Fatigue Crack Growth : Mechanics, Behavior and Prediction [1 ed.] 9781608767700, 9781606924761

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Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

FATIGUE CRACK GROWTH: MECHANICS, BEHAVIOR AND PREDICTION

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No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in rendering legal, medical or any other professional services.

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved. Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

FATIGUE CRACK GROWTH: MECHANICS, BEHAVIOR AND PREDICTION

ALPHONSE F. LIGNELLI

Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works. Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication.

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This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA Fatigue crack growth : mechanics, behavior, and prediction / editor, Alphonse F. Lignelli. p. cm. Includes index. ISBN 978-1-60876-770-0 (E-Book) 1. Materials--Fatigue. 2. Fracture mechanics. I. Lignelli, Alphonse F. TA418.38.F3677 2009 620.1'126--dc22 2009000612

Published by Nova Science Publishers, Inc.

New York

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

CONTENTS Preface

vii

Research and Review Studies

1

Chapter 1

Real-time Fatigue Damage Monitoring via In Situ Ultrasonic Sensing Shalabh Gupta and Asok Ray

3

Chapter 2

Multiscale Approaches to Fatigue Crack Growth from the Debonding of Particle/Ductile-Matrix Interfaces Yong X. Gan

49

Chapter 3

Advances in the Numerical Modelling of Fatigue Crack Closure Using Finite Elements J. Zapatero and A. Gonzalez-Herrera

83

Chapter 4

Textural Fractography of Fatigue Fractures Hynek Lauschmann and Noel Goldsmith

125

Chapter 5

A Novel Fractography for Investigation of the Fatigue Fracture Process in Materials Manabu Tanaka and Ryuichi Kato

167

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Short Communications

191

Fatigue Behaviour Model for Short Fibre Glass Reinforced Polyamides J.A. Casado and F. Gutiérrez-Solana

193

The Effect of Overload Plastic Zone on Fatigue Crack Propagation after Overloading Chobin Makabe and Anindito Purnowidodo

211

Fatigue Crack Growth Prediction in Asphalt Concrete Materials with Damage Mechanics Model Wing Gun Wong and Zhi Suo

227

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vi

Contents Fracture Strength Evaluation of Center-Crack Tensile Specimens Made of Heat-Treated Wrought Aluminium Alloys M. Jeyakumar and T. Christopher

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Index

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239

255

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PREFACE Fatigue is one of the primary reasons for the failure of structural components. The life of a fatigue crack has two parts, initiation and propagation. Dislocations play a major role in the fatigue crack initiation phase. It has been observed in laboratory testing that after a large number of loading cycles dislocations pile up and form structures called persistent slip bands (PSB).Fatigue crack growth is a crucial subject in the aircraft, railroad and other transportation industries as well as in the biomedical industry.This new book presents the lates research in the field. Estimation of structural damage and quantification of structural integrity are critical for safe and reliable operation of human-engineered complex systems. Fatigue damage is one of the most commonly encountered sources of structural degradation in mechanical systems. Detection of incipient fatigue damage is essential for averting widespread crack growth that leads to catastrophic failures. Chapter 1 presents online in situ monitoring of fatigue damage using the ultrasonic sensing technique that is sensitive to small microstructural changes, robust to measurement noise, and also suitable for real-time applications. A recently reported information-theoretic method of data-driven pattern recognition, called Symbolic Dynamic Filtering (SDF), has been used for real-time analysis of ultrasonic data, where the time series data in the fast scale of process dynamics are analyzed at discrete epochs in the slow scale of fatigue damage evolution. SDF includes preprocessing of ultrasonic data using wavelet transform, which is well suited for timefrequency analysis of non-stationary signals and enables noise attenuation in raw data. The wavelet-transformed data is partitioned using the maximum entropy principle to generate symbol sequences, such that the regions of data space with more information are partitioned finer and those with sparse information are partitioned coarser. Subsequently, statistical patterns of evolving damage are identified from these sequences by construction of a (probabilistic) finite-state machine that captures the dynamical system behavior by information compression. A computer-controlled fatigue test apparatus, equipped with ultrasonic sensors and an optical microscope, has been used to experimentally validate the concept of ultrasonic based real-time monitoring of fatigue damage in polycrystalline alloys. The task of fatigue damage monitoring is formulated as: (i) forward problem of pattern recognition for (offline) characterization of the statistical behavior of fatigue damage evolution and (ii) inverse problem of pattern identification for (online) estimation of the remaining useful life based on the real time ultrasonic data and the statistical information generated offline.

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viii

Alphonse F. Lignelli

Fatigue crack initiation and propagation due to the debonding of particle/ductilematrix interfaces were investigated at both macro- and micro-scales. First, macroscale approach was used to obtain the stress field in the elastic-plastic region within the matrix in front of the debonded particle. Continuum mechanics is applied in the derivation. Stress intensity solutions were obtained in the tension and compression stages in a fatigue cycle. In the plastic zone just ahead of the debonded particle, a microscale approach was used to get the stress solutions. In classical plasticity theory, the material property at the crack tip is considered to be isotropic and the maximum stress in the plastic zone is assumed to be the yield strength of the material. Our recent examination on the crack tip of ductile materials containing particle inclusions indicates that the deformation state at the crack tip is highly anisotropic. In order to describe such behavior, the deformation mechanisms of single crystal plasticity are enforced in Chapter 2. That is, the deformation of the material in the crack tip region is due to the motion of dislocation on different slip systems. Based on such a consideration, we assume that the stresses at the boundary between the elastoplastic region and the plastic zone propagate into the plastic zone. The magnitudes of the stress components are determined using the formulation of slip line theory. The primary slip lines are collinear with the dislocation motion directions. The second part of this work is specifically on the fatigue crack propagation. Once a short crack from the interface debonding starts growing, how to characterize the fatigue crack growth resistance becomes an important issue. We use a simulated crack (a through thickness notch) to study the fatigue crack growth. The specific energy of damage, a parameter which may be used to characterize the fatigue crack growth resistance of the material, is obtained. The relationship between fatigue fracture surface morphology and the specific energy of damage is discussed as well. The accurate determination of fatigue crack closure has been a complex task for years. It has been approached by means of experimental and numerical methods. Experimental methods are controversial, the results obtained may have low precision, high scatter, and are subject to problems of interpretation. Finite element method has been an alternative for the study of fatigue crack closure. However, such analyses are complex and computationally expensive. Plasticity-induced crack closure is the main mechanism causing fatigue crack closure. It occurs when the flanks of the crack contact with a load above the minimum load in fatigue; as a result, the crack is subject to a smaller crack driving force. Difficulties on the accurate modelling of this phenomenon derive from the need to simulate the preceding fatigue cycles, joined to the high plasticity induced at the crack tip, this results in high computational costs. There exist several parameters which must be strictly controlled to avoid its influence over the results. Chapter 3 summarizes the main recommendations derived from a comprehensive study of the variables that influence the model accuracy. A special attention is paid to the influence of the minimum element size and the effect of the length of the simulated crack wake. It also has been studied issues as the node release scheme, the material yielding model or the significance of different criterion to establish the opening or closure loads. Finally, the main results and findings regarding fatigue crack closure are summarized. Both, 2D and 3D cases are shown. Bi-dimensional studies permit the analysis of the influence of different parameters as crack length, maximum load or stress ratio R. The tri-dimensional model allows the detection of the presence of closure in a small external area of the specimen

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Preface

ix

and interesting information is obtained regarding the shape an extension of the plastic zone size. As explained in Chapter 4, the reconstitution of the history of a fatigue crack growth is based on the knowledge of correspondences between the morphology of the crack surface and the crack growth rate - CGR. Within conventional fractography, striations and/or beach lines are the main source of information. However, striations are often not present or not distinguishable, and the interpretation of beach lines is strictly limited to the particular case. As an alternative information source, applications of image textural analysis were developed, called overall Textural fractography. Mezoscopic SEM magnifications (30 to 500x) are used, under which images of fracture surfaces contain complicated (image) textures without distinct borders. The aim is to find any characteristics of this texture which correlate with CGR. Preprocessing of images is necessary to obtain a homogeneous texture. Six methods of image textural analysis have been developed and realized as computational programs: application of 2D Fourier and 2D Wavelet transformations, fractal analysis, Gibbs random field (GRF) model, idealization of light objects into a fiber structure, and auto-shape decomposition (which is a fully original method developed for fractographic applications). By using any of these methods, images are characterized by a set of numerical characteristics - image feature vector. The relation between image feature vectors and crack growth rates is expressed by means of a multivariate statistical model or a neural network. Solutions of cases of crack growth under constant cycle loading are presented. For cases of different variable cycle loadings, the conventional definition of fatigue crack growth rate (CGR) itself is shown to be fractographicly confusing. A new concept of reference crack growth rate (RCGR) is proposed to cover all fatigue cracks regardless of the type of loading. RCGR may be estimated directly from images of fracture surfaces by means of the methods of textural fractography. Physical interpretation is suggested on the basis of cycle-by-cycle crack growth description. Both approaches are compared in application – quantitative fractography of three sets of specimens (from two aluminium alloys and a steel one) loaded by constant cycle, periodic and random blocks, respectively. Common morphologic features of all fracture surfaces are found that are closely related to RCGR. Chapter 5 discusses a novel fractography, based on fractal analysis using the threedimensional data of fracture surfaces, which was developed for the investigation of the fatigue fracture process in materials. Change in the fracture surface morphology with crack growth was estimated by the fractal dimension of small regions on the fracture surface of high-strength 17-7PH (AISI631) stainless steel fatigued by repeated bending, in which characteristic fatigue fracture surface patterns, such as striations, were not observed. The fractal dimension represented microstructures such as microcracks, dimples and small steps on the fracture surface, and the fractal dimension of the small regions was displayed as the fractal dimension map (FDM). Fatigue cracks were initiated on the specimen surface and grew into the interior of the specimen. Quasi-cleavage fracture with the smaller fractal dimension was the dominant fracture mechanism in this steel, while ductile fracture with the larger fractal dimension was also observed near the specimen surface fatigued at the lower values of the maximum total strain range. The fractal dimension increased with increasing distance from the specimen surface, irrespective of fracture mechanisms. This was attributed to the fact that the fracture surface formed in the earlier stage was more damaged and more simplified by cyclic compressive load than that formed in the later stage in the fatigue fracture process, and the total strain range (the cyclic compressive load) decreased with

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Alphonse F. Lignelli

increasing distance from the specimen surface. The fatigue fracture process of materials was examined numerically and visually by the three-dimensional fractal analysis using FDM. A new fractography in this study is applicable to the investigation of the fatigue fracture process in many kinds of materials, including non-metallic materials such as polymers. The first Short Communication shows the existence of three states in the fatigue behaviour of short fibre-glass reinforced polyamide up to the moment of failure. In the state I creep transient deformation processes are produced. If temperature, associated to the dynamic process, stabilises, the material is able to withstand an unlimited number of cycles. On the other hand, if temperature grows continuously the material takes a critical strain for which a transition to the state II is reached. This new state shows a constant material´s deformation rate and a homogeneous nucleation of crazing phenomena. The hysteresis energy per cycle grows in a constant way and the material´s temperature increases. These conditions lead to the final instability that starts when the material´s deformation reaches a new critical value for which a second transition to the third state takes place. Under these circumstances local instabilities precede material´s final fracture. This is due to the quantity of crazes and their sizes, which are important enough to origin high concentration of mechanical effects to produce a material´s accelerated growth deformation to obtain its fracture. The retardation of fatigue crack propagation has been observed after applying single overload. However, it was found that the fatigue crack growth rate actually accelerated after a tensile overload under a negative value of baseline stress ratio. This type of crack propagation behavior was related to the change of the compression residual stress to tensile residual stress distributed in the vicinity of the crack tip after an overload. In the second Short Communication, it was discussed that the number of delay and acceleration cycles of crack propagation were associated with creating mechanisms of static and inverse plastic zones, which influenced the residual stress state in front of the crack. The extent of fluctuation of the crack opening stress level is related to the extent of the overload-affected zone. Also, it was discussed that the residual fatigue crack propagation life was influenced by the applied overload level and the conditions of applied cyclic stress after overloading. Fatigue crack has been recognized as one of the main forms for structural damage in flexible pavements. Under the action of repeated vehicular loading, deterioration of the asphalt concrete (AC) materials in pavements, caused by the accumulation and growth of micro and macro cracks, gradually takes place. Existing prediction models in asphalt concrete pavement do typically no take the interaction and dependencies between micro and macro mechanics into account. In the third Short Communication, the Indirect Tensile Fatigue Tests (ITFT) and Indirect Tensile Stiffness Modulus (ITSM) tests were carried out on three kinds of asphalt concrete materials to establish fatigue damage models and failure criteria. Fatigue damage model, based on continuum damage mechanics, to describe the formation of microcracks and crack propagation were developed for the wearing course materials. With the fatigue damage models, finite element analysis was carried out to study the crack resisting performance of the three wearing course materials in a flexible pavement structure. It is expected that this method will be a useful tool for predicting service life for new pavements, as well as for assessing residual life of existing ones. Modifications are made in one of the stress fracture criteria of Whitney and Nuismer (viz., the point stress criterion) to improve fracture strength estimations of cracked bodies. To take in to account the variation of the characteristic length with the crack size in fracture model, a simple and realistic relation is established for the characteristic length in terms of the

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fracture strength and ultimate tensile strength of the material. An empirical relation is presented in the fourth Short Communication for the generation of failure assessment diagram from the improved point stress criterion. Modifications made in the fracture model correlate well with test data of heat-treated wrought aluminium alloys.

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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RESEARCH AND REVIEW STUDIES

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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In: Fatigue Crack Growth... Editor: Alphonse F. Lignelli, pp. 3-48

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Chapter 1

R EAL - TIME FATIGUE DAMAGE M ONITORING VIA I N S ITU U LTRASONIC S ENSING∗ Shalabh Gupta†and Asok Ray‡ Mechanical Engineering Department, The Pennsylvania State University University Park, PA 16802, USA

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Abstract Estimation of structural damage and quantification of structural integrity are critical for safe and reliable operation of human-engineered complex systems. Fatigue damage is one of the most commonly encountered sources of structural degradation in mechanical systems. Detection of incipient fatigue damage is essential for averting widespread crack growth that leads to catastrophic failures. This chapter presents online in situ monitoring of fatigue damage using the ultrasonic sensing technique that is sensitive to small microstructural changes, robust to measurement noise, and also suitable for real-time applications. A recently reported information-theoretic method of data-driven pattern recognition, called Symbolic Dynamic Filtering (SDF ), has been used for real-time analysis of ultrasonic data, where the time series data in the fast scale of process dynamics are analyzed at discrete epochs in the slow scale of fatigue damage evolution. SDF includes preprocessing of ultrasonic data using wavelet transform, which is well suited for timefrequency analysis of non-stationary signals and enables noise attenuation in raw data. The wavelet-transformed data is partitioned using the maximum entropy principle to generate symbol sequences, such that the regions of data space with more information are partitioned finer and those with sparse information are partitioned coarser. Subsequently, statistical patterns of evolving damage are identified from these sequences by construction of a (probabilistic) finite-state machine that captures the dynamical system behavior by information compression. ∗

This work has been supported in part by the U.S. Army Research Laboratory and the U.S. Army Research Office (ARO) under Grant No. W911NF-07-1-0376 and by NASA under Grant No. NNX07AK49A. Any opinions, findings and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of the sponsoring agencies. † E-mail address: [email protected] ‡ E-mail address: [email protected]

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Shalabh Gupta and Asok Ray A computer-controlled fatigue test apparatus, equipped with ultrasonic sensors and an optical microscope, has been used to experimentally validate the concept of ultrasonic based real-time monitoring of fatigue damage in polycrystalline alloys. The task of fatigue damage monitoring is formulated as: (i) forward problem of pattern recognition for (offline) characterization of the statistical behavior of fatigue damage evolution and (ii) inverse problem of pattern identification for (online) estimation of the remaining useful life based on the real time ultrasonic data and the statistical information generated offline.

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1.

Introduction

Gradually evolving changes in the structural parameters of a mechanical system over its service life may generate uncertainties in both transient and stationary behavior. This problem is often addressed by overly conservative estimates of the design parameters due to lack of available information. Consequently, the engineering design of mechanical systems suffers from enforcement of large safety factors and results in manufacture of cumbersome and unnecessarily expensive machinery. The alternative is to have expensive and time-consuming inspections. In the current state-of-the-art maintenance actions are based on fixed usage intervals. On-line sensing of damage would allow re-evaluation and extension of service life and inherent protection against unforeseen early failures to reduce the frequency of inspections and increase the mean time between major maintenance actions on serviceable structures. Prediction of structural damage and quantification of structural integrity are critical for safe and reliable operation of human-engineered complex systems. Fatigue damage is one of the most commonly encountered sources of structural degradation during both nominal and off-nominal operations of such systems [1]. Detection of fatigue damage at an early stage is essential because the accumulated damage could potentially cause catastrophic failures in the system, leading to loss of expensive equipment and human life [1]. Therefore, it is necessary to develop diagnosis and prognosis capabilities for reliable and safe operation of the system and for enhanced availability of its service life. In the current state-of-theart, direct measurements of fatigue damage at an early stage (e.g., crack initiation) are not feasible due to lack of appropriate sensing devices and analytical models. This chapter attempts to address this inadequacy by taking advantage of the sensitivity of the ultrasonic impedance on small changes that occur inside the material during the early stages of fatigue damage [2]. Since a vast majority of structural components that are prone to fatigue damage are made of ductile alloys, this chapter dwells on fatigue damage sensing and prediction for such materials. Sole reliance on model-based analysis for structural damage monitoring is infeasible because of the difficulty in achieving requisite accuracy in modeling of fatigue damage evolution. Many model-based techniques have been reported in recent literature for structural health monitoring, remaining life estimation and prediction of damage precursors [3][4][5][6][7][8]. Apparently, no existing model, solely based on the basic fundamental principles of physics [9], can adequately capture the dynamical behavior of fatigue damage at the grain level. In general, these models are critically dependent on the initial defects in the materials, which may randomly form crack nucleation sites and identification of exact

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Real-time Fatigue Damage Monitoring via In Situ Ultrasonic Sensing

5

initial and boundary conditions is not feasible. As such, these defects are difficult to identify and model [10][11]. Small deviations in the distribution of initial defects may produce large variations in the evolution of fatigue damage for (apparently) identical specimens under the same loading and environmental conditions [11]. Specifically in the short crack region the appearance of many crack nucleation sites can be treated as random events. This random distribution of micro-structural flaws produce a wide uncertainty in the crack initiation phase [1][11]. For example, inclusions, casting defects and machining marks originating during fabrication may cause stress augmentation at certain locations. These surface and sub-surface defects that are largely unavoidable constitute integral parts of the material microstructure of the operating machinery. In addition, fluctuations in usage patterns (e.g., random overloads) and environmental conditions (e.g., temperature and humidity) may adversely affect the performance and service life of mechanical systems leading to unanticipated failures. As such, evolution of fatigue damage is considered as a stochastic phenomenon [1][11][12]. A stochastic measure of fatigue crack growth is proposed in recent literature [13]. The stochastic phenomenon of fatigue damage evolution makes the maintenance efforts more conservative, difficult and expensive. With heavy usage and stringent safety requirements, as the machinery ages, the frequency of major maintenance increases and leads to premature replacements of the critical components. However, in general, both safety and economics suffer as no good compromise can be achieved without systematic analysis of the situation. This problem motivates the research, since one of the most fundamental solutions to this problem is on-line failure diagnosis as well as on-line prognosis that allows remaining life prediction for the critical structural components of operating machinery, under anticipated load profiles. The above discussion evinces the need for online updating of information using sensing devices (e.g., ultrasonics, acoustic emission and eddy currents) which are sensitive to small microstructural changes and can provide useful and reliable estimates of the anomalies at early stages of fatigue damage evolution [2]. This chapter presents real-time fatigue damage monitoring using the ultrasonic sensing technique to examine small microstructural changes in polycrystalline alloys during both fatigue crack initiation and propagation phases. Consequently, the analysis of time series data from available sensors is essential for monitoring the evolving fatigue damage in real time [14]. The theme of data-driven pattern recognition and anomaly detection, formulated in this chapter, is built upon the concepts of Symbolic Dynamics [15], Finite State Automata [16], Statistical Mechanics [17], and Information Theory as a means to qualitatively describe the dynamical behavior in terms of symbol sequences [18][19][20][15]. The chapter presents symbolic dynamic filtering (SDF ) [18][21][17][22][23] to analyze time series data of sensors (e.g., ultrasonic) for detection and identification of gradually evolving fatigue damage. To this end, a computer-controlled fatigue test apparatus, equipped with multiple sensing devices (e.g., ultrasonics and optical microscope), has been used to experimentally validate the concept of real-time fatigue damage monitoring. The sensor information is integrated with a software module consisting of the SDF algorithm for real-time monitoring of fatigue damage. Experiments have been conducted under different loading conditions on specimens constructed from the ductile aluminium alloy 7075 − T 6. This chapter is organized in eight sections and one appendix. Section 1. provides a brief

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Shalabh Gupta and Asok Ray

discussion of the problem statement of real-time fatigue damage monitoring using sensing devices such as the ultrasonics. Section 2. provides the background of fatigue damage sensing methods and motivation of the current research. Section 3. presents the details of experimental apparatus used for validating the concept of real-time fatigue damage monitoring. Section 4. formulates the problem of fatigue damage monitoring in the setting of two-time-scales and briefly describes the procedure of a recently reported data-driven pattern recognition tool, called symbolic dynamic filtering (SDF ). Section 5. provides an overview of the forward and the inverse problems of anomaly detection in complex dynamical systems. Section 6. presents the details of experimental procedure, SDF -based pattern recognition and results of real-time fatigue damage detection using ultrasonic sensing technology. Section 7. presents the solution procedure and results derived from both the forward and the inverse problems for real-time fatigue damage estimation. Section 8. summarizes and concludes the chapter with recommendations for future research. Appendix Appendix A. provides a brief overview of symbolic dynamic filtering (SDF ) including the concepts of symbolic dynamic encoding, wavelet based partitioning, probabilistic finite state machine construction and pattern identification.

2.

Fatigue Damage Sensing Techniques

Several techniques based on different sensing devices (e.g., ultrasonics, acoustic emission and eddy currents) have been proposed in recent literature for fatigue crack monitoring [24][25][26]. The capabilities of electrochemical sensors [27] and thermal imaging techniques [28] have also been investigated for structural failure analysis. A review of different vibration based damage detection and identification methods is provided by Doebling et. al [29]. This section presents a brief review of fatigue damage sensing methods as below.

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2.1.

Acoustic Emission

Acoustic emissions are the stress waves that are produced due to sudden redistribution of the stress inside the material structure. Some of the possible causes of the changes in the internal structure of the material can be dislocation movement, crack initiation and growth, and crack opening and closure. Since the primary sources of acoustic emissions are damagerelated, the detection and monitoring of these emissions are commonly used to predict and estimate material failure. As such, acoustic emission technique is commonly used to monitor defects and causes of failures in structural materials. This technique has been employed and tested for fault diagnosis in numerous applications including polycrystalline alloys, composite materials, and also in the study of mechanical behavior of ceramics and rocks. The traditional analysis methods using acoustic emission technique include monitoring the acoustic-emission counts, the peak levels, and the energy of the signal. These parameters are used for correlation with the defect formation mechanisms and for providing a quantified estimate of faults. Acoustic emission technique has been investigated by several researchers for early detection of fatigue and fracture failures of materials [30][31][32][33][34][35][36]. Acoustic emission technique has also been widely used for detection of faults or leakage in pressure vessels, tanks, and piping systems and for monitoring the welding and corrosion progress

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Real-time Fatigue Damage Monitoring via In Situ Ultrasonic Sensing

7

in materials. One of the advantages here is that acoustic emissions are sensitive to the activities occurring inside the material microstructure. Moreover, acoustic emission sensors are compact and can be easily mounted on the surface of a specimen being examined for online testing and for continuous monitoring of evolving damage. The major drawback of acoustic emission technique is that the acoustic emission signals are usually very weak and give poor performance in noisy environments where signal-noise separation becomes a difficult task.

2.2.

Eddy Currents

The other common sensing technique used for fault diagnosis in structural materials is the eddy current technique that is based on the principal of electromagnetism. Eddy currents are produced through a process called electromagnetic induction. When a source of alternating current is supplied to a conducting material, such as a copper wire, a magnetic field develops in and around the material. Eddy currents are induced electrical currents that are produced in another electrical conductor that is brought into a close proximity of this magnetic field. The presence of a crack or detriment in the material affects the flow pattern of the eddy current, which can be detected for prediction and estimation of the structural damage [37][38][39]. The advantages of eddy current inspection technique include sensitivity to small cracks and other defects, portability of sensor equipment, minimum part preparation, and non-contact evaluation. However, there are certain limitations of the eddy current inspection technique such as the depth of penetration is limited, only conductive materials can be inspected, and surface finish and roughness may interfere. Since eddy currents tend to concentrate at the surface of a material, they can only be used to detect surface and near surface defects.

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2.3.

Ultrasonics

Another common method for fault diagnosis in structural materials is using the ultrasonic sensing technique. The ultrasonic flaw detector functions by emitting high frequency ultrasonic pulses that travel through the specimen and return back through the receiver transducers. As with the propagation of any wave, it is possible that discontinuities in the propagation media will cause additive and destructive interference. Since material characteristics (e.g., voids, dislocations and short cracks) influence the ultrasonic impedance, a small fault in the specimen is likely to change the signature of the signal at the receiver end. Therefore, the signal can be used to capture some of the minute details and small changes during the early stages of fatigue damage, which may not be possible to detect by an optical microscope [2]. Ultrasonic sensing methodology has been effectively utilized for microstructural analysis in polycrystalline alloys to examine the fatigue phenomenon [40][41]. Impedance of the ultrasonic signals has been shown to be sensitive to small microstructural changes occurring during the early stages of fatigue damage [2][42][43][44]. Moreover, ultrasonic sensing is applicable to real-time applications and the sensing probes can be easily installed on the specimen. Ultrasonic sensing technique is also robust to noisy environments since the externally excited waves are of very high frequency and they do not interfere with small disturbances. As such, the research in this chapter is based on ultrasonic sensing technique to examine small microstructural changes during early stages of fatigue damage evolution [44][45][46].

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3.

Shalabh Gupta and Asok Ray

Fatigue Damage Test Apparatus

This section presents the description of an experimental apparatus that is designed to study the growth of fatigue damage in mechanical systems [47]. The main content of this section include the description of the fatigue damage testing apparatus that is equipped with different sensing devices for process control and real-time monitoring of fatigue damage. The primary objective of the fatigue test apparatus is to demonstrate online sensing and prediction of fatigue damage. As such, the requirements of the apparatus are: • Capability to operate under cyclic loading with multiple sources of input excitation • Provision of a failure site such that the damage accumulation takes place within a reasonable period of time in the laboratory environment with negligible damage to other components of the test apparatus • Capability of real-time data acquisition from appropriate sensing devices

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• Accommodation of online data analysis tools for monitoring the evolution of fatigue damage in real time. The experimental apparatus, shown in Figure 1, is a special-purpose uniaxial fatigue testing machine that operates on the hydraulic power supplied by a hydraulic pump device, which moves under load control at speeds up to 12.5 Hz; a detailed description of the apparatus and its design specifications are reported in [47]. The test apparatus is also connected to three computers dedicated for the tasks of data acquisition and control. The test specimens are subjected to tensile-tensile cyclic loading by a hydraulic cylinder under the regulation of computer-controlled electro-hydraulic servo-valves. The feedback signals that are generated from the load cell and the extensometer are processed by signal conditioners that include standard amplifiers and signal processing units. These signals are passed to the controller that governs the hydraulic servo-valve for operation under specified load and position limits. The image data of the specimen surface from the optical microscope and the sensor data from the ultrasonic transducers are passed to the data analysis and damage estimation subsystem. The information from the optical microscope is analyzed to determine the fatigue crack length on the specimen surface. Data sets from the ultrasonic sensors are analyzed using symbolic dynamic filtering (SDF ) algorithm for fatigue damage estimation even before the optical microscope detects a surface crack. A brief description of the associated computer hardware, process instrumentation and the control module of the fatigue test apparatus is provided below. • Subsystem for Closed Loop Servo-Hydraulic Unit and Controller: The instrumentation and control of the computer-controlled uniaxial fatigue test apparatus includes a load cell, an extensometer, an actuator, the hydraulic system, and the controller. The servo-hydraulic unit can excite the system with either random loads or random strains at variable amplitudes. The control module is installed on a computer which is dedicated to machine operation. The controller operates the machine according to a schedule file that contains the specifications of the loading profile and the number

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Figure 1. Fatigue Damage Test Apparatus.

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of load cycles for each type of test in the profile. The real time data from the extensometer and the load cell are supplied to the controller for operation under specified position and load limits. • Subsystem for data acquisition, signal processing, and engineering analysis: In addition to the computer for controlling the load frame, a second computer is used for real-time image data collection from the microscope to monitor the growth of surface cracks. The instrumentation for the ultrasonic flaw detection scheme is connected to a third computer. The real time ultrasonic data collected on this computer is transferred at regular intervals to a fourth computer on which the data analysis algorithm is installed. The algorithm based on symbolic dynamic filtering (SDF ) generates the information about fatigue damage in terms of anomaly measures at different time epochs and the corresponding plots are displayed on the screen in real time. These laboratory computers are interconnected by a local dedicated network for data acquisition, data communications, and control. An Ethernet network and an RS − 232 serial data line connect the computers. The main elements of the hydraulic pump system are a 3-phase induction motor driven pump, the oil supply manifold, and the cylinder. The purpose of the supply manifold is to properly sequence the system pressure and to accommodate the accumulators. The accumulators maintain the system pressure when the instantaneous demand from the servo valve/cylinder is higher than the flow rate available from the pump. A small solenoid valve is used under most shutdown conditions to bleed the accumulator pressure slowly. This

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Shalabh Gupta and Asok Ray

prevents a high-pressure spike from causing the return filters to fail. The hydraulic cylinder has a 6 inch (∼ 152 mm) bore and a 2.5 inch (∼ 64 mm) stroke length, and is double ended (i.e., the rod extends through seals at both ends of the cylinder). The software effectively has two threads of operation, the main program and the controller program. A detailed description of the software and the associated hardware is provided in [47]. The software programs for control and data acquisition are written in C++ programming language and are installed in the real-time Linux operating environment. The main program is the user space that enforces the test sequence. The controller is installed in the kernel space which is an Interrupt Service Routine (ISR) that generates the Direct Memory Access (DM A) completion interrupt signals. The analog-to-digital (A/D) board is initialized to take 20 readings per frame, which represents 10 readings each for 2 channels. These channels are connected to the load cell output and Linear Variable Differential Transformer (LV DT ) output of cylinder position. The DM A controller on the personal computer motherboard is programmed to read 20 single 16 bit words and store them sequentially in a given memory location for each transfer. It is also programmed to reload the initial address for the next transfer after each transfer of 20 readings is complete. When a reading is taken, the result is put into a First-in-First-out (F IF O) on the A/D board and a DM A request is issued. The DM A controller on the motherboard retrieves the data and stores it in system ram (in sequence). When the 20th reading is stored, the DM A controller asserts a signal that is looped back to an interrupt line by the A/D board. At this point, control is given to the Controller ISR. The controller sends 5 packets of data: a packet to transmit maximum load reading; a packet to transmit the minimum load reading; a packet to trigger an ultrasonic reading at low load, a packet to trigger video at high load and a packet to trigger ultrasonic readings at high load. The test apparatus is equipped with four different sensing devices including: a) A travelling optical microscope for monitoring surface cracks during fatigue damage evolution; b) An Ultrasonic flaw detector for detection of microstructural damage during early stages of fatigue damage;

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c) An Extensometer (LVDT) for position measurements; and d) A Load cell for load measurements.

3.1.

Geometry of the Test Specimens

The specimens used in the experimental apparatus are typical hourglass shaped flat plates that have a machined notch for a stress riser to guarantee crack propagation at the notch end. Specimens used in this study are made of 7075-T6 aluminium alloy. In this chapter, different specimens with either a center notch or a side notch geometry are used for fatigue experiments. These specimens with local stress concentration regions are designed to break in a reasonably short period of time to enhance the speed of the experiments. The geometries of the two different specimens are presented below:

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Figure 2. Cracked specimens: a) Left image- specimen with a center notch and b) Right image- specimen with a side notch.

• Specimens with center notch geometry: Figure 2 shows a typical center notched specimen (left image) used for testing in the fatigue damage test apparatus. This specimen has a notch at the center that is made to increase the stress concentration factor to ensure crack initiation and propagation at the notch ends. The specimens of this configuration are 3 mm thick and 50 mm wide, and have a slot of 1.58 mm × 4.5 mm at the center. • Specimens with side notch geometry: Figure 2 also shows a typical compact specimen (right image) with a notch on one side. The specimens of this configuration are 3 mm thick and 50 mm wide with a slot on one side of 1.58 mm diameter and 4.57 mm length.

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The test specimens are subjected to sinusoidal loading under tension-tension mode (i.e., with a constant positive offset) at a frequency of 12.5 Hz. The direct component (DC) offset is provided in the load cycling to ensure that the specimen is always under tension. Since inclusions and flaws are randomly distributed across the material small cracks appear at these defects and propagate and join at the machined surface of the notch even before microscopically visual cracks appear on the surface. Table 1 provides the material properties of 7075-T6 aluminium specimens.

3.2.

Sensors for Damage Detection

The fatigue damage testing apparatus is equipped with a variety of damage sensors [47]. Two types of sensors that have been primarily used for damage detection are: a) the travelling optical microscope and b) the ultrasonic flaw detector. 3.2.1.

Travelling Optical Microscope

The travelling optical microscope, shown as part of the test apparatus in Figure 1, provides direct measurements of the visible part of a crack. The primary instrument for measuring crack length is a Questar QM100 Step Zoom Long distance microscope. This microscope

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Table 1. Properties of 7075-T6 specimens Physical Property

Value

Tensile Strength, Ultimate

570 MPa

Tensile Strength, Yield

505 MPa

Modulus of Elasticity

72 GPa

Fatigue Strength

160 MPa

Poisson’s Ratio

0.33

Shear Modulus

26.9 GPa

Shear Strength

330 MPa

is mounted on a 3-axis stepper motor driven precision stage. The resolution of the optical microscope is about 2 microns at a working distance of 10 to 35 cm. The microscope can also be focused at different magnifications. The images are taken at a magnification of 75x. The long distance between the microscope and the specimen is a key feature, because it allows mounting of sensor probes without interference from the microscope optics. The microscope is used as the lens for a digital 8-bit monochrome progressive scan (non-interlaced) video camera that is capable of asynchronous operation (i.e., is not limited to a constant frame rate like a normal video camera). Since the crack tip moves out of the field of view of the microscope during the test, the motorized stage is used to move the microscope in coordination with the progress of the crack. The growth of surface crack is monitored continuously by the microscope which takes the images of the specimen surface at regular intervals. In order to take pictures, the controller slows down the machine to less than 5 Hz to obtain a better resolution of the images. In the experiments with a center notch geometry, the microscope shifts from left to right side of the central notch and vice versa after every 200 cycles to track crack growth on both sides of the notch. The data acquisition software also allows for manual operation and image capture at the desired moment. Figures 3 shows typical images of a broken specimen. The semi-circular region visible in Figure 3 is the notch. Two different stages of fatigue crack growth are shown: a) first appearance of a crack on the surface of the specimen and b) almost broken specimen with a very large crack growth. One observation from these figures indicates that the profile of crack growth is not always a straight line. The crack tip tends to propagate in the direction where there is the least resistance to crack growth. The crack length can be measured by moving the microscope with the help of a user friendly software interface such that a cursor superimposed on the microscope image is over the crack tip. The traveled distance is denoted as the crack length measured from the notch end. The operator records the initial position of the notch edge(s). As the crack propagates, the operator periodically moves the microscope until the cursor is over the crack tip, saves a microscope image, and saves the motion stage position, i.e. crack tip position. The saved

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Figure 3. Images of a specimen as captured by the optical microscope indicating the first appearance of a surface crack (left image) and a fully developed large crack (right image).

images are displayed in a separate window and it is possible to review the images during the test. The position of the crack tip is saved so that the motion controller can be used to quickly return to the crack tip location from anywhere in the range of motion of the stage. The position of all three stages is recorded so that the proper focus and elevation are maintained. The major weakness of the optical method is that the crack must be visible on the side of the specimen that is being observed, or for convenience, the front. One unfortunate circumstance is that a crack initiates on the back side of the specimen, and propagates through the specimen’s entire thickness. Furthermore, the microscope is unsuitable for most field applications.

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3.2.2.

Ultrasonic Flaw Detector

As a ductile alloy structure is commonly subjected to fatigue failure, a large portion of its service life is spent in crack initiation and in the presence of very small cracks. The objective of this research is to acquire knowledge of the damage evolution during the major part of service life at the crack initiation stage, and not simply at the crack propagation stage when the life is largely expended. There are very few methods for detection of microstructural flaws and extremely small cracks in ductile materials. There are fewer still that are suitable for installations outside of the laboratory in actual field operations. Ultrasonic flaw detection meets the requirements of real time damage sensing on structures in service. Ultrasonic flaw detectors are commonly used in the aerospace and nuclear power industries to detect flaws in structures, and have been used by many researchers for crack length measurements in laboratory environment [48]. The ultrasonic flaw detector functions by emitting high frequency ultrasonic pulses that travel through the specimen and return back through the receiver transducers. A piezoelectric transducer is used to inject ultrasonic waves in the specimen and an array of receiver transducers is placed on the other side of notch to measure the transmitted signal, as seen in Figure 4. A Matec TB1000 Gated Amplifier PC add-in card drives a piezoelectric transducer with a sine wave with amplitude of 300V. To be more explicit, the excitation signal consists of short bursts of a sine wave of constant amplitude interrupted by relatively long periods of inactivity at 0V. The wedges, used for the transducers in the tests, have a high enough slope angle that the signal takes multiple paths through the test article and reach the pickup transducers. In the experiments for central notched specimens, an array of 2 receiver transducers is

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Figure 4. Ultrasonic flaw detection scheme.

placed below the notch to detect faults on both left and right side of the notch. In case of the compact specimens, a single receiver transducer is placed under the notch to detect fatigue damage at the notch tip. The ultrasonic waves are generated as 10MHz sine wave signals. The ultrasonic system is synchronized with the load cycling such that the waves are emitted during a very short portion at the peak of every load cycle (∼12.5Hz). Ultrasonic measurements are taken at stress levels that exceeded the crack opening stress. The crack is open when the specimen is under maximum stress at the peak of a load cycle and this causes maximum attenuation of the ultrasonic waves. Note that if the crack closure occurs at low loads, then an alternative method would be needed to detect anomalies. The sender and receiver ultrasonic transducers are placed on two positions, above and below the central notch, so as to send the signal through the region of crack propagation and receive it on the other side, as seen in Figure 4. As with the propagation of any wave, it is possible that discontinuities in the propagation media will cause additive and destructive interference. Since material characteristics (e.g., voids, dislocations and short cracks) influence the ultrasonic impedance, a small fault in the specimen is likely to change the signature of the signal at the receiver end. Therefore, the signal can be used to capture some of the minute details and small changes during the early stages of fatigue damage, which may not be possible to detect by an optical microscope [2]. Prior to the appearance of a crack on the surface of the specimen as detected by the optical microscope, deformations (e.g., dislocations and short cracks) inside the specimen cause detectable attenuation and/or distortion of the ultrasonic waves [42]. Recent literature has also shown nonlinear modelling approaches of the ultrasonic wave interference with the material micro-structures [49] [50]. An elaborate description of the properties of ultrasonic waves in solid media is provided by Rose [51]. It is observed that cracks always start at the stress-concentrated region near the notch but the exact site of crack nucleation can be treated as a random event. Formation of very small cracks is difficult to detect and model due to large material irregularities. The ultrasonic technique is easy to install at a potential damage site and is capable of detecting incipient fatigue damage before the onset of widespread fatigue crack propagation. In contrast, an optical microscope is only capable of detecting cracks when they appear on the front surface of the specimen. The ultrasonic instrument is more effective than optical microscopy in measuring the condition of the specimen, since the microscope can only capture the condition on one face of the specimen and the ultrasonic measurements are affected throughout the cross section of the crack. This is particularly the case when the crack is small, because then the 2-D geometry of the crack is not well represented by a measurement on the surface. The study in this chapter is based on analyzing the ultrasonic data for monitoring fatigue

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Figure 5. Crack surface showing the macro stages of crack growth [47].

damage during both crack initiation and crack propagation stages.

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3.3.

Progression of Fatigue Damage

This section discusses the progression of crack growth as observed in the specimens designed for this study [47]. Although it may seem formulistic for practitioners in fracture mechanics, six stages of crack growth are considered in this research. The first stage is crack initiation. The second stage is two-dimensional (2D) crack growth under plane strain. The third stage is the propagation of a through crack under plane strain. The fourth stage of fatigue growth is a transition from plane strain to plane stress crack growth. The fifth stage of fatigue growth is pure plane stress crack growth. The final stage is rupture, where there is so little residual strength left that the remaining ligament may fail in only a few stress cycles. Figure 5 shows the crack surface of a specimen with different macro stages of crack growth. The crack initiation stage occurs prior to the formation of a well defined crack and involves mechanisms at a microstructural level. Optical microscopy does not adequately resolve the features of this stage of crack formation, most of which are too small and are not clearly observable on the surface of the specimen. Thus, there is no way to correlate the progression of crack initiation with measurements from other instruments. The other problem is a lack of usable models. Strain-Life is commonly used to model initiation. Strain-life models essentially allocate the observed period of initiation in a fixed pattern. However, initiation is highly uncertain and therefore the models should have a stochastic structure. Thus, however good an analytical model is, it may not be adequate to predict the crack initiation behavior of any given specimen. Since inclusion and flaws are randomly distributed throughout the material, small cracks that form at these defects propagate and join on the machined surface of the notch for a considerable number of cycles before even microscopically small cracks appear on the surface. Figure 6 shows the origin of several small cracks at the edge of the notch as indicated by blue regions. The image is taken by a surface interferometer from ZY GO that provides noncontact three-dimensional quantitative surface topography measurement of the specimen. Using ZYGO’s technology of phase-shifting interferometry, the interferometer measures a wide range of surfaces and provides precise 3D

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Figure 6. Three dimensional surface profile of a broken specimen. The notch edge is indicated in the figure. The image is taken by a surface interferometer from ZYGO. The colors shown in the legend indicate different depth levels on the surface. As seen in the figure, there are probably three different regions where small cracks developed near the notch.

profiling and is ideally suited for high resolution measurements on various smooth surfaces. The 2D stage of crack growth is important for a number of reasons as it is a large source of variability in the life of a structure. It is possible that 2D cracks have approximately equal length through the thickness and along the width of the specimen. Thus, it might be possible that the 2D stage will last until the surface crack length is equal to the thickness of the specimen. There are three major scenarios for the 2D crack progression on a given side of the notch. The first scenario is that a corner crack forms on both corners of the notch at approximately the same time. These cracks grow independently in roughly semicircular shape until they join in the middle. The relatively small ligament left between the cracks cracks very quickly forming a through crack. This is ideal because the visible portion of the crack on either surface of the specimen is a very good measure of the progress of crack growth for the entire test. A somewhat less observed scenario is that a penny shaped crack forms in the center of the notch and grow outwards to the edge of the specimen. This phenomenon may result in fairly large crack growth in the center of the specimen before it propagates outward to the surface. However, it is very likely that surface cracks form and join the interior crack before an interior crack propagates to the surface. The most problematic scenario is that a corner crack forms on one surface and propagates through to the other side of the specimen. This results in a crack that is longer on one surface than the other for most part of the test. In any given test, a mixture of these scenarios will undoubtedly occur. Once one side of the notch has cracked through from one surface to another the third stage of crack growth is started. In center notch specimens, it is extremely unlikely that the crack on one side of the notch will form a through crack at the same time as the other side of the notch. When the cracking of both sides of the notch has formed a through crack, the total crack length has been observed to grow as a center cracked specimen. For a large majority of specimens when the second through crack is formed, it grows much more quickly than the other side. Assuming reasonable stress levels, the bulk of the specimen is in a state of plane strain

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through the stage where a through center-crack has formed for a considerable portion of the propagation afterwards. The material at the surfaces of the specimen is always in plane stress since it is not constrained in the through-thickness direction. For this stage of crack growth, any plasticity in the bulk of the material is constant through the thickness of the specimen as the material is constrained by the material around it. As the localized stress becomes high enough, the stress state of the specimen undergoes a transition from plane strain to plane stress. This is evidenced by the formation of angled area on the surface of the specimen. As this stage progresses, the angled areas grow larger, and the area in the center of the specimen that is still in plane strain gets smaller and eventually disappears. This transition is observable on the surface, since the crack grows in a reasonable facsimile to a straight line when it is in plane strain. Observing the specimen after it has failed all the way through, it can be seen that the plane strain area in the center of the transition from plane strain to plane stress remains in plane with the crack that formed fully in plane strain. Thus, when the transition from plane strain to plane stress occurs, the crack on the surface starts to move up or down on the surface of the specimen. The final stage of crack growth is rupture, which is a failure of the remaining ligaments in a few cycles.

4.

Problem Formulation

This section presents the problem formulation for pattern recognition and anomaly detection based on symbolic dynamic filtering (SDF ) in complex dynamical systems. The underlying concepts and essential features of SDF [21][17] are presented in the appendix.

4.1.

Concept of Two Time Scales for Damage Monitoring

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Fatigue damage detection is formulated as a two-time-scale problem as explained below. • The fast time scale is related to the response time of process dynamics. Over the span of a given time series data sequence, the behavioral statistics of the system are assumed to remain invariant, i.e., the process is assumed to have statistically stationary dynamics at the fast time scale. In other words, statistical variations in the internal dynamics of the system are assumed to be negligible on the fast time scale. • The slow time scale is related to the time span over which the process may exhibit non-stationary dynamics due to (possible) evolution of anomalies. Thus, an observable non-stationary behavior can be associated with anomalies evolving at a slow time scale. A pictorial view of the two time scales is presented in Figure 7. In general, a long time span in the fast time scale is a tiny (i.e., several orders of magnitude smaller) interval in the slow time scale. For example, fatigue damage evolves on a slow time scale, possibly in the order of months or years, in machinery structures that are operated in the fast time scale approximately in the order of seconds or minutes. Hence, the behavior pattern of fatigue damage is essentially invariant on the fast time scale. Nevertheless, the notion of fast and

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Figure 7. Pictorial view of the two time scales: 1) slow time scale where anomalies evolve and 2) fast time scale where data acquisition is done.

slow time scales is dependent on the specific application, loading conditions and operating environment. As such, from the perspective of fatigue damage monitoring, the sensor data acquisition is done on the fast time scale at different slow time epochs separated by uniform or non-uniform intervals on the slow time scale.

4.2.

Methodology

The problem of fatigue damage monitoring is formulated to achieve the following objectives: • Information-based identification of damage progression patterns - The possible sources of information can include time series data of appropriate sensors (e.g., ultrasonic) mounted on critical components of the system; • Real-time execution - The analytical tools must be computationally efficient and have the capability of real-time execution on commercially available inexpensive platforms;

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• Capability of small change detection - The pattern recognition methodology for anomaly detection must be sensitive to small changes and have the capability of providing early warnings of incipient faults. The methodology must also be capable of estimating fault precursors to formulate a decision and control policy for damage mitigation and life extension; • Robustness to measurement noise and disturbances - The pattern recognition tool must be robust to noise and disturbances and must have low probability of false alarms. Once the appropriate sensor selection is done, the next task is development of analytical tools for analysis of time series data [14]. Various signal processing applications deal with data analysis and attempts have been made to extract maximum useful information from the ensemble of sensor data. The problem of feature extraction from time series data for damage monitoring has been recently addressed by many researchers [52] [53] [54]. The tools of statistical pattern recognition, auto-regressive model analysis, and wavelet analysis were applied to classify faults by different data patterns. However, the critical issue of early

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detection of gradually evolving faults in a real time setting were not addressed. Moreover, no quantifying measure was provided for damage accumulation and growth rate based on statistical information. Recently, some techniques of nonlinear dynamics have also been applied for damage monitoring [55] [56] [57], which are primarily based on the concepts of attractor-based cross-prediction error between the measured signal and its predicted value. However, since the dimensions of the phase space may grow unbounded for noisy data, this analysis could be computationally expensive and infeasible for real-time applications. Furthermore, dealing with high dimensions might lead to spurious results and dimension reduction may lead to loss of vital information. To alleviate these difficulties, this chapter has adopted a novel method of wavelet-based partitioning [21] [58]. Based on this partitioning, the pertinent information is extracted from time series data sets in the form of probability distributions. Slight deviations in these distributions from that under the nominal condition is captured to identify the damage pattern. This chapter presents symbolic dynamic filtering (SDF ) [21][17][22][23] to analyze time series data of sensors (e.g., ultrasonic) for detection of precursors leading to crack initiation and eventual widespread fatigue.

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The core concept of SDF is based on appropriate phase-space partitioning of the dynamical system to yield an alphabet to obtain symbol sequences from time series data [59]. The time series data of appropriate sensors (e.g., ultrasonic) are processed and subsequently converted from the domain of real numbers into the domain of (discrete) symbols [15][18][17]. The resulting symbol sequence is a transform of the original time series sequence such that the loss of information is minimized in the sense of maximized entropy. The chapter has adopted wavelet-based partitioning approach for symbol sequence generation [21] [58]. Wavelet based partitioning approach is robust and is particularly effective with noisy data [58]. Subsequently, tools of Computational Mechanics [21][17][60][61] are used to identify statistical patterns in these symbolic sequences through construction of a (probabilistic) finite-state machine [21][16]. Transition probability matrices of the finite state machines, obtained from the symbol sequences, capture the pattern of the system behavior by means of information compression. For anomaly detection, it suffices that a detectable change in the pattern represents a deviation of the nominal pattern from an anomalous one. The state probability vectors, which are derived from the respective state transition matrices under the nominal and an anomalous condition, yield a statistical pattern of the anomaly. The concept of SDF is illustrated in Figure 8. Symbolic dynamic filtering (SDF ) for anomaly detection is an information-theoretic pattern recognition tool that is built upon a fixed-structure, fixed-order Markov chain, called the D-Markov machine [21][17]. Recent literature [44] [62] has reported experimental validation of SDF -based pattern recognition by comparison with other existing techniques such as Principal Component Analysis (P CA) and Artificial Neural Networks (AN N ); SDF has been shown to yield superior performance in terms of early detection of anomalies, robustness to noise [58], and real-time execution in different applications such as electronic circuits [62], mechanical vibration systems [63], and fatigue damage in polycrystalline alloys [44].

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Shalabh Gupta and Asok Ray

α

8

χ δ

6

Phase Trajectory

4 2

ε

γ

η

0.5 0 -0.5

-1 -1

-0.5

0

Re fe re nc e Dist ribut ion

0

1

2

β

γ

3

State Probability Histogram

2 0.5

η

φ

0

φ

0 1

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Symbol Sequence ……φ χ γ η δ α δ χ……

β

10

3

α δ

1

χ, ε

1

Finite State Machine (Hidden Markov Model)

0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00

Curre nt Dist ribut ion

0

1

2

3

State Probability Histogram

Figure 8. Conceptual view of symbolic dynamic filtering.

4.3.

Procedure for Anomaly Detection

The SDF -based anomaly detection requires the following steps:

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• Time series data acquisition on the fast time scale from appropriate sensors - Collection of data sets is done at different slow time epochs. As stated in the previous subsection, the choice of time scales is dependent on the application and requires an approximate a priori knowledge about the time period of evolution of anomalies (e.g., fatigue crack growth). • Transformation of time series data from the continuous domain to the symbolic domain - This is done by partitioning the data (e.g., ultrasonic) into finitely many discrete regions to generate symbol sequences at different slow time epochs [15][18]. The chapter has presented a wavelet-based partitioning scheme for symbol sequence generation. • Construction of a finite state machine - The machine is constructed from the symbol sequence generated at the nominal condition • Calculation of the pattern vectors at different slow time epochs - The elements of these pattern vectors consist of the visiting frequencies of the finite state machine states

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• Identification of behavioral changes- Fatigue damage detection is based on the information derived from the evolution of the pattern vector at different slow time epochs with respect to the one at the nominal condition

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5.

Forward and Inverse Problems

Fatigue damage monitoring is formulated as a solution of two interrelated problems [21]: (i) forward problem of Pattern Recognition for (offline) characterization of the anomalous behavior, relative to the nominal behavior and (ii) inverse problem of Pattern identification for (online) estimation of parametric or non-parametric changes based on the knowledge assimilated in the forward problem and the observed time series data of quasi-stationary process response. The forward problem consists of prediction of outcomes, given a priori knowledge of the underlying model parameters. In absence of an existing model this problem requires generation of behavioral patterns of the system evolution through off-line analysis of an ensemble of the observed time series data. The objective of the inverse problem is to infer the anomalies and to provide the estimates of the remaining useful life from the observed time series data in real time based on the information generated during the forward problem. Inverse problems arise in different engineering disciplines such as geophysics, structural health monitoring, weather forecasting, and astronomy. Inverse problems often become ill-posed and challenging due to the following reasons: (a) high dimensionality of the parameter space under investigation and (b) in absence of a unique solution where change in multiple parameters can lead to the same observations. That is, it may not always be possible to identify a unique anomaly pattern based on the observed behavior of the dynamical system. Nevertheless, the feasible range of parameter variation estimates can be narrowed down from the intersection of the information generated from inverse images of the responses under several stimuli. In presence of sources of uncertainties, any parameter inference strategy requires estimation of parameter values and also the associated confidence intervals, or the error bounds, to the estimated values. As such, inverse problems are usually solved using the Bayesian methods that allow observation based inference of parameters and provide a probabilistic description of the uncertainty of inferred quantities. A good discussion of inverse problems is presented by Tarantola [64]. The algorithms of SDF can be implemented to solve both these problems. In context of fatigue damage monitoring, the tasks and solution steps of these two problems as followed in this chapter are discussed below.

5.1.

Forward Problem

The primary objective of the forward problem is identification of changes in the behavioral patterns of system dynamics due to evolving anomalies on the slow time scale. Specifically, the forward problem aims at detecting the deviations in the statistical patterns in the time series data, generated at different time epochs in the slow time scale, from the nominal behavior pattern. The solution procedure of the forward problem requires the following steps:

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Shalabh Gupta and Asok Ray

F1. Collection of time series data sets (at fast time scale) from the available sensor(s) at different slow time epochs; F2. Analysis of these data sets using the SDF method as discussed in earlier sections to generate pattern vectors defined by the probability distributions at the corresponding slow time epochs. The profile of anomaly measure (see Appendix A.1.) is then obtained from the evolution of this pattern vector from the nominal condition; F3. Generation of a family of such profiles from multiple experiments performed under identical conditions to construct a statistical pattern of damage growth. Such a family represents the uncertainty in the evolution of fatigue damage due to its stochastic nature. The uncertainty arises from the random distribution of microstructural flaws in the body of the component leading to a stochastic behavior [11].

5.2.

Inverse Problem

The objective of the inverse problem is to infer the anomalies and to provide estimates of system parameters from the observed time series data and system response in real time. The decisions are based on the information derived in the forward problem. For eg., in the context of fatigue damage, identical structures operated under identical loading and environmental conditions show different trends in the evolution of fatigue due to surface and sub-surface material uncertainties. Therefore, as a precursor to the solution of the inverse problem, generation of an ensemble of data sets is required during the forward problem for multiple fatigue tests conducted under identical operating conditions. Damage estimates can be obtained at any particular instant in a real-time experiment with certain confidence intervals using the information derived from the ensemble of data sets of damage evolution generated in the forward problem [21]. The solution procedure of the inverse problem requires the following steps:

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I1. Collection of time series data sets (in the fast time scale) from the available sensor(s) at different slow time epochs up till the current time epoch in a real-time experiment as in step F1 of the forward problem; I2. Analysis of these data sets using the SDF method to generate pattern vectors defined by probability distributions at the corresponding slow time epochs. The value of anomaly measure at the current time epoch is then calculated from the evolution of this pattern vector from the nominal condition (see Appendix A.1.). The procedure is similar to the step F2 of the forward problem. As such, the information available at any particular instant in a real-time experiment is the value of the anomaly measure calculated at that particular instant; I3. Detection, identification and estimation of an anomaly (if any) based on the computed anomaly measure and the statistical information derived in step F3 of the forward problem. A schematic of the overall framework for the fatigue damage monitoring problem in mechanical systems is shown in Figure 9. As shown in Figure 9, the forward problem section

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Figure 9. Framework of SDF based fatigue damage detection in mechanical systems. involves the generation of ultrasonic data sets from fatigue experiments which are analyzed using the SDF method to produce a profile of the anomaly measure (see Appendix) that represents the evolution of fatigue damage. Following the same procedure, several experiments are conducted under identical conditions to generate a family of anomaly measure profiles. Such a family represents the stochastic behavior of the fatigue damage evolution on a slow time scale (see Section 7.1.). This family of anomaly measure profiles is analyzed in the inverse problem section to generate the requisite statistical information (see Section 7.2.). The information available in real time is the value of the anomaly measure obtained from the analysis of ultrasonic data at any particular time epoch. This information is entered in the inverse problem section that provides the estimates of the expended life fraction. The estimates can only be obtained within certain bounds at a particular confidence level. The online statistical information of the damage status is significant because it can facilitate early scheduling for the maintenance or repair of critical components or to prepare an advance itinerary of the damaged parts. The information can also be used to design control policies for damage mitigation and life extension.

6.

Real-time Fatigue Damage Detection

The fatigue tests were conducted using center notched specimens, made of the Aluminum alloy 7075-T6, at a constant amplitude sinusoidal load for a low-cycle fatigue, where the maximum and minimum loads were kept constant at 87MPa and 4.85MPa. A significant amount of internal damage occurs before the crack appears on the surface of the specimen when it is observed by the microscope [65]. However, it is also possible that the crack appears on the other surface of specimen or on the surface of the notch.

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Shalabh Gupta and Asok Ray

This surface or sub-surface damage caused by multiple small cracks and microstructural damage affects the ultrasonic waves when they pass through the region where these faults have developed. This phenomenon causes signal distortion and attenuation at the receiver end. The crack propagation stage starts when this internal damage eventually develops into a single large crack. Subsequently, the crack growth rate increases rapidly and when the crack is sufficiently large, complete attenuation of the transmitted ultrasonic signal occurs, as seen at the receiver end. This sudden sharp change in the rate of progression of fatigue damage is clearly visible after the crack appears on the surface. The rapid change in the statistical patterns of the ultrasonic data also indicate the onset of crack propagation. After the crack appears on the surface, fatigue damage growth can be easily monitored by the microscope but the ultrasonics provide early warnings even during the crack initiation phase.

6.1.

Experimental Procedure

The ultrasonic sensing device is triggered at a frequency of 5 MHz at each peak of the (∼12.5 Hz) sinusoidal load. The slow time epochs were chosen to be 3000 load cycles (i.e., ∼240 sec) apart. At the onset of each slow time epoch, the ultrasonic data points were collected on the fast time scale of ∼8 sec, which produced a string of 10,000 data points. It is assumed that during this fast time scale, no major changes occurred in the fatigue crack behavior. The nominal condition at the slow time epoch t0 was chosen to be 5.0 kilocycles to ensure that the electro-hydraulic system of the test apparatus had come to a steady state and that no significant damage occurred till that point. The anomalies at subsequent slowtime epochs, t1 , t2 , ....tk ..., were then calculated with respect to the nominal condition at t0 .

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6.2.

Data Analysis using Symbolic Dynamic Filtering (SDF )

Following the SDF procedure for anomaly detection, the alphabet size for partitioning was chosen to be |Σ| = 8 and window length of D = 1, while the mother wavelet chosen to be ‘gaus2’ [66]. (Absolute values of the wavelet scale series data were used to generate the partition because of the symmetry of the data sets about their mean.) The wavelet basis, ‘gaus2’, provides better results than the wavelet bases of the Daubechies family [67] because the ‘gaus2’ wavelet base closely matches the shape of the ultrasonic signals [58]. This combination of parameters was capable of capturing the anomalies earlier than the optical microscope. Increasing the value of |Σ| further did not improve the results and increasing the value of D created a large number of states of the finite state machine, many of them having very small or zero probabilities, and required larger number of data points at each time epoch to stabilize the state probability vectors. State probability vector p0 was obtained at the nominal condition of time epoch t0 and the state probability vectors p1 , p2 , . . . pk .... were obtained at other slow-time epochs t1 , t2 , . . . tk ..... It is emphasized that the anomaly measure is relative to the nominal condition which is fixed in advance and should not be confused with the actual damage at an absolute level.

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6.3.

25

Results and Discussion

The six triplets of plates in Figure 10 show two-dimensional images of a specimen surface, ultrasonic data and histograms of probability distribution of automaton states at six different time epochs, approximately 5, 30, 40, 45, 60 and 78 kilocycles, exhibiting gradual evolution of fatigue damage [68]. In each triplet of plates from (a) to (f) in Figure 10, the top plate exhibits the surface image of the test specimen as seen by the optical microscope. As exhibited on the top plates, the crack originated and developed on the right side of the notch at the center. Histograms in the bottom plates of six plate triplets in Figure 10 show the evolution of the state probability vector corresponding to fatigue damage growth on the test specimen at different slow time epochs, signifying how the probability distribution gradually changes from uniform distribution (i.e., minimal information) to delta distribution (i.e., maximum information). The middle plates show the ultrasonic time series data collected at corresponding slow time epochs. As seen in Figure 10, the visual inspection of the ultrasonic data does not reveal much information during early stages of fatigue damage but the statistical changes are captured in the corresponding histograms. The top plate in plate triplet (a) of Figure 10 shows the image at the nominal condition (∼5 kilocycles) when the anomaly measure is taken to be zero, which is considered as the reference point with the available information on potential damage being minimal. This is reflected in the uniform distribution (i.e., maximum entropy) as seen from the histogram at the bottom plate of plate pair (a). Both the top plates in plate triplets (b) and (c) at ∼30 and ∼40 kilocycles, respectively, do not yet have any indication of surface crack although the corresponding bottom plates do exhibit deviations from the uniform probability distribution. This is an evidence that the analytical measurements, based on ultrasonic sensor data, produce damage information during crack initiation, which is not available from the corresponding optical images. The top plate in plate triplet (d) of Figure 10 at ∼45 kilocycles exhibits the first noticeable appearance of a ∼300 micron crack on the specimen surface, which may be considered as the boundary of the crack initiation and propagation phases. This small surface crack indicates that a significant portion of the crack or multiple small cracks might have already developed underneath the surface before they started spreading on the surface. The histogram of probability distribution in the corresponding bottom plate shows further deviation from the uniform distribution at ∼5 kilocycles. The top plate in plate triplet (e) of Figure 10 at ∼60 kilocycles exhibits a fully developed crack in its propagation phase. The corresponding bottom plate shows the histogram of the probability distribution that is significantly different from those in earlier cycles in plate triplets (a) to (d), indicating further gain in the information on crack damage. In this case, the middle plate also shows significant drop in the amplitude of ultrasonic signals due to development of a large crack. The top plate in plate triplet (f) of Figure 10 at ∼78 kilocycles exhibits the image of a completely broken specimen. The corresponding bottom plate shows delta distribution indicating complete information on crack damage. The middle plate shows a complete attenuation of the ultrasonic signals. The normalized anomaly measure curve in Figure 11 shows a possible bifurcation where the slope of the anomaly measure changes dramatically indicating the onset of crack propagation phase. First appearance of a fatigue crack on the surface of the specimen was

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Figure 10. Pictorial view of the evolving fatigue crack damage, corresponding ultrasonic data and histograms of probability distribution [68].

detected by the optical microscope at approximately 45 kilocycles, which is marked by the dashed vertical line in Figure 11. The slope of the anomaly measure represents the anomaly growth rate while the magnitude indicates the changes that have occurred relative to the nominal condition. An abrupt change in the slope (i.e., a sharp change in the curvature) of anomaly measure profile provides a clear insight into a forthcoming failure. The critical information lies in the region to the left of the vertical line where no crack was visible on the

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1

Normalized Anomaly Measure

0.9 0.8 0.7 0.6 0.5 0.4

First Crack Detection by Microscope

0.3 0.2 0.1 0 0

10

20

30

40

50

60

70

80

Load Kilocycles

Figure 11. Fatigue damage detection.

surface. The slope of anomaly measure curve showed a clear trend of growth of anomaly right after ∼15 kilocycles. This was the region where multiple small cracks were possibly formed inside the specimen, which caused small changes in the ultrasonic signal profile. Fatigue damage detection using SDF of ultrasonic data has been successfully implemented in real time [44].

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6.4.

Real-time Implementation

Fatigue damage detection vis symbolic dynamic filtering (SDF ) has been successfully implemented in real time. The nominal condition is chosen after the start of the experiment at time epoch t0 , when the system attains a steady state and is considered to be in the healthy condition with zero anomaly measure. The function module for SDF is triggered at this point. The D-Markov machine states are fixed in advance using a priori determined values of the parameters: alphabet size |Σ| and window length D. The tasks of wavelet space partitioning and D-Markov machine construction are performed based on the time series data at the slow-time epoch t0 (nominal condition). The state probability vector p0 at time epoch t0 is stored for computation of anomaly measures at subsequent slow time epochs, t1 , t2 , ..., tk , .., which are chosen to be separated by uniform intervals of time in these experiments. The ultrasonic data acquisition software has a subroutine that writes the time series data of ultrasonic signals into text files at time epochs t0 , t1 , ..., tk , ... These text files are then transferred to the function module for anomaly detection such that the ST SA-based algorithm can read the data from the text files to calculate the anomaly measure at the specified time epochs. The algorithm is computationally very fast (i.e., several orders of magnitude faster relative to slow-time-scale damage monitoring) and the results can be easily plotted on the screen such that the evolution of anomaly measure is exhibited in real time. The plot is updated with the most recent value of anomaly measure at each (slow-time) epoch. Thus, the SDF algorithm enables on-line health monitoring and is capable of issuing early warnings of incipient failures.

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7.

Real-time Estimation of Remaining Useful Life

The analytical tool for fatigue damage estimation is based on an ensemble of stochastic data of ultrasonic signals that is generated from a set of identical experiments [46]. The stochastic data represents the behavioral pattern of fatigue damage evolution. A statistical analysis procedure of this stochastic data is developed to obtain the estimates of the fraction of used fatigue life that in turn can provide the estimate of the remaining useful life in real time. A set of experiments have been conducted and the results are provided for application and validation of the proposed statistical approach. As discussed in Section 5., the fatigue damage monitoring problem including damage detection and estimation of remaining useful life, is partitioned into two sub-problems: 1) the forward (or analysis) problem and 2) the inverse (or synthesis) problem. The previous section presented the results of real-time fatigue damage detection that only requires detection of any deviation in the statistical patterns of ultrasonic data from the nominal condition. However, for real-time estimation of the remaining useful life a complete solution of both the forward and the inverse problems is required. This section presents the results of realtime fatigue life estimation.

7.1.

Solution of the Forward Problem

This subsection presents a detailed description of the solution procedure of the forward problem. As discussed earlier, the primary objective of the forward problem is to identify the behavioral pattern of damage evolution in a complex dynamical system involving the uncertainties (if any) which can be both parametric or non-parametric in nature. 7.1.1.

Sources of Uncertainties

In case of fatigue damage, the sources of uncertainties include: a) material inhomogeneities such as voids or inclusions,

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b) surface defects including finishing marks that usually develop from the machining process, nonuniform polishing and other deformities, c) sub-surface defects originating due to random distribution of microstructural flaws like dislocations and grain boundaries, d) Variations in the critical dimensions of the components resulting from the non-zero tolerances of the cutting tools used in the fabrication process, e) small fluctuations in the environmental conditions such as humidity and temperature, f) small fluctuations in the operating conditions due to noisy environment and finite precision of the mechanical system. In the presence of above uncertainties, a complete solution of anomaly detection problem cannot be obtained in the deterministic setting because the profile of anomaly progression would not be identical for similarly manufactured components. In that case, the

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problem can be represented in the stochastic setting, where a family of anomaly progression profiles are generated from multiple experiments conducted under identical conditions [69]. As such, the requirement of the forward problem is to generate a pattern that consists of a family of anomaly progression profiles. Each member of this family represents the anomaly measure profile of a particular sample. This profile is generated from a fatigue test that is conducted to observe the entire service life of the specimen from the healthy condition to the eventual failure. 7.1.2.

Experimental Procedure

The fatigue tests were conducted on 7075-T6 aluminum specimens at 12.5 Hz frequency. In this case the compact specimens were used. The specimen were subjected to a sinusoidal load cycling where the maximum and minimum loads are 89.3MPa and 4.85MPa at the nominal condition. Ultrasonic waves with a frequency of 10 MHz are triggered at the peak of each sinusoidal load cycle where the stress is maximum and the crack is open causing maximum attenuation of the ultrasonic waves. Since the ultrasonic frequency is much higher than the load cycling frequency, data collection is performed for a very short interval in the time scale of load cycling. The slow time epochs have been chosen to be 1000 load cycles (i.e., ∼80 sec) apart. At the onset of each slow time epoch, the ultrasonic data points are collected on the fast time scale of 50 cycles (i.e., ∼4 sec), which produced a string of N = 15, 000 data points. It is assumed that during the fast time scale of 50 cycles, the system remains in a stationary condition and no major changes occur in the fatigue damage behavior. These sets of time series data points collected at different slow time epochs are analyzed using the SDF method (see Appendix) to calculate the anomaly measures at those slow time epochs.

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7.1.3.

Data Analysis using Symbolic Dynamic Filtering (SDF )

The nominal condition at the slow time epoch t0 is chosen to be ∼ 0.5 kilocycles to ensure that the electro-hydraulic system of the test apparatus had come to a steady state and it is assumed that no significant damage occurred till that point. This nominal condition is chosen as a benchmark where the anomaly measure is chosen to be zero. The anomalies at subsequent slow time epochs, t1 , t2 , ...tk .., are then calculated using SDF to yield a profile of anomaly measure representing the progression of fatigue damage on the slow time scale. The data collection is stopped at a time epoch tf considered as the final failure point where the ultrasonic energy is attenuated to 2% of the nominal condition. The energy of the signal is defined as: E=

N X

|s(i)|2

i=1

where |s(i)| is the magnitude of the ith data point of the ultrasonic signal. Once the failure point is reached the specimen is already under crack propagation stage and a sufficiently large crack has developed such that it is no longer useful and is considered as broken. Following the above procedure, a family of profiles is generated for multiple experiments conducted under identical experimental conditions.

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For the SDF procedure, the alphabet size for partitioning has been chosen to be |Σ| = 8 and window length of D = 1, while the wavelet basis is chosen to be ‘gaus2’ [66]. The algorithm is readily implemented in real-time and is computationally very fast in the sense that the code execution time is several orders of magnitude smaller than the interval between two adjacent slow time epochs. 7.1.4.

Generation of Statistical Patterns

Similar to the procedure described above, ultrasonic time series data are generated under both nominal and anomalous conditions at different slow time epochs for multiple experiments conducted on identically manufactured specimens under identical experimental conditions. SDF based analysis of the data from each of these experiments produce a profile of anomaly measure thereby generating an ensemble of anomaly measure profiles for multiple experiments. This family of profiles represents a stochastic pattern of the progression of fatigue damage under identical experimental conditions. To this effect, ℓ = 40 experiments have been conducted and the profiles of anomaly measures are shown in Figure 12. The family of the anomaly measure profilesof these  experiments is plotted versus a normalized t−t0 variable, expended life fraction, τe = tf −t0 , where t is the actual number of cycles, t0 is the nominal condition chosen to be ∼ 0.5 kilocycles for each experiment and tf is the final time of failure for each experiment as described in the previous section. [Note: The expended life fraction τe is normalized between 0 and 1.] 0.6

Anomaly Measure

Family of fatigue damage profiles generated by 40 0.5 samples under identical loading conditions 0.4

0.3

0.2

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0.1

0 0

0.2

0.4

0.6

0.8

1

Expended life fraction

Figure 12. Statistical behavior of fatigue damage represented by a family of anomaly measure profiles generated by 40 identical experiments [46]. For each individual experiment, the state probability vector p0 is generated at the nominal condition t0 by partitioning the wavelet domain using the maximum entropy principle [58]. As a consequence, p0 has uniform distribution, i.e. each element has equal probability. In contrast, for the completely broken stage of the specimen, the entire probability

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distribution is concentrated in only one element of the state probability vector, i.e. delta distribution, which indicates a very large attenuation of the ultrasonic signal [44]. Therefore, as the fatigue crack damage evolves, the uniform distribution (i.e., maximum entropy) under nominal condition degenerates toward the delta distribution (i.e., zero entropy) for the broken specimen. Since, p0 has uniform distribution for all experiments, the statistical property is identical for all experiments at the nominal condition. As seen in Figure 12, each profile has a smaller slope of the anomaly measure during the initial period of fatigue damage, i.e., the crack initiation region. Anomaly measure gradually increases during this period where small microstructural damage occurs in the specimen. During the end stage of this period small micro cracks eventually develop into a single large crack leading to a transition from the crack initiation stage to the crack propagation stage (approx from τe = 0.5 to τe = 0.7). This phenomenon is observed by a sharp change in the slope of the anomaly measure profile of each sample. Once the crack propagation stage starts the fatigue damage occurs rapidly eventually leading to the final failure.

7.2.

Solution of the Inverse Problem

The objective of the inverse problem is identification of anomalies and estimation of the fault parameters based on the family of curves generated in the forward problem [46]. It is essential to detect the evolving fatigue damage and to estimate the remaining useful life during the operating period of the mechanical system, so that appropriate remedial action(s) can be taken before the onset of widespread fatigue propagation leading to complete failure. Therefore, estimation of fatigue damage is crucial for scheduled maintenance.

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7.2.1.

Generation of the Pattern Matrix

In an online experiment, time series data sets (at fast time scale) of the ultrasonic sensors are generated at different slow time epochs up till the current time epoch. These data sets are analyzed using SDF method as discussed in Appendix Appendix A. to generate the probability distributions at the corresponding slow time epochs. The value of anomaly measure at the current time epoch is then calculated from the evolution of this probability vector from the nominal healthy condition. As such, the information available at any particular instant in a real time experiment is the value of anomaly measure calculated at that particular instant. Based on this derived value of the anomaly measure the exact determination of the expended life fraction (τe ) is not possible due to the variations observed in the statistical family as seen in Figure 12. Therefore, due to uncertainty in determining its exact value at a particular value of anomaly measure, τe can be treated as a random variable [65]. The range of anomaly measure (i.e. the ordinate in Figure 12) is partitioned into h = 100 uniformly spaced levels. A pattern matrix T of dimension ℓ×h is then derived from the anomaly measure profiles shown in Figure 12. The elements of T are derived such that each column of T corresponds to the values of τe measured for ℓ samples at the corresponding anomaly measure. As such, the elements of each column of T describe a distribution of the random variable τe .

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32 7.2.2.

Shalabh Gupta and Asok Ray Estimation of the Expended Life Fraction

In order to estimate the value of τe by statistical means, a two-parameter lognormal distribution [13] [70] is hypothesized for each column of T . Lognormal distribution is obtained for each column of T over the mean and the variance of τe . The goodness of fit is examined by both χ2 and Kolmogorov−Smirnov tests [71]. The number of bins were taken to be r = 8 for the data set of each column of T . With f = r − 2 − 1 = 5 degrees of freedom, the χ2 -test shows that, for each of the h data sets, the hypothesis of the two-parameter lognormal distribution passed the 20% significance level [71] which suffices the conventional standard of 5% significance level. Also, for each of the h data sets, the hypothesis passed the 20% significance level of the Kolmogorov−Smirnov test which again suffices the conventional standard of 5% significance level. A good discussion of these statistical tests is provided in reference [71]. Once the lognormal distributions are obtained, the confidence intervals bounds at different confidence levels can be computed from the properties of the distribution using elementary statistics [71] [72]. Confidence level signifies the probability that the estimated parameter will lie within the corresponding confidence interval. As an example, for a confidence level of 95%, the probability that the actual parameter will lie between the specified confidence interval is 95%. Figure 13 provides the plots of confidence interval bounds at three different confidence levels of 95%, 85% and 75%. As an illustration in Figure 13, the confidence interval bounds at 95% confidence level are shown for an arbitrary value of anomaly measure equal to 0.225 (lower bound=0.6610 and upper bound=0.8464). The estimate τbe of the expended life fraction τe can be obtained at the point of highest probability, i.e. the mean of the distribution. The other useful parameter is the remaining life fraction whose estimate τbr is obtained at any instant as: τbr = 1 − τbe . The information on the remaining life estimate in a real-time experiment is useful for development of life extending control and resilient control strategies for prevention of widespread structural damage and catastrophic failures.

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7.2.3.

Experimental Validation and Results

The proposed methodology is validated by fatigue experiments on three new test specimens. The profiles of anomaly measure are computed using the SDF method for these three test specimens. Figure 13 also shows the profiles of the three test samples along with the plots of confidence interval bounds derived from the statistical ensemble. The estimates of the mean τbe of the expended life fraction with the standard deviation σ b are obtained at (arbitrary) different values of the anomaly measure using the procedure described in the previous section. The results are interpolated for values of the anomaly measure that lie in between the two columns of the pattern matrix T . Confidence interval bounds are obtained at three different confidence levels of 95%, 85% and 75%. Figure 13 shows that the uncertainty in the fatigue damage is higher in the crack initiation phase as indicated by the width of confidence intervals for any particular value of the anomaly measure. Subsequently, upon onset of the crack propagation phase, the confidence intervals are significantly more tight than those in the crack initiation phase. This observation is explained by the fact that the uncertainty in the crack initiation phase depends on the random distribution of flaws in the specimen [69]. During this crack initiation phase,

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0.6

Anomaly Measure

0.5

95% confidence interval 85% confidence interval 75% confidence interval Mean

Validation Sample 3

Validation Sample 2

0.4

0.3

Anomaly Measure 0.225

Validation Sample 1

Mean 0.7495

0.2 95% Confidence Interval 0.6610−0.8464

0.1

0 0

0.2

0.4

0.6

0.8

1

Expended Life Fraction

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Figure 13. Plots of confidence interval bounds are shown at three different confidence levels of 95%, 85% and 75%. Profiles of anomaly measure are also shown for three new validation test specimens [46].

small cracks originate from the microstructural damage (eg. dislocations, voids and inclusions) at multiple sites in the entire body of the material structure causing a high uncertainty in fatigue damage behavior. These multiple small cracks eventually develop into a single large crack leading to the onset of crack propagation phase. Therefore, the uncertainty in the crack initiation phase relates to the inhomogeneity in the material and non-uniform distribution of the initial conditions in the specimen causing stress augmentation at certain locations which directly affects the formation of small cracks. As such, the information that is derived during the crack initiation phase can act as an early warning of the onset of widespread fatigue in the crack propagation phase. The information from Figure 13 (including the estimate of τe and different confidence intervals) can be utilized for real-time monitoring of the fatigue damage and for development of probabilistic robust control strategies for damage mitigation and prevention of catastrophic failures.

8. 8.1.

Summary, Conclusions and Recommendations for Future Work Summary and Conclusions

The main contribution of this chapter is real-time monitoring of fatigue damage in polycrystalline alloys that are commonly used in mechanical structures. The chapter has demonstrated the capabilities of ultrasonic sensing technique for detection of small microstructural changes during early stages of fatigue damage. The chapter has adopted sensor based

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anomaly detection methodology because of the difficulty in achieving requisite accuracy in developing structural models of failure mechanisms at the microstructural level based on the fundamentals of physics. As such, an alternative approach is presented which relies on information based real time sensing of fatigue damage in mechanical systems. This chapter presents a recently reported information-theoretic technique, called Symbolic Dynamic Filtering (SDF), for real-time analysis of ultrasonic data. The underlying concept of SDF is built upon the principles of Symbolic Dynamics, Information Theory, and Statistical Signal Processing, where time series data from selected sensor(s) (e.g., ultrasonics) in the fast time scale of the process dynamics are analyzed at discrete epochs in the slow time scale of fatigue damage evolution. SDF includes pre-processing of ultrasonic data using the wavelet analysis, which is well suited for time-frequency analysis of nonstationary signals and enables noise attenuation from raw data. The wavelet-transformed data is partitioned using the maximum entropy principle to generate symbol sequences, such that the regions of data space with more information are partitioned finer and those with sparse information are partitioned coarser. Subsequently, robust statistical patterns of evolving damage are identified from these sequences by construction of a (probabilistic) finite-state machine that captures the dynamical system behavior by means of information compression. Furthermore, the problem of fatigue damage monitoring is constructed into two subproblems: (i) Forward problem of Pattern Recognition for characterization of the anomalous behavior, relative to the nominal behavior; and (ii) Inverse problem of Pattern Identification for estimation of parametric or non-parametric changes based on the knowledge assimilated in the forward problem and the observed time series data of quasi-stationary sensor measurements. In this regard, this chapter has presented a statistical approach for estimation of the remaining useful life. To this effect, a stochastic data base of ultrasonic measurements has been generated from several experiments conducted under identical loading conditions. This data base has been analyzed to derive the behavioral pattern of fatigue damage under identical loading conditions which is subsequently used to provide the estimates of used life fraction in a fatigue experiment. The codes of SDF are executable in real time and have been demonstrated in the laboratory environment for on-line monitoring of fatigue damage, based on the analysis of ultrasonic sensor signals, before any surface cracks are visible through the optical microscope in a special-purpose fatigue testing apparatus. In this research, the experiments have been conducted on the specimens fabricated from the aluminum alloy 7075-T6.

8.2.

Recommendations for Future Work

The reported work is a step toward building a reliable instrumentation system for early detection of fatigue damage in polycrystalline alloys; further theoretical and experimental research is necessary before its usage in industry. While there are many research issues that need to be addressed, the following topics are being currently pursued and are recommended for future research: 1. Investigation of other sensing techniques- The research work reported in this chapter is based on an ultrasonic sensing technique. Future research would require inves-

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tigation of the potential capabilities of other sensing techniques, such as acoustic emission, and eddy currents for early detection of fatigue damage and their real time implementation; 2. Development of stochastic measures of fatigue crack growth in compact specimensFuture work requires development of stochastic models of fatigue crack damage under different loading conditions. Furthermore, the stochastic data bases crack growth can be used for studying complex phenomenon such as fractal behavior of fatigue damage evolution; 3. Investigation of fatigue damage sensing under different loading conditions- The results in this chapter are primary based on constant-amplitude low cycle loading conditions. Future work requires validation of the SDF technique for early detection of fatigue damage under different conditions, such as high-cycle loading, variableamplitude loading, and spectral loading; 4. Study of microstructural changes- The work reported in this chapter is based on a dynamical data-driven approach that relies on ultrasonic sensing due to lack of accurate physics-based models during early stages of fatigue damage. Future work would require investigation of small microstructural changes during fatigue damage evolution. Analytical models of microstructural changes need to be formulated using advanced experimental devices such as the atomic force microscope and the scanning electron microscope; 5. Investigation of surface deformities- Future work requires to study the surface deformities occurring during the early stages of fatigue damage using the surface interferometer;

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6. Development of real time life extending control policies- The information extracted from time series data using the SDF method can be used for development of control strategies for real time life extension and damage mitigation.

Appendix A.

Symbolic Dynamic Filtering Concept

This section presents a brief summary of the underlying concepts and essential features of a recently reported data-driven pattern identification tool called symbolic dynamic filtering (SDF ) [21]. The concept of SDF is built upon the principles of several disciplines including Symbolic Dynamics [15, 20], Statistical Pattern Recognition [73], Statistical Mechanics [17], Information Theory [74] and Probabilistic Finite State Machines [16]. While the details are reported in recent publications [21, 22, 58, 17], the essential concepts of space partitioning, symbol sequence generation, construction of a finite-state machine from the generated symbol sequence and pattern recognition are consolidated here and succinctly described for self-sufficiency, completeness and clarity of the chapter.

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Appendix A.1.

Shalabh Gupta and Asok Ray

Symbolic Dynamic Encoding

The continuously-varying finite-dimensional model of a dynamical system is usually formulated in the setting of an initial value problem as: dx(t) = f (x(t), θ(ts )); x(0) = x0 , dt

(A.1)

where t ∈ [0, ∞) denotes the (fast-scale) time; x ∈ Rn is the state vector in the phase space; and θ ∈ Rℓ is the (possibly anomalous) parameter vector varying in (slow-scale) time ts . The gradual change in the parameter vector θ ∈ Rℓ due to possible evolution of anomalies can alter the system dynamics and hence change the state trajectory. Let Ω ⊂ Rn be a compact (i.e., closed and bounded) region, within which the trajectory of the dynamical system, governed by Eq. (A.1), is circumscribed as illustrated in Fig. 8. The region Ω is partitioned as {Φ0 , · · · , Φ|Σ|−1 } consisting of |Σ| mutually exclusive (i.e., S|Σ|−1 Φj ∩ Φk = ∅ ∀j 6= k), and exhaustive (i.e., j=0 Φj = Ω) cells, where Σ is the symbol alphabet that labels the partition cells. A trajectory of the dynamical system is described by the discrete time series data as: {x0 , x1 , x2 , · · · }, where each xi ∈ Ω. The trajectory passes through or touches one of the cells of the partition; accordingly the corresponding symbol is assigned to each point xi of the trajectory as defined by the mapping M : Ω → Σ. Therefore, a sequence of symbols is generated from the trajectory starting from an initial state x0 ∈ Ω, such that: x0 ֌ s0 s1 s2 . . . sj . . . (A.2) where sk , M(xk ) is the symbol generated at the (fast scale) instant k. The symbols sk , k = 0, 1, . . . are identified by an index set I : Z → {0, 1, 2, . . . |Σ| − 1}, i.e., I(k) = ik and sk = σik where σik ∈ Σ. Equivalently, Eq. (A.2) is expressed as:

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x0 ֌ σi0 σi1 σi2 . . . σij . . .

(A.3)

The mapping in Eq. (A.2) and Eq. (A.3) is called Symbolic Dynamics as it attributes a legal (i.e., physically admissible) symbol sequence to the system dynamics starting from an initial state. The partition is called a generating partition of the phase space Ω if every legal (i.e., physically admissible) symbol sequence uniquely determines a specific initial condition x0 . In other words, every (semi-infinite) symbol sequence uniquely identifies one continuous space orbit [19]. Symbolic dynamics may also be viewed as coarse graining of the phase space, which is subjected to (possible) loss of information resulting from granular imprecision of partitioning boxes. However, the essential robust features (e.g., periodicity and chaotic behavior of an orbit) are expected to be preserved in the symbol sequences through an appropriate partitioning of the phase space [18]. Figure 8 pictorially elucidates the concepts of partitioning a finite region of the phase space and the mapping from the partitioned space into the symbol alphabet, where the symbols are indicated by Greek letters (e.g., α, β, γ, δ, · · · ). This represents a spatial and temporal discretization of the system dynamics defined by the trajectories. Figure 8 also shows conversion of the symbol sequence into a finite-state machine and generation of the

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state probability vectors at the current and the reference conditions. The states of the finite state machine and the histograms in Fig. 8 are indicated by numerics (i.e., 1, 2, 3 and 4). Although the theory of phase-space partitioning is well developed for one-dimensional mappings [19], very few results are known for two and higher dimensional systems. Furthermore, the state trajectory of the system variables may be unknown in case of systems for which a model as in Eq. (A.1) is not known or is difficult to obtain. As such, as an alternative, the time series data set of selected observable outputs can be used for symbolic dynamic encoding as explained in the following subsection.

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Appendix A.2.

Wavelet Space Partitioning

As described earlier, a crucial step in symbolic dynamic filtering (SDF ) is partitioning of the phase space for symbol sequence generation [20]. Several partitioning techniques have been reported in literature for symbol generation [14][59], primarily based on symbolic false nearest neighbors (SF N N ). These techniques rely on partitioning the phase space and may become cumbersome and extremely computation-intensive if the dimension of the phase space is large. Moreover, if the time series data is noise-corrupted, then the symbolic false neighbors would rapidly grow in number and require a large symbol alphabet to capture the pertinent information on the system dynamics. The wavelet transform [67] largely alleviates these shortcomings and is particularly effective with noisy data from high-dimensional dynamical systems [58]. As such, this chapter has used a waveletbased partitioning approach [21][58] for construction of symbol sequences from time series data. In wavelet-based partitioning approach, time series data are first converted to wavelet domain, where wavelet coefficients are generated at different time shifts. The choice of the wavelet basis function and wavelet scales depends on the time-frequency characteristics of individual signals. Guidelines for selection of basis functions and scales are reported in literature [58]. The wavelet space is partitioned with alphabet size |Σ| into segments of coefficients on the ordinate separated by horizontal lines. The choice of |Σ| depends on specific experiments, noise level and also the available computation power. A large alphabet may be noise-sensitive while a small alphabet could miss the details of signal dynamics [58]. The partitioning is done such that the regions with more information are partitioned finer and those with sparse information are partitioned coarser. This is achieved by maximizing the Shannon entropy [74], which is defined as: |Σ|−1

S=−

X

pi log(pi )

(A.4)

i=0

where pi is the probability of a data point to be in the ith partition segment. Uniform prob1 for i = 0, 1, . . . , |Σ| − 1, is a consequence of maximum ability distribution, i.e., pi = |Σ| entropy partitioning [58]. Each partition segment is labelled by a symbol from the alphabet Σ and accordingly the symbol sequence is generated from the wavelet coefficients. The structure of the partition is fixed at the nominal condition, which serves as the reference frame for symbol sequence generation from time series data at anomalous condition(s).

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Shalabh Gupta and Asok Ray

Recently, an alternative method called Analytic Signal Space Partitioning (ASSP ) [75] has been reported for symbolic time series analysis. The underlying concept of ASSP is built upon Hilbert transform of the real-valued data sequence into corresponding complexvalued analytic signal sequence that, in turn, is partitioned in the 2-dimensional plane.

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Appendix A.3.

Probabilistic Finite State Machine (P F SM )

Once the symbol sequence is obtained, the next step is the construction of a Probabilistic Finite State Machine (P F SM ) and calculation of the respective state probability vector as depicted in the lower part of Fig. 8 by the histograms. The partitioning is performed at the nominal condition that is chosen to be the healthy state having no anomalies. A P F SM is then constructed at the nominal condition, where the states of the machine are defined corresponding to a given alphabet set Σ and window length D. The alphabet size |Σ| is the total number of partition segments while the window length D is the length of consecutive symbol words [21], which are chosen as all possible words of length D from the symbol sequence. Each state belongs to an equivalence class of symbol words of length D, which is characterized by a word of length D at the leading edge. Therefore, the number n of such equivalence classes (i.e., states) is less than or equal to the total permutations of the alphabet symbols within words of length D. That is, n ≤ |Σ|D ; some of the states may be forbidden, i.e., these states have zero probability of occurrence. For example, if Σ = {α, β}, i.e., |Σ| = 2 and if D = 2, then the number of states is n ≤ |Σ|D = 4; and the possible states are words of length D = 2, i.e., αα, αβ, βα, and ββ. The choice of |Σ| and D depends on specific applications and the noise level in the time series data as well as on the available computation power and memory availability. As stated earlier, a large alphabet may be noise-sensitive and a small alphabet could miss the details of signal dynamics. Similarly, while a larger value of D is more sensitive to signal distortion, it would create a much larger number of states requiring more computation power and increased length of the data sets. Applications such as two-dimensional image processing, may require larger values of the parameter D and hence possibly larger number of states in the P F SM . Using the symbol sequence generated from the time series data, the state machine is constructed on the principle of sliding block codes [15]. The window of length D on a symbol sequence is shifted to the right by one symbol, such that it retains the most recent (D-1) symbols of the previous state and appends it with the new symbol at the extreme right. The symbolic permutation in the current window gives rise to a new state. The P F SM constructed in this fashion is called the D-Markov machine [21], because of its Markov properties. Definition 1.1 A symbolic stationary process is called D-Markov if the probability of the next symbol depends only on the previous D symbols, i.e., P (sj |sj−1 ....sj−D sj−D−1 ....) = P (sj |sj−1 ....sj−D ). The finite state machine constructed above has D-Markov properties because the probability of occurrence of symbol σ ∈ Σ on a particular state depends only on the configuration of that state, i.e., the previous D symbols. The states of the machine are marked with the

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39

corresponding symbolic word permutation and the edges joining the states indicate the occurrence of a symbol σ. The occurrence of a symbol at a state may keep the machine in the same state or move it to a new state. Definition 1.2 Let Ξ be the set of all states of the finite state machine. Then, the probability of occurrence of symbols that cause a transition from state ξj to state ξk under the mapping δ : Ξ × Σ → Ξ is defined as: X πjk = P (σ ∈ Σ | δ(ξj , σ) → ξk ) ; πjk = 1; (A.5) k

Thus, for a D-Markov machine, the irreducible stochastic matrix Π ≡ [πij ] describes all transition probabilities between states such that it has at most |Σ|D+1 nonzero entries. The definition above is equivalent to an alternative representation such that, πjk ≡ P (ξk |ξj ) =

P (σi0 · · · σiD−1 σiD ) P (ξj , ξk ) = P (ξj ) P (σi0 · · · σiD−1 )

(A.6)

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where the corresponding states are denoted by ξj ≡ σi0 · · · σiD−1 and ξk ≡ σi1 · · · σiD . This phenomenon is a consequence of the P F SM construction based on the principle of sliding block codes described above, where the occurrence of a new symbol causes a transition to another state or possibly the same state. For computation of the state transition probabilities from a given symbol sequence at a particular slow time epoch, a D-block (i.e., a window of length D) is moved by counting occurrences of symbol blocks σi0 · · · σiD−1 σiD and σi0 · · · σiD−1 , which are respectively denoted by N (σi0 · · · σiD−1 σiD ) and N (σi0 · · · σiD−1 ). Note that if N (σi0 · · · σiD−1 ) = 0, then the state σi0 · · · σiD−1 ∈ Ξ has zero probability of occurrence. For N (σi0 · · · σiD−1 ) 6= 0, the estimates of the transitions probabilities are then obtained by these frequency counts as follows: N (σi0 · · · σiD−1 σiD ) (A.7) πjk ≈ N (σi0 · · · σiD−1 ) where the criterion for convergence of the estimated πjk , is given in [22] as a stopping rule for frequency counting. The symbol sequence generated from the time series data at the nominal condition, set as a benchmark, is used to compute the state transition matrix Π using Eq. (A.7). The left eigenvector p corresponding to the unique unit eigenvalue of the irreducible stochastic matrix Π is the probability vector whose elements are the stationary probabilities of the states belonging to Ξ [21][17]. The partitioning of time series data and the state machine structure should be the same in both nominal and anomalous cases but the respective state transition matrices could be different.

Appendix A.4.

Pattern Identification Procedure

Behavioral pattern changes are quantified as deviations from the nominal behavior (i.e., the probability distribution at the nominal condition). The resulting anomalies (i.e., deviations of the evolving patterns from the nominal pattern) are characterized by a scalar-valued

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Shalabh Gupta and Asok Ray

function, called Anomaly Measure ψ that is quasi-static in the fast time scale and is monotonically non-decreasing in the slow time scale. The state probability vector at any time instant corresponds to a singleton point on the unity-radius hypersphere. During fatigue damage evolution, the tip of the probability vector moves along a path on the surface of this hypersphere. The initial starting point of the path is the probability vector with uniform distribution obtained with maximum entropy partitioning (see Section Appendix A.2.). As the damage progresses, the probability distribution changes; eventually when a very large crack is formed, complete attenuation of the ultrasonic signal occurs and consequently the tip of the probability vector reaches a point where all states have zero probabilities of occurrence except one which has a probability one (i.e., a delta-distribution); this state corresponds to the partition region where all data points are clustered due to complete attenuation of the signal. In the context of fatigue damage, the anomaly measure is formulated on the following assumptions. • Assumption #1: The damage evolution is an irreversible process (i.e., with zero probability of self healing) and implies the following conditions. ψ k ≥ 0; ψ k+ℓ − ψ k ≥ 0 ∀ℓ ≥ 0 ∀k

(A.8)

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• Assumption #2: The damage accumulation between two time epochs is a path function, i.e., dependent on the path traversed to reach the target state from the initial state. In the context of fatigue damage in polycrystalline alloys at room temperature, the crack length is traditionally defined by a straight line joining the starting point to the tip of the crack but, in reality, the actual crack follows a complicated path (possibly fractal in ductile materials). In fact, at the initial stages of fatigue damage, there can be multiple short cracks oriented in different directions. Therefore, crack length alone does not provide complete information on fatigue damage evolution. Since ultrasonic signals are highly sensitive to small micro-structural changes, signal distortion is a good index of anomaly growth. The tip of the probability vector, obtained through symbolic dynamic filtering (SDF ) method, moves along a curved path on the surface of the unity-radius hypersphere between the initial point p0 (i.e., uniform distribution obtained under maximum entropy partitioning) and the final point at very large crack formation pf (i.e., δ-distribution due to complete attenuation of the signal). The phenomenon such as piling up of dislocations, strain hardening or reflections from multiple crack surfaces affect the ultrasonic signals in a variety of ways. An increase of the ultrasonic amplitude is also observed during very early stages of fatigue damage due to hardening of the material. On the other hand, ultrasonic signals attenuate sharply at the crack propagation stage upon development of a large crack. As such, distortion of ultrasonic signals at a single time epoch may not uniquely determine the state of fatigue damage. The rationale is that two signals may exhibit similar characteristics but, in terms of actual incurred damage, the states are entirely different. Consequently, fatigue damage is a path function instead of being a state function. This assessment is consistent with assumption #1 implying that the damage evolution is irreversible. That is, at two different time epochs, the damage cannot be identical unless the net damage

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increment is zero. Consequently, by assumption #2, the anomaly measure should follow the traversed path of the probability vector, not the straight line joining the end points (i.e., the tips of the probability vectors). The anomaly measure, based on the path between the nominal state to the completely damaged state, can be different even for identical test samples and under the same loading conditions because of the stochastic nature of fatigue phenomena. As such, analysis of a stochastic data set collected under identical experimental conditions is essential for identification of variations in different data sets. The following distance function is derived between probability vectors at two time epochs: d(pk , pl ) ≡

q T (pk − pl ) (pk − pl )

(A.9)

The algorithm for computation of the anomaly measure ψ compensates for spurious measurement and computation noise in terms of the sup norm which is defined as k e k∞ ≡ max(|e1 |, · · · , |em |) of the error in the probability vector (i.e., the maximum error in the elements of the probability vector). The algorithm is presented below. e = p0 ; k = 1; i) ψ 0 = 0; δψ 1 = 0; p

e ||∞ > ǫ then δψ k = d(pk , p e ) and p e ← pk ; ii) if ||pk − p

iii) ψ k = ψ k−1 + δψ k ;

iv) k ← k + 1; δψ k = 0; go to step (ii). The real positive parameter ǫ, is associated with the robustness of the measure against measurement and computation noise and is identified by performing an experiment with a sample with no notch. Since there is no notch there is practically no stress augmentation and relatively no fatigue damage. As such, the parameter ǫ is estimated as: ǫ ≈ max (||pl+1 − pl ||∞ )

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l∈{1,..N }

(A.10)

from N consecutive observations with N ≫ 1. e is initialized to the The algorithm works in the following fashion: the reference point p starting point p0 and anomaly measure ψ 0 is set to 0. At any slow time epoch tk if the state probability vector moves such that the distance travelled in any particular direction (i.e. the sup norm || • ||∞ ) is greater than ǫ as specified in step (ii), then the anomaly measure is e ) and the reference point is shifted to the current point pk . incremented by δψ k = d(pk , p The procedure is repeated at all slow time epochs. As such, the total path travelled by the tip of probability vector represents the deviation from the nominal condition and the associated damage.

Appendix A.5.

Summary of SDF -based Pattern Recognition

The symbolic dynamic filtering (SDF ) method of statistical pattern recognition for anomaly detection is summarized below.

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42

Shalabh Gupta and Asok Ray • Acquisition of time series data from appropriate sensor(s) variables at a nominal condition, when the system is assumed to be in the healthy state (i.e., zero anomaly measure) • Generation of the wavelet transform coefficients of the data obtained with an appropriate choice of the wavelet basis and scale [58] • Maximum entropy partitioning in the wavelet domain at the nominal condition (see Appendix A.2.) and generation of the corresponding symbol sequence • Construction of the D-Markov machine and computation of the state probability vector p0 at the nominal condition • Generation of a time series data sequence at another (possibly) anomalous condition and conversion to the wavelet domain to generate the respective symbolic sequence based on the partitioning constructed at the nominal condition • Computation of the corresponding state probability vector p using the finite state machine constructed at the nominal condition • Computation of scalar anomaly measure µ.

Capability of SDF has been demonstrated for anomaly detection at early stages of gradually evolving anomalies by real-time experimental validation. In this regard, major advantages of SDF are listed below: i. Robustness to measurement noise and spurious signals [58] ii. Adaptability to low-resolution sensing due to the coarse graining in space partitions [21] iii. Capability for small change detection because of sensitivity to signal distortion [44] and

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iv. Real-time execution on commercially available inexpensive platforms [62][44].

Appendix A.6.

Stopping Rule for Determining Symbol Sequence Length

This appendix presents a stopping rule that is necessary to find a lower bound on the length of symbol sequence required for parameter identification of the stochastic matrix Π. The stopping rule [22] is based on the properties of irreducible stochastic matrices [76]. The state transition matrix, constructed at the rth iteration (i.e., from a symbol sequence of length r), is denoted as Π(r) that is an n × n irreducible stochastic matrix under stationary conditions. Similarly, the state probability vector p(r) ≡ [p1 (r) p2 (r) · · · pn (r)] is obtained as ri pi (r) = Pn

j=1 ri

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(A.11)

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43

where ri is the number of
D-blocks (i.e., symbol strings of length D) representing the ith Pn state such that j=1 rj + D − 1 = r is the total length of the data sequence under symbolization. The stopping rule makes use of the Perron-Frobenius Theorem [76] to establish a relation between the vector p(r) and the matrix Π(r). Since the matrix Π(r) is stochastic and irreducible, there exists a unique eigenvalue λ = 1 and the corresponding left eigenvector p(r) (normalized to unity in the sense of absolute sum). The left eigenvector p(r) represents the state probability vector, provided that the matrix parameters have converged after a sufficiently large number of iterations. That is, under the hypothetical arbitrarily long sequences, the following condition is assumed to hold. p(r + 1) = p(r)Π(r) ⇒ p(r) = p(r)Π(r) as r → ∞

(A.12)

Following Eq. (A.11), the absolute error between successive iterations is obtained such that k (p(r) − p(r + 1)) k∞ =k p(r) (I − Π(r)) k∞ ≤

1 r

(A.13)

where k • k∞ is the max norm of the finite-dimensional vector •. To calculate the stopping point rstop , a tolerance of η (0 < η ≪ 1) is specified for the relative error such that: k (p(r) − p(r + 1)) k∞ ≤ η ∀ r ≥ rstop k (p(r)) k∞

(A.14)

The objective is to obtain the least conservative estimate for rstop such that the dominant elements of the probability vector have smaller relative errors than the remaining elements. Since the minimum possible value of k (p(r)) k∞ for all r is n1 , where n is the dimension of p(r), the least of most conservative values of the stopping point is obtained from Eqs. (A.13) and (A.14) as:   n rstop ≡ int (A.15) η where int(•) is the integer part of the real number •.

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References [1] S. Ozekici, Reliability and Maintenance of Complex Systems, vol. 154. NATO Advanced Science Institutes (ASI) Series F: Computer and Systems Sciences, Berlin, Germany, 1996. [2] E. E. Keller and A. Ray, “Real time health monitoring of mechanical structures,” Structural Health Monitoring, vol. 2(3), pp. 191–203, 2003. [3] M. A. Meggiolaro and J. T. P. Castro, “Statistical evaluation of strain-life fatigue crack initiation predictions,” International Journal of Fatigue, vol. 26, pp. 463–476, 2004. [4] P. Johannesson, T. Svensson, and M. D. Jacques, “Fatigue life prediction based on variable amplitude tests-methodology,” International Journal of Fatigue, vol. 27, pp. 954–965, 2005.

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[5] S. Ishihara and A. J. McEvily, “Analysis of short fatigue crack growth in cast aluminium alloys,” International Journal of Fatigue, vol. 24, pp. 1169–1174, 2002. [6] C. Bjerk´ en and S. Melin, “A tool to model short crack fatigue growth using a discrete dislocation formulation,” International Journal of Fatigue, vol. 25, pp. 559–566, 2003. [7] M. D. Chapetti, “Fatigue propagation threshold for short cracks under constant amplitude loading,” International Journal of Fatigue, vol. 25, pp. 1319–1326, 2003. [8] D. V. Ramsamooj, “Analytical prediction of short to long fatigue crack growth rate using small- and large-scale yielding fracture mechanics,” International Journal of Fatigue, vol. 25, pp. 923–933, 2003. [9] R. K. Pathria, Statistical Mechanics. Elsevier Science and Technology Books, 1996. [10] E. Ott., Chaos in Dynamical Systems. Cambridge University Press, 1993. [11] K. Sobczyk and B. F. Spencer, Random Fatigue: Data to Theory. Academic Press, Boston, MA, 1992. [12] N. Scafetta, A. Ray, and B. J. West, “Correlation regimes in fluctuations of fatigue crack growth,” Physica A, vol. 59, no. 3. [13] A. Ray, “Stochastic measure of fatigue crack damage for health monitoring of ductile alloy structures,” Structural Health Monitoring, vol. 3, no. 3, pp. 245–263, 2004. [14] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis, 2nd ed. Cambridge University Press, United Kingdom, 2004. [15] D. Lind and M. Marcus, An Introduction to Symbolic Dynamics and Coding. Cambridge University Press, United Kingdom, 1995.

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[16] H. E. Hopcroft, R. Motwani, and J. D. Ullman, Introduction to Automata Theory, Languages, and Computation, 2nd ed. Addison Wesley, Boston, 2001. [17] S. Gupta and A. Ray, “Statitical mechanics of complex systems for pattern identification,” Journal of Statistical Physics, vol. 134, no. 2, pp. 337–364, 2009. [18] R. Badii and A. Politi, Complexity hierarchical structures and scaling in physics. Cambridge University Press, United Kingdom, 1997. [19] C. Beck and F. Schl¨ ogl, Thermodynamics of chaotic systems: an introduction. Cambridge University Press, United Kingdom, 1993. [20] C. S. Daw, C. E. A. Finney, and E. R. Tracy, “A review of symbolic analysis of experimental data,” Review of Scientific Instruments, vol. 74, no. 2, pp. 915–930, 2003. [21] A. Ray, “Symbolic dynamic analysis of complex systems for anomaly detection,” Signal Processing, vol. 84, no. 7, pp. 1115–1130, 2004.

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[22] S. Gupta and A. Ray, Symbolic Dynamic Filtering for Data-Driven Pattern Recognition, Chapter 2 in Pattern Recognition: Theory and Application. Nova Science Publisher, Hauppage, New York, Editor- Erwin A. Zoeller, 2007. [23] S. Gupta and A. Ray, “Pattern identification using lattice spin systems: A thermodynamic formalism,” Applied Physics Letters, vol. 91, no. 19, p. 194105, 2007. [24] S. Grondel, C. Delebarre, J. Assaad, J. P. Dupuis, and L. Reithler, “Fatigue crack monitoring of riveted aluminium strap joints by lamb wave analysis and acoustic emission measurement techniques,” N DT &E international, vol. 35, pp. 339–351, 2006. [25] D. A. Cook and Y. H. Berthelot, “Detection of small surface-breaking fatigue cracks in steel using scattering of rayleigh waves,” N DT &E international, vol. 34, pp. 483– 492, 2001. [26] V. Zilberstein, K. Walrath, D. Grundy, D. Schlicker, N. Goldfine, E. Abramovici, and T. Yentzer, “Mwm eddy-current arrays for crack initiation and growth monitoring,” International Journal of Fatigue, vol. 25, pp. 1147–1155, 2003. [27] A. Witney, Y. F. Li, J. Wang, M. Z. Wang, J. J. DeLuccia, and C. Laird, “Electrochemical fatigue sensor response to ti-6wt% al-4wt% v and 4130 steel,” Philosophical Magazine, vol. 84, no. 3-5, pp. 331–349, 2004. [28] B. Yang, P. K. Liaw, G. Wang, W. H. Peter, R. Buchanan, Y. Yokoyama, J. Y. Huang, R. C. Kuo, J. G. Huang, D. E. Fielden, and D. L. Klarstrom, “Thermal-imaging technologies for detecting damage during high-cycle fatigue,” Metallurgical and Materials Transactions A, vol. 35A, pp. 15–24, 2004. [29] S. W. Doebling, C. R. Farrar, and M. B. Prime, “A summary review of vibrationbased damage identification methods,” The Shock and Vibration Digest, vol. 30, no. 2, pp. 91–105, 1998. [30] A. Almeida and E. V. K. Hill, “Neural network detection of fatigue crack growth in reveted joints using acoustic emission,” Mat. Eval., vol. 53, no. 1, pp. 76–82, 1995.

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[31] J. Baram, “Fatigue-life prediction by an order statistics treatment of acoustic-emission signals,” Exper. Mech., vol. 33, pp. 189–194, 1993. [32] M. N. Bassim, S. S. Lawrence, and C. D. Liu, “Detection of the onset of fatigue crack growth in rail steels using acoustic emission,” Eng. Fracture Mech., vol. 47, no. 2, pp. 207–214, 1994. [33] D. O. Harris and H. L. Dunegan, “Continuous monitoring of fatigue-crack growth by acoustic-emission techniques,” Exper. Mech., vol. 14, pp. 71–80, 1974. [34] K. Y. Lee, “Cyclic ae count rate and crack growth rate under low cycle fatigue fracture loading,” Eng. Fracture Mech., vol. 34, no. 5/6, pp. 1069–1073, 1989. [35] M. V. Lysak, “Development of the theory of acoustic emission by propagating cracks in terms of fracture mechanics,,” Eng. Fracture Mech., vol. 55, no. 3, pp. 443–452, 1996.

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[36] M. Scala and S. M. Cousland, “Acoustic emission during fatigue crack propagation in the aluminium alloys 2024 and 2124,” Mat. Sci. and Eng., vol. 61, pp. 211–218, 1983. [37] V. Zilberstein, D. Grundy, V. Weiss, N. Goldfine, E. Abramovici, J. Newman, and T. Yentzer, “Early detection and monitoring of fatigue in high strength steels with mwm-arrays,” International Journal of Fatigue, vol. 27, pp. 1644–1652, 2005. [38] H. S. Bai, L. Y. Yu, and J. W. He (Ho), “A monitoring system for contact fatigue crack testing,” NDT International, vol. 22, no. 3, pp. 162–167, 1989. [39] N. Yusa, L. Janousek, M. Rebican, Z. Chen, K. Miya, N. Chigusa, and H. Ito, “Detection of embedded fatigue cracks in inconel weld overlay and the evaluation of the minimum thickness of the weld overlay using eddy current testing,” Nuclear Engineering and Design, vol. 236, no. 18, pp. 1852–1859, 2006. [40] L. W. Anson, R. C. Chivers, and K. E. Puttick, “On the feasibility of detecting precracking fatigue damage in metal matrix composites by ultrasonic techniques,” Composites science and technology, vol. 55, pp. 63–73, 1995. [41] S. Vanlanduit, P. Guillaume, and G. V. D. Linden, “Online monitoring of fatigue cracks using ultrasonic surface waves,” N DT &E international, vol. 36, pp. 601–607, 2003. [42] S. I. Rokhlin and J. Y. Kim, “In situ ultrasonic monitoring of surface fatigue crack initiation and growth from surface cavity,” International journal of fatigue, vol. 25, pp. 41–49, 2003. [43] S. Kenderian, T. P. Berndt, R. E. Green, and B. B. Djordjevic, “Ultrasonic monitoring of dislocations during fatigue of pearlitic rail steel,” Material Science and Engineering, vol. 348, pp. 90–99, 2003.

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[44] S. Gupta, A. Ray, and E. Keller, “Symbolic time series analysis of ultrasonic data for early detection of fatigue damage,” Mechanical Systems and Signal Processing, vol. 21, no. 2, pp. 866–884, 2007. [45] S. Gupta, A. Ray, and E. Keller, “Online fatigue damage monitoring by ultrasonic measurements: A symbolic dynamics approach,” International Journal of Fatigue, vol. 29, no. 6, pp. 1100–1114, 2007. [46] S. Gupta and A. Ray, “Real-time fatigue life estimation in mechanical systems,” Measurement Science and Technology, vol. 18, no. 7, pp. 1947–1957, 2007. [47] E. E. Keller, Real time sensing of fatigue crack damage for information-based decision and control. PhD thesis, Department of Mechanical Engineering, Pennsylvania State University, State College, PA, 2001. [48] M. T. Resch and D. V. Nelson, “An ultrasonic method for measurement of size and opening behavior of small fatigue cracks,” Small-Crack Test Methods, ASTM STP 1149, J.M. Larsen and J.E. Allison, Eds., American Society for Testing and Materials, Philadelphia, pp. 169–196, 1992.

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[49] P. B. Nagy, “Fatigue damage assessment by nonlinear ultrasonic materials characterization,” Ultrasonics, vol. 36, pp. 375–381, 1998. [50] J. H. Cantrell and W. T. Yost, “Nonlinear ultrasonic characterization of fatigue microstructures,” International Journal of fatigue, vol. 23, pp. 487–490, 2001. [51] J. L. Rose, Ultrasonic waves in solid media. Cambridge university press, 2004. [52] H. Sohn, C. R. Farrar, N. F. Hunter, and K. Wordan, “Structural health monitoring using statistical pattern recognition techniques,” Journal of Dynamic Systems, Measurement and Control, vol. 123, pp. 706–711, December 2001. [53] Z. K. Peng and F. L. Chu, “Application of the wavelet transform in machine condition monitoring and fault diagnosis: a review with bibliography,” Mechanical Systems and Signal Processing, vol. 18, pp. 199–221, 2004. [54] X. Lou and K. A. Loparo, “Bearing fault diagnosis based on wavelet transform and fuzzy interference,” Mechanical Systems and Signal Processing, vol. 18, pp. 1077– 1095, 2004. [55] J. M. Nichols, M. D. Todd, M. Seaver, and L. N. Virgin, “Use of chaotic excitation and attractor property analysis in structural health monitoring,” Physical Review E, vol. 67, no. 016209, 2003. [56] W. J. Wang, J. Chen, X. K. Wu, and Z. T. Wu, “The application of some non-linear methods in rotating machinery fault diagnosis,” Mechanical Systems and Signal Processing, vol. 15, no. 4, pp. 697–705, 2001. [57] L. Moniz, J. M. Nichols, C. J. Nichols, M. Seaver, S. T. Trickey, M. D. Todd, L. M. Pecora, and L. N. Virgin, “A multivariate, attractor-based approach to structural health monitoring,” Journal of Sound and Vibration, vol. 283, pp. 295–310, 2005. [58] V. Rajagopalan and A. Ray, “Symbolic time series analysis via wavelet-based partitioning,” Signal Processing, vol. 86, no. 11, pp. 3309–3320, 2006.

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[59] M. Buhl and M. Kennel, “Statistically relaxing to generating partitions for observed time-series data,” Physical Review E, vol. 71, no. 4, p. 046213, 2005. [60] J. P. Crutchfield and K. Young, “Inferring statistical complexity,” Physical Review Letters, vol. 63, pp. 105–108, 1989. [61] J. P. Crutchfield, “The calculi of emergence: Computation dynamics and induction,” Physica, vol. D, no. 75, pp. 11–54, 1994. [62] C. Rao, A. Ray, S. Sarkar, and M. Yasar, “Review and comparative evaluation of symbolic dynamic filtering for detection of anomaly patterns,” Signal, Image and Video Processing, DOI 10.1007/s11760-008-0061-8, 2008. [63] A. Khatkhate, S. Gupta, A. Ray, and E. Keller, “Life extending control of mechanical structures using symbolic time series analysis,” in American Control Conference, (Minnesota, USA), 2006.

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[64] A. Tarantola, Inverse Problem Theory. Society for Industrial and Applied Mathematics, 2005. [65] S. Suresh, Fatigue of Materials. Cambridge University Press, Cambridge, U.K., 1998. [66] MATLAB Wavelet Toolbox. Mathworks Inc, 2006. [67] S. Mallat, A Wavelet Tour of Signal Processing 2/e. Academic Press, 1998. [68] S. Gupta, Behavioral Pattern Identification for Structural Health Monitoring in Complex Systems. PhD thesis, Department of Mechanical Engineering, Pennsylvania State University, State College, PA, 2006. [69] M. Klesnil and P. Lukas, Fatigue of Metallic Materials. Material Science Monographs 71, Elsevier, 1991. [70] J. L. Bogdanoff and F. Kozin, Probabilistic models of cumulative damage. John Wiley, New York, 1985. [71] H. D. Brunk, An introduction to mathematical statistics,3rd Edn. Xerox Publishing,Lexington, MA., 1995. [72] G. W. Snedecor and W. G. Cochran, Statistical Methods. Eighth Edition,Iowa State University, Press, 1989. [73] R. Duda, P. Hart, and D. Stork, Pattern Classification. John Wiley & Sons Inc., 2001. [74] T. M. Cover and J. A. Thomas, Elements of Information Theory. John Wiley, New York, 1991. [75] A. Subbu and A. Ray, “Space partitioning via hilbert transform for symbolic time series analysis,” Applied Physics Letters, vol. 92, p. 084107, February 2008.

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[76] R. B. Bapat and T. E. S. Raghavan, Nonnegative Matrices and Applications. Cambridge University Press, 1997.

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Chapter 2

M ULTISCALE A PPROACHES TO FATIGUE C RACK G ROWTH FROM THE D EBONDING OF PARTICLE /D UCTILE -M ATRIX I NTERFACES Yong X. Gan∗ Department of Mechanical, Industrial and Manufacturing Engineering, College of Engineering, The University of Toledo, 2801 West Bancroft Street, Toledo, OH 43606, USA.

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Abstract Fatigue crack initiation and propagation due to the debonding of particle/ductilematrix interfaces were investigated at both macro- and micro-scales. First, macroscale approach was used to obtain the stress field in the elastic-plastic region within the matrix in front of the debonded particle. Continuum mechanics is applied in the derivation. Stress intensity solutions were obtained in the tension and compression stages in a fatigue cycle. In the plastic zone just ahead of the debonded particle, a microscale approach was used to get the stress solutions. In classical plasticity theory, the material property at the crack tip is considered to be isotropic and the maximum stress in the plastic zone is assumed to be the yield strength of the material. Our recent examination on the crack tip of ductile materials containing particle inclusions indicates that the deformation state at the crack tip is highly anisotropic. In order to describe such behavior, the deformation mechanisms of single crystal plasticity are enforced in this work. That is, the deformation of the material in the crack tip region is due to the motion of dislocation on different slip systems. Based on such a consideration, we assume that the stresses at the boundary between the elastoplastic region and the plastic zone propagate into the plastic zone. The magnitudes of the stress components are determined using the formulation of slip line theory. The primary slip lines are collinear with the dislocation motion directions. The second part of this work is specifically on the fatigue crack propagation. Once a short crack from the interface debonding starts growing, how to characterize the fatigue crack growth resistance becomes an important issue. We use a simulated crack (a through thickness notch) to study the fatigue crack growth. The specific energy of damage, a parameter which may be used ∗

E-mail address: [email protected]. Tel: +1-419-530-6007; Fax: +1-419-530-8206; Corresponding author.

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Yong X. Gan to characterize the fatigue crack growth resistance of the material, is obtained. The relationship between fatigue fracture surface morphology and the specific energy of damage is discussed as well.

Keywords: particle/ductile matrix interface, slip bands, debonding, crack initiation, propagation kinetics, crack layer, energy dissipation, fatigue damage tolerance.

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1.

Introduction

The pre-mature failure of structures in many engineering fields have drawn attention to the behavior of materials under fatigue loading. Previous work has shown that the fatigue property of composite materials is very sensitive to the interface bonding conditions. For example, Kawaguchi and Pearson [1] have studied the fatigue behavior of glass particle and fiber reinforced epoxy composites. They investigated the role of adhesion promoters in fatigue crack propagation by treating the glass reinforcements with silane-based adhesion promoters. It was shown that the type of adhesion promoters had a significant influence on the toughening mechanisms including interface dedonding. Under the moisture exposure conditions, glass spheres treated with n-butyltrimethoxysilane exhibited crack tip shielding mechanisms such as microcracking, shear yielding, fiber bridging and pull-out, and debonding. It is found that the matrix shear yielding is the prevalent toughening mechanism. The effect of bonding conditions on the performance of particle filled composite materials is still not fully understood. Kawaguchi and Pearson [2] reported that strong matrixparticle adhesion may lower the fatigue crack propagation resistance. While the studies on Si3 N4 nanoparticle filled epoxy composites under sliding wear conditions showed that the strong interfacial adhesion between Si3 N4 nanoparticles and the matrix reduced the wear rate of the composites [3]. It should be noted that severe wear is observed in unfilled epoxy dominated by fatigue-delamination mechanism. Nano-Si3 N4 enhanced the resistance to thermal distortion of the composites, and slowed down the tribochemical reactions that facilitate the removal of materials from the wear surfaces. Debonding of coated fiber reinforced composites under tension-tension cyclic load was investigated [4]. The bi-interfacial debonding (fiber/coating and coating/matrix) behavior was analyzed using a double shear-lag model. Based on this model, the debond growth rate and strain energy were calculated by finite element method. Non-uniform damage of coating materials was accounted in the analysis. There exists two-interface coupling in debonding. It was found that the strength and thickness of coating materials are the major factors controlling the bi-interfacial crack growth. Numerical simulation of progressive damage evolution in fiber reinforced composites was performed to understand interface stress statistics and the fiber debonding paths development [5]. A meso cell including several hundred inclusions was used to account for the micro structure statistics of composite. Both the local stress and effective elastic moduli of disordered fibrous composites were computed. Micromechanics based approaches have been used for debonding analysis [6-10]. Cavallini, Bartolomeo, and Iacoviello [6] investigated the fatigue damaging micromechanisms in three different ferritic-pearlitic ductile cast irons with the main focus on graphite nodules debonding. Chan, Lee and Nicolella et al. [7] studied the near-tip fracture processes of

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dental nanocomposites under cyclic loads. It is found that particle bridging, debonding at the poles of particle/matrix interface, and crack deflection around the particles are the major micromechanics responses to the fatigue loading. The effect of environmental conditions on the subcritical debond-growth rates was examined by Sharratt, Wang and Dauskardt [8]. Temperature and relative humidity are sensitive factors. Long term exposure to a moist environment resulted in a time-dependent decrease in adhesion between bisphenol polymer and SiNx or SiO2 substrate. A stress-dependent transport model was proposed to describe the moisture diffusion mechanism in the interface region. In Shi, Cui and Zhou’s work [9], three different interfacial degradation models including the shear lag model, the linear degradation model and a modified power degradation model were used to describe the bond decay at steel/concrete interface. The role of interfacial friction in resisting interfacial debonding was also addressed. Micro-level damage in discontinuous fiber reinforced composites were described by fiber/matrix interfacial debonding and fiber failure [10]. The Weibull damage law was used to predict the microscopic damage behavior of composites with different fiber contents and orientations. Crack initiation or small crack growth plays a critic role in interface debonding as shown by Okazaki and Yamano’s studies on the early growth of debonding crack in the interface between a Ni-based superalloy and ZrO2 thermal barrier layer [11]. In small crack growth, plasticity-induced crack closure was observed, but the effect of crack closure in fatigue crack growth predictions was less than the estimation by the classical approaches, which was demonstrated through numerical simulations by Jiang, Feng and Ding [12]. In addition to crack closure, the shear deformation of matrix ahead of a small crack slows down the interfacial debonding rate [13]. Small crack growth behavior is dependent on the stress level as shown by the studies of SiCp /Al composites under fatigue loading [14]. The incorporation of SiC particles increased the fatigue crack growth resistance of aluminium at a low applied stress. The composite reinforced with bigger sized particles (60 µm SiCp ) has higher fatigue crack growth resistance than the composite with smaller sized particle (5 µm SiCp ). At a high applied stress, the 60 µm SiCp /Al composite showed lower fatigue resistance than the 5 µm SiCp /Al composite. Such a converse trend is explained by the interaction and coalescence of multiple cracks. In the composite with finer particles, small cracks grew avoiding particles at the high stress level. There are less particles appearing on the fracture surfaces in small crack size region, indicating that the particle/interface debonding resistance is higher for the 5 µm SiCp /Al composite at high stress levels. Foley, Obaid, and Huang et al. [15] determined the microdebonding in E-glass/vinyl ester composites in the loading rate range from quasi-static to 50 mm/s. The average shear strength and energy absorbed during the microdebonding and frictional sliding processes were obtained. The results showed that the interface debonding is sensitive to the loading rate. The loading rate also affects the debonding of the γ/δ interface in stainless steels. The loading rate was changed by holding the specimens under continuous fatigue and creepfatigue conditions [16]. Under continuous fatigue loading condition, the crack initiation and growth mechanisms are varied depending on the directionality of the δ-ferrite fibers with respect to the loading direction. If the loading direction is parallel to the δ-ferrite fibers, dislocation motion along the slip lines induces crack initiation from the ferrites. If the loading direction is perpendicular to the δ-ferrite fibers, the debonding of the δ/γ interface becomes the major crack nucleation mechanism. Under creep-fatigue conditions,

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microcrack initiated via the γ/δ interface debonding. Microdebonding or subcritical debonding behavior is also dependent on surface chemistry [17] and temperature [18, 19]. To evaluate the surface chemistry effect, subcritical debonding of thin polymer layers from inorganic dielectrics was studied using selected amino- and vinyl-functional silane adhesion promoters [17]. Due to the surface modification, the failure occurs not at the interface but in a region very close to the interface. The effect of temperature on debonding is especially significant in metal matrix composite materials [20-34]. At elevated temperatures, thermomechanical fatigue accounts for the failure of these materials. Alternating plastic shearing of the interface takes place under combined mechanical and thermal stresses [18]. At low temperatures, metal matrices such as Al typically shows an initial hardening process, while at high temperatures, only cyclic softening is found [19]. Fatigue tests on reinforced titanium composites revealed various interface damage mechanisms [20-27]. Shear frictional sliding [20], interfacial debonding [21], fibre bridging [22], surface embrittlement [23], matrix ligament premature ductile shear [24], and crack deflection [25] are typical damage mechanisms observed. These damage mechanisms could occur simultaneously depending on loading modes, but bonding always exists and is considered as the major mechanism. A stress-based criterion for predicting the debonding behavior was proposed [22]. Rios, Rodopoulos and Yates [26] assessed the initial and final damage states caused by interface debonding and fiber bridging to determine the damage accumulation rates in SiC fiber reinforced titanium composite. Their method was used for damage tolerant fatigue design. Residual stiffness and the post-fatigued tensile strength as a function of microstructural damage were obtained through computer simulation, and the interfacial frictional stress and the critical crack length were also calculated [27]. Under combined thermal and mechanical fatigue loading, carbon fiber/Al and SiC fiber/Al composites were found to fail by a ratchetting mechanism, which is characterized by the progressive plastic deformation increasing with the number of cycles, even at stress levels far below the yield stress [28]. It is further found that the main phenomenon leading to composite failure is ratchetting at high load levels and interface degradation at low load levels. Short crack growth in a steel containing different particle inclusions including Al2 O3 , MnS and Ti3 N4 was studied by finite element method [29]. Crack-tip displacements and energy release rates were taken as the driving forces. It was found that the energy release rate is the highest for the Al2 O3 inclusion case with a short through thickness crack. Li and Ellyin [30] studied the fatigue damage and the localization in Al2 O3 particulate reinforced aluminum composites. The primarily damage forms are particle debonding, fractured particles and matrix cracks. Mesoscale reinforcement defects, such as a clump of large particles, were also found causing damage localization. These defects were assumed to be the reason for short crack initiation and extension. In Murtaza and Akid’s work on steel [31], it is reported that debonding at the matrix/inclusion interface is the major mechanism for the formation of short cracks. Stress redistribution at interfaces in alumina/aluminum multilayered composites was investigated [32]. The effects of interfacial debonding or of plastic slip in the metal phase adjacent to strongly bonded interfaces were considered. The results of stress measured around the crack reveal that debonding is much more effective than slip in reducing the stress ahead of the crack. Interaction of short fatigue crack with different

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types of particles was studied. Stronger interaction of fatigue crack with Si particles, as compared to SiC particles, was observed in particle reinforced A356 casting alloy [33]. Modeling fatigue debonding can be found in many literatures, e.g. [34-39]. In Gradin and B¨ acklund’s work [34], a unit cell model containing a steel bar and a co-centric epoxy cylinder was used to study the progressive de-bonding between the fibre and the matrix. Energy release rate was correlated to the interfacial debonding length. While in the work shown in [35-37], void formation and growth due to fatigue loading was characterized by the tensile stress at the interface. Three distinguishable debonding stages, two transient ones separated by a steady stage, were defined by Botsis and Zhao [38]. Stress intensity factor may be used to distinguish the steady and the transient stages because the total stress intensity factor was found to be approximately constant at the steady state. Debonding under different loading modes including mode I, mode II and mixed mode (I & II) was studied by Dessureautt and Spelt [39]. It was observed that the debonding rate was the greatest under mixed-mode conditions.

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In this work, the emphasis will be put on the fatigue crack initiation and propagation from the debonding of particle/ductile-matrix interfaces. Both macro- and micro-scale analysis will be performed. The macroscale approach based on continuum mechanics will be used to obtain the stress field in the elastic-plastic region within the matrix in front of the debonded particle. Treating the debonded region as a crack, the stress intensity solutions in the tension and compression stages in a fatigue cycle will be derived. In the plastic zone just ahead of the debonded particle, the microscale approach will be used to find the stress solutions. In the classical plasticity theory, the material property at the crack tip is considered to be isotropic and the maximum stress in the plastic zone is assumed to be the yield strength of the material. In the first part of this work, our recent examination on the crack tip of ductile materials containing particle inclusions will be presented. As will be discussed later, the deformation state at the crack tip is highly anisotropic. In order to describe such a behavior, the deformation mechanisms of single crystal plasticity will be enforced in this work. The deformation of the material in the crack tip region due to the motion of dislocation on different slip systems will be described. Based on such a consideration, we assume that the stresses at the boundary between the elastoplastic region and the plastic zone propagate into the plastic zone. The magnitudes of the stress components will be determined using the formulation of the slip line theory. The primary slip lines are assumed to be collinear with the dislocation motion directions. The second part of this work will be specifically on the fatigue crack propagation. Once a short crack from the interface debonding starts growing, how to characterize the fatigue crack growth resistance becomes an important issue. We will use a simulated crack (a through thickness notch) to study the fatigue crack growth kinetics. The specific energy of damage, a parameter which may be used to characterize the fatigue crack growth resistance of the material, will be evaluated. The relationship between energy release rate and the specific energy of damage will be discussed as well.

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2.

Crack Initiation Stage: Microscale Approach

2.1.

Model Development

A material containing particle under Mode I cyclic loading is shown in Figure 1(a). As well known, the major failure type is the crack propagation in the plane perpendicular to the loading direction. Considering an element containing just one particle as shown in Figure 1(b), this plane is coincidence with xoy plane. Here we treat the matrix as a considerably ductile material so that in-plane slip is the prevailing plastic deformation mechanism. S is the unit vector parallel to the slip direction. N is the unit vector along the slip plan normal. The evidence for such a case is shown in Figure 2. In this scanning electron micrograph, a MnS particle embedded in a pearlitic steel matrix can be seen. The particle was debonded from the matrix under tension-tension fatigue loading. Along the main crack propagation direction (marked as x-direction), two distinct slip regions are found. These regions are denoted as Region I and Region II. In each of these regions, persistent slip lines are clearly defined. What we are interested in is how the debonding occurred and what is the criterion for the debonding initiation.

Cyclic loading direction

z y φ

S

N

o Particle filled material

x

icle

Part

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Matrix

(a)

(b)

Figure 1. Sketch of cyclic loading on particle filled materials: (a) cyclic loading configuration, (b) the controlled volume containing a particle embedded in the matrix. Although there are also some other slip zones around the particle, the predominant slip activities that determine the main crack speed are from Region I and Region II. Therefore, in the following analysis, we only examine these two regions. To better understand the

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Region II

Region I

x

o

20 µm y

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Figure 2. An SEM image showing the slip zones around the debonded particle. The main crack propagation direction is coincidence with positive x-direction. debonding, we differentiate the partial debonding and the fully debonded states, as shown in Figures 3(a) and 3(b), respectively. In order to find the stress solution, the debonded region is considered as a crack. A stress intensity approach is applied here to find an approximate solution. Figures 3(c) shows both the global and the local coordinates for deriving the stress solutions in the slip regions, as will be detailed later.

2.2.

Analytical Solutions

In a tension-tension cycle, supposing that plane-strain conditions hold, the non-zero components of the stress field ahead of the particle in debonding are KI cos σxx = √ 2πr

      θ θ 3θ 1 − sin sin , 2 2 2

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(1a)

56 τxz σzz

Yong X. Gan       θ 3θ KI θ sin sin , = τzx = √ cos 2 2 2 2πr       KI θ θ 3θ =√ cos 1 + sin sin , 2 2 2 2πr

(1b) (1c)

σyy = µ(σxx + σzz )

(1d)

where KI is the stress intensity factor related to the particle shape and the crack length. In the simplest case, the particle can be considered in spherical shape. Thus, the stress p intensity factor for the partial debonding case as shown in Figures 3(a) is KI = 1.12 π √ π(a − c)σ∞ , while for the complete debonded case as shown in Figures 3(b) KI = 1.12 πaσ∞ , where σ∞ is the remote stress. π Other stress components are zeros, i.e. τxy = τyx = τyz = τzy = 0. If part of compression loading exists in a complete fatigue cycle, we assume that during the compression loading, arrest of fatigue crack occurs, which would not contribute to the particle/matrix interface debonding. Assuming the crack tip is in the fully plastic state, then the following yielding criterion holds NΣS = ±τi , (2) where τi is the shear strength of the ith slip system, τi = τI for Region I and τi = τII for Region II. N is the surface normal of the slip plane, S is a unit vector along the slip direction. If the dislocation motion is along positive S, the right hand side takes positive τi , while in the case that the slip occurs along negative S, the negative sign is kept on the right y

y

ψ

ξ

y

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c 2b

I

I

x 2b

o a

II

x I

o II

a

S

N

φI

o

x II

(a)

(b)

(c)

Figure 3. Sketches for particle/matrix interface debonding analysis: (a) partial debonding, (b) complete debonding, (c) configuration of global, local coordinates related to the slip direction and slip plane normal vectors.

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hand side. Σ is the stress tensor, which is given by:   σxx 0 τxz Σ =  0 σyy 0  . τzx 0 σzz The components of N are Nx , Ny and Nz , and S has the components: Sx , Sy and Sz . Since only the in-plane slip is considered in this work, the z-components for both N and S are zeros. Therefore, the yield condition can be rewritten as    σxx 0 τxz Sx (Nx Ny 0)  0 σyy 0   Sy  = ±τi . τzx 0 σzz 0 From the above expression, we have the long form Nx σxx Sx + Ny σyy Sy = ±τi ,

(3)

The components of N and S can be represented by the direction cosine of the slip plane normal or the direction cosine of the slip direction, i.e. Sx = cos φi ,

(4a)

Sy = sin φi ,

(4b)

Nx = − sin φi ,

(4c)

Ny = cos φi ,

(4d)

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where φi = φI for Region I and φi = φII for Region II. Substituting these relations into Eq. (3) yields σxx − σyy − sin(2φi ) = ±τi , (5) 2 Eq. (5) provides the yield function related to the slip angle and the stress field when the material is in a fully-plastic state. For the partial debonding case, along the radial line θ = 0, the in-plane stress state is expressed as KI σxx = √ , 2πr∗

(6a)

τxy = 0,

(6b)

2τi + σxx . sin(2φi )

(6c)

σyy = ±

∗ where p r is the distance from the origin to an arbitrary point on the θ = 0 radial line, KI = π(a − c)σ∞ . In a tension cycle, σ∞ takes positive values, while in a compression cycle, σ∞ takes negative values. For a completely debonded case, the stress intensity factor √ needs to be modified, i.e. KI = 1.12 πaσ∞ . Since the stress field along the radial line π θ = 0 is obtained, it is fairly straightforward to find the stress state within the slip region. 1.12 π

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Yong X. Gan

One of the ways is to use the slip line analysis[40] to solve the stress components in slip Region I and Region II. Referring to Figures 3(c) and (4), the local coordinates are chosen so that the ξ-axis is parallel to the α-lines and the ψ-axis is parallel to the β-lines. It is noted that the two family of slip lines, α-lines and β-lines, are mutually perpendicular with each other. For any arbitrary point, P in Region I, the coordinates are (xP , yP ) or (r, θ). The α-line passing through point P intercepts the x-axis at point Q and the β-line intercepts point R. The distance of point Q from the origin O is xQ and the distance of point R from the origin O is xR . For single slip, the stress within a slip zone may be determine by enforcing the equilibrium conditions. This allows us to find the stress state inside Region I through the propagation of the boundary conditions at the line segment OR into the slip zone. In other words, the local stress component σξξ at point P takes that same value of that at the boundary point Q and σψψ at point P comes from the value of that at the boundary point R. The stresses at point Q and R in ξ-ψ coordinates may be found by the following transformation σxx + σyy σxx − σyy σξξ = + cos 2φI + σxy sin 2φI , (7a) 2 2 σxx − σyy σxx + σyy − cos 2φI − σxy sin 2φI , (7b) σψψ = 2 2 σxx − σyy σξψ = − sin 2φI + σxy cos 2φI . (7c) 2 Substituting Eq. (6) into Eq. (7) yields     σxx − σyy σxx + σyy + cos 2φI + (σxy )Q sin 2φI , (σξξ )Q = 2 2 Q Q     σxx + σyy σxx − σyy (σψψ )R = − cos 2φI − (σxy )R sin 2φI , 2 2 R R

(8a)

(8b)

σxx − σyy sin 2φI + σxy cos 2φI . (8c) 2 Considering the case that the slip is along positive S direction so that the right hand side of Eq. (2) takes a positive value, Eq. (9) may be simplified as, Copyright © 2009. Nova Science Publishers, Incorporated. All rights reserved.

σξψ = −

KI (1 − cos 2φI ), (σξξ )Q = p 2πxQ (σψψ )R = √

KI (1 + cos 2φI ), 2πxR

(σξψ )Q = (σξψ )R = τI .

(9a)

(9b) (9c)

Since the line equations for the two lines P Q and P R are y − yP = tan(φI )(x − xP ) and y − yP = − cot(φI )(x − xP ), respectively. The x coordinates for point Q and R can be expressed as:   cos(φI + θ) sin θ = xQ = r cos θ − r, (10a) tan φI cos φI

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cos(φI − θ) r. cos φI

59 (10b)

The stress state at point P is given by (σξξ )P = (σξξ )Q , (σψψ )P = (σψψ )R , and (σξψ )P = (σξψ )Q = (σξψ )R , which can be obtained by substituting Eq. (10) into Eq. (9) and expressed as √ KI cos φI

√ 2KI cos φI

√ KI cos φI

√ 2KI cos φI

(1 − cos 2φI ) = p sin2 φI , (σξξ )P = p 2πr cos(φI + θ) 2πr cos(φI + θ)

(11a)

(σψψ )P = p (1 + cos 2φI ) = p cos2 φI , 2πr cos(φI − θ) 2πr cos(φI − θ)

(11b)

(σξψ )P = τI .

p 1.12

(11c)

1.12 √

where KI = π π(a − c)σ∞ for partial debonding and KI = π πaσ∞ for complete debonding. The stress components can be expressed in the global coordinate system through the tensorial transformation σrr = σθθ =

σξξ − σψψ σξξ + σψψ + cos[2(θ − φI )] + σξψ sin[2(θ − φI )] 2 2

σξξ + σψψ σξξ − σψψ − cos[2(θ − φI )] − σξψ sin[2(θ − φI )] 2 2 σrθ = −

σξξ − σψψ sin[2(θ − φI )] + σξψ cos[2(θ − φI )]. 2

(12a) (12b) (12c)

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The some formulism can be used to find the stress state in slip Region II, but τII (the critical shear stress for materials in this region) and φII (the slip angle in Region II) should be used. Referring to Figure 5, for any point U (xU , yU ) in slip Region II, the polar coordinates are (r, θ). The α-line passing through an arbitrary point U in this slip region intercepts the slip boundary at point V and the β-line intercepts x-axis at point W . Since the stress component (σξξ )U takes the same value of that at point V , and (σψψ )U = (σψψ )W . (σψψ )W = √

KI (1 + cos 2φII ). 2πxW

(13)

(σξξ )V can be evaluated from (σxx )V , (σxy )V , and (σyy )V . Assuming that the angle sector boundary and x-axis is equal to θII , substituting θ = θII into Eq. (1a) yields       KI θII 3θII θII (σxx )V = √ 1 − sin sin (1 − cos 2φII ). (14) cos 2 2 2 2πxV In the global coordinates, (τxy )V = 0. (τyy )V is found from the yield condition, i.e. (σyy )V = ±

2τII + (σxx )V . sin(2φII )

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60

Yong X. Gan

Again, we assume that the dislocation moves along positive S direction, the first term on the right hand side of Eq. (15) takes positive sign. In local coordinate systems, (σξξ )V is expressed as: KI (σξξ )V = √ cos 2πxV



θII 2

     θII 3θII 1 − sin sin . 2 2

(16)

The equations for the two lines, U V and U W in Figure 5 are y − yU = − tan(φII )(x − xU ) and y − yU = cot(φII )(x − xU ), respectively. The x coordinates for point V can be found from the intercept of line U V and the slip boundary, and xR can be found from the intercept of line U W on x-axis, i.e. xV =

yU + xU tan φII sin(φII + θ) = r, tan φII − tan θII cos φII (tan φII − tan θII ) xW = xU − yU tan(φII ) =

(17a)

cos(φII − θ) r. cos φII

(17b)

Therefore, the stress components for point U in parametric forms are: KI cos (σξξ )U = √ 2πxV



θII 2

(σψψ )U = √



1 − sin



θII 2



sin



3θII 2



,

KI (1 + cos 2φII ), 2πxW

(σξψ )U = τII .

(18a)

(18b) (18c)

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The stress components in Region II can be expressed in the global coordinate system through the following tensorial transformation σrr =

σξξ + σψψ σξξ − σψψ + cos[2(θ + φII )] + σξψ sin[2(θ + φII )] 2 2

(19a)

σθθ =

σξξ + σψψ σξξ − σψψ − cos[2(θ + φII )] − σξψ sin[2(θ + φII )] 2 2

(19b)

σrθ = −

σξξ − σψψ sin[2(θ + φII )] + σξψ cos[2(θ + φII )]. 2

(19c)

Following the above derivations, the general forms of the stress state ahead of a debonded particles can be determined. For the specific case as shown in Figure 2, the slip angle for Region I, φI , is about 45◦ . The slip sector boundary is defined by the line equation θ = 30◦ . In the second slip region, Region II, the slip angle, φII , is only 15◦ . The slip sector boundary is along the radial line, θ = −30◦ . Substituting these parameters into Eqs. (11) and (18) respectively, the numerical values of stresses at point P in Region I and point Q in Region II can be determined.

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y ψ

ξ

S α− lin

e

N Slip boundary

e lin β−

r P

o

φI

θ Q

R

x

Figure 4. Sketches for finding the analytical solutions to stress in Region I.

y

ψ

2π−θ

r V

N

W

x

φ II U

S

Slip boundary

β−lin e

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o

α−lin e

ξ

Figure 5. Sketches for finding the analytical solutions to stress in Region II.

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3.

Yong X. Gan

Propagation Stage: Macroscale Approach

In the propagation stage, the main focus will be on the crack growth kinetics. The debonding was assumed to fully develop into a through thickness crack located at one edge of the specimens. In order to examine the shear shear strength effect, the materials tested are cut from different layers in a railhead. The railhead material has pearlite structure with particle inclusions. Interface debonding of particles such as MnS from the pearlite matrix is a major failure problem. In the crack initiation stage, the deformation of the matrix follows plastic shear mechanisms and slip occurs along the interface of those laminar ferrite and cementite, which can be characterized using the microscale approach as introduced in the previous part of this work. The fatigue damage tolerance of these layers was evaluated as the value of the specific energy of damage, γ, generated from the modified crack layer (MCL) theory. It will be demonstrated that the fatigue crack growth kinetics and the fatigue damage tolerance of the rail are sensitive to the location of the layers. The top layer with head hardening has the smallest value of γ (1020 kJ/m3 ), while the crack growth speed is the highest among the three layers The middle layer and the bottom layer have very closed values of γ, 1300kJ/m3 and 1350kJ/m3 , respectively. Microscopic examination on the fracture surface of the specimens from these layers reveals two distinctive regions which correspond to the two stages of crack growth kinetics; stable crack growth and unstable crack growth. The fatigue fracture mechanisms are also dependent on the position of the layers, which will be revealed by damage species on the fractured surface of the specimens from different layers of the rail. The fracture mechanisms change from more brittle-like for specimens from the top layer to ductile for specimens from both the middle layer and the bottom layer. Fatigue is one of the most common reasons resulting in the premature failure of rails which may cause very serious accidents such as derailment. To prevent fatigue failure, materials selection and process control are very important. Studies have demonstrated that different manufacturing procedures such as quenching, hot rolling etc. can change the mechanical properties of rails [41]. Also, the chemical composition of the steels has very important effect on the properties of rails. There are various materials which can be used for rail manufacturing. Copper alloys may be used as rail materials [42] to meet the requirements of high electrical conductivity for trolley bus tracks. In most cases, steels are widely used for railway track manufacturing because steels are cost-effective. Most rails are rolled from open-hearth basic steel, and contain from 0.50 to 0.60% in weight of carbon, 0.90 to 1.20% manganese, 0.10 to 0.30% silicon, and no more than 0.06% of sulphur and phosphorous. Alloy steels have been used for heavy duty rails at points in crossings and on sharp curves, etc. It is also true for the manufacturing of some special parts for rail stabilization. Some of these parts include bearing-plate, fish-plates, spikes, fang-bolts, hook-bolts, Stewart’s key, chair, open coiled key etc. Most of the common used alloy steels can be divided into three classes. They are medium manganese steel, high manganese steel, and chromium steel. A typical medium manganese steel has the composition as follows. Manganese content is from 1.10 to 1.40 wt%. Carbon is in the range from 0.45 to 0.55 wt%. The content of silicon is bigger than 0.05 wt%, while the content of phosphorous and sulphur is less than 0.05 wt%. Recent development in rail materials have resulted the widely use of premium rail steels. As compared with other rail steels, the carbon content of the premium rail steel is in the

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range from 0.72% to 0.82%. The microstructure of the steel is mainly pearlite which is favorable for increasing the wear life of the rails. The premium steel can be made by open hearth, basic oxygen, or electric furnace followed by a continuous casting process into hot topped ingots. In order to remove any injurious segregation and pipe, rolling has to be used in the manufacturing processes. Another important procedure in manufacturing premium rail is the head hardening heat treatment. The as-rolled rails will be on-line hardened to achieve head hardening effect. Such heat treatment can introduce a microstructural gradient along the vertical direction of the rail head. The yield strength of the top layer of the rail head increases after on-line air quenching head hardening heat treatment. The tendency of stress softening can be prevented. Thus, the deformation and wear resistance of the rail under service loading conditions can be effectively improved. It is expected that the heat treatment will increase the wear life of rails by about a factor of three. Beynon, Garnham, and Sawley [43] have studied three pearlitic rail steels to examine the head hardening effect on the wear behavior of the steels. Crack networks were formed in the specimens tested at nominal contact stresses of 1200 MPa and 1500 MPa. Isolated cracks were produced in the specimens tested at 1800 MPa. The head-hardened grade eutectoid steel had the best resistance to rolling contact fatigue. The naturally hard eutectoid steel was the next best, whilst the lowest strength steel gave the worst performance. The effect of heat treatment on fatigue damage is very complicated. Head hardening treatment may exert adverse effect on the fatigue properties of the rail steels as reported by Hellier and Merati [44]. In their work, the fatigue testing was conducted on single edge notched rectangular bar specimens machined from the head of a length of virgin head hardened heavy haul railway rail, manufactured using an ingot route. An increasing load technique was employed for threshold determination with small positive stress ratio R. The √ tests resulted in a fatigue threshold of 11.4 MPa m for the stress ratio R = 0.11 which is √ lower than the established fatigue threshold for standard carbon rail steel of 12-14 MPa m. In shear deformation dominated fatigue, the heat treatment displayed propitiate acts on increasing the fatigue damage tolerance. Rail rolling contact fatigue studies have been investigated based on the performance of naturally hard and head hardened rails in track. It is expected that rolling contact fatigue can be reduced substantially by the use of harder rails. Fatigue defects occurred in the form of head splitting on the high rails of the curves; the phenomenon was much more pronounced for the non-heat treated steel (natural hard). Spalling defects also occurred in the case of the natural hard steel. The natural hard steel exceeds its ductility limit earlier because of pronounced cold deformation so that plastic shear deformations and ultimately cracking and spalling occur. On contrary, the head hardened rails prove far more resistant to contact fatigue damage due to their higher yield point. The service life of head hardened rails can therefore be expected to be longer than that for the natural hard rails. However, for defective rails containing debonded particles, the lifetime of the fatigue is controlled by the propagation rather than the initiation. The crack growth behavior is generally very different from the initiation behavior. It is necessary to evaluate the microstructure related fatigue damage tolerance for defective rails. This will be helpful for the prediction in the service lifetime of the rails. Criteria such as fracture toughness and the J-integral are often used for the evaluation of fracture resistance of various engineering materials. They are also very important in understanding the fatigue crack initiation behavior and have found engineering applications

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Yong X. Gan

under various service conditions. In this part of our work, evaluation of the fatigue crack propagation behavior based on experimentally determined irreversible work and energy dissipation will be presented. The energy dissipated into damage formation will be considered as the crack growth driving force. A materials parameter, the specific energy of damage will be used as the fatigue damage tolerance criterion. Correlation between the fatigue damage tolerance and the microstructure of the rail will also be made.

3.1.

Model Development and Fatigue Damage Tolerance Formulation

In this part, we follow the framework of the modified crack layer (MCL) model [45] to establish a criterion for evaluating the fatigue damage tolerance in the propagation stage. The MCL model has been applied for analysis of fatigue crack propagation behavior of various materials including metallic alloys, polymers, adhesive joint bonds and cement materials [46-48]. Considering the fatigue crack from particle/matrix interface debonding and its surrounding active zone in the material as a thermodynamic entity, the crack layer and fatigue damage zone are aligned within the xoy plane as shown in Figure 1(b). The following relationship can be obtained based on entropy and energy balance considerations. T S˙ = (J ∗ − aγ)

da + D. dN

(20)

where T is the ambient temperature and S˙ is the rate of change of the entropy of the system comprising the crack and the surrounding damage zone. J ∗ is the energy release rate. γ is the specific damage of energy. a is the crack length. da/dN is the crack speed. N is the number of cycles. D is the rate of energy dissipation into damage formation associated with the active zone evolution. At minimum entropy, T S˙ = 0. Eq. (20) can be rearranged as da D = . (21) dN aγ − J Under the load control fatigue testing, the energy release rate J ∗ can be evaluated by 1 ∂P . (22) B ∂a where P is the potential energy (area above the unloading curve) at each crack length a, and B is the specimen thickness. The cyclic rate of energy dissipation, D associated with active zone evolution can be evaluated by the difference between the hysteresis energy related to crack propagation and the hysteresis energy dissipated into the bulk of the material. It can be expressed as: 1 (23) D = (Hn − Hu ). B where Hn is the hysteresis energy for the notched specimen at any crack length and Hu is the hysteresis energy for the unnotched specimen with the same dimension as that of the notched one. Rearranging Eq. (21) yields the important equation of the MCL model as follows:

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J∗ =

J∗ D = γ − da . a a dN

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(24)

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65

The quantities J ∗ , da/dN , and a, can be obtained from fatigue crack propagation experiments. The relationship expressed in Eq. (24) will be plotted in a two dimensional domain, directly giving the value of the specific energy of damage γ which is the intercept of the straight line. Calculating the specific energy of damage γ from the change in work approach was given in [45, 49]. The step-wise analysis of data is similar to the approach as shown in [45, 49]. γ is a material property related parameter which is obvious by examining the basic formulation of the MCL model as shown in Eq. (24). As the fatigue crack propagates, the energy release rate increases, thus the change of the left term J ∗ /a in the MCL model as shown in Eq. (24) can be leveled by both the increasing of J ∗ and the crack length, a. The variation of the term in the right side of Eq. (24), D/[a(da/dN )], depends on several factors. These are the crack length, a, the crack speed, da/dN and D, the cyclic rate of energy associated with the damage formation. On one hand, it is clear that the crack speed changes with the crack length. Since the fatigue is under stress control condition, the larger the crack length, the higher the crack speed. On the other hand, there exists some relationship between D and crack length, a. From energy balance analysis, it is clear that the value of D will increase with the increase in the crack length, a. Thus, the variation of D can be well balanced by the change in both a and da/dN . If D takes a higher value, it corresponds to a larger crack length and a higher crack speed. A higher value of D indicates an increase in the irreversible energy into the active zone of the material for damage formation, which means more energy is absorbed by the specimen. The more the absorbed energy, the faster the crack moves. Thus, increasing in D results in the increase of both a and da/dN . This would keep the term D/[a(da/dN )] not to change. Instead, it varies coherently with the left term, J ∗ /a, in Eq. (24) so that the summation of J ∗ /a and D/[a(da/dN )] can be kept as a constant in the entire energy release rate range. Thus, on the J ∗ /a vs D/[a(da/dN )] plot, a straight line which is almost parallel to the D/[a(da/dN )] axis can be obtained.

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3.2.

Material and Experimental

The premium rail steel used in this work was a medium carbon steel which contains 0.72 0.78% C, 0.60 1.0% Mn and small amount of other alloy elements. The rail was form by hot rolling of the ingot followed by air quenching to achieve head hardening effect. The rail head was sliced longitudinally into thin layers for the preparation of test specimens. Totally fourteen layers with the same thickness of 2 mm were sliced from the entire rail head among which three slices from representative locations were chosen for studies. One of these was the top layer (Layer A) containing the head hardening structure. Another layer was from the middle of the head (Layer B). The third layer (Layer C) was from the bottom of the head which is very close to the web. The schematic representation of the locations of these layers is shown in Figure 6. Both notched and unnotched specimens from Layers A, B, and C were prepared for static and fatigue tests. Rectangular unnotched specimens with 75 mm length, 18 mm width and 2 mm thickness were machined. At the center of one free edge of the specimens, a 60◦ notch was introduced using a very sharp triangle file. The notch depth was about 1.8 mm so that the notch depth to sample width ratio (a/w) was 0.1. Tension-tension fatigue pre-cracking was performed until the crack length reached 3.0 mm. This type of notch is used to simulate the defect caused by the coalescence of many entirely debonded particles. The fatigue crack propagation tests were conducted at ambient

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Yong X. Gan

temperature of 25◦ under load control conditions using a frequency of 1 Hz. The maximum stress is 200 MPa, and the ratio of minimum stress to maximum stress is 0.1. The change in stress was kept in sinusoidal wave form. The crack length at various intervals of number of cycles was recorded during the tests. A video camera with a zoom lenz was used to view the crack tip region and measure the crack length. For each layer, nine repeated tests were performed and data from typical specimen was used for the fatigue crack propagation analysis and fracture surface examination. The fracture surfaces of fatigue fractured specimens were examined using a Hitachi S-2150 scanning electron microscope (SEM).

3.3. 3.3.1.

Results and Discussion Shear Strength

Static tests on specimens cut from layers A, B and C were performed to estimate the critical shear stress of each layer. From the stress strain curves, we determined the yield strength of each layer. It is found that for layer A, the yield strength is 640 MPa. Layer B showed the yield strength of 420 MPa, and layer C has the yield strength of 440 MPa. The difference in the yield strength from layer to layer can be explained is due to the heat treatment of the rail. Since the top layer, layer A, has a much higher cooling rate than that of the middle layer, layer B, or the bottom layer, layer C, hardening from the air flow cooling is more obvious in layer A. Consequently, it has a significantly higher yield strength than other layers. Layer B and layer C do not show too much difference in the yield strength. From the yield strength of these layers, their critical shear strength, τc , may be approximated by τc = σys /2, where σys stands for the yield strength. Therefore, the critical shear strength for layer A is about 320 MPa. For layer B, τc =210 MPa, and for layer C 220 MPa. Layer A Rail Head Layer B

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Layer C

Web

Rail Base

Figure 6. Schematic representation of the rail head and the locations of the three layers (Layer A, B and C) for test specimen preparation.

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67

Fatigue Lifetime

A plot of the fatigue crack length, a, versus the number of loading cycles, N , for the three layers is shown in Figure 7. From the results shown in Figure 7, it can be seen that the total fatigue lifetime of the top layer (Layer A) is approximately 16,000 cycles. The lifetime of the middle layer (Layer B) is over 24,000 cycles. The bottom layer (Layer C) has the longest lifetime of 32,000 cycles among all the three layers. The critical value of the fatigue crack for the top layer is about 10 mm. The middle layer displayed a critical fatigue crack length of 12.8 mm. The critical crack length for the bottom layer is about 11 mm. It is evident that it takes less cycles for the crack in the top layer to grow from 3 mm to the critical length. Such a tendency can be explained by the change in microstructure from the top layer to the inner layers.

14

Layer B 12

Layer C

Crack Length, a (mm)

Layer A 10

8

6

4

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2

0

1

2 Number of Cycles, N

3

4 4

x 10

Figure 7. Fatigue crack length, a, versus the number of cycles, N , for the specimens from the three layers of the rail head. In the head hardening quenching process, there exists difference in cooling speed along the vertical direction of the rail head. Faster cooling normally results in finer pearlite grains and smaller space between the sub-structure of ferrite and cementite in the pearlite crystals. Since the cooling speed in the middle layer and the bottom layer was significantly lower than that of the top layer during the head hardening process, a more supercooling state can be established inside the top layer. The tendency of crystal initiation in the top layer is

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much higher than that in the middle layer. However, the equilibrium growth process can not be finished due to the fast cooling. Thus the grain size in the top layer is smaller than that in the middle layer. Such a microstructure profile may be ideal for the increase of the crack initiation lifetime of the top layer due to the less tendency of stress softening of this layer. However, the top layer with higher hardness has less capability to withstand inelastic deformation. Thus shorter fatigue propagation lifetime can be found for the top layer, as shown in Figure 7. 3.3.3.

Fatigue Crack Propagation Speed

The slope of the curves in Figure 7 is taken as the crack speed at each crack length. The relationship of crack speed, da/dN , and crack length, a, is shown in Figure 8. The curves illustrated in this figure demonstrate change in crack growth kinetics with the increasing of the fatigue crack length. In the stage of pre-cracking where the crack length is less than 3 mm, crack initiation and threshold behavior can be observed. After the threshold stage, the fatigue crack propagation (FCP) kinetics can be divided into two different stages. The first stage is the crack stable growth stage. In the second stage, an unstable crack propagation kinetics was observed. All the three curves approached asymptotic values in the second stage.

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For the specimens from the three layers; the crack growth speed varied appreciably. The top layer displayed the highest crack speed in both stages. The crack speed in the very beginning of the first stage for the top layer is about 5.0×10−8 m/cycle. In the entire crack stable propagation stage the crack speed ranges from 5×10−8 m/cycle to 6×10−6 m/cycle. In the second stage, the specimen from the top layer exhibits crack speed larger than 1×10−5 m/cycle. In the beginning of the first stage, the crack speed for the middle layer is very closed to that of the top layer. However, with the increasing of the crack length, the difference in crack speed between the top layer and the middle layer becomes evident. The middle layer displayed a lower crack speed than that of the top layer. The bottom layer demonstrated the threshold behavior in the very beginning of the first stage. In most of the fatigue crack growth range, the crack speed of the bottom layer is lower than that of the middle layer.

3.3.4.

Energy Release Rate

The potential energy, P , was calculated from the loading and unloading curves recorded at intervals of number of cycles as the area above the unloading curve. On this basis, the relationship between the potential energy and the fatigue crack propagation length, a, was established. The relationship between potential energy and crack length was used for the determination of energy release rate using Eq. (22). Figure 9 illustrates the value of energy release rate, J ∗ , as a function of the FCP crack length for the three layers from the premium rail steel. With the increase of crack length, a, the values of J ∗ for these specimens from the three layers are also increase. A critical value of the energy release rate Jc∗ for each layer of the rail can be determined from the results shown in this figure. For the top layer, this value is about 12 kJ/m2 . The middle layer and the bottom layer displayed very closed values of the critical energy release rate.

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−4.5 Logarithmic Crack Speed, lg(da/dN) (m/cycle)

Layer A −5

Layer B

−5.5

Layer C

−6

−6.5

−7

−7.5

−8

2

4

6 8 10 Crack Length, a (mm)

12

14

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Figure 8. Fatigue crack propagation speed, da/dN , versus crack length, a, for the specimens from the three layers of the rail head. The value of Jc∗ for the middle layer and the bottom layer is 17 kJ/m2 approximately. This indicates that due to the increase in hardness, the top layer has a lower value of the energy release rate at any given crack length than the middle layer and the bottom layer have. The effect of microstructure on the energy release rate displayed very similar trend as found in the change of the fracture toughness with the microstructure [50]. The top layer has the lowest fracture toughness due to the head hardening heat treatment, while the middle layer and the bottom layer demonstrated almost the same values of the fracture toughness [50]. It is reasonable to say that the layer with higher fracture toughness also demonstrated a higher dynamic energy release rate for the premium rail steel. 3.3.5.

Hysteresis Energy

The quantity of hysteresis energy for both notched and unnotched specimens, Hn and Hu respectively, is calculated from the area of hysteresis loops recorded during the fatigue experiments. In this study, the hysteresis area was measured, using a planimeter, as the area within the respective hysteresis loops. The hysteresis energy corresponding to each cycle was then calculated. In each of the specimen from the three different layers of the rail steel, Hn includes the energy expended on damage processes associated with crack growth and

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20

Energy Release Rate, J* (J/m2)

18

Layer C

16

Layer B

14 12

Layer A

10 8 6 4 2

4

6

8 10 Crack Length, a (mm)

12

14

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Figure 9. Energy release rate, J ∗ , versus the crack length, a, for the specimens from the three layers of the rail head. energy dissipated into the bulk of the material. The hysteresis energy calculated based on the unnotched specimen, Hu , comprises the only exponent of energy dissipated into the bulk of the materials. Both Hn and Hu are irreversible work related to the fatigue crack propagation. The relationship between hysteresis energy and the number of cycles, N , for both notched specimens and unnotched specimens is shown in Figures 10(a), (b) and (c) respectively for the top layer, the middle layer and the bottom layer. Hn for the notched specimens is higher than that of Hu , for the unnotched specimens from the same layer of the rail. In the given range of cyclic stress, the relationship between Hu and N is linear, while for the notched specimen, the hysteresis energy, Hn , increases remarkably with the increasing in the number of cycles. It is also noted that the microstructure and the related mechanical properties determine the energy dissipation behavior in the fatigue crack propagation process. This can be proven by examining the relationship between the tensile strength and the hysteresis energy for the unnotched specimens from different layers. As shown in [50], the tensile strength of the top layer is slightly higher than that of the middle layer and the bottom layer. Correspondingly, the hysteresis energy of the top layer is also lower than that of the middle layer and the bottom layer in the entire fatigue cycle range. A possible explanation for such behavior is that the higher the strength of the layer, the more elastic response involved in the given

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stress cycling range. Thus, the irreversible process such as inelastic response was reduced and lower hysteresis energy can be found for the layer.

3.3.6.

Cyclic Rate of Energy Dissipation into Active Zone Evolution

Based on the value of hysteresis energy for both notched and unnotched specimen and the relationship between a versus N , the quantity of D, the cyclic rate of energy dissipation into active zone evolution was determined. Eq. (23) was used for the calculation of D. The relationship of D and the crack length, a, for each layer of the premium rail steel is shown in Figure 11. The top layer, Layer A, displayed much higher value of the cyclic rate of energy dissipation into the active zone evolution. The other two layers, Layer B and Layer C demonstrated very similar behavior as can be found in Figure 11. For all of the three layers, it is evident that with the increase in crack length, the value of D increases monotonically. In the very beginning of the crack stable propagation stage, corresponding to the crack length of about 3.0 mm, the value of D is very small; which is only 3 (J/m·cycle). In the succeeding crack stable propagation stage, which corresponds to crack length range from 3.0 mm up to approximately 9 mm, change in the value of D was obvious especially for the top layer. This value changed from 3 (Joule/m·cycle) to about 25 (Joule/m·cycle). The value of D for the middle layer and the bottom layer did not increase so drastically as the top layer did. For both layers, D was doubled from 3 (Joule/m·cycle) to 6 (Joule/m·cycle). In the second stage corresponding to unstable crack propagation, the increase of D with crack length is much faster than that in the first stage (crack stable propagation stage). In this study, the fatigue test was under stress control, the bigger the D, the more the irreversible energy. Consequently, the rail steel specimen absorbed more energy in the fatigue damage process for the evolution of active zone. Thus, the damage zone in the top layer developed fast than that in the other two layers.

Notched

Hysteresis Energy, H

i

50 40

Unnotched 30 20 10

0

(a)

5 Number of Cycles, N

x 10

−3

50 40

Unnotched

30 20 10

10

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60

Hysteresis Energy, H i(J x 10

(J x 10 −3)

) −3

(J x 10

i

Hysteresis Energy, H

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60

70 )

70

70

0

4

(b)

5 Number of Cycles, N

50 40 30

x 10

Unnotched

20 10

10

Notched

60

0

4

5 Number of Cycles, N

10 x 10

4

(c)

Figure 10. Hysteresis energy versus the number of fatigue loading cycles, N , for both unnotched and notched specimens from the three layers of the rail head: (a) the top layer (Layer A), (b) the middle layer (Layer B), (c) the bottom layer (Layer C).

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Cyclic Rate of Energy Dissipation, D (J/m)

30

25

Layer A 20

Layer B

15

10

Layer C 5

0

2

4

6 8 10 Crack Length, a (mm)

12

14

Figure 11. The cyclic rate of energy dissipation, D, versus the crack length, a, for the specimens from the three layers of the rail head.

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3.3.7.

Specific Energy of Damage

The fatigue damage tolerance is evaluated by the specific energy of damage γ. The parameters γ was calculated using the MCL theory along with the experimental data previously generated including a, da/dN , J ∗ , and D. If the experimental results are in agreement with the MCL model of Eq. (24), a plot of J ∗ /a versus D/[a(da/dN )] should give a straight line. Indeed, based on the results presented in Figure 12, three straight lines which are almost parallel to the horizontal axis were obtained for the three layers with different microstructure from different locations inside the premium rail steel. The intercepts of the three lines give the values of γ for each layer. From the results shown in Figure 12, the applicability of the MCL model to describe the FCP behavior of the rail material has been demonstrated. Also, the value of γ, being a material property related parameter, is suitable for characterizing the fatigue damage tolerance. Due to the microstructure change from the top layer to the middle layer and to the bottom layer [50], the value of the specific energy for each layer is different. The top layer (Layer A), with head hardening heat treatment, demonstrated the lowest value of γ, while the bottom layer (Layer C) which is located very closed to the web, displayed the highest value of γ. The specific energy of damage of the middle layer, which has the lowest air

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cooling speed during heat treatment, is very closed to that of the bottom layer. For the top layer, the value of the specific energy of damage is about 1020 kJ/m3 . The middle layer which is a representative of the base material of the premium rail steel has a value of γ 1300 kJ/m3 . The values of γ of the bottom layer, 1350 kJ/m3 , is slightly higher than that of the middle layer. Since γ is a parameter characteristic of the fatigue damage tolerance, it can be used for evaluating the resistance to FCP of the different layers from the rail head. A smaller value of γ indicates lower resistance to FCP since less energy is required to cause a unit volume of the material to change from an undamaged state to a damaged state. Obviously, the top layer with the lowest value of γ, has the least resistance to the fatigue crack propagation under the given load control conditions. This can explain that if a defect exists inside in a corner or in the middle of the railhead, it will cause crack propagation towards the top of the head and could result in catastrophic fracture of the track. The dependency of the fatigue damage tolerance on the location inside the railhead provides a guideline for predicting the crack move direction and the lifetime of the rail track. It is also evident that the higher the shear strength of the layer, the lower the γ value because slip is more difficult. 3.3.8.

Fatigue Crack Propagation Kinetics

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2500

3

2000 1500 1000 500 0

5

6

7

2000 1500 1000 500 0

8

lg(D/[a(da/dN)]), (J/m 3 )

(a)

2500

3

Specific Energy of Damage, J*/a (J/m )

Specific Energy of Damage, J*/a (J/m )

2500

3

Specific Energy of Damage, J*/a (J/m )

The fatigue crack propagation speed versus the energy release rate for the three layers from the rail head is shown in Figure 13. The crack growth kinetics depends on the location of the layers. The top layer displayed the highest crack growth speed in the entire energy release rate range. In most part of the energy release rate range, for say, J∗ less than 12 kJ/m2 , the middle layer and the bottom layer have the crack speed very close to each other. In the energy release rate range of above 12 kJ/m2 , the middle layer has higher crack speed than the bottom layer has. It can also be seen from Figure 13 that the three curves display the similar two-stage crack growth behavior which are corresponding to the stable crack growth stage and the unstable crack growth stage of the specimens from the three layers. Such a phenomena of the fatigue crack propagation kinetics is basically in agreement with

5

6

7

2000 1500 1000 500 0

8

lg(D/[a(da/dN)]), (J/m 3 )

(b)

5

6

7

8

lg(D/[a(da/dN)]), (J/m 3 )

(c)

Figure 12. Plot for determining the specific energy of damage, γ, of the three layers of the rail head: (a) the top layer (Layer A), (b) the middle layer (Layer B), (c) the bottom layer (Layer C).

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the calculated results of crack speed, da/dN , versus crack length, a as shown in Figure 8. A threshold stage was observed only in the pre-crack initiation stage for the top layer and the middle layer. But it extended to the beginning of the stable crack propagation stage for the specimens from the bottom layer. In the stable crack propagation stage, the decreased acceleration in crack speed is an indicative of material damage within the area in front of the crack tip associated with fatigue crack propagation. This is in contrast with the Paris law in which the relationship between lg(da/dN ) and ∆K displayed a linear behavior in the entire stress intensity factor range. In the following section, fatigue damage evaluation will be presented based on the morphological examination.

−4.5 Logarithmic Crack Speed, lg(da/dN) (m/cycle)

Layer A −5

Layer B

−5.5

Layer C

−6

−6.5

−7

−7.5

−8

5

10

15 *

20

2

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Energy Release Rate, J (J/m )

Figure 13. The fatigue crack propagation speed versus the energy release rate for the specimens from the three layers of the rail head.

3.3.9.

Fatigue Fracture Surface Morphology

Fracture surface morphology examination was performed on the specimens from the three layers to identify the fatigue damage species. The fatigue fracture surface for a typical sample is schematically shown in Figure 14. The notch is at the left hand side of the sample. The fatigue crack propagation direction is from left to right. The fracture surface can be divided into two distinct zones according to the morphological features. Zone 1 is the crack stable propagation region. Zone 2 is featured by the unstable crack propagation. These two

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zones are related to the two stages of fatigue fracture kinetics, crack stable propagation and fast crack growth. Since fatigue damage is captured mainly in the stable crack propagation stage and the fracture surface morphology in the fast crack propagation region of the specimens from the three layers displayed very similar features to that of the specimens fractured under monotonic loading conditions, the analysis of fatigue damage mechanisms in the following section is on the morphological examination of the stable crack propagation region.

Debonded region

Fatigue crack propagation direction

M

Stable carck growth region

Unstable carck growth region

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Figure 14. Schematic representation of the fatigue fracture surface showing the two zones and the location for microscopic examination. The length of the stable crack growth region of a typical specimen from the top layer (Layer A) is around 5 mm. This region contains the fatigue damage species associated with the stable crack propagation. Typical features in this region include the debonding of inclusions, limited pulled-up ferrite strips along grain boundaries, and river patterns consisting cleavage facets, as shown in Figure 15, an SEM micrograph at 300X taken from location ”M” as shown in Figure 14. These features indicate a ductile-brittle mixed fracture mechanism related to the first stage of stable crack growth. Intergranular separation can be found in the left hand side of this picture. On the top section of this micrograph, well-pronounced river pattern can be seen. The cleavage was originated from the top site and swept over the rest area in the right part of this micrograph. The change in orientation of the pearlite columns may served as a mechanism to change the path of the cleavage propagation. The global view of the fatigue fractured sample from the middle layer can also be illustrated as Figure 14. The first region for this specimen is around 6 mm in length which is a little bit longer than that of the specimen from the top layer. Unlike the top layer, the middle layer (Layer B) demonstrates more ductile fracture surface features in this region, as shown in Figure 16, an SEM micrograph at 300X taken at location “M” in Figure 14. Generally, the fatigue damage species associated with the ductile crack growth events include ductile tearing, pulled-up pearlite lamella, limited microcracks and micro-voids coalescence as found in a fully pearlitic eutectoid steel and some metallic alloys [51-53]. From the micrograph

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Fatigue crack propagation direction

100 µm

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Figure 15. An SEM micrograph at 300X taken from the beginning of the stable crack propagation region of the specimen from the top layer, showing a ductile-brittle mixed fracture mechanism characterized by particle debonding, intergranular separation, cleavage and river pattern.

shown in Figure 16, the morphological features such as discontinuity inside grains, separation between cementite and ferrite laminae, inter-granular tearing ridges and secondary microcracks can be well identified. These features reflect the crack deceleration and indicate a considerably high energy consuming process associated with the crack propagation. By comparison on the surface morphological features shown in Figures 15 and 16. It is evident that a more severe fatigue damage mechanism and thus a higher energy consuming process related to the first stage of crack growth in the specimen from the middle layer than that from the top layer. That is why the middle layer has a higher fatigue damage tolerance (the specific energy of damage) than the top layer. Very similar to the middle layer (Layer B), the bottom layer (Layer C) demonstrates slip and debonding dominated fatigue failure mechanisms, which is shown in Figure 17, at 300X. The entire fracture surface of this specimen can still be divided into two regions according to the fatigue damage features. The first region (stable crack growth region) is around 7 mm in length. Figure 17, taken from location ”M” as shown in Figure 14, shows

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Fatigue crack propagation direction

100 µm

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Figure 16. An SEM micrograph at 300X taken from the beginning of the stable crack propagation region of the specimen from the middle layer, showing a ductile fracture mechanism characterized by ductile tearing, discontinuity inside grains, separation between cementite and ferrite laminae, pulled-up pearlite lamellae, secondary microcracks and micro-voids coalescence.

several large ductile tearing ridge lines aligned along the crack propagation direction. In addition, pulled up ferrite strips, microcracks and micro-voids coalescence can be found in Figure 17. These features reveal a considerable slip controlled plastic deformation and ductile fracture mechanism related to the first stage of stable crack growth. Like those features found in the first region of fatigue fractured specimen from the top layer, debonding, discontinuity inside grains, separation between cementite and ferrite laminae, inter-granular tearing ridges and secondary microcracks take most of the areas in this region. Obviously, such features related to the plastic deformation induced cracking deceleration and indicate a considerably high energy consuming process associated with the crack growth. This is the explanation for the lower crack speed and higher fracture resistance of the bottom layer than that of the top layer (Layer A), but very close to the value of that for the middle layer (Layer B).

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Fatigue crack propagation direction

100 µm Figure 17. An SEM micrograph at 300X taken from the beginning of the stable crack propagation region of the specimen from the bottom layer, showing a ductile fracture mechanism characterized by microcracks, pulled up ferrite strips and tearing ridges.

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4.

Conclusion

1. Stress field around debonded particles is obtained through the microscale approach incorporating both stress intensity analysis and slip line theory. The closed form solutions for two slip regions are derived. In an ideally plastic state, the stress conditions at the slip boundaries propagate into the slip regions. 2. Fatigue crack growth kinetics due to interface debonding and slip deformation is dependent on the heat treatment conditions. The material with a higher air cooling rate (the top layer in rail head in this study) shows higher crack speed than that without air quenching (both the middle layer and the bottom layer of the rail). For fatigue fractured specimens from the three layers, two distinctive zones, stable crack propagation zone and unstable cracking zone are found through fracture surface morphology analysis. 3. The fatigue damage tolerance of a material may be expressed as the specific energy

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of damage, a parameter characteristic of the fatigue fracture resistance of the material. The value of the specific energy of damage can be obtained using the formulism of the modified crack layer (MCL) theory. In this work we demonstrated that the fatigue damage tolerance of different layers within the rail head varies due to the difference in the harden state and shear strength. The top layer with the most severe head hardening heat treatment displays the least value of the specific energy of damage γ, 1020 kJ/m3 . Due to the less hardening heat treatment, the middle layer has a higher value of γ, 1300 kJ/m3 , than the top layer. This value is very closed to that of the bottom layer, 1350 kJ/m3 . 4. The fatigue damage species on the fractured surface of the specimens from different layers of the rail revealed change in fracture mechanism from brittle-ductile mixed type for specimens from the top layer to ductile for specimens from both the middle layer and the bottom layer. In the stable crack growth region, the fracture surface of the specimen from the top layer showed cleavage and inter-granular separation, revealing some brittlelike fracture mechanisms. The more ductile fracture surface features such as drawn-out pearlite lamella, tearing ridges and micro-cracks are found on the fracture surface of the specimens from both the middle layer and the bottom layer.

Acknowledgment This work is supported in part by the Office of Research and Development, University of Toledo through 2008 Summer Faculty Research fellowships. The author also acknowledges the support from The Center for Teaching and Learning at The University of Toledo through 2008 Summer Faculty Teaching Fellowships.

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Biosketch of the Author Dr. Yong X. Gan, joined the Department of Mechanical, Industrial and Manufacturing Engineering at The University of Toledo (UT) in 2007 as Assistant Professor. Before he came to UT, he was with Department of Mechanical Engineering at Cooper Union as Assistant Professor from September 2005 to August 2007. He received his undergraduate degree in Chemical Engineering in 1984 from Hunan University, Changsha, China. He received his MS and D.Eng. in Materials Science and Engineering from Beijing University of Aeronautics and Astronautics (BUAA), Beijing, China. He received his M.Phil. and Ph.D. in Mechanical Engineering in 2004 and 2005, respectively, from Columbia University. His major teaching and research activities are on materials processing, and microstructure and property characterization. Before came to USA, he was a faculty member in the Department of Materials Science and Engineering at Beijing University of Aeronautics and Astronautics and he took part in metal, ceramics and composite processing laboratory development. As a Research Scholar at Auburn University, Alabama, he worked on synthesis of conductive polymeric composite materials He also worked at Tuskegee University, Alabama, in the field of fracture, fatigue and failure analysis of polymers and advanced polymeric composite materials. He published more than fifty papers and book chapters in materials research and development. He is a registered Professional Engineer (P.E.) in the State of Alabama, USA.

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References [1] Kawaguchi, T.; Pearson, R. A. Composites Sci. Technol. 2004, 64, 1991-2007. [2] Kawaguchi, T.; Pearson, R. A. Composites Sci. Technol. 2004, 64, 1981-19889. [3] Shi, G.; Zhang, M. Q.; Rong, M. Z.; Wetzel, B.; Friedrich, K. Wear 2003, 254, 784-796. [4] Zhang, R.; Shi, Z. Int. J. Fatig. 2008, 30, 1074-1079. [5] Kushch, V. I.; Shmegera, S. V.; Mishnaevsky, L. Int. J. Solids Struct. Technol. 2008, 45, 2758-2784. [6] Cavallini, M.; Bartolomeo, O. D.; Iacoviello, F. Eng. Fract. Mech. 2008, 75, 694704. [7] Chan, K. S.; Lee, Y. D.; Nicolella, D. P.; Furman, B. R.; Wellinghoff, S.; Rawls, R. Eng. Fract. Mech. 2007, 74, 1857-1871. [8] Sharratt, B. M.; Wang, L. C.; Dauskardt, R. H. Acta Mater. 2007, 55, 3601-3609. [9] Shi, Z.; Cui, C.; Zhou, L. Europ. J. Mech. 2006, 25, 808-818. [10] Kabir, M. R.; Lutz, W.; Zhu, K.; Schmauder, S. Comp. Mater. Sci. 2006, 36, 361-366. [11] Okazaki, M.; Yamano, H. Int. J. Fatig. 2005, 27, 1613-1622. [12] Jiang, Y.; Feng, M.; Ding, F. Int. J. Plast. 2005, 21, 1720-1740. [13] Shi, Z.; Chen, Y.; Zhou, L. Composites Sci. Technol. 2005, 65, 1203-1210. [14] Chen, Z. Z.; Tokaji, K. Mater. Lett. 2004, 58, 2314-2321. [15] Foley, M. E.; Obaid, A. A.; Huang, X.; Tanoglu, M.; Bogetti, T. A.; McKnight, S. H.; Gillespie, J. W. Composites A: Appl. Sci. Manufact. 2002, 33, 1345-1348.

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[16] Hong, H. U.; Rho, B. S.; Nam, S. W. Int. J. Fatig. 2002, 24, 1063-1070. [17] Snodgrass, J. M.; Pantelidis, D.; Jenkins, M. L.; Bravman, J. C.; Dauskardt, R. H. Acta Mater. 2002, 50, 2395-2411. [18] Zhang, J.; Wu, J.; Liu, S. Composites Sci. Technol. 2002, 62, 641-654. [19] Biermann, H.; Kemnitzer, M.; Hartmann, O. Mater. Sci. Eng. A 2001, 319-321, 671674. [20] Tanaka, Y.; Kagawa, Y.; Liu, Y. -F.; Masuda, C. Mater. Sci. Eng. A 2001, 314, 110117. [21] Rodopoulos, C. A.; Yates, J. R.; Rios, E. R. Theor. Appl. Fract. Mech. 2001, 35, 59-67.

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[22] Warrier, S. G.; Maruyama, B.; Majumdar, B. S.; Miracle, D. B. Mater. Sci. Eng. A 1999, 259, 189-200. [23] Foulk, J. W.; Allen, D. H.; Helms, K. L. E. Mech. Mater. 1998, 29, 53-68. [24] Doel, T. J. A.; Cardona, D.C.; Bowen, P. Int. J. Fatig. 1998, 20, 35-50. [25] Warrier, S. G.; Majumdar, B. S.; Miracle, D. B. Acta Mater. 1997, 45, 4969-4980. [26] Rios, E. R.; Rodopoulos, C. A.; Yates, J. R. Int. J. Fatig. 1997, 19, 379-387. [27] Wang, P. C.; Jeng, S. M.; Yang, J. -M.; Russ, S. M. Acta Mater. 1996, 44, 3141-3156. [28] Ghorbel, E. Composites Sci. Technol. 1997, 57, 1045-1056. [29] Melander, A. Int. J. Fatig. 1997, 19, 13-24. [30] Li, C.; Ellyin, F. Mater. Sci. Eng. A 1996, 214, 115-121. [31] Murtaza, G.; Akid, R. Int. J. Fatig. 1995, 17, 207-214. [32] Shaw, M. C.; Marshall, D. B.; Dalgleish, B. J.; Dadkhah, M. S.; He, M. Y.; Evans, A. G. Acta Metall. Mater. 1994, 42, 4091-4099. [33] Wang, Z.; Zhang, R. J. Acta Metall. Mater. 1994, 42, 1433-1445. [34] Gradin, P. A.; B¨ acklund, J. Int. J. Adhesion Adhesives 1981, 1, 154-158. [35] Horst, J. J.; Salienko, N. V.; Spoormaker, J. L. Composites A: Appl. Sci. Manufact. 1998, 29, 525-531. [36] Horst, J. J.; Spoormaker, J. L. Pol. Eng. Sci. 1996, 36, 2718-2726. [37] Horst, J. J.; Spoormaker, J. L. J. Mater. Sci. 1997, 32, 3641-3651. [38] Botsis, J.; Zhao, D. Composites Sci. Technol. 1997, 28, 657-666.

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[39] Dessureautt, M.; Spelt, J. K. Int. J. Adhension Adnesives 1997, 17, 183-195. [40] Kysar, J. W.; Gan, Y. X.; Mendez-Arzuza, G. Int. J. Plasticity. 2005, 21, 1481-1520. [41] Liu; Y. Z. J. Mater. Proc. Tech. 1997, 63, 542-545. [42] Persad, C.; Raghunathan, S. IEEE Trans. Mag. 1995, 31, 740-745. [43] Beynon, J. H.; Garnham, J. E.; Sawley, K. J. Wear 1996, 192, 94-111. [44] Hellier, A. K.; Merati, A. A. Int. J. Fatigue 1998, 20, 247-249. [45] Aglan, H.; Chudnovsky, A.; Moet, A.; Stalnaker, D. Int. J. Fract. 1990, 44, 167-178. [46] Aglan, H.; Gan, Y. X. J. Mater. Sci. 2001, 36, 389-397. [47] Aglan, H. J. of Elas. and Plas. 1993, 25, 307-21.

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[48] Aglan, H.; Abdo, Z. J. Adhesion Sci. Tech. 1996, 10, 183-198. [49] Aglan, H.; Gan, Y.; El-Hadik, M.; Faughnan, P.; Bryan, C. J. of Mater. Sci. 1999, 34, 83-97. [50] Khourshid, A. M.; Gan, Y. X.; Aglan, H. J. Mater. Eng. Perform. 2001, 10, 331-336. [51] Lewandowski, J. J.; Thompson, A. W. Metal. Trans. A 1986, 17, 1769-1783. [52] Aglan, H. A.; Gan, Y. X.; Chin, B. A.; Grossbeck, M. L. J. Nucl. Mater. 2000, 278, 186-194.

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[53] Aglan, H. A.; Gan, Y. X.; Chin, B. A.; Grossbeck, M. L. J. Nucl. Mater. 1999, 273, 192-201.

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In: Fatigue Crack Growth… Editor: Alphonse F. Lignelli, pp. 83-124

ISBN: 978-1-60692-476-1 © 2009 Nova Science Publishers, Inc.

Chapter 3

ADVANCES IN THE NUMERICAL MODELLING OF FATIGUE CRACK CLOSURE USING FINITE ELEMENTS J. Zapatero1 and A. Gonzalez-Herrera2 Departamento de Ingeniería Civil, de Materiales y Fabricación, ETSI Industriales, Universidad de Málaga, Plaza el Ejido s/n, 29071 Málaga, Spain

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Abstract The accurate determination of fatigue crack closure has been a complex task for years. It has been approached by means of experimental and numerical methods. Experimental methods are controversial, the results obtained may have low precision, high scatter, and are subject to problems of interpretation. Finite element method has been an alternative for the study of fatigue crack closure. However, such analyses are complex and computationally expensive. Plasticity-induced crack closure is the main mechanism causing fatigue crack closure. It occurs when the flanks of the crack contact with a load above the minimum load in fatigue; as a result, the crack is subject to a smaller crack driving force. Difficulties on the accurate modelling of this phenomenon derive from the need to simulate the preceding fatigue cycles, joined to the high plasticity induced at the crack tip, this results in high computational costs. There exist several parameters which must be strictly controlled to avoid its influence over the results. This chapter summarizes the main recommendations derived from a comprehensive study of the variables that influence the model accuracy. A special attention is paid to the influence of the minimum element size and the effect of the length of the simulated crack wake. It also has been studied issues as the node release scheme, the material yielding model or the significance of different criterion to establish the opening or closure loads. Finally, the main results and findings regarding fatigue crack closure are summarized. Both, 2D and 3D cases are shown. Bi-dimensional studies permit the analysis of the influence of different parameters as crack length, maximum load or stress ratio R. The tri-dimensional model allows the detection of the presence of closure in a small external area of the specimen

1 2

E-mail address: [email protected] E-mail address: [email protected]; Tel. +34 952 132967; Fax +34 952 131371. Corresponding author

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J. Zapatero and A. Gonzalez-Herrera and interesting information is obtained regarding the shape an extension of the plastic zone size.

Keywords: Fatigue, crack closure, finite elements, crack growth, crack wake.

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1. Introduction The identification of the crack closure phenomenon by Elber [1] in the early 1970s provided light and an explanation over some complex effects observed during fatigue crack growth. This in turn allowed crack growth rates to be more accurately calculated by replacing the traditional load range with a new, “effective” range corresponding to the cycle portion where the crack was open. This new approach also facilitated the analysis of some previously unexplained findings such as the delay caused by overloads. However, the precise determination of the crack opening or closure point soon met with considerably difficulties. Thus, the complexities of their experimental determination [2, 3] precluded their accurate quantification —and hence their correlation with the crack propagation rate. Some issues such as the influence of the maximum load, stress ratio, material properties, specimen thickness and crack length were difficult to assess. Distinguishing closure mechanisms according to whether they were induced by crack wake plasticity, roughness or the presence of oxide in the contact surfaces, allowed a variety of crack propagation situations to be examined and specific explanations provided. Nevertheless, the mechanism behind plasticity-induced crack closure has proved the most significant in a wide variety of practical settings —particularly in relation to crack growth beyond the threshold region. The most severe hindrance to acquiring a deeper knowledge about this phenomenon is the complexity of the experimental testing involved. Only indirect methods based on measuring and processing related variables —some of which can only be quantified at long distances from the crack tip— have so far allowed it to be identified [3]. One alternative to experimental testing is numerical analysis, especially with the finite element method, which has been extensively refined since its inception in the 1970s. However, this method is also subject to some constraints; the most serious is the high computational cost incurred in analysing a problem that only occurs after a large number of load cycles and in a zone several orders of magnitude smaller than the scale of the problem. This shortcoming has been gradually lessened thanks to the availability of increasingly powerful computers; however, it is still quite substantial in relation to the 3D problem. The other major constraint of numerical methods is the difficulty of constructing computational models capable of accurately representing the real-life situation. Ever since the earliest fatigue crack closure calculations based on the finite element method were reported, the results have exhibited a strong dependence on the particular model used [4]. While a sort of recommendation for simulating the problem has gradually been compiled, there remain a number of questionable, obscure points. The difficulty of checking the numerical results against reliable experimental data makes this work especially valuable on account of the modelling procedure used. A number of authors have modelled fatigue crack growth including the effect of fatigue crack closure using the finite element method. The most important references, which were

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used as the starting point for these works, are discussed below. Interested readers can find additional information elsewhere [4–7]. Let us first cite the early work by Newman [8, 9], who developed the first plane stress fatigue propagation model considering the effect of crack closure. His work was extended to plane strain problems by Fleck and Newman [10], and to tri-dimensional cases by Chermahini and co-workers [11, 12]. It was the first time that a computational and result analysis methodology was developed and used to derive expressions to obtain crack closure under plane stress and plane strain conditions, as well as for several specimen thicknesses in the 3D case. The authors used models consisting of perfect elastic–plastic materials and contact between nodes as the closure criterion. Contact was simulated by means of appropriately positioned springs. One other important, somewhat more recent work line, was that followed by Sehitoglu, McClung and co-workers [5, 6, 13–17], who established specific methodologies in order to clarify the influence of various modelling parameters and defined other, highly significant ones such as the crack opening point. In addition, they addressed various previously unexplored aspects such as the size of the plastic zone, closure outside the small scale yielding range, and threshold conditions, among others. Worth special note in this context is the work of McClung and Sehitoglu in 1989 [5, 6]. These authors reviewed existing methods for modelling process and compiled available recommendations regarding the minimum element size. Also worth note is the work of Sehitoglu and Sun [16, 17], who developed a new criterion to identify the crack opening or closure point as a function of the stress status in the vicinity of the crack tip rather than in terms of contact of nodes. These studies were followed by much research that was facilitated by easier access to increasingly powerful computers. Thus, Eyllin and Wu [18, 19] carefully examined the problem by using plasticity models for materials involving substantial hardening. Also, Wei and James [20] reported numerical estimates based on finite element models and their experimental counterparts, and noted their difficult in converging. Similarly, Dougherty et al. [21, 22] sought to correlate experimental measurements with numerical computations. Some recent papers can be cited as those of Solanki et al. [7, 23, 24] who report a new criterion for determining crack opening or the work of Antunes et al. [25, 26]. Finally, regarding 3D models, we can cite the works of Zhang et al. [27, 28] and Dodds et al. [29-31] which analyzes closure in three dimensional cracks. Recently, Alizadeh et al. [32] present results with a 3D FE model in a centre-cracked plate, the number of thickness divisions is increased up to 10. As noted earlier, the results have been rather disparate as a consequence of using materials and specimens with different properties and geometries in some cases, but, especially of the specific methodologies used to simulate the problem or the procedures followed to interpret the results. The most crucial choice in simulating the problem is the minimum element size in the vicinity of the crack tip; this allows the plastic zone to be more or less finely meshed — smaller elements facilitate the detection of strain and stress gradients in such a zone, although at the expense of increased computational costs. Also, the minimum element size influences the accuracy with which crack growth in each cycle can be simulated as crack propagation is modelled by changing the boundary conditions of a single element in each cycle. The most widely debated subject as regards interpretation of the results is the criterion to be used in order to define crack closure or opening. Initially, contact at the last node was

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taken (Knc) to be the point signalling closure or opening; subsequently, the sign change in crack tip stress (Ktt) was thought to be more appropriate for this purpose. In this chapter, we shall henceforth use subscripts nc (node contact) and tt (tip tensile) to denote the former and latter criteria. Thus, Kncop and Kttop denote the opening loads determined under each criterion, and Knccl and Kttcl the corresponding closure loads. The main body of the chapter is structured in four section, two sections will focus on methodological aspects (section 2 where the main aspects will be described and discussed and section 3 where more specific issues will be studied) and the other two will be dedicated to shown the result obtained, bi-dimensional results in section 4 and tri-dimensional results in section 5. Finally the main findings and conclusions will be outlined.

2. Key Issues in the Numerical Modelling Process In this section, the main aspects of the modelling process are going to be described. A brief presentation of the commonly accepted recommendations in the bibliography will be made first, and secondly it will be completed with the results of ours own works. Some of the most critical issues which have been object of more specific studies will be left for the next section. P

y

a

Crack tip

x Plastic zone

w

Plastic wake Plastic wake simulated

P

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Figure 1. Scheme of the C(T) specimen. Crack tip, plastic zone and plastic wake.

Figure 1 shows a scheme of the specific problem addressed. Most of the calculation has been made with an aluminium compact tension C(T) specimen. As can be seen, the most critical zone is the crack tip, where stress concentration is maximal, and so stress and strain gradients. The model would be incomplete if the plastic wake left by the crack while growing it is not modelled. In Figure 1, a coordinate reference system referred to the crack tip has been defined in order to plot the results. Its origin was made to coincide with the crack tip and x values were positive for the points in front of the crack and negative for those behind it. Computations presented here have been done by means of several versions of the FE commercial software ABAQUS and ANSYS.

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Due to the symmetry, only one half is modelled. Crack closure or crack lip contact was modelled by using a rigid contact surface that prevented penetration beyond the symmetry plane of the crack. Bi-dimensional models used a four node isoparametric quadrilateral elements. With few exceptions —and only in areas well apart from the plastic zone—, no higher-order elements were used in order to avoid additional numerical difficulties in modelling node contact and node release. Although the main conclusions regarding element size and modelling parameter have been obtained to linear elements, they should be easy to transpose to similar quadratic elements. The use of constant triangular elements was limited and only zones well apart from the crack tip —which would exhibit elastic behaviour throughout the process— were allowed in order to avoid problems in meshing transition zones. This in turn avoided major distortions and the presence of a large number of elements in zones not requiring it, which would have increased the computational cost without increasing precision.

Plastic zone Crack tip

Plastic wake

Figure 2. Bi-dimensional meshing of the C(T) specimen.

Figure 2 shows a typical mesh for one half of the C(T) specimen (with w = 50 mm) in ABAQUS. It clearly shows the different zones used to accurately model the element size transitions between zones. As can be seen, the vicinity of the crack tip required very fine meshing —this is described in detail later on. The most difficult step was the mesh of the plastic zone. Traditionally, a uniform mesh covering the whole crack wake and plastic zone

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with highly regular elements has been used to avoid numerical errors. This is schematically depicted in Figure 2. This uniform mesh requires a large number of elements —roughly half of all. The strong impact of this computational requirement led us to conduct a careful study in order to optimize the mesh. The main difficulty was that, during the crack growing process, every node in the simulated crack wake acted as crack tip at some point in the computations and was thus influenced by the minimum mesh size used at the tip as shown below. Based on the results, it was proved that a fine mesh for the whole plastic zone was unnecessary; it also was possible a slight transitions in element size with increasing distance from the final crack tip. Therefore, the use of this meshing scheme should be appropriately checked. The outcome was an extremely reduced number of elements in the 2D models and hence substantially shorter computation time. In fact, the initial number of elements (9000) was reduced below 1500 in bi-dimensional models. Another important difficulty arises from the inability to compute the thousands of load– unload cycles actually involved in the fatigue crack propagation process. This entails reproducing the phenomenon by computing only a discrete number of cycles. Between them the crack length must be increased by a factor greater than that actually observed and leaving tens or even hundreds of cycles which are not computed in between. Crack growth is assimilated to the release of a node; therefore, the extent by which a crack grows is assumed to be similar to the mesh dimension. This provides an envelope that reproduces the real-life situation more or less accurately. All this entails precisely defining the conditions for simulating the crack, load type and cycles, contact modelling scheme, plastic wake and crack propagation, among others. In the following subparts, reported experience in this topic is analysed, its influence on the results exposed and the way it was dealt with in our work discussed.

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2.1. Meshing Scheme and Element Size This is the most immediate determining factor in a finite element model. The principal issue in this case is that, in order to capture plasticity around the crack lips, this area must be meshed with very small elements (Figure 2). Also, in order not to overly increase the computational cost, one must use a great transition from these areas to the most remote ones; this results in large distortions between elements close to the crack tip and those well apart from it. Finally, a third zone exists that coincides with the area where loads are applied and thus requires special care. The scale reference to define the minimum element size is the plastic zone size. In all of ours works, the Dugdale expression for the plastic zone at the crack front has been used:

rpD

π ⎛ K ⎜ = 8α ⎜⎝ σ Yield

⎞ ⎟⎟ ⎠

2

(1)

where α is a constraint factor equal to 1 for plane stress and 3 for plane strain. To establish the minimum element size around the crack tip properly, different authors have used different meshing schemes; however, there are several commonly accepted criteria

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for mesh refining [5, 8, 23]; the best-known and widely used was that proposed by Sehitoglu and McClung [5]. These authors recommend dividing the plastic zone into a number of elements ranging from 20 triangular elements to 10 linear elements. This recommendation seeks to ensure accurate reproduction of the reversed plastic zone under compression stress and has been followed in many subsequent studies. This has been the starting point to our approach to the problem. However, the increased computational power of current computers led us to check its accuracy by analysing changes in opening and closure stress as the size of the mesh elements in the plastic zone was reduced. This comprehensive study will be presented in the next section and led us to propose a new mesh recommendation and methodology [4]. Regarding the meshing of the load application zone, care must be taken to distribute loads between a large enough number of elements in order to avoid material yielding by a huge punctual load applied at a node. The results were essentially identical irrespective of the particular load distribution scheme used provided the elastic regime prevailed.

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2.2. Load Cycles and Crack Growth The results reported in the present text were obtained by simulating fatigue crack growth under constant amplitude loading. Each cycle ramp required solving a non-linear problem. This makes the finite element method useless for iterative calculations of fatigue life, due to computational costs.Calculations for a variety of loading schemes involving several values of Kmax and crack length at stress ratios (R = Kmin/Kmax) ranging from 0 to 0.9 were performed. The stress intensity factor in mode I, KI, is used throughout to refer to load levels; this makes the problem non-dimensional and cancels the influence of the actual specimen size. In these calculations, the crack propagation procedure is very important. It must be considered that, due to the mesh size used, the crack propagates faster in simulations than in real life. Growth here is simulated by modifying the boundary conditions. Alternative approaches (e.g. releasing several nodes at once) have proved less efficient than releasing a single node in each cycle. There has been much debate about the most suitable position along the load cycle where the boundary conditions should be changed. Usually, the node is released at the start or end of the load cycle. Palazotto [33] determined the best choice by examining various intermediate positions in both the loading and unloading process, and concluded —mainly in terms of numerical convergence— that the best results were obtained by releasing the node when 98% of the load has been applied. Part of our work was dedicated to elucidate this aspect and further details are given later on. Nevertheless it was found that this is not a determining factor provided the mesh size used in the vicinity of the crack tip was small enough, as it would be see in next section. Most of the results given here were obtained by releasing nodes at the maximum load point. In addition to greater physical significance —the crack grow rate increases with increasing load—, this choice is subject to less numerical instability as the released node is under tensile stress and will therefore tend to depart from its contact. It should be noted that the specific material behaviour model used is especially important here. In those materials with weak hardening rules, stabilization is immediate and, based on our own experience, the results are scarcely affected.

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As regards load application, convergence during unloading was found to be facilitated by introducing an additional step involving a constant load to allow the new boundary conditions to settle before the unloading cycle starts.

2.3. Contact Simulation Crack closure phenomenon implies the contact between two surfaces due to the plastic wake and then the delay in the crack growth rate. Contact introduce an abrupt change in boundary conditions and hence a substantial non-linearity. That is why the simulation of this process must be carefully examined. Traditionally, modelling procedures have been kept simple in order to avoid high computational costs. It is the case of inserting springs into potential contact zones where the stiffness constants change, decreasing from a large finite value to zero as a function of the node position [8, 10]. Today, most FE commercial software allows contact to be very easily simulated, so only due care must be taken to avoid numerical convergence problems. ABAQUS and ANSYS were used; both of them are highly effective in this respect. An improved numerical convergence is obtained when a softened contact model is employed. In these works an exponential pressure-clearance relationship has been used with a maximum allowable penetration under 10-11 m (very below other physical magnitudes as the roughness).

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2.4. Plastic Wake Accurately simulating the plastic wake is essential in order to precisely study fatigue crack closure; the effect of the simulated wake should be consistent with that of the actual wake. When the first numerical load cycle is applied, the material is subject to no residual stress and in a rather unrealistic condition —at least in a fatigue problem involving constant amplitude loading. Depending on how far the crack progresses initially, the effect of the simulated wake on crack behaviour will be more or less close to the real-life situation. Therefore, any factors influencing plastic strain in the cycles preceding data collection and closure quantification point is crucial. In fact, the wake is the path along which a crack grows, via hundreds or thousands of cycles, during the fatigue process. Therefore, the whole wake cannot be simulated numerically and one must focus on a limited portion instead. The wake length to be simulated has been studied by many authors. Most, however, failed to check whether the simulated wake had the same effect as the actual wake; they only sought to stabilize their numerical results and took them for granted once obtained. As described in detail later on this chapter, a different approach has been used in our work. A unique model meshing scheme was established to collect data from the same fixed node every time. Changing the starting node, the simulated crack wake length was variable. This procedure cancelled the potential influence of meshing errors in the strain field, irrespective of the specific simulated wake length. It also allowed the above-described homogeneity and uniformity changes in the meshing strategy for the plastic zone to be made —which is highly effective in order to decrease the number of elements required.

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2.5. Modelling the Material Plastic Properties Due to the large plastic strains involved —reversible in the zone near the crack tip— the plastic material model is an important issue. If correct results are to be obtained an accurate and realistic model must be used. A number of studies involving various materials have been reported. The material modelled has been an Al-2024-T351 aluminium alloy that shows weak hardening (E = 73.5 GPa, σyield = 425 MPa, K’ = 685 MPa, n’ = 0.073, being K’ and n’ the parameters in the Ramberg-Osgood yielding model). This material has been numerically modelled in different ways with essentially identical results. Initially, both perfect elastic– plastic and isotropic hardening models were used. The perfect elastic–plastic model revealed that increasing the number of elements in the vicinity of the crack tip resulted in some uncertainty in the strain estimated by the model; this led to variability in the opening and closure stress values [4] —which are highly sensitive to the resulting plastic strain field. The yielding model should therefore include some hardening, so isotropic hardening models were used, some of which where represented on a bi-linear diagram (denoted by Bih in the figures) and others on a tri-linear diagram (denoted by Tih); the two provided essentially identical results as it will be see later on. The final results presented in this chapter has been calculated with a model that uses isotropic hardening with H/E = 0.003 (where H=dσ/dεp is the plastic modulus and E is the Young’s modulus) that fits well the actual cyclic behaviour of aluminium Al-2024-T351. It has been modelled with a tri-linear representation (Tih). 1

Plane stress, R = 0.3, Isotropic hardening 0.8

K/Kmax

0.6

0.4

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Kttop Kttcl

0.2

Kncop Knccl

0 0

0.02

0.04

H/E

0.06

0.08

Figure 3. Influence of the hardening level.

Nevertheless, the potential influence on the results of the material hardening level and model type used was also examined. Figure 3 shows the opening and closure stress values

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obtained in plane stress calculation with R = 0.3 and different hardening level. As can be seen, both parameters decreased with increasing hardening; however, the results were quite similar in the zone with H/E < 0.01, so it can be concluded that any influence can be considered negligible at low hardening levels. 1

Plane stress, R = 0.3, Hardening model effect

Ktt /Kmax

0.8

0.6

0.4 ISOT Kttop 0.2 KIN Kttop 0 0

0.02

0.04

H/E

0.06

0.08

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Figure 4. Influence of the hardening rule.

The effect of using kinematic or isotropic hardening rules was also examined and it was found appreciable differences again at H/E ratios above 0.03. Figure 4 shows the results obtained at R = 0.3 by using similar computations load case but different hardening rules (denoted ISOT, isotropic and KIN, kinematic). At low H/E ratios, result convergence was independent of the specific hardening model used. Higher H/E ratios required correctly fitting the model to the real-life situation. In addition, the potential influence of some hardening-related phenomena such as the replication of hardening cycles with no crack progress were examined, both at intermediate steps (when the wake was created) and at the end (before the situation at the crack tip was validated after the last node has been released). At moderate hardening levels, the effect was negligible except in terms of numerical calculations, so the methodology used can be considered valid.

2.6. Determining the Crack Opening (or Closure) Stress Identifying the crack opening and closure points was the ultimate aim of this work as this was the information required to determine the effective crack driving force. Although the

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discussion will refer to the determination of the opening stress only, the conclusions also hold for the closure stress.

uy, Load Cycle

6E-6

Kmax

uy (m)

4E-6

Kncop

2E-6

Kmi n

0E+0 0

0.0001

x (m)

0.0002

0.0003

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Figure 5. Displacement (uy) profile of the crack during a load cycle. Identification of crack opening by mean of the node contact criterion (Kncop).

Although some alternative procedures exist [24], the crack opening stress can be readily determined by monitoring the loading process stepwise. The key to an accurate identification is the criterion used to establish it from changes in the monitored variable. There are two types of widely endorsed methods for this purpose that assume crack opening to occur when the free surfaces lose contact, denoted Knc in our case, this translates into the release of the last node in contact. It can be visualised in Figure 5 where the evolution of the crack displacement profile along the cycle shows such a point. In the other method category, a crack is assumed to open when the crack tip, compressed in the preceding cycle, ceases to be under compression stress and starts to progress under tensile stress (Ktt). Figure 6 shows the variation of stress normal to the crack plane (σy) during loading. The crack tip lies at x = 0 and negative x values represent the inside of the specimen (as per the references established in Figure 1). This last interpretation is widely subscribed and seems the most reasonable as the significance of the opening (or closure) phenomenon lies in the fact that it reduces the length of the cycle segment where it operates. This was first defined and studied in depth by Sehitoglu and Sun [16,17]. First, the crack opening point was taken to be that where the whole crack plane was placed under tensile stress (Kt, tensile) [16]. In subsequent work [17], the corresponding time for the crack tip (Ktt, tip tensile) was determined. This facilitated analysis of the results, particularly with plane strain, where contact is difficult to evaluate. One advantage of this criterion is that it is less markedly influenced by the mesh size used in the vicinity of the crack tip by effect of variable σy being determined from element integration

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points that are closer to the crack tip than is the first crack node —from which variable uy is evaluated—, which is one element apart from the tip. 8E+8

ٛ y, Load Cycle

Kmax

6E+8 4E+8

Kttop

2

Kmin

0E+0 -0.0016

-0.0012

-0.0008

x (m)

ٛ y (N/m )

2E+8

-0.0004

0

-2E+8 -4E+8 -6E+8 -8E+8

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Figure 6. Variation of the stress σy at the crack tip during a load cycle. Identification of crack opening by using the tip tensile stress criterion (Kttop).

As shown in the following section, the results obtained in various studies [34] suggest that both criteria converge and are less precise in terms of Knc in the case of crack opening. As also shown below in relation to closure stress, the two criteria reflect different phenomena in relation to crack behaviour; close examination of the differences can no doubt shade valuable light on this phenomenon. There are alternative criteria based on the last node under compression stress or the switch to plastic tensile stress [35]. It is interesting to identify the most accurate among them and, especially, that best fitting the experimental crack opening point —which has been the subject of much debate and controversy. In our work only node contact (nc) and tip tensile (tt) criterions has been employed and studied. Finally, due mention should also be made of another potential source of error in interpreting the results, whichever criterion is used. Because the crack opening point is identified from cycle data, the discretization level used will have a direct impact. Also, because a load cycle is resolved iteratively in sub-steps, information at additional times will also be available (considering time in computational terms, not in physical terms). Computational software generally uses automatic time intervals suited to the specific problem convergence. It is therefore unlikely that a calculated point will coincide with the crack opening or closure point. This entails interpolation, which can introduce considerable errors (in the region of 25% or higher) and produce absolutely spurious information. In our case, the last load and unload cycles were computed by using a preset sub-time step —mostly from

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1/25 to 1/50—; this increased the computation time, but ensured that the interval would lie within the reading error range. 8E+08

2E-06

R = 0.3, Plane stress

R = 0.3, Plane stress

uy

σy

cycle

1

1.5

2

Ktt

2

Knc

0.5

ٛ y (N/m )

uy (m)

0E+00 0

cycle

-1E-08 0

0.5

1

1.5

2

Load (0-1) - Unload (1-2)

(a)

-8E+08

Load (0-1) - Unload (1-2)

(b)

Figure 7. Procedure used to determine Kncop (a) and Kttop (b) by interpolation over a cycle.

Also, Ktt and Knc require two different interpolation schemes. The determination of Ktt is subject to little error as the trend between the time preceding and following opening (or closure) of the crack is well-defined and linear —a sign change in σy. However, contact involves an abrupt change in the displacement variable that governs it. In a time the node is still in contact and in the subsequent time it is not; therefore, interpolation can only rely on subsequent data and will therefore be more imprecise. Figure 7 provides a schematic depiction of the changes in each variable and the way the sought values are interpolated.

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2.7. Specific 3D Modelling Aspects Tri-dimensionally modelling of fatigue crack closure implies additional aspects and requirements to consider. They are intimately joined to difficulties which arise with the mesh through the thickness of the specimen. The difference of the element size at different parts of the model in the bi-dimensional mesh becomes a source of problems when it is extended to the third dimension (z) along the thickness. As the larger elements are 1000 times the smaller one, a constant through thickness number of element can not be established. So an additional problem is summed to the great element size transition presented at the 2D model causing more distortion in the mesh. In the present work, a compact C(T) aluminium specimen has been modelled with three different thicknesses (b = 3, 6 and 12 mm), a range of loads from Kmax = 20 to 30 MPa·m1/2 and several stress relations R = Kmin/Kmax (R = 0.1 to 0.7). The minimum element size (sme) at the crack tip is established by the recommendations obtained in 2D (as it will be see in next section), we consider above 90 elements in the plastic zone. In order to keep an acceptable element shape ratio, that requirement imposes a maximum element height to the prism shaped elements placed at the crack tip, and then a minimum number of the thickness divisions. Depending on the specimen thickness, this number of division ranged from 35 to 100, far above previous works in 3D; Chermahini [11, 12] used 4 divisions, Dodds et al. [29-31] 5 divisions and Alizadeh et al. [32] 10 divisions, as stated previously.

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y

x

(a)

tied contact surface

Number of divisions of the thickness

d

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(b)

(c) Figure 8. Tri-dimensional meshing of the C(T) specimen.

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This large number of through-thickness division can not be admitted to the main body of the specimen because it would generate an extremely large number of elements in the whole model, being impossible to compute. So a transition in the through-thickness mesh has been done. It has been done with ABAQUS by means of tied contact surfaces. These surfaces inside the body of the model are fully constrained contact surfaces and are a simple way to produce an easy mesh transition. In the present models it has been used to divide it in a semicylindrical volume (Figure 8) in the region close to the crack tip, meshed with hexahedral prism shaped elements, and in the outside zone, where 8 node tetrahedral elements were employed in order to facilitate the transition and decrease the total number of elements. This area it is not subject to the high gradient in strain and stress which affect the plastic zone. Linear elements in the area close to the crack tip have been chosen in order to facilitate the contact computing convergence. The minimum element size has been 10 μm, which satisfied the requirement previously stated for minimum element size for the smaller load case calculated (Kmax = 20 MPa·m1/2). The number of through-thickness division was established in order to limit the element height, avoiding very thin elements close to the crack tip where the high plastic gradient could lead to erroneous results. So a shape ratio below 6 to 1 was imposed. As a consequence a number of through-thickness divisions of 35, 50 and 100 for the specimen thickness b = 3, 6 and 12 mm respectively, was used. Despite of this, the number of element of the model is large and the final calculation is close to the limit of the computation device employed in this work (two parallel computing CPUs with 2GB RAM memory). More details can be found in reference [36].

3. Specific Issues Studied

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There are three specific aspects of the modelling process with a strong influence on the results: The minimum element size at the crack tip, the length and the way in which the plastic wake is simulated and the node liberation position in the load cycle. An additional problem to determine the effect of these issues is that there exist important crossed influences between them. In this section, the effort made to clarify this problem and the modelling recommendations proposed are presented.

3.1. Minimum Element Size Correctly determining the minimum element size to be used in meshing the vicinity of the crack tip is crucial to ensure accurate results at an acceptable computational cost. In terms of computational cost, the minimum element size not only determines the total number of elements in the model, but also, due to its influence on the plastic wake length to be simulated, dictates the number of load and unload cycles to be computed. A comprehensive study has been performed in order to analyse this influence and determine an optimum minimum size to be recommended. This study has been done by using a non-dimensional term η = rpD/sme, where rpD is the plastic zone size in the Dugdale equation (eq.1) and sme is the minimum element size used at the crack tip. This parameter represents the number of elements into which the plastic zone would be divided in a mesh consisting of

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elements identical to the smallest one (as shown in Figure 2). We shall also use δ, which is the inverse of η and represents the normalized minimum element size (minimum element size divided by the plastic zone size and then independent from Kmax). The influence of this parameter was studied by examining a large number of calculated plane stress and plane strain values [4]. R values ranging from 0 and 0.7 were used, but focussing on R = 0.1 and R = 0.3, considered as representatives values. Also, the influence of both the loading level and the simulated wake length were examined. To this end, we used Kmax values from 10 to 35 Mpa·m1/2 and simulated crack lengths from 0.3w to 0.6w in the former case. No influence of these parameters was detected. A series of models where the minimum element size was gradually reduced was constructed, and the resulting opening and closure stress values were determined, in accordance with the two above-described criteria (Knc and Ktt). Figure 9 shows the opening stress values obtained as a function of η, using plane stress at R = 0.3. Each mark represents a computation with the stated mesh —lines are only intended to illustrate trends. Figure 10 shows the corresponding closure stress values at the specific R level used. The curves exhibit a clear-cut variation pattern. Thus, the first trend is an initial rising portion that corresponds to McClung's recommendation and what it seems an asymptotic behaviour at η values above 30. However, as the number of divisions continues to grow, the first expected asymptotic pattern clearly vanishes and values tend to a different final limit corresponding to a mesh size one order of magnitude smaller than that commonly accepted at that date. McClung's recommendation was aimed at obtaining an accurate representation of the plastic zone strain field; however, the closure stress is strongly influenced by the degree of local plasticity at the crack tip —where a very steep gradient of plastic strain exists—; this makes it especially important to mesh this zone rather than the whole plastic zone. 1

Plane stress, R = 0.3

K/K max

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0.8

0.6

0.4 Kttop

0.2

Kncop McClung, 1989

0 0

30

60

ٛ

90

Figure 9. Crack opening values in terms of η. R = 0.3. Plane stress.

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Plane stress, R = 0.3

K/K max

0.8

0.6

0.4 Kttcl

0.2

Knccl McClung, 1989

0 0

30

60

ٛ

90

120

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Figure 10. Crack closure values in terms of η. R = 0.3. Plane stress.

Figure 9 exposes another interesting phenomenon as element size is decreased: the Kttop and Kncop criteria, represented by dashed lines, converge on a single value, so it is indifferent to use either in relation to crack opening. This is quite reasonable as the distance between the points that define crack opening in the two criteria will decrease with decreasing distance between the last node in contact and that at the crack tip. The closure stress values of Figure 10 reveal differences between the results provided by the two criteria. Thus, the values obtained with Kttcl tend to 0.9, whereas those provided by Knccl tend to 0.5. This seemingly abnormal behaviour is confirmed by the results obtained with the different material models and computational procedures, and is inherent in the deformation process involved in the crack closure mechanism. It can be understood observing Figure 7. The crack tip is placed under tensile stress (Kttop) as soon as the whole pressure of the wake is released (i.e. when the last contact is lost, Kncop) during loading, nevertheless the situation is different for crack closure. When unloading starts, the crack tip evolves rapidly from tensile stress to compression stress (Kttcl) through its linear range; however, contact with the wake only occurs once the crack tip has been sufficiently compressed to offset tensile plasticity in the previous load cycle. This prevents immediate contact and delays it until Knccl is reached. This could never previously be exposed due to mesh size restrictions which prevent the observation of substantial differences in this respect. Based on the trends illustrated in Figures 9 and 10, we would still be far away from the level of discretization required for the curves to clearly exhibit horizontal asymptotic behaviour. Therefore, the recommended mesh should consist of 1000 or 10000 divisions of the plastic zone size and would involve rather cumbersome computations in 2D and absolutely unfeasible calculations in 3D. It must be considered not only the fine mesh required, but also the large number of cycles to be simulated. That questions the use of so small elements.

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However, a plot of the results as a function of δ (i.e. the inverse of η) exposes this trend more clearly and allows develop a working method to identify crack opening or closure without the need to use a computationally unfeasible model. 1

Plane stress, R = 0.3

K/K max

0.8

0.6

Bih Kttop

0.4

Bih Kttcl Tih Kttop 0.2

Tih Kttcl McClung, 1989

0 0

0.05

0.1

0.15

ٛ

0.2

Figure 11. Kttop/Kmax and Kttcl/Kmax ratios obtained under plane stress at R = 3 as a function of δ.

1

Plane stress, R = 0.3

K/Kmax

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0.8

0.6

Bih Kttop Bih Kttcl Tih Kttop Tih Kncop Tih Kttcl

0.4

0.2

0 0

0.005

0.01

δ

0.015

0.02

Figure 12. Extrapolation to δ = 0 of .Kttop/Kmax and Kttcl/Kmax ratios obtained under plane stress at R = 3.

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Figure 11 shows the results corresponding to R = 0.3 in the previous Figure 9, although as a function of δ. As can be seen, as δ (non-dimensional size of the elements) decreases, a point δ < 0.04 is reached where the results align (i.e. δ exhibits a linear relationship with the different crack opening and closure criteria). Expanding the horizontal scale (Figure 12) confirms this assertion. In this new figure, the results were graphically interpolated in order to obtain a value at δ = 0, which would correspond to the horizontal asymptote sought in the previous figures. This is the value which would be obtained by using an ideally mesh of zerosize elements, i.e. for a continuum —an infinite number of divisions. Also, Figure 12 confirms that computations performed using two different types of material representation (bi-linear (Bih) and tri-linear (Tih) curves) led to the same final result, even though they provided slightly different values at a given mesh size δ. Also, it can be seen that the Kncop and Kttcl criteria clearly converge. Based on the foregoing, a method were developed for accurately estimating the crack opening or closure stress by using several models (2 or more) with different number of elements and various mesh sizes in the vicinity of the crack tip. The minimum element size of this mesh must be above a threshold level corresponding to the inflection points in the curves of Figure 11 [4]. This is consistent with the plane stress and plane strain results obtained at different R values [34]. This methodology has been applied for all de bi-dimensional results presented in this chapter. Regarding tri-dimensional models, the computation of three problems (with a computation time ranging from 1 to 3 weeks, each one) can be considered unacceptable. Nevertheless, these Figures provide an indication of the element size necessary to minimize the error and an estimation of this.

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3.2. Modelling the Plastic Wake As noted earlier, modelling the plastic wake is a key issue in order to the properly and accurately simulation of crack closure; a numerical simulated plastic wake smaller than the actual wake will yield spurious results, while an exceedingly long wake will raise computational costs excessively and unnecessarily. An additional difficulty in the numerical modelling of the plastic wake is that it is associated to the problem of determining the minimum mesh size as described in the previous section. This forced us to perform various tests in order to support the ensuing conclusions. The first was made with 30 divisions of the plastic zone as mesh size in order to examine the effect of the simulated wake length on the results; this mesh size exceeded McClung's recommendation but could be judged deficient according to the conclusions obtained from the study of the effect of the minimum element size. A single numerical model where the final crack tip was invariably at the same point was used. Various computations involving application of load cycles at different starting points were performed. In this way, the results were only dependent on the simulated wake length (Δaw). This study was done under both plane stress and plane strain conditions, using variable degrees of crack propagation at an identical maximum load, Kmax, and different load ratios (R). In Figures 13, for plane stress, and Figure14 for plane strain, the results are plotted as a function of the simulated wake length normalized by the plastic zone size (rpD). Only opening

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J. Zapatero and A. Gonzalez-Herrera 1

Plastic wake effect, Plane Stress

Kttop/Kmax

0.8

0.6

0.4

R= 0 R = 0.1

0.2

R = 0.3 R = 0.5

0

0

0.2

0.4

0.6

ΔaW /rpD

0.8

1

Figure 13. Influence of the plastic wake length (Δaw/rpD) on Kttop/Kmax at different R values. Plane stress.

1

Plastic wake effect, Plane Strain

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Kttop/Kmax

0.8

0.6

0.4

0.2 R=0 R = 0.3

R = 0.1 R = 0.5

R = 0.2

0 0

0.2

0.4

0.6

Δ aW /rpD

0.8

1

Figure 14. Influence of the plastic wake length (Δaw/rpD) on Kttop/Kmax at different R values. Plane strain.

stress as determined using the Ktt criterion was examined. The Knc criterion was found to provide similar results. The closure process was thus not addressed. As can be seen from

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these Figures, an increase in simulated wake length resulted in an increased opening stress; as a consequence, the effect of not simulating the whole wake was negligible above a given threshold. In the most extreme case (R = 0), opening stress values were essentially identical at wake lengths exceeding the Dugdale plastic zone size. Beyond such a distance, the wake was completely formed and its effect was the same as that observed under constant amplitude loading. As a general conclusion, the crack length to be simulated is directly related to R. Figure 13 exposes a horizontal asymptotic variation. Numerically fitting these curves allows one to determine the corresponding asymptotes and the wake length that can be expected to cause acceptable errors. Also, the curves are highly useful with a view to estimating the errors made by using models where the variation of some parameter may question the accuracy of the results by virtue of this effect. In plane strain (Figure 14), however, the plastic wake has a much less influence (in the range of ΔaW/rpD = 0.1). 1

Kttop, R = 0.1, Plane stress

0.8

0.8

0.6

0.6

η = 52

0.4

Kncop/Kmax

Kttop/Kmax

1

Kncop, R = 0.1, Plane stress

η = 52

0.4

η = 93 0.2

η = 93 0.2

η = 146

0

0 0

0.5

1

1.5

Δ aW /rpD

2

0

2.5

0.5

1

1

0.8

0.8

0.6

0.6

η = 52

0.4

1

1.5

Δ aW /rpD

2

η = 52

0.4

η = 93 0.2

η = 146

0

η = 93 0.2

η = 146

0 0

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2.5

Knccl , R = 0.1, Plane stress

Kttcl , R = 0.1, Plane stress

Knccl /Kmax

Kttcl /Kmax

η = 146

0.5

1

1.5

ΔaW /rpD

2

2.5

0

0.5

1

1.5

ΔaW /rpD

2

2.5

Figure 15. Influence of the plastic wake length (Δaw/rpD) with different minimum element sizes (η).

This study was completed by another that involved previously unexplored aspects. Such was the case with the performance of the closure criteria and the way they are influenced by the minimum mesh size used. To this end, the previous study was repeated at different mesh sizes, using 52, 93 and 146 divisions of the plastic zone, respectively, with both plane stress and plane strain. Also, the study included the effect of the wake on crack closure, which was unclear. Both crack opening and closure were examined using both criteria. Figure 15 shows the results obtained as regards opening and closure at R = 0.1 under plane stress, using the two criteria (nc and tt) and three mesh sizes.

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The most immediate conclusion is that, for crack opening with the two criteria and for closure with Knccl, the trend is similar to that observed in the previous study (Figures 13 and 14); all are affected by the simulated wake length. The most important difference is the horizontal asymptote (final value), which follows the trend described in the previous section in relation to the minimum mesh size (Figures 9 and 10). Closure point determined according to the Kttcl criterion under plane stress is independent of the wake length. This reflects the null significance of this criterion in crack closure as it will be confirmed lately with the experimental validation (next section).

3.3. Node Release Scheme One of the most widely debated issues in this context is the choice of the node release point in each cycle. In the light of the above-described results, it is interesting to revisit the arguments on the node release point and check its influence on the results, with special emphasis on the minimum mesh size. The widespread assumption exists that the maximum load point is appropriate and results in little difference from alternative release points. Therefore, the previous results were calculated by assuming node release to occur at the maximum load point. However, the effect of releasing nodes at the minimum load point has also been examined. Calculations were done mainly under plane stress at R = 0.3. 0.8

Plane stress, R = 0.3

K/K max

0.6

Kttop (at Kmin) Kncop (at Kmin) Kttop (at Kmax) Kncop (at Kmax) McClung, 1989

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0.4

0.2 0

50

100

150

ٛ

200

250

300

Figure 16. Influence of the node release point. Variation of Kttop/Kmax and Kncop/Kmax with η under plane stress at R = 0.3.

Figure 16 compares the η dependence of the crack opening values as obtained by releasing the node at the minimum load level with those previously obtained (releasing at

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maximum load) using a tri-linear isotropic hardening (Tih) yielding model. As can be seen, the Kttop curves exhibit a substantial difference that decreases with increasing number of divisions. However, closer analysis of this trend reveals that convergence on the same objective value is faster above η = 100, where a horizontal asymptote is clearly observed. Also, the Kncop criterion converges similarly as in the previous case. The Kttcl results for crack closure are virtually identical; also, the asymptote is even more marked than if the node is released at the maximum load point. Similar conclusions can be drawn regarding Knccl for crack closure. 1

Plane stress, R = 0.3

K/K max

0.8

0.6

0.4

0.2

Kttop (at Kmin)

Kttop (at Kmax)

Kncop (at Kmin)

Kncop (at Kmax)

Kttcl (at Kmin)

Kttcl (at Kmax)

0 0

0.005

0.01

ٛ

0.015

0.02

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Figure 17. Influence of the node release point. Variation of Kttop/Kmax and Kncop/Kmax with δ under plane stress at R = 0.3.

These results confirm the accuracy of the proposed methodology and support the assumption that the outcome is independent of the node release point used provided a small enough minimum element size is used at the crack tip. Figure 17 confirms this consistency in terms of δ and shows that the same objective value can be reached whichever node release point is chosen. Also, the wake length to be simulated is not influenced by this choice, similarly as in the case described in the previous section. Once the methodology description has been done, the two following section will expose the main results obtained.

4. Bi-dimensional Results The most important results derived from the bi-dimensional models are those concerning fatigue crack opening and closure values. Nevertheless, apart of this and of descriptive results

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of the stress and strain field around the crack tip, the lesser computational cost associated to bi-dimensional models permits the study of the influence of several parameters which could be considered of importance in fatigue crack closure. In this section, the influence of maximum load and the crack length will be described first, and then the analysis of the effect of the stress ratio R will be presented. Finally fatigue crack closure results will be shown and its correlation with experimental tests.

4.1. Influence of Maximum Load The analysis of the maximum load applied effect over crack opening and closure has been studied in terms of the stress intensity factor in mode I, KI. A broad range of loads had been applied and calculated ranging from Kmax = 6 Mpa·m1/2 to Kmax = 40 Mpa·m1/2. Above the threshold load and below the stress intensity factor corresponding to the fracture toughness of aluminium, in order to keep the small scale yielding condition. Different studies were made and no differences were found in all cases, either for plane stress or plane strain conditions. Some results can be seen in Figure 18 which correspond to a study performed for four different Kmax levels but with the same conditions in the numerical models. The number of divisions of the plastic zone was 50 (η = 50) for all cases. It can be seen that the values obtained are absolutely coincident for any of the models, for every criterion used and either opening or closure. 1

Kmax effect, R = 0.3, ٛ = 50

K/Kmax

0.8

0.6

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0.4

0.2

Kttop

Kttcl

Kncop

Knccl

0 7

11

1/2

Kmax (MPa·m )

15

Figure 18. Influence of Kmax, plane stress, R = 0.3, η= 50.

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According to these results, it could be concluded that the maximum load level has no effect over crack closure. Nevertheless, it must be said that this conclusion is related to the present numerical model in which no other mechanism to induce crack closure is included and small scale yielding conditions are supposed. Features such as roughness or oxideinduced crack closure imply a scale factor intimately related to the maximum load by means of the plastic zone size and they have not been included in the numerical model.

4.2. Crack Length Effect, a Several analyses have been carried out in order to study the effect of crack length on closure. Initially C(T) specimens were modelled with crack length from a = 0.3w (a = 15 mm, for w = 50 mm specimen) to a = 0.5w (a = 25 mm). In a preliminary study, no clear trend was observed which resulted either in closure or in opening levels and regardless of which definition of crack closure could be used. An additional work was developed with different specimen geometry (CCT) which permits a broader valid range of crack length to be studied. In this case, a CCT aluminium 6082-T8 specimen with weak cyclic hardening rule was utilised. The crack length ranged from a = 0.05w to a = 0.5w. The results obtained for R = 0.3 can be seen in Figure 19 for plane stress. Negligible differences are shown for Ktt definition, and more scatter is presented in the case of Knc, especially for the smaller crack size due to the difficulty of determining the point of node contact using this criterion. 1

Crack length effect, R = 0.3, Plane Stress

0.8

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K/Kmax

0.6

0.4

0.2

Knco p

Knccl

Kttop

Kttcl

0 0

10

20

a/w %

30

40

Figure 19. Influence of crack length, a, CCT specimen, plane stress, R = 0.3.

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It can be observed that crack closure maintain a constant value above a = 0.1w, above the range of what it would be considered small crack. Thus it can be concluded that for the crack length range of interest, this parameter (crack length, a) has no influence over crack closure or opening values.

4.3. Stress Ratio Effect, R As a consequence of the effect of Kmin in the development of the crack wake, the stress ratio, R, becomes a fundamental parameter, presenting a great influence over closure and opening stress. A comprehensive study of the effect of R was performed. The numerical study consisted in the determination of opening and closure values, using both criteria, node contact (nc) and tip tensile (tt), in plane stress and plane strain, for a broad range of R from 0 to 0.9 (with a stepping of 0.05). That requires a huge number of calculations. The results were obtained with three different minimum element sizes and then extrapolated to δ = 0 with the methodology developed previously in order to minimise the numerical error due to minimum element size. This method has been of great interest to obtain valid results for high R ratios. More details can be consulted in reference [34]. 1

Plane Stress

0.8

K/Kmax

0.6

0.4

Kttop (Plane Stress )

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Kttcl (Plane Stress ) Kncop (Plane Stres s) Knccl (Plane Stress )

0.2

Elber Schijve Experimental tes t

0 0

0.2

0.4

R

0.6

0.8

Figure 20. Plane stress opening and closure stress in terms of R.

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109

Plane Strain

0.8

K/Kmax

0.6 Kttop (Plane Stress ) Kttop (Plane Strain)

0.4

Kttcl (Plane Strain) Elber Schijve

0.2

Experimental tes t

0

0

0.2

0.4

R

0.6

0.8

1

Figure 21. Plane strain opening and closure stress in terms of R.

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Some of the results obtained are plotted in Figures 20 and 21. The different symbols represent a final calculation for every value of R. They are accompanied by several curves corresponding to the curve fitted for every criterion. Experimental correlation for aluminium alloys presented by Elber [1] (eq.2) and Schjive [37] (eq.3) and some experimental results are also plotted as a reference to compare.

U=

ΔK eff ΔK

= 0.5 + 0.4 R

U = 0.55 + 0.33R + 0.12R 2

(2)

(3)

In Figure 20, opening and closure values for plane stress case are represented. In plane strain, due to the numerical difficulties to obtain Knc, only the tip tensile definition (Ktt) has been considered valid so it is represented together with the same results obtained in plane stress in Figure 21.

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J. Zapatero and A. Gonzalez-Herrera

It can be seen that Kttcl results (either plane stress and plane strain) presents an almost constant value, about 0.9 for plane stress and 0.83 for plane strain. So no influence of R can be inferred from this definition of closure. The reduced effect that the plastic wake presents over this criterion has also been seen previously so it can be inferred that this definition of closure, being an event in the crack cycle evolution, it is not related to the phenomenon of crack closure stated by Elber and no physical sense is associated to them. This aspect has been experimentally correlated as it will be see later. In the case of Kttop, in plane stress (Figure 20), the evolution is parallel to the expression proposed by Elber (eq.2) and Schijve (eq.3), but with a higher opening point. It also must be said that for high R ratios, opening is well defined and it is significant. Moreover, as pointed out previously, the convergence between both definitions for opening is clear, the results obtained for Kttop and Kncop are hardly the same up to R = 0.5. Focussing in closure and the node contact definition Knccl, for plane stress, the values are lower than those corresponding to opening and a great agreement is presented in terms of R with the expression proposed by Schijve (eq.3). In plane strain (Figure 21), two tendencies can be observed, for low R, the results are quite lesser than those obtained in plane stress and lesser than the experimental correlations. However, for R above 0.5, the results for plane stress and plane strain are quite coincident. The results obtained have been correlated and curve fitted providing expressions which summarize the results for every criterion used, in plane stress and plane strain [34]. The expression obtained for Kttop (eq.4) can be considered the most representative of crack opening due to the lesser error.

U ttop = 0.426 + 0.6058R − 0.7177 R 2

(4)

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4.4. Experimental Correlation There exist several methods to measure crack closure from experimental tests (e.g. the compliance method). However, these methods are controversial; the results obtained present a high scatter and are subject to problems of interpretation. So the experimental validation of these numerical results becomes rather complex. In this work, two ways for the experimental validation of the numerical results are presented. In one hand, direct comparison with experimental crack closure measurement by means of the compliance method, on the other hand, a correlation of the different numerical results with experimental crack growth rate curves has been performed in order to evaluate its ability to characterise crack closure. The analysis of the correlation coefficient of the fit to the experimental data with a power function for different effective driving force (ΔKeff) derived from the numerical results, provided important conclusions regarding the effect of R on closure and the interpretation and physical significance of each definition of closure. For the case of direct comparison, experimental results have also been included in Figure 20 and 21; they are represented as dot marks. This experimental test was performed with C(T) specimen, with w = 50 mm and thickness b = 12 mm and the same aluminium alloy Al-2024T351. R ranged from 0.1 to 0.33. The strain was measured by means of a back face strain gage and the compliance method was employed to determine crack closure. No differences

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were found between opening and closure. Different numerical techniques were applied to increase the accuracy of the results and decrease the variability. A general good agreement can be observed with the previous experimental correlation proposed by Elber (eq.2) and Schijve (eq.3). In the range from R = 0.1 to 0.2, the results fits properly the Schijve´s expression. However, with higher R, the tests provide values above both correlations and there is an increased dispersion due to difficulties related to the experimental methodology. Comparing with the numerical results two different interpretations can be made. On the one hand, from the observation of plane stress results in Figure 20, it can be seen that all the results lie between opening and closure values, as maximum and minimum limit. The first group, from R = 0.1 to 0.2, is absolutely coincident with the numerical crack closure using the node contact criterion. Nevertheless, the other group (from R = 0.2 to 0.33) is close to the opening values with both criteria. On the other hand, focussing on Figure 21, the experimental results are plotted lying at an intermediate position between plane strain and plane stress numerical curves as it would be expected for a 12 mm thick specimen. In general, taking into consideration the difficulties involved, it can be said that the experimental test results provide confidence to the numerical results. It also confirms the null relevance of the Kttcl criterion in order to characterise plasticity induced crack closure. Anyway, due to the variability and lack of accuracy of the experimental results they must be considered as a reference but no more conclusions can be drawn. Further work should be done in this sense to correlate directly numerical and experimental determination of crack closure. An alternative procedure based on constant amplitude fatigue crack growth tests was developed. In order to accurately reproduce the case modelled numerically, a constant amplitude load Pmax test was carried out, instead of constant Kmax. The load and validity of the results were limited to the Paris range of the fatigue crack growth rate curve, so that small scale yielding condition applied and crack growth was in the regime where plasticity-induced crack closure would be the dominant mechanism. Three different sets of tests were made with different specimen thickness (b = 4, 8 and 12 mm, denoted B4, B8 and B12 respectively) and a range of different stress ratios were tested (R = 0.1, 0.3, 0.5 and 0.7). Three tests were repeated for each stress ratio. The crack growth was measured by means of the ACPD technique. An Al-5383-H321 aluminium alloy C(T) specimen with w = 56 mm corresponds to B8, and Al-2024-T351 and w = 50 mm in the case of B4 and B12. Additional details of this work can be consulted in reference [38]. The crack growth rate curves obtained were fitted with a power function of ΔK for ΔK applied (ΔKappl) and various ΔK effective (ΔKeff) with the expression corresponding to the Paris law:

da = C ΔK m dN

(5)

A correlation coefficient r2 was used to evaluate the accuracy of the curve fitting process. This coefficient is defined as follows:

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r = 1−

∑(y

2

(∑ y

−~ yi )

2

i

(6)

) − (∑ y )

2

2 i

i

n

where yi is the real value and ~yi correspond to the curve fitted. The procedure established to validate the numerical results consisted on the application of different definitions of fatigue crack closure to obtain several ΔK effective in terms of R. As a reference, crack closure has also been evaluated using the previously established expressions for crack closure in aluminium alloys proposed by Elber (eq.2) and Schijve (eq.3). The variation of the correlation coefficient, r2, from those previously obtained provides an indication of the significance of each definition in terms of crack closure and it will indicate if the numerical results increase the accuracy of these accepted expressions. Several definitions for ΔK effective have been tried using the numerical results relevant to the crack opening or closure definition. Five definitions in plane stress were used (ΔKop, ΔKttcl, ΔKnccl, ΔKtt, ΔKnc) and three in plane strain (where no node contact definition, Knc, was used due to its poor accuracy). In the case of ΔKop, due to the convergence observed in both criterions Kncop and Kttop, they have been considered as the same value (and denoted Kop). ΔKtt and ΔKnc correspond to an averaged definition of fatigue crack closure taking in consideration the loading and unloading portions of the cycle. They are defined as follows:

ΔK tt = K max −

ΔK nc = K max −

K ttop + K ttcl 2

K ncop + K nccl 2

= K max −

= K max −

K op + K ttcl

(7)

2

K op + K nccl

(8)

2

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Table 1. Correlation coefficient, r2. Bi-dimensional results Crack Drive ΔKapplied ΔKElber ΔKSchjive ΔKop(a) ΔKttcl(a) ΔKnccl(a) ΔKtt(a) ΔKnc(a) ΔKop(b) ΔKttcl(b) ΔKtt(b)

B4 0.821 0.894 0.888 0.878 0.149 0.885 0.966 0.973 0.543 0.158 0.833

B8 0.88 0.951 0.95 0.8 0.246 0.946 0.96 0.966 0.709 0.28 0.888

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B12 0.805 0.901 0.906 0.795 0.092 0.922 0.955 0.972 0.511 0.077 0.845

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In Table 1 the correlation coefficients for all these criteria are resumed. The results corresponding to plane stress condition are denoted with a suffix (a) and those corresponding to plane strain with a suffix (b). The first issue that should be pointed out is that the same pattern was observed for r2 in term of the different ΔKeff, independently of the specimen thickness. This means that the correlation coefficient is directly related to the definition employed to determine ΔK effective. Focussing on the numerical values of r2, it can be seen that a poor correlation is observed for some criteria (ΔKttcl(a), ΔKop(b) and ΔKttcl(b)) and can be discarded to explain the fatigue crack closure phenomenon. Others (ΔKop(a), ΔKnccl(a) and ΔKtt(b)) show a good agreement but they are not better than previous definitions of closure. They do not present any additional information to that obtained from the accepted experimental correlation (eq.2 and eq.3). Finally, the definition computed from averaging the loading and unloading driving force, ΔKtt(a) and ΔKnc(a) in plane stress, present a better correlation coefficient. ΔKnc(a) is very close to the Elber curve, diverging at values of R above 0.4, so the higher correlation coefficient is due to a better representation of crack closure at high R ratios by means of FE models than found in the experimental expression.

5. Tri-dimensional Results As in previous section, the most important results in tri-dimensional model are fatigue crack closure values, specially its evolution along the thickness. Nevertheless, it is also important to take an overview of the results obtained in terms of stress and strain field. First section will be dedicated to this. Following, fatigue crack closure results will be presented and finally they will be validated with experimental tests.

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5.1. Plastic Zone and Crack Closure A huge number of results could be reported, nevertheless only the more representative are going to be presented here. Those are the crack opening displacement (uy, displacement normal to the crack plane) and the plastic zone and the plastic wake at the crack. More information can be found in references [36, 39]. The displacement normal to the crack plane is represented (uy) in Figure 22, corresponds to the crack profile at the instant in which minimum load is applied (Kmin), once the fatigue crack has been developed and the effect of the crack wake is present. The calculation parameters for this cases are R = 0.3 and Kmax= 25 MPa·m1/2. The specimen thickness is b = 3 mm. Only the elements belonging to the volume of interest surrounding the crack tip have been represented (see Figure 8b). The lateral face that can be seen it is the external face. The crack tip has been highlighted with a doted line. The graphical contour scale has been adopted in order to magnify the small displacements close to the crack tip. It ranges from 0 (red in the colour version, the darkest colour in grey scale) to 2 μm. The area in the zone in front of the crack tip, corresponding to the value uy = 0, it is the contact area at minimum load. Only one node ahead of the crack tip is in contact in the interior zone. In contrast, very close to the external face, the area where the plastic wake has been properly developed is completely

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J. Zapatero and A. Gonzalez-Herrera

closed. The evolution of the zone in contact from the exterior to the inner region is very pronounced. At a very short distance from the exterior face, the midplane results are achieved. Comparing results for different thickness and the same load case it can be concluded that the size of the portion of the thickness in which the contact is dominant is equal in absolute terms. This conclusion is the same in different load cases computed and is confirmed with the analysis and quantification of the presence of fatigue crack closure as it will be seen in next section.

Exterior face

Crack tip Crack advance Contact area

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Figure 22. Displacement normal to the crack plane (uy) at minimum load (Kmin), R = 0.3 and Kmax= 25 MPa•m1/2, Δaw/ rpD = 0.38. Specimen thickness b = 3 mm.

It is also illustrative to observe the volume where the material has been plasticized; it corresponds to the plastic wake, in front of the crack tip, and the plastic zone, behind. In Figure 23, the whole plastic region is represented; the elements in elastic range have been removed. Figure 23a corresponds to R = 0.3, b = 6 mm and Kmax = 20 MPa·m1/2, the point of view is opposite to the previous in Figure 22, so the centreplane it can be seen. The plastic zone corresponding to R = 0.3, b = 12 mm and Kmax = 20 MPa·m1/2, is represented at Figure 23b, the external face is viewed in this case. The crack tip is highlighted and the plastic zone and the plastic wake are shown. If we observe the evolution of the plastic zone along the thickness, at the interior face, it is similar to those obtained in plane strain conditions. It can be seen that it starts to grow and it should be expected that something similar to plane stress condition would be achieved at the external face. Nonetheless, the trend is inverted near the external face and the plastic zone size start to decrease. The area in which this change is developed is coincident with the area affected by contact and crack closure as it will be seen later. The plastic zone size has been quantified along the thickness for different load case and has been reported in reference [40].

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centreplane Plastic zone

Plastic wake Crack tip Crack advance

(a)

Exterior face

Plastic wake Crack tip

Crack advance

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Plastic zone

(b) Figure 23. Plastic zone and plastic wake. R = 0.3 and Kmax = 20 MPa•m1/2, Δaw/ rpD = 0.6, (a) b = 6 mm and (b) b = 12 mm.

Finally another interesting parameter to evaluate is the stress in normal direction to the crack plane (σy). At minimum load (Kmin) it can be useful to estimate the size of the reversed plastic zone. In Figure 24, it is represented for Kmax = 25 MPa·m1/2, R = 0.3 and b = 3 mm, at Kmin. Focusing on the stress level around the crack tip, the reversed plastic zone corresponds

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J. Zapatero and A. Gonzalez-Herrera

approximately with the region in which σy is below –σyield, as it has been marked in the Figure. It is almost constant along the thickness and very slightly bigger close to the free surface due to the effect of the contact area. It must be said that quantitatively the reversed plastic zone size is similar to those obtained in 2D finite element plane strain calculations.

Exterior face

Crack tip Reversed plastic zone Crack advance

Figure 24. Normal stress (σy), Kmax = 25 MPa•m1/2, R = 0.3 and b = 3 mm, at minimum load (Kmin). Estimated reversed plastic size.

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5.2. Fatigue Crack Closure As it has been said, the possibility of determining different values for fatigue crack closure and opening loads along the thickness is the main issue derived from tri-dimensional modelling of this problem. This information only has been approached with complex experiment with polycarbonate specimens, while experimental tests provide a unique value corresponding to the change in the response of an indirect parameter (compliance method). In this section, we are going to focus on the response of the crack tip along the thickness and the presence of closure. Figures 25 show crack opening and closure values along the thickness (in terms of variable d, distance to the exterior face, defined in Figure 8, due to symmetry only one half of the specimen thickness is plotted). They correspond to a constant maximum load (Kmax = 25 MPa·m1/2) and constant specimen thickness (b = 3 mm) and can be considered representative of the whole matrix of calculations made. They are plotted separately for four different values of R (0.1, 0.3, 0.5 and 0.7). In every figure, both criteria are included (Ktt and Knc) and opening and closure values are represented.

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1 1/2

1/2

K = 25 Mpam , R = 0.1, b = 3 mm

K = 25 Mpam , R = 0.5, b = 3 mm

0.8

0.8

0.6

0.6

K/Kmax

K/Kmax

117

0.4

0.4

0.2

0.2 Kttop

Kttcl

Kncop

Knccl

Kttop

0

Kttcl

Kncop

Knccl

0 0

0.5

d (mm)

1

1.5

0

0.5

(a)

d (mm)

1

1.5

(c) 1

1 1/2

K = 25 Mpam1/2, R = 0.7, b = 3 mm

K = 25 Mpam , R = 0.3, b = 3 mm

0.6

0.6

K/Kmax

0.8

K/Kmax

0.8

0.4

0.4

0.2

0.2 Kttop

Kttcl

Kncop

Knccl

0

Kttop

Kttcl

Kncop

Knccl

0 0

0.5

d (mm)

(b)

1

1.5

0

0.5

d (mm)

1

1.5

(d)

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Figure 25. Opening and closure stress along the thickness. Kmax = 25 MPa•m1/2 and b = 3 mm, for four different values of R: (a) R = 0.1, (b) R = 0.3, (c) R = 0.5 and (d) R = 0.7.

In these Figures, it can be seen that the effect of crack closure is more prominent close to the exterior zone, as it was expected. There is an exterior “transition zone” in which crack closure is higher, achieving then a constant value up to the midplane. It also can be seen that the same trend is observed for the different criterion employed. Qualitatively, the transition zone size is different for every calculation case and criterion employed. The extension of the transition zone is lesser at higher R. At R = 0.1 (Figure 25a) it is close to 1 mm, while at R = 0.7 it is about 0.1 mm (Figure 25d). An exception to this is the behaviour of Kttcl curves (black line), whose transition zone is hardly the same independently of R. Previously had been pointed out the poor significance of this criterion in order to characterise the crack closure delay effect [4, 34, 38], these Figures confirm this observation.

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K = 20 Mpam , R = 0.3, b = 3, 6 and 12 mm 0.8

K/Kmax

0.6

0.4 Kttop 3 mm Kttcl 3 mm Kncop 3 mm Knccl 3 mm

0.2

Kttop 6 mm Kttcl 6 mm Kncop 6 mm Knccl 6 mm

Kttop 12 mm Kttcl 12 mm Kncop 12 mm Knccl 12 mm

0 0

1

2

3

d (mm)

4

5

6

(a) 1 1/2

K = 20 Mpam , R = 0.3, b = 3, 6 and 12 mm 0.8

K/Kmax

0.6

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0.4 Kttop 3 mm Kttcl 3 mm Kncop 3 mm Knccl 3 mm

0.2

Kttop 6 mm Kttcl 6 mm Kncop 6 mm Knccl 6 mm

Kttop 12 mm Kttcl 12 mm Kncop 12 mm Knccl 12 mm

0 0

0.1

0.2

0.3

d (mm)

0.4

0.5

0.6

(b) Figure 26. Opening and closure stress along the thickness. (a) Kmax = 20 MPa•m1/2 and R = 0.3 for b = 3, 6 and 12 mm. (b) external transition zone enlarged.

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In Figures 26, an interesting characteristic of the transition zone arises when the results for different specimen thickness but with the same load case are plotted together. The load case is Kmax = 20 MPa·m1/2 and R = 0.3 for b = 3, 6 and 12 mm. As the horizontal scale is the distance in absolute terms, it ranges from 0 to 6 mm, the results corresponding to the specimen thickness b = 3 mm and 6 mm, are only plotted in their valid ranges. Remarkable and an interesting aspect to observe is how the results obtained are almost the same independently of the specimen thickness. They seem to be superimposed. So the transition zone is approximately equal (about 0.3 mm), and the behaviour of the crack tip along the thickness can be characterised independently of the specimen thickness. In order to quantify the length of the transition zone, a parameter λ has been defined as the distance from the external face in which the value obtained is 99% those obtained at the centerplane (asymptotic value). A curve fitting process has been done to evaluate λ in terms of the variable U(d) = (KmaxKop/cl)/(Kmax-Kmin). In this case d = λ corresponds to the value U(λ) = 0.99Umax in the fitted curve (eq. 9). Umax is the asymptotic value obtained at the centerplane, c and q are constants to be fitted.

U ( d ) = U max −

1 (d + c )q

(9)

In Figure 27, the ratio between the transition zone λ for two different specimen thickness (b = 3 mm and 6 mm, λ3mm/λ6mm) for different values of R (0.1, 0.3, 0.5 and 0.7) are plotted. It correspond to Kmax = 20 MPa·m1/2 and Kmax = 25 MPa·m1/2. It can be seen that this values are close to unity, considering some scatter due to the large numerical process involved to obtain this ratio. 1.5

1.5

1/2

Kmax = 20 Mpam

1.0

ٛ 3mm/ٛ 6mm

ٛ 3mm/ٛ 6mm

1.0

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1/2

Kmax = 25 Mpam

0.5

0.5

Uttop

Uttcl

Uncop

Unccl

R = 0.5

R = 0.7

0.0

Uttop

Uttcl

Uncop

Unccl

R = 0.3

R = 0.5

R = 0.7

0.0 R = 0.1

R = 0.3

R = 0.1

Figure 27. Transition zone ratio for (a) Kmax = 20 MPa•m1/2 and (b) Kmax = 25 MPa•m1/2.

These results suggest that the transition zone it is not affected by the specimen thickness. Nevertheless, it can not be concluded that it is irrelevant in fatigue crack closure and then in the delay effect of the fatigue crack growth rates. The external constant transition zone would

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J. Zapatero and A. Gonzalez-Herrera

affect this parameter but balanced by the effect of the inner area which is directly dependent of the specimen thickness. From this, the issue of how this external contact area affects the whole crack growth rate becomes the key question. The curvature observed in experimental tests is consistent with these results and can be justified as the internal zone of the crack grows a little bit faster than the exterior zone due to a reduced effect of crack closure. The curvature will probably contribute to reduce and balance the observed effect being an interesting issue to be studied in future. Experimental evidence has been reported regarding the effect of contact in this external area in aluminium alloys [41, 42]. Experimental tests have been performed in which part of the external layer of the specimen surface is machined away causing the effect of fatigue crack closure delay decrease or even cease.

5.3. Experimental Correlation Once we have obtained different values for crack closure along the thickness the question is which of them is valid or significantly related to the crack growth delay effect due to fatigue crack closure. This is a complex question and it is not easy to answer. In this work, only an attempt to check if the influence of crack closure is related to the thickness of the specimen or to the external transition zone is made. The method described in section 4.4 for the bi-dimensional case is applied again. In that case, several definitions for ΔK effective were tried using the numerical results relevant to the crack opening or closure definition. Finally, the definition computed from averaging the loading and unloading driving force, ΔKnc, obtained from bi-dimensional FE plane stress calculations, presented the highest correlation coefficient being shown to be the best representative of fatigue crack closure. So, only this criterion is going to be used here. In Table 2 the correlation coefficients obtained for this criterion in plane stress, for ΔKappl, and for the reference values (ΔKElber, ΔKSchjive) are summarized. Focussing on the numerical values of r2, it can be seen that an improved correlation was obtained for ΔKnc(a) (2D plane stress). The same pattern it was shown in the three different sets of tests. Table 2. Correlation coefficient, r2. Reference values

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Crack Drive

B4

B8

B12

ΔKapplied

0.821

0.88

0.805

ΔKElber

0.894

0.951

0.901

ΔKSchjive

0.888

0.95

0.906

ΔKnc(a) 2D plane stress

0.973

0.966

0.972

In order to evaluate the influence of the different values obtained along the thickness, the same process was applied with the tri-dimensional results at the interior plane, at the exterior surface and at different distance from the exterior. A comparison among results obtained at different relative distance (defined as d/b) was made first; maximum correlation coefficient was present at different position depending of the specimen thickness. This suggested that the effect of crack closure would be related to the thickness in absolute terms.

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In table 3, results are shown in terms of the absolute distance to the exterior. In this case it can be observed that the maximum value it is at d = 0.3 mm, for the three specimen thickness. The same pattern for all the thicknesses can be observed. Maximum values are present in the same area, ranging from d = 0.2 to 0.4. It also can be seen that the results obtained close to the exterior surface have a good correlation. On the other hand, the interior values present a poor correlation. Table 3. Correlation coefficient, r2. Absolute distance Crack Drive

B4

B8

B12

ΔKnc(3D exterior)

0.905

0.945

0.938

ΔKnc(3D d = 0.1 mm)

0.617

0.813

0.595

ΔKnc(3D d = 0.3 mm)

0.968

0.971

0.969

ΔKnc(3D d = 0.5 mm)

0.883

0.913

0.851

ΔKnc(3D interior)

0.704

0.804

0.674

No final conclusion can be obtained from these results, nevertheless the idea of the dominance of the exterior area in PICC and that this influence is not dependent of the specimen thickness can be reinforced. The fact that the correlation is the best when d = 0.3 mm it is indicative of this idea but, due to de different values obtained for fatigue crack closure or opening load along the thickness, additional work must be done in order to formulate a criteria to obtain a significant single value.

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6. Conclusion As a resume of the question stated in this chapter, several conclusions will be outlined regarding three different issues, those are: numerical modelling procedure, bi-dimensional results and tri-dimensional results. Regarding the methodology for modelling fatigue crack closure, the key aspects of the process were carefully examined and appropriate solutions developed for the specific problem addressed. It was found of a great importance the effect of mesh refining on result convergence and the need to reduced mesh size to unprecedented levels and substantially expand previous recommendations. The results were found to depend strongly on the way the vicinity of the crack tip is meshed. An alternative method to obtain the target value without using an unnecessarily fine mesh has been proposed and used. It involves the calculation of two or more finite element models involving variable mesh sizes above a preset value around the crack tip and subsequently extrapolating the results in order to obtain the value for a continuous mesh. It also were determined the minimum wake length to be simulated in the numerical model in order to ensure it represent the real wake. The length necessary varied with R and was quantified in order to facilitate the subsequent modelling of the problem. The effect was less marked with plane strain (only one-tenth of the plastic size).

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The Kttop and Kncop criteria were found to converge. Nevertheless, Kttcl and Knccl were found to diverge to rather disparate final values in crack closure. Kncop and Knccl exhibit a nearly constant difference of 0.1. The influence of the type of plastic material model and hardening rule (isotropic or kinematic) was found to be negligible at low hardening ratios. At higher levels, however, the choice was critical as it led to highly disparate results. The use of hardening improved convergence and stabilized the results with respect to the widely used perfect elastic–plastic model. The influence of the point used to release nodes during crack growth was also examined and it was found that it was indifferent provided a sufficient minimum element size was used. The bi-dimensional results show that the stress ratio R is the principal factor which affects fatigue crack closure. Kmax and the crack length, a, present no influence over fatigue crack closure in the range of interest of this study. Nevertheless R is directly related to the value obtained of opening and closure. This relationship has been established by means of a great deal of calculation in plane stress and plane strain. The effect of R has been compared with the previous established relation and with values obtained from experimental tests and it has been found a general good correlation between numerical results and the experimental crack growth rate in terms of R. ΔKnc and ΔKtt in plane stress present the best correlation between numerical results and experimental data, it is better than those obtained using either Elber or Schijve relationships. A better representation of crack closure at high R ratios seems to be the cause of the improved correlation observed with the numerical results. Finally, the 3D fatigue crack closure modelling results reported show the crack behaviour through the thickness. The large number of elements employed in the mesh of the present finite element model has supposed an increase in the degree of definition and accuracy in relation to previous 3D model. The plastic zone has been visualized and quantified and crack closure and opening has been evaluated along the crack tip for a great set of calculation cases. An analysis of the plastic zone size has shown that it does not correspond to the classical “bone” shape. Results in the interior surface are similar to those obtained in 2D plane strain conditions and a reduced effect of closure is observed. However, close to the external surface, 2D plane stress results are not reproduced, the plastic zone size is lower and an important change is observed. This transition is produced inside a thin external slice of the specimen and it can only be captured if a fine mesh along the thickness is done. It has been found that crack closure is relevant only in this area. This conclusion is in accordance with experimental observations. The transition zone size has been computed for different specimen thicknesses and has turned out to be independent of the specimen thickness. Experimental correlation of fatigue crack growth rates curves in terms of R with the results obtained along the thickness also shown to be dependent of the distance to the exterior surface and independent of specimen thickness. Compressive yielding of this external area seems to be partly the cause of a reduction in the plastic zone extension. Nevertheless, another reason is the stress distribution due to the stress state and constraints condition. This is a complex relation in which the higher interior stiffness “transfers” load from the exterior face to the interior. As a consequence it could be said that if we evaluate Kmax at different planes, a lower net value would be obtained at the exterior surface compared with those

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applied to the specimen and the interior plane, resulting in a smaller plastic zone size. This observation is only qualitative and should be validated in subsequent work.

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References [1] Elber W. In Damage tolerance in aircraft structures; ASTM STP, Philadelphia, PA, 1971; Vol. 486, pp 230–47. [2] Donald, J.K., In ASTM STP 982, Philadelphia, PA; 1988; Vol. 982, pp 222–229. [3] Fleck, N.A., In Fatigue crack measurement: Techniques and applications; Ed. Marsh, K.J., Smith, R.A., Ritchie, R.O., 1991, pp 69–93. [4] Gonzalez-Herrera, A., Zapatero, J., Engng Fract Mech 2005, 72, 337–355. [5] McClung, R.C., Sehitoglu, H. Engng Fract Mech 1989, 33, 237–252. [6] McClung, R.C., Sehitoglu, H. Engng Fract Mech 1989, 33, 253–272. [7] Solanki, K., Daniewicz, S.R., Newman, J.C. Engng Fract Mech 2004, 71, 149–171. [8] Newman, J.C. PhD Thesis, VPI&SU, Blacksburg, VA, 1974. [9] Newman, J.C. In ASTM STP 590, Philadelphia, PA, 1976; Vol 590, pp 281–301. [10] Fleck, N.A., Newman, J.C. In ASTM STP 982, Philadelphia, PA; 1988; Vol. 982, pp 319–341. [11] Chermahini, R.G. Ph.D. Thesis, Old Dominion University, Norfolk, VA, 1986. [12] Chermahini, R.G., Shivakumar, K.N., Newman, J.C. In ASTM STP 982, Philadelphia, PA, 1988; Vol. 982, pp 398–413. [13] McClung, R.C. Fatigue Fract Engng Mater Struct 1991, 14, 455–468. [14] McClung, R.C., Thacker, B.H., Roy, S. Int J Fract 1991, 50, 27–49. [15] McClung, R.C. Int J Fract 1992, 52, 145–157. [16] Sehitoglu, H., Sun, W. ASME J Engng Mater Technol 1991, 113, 31–41. [17] Sun, W., Sehitoglu, H. Fatigue Fract Engng Mater Struct 1992, 15, 115–128. [18] Wu, J., Ellyin, F. Int J Fract 1996, 82, 43–65. [19] Ellyin, F., Wu, J. Fatigue Fract Engng Mater Struct 1999, 22, 835–847. [20] Wei, L.W., James, M.N. Engng Fract Mech 2000, 66, 223–242. [21] Dougherty, J.D., Padovan, J., Srivatsan, T.S. Engng Fract Mech 1997, 56(2), 167–187. [22] Dougherty, J.D., Padovan, J., Srivatsan, T.S. Engng Fract Mech 1997, 56(2), 189–212. [23] Solanki, K., Daniewicz, S.R., Newman, J.C. Engng Fract Mech 2003, 70, 1475–1489. [24] Solanki, K., Daniewicz, S.R., Newman, J.C. Engng Fract Mech 2004, 71,1185–1195. [25] Antunes, F.V., Borrego, L.F.P., Costa, J.D., Ferreira, J.M. Fatigue Fract Engng Mater Struct 2004, 27, 825–835. [26] Antunes, F.V., Rodrigues, D.M., Engng Fract Mech 2008, 75, 3101-3120. [27] Zhang, J.Zhen., Zhang, J.Zhong., Shan Yi Du, Engng Fract Mech 2001, 68, 1591–1605. [28] Zhang, J.Z., Bowen, P. Engng Fract Mech 1998, 60(3), 341–360. [29] Roychowdhury, S., Dodds, R.H., Engng Fract Mech 2003, 70, 2363–2383. [30] Roychowdhury, S., Dodds, R.H., Int J Solid Struct 2004, 41, 2581–2606. [31] Carlyle, A.G., Dodds, R.H. Engng Fract Mech 2007, 74, 457-466. [32] Alizadeh, H., Hills, D.A., de Matos, P.F.P., Nowell, D., Pavier, M.J., Paynter, R.J., Smith, D.J., Simandjuntak, S. Int J Fatigue 2007, 29, 222-231. [33] Palazotto, A. N., Mercer, J.C., Engng Fract Mech 1990, 35, 967–986. [34] Zapatero, J., Moreno, B., Gonzalez-Herrera, A., Engng Fract Mech 2008, 75, 41-57.

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[35] Tsukuda, H., Ogiyama, H., Shiraishi, T., Fatigue Fract Engng Mater Struct 1995, 18, 503–514. [36] Gonzalez-Herrera, A., Zapatero, J. Engng Fract Mech 2008, doi:10.1016/j.engfracmech. 2008.04.024, in press. [37] Schijve J. Engng Fract Mech 1981, 14, 461-465. [38] Gonzalez-Herrera, A., Moreno, B., James, M.N., Zapatero, J., In Proceeding of the 9th International Fatigue Congress, Fatigue 2006, Atlanta GA (USA), 2006; p. O150. [39] Gonzalez-Herrera, A., Ph.D. thesis, University of Malaga, 2004. (in Spanish) [40] Gonzalez-Herrera, A., Garcia-Manrique, J., Cordero, A., Zapatero, J. Key Engng Mater 2006, 324-325(1), 555-558 [41] Telesman, J., Fisher, D.M., In ASTM STP 982, Philadelphia, PA.1988. pp. 568-582. [42] Matsuoka, S., Tanaka, K. Engng Fract Mech 1980, 13, 293–306

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In: Fatigue Crack Growth… Editor: Alphonse F. Lignelli, pp. 125-166

ISBN: 978-1-60692-476-1 © 2009 Nova Science Publishers, Inc.

Chapter 4

TEXTURAL FRACTOGRAPHY OF FATIGUE FRACTURES Hynek Lauschmann1 and Noel Goldsmith2 1

Czech Technical University, Faculty of Nuclear Sciences and Physical Engineering, Dept. of Materials, Trojanova 13, 12000 Prague 2, Czech Republic 2 Defense Science & Technology Organisation, Melbourne, Australia

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Abstract The reconstitution of the history of a fatigue crack growth is based on the knowledge of correspondences between the morphology of the crack surface and the crack growth rate CGR. Within conventional fractography, striations and/or beach lines are the main source of information. However, striations are often not present or not distinguishable, and the interpretation of beach lines is strictly limited to the particular case. As an alternative information source, applications of image textural analysis were developed, called overall Textural fractography. Mezoscopic SEM magnifications (30 to 500x) are used, under which images of fracture surfaces contain complicated (image) textures without distinct borders. The aim is to find any characteristics of this texture which correlate with CGR. Pre-processing of images is necessary to obtain a homogeneous texture. Six methods of image textural analysis have been developed and realized as computational programs: application of 2D Fourier and 2D Wavelet transformations, fractal analysis, Gibbs random field (GRF) model, idealization of light objects into a fiber structure, and auto-shape decomposition (which is a fully original method developed for fractographic applications). By using any of these methods, images are characterized by a set of numerical characteristics - image feature vector. The relation between image feature vectors and crack growth rates is expressed by means of a multivariate statistical model or a neural network. Solutions of cases of crack growth under constant cycle loading are presented. For cases of different variable cycle loadings, the conventional definition of fatigue crack growth rate (CGR) itself is shown to be fractographicly confusing. A new concept of reference crack growth rate (RCGR) is proposed to cover all fatigue cracks regardless of the type of loading. RCGR may be estimated directly from images of fracture surfaces by means of the methods of textural fractography. Physical interpretation is suggested on the basis of cycle-by-cycle crack growth description. Both approaches are compared in application – quantitative fractography of three sets of specimens (from two aluminium alloys and a steel one) loaded by constant cycle, periodic and random blocks,

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Hynek Lauschmann and Noel Goldsmith respectively. Common morphologic features of all fracture surfaces are found that are closely related to RCGR.

Keywords: fatigue, fractography, reconstitution of crack growth, variable cycle loading, image analysis, image texture.

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Introduction One of the achievements of quantitative fractography is the reconstitution of the history of fatigue crack growth process. For the input information, specimens of the material must be loaded in the laboratory under service conditions. Simultaneously, crack growth process is to be recorded so that an estimate of macroscopic crack growth rate (CGR) may be later assigned to each locality of crack surface (here CGR is understood generally as a vector defined by size and direction). Fracture surfaces are documented by SEM and mean CGR are assessed to single images. Then the set of pairs [image, CGR] is analyzed with the aim to relate some information present in the morphology of the crack surface to the CGR. So a basis is obtained on which an unknown CGR can be estimated from images of a fracture surface which arose without control. Finally, the crack growth process may be reconstituted using integration of CGR along a crack growth line. The field of application is especially the industrial lifetime tests. Unfortunately, service cracks are usually squeezed, corroded, etc. Cases when their morphology can be related to a standard are very rare. The main and typical feature of fatigue cracks are striations, fine equidistant grooves in the fracture surface. Under constant cycle loading (repeating the same cycle in force or deformation), striation spacing as well as CGR increase monotonically with increasing crack length. An unambiguous relation between striation spacing and CGR exists, offering a basis for the reconstitution, [1]-[4]. The method cannot be used when striations are not visible, typically due to corrosion [5]-[7]. An alternative feature would be desirable not only for such cases, but also as an independent source of information to check results of analysis from striations. However, cyclic loading in practical service is usually variable (a sequence of variable loading cycles). Morphologies of fracture surfaces are diversiform and deeply dependent on the loading regime. CGR locally decreases and increases, changing not only striation spacing, but even their occurrence. Therefore, striations cannot be used as a quantitative feature. In this case, the main information source for quantitative fractography are beach lines (called also crack arrest lines) - visible traces of crack front which are joined with significant changes in loading. It makes the reconstitution of crack growth possible only in cases when loading history is known. Then the mutual relations between beach lines and special loading events may be studied. Instead of loading cycles in previous case, loading events creating beach lines serve as significant time points for the reconstitution of crack growth [8-11]. Many loading sequences do not create distinguishable beach lines. To enable accurate reconstitutions of the history of crack growth within fatigue testing, special short sequences of loading cycles were inserted into the loading programs to make readable traces ("marking fracture surfaces", [12-16]). The time reconstitution of fractures from practical service was possible only individually in special cases, and its reliability was limited. Consequently, a general, unifying approach for all cases was highly desirable.

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Textural Fractography Striations and beach lines belong to fractographic features - strictly defined measurable objects in the morphology of the fracture surface. Exploitation of such features means to distinguish their individual occurrences, while required characteristics are measured. A methodological alternative to localized individual features is integral analysis of the image as a whole. This idea has been developed in Dept. of Materials, FNSPE CTU Prague, since about 1990, having later obtained the overall title Textural fractography [17]. In the contrary to conventional methods, structures in images of fracture surfaces are studied as image textures. The term image texture - originating from computer image analysis - denotes a random 2D structure of similar elements with some kind of ordering. A special character of images of fatigue fracture surfaces was met, with a continuous brightness scale and an absence of distinct borders of textural elements. It lead us to re-define image texture for fractographic purposes generally as a homogeneous 2D random field. (Homogeneous means stationary or transition invariant - the character of randomness in regularity and regularity in randomness is supposed to be the same within the whole image). Information in the form of integral parameters of the whole image may be of two types: •

•

Estimates of statistical or model parameters directly from gray-scale images without analysis of textural elements. As examples, decomposition methods (Fourier, wavelet), analysis of gray-level coincidences (Gibbs random field model) and fractal analysis will be presented. Structural analysis - extraction of textural elements followed by the application of binary random field models. A method aimed at fiber similar objects will be described.

SEM Magnification

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For the application of the textural method, the mezoscopic dimensional area with SEM magnifications between macro- and microfractography (about 30 to 500 x) is especially suitable. These magnifications were not used very often in the past for the absence of measurable objects in images. The magnification to be used is limited by several conditions touching different scales [17]: 1. Condition on the whole set of images: A common type of image texture is dominant in all images. (For example, images without and with distinguishable striations - as the dominant textural element - should not be analyzed together.) 2. Conditions on individual images: a) the change of CGR is negligible within single images (CGR can be characterized by a constant), b) the texture is approximately homogeneous (the change with increasing CGR is negligible), c) the number of textural elements is representative enough to characterize the texture. 3. Condition on discretization: Textural elements which are expected to be the main source of information are represented with a sufficient resolution.

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Conditions 1, 2c and 3 tend to smaller magnification, conditions 2a and 2b to greater one. Therefore, the choice of magnification is a compromise. SEM magnification 200 x with field of view 0.6 x 0.45 mm has been proved as suitable in numerous applications. However, the experience shows that textural image analysis selects features which correlate with CGR under all reasonable SEM magnifications.

Pre-processing of Images SEM images of fracture surfaces often contain significant fluctuations of mean brightness and contrast. These fluctuations are caused by morphological aspects (height and slope of crack surface) with characteristic dimensions corresponding to the image size or larger, and must be removed to prevent their withering effect on results. A suitable method - normalization [18] was derived by generalization from one-dimensional stochastic processes. A moving algorithm corrects the original brightness x by local mean value mS and standard deviation sS computed in surrounding S, called mask. By applying the formula

xi′, j =

0 for u < 0 u for 0 ≤ u ≤ 255 255 for u > 255

,

u = 128 + 50

xi , j − mS ( x) sS ( x )

,

(1)

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the original image is transformed to a homogeneous texture with mean value 128 and standard deviation 50 (for an 8-bit range). In contrast to the generally used equalization [41], the shape of textural elements is conserved. The size of the mask S is a very important parameter fundamentally affecting the results. A too small mask destroys textural elements to smaller ones, while a too large mask does not homogenize images sufficiently. In some degree, the size of mask S defines the morphologic substructure which creates input of further analysis.

Figure 1. Original and normalized image (section 600 x 450 pixels)

Modelling of the Relation between Image Textures and CGR Within this chapter, solution for the case of constant cycle loading will be presented. In other words, method described is an alternative to analysis based on striations.

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Many unsuccessful attempts proved that no scalar image characteristic of any type is robust enough to characterize the relation between CGR and the crack morphology. Although the dependence can be very close in images of individual crack surfaces, within different testing specimens the same parameter can depend on CGR in a different way. Therefore, images must be characterized by a set of characteristics, called feature vector. Usually characteristics of one type (e.g. wavelet) are used as image features. Six methods of composing feature vectors of image textures are commented in Appendix. Let us have a set of q images with assessed macroscopic crack rates (CGRs) vi , i=1,2,…,q, and characterized by a set of k textural parameters fij , j=1,2,…,k, creating feature vector fi.=[fi1, fi2 , ... , fik]. Due to the range of crack rate covering several decadic orders, logarithm log10v is considered in computations. The simplest model expressing the CGR as a function of a set of image parameters is a multilinear function [17],[19] resulting into a system of regression equations k

log(vi ) ≈ ∑ c j fij + ck +1 , i = 1,… , q .

(2)

j =1

The system of equations may be formalized in a matrix form

L = F*C ,

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⎡ f11 ⎢ F=⎢ ⎢ f q1 ⎣

f12 fq 2

(3)

f1k f qk

1⎤ ⎥ . ⎥ 1⎥⎦

(4)

The i-th row of matrix F is feature vector fi completed by constant one, [fi1, ... , fik, 1], L is a column vector of logs of crack rates and C is a column vector of unknown regression parameters cu. The solution for parameters cu is estimated by the least squares method. The system must be strongly overdetermined - the number of equations must be significantly greater then the number of estimated constants, q>>k+1. The number of equations matches the number of images, and the number of estimated constants k+1 is given by the number of image features k. Therefore, it may be simply said: for a given set of images, feature vector of any type may be used but the number of its components should be much smaller then the number of images. If this condition was not satisfied, final model would include individual random fluctuations which are specific for input images, and would not describe the relation between the morphology of fracture surface and CGR in the given material generally. Sometimes matrix F is singular - its columns representing single features are mutually dependent. Typical case is that pairs of features are significantly correlated - one of them may be excluded. A general way of solution of such cases is analysis of principal components (Karhunen-Loewe expansion). Not all characteristics fu predicate the CGR. An instrument for testing the significance is the test of the zero value of the estimated coefficients cj , j=1,…, k+1 . We test the hypothesis

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H0: cj = 0 against the alternative H1: cj ≠ 0. The test criterion is a Student’s t-distributed statistic [20]

(

)(

2 ⎛ log v − log v ∑ ⎜ (F′F )−1 tj = cj ⎜ q−k −2 ⎝

)

⎞ ⎟ j, j ⎟ ⎠

−1 / 2

.

(5)

If the absolute value of tj is lower than the critical value at the selected level of significance α and q-k-2 degrees of freedom, t j < t1−α / 2 (q − k − 2 ) , hypothesis H0 cannot be rejected and

j-th textural feature f.j should be excluded. Selection of features which compose the final model may be interpreted - more or less according to the type of features - with respect to textural components in images, which can be immediately related to morphologic aspects of fracture surface. On the other hand, a handicap regarding the final model originates from the multiparametric character of the method. Model counterbalances many increments from different sources of information. Therefore, its fractographic interpretation is hardly possible.

Crack growth rate [μm/cycle]

1 0.5

OUTPUT (from images)

0.2 C16 C17 C18 C19

0.1

INPUT (from experiment)

0.05 0.05

0.1

0.2

0.5

1

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Figure 2. Example of graph Input CGR - Output CGR. Points correspond with single images.

The quality of model obtained may be simply visually checked by a graph comparing input and output CGRs, as shown in Fig. 2. While input CGRs must be known from experiment and serve for estimation of model parameters cj, output CGRs are computed from the equation (2) with estimated parameters cj. So, in fact, output values are model estimates of input values, and their difference is a measure of the quality of the model. Therefore, we expect points [input CGR, output CGR], representing single images, to be spaced along the diagonal line (y = x) of graph. Simultaneously, the smaller is width of strip of points, the better expression of the relation between CGR and crack surface morphology was found. Within numerous applications on specimens of different materials, width of data strip varied from about 0.1 to 0.2 (in vertical direction, for log10v). In some cases, multilinear model does not provide satisfactory results - the strip of points in graph is too wide or deflected. A well-tried method is to supplement or replace feature vector by

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transformed features - log f ui , f ui , j ≠ 0, etc., while the result of multilinear regression is j

checked. A universal method how the relation between image features and CGRs may be expressed is an application of neural networks.

Solution for Variable Cycle Loading A general approach to analysis of cracks caused by variable cycle loading must be started with focusing on the definition of crack growth rate.

Reference Crack Growth Rate (RCGR) Crack growth rate (CGR) is conventionally defined as v = Δa/ΔN,

(6)

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where a denotes crack length and N is number of loading cycles. It is not generally known that this definition cannot be used for crack growth under variable cycle loading. The problem will be obvious from a comparison of two particular loading sequences shown in Fig. 3 [21]. In case 2, “small” cycles 2,4,... are inserted between cycles of loading 1. Due to the effect of retardation after an overload, inserted cycles are not active. The morphology of the crack surface is the same in both cases with respect only to the number of “great” cycles. However, all cycles are counted within the conventional definition of CGR - active "great" loading cycles as well as non active “small” loading cycles. Therefore, crack growth rates assigned in both cases to the same crack morphology, are v and v/2, respectively. Moreover, an arbitrary discrepancy could be obtained: inserting n “small” cycles between each two “great” results into crack growth rate v/(n+1).

Figure 3. Fractographic ambiguity of the conventional definition of crack growth rate (CGR): loadings creating the same crack surface morphology at CGR values v and v/2, respectively.

It is evident that conventional definition of CGR is limited only to fractures under loading by a constant cycle (in force, deformation, ΔK, etc.). Then CGR is unambiguously related to the crack surface morphology, which is the main presumption of quantitative fractographic analyses. So the fact, that cases of fatigue under variable cycle loading are fractographicly "timed" not by loading cycles, but by significant events in loading, is not caused only by selection of beach lines as fractographic feature. Counting of cycles cannot be used, and another general

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measure of time does not exist. However, counting of time by selected loading events makes from each type of loading a particular case without any connection to other cases. The absence of a generally valid definition of crack growth rate, which could be related to crack surface morphology, is the main reason why no universal methodology has been developed for the fractographic reconstitution of a fatigue crack growth under variable cycle loading. In following paragraphs cited from [21] we attempt to establish a generalized definition of crack growth rate, unifying the description of the fatigue crack growth under all types of loading. The conventional presumption of quantitative fractography, valid only for a constant cycle loading, is: "The crack surface morphology is unambiguously related to the (conventional) CGR." Let us generalize the presumptions as follows: 1) We suppose the possibility of defining the reference crack growth rate (RCGR, vref), 2) so that it would be unambiguously and without any respect to the type of loading 3) related to selected features of the fracture surface morphology. To hold the continuity with the conventional approach, we will require the reference concept to be identical with the conventional in the case of a constant cycle loading: vref,cc = v. In fact it means that RCGR concept relates all cases of fatigue to the archetypal reference case, fatigue crack growth under constant cycle loading. Proposed generalization of presumed facts touches both sides of the original relation – the definition of crack growth rate as well as the morphology of crack surface: •

The reference crack growth rate vref must hold the general character of velocity - the ratio increment of distance / increment of time. In the case of fatigue, the distance is measured by the crack length, and so its increment is fully determined. Then, only time remains to be defined differently. Conventionally it is given by counting cycles: Δt = 1 for each cycle. Therefore, the concept of RCGR implies a more general definition of time – the reference time tref . It follows

vref = Δa/Δtref .

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•

(7)

Regarding the crack surface morphology, some features are presumed, which are typical for the fatigue damage generally and always present regardless of the character of loading. That means: Although crack surfaces created by different loading sequences are more or less different, they are expected to exhibit some common morphologic features. These features should be (quantitatively – by changing shape, dimensions, counts, etc.) related to RCGR.

The RCGR concept may seem to be absolutely vague: it is only an idea dealing with two “great unknowns”: another definition of crack growth rate, and some morphologic features related to it. The idea itself does not give any guide how this couple should be found. Two solutions making use of the advantages of textural fractography were proposed.

Direct Fractographic Solution - Generalization of the Textural Method Following theory will be limited to cyclic loadings with variable, sufficiently stationary parameters. Here "sufficiently" is meant with respect to the morphology of fracture surfaces:

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it should be "calm", without striking local differences. Especially overloads should be distributed more or less regularly. Under such conditions, the ratio between reference and conventional CGR can be approximately expressed as a constant [21]

RCGR vref = =B. CGR v

(8)

The value of B is linked only to the given type of loading (in the case of constant cycle loading particularly the ratio will be Bcc = 1). This idea is intuitively based on following experience: inserting regular overloads into a constant cycle loading lowers crack rate by a more or less - constant multiplicative factor in the whole range of technical interest. However, the formula (8) is not a presumption but a definition. Only results of the application to experimental data can show whether it gives a demonstrative output. From eq. (8), RCGR may be expressed as vref = v.B,

(9)

log(vref) = log (vB) = log(v) + log(B).

(10)

that means

After having denoted log(B) = b, we have

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log(vref) = log(v) + b.

(11)

Figure 4. Schematic plot of the definition of RCGR by equation (8). The switch from CGR to RCGR is equivalent with shifting sets of images along axis log(v) by log(B).

Realizing (11) we may illustrate the idea by a scheme shown in Fig. 4. Now it is also clear that the special case of inactive small loading cycles inserted into greater loading from Fig. 3 is solved by B = 2 for case 2: for RCGR = 2CGR we have

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Hynek Lauschmann and Noel Goldsmith

vref = 2(v/2) = v, what means that the same value of RCGR is related with the same morphology of crack surface in both cases 1 and 2. When we apply multilinear model (eq. (2)) to RCGR, we obtain [21] k

log(vi ,ref ) = log(vi ) + b ≈ ∑ c j fij + ck +1 , i = 1,… , n.

(12)

j =1

The new unknown parameter b can be transferred to the right side of the equation as an additional regression constant: k

log(vi ) ≈ ∑ c j fij + ck +1 − b, i = 1,… , q.

(13)

j =1

Let us remember: b is a characteristic of loading - it is expected to be equal for all specimens that have been loaded by the same loading type. Particularly, in the case of a constant cycle loading, factor b disappears: bcc = log(1) = 0. Matrix form of the system of equations (13) is the same as in previous case (3), L = F * C,

(14)

where L is the column vector of logs of conventional crack growth rates and C is the column vector of constants [c1, ..., ck+1, b2, b3, ...] to be estimated. F is the matrix of image features, completed with new columns in the following manner:

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⎡ f11 ... ⎢ f ⎢ 21 ... ⎢ ⎢ F = ⎢ fi1 … ⎢ f i +1,1 ... ⎢ ⎢ ⎢ f ⎣ q1 ...

f1k f2k f ik fi +1,k f qk

0⎤ 0 ⎥⎥ ⎥ ⎥ 1 −1 ... 0 ⎥ 1 −1 ... 0 ⎥ ⎥ ⎥ 1 0 ... −1⎥⎦

1 1

0 0

... ...

for constant cycle loading, for constant cycle loading, for type 2 of loading,

(15)

for type 2 of loading, for the last type of loading.

Each row of matrix F represents one image, fi1, ... , fik are image features. The next column of ones is for ck+1 in eq. (13). Then one column is added for each type of the variable loading, containing -1 in relevant rows for -b in eq. (13). The system is solved by the least squares method, while new parameters bp, p = 2,3,... are estimated together with parameters cj of the multilinear model. The case of constant cycle loading must be included. All comments to the solution in case of constant cycle loading are valid also here. The ratio Bp defining RCGR (8) for the p-th type of loading is estimated as b

B p = 10 p .

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

(16)

Textural Fractography of Fatigue Fractures

135

So the constants Bp defining the RCGR for the p-th type of loading, p = 2,3,…, are estimated simultaneously while solving the standard system of equations for the textural method. In this manner, the definition of RCGR was realized as a simple computational task. The second "great unknown" of the RCGR concept – specifying the morphologic features which correspond to RCGR - is solved by testing the significance of contributions of single image features according to eq. (6). After having excluded image features that do not contribute to the model, we receive a final selection of image features • •

reflecting RCGR, and joined with the morphological aspects of fracture surfaces that are common for all images, regardless of the type of loading.

In this way, the image textural analysis automatically extracts desired common features, also from images that are visually very different.

Physical Solution The direct fractographic solution shows a purely phenomenological, descriptive approach. However, we would like to give our considerations a physical background. As it has been said above, the idea of RCGR consists in relating fatigue crack growth under different loadings by variable cycles to the basic case of constant cycle loading. The principle [23] is a correction in counting time: for time increment Δtref,n of n-th cycle, a special value will be proposed instead of ΔNn = 1 in conventional approach. Then, according to eq.(7), reference crack growth rate is defined as

RCGR = vref =

Δa = Δtref

Δa ΔN

∑ Δt

.

(16)

ref , n

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n =1

Simultaneously, equivalence of the reference concept with the conventional CGR definition was required in the basic case of constant cycle loading, what means that reference time increment in n-th cycle within constant cycle loading is Δtref,n,cc = 1, independently of n. Now we must ask, how the reference time increment of one cycles should be defined in case of a general loading. A simple solution is based on following reasoning: The crack increment in a given cycle within a given loading sequence may be expressed as

Δareal = h ⋅ Δacc ,

(17)

where Δareal is real crack increment, and Δacc is crack increment within the constant cycle loading, both in the given cycle at the given crack length. Factor h may be interpreted as effectivity of the given cycle in relation to constant cycle loading. In all cycles of constant cycle loading, h = 1. Under variable cycle loading, h < 1 in cycles with smaller loading after greater ones constituting a tensile overload, and h > 1 in cycles with greater loading after smaller ones or due to acceleration after a pressure overload.

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

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Hynek Lauschmann and Noel Goldsmith

Now, let us consider a sequence of loading cycles, including a number N of cycles characterized by values Δacc and h. The cumulative crack increment of these cycles will be

∑ Δa = N ⋅ h ⋅ Δa

.

cc

(18)

Result is the same as if N⋅h cycles - instead of N - were applied in case of constant cycle loading. In other words, the effect of the factor h may be explained also as a correction of the number of applied cycles. With some degree of abstraction, this idea may be applied also to one cycle. So we come to the solution: For the reference time increment of a given cycle, its effectivity h will be set instead of 1 in conventional approach. Then the definition of RCGR may be written as [21]-[23]

RCGR = vref =

Δa

,

ΔN

(19)

∑h

n

n =1

where hn is the effectivity of n-th cycle - the ratio

hn =

Δareal ,n Δacc , n

.

(20)

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Correction of time increment from one cycle may seem to be very unnatural. However, the effect of interactions of different cycles really has also a time consequence, as it is shown in Fig. 5. The active part of the cycle is between Kop and Kmax. Expressing this effect by effectivity h corresponds with the fact that we do not study fatigue crack opening but its effect on crack growth.

Figure 5. Shortening of active time of a cycle after an overload. K - stress intensity factor (SIF), Kmin, Kmax, Kop - minimum, maximum and opening SIF value in a given cycle.

The effect of retardation after overloads significantly prevails in real loading sequences. Therefore, mostly h ≤ 1 and RCGR ≥ CGR for a variable cycle loading. An overview of the hypothesis is summarized in Table 1.

Fatigue Crack Growth : Mechanics, Behavior and Prediction, edited by Alphonse F. Lignelli, Nova Science Publishers, Incorporated, 2009. ProQuest Ebook Central,

Textural Fractography of Fatigue Fractures

137

Table 1. The conventional and reference concept (physical interpretation) of the fatigue crack growth rate [21]. Concept Loading One cycle

Conventional Constant cycle

Reference time / crack rate General

Crack increment

Δacc

Δacc . h*)

Time increment

ΔN1 = 1

Δtref,1 = h*) ΔN

ΔN

Time interval

Δ t ref = ∑ hn ≤ Δ N

ΔN = ∑ 1 n =1

Δa ΔN

Crack growth rate

v cc =

Comparison

vcc = vref

*)

n =1

(CGR)

v ref =

Δa Δ t ref

v ref ≥

Δa ΔN

(RCGR)

effectivity of the cycle; after overloading: h