp 67-76.

[6] Dowling, N E., Thangjitam, S., Lese, C., and Fash, J w., "Some Comments on Methods ofReducing and Reconstructing Irregular Fatigue Loading Histories," The Rainflow Method in Fa- tigue, Y Murakami, Ed., Butterworth Heineman, 1992, pp 51-60

[7] Sunder, R., Prakash, R v.,and Mitchenko, E I., "Prediction of Notch Root Crack Growth Rate

Under Elastic and Inelastic Loading," Project Document PDST 9218, National Aeronautical

Lab-oratory, Bangalore, India, April 1992 (also appears in this publication, ASTM STP 1211, as

"Calculation of Spectrum Load Notch Root Crack Growth Rate Under Elastic and InelasticConditions," pp 30-44)

[8] Van Dijk, G M and De Jonge, J B., in Proceedings, 8th 1CAF Symposium, InternationalCommittee on Aeronautical Fatigue, Lausanne, Switzerland, 1975, pp 3.61/1-3.61/39.9

SUNDER ON DAMAGE IN NOTCH ROOT CRACK INITIATION 29

[9] Shuetz, D., in Proceedings, Seventh ICAF Symposium, International Conference on Aeronautical

[12] Newman, J c., "A Crack Closure Model for Predicting Fatigue Crack Growth Under Aircraft

Spectrum Loading," in Methods and Models for Predicting Fatigue Crack Growth Under Random

Loading, ASTM STP 748, J B Chang and C M Hudson, Eds., American Society for Testing

and Materials, Philadelphia, 1981, pp 53-84

[13] De Koning, A U., "A Simple Crack Closure Model for Prediction of Fatigue Crack Growth RatesUnder Variable Amplitude Loading," Fracture Mechanics (J3th Conference), ASTM STP 743,

American Society for Testing and Materials, Philadelphia, 1981, pp 63-85

[14] Perret, B H E., "An Evaluation of a Method of Reconstituting Fatigue Loading from Rainflow

Counting," Proceedings, 14th Symposium of the International Committee on Aeronautical Fatigue,

EMAS, 1987, pp 355-402

[15] Schijve, J., Jacobs, F A., and Tromp, P J., "The Effect of Load Sequence on Fatigue CrackPropagation Under Random Loading and Program Loading," NLR TR 710140, National Aero-space Laboratory, Amsterdam, 1971

R Sunder, 1R V Prakash,2 and E I Mitchenko3

Calculation of Spectrum Load Notch Root Crack Growth Rate Under Elastic and

Inelastic Conditions

REFERENCE: Sunder, R., Prakash, R.v., and Mitchenko, E I., "Calculation of SpectrumLoad Notch Root Crack Growth Rate Under Elastic and Inelastic Conditions," Advances in Fatigue Lifetime Predictive Techniques: Second Volume, ASTM STP 1211, M R Mitchell and

R W Landgraf, Eds., American Society for Testing and Materials, Philadelphia, 1993, pp.30-44

ABSTRACT: A hysteretic model of crack opening stress variation forms the basis for spectrumload notch fatigue crack growth analysis The model is based on the observation of a significantdifference between crack opening and crack closure stress, particularly under high stress-straincycling The two stress levels are controlled by the largest cycle in the load sequence Theanalysis uses rainflow cycle count, constant amplitude long crack growth rate constants andcyclic stress-strain curve as inputs apart from closure data Growth rate calculations for 6 mmthick L168 (2014-T651) AI-Cu aHoy coupons under FALSTAFF load spectrum compare weHwith experimental data obtained on a long crack and at notch root crack depth ranging fromunder 50 microns to 2 mm

KEYWORDS: notch root fatigue, crack growth, crack closure and crack opening, spectrumloading, rain flow cycle count

Nomenclature

C, m material constants in crack growth rate expression

E Young's modulus

Fa angular offset correction factor for part-through crack

Fw finite width correction factor for K

K stress intensity factor

K K due to unit applied stress

K, theoretical stress concentration factor

Q crack shape correction factor for K

R stress ratio

S applied stress

W specimen width (single edge notch) or half width (central notch)

a crack length (depth in case of part-through crack)

da/ dH crack growth rate under spectrum loading mm/h

da/dN crack growth rate, mm/cycle

k, strain inelasticity correction factor

r notch radius

] National Aeronautical Laboratory, Bangalore 560 017, India

2Indian Institute of Science, Bangalore 560 012, India

3Institute for Strength Problems, Kiev 252 014, Ukraine

30

Notch root fatigue determines endurance of built-up structures The technology to detectsmall defects at notches and joints, combined with the ability to predict their growth, canplaya crucial role in assuring better design quality and more reliable operation of airframes.Small crack growth behavior can differ noticeably from that of long cracks The difference

is traced largely to two factors: fatigue crack closure and crack driving force

This study is an extension of previous work aimed at formulating simple engineeringprocedures for fatigue analysis under aircraft flight spectrum loading [1,2] It uses the samegoverning equation for fatigue crack growth rate

It is assumed that this equation applies to both long as well as short cracks and that anomolous

behavior of short fatigue cracks can be described by considering sensitivity of arguments inthe equation to notch root conditions

The next section describes a crack closure model valid for notch root fatigue under cyclicinelasticity Analytical expressions from the literature are adapted for stress intensity esti-mates Crack growth rate analysis is described in detail, and material constants required forcalculations are listed Estimates of notch root crack growth rate under FALSTAFF loadspectrum [3] are compared with experimental results

Modeling Notch Root Fatigue Crack Growth

The schematic in Fig 1 describes the proposed framework of fatigue crack growth rateanalysis Apart from crack closure data whose detailed consideration is forthcoming, longcrack constant amplitude growth rate and cyclic stress-strain curve constants form materialdata inputs for analysis The analysis avoids cycle-by-cycle crack growth estimates by op-erating directly on the rainflow cycle count

Loading data are represented by the rainflow cycle count obtained for the given loadspectrum using ASTM E 1049, Practices for Cycle Counting in Fatigue Analysis Transients

in crack driving force and crack closure associated with crack extension are neglected It isassumed that growth rate is sufficiently small to ensure stability of the crack tip stress-strainand crack wake deformation scenario, as influenced by the largest (major) cycle in thespectrum This assumption is supported by experimental results that appear later in thepaper and also by fractographic evidence of absence of transient phenomena [4].

Apart from ignoring transient effects, the procedure is similar in two other respects tothe Constant Closure Zero Threshold (CCZT) model [1] It assumes that a fatigue crackwill grow provided it is open (presence of a threshold stress intensity is ignored) Also, loadinteraction phenomena other than the closure phenomenon are neglected

The CCZT model's applicability was limited to long crack Linear Elastic Fracture chanics (LEFM) conditions We consider major differences between long cracks and smallcracks growing out of a notch root These are associated with notch root inelasticity andassociated stress-strain excursions that distort LEFM and closure response These two aspectsare described in detail in the following They form the framework for a new HystereticClosure Zero Threshold (HCZT) model, which, unlike the CCZT model, can handle thebehavior of both short as well as long cracks

Me-Notch Root Fatigue Crack Closure

A recent study on L165 (2014-T6) AI-Cu alloy sheet notched coupons showed [5], thatunder programmed load sequences involving notch root cyclic inelasticity, crack openingstress in individual cycles can be noticeably different depending on their position with respect

to the rising and falling half of the major cycle Most analytical and numerical models ofclosure [6-9]relate it to crack wake development and thereby, to crack extension in fatigue.Notch root closure behavior on the contrary appears to be highly sensitive to cycle-by-cyclestress-strain excursions Figure 2 provides a schematic description of crack closure under

cyclic loading Figure 2a is the response of a long crack (LEFM conditions) The thick line

represents effective stress intensity variation as a consequence of the closure phenomenon.With the onset of closure on the falling half cycle, stress intensity effectively stops dropping

further The variation in applied stress intensity from Kd to K min evokes a negligible change

inKerrindicated by the drop in effective stress intensity to Kop- The CCZT model effectively