S'IP 1211 Advances in Fatigue Lifetime Predictive Techniques: Second Vo/ume Library of Congress 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 Foreword This publication, Advances in Fatigue Lifetime Predictive Techniques: Second Volume, contains papers presented at the Second Symposium on Advances in Fatigue Lifetime Predictive Techniques, held in Pittsburgh, PA on 4-5 May 1992 The symposium was sponsored 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 and are co-editors of the resulting publication Contents Overview Use of Mechanistic Life Prediction Methods for the Design of Damage-Tolerant Composite Material Systems-K L REIFSNIDER Contribution of Individual Spectrum Load Cycles to Damage in Notch Root Crack Initiation, Short and Long Crack Growth-R SUNDER 19 Calculation of Spectrum Load Notch Root Crack Growth Rate Under Elastic and Inelastic Conditions-R SUNDER,R V PRAKASH,ANDE I MITCHENKO 30 Fatigue Damage Due to Sub-Threshold Load Cycles Between Periodic Overloadsz ZHOUANDF J ZWERNEMAN 45 Exact Determination of l1K.rr and Crack Propagation Prediction for Selected Loading Sequences-so ZHANG,H DOKER,H NOWACK,K SCHULTE,AND K.-H TRAUTMANN 54 FEM Analysis of Cyclic Deformation Around the Fatigue Crack Tip After a Single Overload-v OLIVAANDI KUNES 72 A Creep Cavity Growth Model for Creep-Fatigue Life Prediction of a Unidirectional W/Cu Composite-y.-s KIM,M J VERRILLI,AND G R HALFORD 91 Thermal-Mechanical Fatigue Lifetime Prediction of an Austenitic Stainless SteelH J SHI, C ROBIN,ANDG PLUVINAGE 105 The Cumulative Fatigue Damage Behavior of MAR-M 247 in Air and HighPressure Hydrogen-M A MCGAW,S KALLURI,D MOORE,AND HEINE 117 Crack Density and Fatigue Lifetime of Metals Under Variable Amplitude Loading-H BOMASANDP MAYR Discussion 132 140 Determination of Fatigue Limit Between 10S and 109 Cycles Using an Ultrasonic Fatigue Device-c BATHIASANDJ NI 141 Novel Methodology for Fatigue Lifetime Prediction-A 153 PUSKAR Estimation of Fatigue Propagation Life in Resistance Spot Welds-s D SHEPPARD 169 A Reverse Plasticity Criterion for Specifying Optimal Proof Load LevelsS M TIPTONANDJ R SOREM,JR 186 Fatigue Lifetime Prediction of Angle-Plied Fiber-Reinforced Elastomer Composites as Pneumatic Tire Materials-B L LEE, J P MEDZORIAN,P K HIPPO, D S LIU, ANDP C ULRICH 203 Problems with Current Methodology in Using the Arrhenius Equation to Predict the Long-Term Behavior of Polymeric Materials in Geotechnical Environments-D G BRIGHT 236 Author Index 249 Subject Index 251 Overview Based on the success of the first symposium on this topic in 1990 (published as ASTM 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 methods for 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 establishment of common ground between the various disciplines and interest groups A cursory scan of the contents clearly reveals the breadth of coverage The 15 papers included in the volume cover: • fundamental issues in damage development and crack growth, • behavior in both low- and high-cycle regimes, under constant amplitude and spectrum loading conditions, • the performance of advanced materials in hostile environments, including creep-fatigue and thermomechanical fatigue, and • predictive techniques for the real-world environment It is clear that fatigue problems show no sign of disappearing from our increasingly complex and technologically driven society Thus the need to have in hand more powerful and effective techniques for assuring the mechanical integrity of our structures, machines, and devices would seem more urgent than ever The continuing high level of activity in fatigue related technologies 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 our experimental and computational capabilities and, more frequently, in their productive interplay Indeed, well-conceived and executed experiments establish the behavioral database for intelligent model development and also provide the requisite validation tool for finetuning newly developed analytical tools General approaches to damage accumulation and life prediction are the subject of the first three papers The complexities of composite systems are highlighted in the first paper and a critical element model is presented that provides predictions of effective strength that account for the operative failure mode in a given cyclic environment Of note is the establishment of guidelines to help tailor a composite for a desired performance objective The next 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, striation spacing) are helping to provide new insights into crack closure behavior as it affects retardation or acceleration, or both Particularly encouraging is the development of mechanics models detailing the crack tip deformation responses as an effective means for predicting fatigue performance Material performance in nonambient environments is the subject of three papers Creepfatigue responses in a metal matrix composite is interpreted using a microstructural model 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 austenitic stainless steel under both in-phase and out-of-phase loading modes Finally, a damage curve approach is applied to the cumulative damage analysis of a nickel-base alloy under twolevel and multiblock loading sequences A productive combination of careful experimentation and innovative model development is apparent in these efforts Valuable insights into sequence effects on damage accumulation is provided by a paper detailing studies of crack density development in three alloys under a variety of loading profiles 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 next studied using a unique dynamic strain gage and SEM observations as a means to evaluate the potential of such test techniques for accelerated testing programs Detailed documentation 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 of advanced predictive techniques in engineering practice A design approach for the difficult problem of spot weld fatigue is developed using finite element methods in conjunction with linear elastic fracture mechanics Encouraging results for a variety of practical weld configurations are presented Using a local plasticity analysis for notched components, the next paper 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 real time monitoring techniques, (dynamic creep, temperature rise, and acoustic emission) as indicators 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 earth structures and waste containment is critically examined in the last paper The many problems associated 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 develop more realistic damage models for use in design applications The growing success record of modern 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 combine efforts to deal more effectively with fatigue on an applied level A conscious attempt has been made to perpetuate such alliances through forums of this type Perceived benefits include identification, by consensus, of key problem areas for research planning and, through collaborative efforts, the formulation of improved strategies for technology transfer Finally, the co-editors are pleased to report that the ASTM Committee E-9 Award for Best Symposium Paper for 1992 was presented to Sheri Sheppard and Michael Strange for their paper "Fatigue Lifetime Estimation in Resistance Spot Welds: Propagation Phase." We congratulate the authors for their fine contribution Michael R Mitchell Rockwell International Science Center, Thousand Oaks, CA 91360; symposium co-chairman and co-editor Ronald W Landgraf Virginia Polytechnic Institute & State University, Blacksburg, VA 24061; symposium co-chairman and co-editor 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 these systems 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 of composite materials, especially fiber-reinforced systems The body of literature is especially replete in the technical area of "effective stiffness" models, many of which are sophisticated and 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 element concept," whereby careful laboratory work is used to define representative volumes of material that enclose a "typical" failure mode This representative volume is divided into a "critical element" that controls the final failure event, and "subcritical elements" that alter the local stress state around the critical element The present paper extends this concept to the fiber/matrix level by introducing micromechanical strength models to be used in the critical elements The result of this advance is that mechanistic models that include explicit representations of the parameters that describe the manner in which the material systems are made can be used to estimate remaining strength when those parameters change during the lifetime of the material Moreover, the model can then be used to "design" or tailor a material system for specific long-term performance This last topic is the focus of the present paper The approach will be demonstrated, and the influence of several parameters will be discussed This discussion will then be used to advance several concepts for the rigorous design of material systems for damage tolerance KEYWORDS: composites, durability, life prediction, damage tolerance, material systems Damage Tolerance "Damage" accumulation and growth generally observed during long-term loading are associated with a loss of internal integrity and a degradation of properties and performance "Damage Tolerance," then, is generically defined by the capability of the material to resist degradation processes associated with "damage development." However, in the context of current literature, and in compliance with regulations that control the certification of structures for use by the general public or for military applications, damage tolerance has a much I Alexander Giacco Professor of Engineering Science and Mechanics, Virginia Tech Blacksburg VA 24061-0219 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 in the lifetime of the item in question In these terms, then, damage tolerance is defined as the "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 composite material systems (our present objective), we must devise a mechanistic approach to the determination of the state of stress and state of material at any given instance during the loading history of a component, and construct a philosophy for the interpretation of the relationship between those states of stress and states of material in terms of remaining (or residual) strength Damage Development, and Property and Performance Evolution We are concerned with damage-induced changes in the properties and performance of continuous-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 illustrates the manner in which such damage develops during long-term performance Initial properties may be altered by damage development of a geometric 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, phase changes, or morphological variations as a function of time We take the position that these changes 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 changes may 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 combine in myriad ways to create phenomena that may be cycle-dependent, time-dependent, rate-dependent, or dependent on combinations of those extensive variables Figure enumerates a few of those phenomena that are commonly discussed and described Of particular importance is the fact that many of these phenomena depend directly or indirectly on time Phenomena such as viscoelastic creep, creep rupture, aging, and environmental degradation are usually explicitly timedependent 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 as viscoelastic creep or diffusion-related environmental degradation may be significantly altered by the presence of cyclic mechanical loading, especially if attended by micro mechanical cracking This brings us to a conclusion of paramount importance, namely, that mechanistic modeling of damage development for the purpose of estimating damage tolerance must be concerned not only with the determination of stress states and material states, but also with the correct modeling of the rate of the processes that create changes in those states Since damage tolerance is defined by remaining strength after some history of mechanical, thermal, or chemical loading, time enters the problem directly as a parameter Since time also defines the 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 evolution can be written, ultimately, as constitutive equations which define changes in stiffness, strength, or local geometry in terms of material parameters However, these rate equations for the damage processes of concern to us are quite difficult to determine, and sometimes very 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 BRIGHT ON PROBLEMS USING THE ARRHENIUS EQUATION 237 the application in the beginning and ensure adequate performance over its anticipated lifetime The common modes that change properties of polymeric materials within a geotechnical environment are installation damage, mechanical deformation, weathering, chemical degradation, and biological deterioration Prediction of long-term performance is usually accomplished by the extrapolation of relatively short-term data collected on material response to a particular mode Typically, extrapolation is limited to one order of magnitude, e.g., one year of data to predict ten-year behavior [1] Time frames exceeding ten years necessitate the use of time-temperature-superposition principles or a representing rate expression to extrapolate short-term data Today, there is a proliferation in the use of rate expressions supposedly representing a polymer's response with time to a particular mode Rate parameters are determined experimentally, usually at elevated temperatures so as to expedite collection of sufficient data within a reasonable time frame These parameters are then extrapolated to temperatures common to a specific environment, and the subsequent integration of the rate expression provides prediction of a material's behavior at some futuristic time The objective is to review a common methodology, simplifying assumptions, common fallacies, and consequences associated with using rate expressions incorporating the Arrhenius equation to predict long-term behavior and performance of polymeric materials in geotechnical applications A Methodology and Protocol In the development of a rate expression modeling a particular deteriorative mode with time, the first step is to determine which physical properties correlate with that mode, which chemical structure(s)/group(s)/constituent(s) reflect changes in those properties, which agent( s) in the service environment are likely to attack these structure( s) /group( s) / constituents(s), and in which physical state the polymer resides when responding to a particular mode [2,3] Assume a geosynthetic product composed of polymeric material A An agent B within the surrounding environment reacts with chemical group within A causing a change in a property, e.g., molecular weight A reaction scheme incorporating components A and B can be [4-6] where -fA is the reaction rate of A; d/dt is the time derivative; [ ] represents the residual concentration (quantity) of the constituents A and B; superscripts a and b are the rate order by constituent; k is the kinetic rate constant, and t is the time parameter The second step is to select a factor to intensify/accelerate the mode of deterioration Temperature accelerates exponentially nearly all reactions If all other factors are held constant, then a single mathematical relation, the Arrhenius equation, describing temperature dependency [2,5,6] of a rate expression 238 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES where k kinetic rate constant, ko = pre-exponential kinetic rate constant, E = apparent activation energy, R - universal gas constant, T = absolute temperature, and In - natural logarithm Assumptions associated with the Arrhenius equation are [4-8]: k is only a function of temperature; ko does not affect the temperature sensitivity of the reaction; E remains constant over the time and temperature range for evaluation, extrapolation, and prediction The third step is to determine the value of the fundamental parameters (ko, E, and k) for the selected rate expression Concentrations of constituents are measured with time at various temperatures to determine k as a function of temperature, k;(T,) [5,6] (see Fig 1) If the plot of concentration verses time produces a straight line correlation per temperature (T,), then the chosen rate expression (i.e., Eq 2) represents the kinetic mechanisms [5,6] of that particular mode of deterioration The Arrhenius' parameters are usually determined by isothermal or isoconversion experiments The fourth step is to determine if - EI R in Eq is constant with changes in both temperature and constituent(s) conversion [7-11] For isothermal experiments, In k is plotted as a function of reciprocal temperature, 1I T, to determine ko and - E I R in Eq as shown in Fig If the In k versus liT is a straight line, then k is solely a function of temperature, as it should be for applicability of the Arrhenius equation (i.e., Eq 3) into the rate expression (e.g., Eq 2) [5,7,8] BRIGHT ON PROBLEMS USING THE ARRHENIUS EQUATION 239 Now a rate constant k can be determined at temperature T of a specific environment from Fig 2, the fifth step; and Eq can be integrated to predict the state of polymeric material A at time t To summarize the protocol, first a rate expression is selected relating material and environmental constituents, and their order is assumed Second, a representative function for temperature dependency is assumed; this is usually the Arrhenius equation Verification of the rate expression and validity in using the Arrhenius equation for temperature dependency are the third and fourth steps, respectively The fifth step is the extrapolation of the Arrhenius parameters to a service temperature and then integration of the rate expression for prediction of the state of a polymeric material after exposure to a particular mode of deterioration for a given time Problems with Applying a Common Methodology A Representative Rate Expression Deteriorative processes are quite complex and usually take place within a heterogeneous environment [2,3] such as a solid geosynthetic product attacked by agent(s) dispersed in a soil or fluid medium The mechanisms of heterogeneous phase reactions depend on polymeric properties such as density, crystallinity, and viscoelasticity As temperature is elevated to accelerate the reaction, properties vary with temperature changing the rate limiting step of the kinetics thus changing the representing form of the rate expression [3] The relevancy of initial rate data are beclouded by such complicating factors as the release of absorbed gases, absorbents or solvents, or both, (e.g., water), plasticizers (e.g., water), and the effect on kinetic mechanisms by residual catalysts, antioxidant and stabilizer packages, and general impurities [11] Volatilization of these gases and liquids affect the kinetics of the initial rate process representing the condensed phase but are not directly measurable [9] Thus, rate parameters determined on initial kinetic data will not have the same value when determined after these factors have either dissipated or stabilized At low conversions where - rA - for t - 0, nth order rate expressions can behave like zero order (n = 0) due to the dominance of the Arrhenius equation temperature term, - E/ R/T [12] Thus, it is necessary for the reaction to progress sufficiently so as to acquire 240 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES meaningful data with which to determine parameters for a rate expression with constituent orders greater than zero With time, some deteriorative reactions (e.g., oxidation and hydrolysis) commonly associated with polymers in geotechnical environments become autocatalytic changing the form of the initial rate expression as the reaction progresses Thus, modeling the kinetics of polymer deterioration is difficult and complex [11], and compounded by the fact that a rate expression developed in the short-term may not be representative or valid in the longterm [14] Constituent Representation-In homogeneous, fluid phase reactions, constituent concentration is fairly straightforward, amount per unit volume [4-6] But in solid state reactions, the concept of concentration of polymeric material can take on a much different meaning [15] The reaction's progression may have to be defined as the change in material thickness or weight or residual equivalents with time [15] Deteriorative reactions that become autocatalytic with time change the form of the initial rate expression [11] as discussed earlier By definition, an autocatalytic behavior means the rate of reaction accelerates with the increasing presence (concentration) of a particular product constituent [4-6] such as a constituent C It should now be apparent that constituency and its order within a rate expression has a significant impact on the prediction of the behavior of material A with time Applicability of the Arrhenius Equation With liquid/solid state reactions initially being diffusion limited, the conventional concept of constituent concentration and their order have questionable significance Thus, it may be difficult to determine whether the rate expression exhibits an exponential dependency with temperature [14,15,23] as would be described by the Arrhenius equation A simple temperature dependency should not be assumed for liquid/solid state reactions [15,23] Determination of Arrhenius' Parameters Temperature Range- Rate parameters are determined experimentally at elevated temperature so as to expedite collection of sufficient data within a reasonable time frame There are several potentially serious problems associated with the collection of rate data due to changes in polymeric behavior caused by elevating temperature Within the experimental temperature range, changes from traversing across and within physical states (e.g., viscoelastic, glassy to viscoelastic), in properties (e.g., density, crystallinity), or in activity/mobility of the deteriorative agent(s) or a combination thereof, may be sufficient to alter reaction mechanisms [3] thus causing rate expressions representing the extreme ends of the temperature range to be different Also, a material's thermal history can dramatically affect the kinetic parameters [24] through effects on physical properties such as density and crystallinity At the low end of temperature range (i.e., ambient), diffusion mechanisms commonly control leading to low activation energies (E); whereas at the high end temperatures, chemical reactions dominate leading to high E values [3] Thus, kinetic parameters development at elevated temperatures potentially have no relevant usefulness in service life predictions at ambient temperatures [2] If the elevated temperature range is sufficiently high to traverse a phase transition (e.g., glass transition), a sharp change in rheological and physical properties will occur [2,3] Thermal transitions of polymers used in geosynthetics are given in Table [2,3] These changes will affect kinetic mechanisms and thus rate parameters [2,3] This would invalidate 242 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES TABLE I-Typical Polymer Low Density PE Medium Density PE High Density PE Polypropylene PET pye transition values for polymer used in geosynthetics Glass Transition, °e -80 -80 -80 -10 +70 +80 Melt Range, °e 60-100 80-120 100-140 100-165 200-260 - extrapolation and use of Arrhenius' parameters (k, ko, E) from above or within a phase transition to service temperatures [2,3] Activation Energy-The activation energy (E) represents the temperature dependency of the rate expression Thus, it is important to determine E as precisely as possible to provide meaningful estimates of service life with time after its extrapolation to service temperatures [25] A slight error in E is magnified into a large error through extrapolation of k to service conditions [2] Thus, the greater the temperature difference between service conditions and elevated temperature levels the smaller the error in E must be for a given accuracy at service conditions [25] Assuming constancy in E with time and temperature, the extrapolation from elevated temperatures to service conditions (Ll T) can change rates ( - rA) by factors of 10 -7 to 10- 20 depending on the magnitude of LlT [25] Thus, a relatively small error in E can translate into a very large error in - rA when extrapolated to another temperature For example, a Yz kcallmol (2.093 kJ I mol) error in E will cause a 50% uncertainty at the 90% confidence level for an extrapolation of the -rA from 400 to 25°C [10] Or, an error in E of as small as 1.0 kcallmol (4.186 kJ Imol) will cause an error of ± 100% at the 95% confidence level in extrapolation of -rA from 225 to 25°C [3] Unfortunately, values of E for the degradation of some well-investigated polymers commonly resemble a list of random numbers [24] Any error in E is compounded again by the integration of rate expressions (e.g., Eqs through 10) for life predictions at 25°C on the state of polymeric material after exposure to a mode of deterioration with time Conclusions on Problems with a Common Methodology (1) Modeling polymer degradation with a rate expression can be difficult and complex, and compounded by the fact an expression most likely will not be valid over the longterm service life of the polymer (2) Principal constituents within an expression vary as the mode of deterioration progresses, further complicating representation over the long-term (3) Constituent order can vary significantly as deterioration progresses (4) In heterogeneous phase reactions, applicability of the Arrhenius equation for temperature dependency should not be automatically assumed (5) The rate process should progress sufficiently before collecting any rate data so as to avoid the impact of initial instabilities by additives and stabilizers, and low conversions on establishing net rate order (6) A relatively small error in the Arrhenius activation energy determined at elevated temperatures can translate into a very large error when extrapolated to lower tem- BRIGHT ON PROBLEMS USING THE ARRHENIUS EQUATION 243 peratures for rate predictions and compounded again upon integration of rate expression with time (7) Arrhenius kinetic parameters determined from accelerated polymeric deterioration at elevated temperatures quite possibly have no relevant bearing on service life predictions at ambient temperatures Examples Degradation Studies of Poly(ethylene terephthalate) Poly( ethylene terephthalate) (PET) as a high-tenacity fiber is used in geosynthetic products Being an ester, PET is susceptible to hydrolysis Work by McMahon et al [20] is a data base commonly used to make predictions on the long-term effects of hydrolysis on the physical and mechanical properties of PET The McMahon [20] rate expression is assumed pseudo first order with the principal constituent being the moles of ester links [A] in PET, and the Arrhenius equation represents temperature dependency Principal experimental parameters were film thickness (0.5 and 10 mil), temperature (60, 71, 82, 90 and 99°C), and relative humidity (20, 50, 75, 95, and 100%) [20] The relative humidity (RH) in ground atmospheres typical of geotechnical environment is about 100% [26] The chosen rate expression matches only the data taken at 60 and 71°C for 95 and 100% RH for both film thicknesses See Figs and 4, which are expanded well beyond published figures At temperatures 82°C and higher, the kinetic data show the onset of an autocatalytic behavior for both film thicknesses Thus, the chosen rate expression does not represent all the kinetic data with which to determine the rate parameters within the Arrhenius equation The mean E was reported as being a function of film thickness: 25.7 and 29.7 kcallmol (107.6 and 124.3 kJ/mol) for 0.5 and 10.0 mil, respectively [20] Thus, E is not constant violating a fundamental condition [4-8] for the use of the Arrhenius equation All the kinetic rate data were collected within the glass transition (60 to 80°C) and well into the viscoelastic range (>80°C) Thus, the kinetic parameters developed within a transition and one physical state and then extrapolated into another state would have no meaning in that state [2,3] be.;ause the reaction mechanisms are influenced by changes in physical properties caused by the elevated temperatures Thus, the McMahon data are inappropriate in predicting the long-term performance of PET subject to hydrolytic attack in geotechnical environments Atmospheric Deterioration of Poly(vinyl chloride) At low temperatures and early isoconversions (>0), the deterioration of poly(vinyl chloride) (PVC) is a hydrochloric stripping reaction with high E values At higher temperatures and later isoconversions «1), deterioration is more rapid oxidation reaction with lower E values [10,13.27] Figure shows at least two sets of isoconversion lines (0.05, 0.1, 0.2 versus 0.7 or 0.8 or 0.9) that are not parallel, and consequently represent at least two different E values [10,13,27] indicating a rate mechanism changing with temperature and conversion which is not likely to be represented accurately by a single rate expression or single E value The use of a global E value (unfortunately a common practice) will only compound present inaccuracies upon extrapolation of - rA through k to service temperatures which, in turn, is compounded again upon integration of - rA for prediction of long-term behavior of PVC with time The Thermal Oxidative Stability of High-Density Polyethylene Polymers can degrade in an oxygen atmosphere at elevated temperatures [28] Oxidation occurring from ambient to about 200°C is usually referred to as autoxidation [28] The autocatalytic phase is followed by a period during which the rate appears to be steady [28] Eventually deceleration occurs, and oxidation proceeds at a much slower rate probably due to exhaustion of preferentially reactive sites [28] The morphology, density, temperature level, and physical state affect the oxygen uptake (oxidation) of branched (medium-density) and linear (high-density) polyethylenes (PE) as shown in Fig [28] Morphology and density definitely affect the rate and amount of oxygen uptake by the two different polyethylenes at 100°C which are reported to be below their melt transition [28], and thus are semicrystalline The uptake data clearly suggest that the rate expressions and subsequent kinetic parameters will be quite different for these two grades of polyethylenes After 200 h, the branched PE assumes an autocatalytic behavior, thus necessitating a change in the initial rate expression The oxygen uptake of the same linear PE was monitored at 140°C which is above its melt temperature and thus amorphous (see Fig 6) [28] Oxygen uptake is approximately ten times that at 100°C [28] Thus for 140 and 100°C data, the rate expressions and values of the kinetic parameters are likely to be quite different Also, the initial uptake rates «100 h) vary significantly with temperature accentuating the differences between representing rate expressions Thus, to characterize the oxidative degradation of linear (high) PE in the short-term so as to predict its behavior and state in the long-term, kinetic rate data should be taken below its initial melt temperature General Conclusions Caution should be exercised when using the Arrhenius equation to represent temperature dependency of the deteriorative mechanisms associated with polymeric materials in geo- technical applications Activation energy (E) should be determined as independent of temperature and time as possible Minimizing the extent of temperature elevation to within a common physical state facilitates its accuracy Determine all kinetic parameters with a single rate expression representing the deteriorative mechanism(s) over the life expectancy of polymeric materials, especially as far out as 25 to 100 years References [I] ASCE Manual of Practice No 66, 1985 [2] Flynn, J H., "The Role of Thermal Analysis in the Lifetime Prediction of Polymers," Proceedings of the Second European Symposium on Thermal Analysis, D Dollimore, Ed., Heyden, London, 1-4 September 1981, pp 223-226 [3] Flynn, J., "Lifetime Prediction for Polymeric Materials from Thermal Analytical ExperimentsProblems, Pitfalls, and How to Deal with Some of Them," 46th Annual Technical Conference, Society Plastics Engineers, Atlanta, GA, 18-21 April 1988, pp 930-932 [4] Boudart, M Kinetics of Chemical Processes, Prentice-Hall, Englewood, NJ, 1968 [5] Levenspiel, 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757-760 (12) Flynn, J H and Wall, L A., "Initial Kinetic Parameters from Thermogravimetric Rate and Conversion Data," Journal of Polymer Science, Polymer Letters, Vol 5, No.2, 1967, pp 191196 (13) Flynn, J H., "Thermogravimetric Analysis Kinetics," ACS Polymer Preprints, Vol 22, No.1, 1981, pp 310-312 BRIGHT ON PROBLEMS USING THE ARRHENIUS EQUATION 247 [14] Garn, P D., "Kinetic Investigations by Techniques of Thermal Analysis," CRC Critical Reviews in Analytical Chemistry, Vol 3, No.1, 1972, pp 66-111 [15] Gomes, W., "Definition of Rate Constant and Activation Energy in Solid State Reactions," Nature, Vol 192, No 4805, 1961, pp 865-866 [16] Ravens, D A S and Ward, I M., "Chemical Reactivity of Polyethylene Terephthalate," Transactions Faraday Society, Vol 57, No.1, 1961, pp 150-159 [17] Ravens, D A S and Sisley, J E., "Cleavage Reactions," Chemical Reactions of Polymers, E M Fettes, Ed., Interscience, New York, 1964, pp 551-564 [18] Ravens, D A S., "The Chemical Reactivity of Poly(ethylene terephthalate): Heterogeneous Hydrolysis by Hydrochloric Acid," Polymer, 1960, Vol 1, pp 375-383 [19] Golike, R C and Lasoski, S W, "Kinetics of Hydrolysis of Polyethylene Terephthalate Films," Journal of Physical Chemistry, Vol 64, No.7, 1960, pp 895-898 [20] McMahon, W, Birdsall, H A., Johnson, G R., and Camilli, C T., "Physical Properties Evaluation of Compounds and Materials," Journal of Chemical Engineering Data, VolA, No.1, 1959, pp.5779 [21) Marshall, I and Todd, A., "The Thermal Degradation of Polyethylene Terephthalate," Transactions Faraday Society, Vol 49, 1953, pp 67-78 [22] Risseeuw, P and Schmidt, H M., "Hydrolysis of HT Polyester Yarn in Water at Moderate Temperatures," 4th International Conference on Geotextiles, Geomembranes, and Related Products, G den Hoedt, Ed., The Hague, Netherlands, 28 May-1 June, 1990, pp 691-696 [23] Jost, W, Diffusion in Solids, Liquids, Gases, Academic Press, New York, 1960, pp 367-368 [24] Flynn, J H., "Analysis of the Kinetics of Thermogravimetry: Overcoming Complications of Thermal History," Thermal Analysis in Polymer Characterization, E A Turi, Ed., Heyden, Philadelphia, 1981, pp 43-59 [25] Flynn, J H and Dickens, B., "Applications of New Kinetic Techniques to the Lifetime Prediction of Polymers from Weight-Loss Data," Durability of Macromolecular Materials, R K Eby, Ed., ACS Symposium Series 95, Washington, D.C., 1979, pp 97-113 [26] Jaillous, J.-M and Verdu, J., "Kinetic Models for the Life Predictions in PET Hygrothermal Ageing: A Critical Survey," 4th International Conference on Geotextiles, Geomembranes, and Related Products, G den Hoedt, Ed., The Hague, Netherlands, 28 May-1 June, 1990, p 727 [27] Schneider, H A., Vasile, c., Furnica, D., and Onu, A., "Uber den Einflub der Aufheizgeschwindigkeit auf die Kinetik des Thermogravimetrischen Abbaues von Polymeren," Die Makromolekulare Chemie, Vol 117, No 2783, 1968, pp 41-49 [28] Hawkins, W L., "Thermal and Oxidative Degradation of Polymers," SPE Transactions, Vol 4, No.3, 1964, pp 187-192 Author Index ~o ~D Bathias, c., 141 Bomas, Hubert, 132 Bright, Donald G., 236 Doker, Hubert, 54 Ni, Jingang, 141 Nowack, Horst, 54 Oliva, Vladislav, 72 p H Pluvinage, Go, 105 Prakash, R v., 30 Puskar, Anton, 153 Halford, Gary R., 91 Heine, Jennifer, 117 Hippo, P K., 203 R Reifsnider, Kenneth L., Robin, c., 105 K Kalluri, Sreeramesh, 117 Kim, Yong-Suk, 91 Kunes, Ivan, 72 S Schulte, Karl, 54 Sheppard, Sheri Do, 169 Shi, H Jo, 105 Sorem, James R Jr., 186 Sunder, R., 19,30 L Landgraf, Ronald Lee, Bo L., 203 Liu, D So, 203 w., T-V Tipton, Steven M., 186 Trautmann, Karl-Heinz, 54 Ulrich, P c., 203 Verrilli, Michael J., 91 M Mayr, Peter, 132 McGaw, Michael Ao, 117 Medzorian, J P., 203 Mitchell, Michael R., Mitchenko, E I., 30 Moore, Dennis, 117 Z Zhang, Shi Jie, 54 Zhou, Zhendong, 45 Zwerneman, Farrel J., 45 249 Subject Index Crack closure, 19 cyclic deformation, 72 loading sequences, 54 spectrum loading, 30 Crack density, metals under variable amplitude loading, 132 Crack growth cyclic deformation, 72 loading sequences, 54 resistance spot welds, 169 spectrum load notch fatigue, 30 sub-threshold load cycles, 45 Crack initiation notch root, 19 resistance spot welds, 169 Crack opening, 19 loading sequences, 54 spectrum loading, 30 Crack opening stress hysteretic model, 30 intensity, loading sequences, 54 Crack rate prediction, cyclic deformation, 72 Crack tip, cyclic deformation around, 72 Creep cavity model, 91 growth model, 91 Creep-fatigue life prediction, 91 Critical element concept, Critical strain energy density, low-cycle fatigue tests, 72 Cumulative fatigue damage, 19 Cyclic loading, fatigue lifetime prediction, 153 Cyclic plastic deformation, finite element method, 72 Cyclic strain, cord-rubber composites, A Acceleration, sub-threshold load cycles, 45 Acoustic emission, 45 cord-rubber composites, 203 Addition model, 132 Air, MAR-M 247 fatigue damage behavior, 117 Aircraft tires, life prediction, 203 AI-Cu alloy, spectrum load cumulative damage, 19 notch root crack growth, 30 Alloys, fatigue behavior, 141 Aluminum alloy edge fatigue crack growth, 72 effective stress intensity factor range and crack propagation, 54 Amplitude loading, variable, 54 life prediction, 132 Arrhenius equation, long-term behavior prediction, 236 B Bauschinger effect, 186 Block loading, cumulative fatigue damage behavior, MAR-M 247, 117 C Carbon steel, fatigue lifetime prediction, 153 Cast nickel-base superalloy, cumulative fatigue damage behavior, 117 Coffin/Manson curve, 153 Composite material systems, damagetolerant, Copper matrix composite, tungsten fiber-reinforced, creep-fatigue life prediction, 91 Cord-rubber composites, life prediction, 203 Cyclic stress-strain curve, fatigue lifetime prediction, 153 D Damage Curve Approach, 117 Damage tolerance, composite material systems, 203 251 252 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES Delamination, cord-rubber composites, 203 Durability, composite material systems Dynamic creep, cord-rubber composites, 203 Dynamic strain gages, miniature, 141 High-strength low-alloy steel, resistance spot welds, 169 Hydrogen, high-pressure, MAR-M 247 fatigue damage behavior, 117 Hysteretic model, crack opening stress, 30 I E Earth structures, polymeric materials, life prediction, 236 Effective stiffness models, Effective stress intensity factor range, loading sequences, 54 Elastic modulus, degradation, 153 Elastomer composites, angle-plied fiber reinforced, life prediction, 203 Energetic theory, 72 Exceedance curve, 19 F Failure mechanism, 141 FALSTAFF, 19,30 FASTRAN II model, 54 Fatigue behavior study, ultrasound, 141 Fatigue damage austenitic stainless steel, 105 cord-rubber composites, 203 cumulative, MAR-M 247, 117 sub-threshold load cycles, 45 Fatigue life equation, 153 Fiber-reinforced systems, design, Finite element method cyclic deformation, 72 life prediction, cord-rubber composites, 203 Fractography, loading sequences, 54 G Geosynthetics, life prediction, 236 Geotechnical environments, polymeric materials, long-term behavior prediction, 236 Grain boundary cavitation, 91 H Heat affected zone, resistance spot welds, 169 Heat generation, cord-rubber composites, 203 Interply shear, cord-rubber composites, 203 K Kinematic hardening, 105 L Life prediction angle-plied fiber-reinforced elastomer composites, 203 creep-fatigue, 91 loading sequences, 54 mechanistic methods, metals under variable amplitude loading, 132 novel methodology, 153 polymeric materials in geotechnical environments, 236 resistance spot welds, 169 sub-threshold load cycles, 45 thermal-mechanical fatigue, austenitic stainless steel, 105 Linear elastic fracture mechanics, 169 Loading frequency, ultrasonic, 141 Loading ramp, 153 Load interaction, sub-threshold load cycles, 45 Load sequence effects, crack propagation, 54 Long crack growth, 30 spectrum load notch fatigue, 19 M MAR-M 247, cumulative fatigue damage behavior, 117 Metal matrix composites, creep-fatigue life prediction, 91 Metals, variable amplitude loading, life prediction, 132 Micromechanical strength models, Microstructural model, creep-fatigue life, 91 Miner's rule, 132 SUBJECT INDEX N Near threshold crack growth method, 54 Neuber's Rule, 186 Notch root crack initiation, 19 fatigue, 19, 30 Notch strain analysis, proof loading, 186 o Overload periodic, 45 reverse plasticity criterion, 186 single, cyclic deformation, 72 p Pneumatic tire materials, life prediction, 203 Polymeric materials, geotechnical environments, long-term behavior prediction, 236 Proof loading, reverse plasticity criterion, 186 R RainfIow cycle count, 19, 30 Ramberg-Osgood relation, 186 Range/damage-exceedance diagram, 19 Realized damage sum, 132 Residual fatigue lifetime, prediction, 153 Residual stress, localized plasticity, 186 Resistance spot welds, fatigue propagation life, 169 Resonance vibration, 141 Retardation delayed, 72 sub-threshold load cycles, 45 Reverse plasticity criterion, proof loading, 186 S Short crack growth, spectrum load notch fatigue, 19 253 Spectrum loading, cumulative fatigue damage behavior, MAR-M 247, 117 Spectrum load notch fatigue, 19 root fatigue, elastic and inelastic conditions, 30 Stainless steel, austenitic, lifetime prediction, 105 Strain energy density proof loading, 186 total, austenitic stainless steel, 105 Strain rate effect, 141 Stress amplitude, cord-rubber composites, 203 Structural stress, linear elastic fracture mechanics, 169 Sub-threshold load cycles, fatigue damage, 45 Superalloy, cumulative fatigue damage behavior, 117 T Thermal-mechanical fatigue lifetime prediction, austenitic steel, 105 Threshold, 45 Tungsten fibers, copper matrix composite, life prediction, 91 TWIST, 19 U Ultrasound fatigue lifetime prediction, 153 high-power, fatigue behavior study, 141 Upper and lower bound damage, 19 W Waste containment sites, polymeric materials, life prediction, 236 W ICu composite, unidirectional, creepfatigue life prediction, 91 ... Engineering Science and Mechanics, Virginia Tech Blacksburg VA 24061-0219 ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES more specific meaning In point of fact, damage tolerance is defined... Creepfatigue responses in a metal matrix composite is interpreted using a microstructural model ADVANCES IN FATIGUE LIFETIME PREDICTIVE TECHNIQUES based on cavity growth Next, a nonlinear kinematic... considering a mechanistic representation of damage tolerance in a laminate, using ply-level modeling We will end our paper by indicating recent advances in the use of micro mechanics to bring the