ABSTRACT Title of Thesis: EVALUATION OF INFORMATION ENTROPY FROM ACOUSTIC EMISSION WAVEFORMS AS A FATIGUE DAMAGE METRIC FOR AL7075-T6 Christine Marie Sauerbrunn, Master of Science, 2016 Thesis Directed By: Professor Mohammad Modarres Department of Mechanical Engineering Information entropy measured from acoustic emission (AE) waveforms is shown to be an indicator of fatigue damage in a high-strength aluminum alloy. Several tension- tension fatigue experiments were performed with dogbone samples of aluminum alloy, Al7075-T6, a commonly used material in aerospace structures. Unlike previous studies in which fatigue damage is simply measured based on visible crack growth, this work investigated fatigue damage prior to crack initiation through the use of instantaneous elastic modulus degradation. Three methods of measuring the AE information entropy, regarded as a direct measure of microstructural disorder, are proposed and compared with traditional damage-related AE features. Results show that one of the three entropy measurement methods appears to better assess damage than the traditional AE features, while the other two entropies have unique trends that can differentiate between small and large cracks. EVALUATION OF INFORMATION ENTROPY FROM ACOUSTIC EMISSION WAVEFORMS AS A FATIGUE DAMAGE METRIC FOR AL7075-T6 by Christine Marie Sauerbrunn Thesis submitted to the Faculty of the Graduate School of the University of Maryland, College Park, in partial fulfillment of the requirements for the degree of Master of Science 2016 Advisory Committee: Professor Mohammad Modarres, Chair and Advisor Professor Balakumar Balachandran Professor Hugh Bruck © Copyright by Christine Marie Sauerbrunn 2016 ii Acknowledgements First and foremost, I would like to thank my advisor, Dr. Mohammad Modarres, for the continued support and guidance throughout both my undergraduate and Master’s studies. I am grateful for the research experience, opportunities, and knowledge you have given to me over the past three years. I would also like to extend my appreciation to my other committee members, Dr. Hugh Bruck and Dr. Balakumar Balachandran. Thank you both for your guidance in the classroom and your advice on this work. Many thanks are extended to my coworkers and friends at the Center of Risk and Reliability (CRR). Specifically, thank you to Huisung Yun for being a great research partner to bounce ideas off of, for your positive attitude each time the lab equipment malfunctioned, and for your unwavering commitment to polishing, imaging, and preparing all the test specimens. Your help over the past two years has been truly invaluable. I am also appreciative for my discussions with Dr. Ali Kahirdeh about acoustic emissions and information entropy. You have encouraged me to think critically about my engineering assumptions while also teaching me to be less self-critical. Finally, thank you to those who first welcomed me to CRR as an undergraduate and helped me get accustomed to research practices; Victor Ontiveros, Mehdi Amiri, Azadeh Keshtgar, and Abdullah Al Tamimi. I am also so thankful for my loving and supportive family and friends. Thank you Mom and Dad for always encouraging me to achieve my goals – I could not have done any of this without you. To my sisters, Liz and Steph, thanks for always welcoming me home after the stressful weeks at school. And to my Oma, thank you iii for always reminding me of the importance of education and supporting all of my endeavors. Special thanks goes to my college friends for all of the fun times and distractions from the stress of classes and work. You all have kept me sane these past five years and have made my time at Maryland unforgettable. Our friendships will extend for many years to come. Finally, this research would not have been possible without the financial support of the Office of Naval Research under grant #N00014140453. On behalf of myself and the University of Maryland, thank you for your generous support. iv Table of Contents Acknowledgements ....................................................................................................... ii Table of Contents ......................................................................................................... iv List of Tables ............................................................................................................... vi List of Figures ............................................................................................................. vii List of Abbreviations ................................................................................................... ix Chapter 1 – Introduction ............................................................................................... 1 1.1 Motivation and Background ............................................................................ 1 1.2 Research Objectives and Methodology ........................................................... 4 1.3 Contributions ................................................................................................... 5 1.4 Outline of Thesis ............................................................................................. 5 Chapter 2 – Literature Review ...................................................................................... 7 2.1 Fatigue Damage in Metals ............................................................................... 7 2.1.1 Stages of Fatigue ................................................................................. 7 2.1.2 Literature on Crack Nucleation, Initiation, and Small Cracks ............ 9 2.2 Acoustic Emission Background .................................................................... 12 2.2.1 AE Theory ......................................................................................... 12 2.2.2 AE Instrumentation and Terminology ............................................... 13 2.2.3 Estimating Fatigue Damage with AE ................................................ 17 2.3 Information Entropy Background ................................................................. 21 2.3.1 Understanding Information Entropy ................................................. 21 2.3.2 Application of Information Entropy to Fatigue and AE .................... 25 Chapter 3 – Experimental Procedure .......................................................................... 30 3.1 Specimen Preparation .................................................................................... 30 3.2 Loading Conditions ....................................................................................... 32 3.3 Strain Measurement ....................................................................................... 35 3.4 Crack Monitoring System ............................................................................. 37 3.5 Acoustic Emission Instrumentation ............................................................... 40 3.6 Mechanical Damping Apparatus ................................................................... 40 3.6.1 Why Mechanical Damping? .............................................................. 40 3.6.2 Damping Techniques ......................................................................... 41 3.7 Details of Individual Fatigue Experiments .................................................... 43 Chapter 4 – Post-Processing Methods ........................................................................ 45 4.1 Noise Reduction ............................................................................................ 45 4.2 Instantaneous Elastic Modulus Calculations ................................................. 49 4.3 Discrete Information Entropy Formulations ................................................. 51 4.3.1 Feature Entropy ................................................................................. 53 4.3.2 Updated Entropy ................................................................................ 55 4.3.3 Temporally Weighted Entropy .......................................................... 57 4.3.4 Summary of Entropy Formations ...................................................... 59 Chapter 5 – Results and Discussion ............................................................................ 61 5.1 Crack Growth and Fracture Surface Images ................................................. 61 5.2 Stress-Life Analysis ...................................................................................... 63 v 5.3 Measured Damage from Modulus Degradation ............................................ 65 5.3.1 Modulus Trends ................................................................................. 65 5.3.2 Normalizing Modulus to Produce Measured Damage ...................... 68 5.4 AE Parameter Cyclic Trends ......................................................................... 73 5.4.1 AE Counts ......................................................................................... 73 5.4.2 AE Absolute Energy .......................................................................... 76 5.4.3 Feature Entropy ................................................................................. 77 5.4.4 Updated Entropy ................................................................................ 79 5.4.5 Temporally Weighted Entropy .......................................................... 81 5.5 Normalized AE Parameters Compared to Measured Damage ...................... 83 5.5.1 Cumulative Counts, Energy, and Feature Entropy ............................ 83 5.5.2 Updated and Temporally Weighted Entropy ..................................... 87 5.6 Summary of Results ...................................................................................... 89 Chapter 6 – Conclusions ............................................................................................. 90 6.1 Summary ....................................................................................................... 90 6.2 Contributions ................................................................................................. 91 6.3 Future Work .................................................................................................. 92 Appendices .................................................................................................................. 95 Appendix A – Applied Load vs. Signal Arrival Time Scatter Plots ...................... 95 Appendix B – Code for Modulus Evolution ........................................................ 101 Appendix C – Error Propagation.......................................................................... 104 C.1 Initial Elastic Modulus .................................................................... 104 C.2 Measured Crack Length .................................................................. 105 Appendix D – Code for AE Entropy Formulations ............................................. 107 Appendix E – Code for Matching Modulus and AE Hits Data............................ 111 Bibliography ............................................................................................................. 114 vi List of Tables Table 1: Various definitions of crack phases .............................................................. 10 Table 2: User defined AE parameters and their descriptions [27] .............................. 16 Table 3: Commonly extracted AE features and their descriptions [4, 27] .................. 17 Table 4: Composition and mechanical properties for Al7075-T6 .............................. 30 Table 5: Loading conditions for each experiment ...................................................... 34 Table 6: Experiments for each loading condition ....................................................... 34 Table 7: Strain measurement techniques and associated attributes. Advantages are in green, disadvantages are in red. ...................................................................... 37 Table 8: AE software settings ..................................................................................... 40 Table 9: Experiment details. Tests with any detail in red are discarded..................... 44 Table 10: AE noise behavior for each test with an acceptable test setup ................... 49 Table 11: Measurement uncertainty for 11 experiments ............................................ 51 Table 12: Summary of entropy formulations .............................................................. 60 Table 13: Parameters for theoretical S-N curve for particular Al7075-T6 specimen . 64 Table 14: Fitted models for measured damage ........................................................... 69 Table 15: Error for crack length measurements .......................................................... 71 Table 16: Error in cycles at measured crack lengths .................................................. 72 vii List of Figures Figure 1: Proposed Short-Term Loading Process (STLP) where constant high- amplitude loading is interrupted by small excitation loading to observe response of damage precursor parameter .......................................................... 3 Figure 2: Crack development and phases of fatigue life............................................... 8 Figure 3: AE sensor recording surface displacement waves from damage region ..... 13 Figure 4: Transformation of AE signal due to signal shaping chain. Adapted from [4]. ......................................................................................................................... 15 Figure 5: AE voltage waveform with labelled features .............................................. 15 Figure 6: a) Probability mass function (PMF) of flipping a fair coin and calculated information entropy. b) PMF of a biased coin with probability of flipping tails as 0.70 and calculated entropy. c) PMF of a biased coin with probability of flipping tails as 0.99 and calculated entropy. .................................................. 23 Figure 7: a) PMF of a rolling a fair die and calculated information entropy. b) PMF of a weighted coin with greater probability of rolling a 4. c) PMF of coin 97.5% likely to result in a 4 and calculated entropy. ................................................. 24 Figure 8: Microstructural disorder as defined by fatigue damage evolution .............. 27 Figure 9: Fatigue specimen geometry ......................................................................... 31 Figure 10: Probability distribution of grain diameter for 375 measured grains ......... 32 Figure 11: Servo-hydraulic MTS machine ................................................................. 33 Figure 12: Crack surfaces for 5A26 and 5A18 ........................................................... 35 Figure 13: Circular segment dimensions .................................................................... 38 Figure 14: Optical microscope images during fatigue of specimen 5A26.................. 39 Figure 15: AE noise amplitude with and without mechanical damping ..................... 42 Figure 16: a) Mechanical damping method. b) Complete experimental setup. .......... 43 Figure 17: Likely and unlikely AE hits based on applied load ................................... 45 Figure 18: AE signals at their associated loads versus time for 5A26. No apparent AE noise. ............................................................................................................... 47 Figure 19: Applied load at instant of AE signals at their arrival times for 5A22. Filterable noise. ............................................................................................... 48 Figure 20: Applied load at recorded AE signals and their arrival times for 5A20. Unfilterable AE noise. .................................................................................... 48 Figure 21: Feature entropy calculation from AE signals ............................................ 54 Figure 22: Updated entropy calculation from AE signals .......................................... 56 Figure 23: Temporally weighted entropy calculation from AE signals ...................... 59 Figure 24: Entropy evolution for three examples of AE signals for each formulation60 Figure 25: Crack growth through grains and inclusions ............................................. 61 Figure 26: Brittle and ductile regions of fatigue crack growth ................................... 62 Figure 27: Fatigue striations in the brittle region........................................................ 62 Figure 28: Highly disordered ductile region with no striations .................................. 62 Figure 29: Experimental S-N plot with theoretical and experimental S-N curves ..... 65 Figure 30: Raw modulus degradation for 8 experiments at various damage levels ... 66 Figure 31: 5A25 fracture surface showing two displaced regions at notch ................ 68 Figure 32: Measured damage for 8 experiments at various damage levels ................ 70 Figure 33: Cumulative AE counts versus fatigue cycles until fracture ...................... 74 viii Figure 34: Cumulative AE energy versus fatigue cycles until fracture ...................... 77 Figure 35: Cumulative feature entropy versus fatigue cycles until fracture ............... 78 Figure 36: Updated entropy versus fatigue cycles until a 1 mm crack and fracture ... 81 Figure 37: Temporally weighted entropy versus fatigue cycles until a 1 mm crack and fracture ............................................................................................................ 82 Figure 38: Normalized cumulative counts, energy, and feature entropy with respect to measured damage where a one-to-one relationship is desired ........................ 85 Figure 39: Example of deviations between ideal and experimental trends ................ 86 Figure 40: Deviation factor for AE counts, energy, and feature entropy for all experiments ..................................................................................................... 87 Figure 41: Normalized updated and temporally weighted entropy against measured damage ............................................................................................................ 88 ix List of Abbreviations AE – Acoustic emission CV – Coefficient of variation LEFM – Linear elastic fracture mechanics MaxEnt – Principle of maximum entropy NDE – Nondestructive evaluation PMF – Probability mass function PSBs – Persistent slip bands SHM – Structural health monitoring STLP – Short-Term Loading Process 1 Chapter 1 – Introduction 1.1 Motivation and Background Virtually all engineered structures undergo some form of cyclic fatigue. For example, mechanical gears can experience tooth fracture while rotating mechanical shafts can fail due to repeated torsional loading. Systems with such components are often permitted to fail when the safety risk is minimal and the components can be replaced at a relatively low cost. However, structures such as bridges and aircraft include critical components where extensive safety risks and monetary losses are a concern upon failure. In turn, estimating when a failure may occur is imperative and inspecting these structures for fatigue damage is common practice. Specifically for aircraft, decades of research and experience have produced guidelines for estimating ideal service life. There are two main approaches for determining retirement time of military aircraft referred to as damage tolerant and safe-life. The United States Air Force adheres to the damage tolerant approach which assumes the structure has inherent defects and predicts the instance when these defects grow to become cracks of critical length [1]. In contrast, the safe-life approach assumes a new structure has no flaws and retires an aircraft based on recorded loading data and estimated crack initiation time. The United States Navy uses this approach because of the extreme loading conditions of taking off and landing on aircraft carriers with relatively short runways. Aircraft are retired once a crack is estimated to initiate and extend to a 0.25 mm length based on extensive and time-consuming, full-scale fatigue tests [1]. While the lower damage threshold of the safe-life approach reduces the likelihood of catastrophic failure, the high safety 2 factors tend to lead to premature retirement and a lower return on investment for aircraft owners [2]. Rather than basing retirement time solely on estimated crack initiation and full-scale testing, structural health monitoring (SHM) and nondestructive evaluation (NDE) methods can be used to estimate the actual material damage due to fatigue. While SHM is most desirable in that the entire structural health is continuously monitored, NDE methods that evaluate structural health during discrete inspection periods are more practical and often implemented. Traditional NDE methods include visual, eddy current, ultrasonic, and radiographic testing [3]. Each of these inspection techniques can assess damage within a structure to a degree of certainty when the probable damage location is known. Another technique that can be used is acoustic emission (AE). Acoustic emission is a technique by which elastic stress waves that propagate through a material at sources of stress are recorded as electrical signals [4]. In contrast to the other NDE methods which detect geometric discontinuities when actively positioned in anticipated damage locations, AE is a potential SHM technique that passively and continuously records microstructural movements even prior to a visible crack. Despite this advantage, however, AE is susceptible to extraneous noise. Distinguishing between noise and damage-related signals is a critical obstacle that has limited the practical SHM application of AE. Instead, AE can be used as an NDE method when noise signals can be filtered in a controlled testing environment. While AE research over the past few decades has proven effective in estimating visible cracks and crack growth rate, detecting fatigue damage at the 3 earliest instance using AE is most desirable. This goal motivated the current research which is the first of two stages. The first stage is identifying AE precursors attributed to cyclic slip and microcracks prior to visible damage. These precursors are then characterized as damage increases within a structure during constant high-amplitude loading. Once damage precursors and their behaviors are completely investigated during normal conditions, the second stage seeks to quantify the changes in these behaviors during a short-term, high-frequency excitation loading referred to as a Short-Term Loading Process (STLP). The STLP concept is depicted in Figure 1 where dα/dN is the evolution of the precursor’s behavior. In these experiments, a constant high-amplitude load will be applied to a structure. This loading will then be interrupted at several instances throughout the fatigue life to measure the known changes in the identified fatigue damage precursors due to the STLP. By knowing the evolution of precursor behavior during a STLP at various degrees of damage, the fatigue damage of a structure with unknown loading history could potentially be estimated by a simple STLP. Figure 1: Proposed Short-Term Loading Process (STLP) where constant high-amplitude loading is interrupted by small excitation loading to observe response of damage precursor parameter N cycles Load Fatigue Excitation Damage evolution Lower Amplitude Loading 4 1.2 Research Objectives and Methodology The objectives of this work are listed as the following: 1. Assess the validity of using instantaneous elastic modulus as a measure of the true microstructural damage in metallic structures. 2. Develop quantitative statistics that reflect the information carried within AE signals. 3. Investigate how features and information from AE signals correlate to fatigue damage prior to a visible crack and compare the utility of these various AE damage parameters. 4. Discuss advantages and limitations of AE damage parameters and suggest damage precursors to be used in STLP testing. In order to achieve these objectives, a series of fatigue experiments were performed on Al7075-T6, a commonly used material in aerospace structures. The local strain around a semicircular notch and the acoustic signals emitted from microstructural damage were measured during the experiments. Unlike previous studies in which fatigue damage is easily measured based on visible crack growth, this work assumes that instantaneous elastic modulus can be an estimate of the unobservable microstructural damage. Subsequently, AE features and information from AE signals are correlated to modulus degradation in order to estimate damage prior to a visible crack. Both traditional AE features like AE counts and AE energy are investigated as well as various formulations of information entropy from AE signals to correlate to fatigue damage. Information entropy is the measure of the disorder of any probability distribution. In turn, it is hypothesized that the information entropy from probability 5 distributions of AE signals reflect microstructural disorder and therefore fatigue damage. Several different methods of representing AE signals as probability distributions are developed from which several measures of information entropy are derived. The strengths and weaknesses of each of these measures in regards to correlating with true fatigue damage are discussed. 1.3 Contributions The contributions of this work are listed as the following: 1. Techniques to reduce extraneous AE noise through development of mechanical damping apparatus and justified post-process filtering. 2. A procedure to measure damage based on elastic modulus degradation. 3. Processes to derive three different proposed information entropy measurements from individual AE signals. 4. Comparison of the traditional AE features and information entropy formulations in regards to fatigue damage. 1.4 Outline of Thesis The rest of this thesis is divided into five chapters. First, Chapter 2 details theory, background, and previous research related to fatigue damage in metals, AE, and information entropy. Chapter 3 describes the experimental procedure including specimen design, data measurement systems, the mechanical damping apparatus, and the characteristics of each fatigue experiment performed. Next, Chapter 4 explains the post-processing methods to reduce AE noise, calculate instantaneous elastic modulus, and derive three different formulations of AE information entropy. Then, results pertaining to crack behavior, stress-life analysis, measured damage from elastic 6 modulus degradation, AE features, and AE information entropy metrics are discussed in Chapter 5. Finally, conclusions and recommendations for future work are provided in Chapter 6. 7 Chapter 2 – Literature Review 2.1 Fatigue Damage in Metals 2.1.1 Stages of Fatigue Fatigue is the gradual degradation and eventual failure of a material that occurs due to cyclic loading lower than the material’s static strength. Research pioneered by August Wöhler during the mid-19 th century led to the conclusion that materials can fail due to repeated stresses that are below the yield strength [5]. Since this time, many other instances of fatigue in mechanical components have been documented and inspired subsequent investigations. An extensive history of fatigue in metals between 1837 and 1994 is given by Schütz [6]. Fatigue is often characterized as three phases of damage: crack initiation, stable crack growth, and unstable crack growth until fracture. This fatigue process is illustrated in Figure 2 and thoroughly explained by Schijve [7]. Fatigue damage begins very early in fatigue life as invisible microcracks nucleate at persistent slip bands (PSBs). These PSBs are a result of moving dislocations within the material due to applied stress. Dislocations move along the direction with greatest shear stress, which varies within a material based on size, shape, crystallographic orientation, and elastic anisotropy of grains [7, 8]. Also, because surface grains are less constrained than subsurface grains, slip bands are more likely to occur at surface grains where the shear stress is often more significant and where dislocations move towards. Once a slip band occurs at a surface grain and a new surface is exposed to air, oxygen is absorbed causing local decohesion of the slip step [8]. This phenomenon causes sites for microcrack initiation. 8 Figure 2: Crack development and phases of fatigue life The initiated microcrack will grow depending on the surrounding microstructure. As a crack grows through the first grain, it will approach an adjacent grain with its own crystalline orientation and ideal slip system direction. The crack in the first grain will then deviate from its original propagation direction based on the second grain’s slip system. Because each grain impedes the crack growth to a varying degree, crack growth rate for microcracks is erratic and dependent on grain boundaries. The microcrack will continue to grow through subsequent grains and inclusions changing its propagation path each time. Once the microcrack has created a large crack front spanning numerous grains, the crack will propagate at a more consistent rate. This is when a crack transitions from a small crack to a large crack. This transition is often determined based on whether the crack growth rate can be characterized by Paris’ law [9]. Introduced in 1961, Paris [9] proposed that the crack growth rate is related to the material’s stress intensity factor range by means of a power relationship. This relationship is recounted in Equation (2.1) where da/dN is the crack growth rate, ∆K Dislocation movement, cyclic slip, microcrack nucleation Erratic microcrack growth FASTRAN software can be used Stable macrocrack growth Paris’ law is valid Rapid crack growth Fracture Measured crack length, a Cycles, N Initiation Growth 9 is the stress intensity factor range, and C and m are constants dependent on the material. Once a large crack reaches a certain length, the crack will rapidly propagate and cause fracture. 𝑑𝑎 𝑑𝑁 = 𝐶∆𝐾𝑚 (2.1) 2.1.2 Literature on Crack Nucleation, Initiation, and Small Cracks A main research topic related to fatigue in metals is differentiating between small cracks and large cracks and attempting to model crack initiation and small crack growth. Crack growth rates for large cracks can be predicted by linear elastic fracture mechanics (LEFM) models including Paris’ law. For LEFM to be valid, the plastic zone around the crack tip should be small relative to the entire crack length. Moreover, this plastic zone should enclose a sufficient number of grains and inclusions for the crack growth rate to be stable [10, 11, 12]. Other sources have further differentiated small cracks into microstructurally small and physically small. While the exact differentiation between small crack phases is unclear, most agree that microstructurally small cracks are about the size of a few grain diameters and physically small cracks are those between several grain diameters and 1 mm. Table 1 summarizes multiple sources’ definitions of crack phases. As previously mentioned, the U.S. Navy retires aircraft once a small crack exceeds 0.25 mm [1]. In turn, this work will focus on fatigue damage prior to a 0.25 mm crack to match U.S. Navy criteria as well as until such a crack reaches 1 mm to agree with common small crack length definitions. 10 Table 1: Various definitions of crack phases Source Crack phases [5] Short cracks Long cracks  Length = less than 1 mm  Inclusions and grain boundaries influence growth  Length = greater than 10 mm  Microstructure and loading ratios have little influence [7] Crack nucleation Microstructurally short cracks Mechanically short cracks Macrocracks  Length = 1 nm – 1 µm  Influenced by surface roughness and microstructure  Length = 1 nm – 100 µm  Crack growth is stunted by microstructural barriers  Length = 100 µm – 1 mm  Crack growth is stunted by changes in crack tip stress field  Length = greater than 1 mm  Bulk material phenomenon [8] Microstructurally short cracks Mechanically short cracks Physically short cracks Long cracks  Length = a few grain diameters  Strong influence of microstructure and surface roughness  Length = several grain diameters  Little influence of the microstructure  Large plastic zone ahead of crack  Length = less than 0.5 mm  Small plastic zone ahead of the crack tip relative to crack length  LEFM is applicable  Length = greater than 0.5 mm  Completely developed  Paris’ law is applicable [11] Crack nucleation Microstructurally small cracks Physically small cracks Dominant long cracks  Length = Unspecified  Crack propagation through zone of the micro-notch root influence  Length = less than 3 times the characteristic length scale of microstructural interactions (MS)  Affected by the grain orientation  Length = 300- 800 µm, between 3MS and 10MS  Affected by the grain orientation  Length = less than greater than 20MS  Cyclic plastic zone is small relative to crack length  LEFM is applicable [10, 22, 24] Microstructurally short cracks Physically short cracks Long cracks  Length = similar to grain size  Continuum mechanics is questionable  Length = on the order of a grain or less  Reduced crack-closure effect  Length = greater than 0.5-1 mm  Paris’ law holds [14, 15] Small cracks Long cracks  Length = less than 1 mm  Nonlinear crack growth so LEFM is not applicable  Length = greater than 1 mm  LEFM is applicable [16] Small cracks Long cracks  Length = on the order of a grain or less  LEFM is not applicable  Length = larger than a few grains  LEFM is applicable 11 Researchers have continually attempted to model the erratic behavior of crack initiation and small crack growth. One notable researcher in this field is James C. Newman, a former NASA engineer. He developed the FASTRAN software that predicts fatigue life using plastically-induced crack-closure models [13]. In 1998 and 1999, Newman, Wu, and their coworkers [14, 15] published work on small crack growth, defined as between 10 µm and 1 mm in length, and fatigue life predictions for Al 7075-T6 and LC9cs clad alloy. The objective was to determine a “single analysis method that was applicable to all crack sizes”. Fatigue experiments along with finite element analyses and weight function analyses assessed the validity of estimating small and large crack growth rates with the FASTRAN software. In turn, the proposed model was able to predict fatigue life to an acceptable degree of accuracy. Newman [16, 17, 18] has continued researching small crack growth using FASTRAN models over the past decade. While FASTRAN software has proven to be a vital tool for predicting fatigue life, greater accuracy is achieved when the current microstructural damage is known. FASTRAN is based on fatigue experiments where cyclic loading was paused so that the material’s surfaces could be replicated. The size and shape of small cracks were then identified from scanning electron microscope images of the replicas. This process, however, cannot be implemented on a large scale. Observing fatigue damage on a microstructural level is ideal, but the replica method cannot easily be implemented as an NDE method. This limitation encourages investigations of other practical NDE methods that can estimate the fatigue damage on a microscopic level. 12 Another prominent researcher of fatigue and fracture mechanics of small cracks is Hael Mughrabi. Similar to Newman, he focuses on estimating fatigue damage based on microstructural changes. Several of his papers [12, 19, 20] discuss microstructural fatigue mechanisms such as cyclic irreversibilities and PSBs in α-iron polycrystals and α-brass single crystals. In addition, other authors have been published notable works [10, 21, 22, 23, 24, 25] that discuss microscale fatigue damage in Al7075-T6 and Ti-6Al-4V, both materials used in aerospace structures. Despite progress in microstructure-based models of fatigue, Mughrabi [12] noted that “a full understanding of the underlying damage mechanisms and the relationship to the fatigue life is difficult to achieve and still lacking”. Again, while a microstructurally-based fatigue model for crack nucleation, initiation, and small crack growth is desired, it is best to develop more practical methods of measuring microstructural fatigue damage through NDE techniques. 2.2 Acoustic Emission Background 2.2.1 AE Theory AE has become a recognized NDE method commonly used to detect flaws in mechanically loaded structures. Other NDE procedures such as ultrasonic and radiographic testing measure a component’s response to an actively applied external source. In contrast, AE is a passive method that senses minute surface displacements due to propagating internal elastic waves from sources of stress as shown in Figure 3. Theoretically, acoustic waves due to damage are continuously propagating through a structure when under repeated stress. Once the applied stress causes permanent deformation, stored elastic strain energy is released partly in the form of an acoustic 13 wave. For each instance of deformation, an acoustic wave is emitted. Consequently, AE has the potential to measure fatigue damage from dislocation motion to crack initiation and to large crack propagation. However, the ability to measure AE is limited by noise. Noise can come from many sources including mechanical vibrations from the testing machine and electrical interference with the instrumentation [4]. In turn, an amplitude threshold is often established. While this threshold allows the system to reject unwanted noise signals, AE signals related to damage and below the noise amplitude will not be detected. As a result, AE transducers cannot monitor in- flight aircraft structural health, but they can rather be used as an inspection method used in a controlled environment. Figure 3: AE sensor recording surface displacement waves from damage region 2.2.2 AE Instrumentation and Terminology The measurement of acoustic emissions began with Joseph Kaiser in 1950 [4]. Since then, standardized AE instrumentation including sensors, preamplifiers, and acquisition systems have been developed. Sensing the elastic wave within in the structure is the first step in measuring and recording AE signals. Most AE sensors are piezoelectric transducers which utilize the piezoelectric effect. These sensors contain a piezoelectric crystal that produces a change in electrical voltage when under mechanical strain. Therefore, when these materials are fastened to a deforming surface, the mechanical wave is transformed into an electrical signal. While the F F To preamplifier & frequency filter Crack Surface disp. waves AE sensor Specimen 14 electrical signal’s size and shape is dependent on the elastic wave, it is also influenced by the frequency behavior of the piezoelectric sensor. There are two general types of sensors; resonant and wideband. Resonant sensors operate in a narrow band near the sensor’s resonant frequency while wideband sensors are damped in order to detect a broader range of AE signal frequencies. Resonant sensors are often preferred because they are generally more sensitive and less expensive than wideband sensors [26]. While wideband sensors are able to reconstruct the elastic wave more accurately in regards to frequency spectrum, they are less sensitive. Once an AE piezoelectric sensor converts a surface displacement into an electrical voltage signal, the signal is passed through a preamplifier and band-pass filter. The preamplifier should be set to properly amplify the signals, and frequency filter characteristics should be selected to match the sensor’s attributes. Typically, the preamplifier is set between 20 dB and 60 dB, and the band pass filter is set between 1 kHz and 2 MHz [26]. The filter will be able to reject noise signals that often have a low frequency and capture damage-related AE signals which typically have frequencies in the range of 150-300 kHz [4]. Also, it is important to note that the recorded AE signals will have a drastically different waveform shape compared to the actual elastic wave. This phenomenon is due to the transformations between an AE source and the acquisition module and is referred to as the signal shaping chain as depicted in Figure 4 [4]. The source of the elastic wave is assumed to be one sudden peak with a smooth frequency distribution. The wave within the structure then attenuates and reflects off of the structure’s boundaries, distorting the wave in the time and frequency domains. Then, 15 because the AE sensor has its own time and frequency response, the electrical signal is an altered form of the propagating elastic wave. Finally, the signal is transformed again after being amplified and filtered. Even though the recorded AE signal is not in fact the exact AE wave within the structure, one can deduce aspects of the AE source from the recorded signal since all AE waves go through the same transformation. Figure 4: Transformation of AE signal due to signal shaping chain. Adapted from [4]. An example of an AE waveform is given in Figure 5. As one can see, the beginning of the signal has a voltage that oscillates close to 0 volts identified as background noise. After a damage-related elastic wave propagates to the AE sensor, the AE signal’s amplitude then increases to a peak amplitude. Subsequently, the signal attenuates back to the background noise behavior. Figure 5: AE voltage waveform with labelled features Source Structure Sensor Amplifier/ Filter Time Domain Response Frequency Domain Response Time Volts Threshold Rise time Duration Amplitude Counts 16 Two important duties an AE acquisition needs to perform is 1) to divide the continuous and complex AE signal into individual waveforms that can be processed and 2) to extract relevant features from these waveforms. For the first task, a series of parameters is often set depending on the structure’s material and geometry. These are described in Table 2 [27]. Table 2: User defined AE parameters and their descriptions [27] User-Defined AE Parameters Description Threshold Voltage value in units of dB that records an AE waveform as a hit when the waveform exceeds this value. Variable that controls the sensitivity. Sampling rate The rate at which the acquisition board samples waveforms on a per second basis. Value is described in units of megasamples per second (MSPS) where 1 MSPS means a sample is taken for every 1 µs. Pre-trigger Value that tells the software how long to record the AE waveform prior to the first voltage threshold crossing. Units are in microseconds. Can be set from 0 µs to the hit length divided by the sample rate. Hit length Value that determines the waveform length. This value can range between 1k and 15k where k = 1024 values. The length of time recorded over the hit is found by dividing the hit length by the sampling rate. For example, k for a 1 MSPS sampling rate means 1024 values will be measured, 1 for every µs, and the waveform will span a time of 1024 µs. Peak definition time (PDT) Ensures correct identification of the signal peak for rise time and peak amplitude measurements. For small metal specimens, the recommended value is 300 µs. Hit definition time (HDT) Ensures each AE signal from the structure is reported as one and only one hit. For small metal specimens, the recommended value is 600 µs. Hit lockout time (HLT) Extraneous measurements during the signals decay are excluded based on this value. For small metal specimens, the recommended value is 1000 µs. For the second duty of an acquisition system, numerous AE features are now common place. These features are labeled in Figure 5 and described below in Table 3 [4, 27]. 17 Table 3: Commonly extracted AE features and their descriptions [4, 27] Commonly Extracted AE Features Description Amplitude The largest voltage value present in the waveform and measured in dB using Equation (2.2). For a signal to be recorded, the amplitude needs to be above the threshold. Counts The number of times the voltage signal crosses the threshold. One of the easiest measurements of the signal and often used in analysis. Value usually is between a single count to a few hundreds. Duration Length of time from the first count to the last count and measured in microseconds. A long duration means the signal is drawn-out while a short duration implies a burst-type signal. Rise time The time between the first count and the count with the greatest amplitude. Units are in microseconds. Energy The area under the voltage-time envelope. This feature is a common measure for discussing AE signals in regards to structural damage. Absolute energy Measured value of the squared voltage signal divided by a reference resistance over the duration of the AE waveform used by particular AE acquisition systems. Units are in attojoules (10 −18 joules). Average frequency A calculated feature reported in kHz that measures the number of counts over the duration of the signal. Frequency centroid Reported in kHz, this value is derived in real time from the Fast Fourier Transform of each signal. Peak frequency Frequency of the signal when the maximum amplitude occurs. This frequency feature is also reported in kHz. 𝑑𝐵 = 20 log10 ( 𝑉𝑚𝑎𝑥 10−6 𝑣𝑜𝑙𝑡𝑠 ) − 𝑃𝑟𝑒𝑎𝑚𝑝𝑙𝑖𝑓𝑒𝑟 𝐺𝑎𝑖𝑛 𝑖𝑛 𝑑𝐵 (2.2) Within this section, the theory and instrumentation for the AE method has been detailed. Now that AE background has been covered, recent literature on estimating structural damage with AE can be summarized. 2.2.3 Estimating Fatigue Damage with AE Over the past few decades, researchers have had success correlating AE signals and their features to fatigue crack propagation in metals. Specifically, many people have been able to relate AE count rate to crack propagation rate using a power relationship similar to Paris’ law. This relationship is given as Equation (2.3) where a is crack length, c is AE counts, the derivatives of a and c with respect to N cycles are the crack growth rate and AE count rate, and α1 and α2 are model parameters. 18 𝑑𝑎 𝑑𝑁 = α1 𝑑𝑐 𝑑𝑁 𝛼2 (2.3) In 1973, Morton [28] tested this relationship on Al 2024-T831. He concluded that dc/dN appeared to be better correlated to the stress intensity range, ΔK, than between dc/dN and da/dN or da/dN and ΔK. Bassim [29] performed fatigue tests on several railway steels to test this hypothesis in 1994 while Berkovits and Fang [30] performed their own experiments on Incoloy 901 a year later. Both studies looked to correlate AE features to the earliest instance of fatigue but had more success relating AE hits once a crack was propagating. Berkovits and Fang noted that “conventional test methods, based on the crack propagation test, cannot accurately measure the initial length of the initial crack because of the impossibility of determining the critical initiation onset in real time”. In the end, both studies countered Morton’s claim and supported a strong relationship between dc/dN and da/dN that followed Equation (2.3). Since this time, other researchers [31, 32, 33, 34, 35, 36, 37] have looked to expand on relating AE counts to crack growth rate. In addition to correlating AE counts to fatigue damage, studies have investigated how other AE features such as energy, rise time, duration, and frequency transform as fatigue damage progresses. Results presented by Han [35] show AE hits that occur during stable crack propagation generally have higher rise times and lower amplitudes compared to AE hits during unstable propagation and fracture. Vanniamparambil and coworkers [38] performed tensile tests on Al2024 specimens and reported that amplitude, absolute energy, counts, and count rate were the most sensitive AE features to crack growth. Vanniamparambil [39] continued investigating these AE features and concluded that the onset of a crack is associated with a shift 19 from low frequency to high frequency waveforms. AE signals can also be classified based on rise time, amplitude, energy, and average frequency where all of these features increase near 90% of fatigue life [40]. In addition, Keshtgar [36] proposed that the ratio between the signal amplitude and the threshold amplitude scaled by the number of counts in the signal could be used as another AE damage-related feature. Because several AE features have proven to be able to classify AE signals throughout structural damage, this current study will also consider these features during fatigue tests. While estimating crack growth rate based on AE signals is a profound accomplishment, estimating crack damage at the smallest possible scale is most desirable. This idea has motivated many people to look at possible AE sources at the atomic scale to better understand wave dynamics within measured AE signals. One technique is to perform tensile and compression experiments on crystals and compare results with theoretical wave equations. James and Carpenter [41] were two such researchers who investigated AE count rate during compression tests on several different crystals. They reported AE counts were generated from dislocation breakaway from pinning points and that AE count rate could be related to dislocation velocity. Polyzos and Trochidis [42, 43, 44] modeled the interaction between dislocations and transverse acoustic waves in copper crystals during tensile tests. They concluded that both nonstationary dislocation motion and annihilation of dislocation kink-antikink pairs are mechanisms for AE signals. Researchers at the Institute of Thermomechanics [45, 46] developed molecular dynamic simulations to calculate stress and local kinetic energies during dislocation motion and twinning in 20 body-centered cubic iron crystals. Overall, studies have confirmed that AE waves within structures propagate due to dislocation motion. While observing and modeling acoustic waves within single crystals is vital to understanding AE mechanisms, studying AE sources within impure, polycrystalline materials is far more challenging but more applicable to real world situations. Due to precipitates, inclusions, and various grain orientations in metallic alloys, pinpointing the exact AE mechanism is often hypothesized but rarely proven. Despite the difficultly, many researchers have looked to correlate AE signals to microstructural changes. AE mechanisms within variously-aged 9Cr-1Mo steel during crack initiation and growth were investigated by Chaswal [47]. This work suggested that crack nucleation took place at the precipitate-matrix interfaces, sudden bursts of hundreds of low amplitude AE signals in a short time correspond to micro-cleavage of ligaments, and extended periods of inactivity correspond to dislocation pile up. Rahman [48] performed rolling contact fatigue tests on rail steel with AE and concluded AE count rate can detect incipient damage and is related to damage size and confirming previous findings [49]. Similarly, Elforjani and Mba [50] concluded there is a relationship between AE counts, energy, and amplitude to detecting incipient cracks in slow speed shafts. During tensile tests of Al7075, Lugo and coworkers [51] reported that much AE activity was recorded in the initial stage compared to later stages and hypothesized that most inclusions cracked in the early deformation stages. Finally, other recent studies [35, 36, 39, 52, 53, 54, 55] hypothesized when crack initiation occurred based on AE signals, but they did not 21 directly correlate early damage to a microstructural level. Overall, researchers concluded that AE activity is present during initial damage due to microcracks and dislocation movement, and various AE features can be correlated to damage. Researchers studying AE over the past six decades have investigated how AE signals and their features correlate to microstructural damage. Some promising damage-related features include counts, count rate, amplitude, rise time, energy, and frequency. These features provide information about the structural damage measured by AE. Rather than using AE features as sources of information, one can rely on the actual information content, a scientifically defined term, carried within the signal. The measure of information, referred to as information entropy, from AE signals will be further investigated, and its effectiveness of detecting fatigue damage will be compared with commonly used AE features. In the following section, background and applications of information entropy will be summarized. 2.3 Information Entropy Background 2.3.1 Understanding Information Entropy Entropy refers to the amount of disorder within a system. Often times, a reader’s first exposer to entropy is to thermodynamic entropy which is the amount of energy not available to do work and related to temperature of a system. However, there are other forms of entropy including information entropy. Rather than having origins in thermodynamics, information entropy is founded on probability, statistical, and communication theory. Connections between information entropy and thermodynamic entropy have been researched, but no concrete conclusion has been 22 made [56]. Fundamentals of information theory and their applications can be found from several reference books [57, 58]. Information theory began in 1948 after Claude E. Shannon proposed limits on data compression for transmitting and recording communication signals in his paper, “A Mathematical Theory of Communication” [59]. Within this paper, he proposed information entropy, also referred to as Shannon entropy, as a measure of disorder or uncertainty in a message that is calculated based on Equation (2.4). In this equation, H is the information entropy, K is a constant that dictates the units, and p(xi) is the probability a certain value, xi, present within the message with n possible values. It should be noted that the probabilities need to sum to 1 meaning a probability distribution should be defined for the signal. In addition, we will let K = 1/log(2) so that the logarithm will have a base of 2 to yield entropy in units of bits [58]. 𝐻 = −𝐾∑ 𝑝(𝑥𝑖) ∗ log (𝑝(𝑥𝑖)) 𝑛 𝑖=1 (2.4) To better understand the information entropy of a signal, it’s best to see a few examples of calculating information entropy from various probability distributions. Two examples will be presented: 1) flipping fair and biased coins and 2) rolling fair and weighted dice. The examples are depicted in Figure 6 and Figure 7, respectively. First, consider the probability distribution of flipping a fair coin where flipping heads or tails is equally likely. A person flipping this coin has little information and is very uncertain about whether the outcome will be heads or tails. 23 Figure 6: a) Probability mass function (PMF) of flipping a fair coin and calculated information entropy. b) PMF of a biased coin with probability of flipping tails as 0.70 and calculated entropy. c) PMF of a biased coin with probability of flipping tails as 0.99 and calculated entropy. Let heads be outcome x1, tails be outcome x2, and the probabilities for each outcome be p(x1) = 0.5 and p(x2) = 0.5. From this probability distribution, the information entropy can be calculated based on Equation (2.4). Subsequently, the entropy is found to be 1.000 bits. This is depicted in Figure 6a. Now consider a biased coin that results in tails 70% of the time. A person flipping the coin now has a bit more information and is less uncertain about the possible outcomes. A new probability distribution can be constructed to reflect the biased coin and the entropy of this distribution is less than before at 0.8813 bits as recorded in Figure 6b. Finally, consider a biased coin that is tails 99% of the time when flipped. A person flipping the coin is now almost certain that the outcome will be tails. This probability distribution is represented in Figure 6c and results in an even lower entropy of 0.0808 bits. This scenario exemplifies that more uniform probability distributions representing a highly disordered and more uncertain variable will result in a higher 24 value for information entropy. In other words, the greater the uncertainty and disorder, the greater the entropy will be. Figure 7: a) PMF of a rolling a fair die and calculated information entropy. b) PMF of a weighted coin with greater probability of rolling a 4. c) PMF of coin 97.5% likely to result in a 4 and calculated entropy. Similarly, one can see how information entropy varies with different probability distributions of fair and weighted dice. First, consider a fair die where rolling any number 1 through 6 is equally likely with probability of 1/6. This distribution is pictured in Figure 7a and results in an entropy value of 2.5850 bits. This value is greater than the entropy for a fair coin. This is because a die with 6 outcomes has greater disorder and uncertainty than a coin with 2 outcomes. Therefore, it can be concluded that the entropy value is dependent on the number of possible outcomes; when more outcomes are possible, entropy will be greater. Now, consider an unevenly weighted die that has the following probability distribution: p(1) = p(2) = p(5) = p(6) = 1/12, p(3) = 1/6, p(4) = 1/2 meaning 3 and 4 are more favored outcomes. The entropy calculated from this distribution is 2.1258 bits, a 25 lower value than if all outcomes were equally likely and reflects less disorder. This is shown in Figure 7b. Lastly, for a die that results in a 4 for 97.5% of the rolls, the person rolling is almost certain that the outcome will be a 4. Therefore, the entropy should be lower and is calculated to be 0.2267 bits as shown in Figure 7c. In the end, given any variable or signal that is represented by a probability distribution, one can calculate information entropy using Equation (2.4) as a measure of the disorder or uncertainty carried within this signal. The maximum entropy will be from a distribution with equally likely outcomes while the minimum entropy of 0 bits will be from a distribution when only one outcome from a sample space is possible. 2.3.2 Application of Information Entropy to Fatigue and AE The disorder measured by information entropy can be calculated from any probability distribution and has found applications in signal processing and communication [57, 60]. In addition, attempts have been made to apply information entropy techniques to fatigue damage. Several studies have estimated fatigue damage models with information entropy since it is a measure of uncertainty. One common idea in the modeling field is the principle of maximum entropy referred to as MaxEnt. First pioneered by Jaynes [61, 62] in 1957, MaxEnt suggests that when selecting a model to fit a probability distribution, the one that best represents the current knowledge is the one with maximum entropy. The model with maximum entropy is the one with the most uncertainty, is the “least biased given the information”, and “is maximally noncommittal with regard to missing information” [61]. Recently, fatigue damage 26 prognosis models with limited or uncertain data have been developed by utilizing the MaxEnt principle and Bayesian inference with much success [63, 64, 65]. Other researchers have implemented information entropy techniques to estimate structural damage unrelated to MaxEnt. Li [66] identified damage in infrastructure using artificial neural networks, Dempster-Shafer evidence theory, and information entropy. Unlike MaxEnt, Li selected decisions within the neural network that had smallest entropy and the lowest uncertainty. In the end, the accuracy improved when this information entropy technique was employed. Another study [67] implemented information entropy in order to estimate optimal maintenance time of pipeline systems. For a new pipe, the maximum pressure the pipe can accommodate is often known with little uncertainty. As a pipeline corrodes, however, the burst pressure decreases and the uncertainty about this critical value increases. In turn, information entropy was used to measure this uncertainty and led to deriving the optimal maintenance intervals during instances with maximum entropy. So far, several papers have been presented that use information entropy as a measure of model uncertainty in structural damage prognosis. These methods, however, do not measure the disorder within a structure that is subjected to damage but only model uncertainty. First, microstructural disorder should be defined in terms of fatigue damage. As previously explained, fatigue damage progresses as dislocations move and create microcracks near inclusions which coalesce to form macrocracks. A structure with minimal damage prior to fatigue loading would then have a series of defects. These defects and microcracks are considered to be disorder within the structure and 27 thus, fatigue damage will be synonymous with microstructural disorder throughout this work. This correlation is exemplified in Figure 8 where microstructural disorder is equated to fatigue damage. Initially, a metallic material has minimal defects among its grains, inclusions, and precipitates. Then, as more cyclic stresses are applied, inclusions crack, microcracks nucleate, and microcracks coalesce to form macrocracks. In the end, the microstructural disorder increases as fatigue damage increases and is attempted to be quantified by AE signals in this work as well as in previous studies. Figure 8: Microstructural disorder as defined by fatigue damage evolution A few researchers have looked into estimating information entropy of AE signals during fatigue experiments. Unnthorsson and coworkers [68] estimated two time-domain entropies and two frequency-domain entropies from AE signals recorded during composite fatigue tests. The entropies were calculated from discrete probability distributions of the amplitude and frequency measured for 1 fatigue cycle Grains and impurities, minimal damage Intergranular and inclusion microcracks Microcrack nucleation Microcracks coalesce to form macrocracks Fatigue Damage Evolution Growing Microstructural Disorder 28 every 5 minutes of testing. In the end, the entropy evolutions were similar to AE count trends, and it was proposed that entropy from AE signals could be used a measure of microstructural disorder. Qi [69] also focused on deriving entropy from AE amplitude distributions to be used as a measure for microstructural damage. Tensile tests were performed on various cements and the updating distributions of AE amplitudes for recorded signals were used when calculating entropy. In this study, the entropy value was calculated based on Equation (2.4) without the negative coefficient before the summation. Therefore, entropy values were reported as negative with a minimum possible value of -2.3 when all AE signals are of a single amplitude and a maximum value of 0 when amplitudes are uniformly distributed. Results showed that the entropy from the AE amplitude spectrum became less negative as the stress increased and then remained constant or decreased slightly near fracture. This is because AE signals often have low amplitudes initially and then more signals at various and higher amplitudes are recorded as damage progresses. Thus, this study proved that measured disorder from AE amplitude distributions increases as damage also increases. Modarres and coworkers [70,71] built off of these studies to estimate entropy from AE counts during fatigue tests of aluminum and titanium alloys. Results showed that information entropy derived from counts mirrored the evolution of counts throughout the test [70], and the cumulative information entropy from counts may be constant at failure [71]. The commonality between this research and Unnthorsson’s and Qi’s work is that the entropy was calculated from AE feature distributions over certain intervals of time where several AE signals have been collected. Rather than 29 using features from several AE hits at a time, one could look at the amplitude distributions of each individual AE waveform. This method would theoretically utilize more information carried in an AE signal compared to summary statistics like the number of counts, the peak amplitude, or the average frequency. This is the fundamental basis of this thesis; to extract information entropy from every individual AE signal during fatigue tests in order to estimate damage evolution. The effectiveness of this method to detect damage both prior to and after an observable crack will be explored. Three different topics were reviewed in this chapter in regards to background and previous literature; fatigue damage in metals, acoustic emission methods and instrumentation, and information entropy as a measure of uncertainty and disorder. The following chapter details the experimental procedure in order to measure the fatigue damage and information entropy from AE signals. 30 Chapter 3 – Experimental Procedure 3.1 Specimen Preparation Specimens used for this study were aluminum alloy 7075-T6 referred to as Al7075-T6 throughout this work. This specific alloy is used in aerospace structures such as stringers, skins, bulkheads, rivets, and extruded sections [3]. The supplied material from Kaiser Aluminum came from the same lot number ensuring maximum microstructural continuity between all fatigue specimens. The material’s composition and mechanical properties were provided by the manufacturer and presented in Table 4. To confirm the mechanical properties such as elastic modulus, yield strength, ultimate yield strength, and percent elongation, two tensile tests were performed. Results from these tests implied that the specimens may have slightly higher strength and more ductility than reported by the manufacturer. Table 4: Composition and mechanical properties for Al7075-T6 Element Al Zn Mg Cu Cr Fe Mn Si Ti V Zr Other Composition (wt %) 89.72 5.7 2.6 1.4 0.20 0.15 0.08 0.06 0.02 0.01 0.01 0.05 Material Property Ultimate Strength (MPa) Yield Strength (MPa) % Elongation Modulus of Elasticity (GPa) Manufacturer- Provided Value 579 513 13.7 Not given Measured Value from Tensile Tests 587 538 24.0 67.8 The raw material was machined into specimens designed according to ASTM standard E466 [72]. Several iterations of the geometries were performed in order to accommodate an extensometer, sensors, and mechanical damping apparatus, which will be explained in subsequent sections. In the end, five different geometries were designed until the final design was selected. The dimensioned geometry is shown in 31 Figure 9. A 1 mm radius edge notch is located at the center of the gauge length and acts as a stress concentrator. This intentional flaw ensures that fatigue damage and crack initiation will occur at this location. Based on a stress concentration handbook [73], the stress concentration factor of the notch is estimated to be 2.61. Within this work, the specimens are referred to as 5A1 through 5A26 where 5 represents the geometry iteration number, A means the grain direction is parallel to loading direction (as opposed to B which means the grain direction is perpendicular to loading direction), and the proceeding numbers are the specimen numbers. Figure 9: Fatigue specimen geometry Once the specimens were machined, one side of the specimen’s gauge length was polished so that grain boundaries could be seen under an optical microscope. In addition, crack initiation is influenced by surface roughness so it is best to ensure all specimens had similar surface roughness values. The polishing process began by sanding the surface with 1000 grit sandpaper followed by 2000 and 3000 grit sandpaper. Then a 3 µm alumina solution was used followed lastly by etching the surface with Keller’s etchant solution. In turn, the grain boundaries could be seen 32 clearly under magnification. The distribution of the grain sizes were estimated by measuring the area of 125 grains from three dogbone specimens and deriving the average diameter for the total 375 grains. Figure 10 displays the distribution where the grains varied between 0.04 mm and 0.27 mm with a mean of 0.124 mm and standard deviation of 0.037 mm. According to ASTM standard E112-12 [74], the grain size number is approximately 3.0. Figure 10: Probability distribution of grain diameter for 375 measured grains 3.2 Loading Conditions A servo-hydraulic Materials Testing System (MTS) machine retrofitted with an Instron 8800 controller was used to perform the fatigue tests. The machine’s maximum load is 100 kN and the load cell used has an uncertainty between 0.13% and 0.54% of the applied load. A computer connected to the load frame is used to specify the loading conditions within the WaveMatrix software program. The load frame is pictured in Figure 11. 33 Figure 11: Servo-hydraulic MTS machine A total of 26 tension-tension fatigue experiments were performed with loading ratio of 0.1 and loading frequency of 5 Hz. The maximum applied load varied between 8 and 15 kN in order to see the differences between low-cycle and mid-cycle fatigue on detecting microstructural damage prior to a visible crack. The theoretical maximum applied stress can be estimated by dividing the load by the gauge-length cross-sectional area, 57.15 mm 2 , and multiplying by the stress concentration factor, 2.61. The maximum applied load and estimated applied stress at the notch for each experiment is listed in Table 5. Similarly, the tests for each loading condition are listed in Table 6. 34 Table 5: Loading conditions for each experiment Specimen Name Max. Applied Load (kN) Est. Notch Stress (MPa) Specimen Name Max. Applied Load (kN) Est. Notch Stress (MPa) 5A1 13 593.7 5A14 8 365.4 5A2 12 548.0 5A15 12 548.0 5A3 11 502.4 5A16 12 548.0 5A4 11 502.4 5A17 8.5 388.2 5A5 10 456.7 5A18 8.5 388.2 5A6 10.5 479.5 5A19 14 639.4 5A7 9 411.0 5A20 15 685.0 5A8 10.5 479.5 5A21 10 456.7 5A9 10.5 479.5 5A22 13 593.7 5A10 11 502.4 5A23 9 411.0 5A11 13 593.7 5A24 12 548.0 5A12 9 411.0 5A25 9 411.0 5A13 9 411.0 5A26 10 456.7 Table 6: Experiments for each loading condition Max. Applied Load (kN) Est. Notch Stress (MPa) Specimen Names 15 685.0 5A20 14 639.4 5A19 13 593.7 5A1, 5A11, 5A22 12 548.0 5A2, 5A15, 5A16, 5A24 11 502.4 5A3, 5A4, 5A10 10.5 479.5 5A6, 5A8, 5A9 10 456.7 5A5, 5A21, 5A26 9 411.0 5A7, 5A12, 5A13, 5A23, 5A25 8.5 388.2 5A17, 5A18 8 365.4 5A14 However, it was found that a crack grew away from the notch for low loading conditions. Crack initiation location can be determined based on the brittle regions of a crack’s surface. A crack will initiate and grow due to brittle fracture until the crack reaches unstable growth where ductile failure will occur. Figure 12a shows the crack surfaces of 5A26 that properly failed at the notch while Figure 12b depicts 5A18 crack surfaces that failed away from the notch. It can be seen the brittle region for 5A26 is at the edge of the notch while the brittle region for 5A18 was near the specimen’s face. This finding suggests the maximum stress was in fact at the face rather than at the notch. 35 Figure 12a: Crack initiated as expected at notch edge for 5A26 Figure 12b: Crack initiated unexpectedly at specimen face for 5A18 Figure 12: Crack surfaces for 5A26 and 5A18 One possible reason to explain this anomaly is that there seems to be is a slight misalignment between the testing grips which most likely applies a small bending moment on all specimens. At higher axial loads, the notch is the predominant stress concentrator and the bending moment has little effect on crack growth location. However, when the axial loading is low, the bending moment causes a greater stress then the notch. The experiments that failed away from the notch (5A14, 5A17, 5A18) are marked in Table 9 and were discarded. 3.3 Strain Measurement In order to measure the strain around the edge notch, an extensometer was used. The Epsilon 3542 extensometer has a gauge length of 25 mm, and can measure between -10% and 10% strain. For the first 3 experiments (5A1, 5A2, 5A3), the Crack initiated Crack initiated 36 extensometer slipped slightly on the specimen because it was not securely fastened. In order to prevent this sliding or rubbing during testing, rubber bands were used to fasten the extensometer to the specimen for subsequent experiments. The extensometer slipped for a few subsequent experiments (5A4, 5A6, 5A10, 5A13). In addition, the extensometer was not applied for 4 other tests (5A8, 5A9, 5A11, 5A12) and confirmed that unusual AE signals were not due to the extensometer sliding on the surface. These tests are also marked in Table 9. One drawback to the extensometer is it measures the strain over 25 mm of gauge length as opposed to exactly and only the area around the notch. Because the notch acts as a stress concentrator, most of the deformation is expected to be around the notch, but the extensometer measures a greater area. In turn, the extensometer is assumed to be less sensitive than a strain measurement technique that is more localized. Other techniques to measure strain were also investigated. First, strain gauges were used and placed near the middle of the notch. While the strain gauges seemed to be quite accurate and had the ability to measure localized strain around the notch, the gauges were susceptible to detaching from the specimen as a crack initiated and grew. In addition, as soon as a strain gauge began to detach, unwanted AE signals were produced. Another technique tested for its effectiveness was Digital Image Correlation (DIC). DIC calculates strain based on how a surface pattern changes on a loaded structure due to applied stress. While this technique had the potential to measure strain around the notch more closely than the extensometer and not cause AE noise signals, the image processing speed of the DIC system used was too slow to capture strain accurately and frequently. Fatigue tests were performed at 5 Hz and the 37 maximum sampling rate of the available DIC was 10 Hz meaning at most 2 measurements of strain were saved per cycle. For this testing, a 10 Hz strain sampling rate was too low compared to the 200 Hz extensometer sampling rate. In addition, the DIC prevented the optical microscope from being used to capture crack initiation and growth. Compared to the optical microscope images, the DIC images of crack initiation were far less clear. The advantages and disadvantages of the strain measurement techniques are summarized in Table 7. Table 7: Strain measurement techniques and associated attributes. Advantages are in green, disadvantages are in red. Extensometer Strain Gauge DIC Strain region around notch +/- 12.5 mm +/- 1mm or greater +/- 1 mm and greater Sampling rate High, 200 Hz used High, 200 Hz used Low, less than 10 Hz Measure strain through crack initiation? Yes No Yes Effect on AE No to little interference High interference No interference Effect on crack monitoring No interference No interference High interference 3.4 Crack Monitoring System In order to monitor the initiation of a crack and its growth, an optical microscope with an attached time-lapse camera captured images throughout testing. An external Meiji dual-arm fiber optic light source illuminated the specimen. The microscope encompassed about a 1.5 mm by 2.0 mm area around the notch where a circular segment of the notch is shown on the left side of each image. In order to measure the small crack length, ImageJ software was used to estimate the picture scales (the number of pixels per millimeter) and the number of pixels the crack spanned based on the circular segment. The geometric relationship between the radius, chord length, and height of the segment was employed to estimate the picture scales. This relationship is depicted in Figure 13. By measuring the chord length, c, 38 and height of the segment, h, in pixels, the radius in pixels can be estimated. Since the radius is known to be 1 mm, the picture scale can therefore be estimated using Equation (3.1). Finally, crack length is then measured in millimeters by measuring the number of pixels spanned and converting to millimeters with the picture scale. Figure 13: Circular segment dimensions 𝑃𝑖𝑐𝑡𝑢𝑟𝑒 𝑆𝑐𝑎𝑙𝑒 = 𝑅 𝑖𝑛 𝑝𝑖𝑥𝑒𝑙𝑠 𝑅 𝑖𝑛 𝑚𝑚 = ( 𝑐2 8ℎ + ℎ 2 ) 𝑝𝑖𝑥𝑒𝑙𝑠 1 𝑚𝑚 (3.1) Pictures were taken every 5 seconds. However, because the specimen was moving during cyclic fatigue loading, about one-half of the images were too blurry and had to be discarded. Figure 14 shows four typical images captured from the optical microscope. Figure 14a shows the area around the notch before cyclic loading was applied. Figure 14b and Figure 14c show slightly blurry and very blurry images, respectively, during the fatigue process before a crack initiated. Finally, Figure 14d is an image once a crack has grown. The crack monitoring system worked well except for one test (5A5) in which the camera stopped recording images. This experiment is marked in Table 9. S c h d R 39 a) Before cyclic loading begins, grain boundaries and inclusions are visible b) Less blurry image during cyclic loading used for crack monitoring c) Very blurry image that cannot be used d) Image after crack has initiated Figure 14: Optical microscope images during fatigue of specimen 5A26 40 3.5 Acoustic Emission Instrumentation AE signals were recorded with a PCI-2 Based AE system supplied by the MISTRAS Group. Two resonant Micro30s AE sensors with a frequency range of 150-400 kHz and resonant frequency of 225 kHz were used. The sensors were mounted to one side of the specimen 23 mm above and below the center of the notch. Ultrasonic gel was used as a couplant, and electrical tape fastened the sensors to the surface. The AE signals passed through a 40 dB preamplifier before reaching the data acquisition module where AEwin software then plotted and extracted the AE signals. The AE acquisition system also received load and extension data as analog inputs from the testing machine. This feature enabled the AE signals to be paired with the applied load and is crucial to post-process filtering. AE signals are recorded as voltage values ranging between -10 and +10 volts with smallest divisions of 0.000305 volts. Other user-defined settings that control how the AE signals are collected are summarized in Table 8. These parameters were selected based on pencil lead break tests [75], a common standard that produces repeatable artificial AE waves with similar characteristics to damage-related AE signals. Table 8: AE software settings Parameter Value Parameter Value Peak definition time (PDT) 300 µs Pre-trigger length 256 µs Hit definition time (HDT) 600 µs Hit length 2048 µs Hit lockout time (HLT) 1000 µs Band pass filter 1 kHz – 3 MHz Sampling rate 1 MSPS 3.6 Mechanical Damping Apparatus 3.6.1 Why Mechanical Damping? A crucial part of the fatigue experiments is collecting AE signals. One of the first AE settings to be prescribed is the amplitude threshold. As previously explained, 41 only AE signals that exceed the threshold will be collected, and the threshold should be set to slightly above the noise amplitude. During initial experiments, the background noise reached amplitudes of 65 dB. Typically, an acceptable AE threshold is about 45 dB while lower thresholds of 40 dB or 35 dB are desirable. In this case, the threshold would have had to be greater than 65 dB meaning most, if not all, AE damage signals would not be detected. Servo-hydraulic machines are known to produce AE background noise of similar frequency and amplitude to damage-related AE signals making filtering the noise a difficult process. Researchers have filtered noise based on signal arrival times [51, 54] and on frequency [35, 52], but neither of these methods proved effective for this particular test setup. Instead, attempts were made to actively limit the noise signals from propagating through the specimens. In turn, a mechanical damping apparatus inspired from Miller’s work [76] and numerous damping configurations was designed. 3.6.2 Damping Techniques When a specimen was fixed to the machine and AE was recorded, the AE amplitude seemed to decrease when two fingers pinched the specimen. In addition, Dr. Ron Miller, an expert in AE and former engineer at the MISTRAS Group, implemented a damping technique for servo-hydraulic fatigue tests during his graduate work at Purdue University [76]. Thus, fastening damping materials to the specimen seemed to be a promising method. Several different configurations and materials were tested and the AE background noise amplitude was observed. In the 42 end, the specimen geometry was altered several times to allow sufficient room for damping material and a final damping design was selected. The final apparatus consists of four sets of clamped, 1/4" thick, styrene- butadiene rubber blocks and four tightly-wrapped, 1/16” thick, neoprene strips attached to areas between the testing grips and specimen gauge length. These elastomers inhibit mechanical vibration and reduced the background noise to below 41 dB for an AE sensor attached on the surface above the notch allowing for a threshold of 45 dB. For these experiments, a second AE sensor was attached on the surface below the notch but experienced noise of 45 dB and the threshold had to be set slightly higher than 45 dB. Only AE signals from the upper sensor were analyzed while the AE signals from the lower sensor were used to validate the upper sensor’s behavior. The amplitude spectrum during 20 seconds of recorded AE noise signals with and without the damping material is depicted in Figure 15 showing a 20 dB decrease in noise amplitude. Figure 15: AE noise amplitude with and without mechanical damping The schematic of the damping apparatus on the specimen is pictured in Figure 16a, and the entire test setup is depicted in Figure 16b. In addition, several of the first 43 experiments did not have the final damping apparatus applied but instead preliminary iterations. Therefore, the damping configurations initially used did not damp the noise amplitude to below 45 dB and the threshold was set to values ranging between 46 and 52 dB. Table 9 lists the AE threshold values applied for each test. Figure 16: a) Mechanical damping method. b) Complete experimental setup. In the future when an AE sensor system is developed for NDE of aircraft structures, reducing and filtering the noise will inevitably be a problem that would need to be solved. In these experiments, the noise was damped by constraining material away from the inspection sites with viscoelastic materials. Likewise, the use of dampers may have to be considered to effectively use AE as an NDE technique. 3.7 Details of Individual Fatigue Experiments Of the total 26 experiments, several had slightly different experimental setups. Three specimens failed away from the notch during low loading, eleven experiments 1. Specimen 2. Clamped styrene- butadiene rubber 3. Wrapped neoprene strip 4. Acoustic sensor 5. Extensometer 6. Optical microscope 7. External light source 8. Testing grip 3 2 4 1 a 5 2 4 1 3 6 7 8 b 44 did not have the extensometer fastened securely or was not attached, and the optical microscope camera failed to record images for one experiment. These 15 experiments were discarded and not used in data analysis. In addition, because of the damping apparatus iterations, the AE threshold setting varied for initial experiments. The AE threshold has an effect on AE results, but tests with an AE threshold greater than 45 dB are discarded due to extensometer performance. The details for each experiment are noted in Table 9. Table 9: Experiment details. Tests with any detail in red are discarded. Specimen Name Max. Applied Load (kN) Est. Notch Stress (MPa) Crack Location Extensometer performance Images of Crack? Upper Sensor AE Threshold 5A1 13 593.7 Notch Slipped Yes 52 dB 5A2 12 548.0 Notch Slipped Yes 52 dB 5A3 11 502.4 Notch Slipped Yes 52 dB 5A4 11 502.4 Notch Slipped Yes 46 dB 5A5 10 456.7 Notch Nominal No 48 dB 5A6 10.5 479.5 Notch Slipped Yes 48 dB 5A7 9 411.0 Notch Nominal Yes 47 dB 5A8 10.5 479.5 Notch Not attached Yes 46 dB 5A9 10.5 479.5 Notch Not attached Yes 47 dB 5A10 11 502.4 Notch Slipped Yes 47 dB 5A11 13 593.7 Notch Not attached Yes 47 dB 5A12 9 411.0 Notch Not attached Yes 47 dB 5A13 9 411.0 Notch Slipped Yes 47 dB 5A14 8 365.4 Grip Nominal Yes 47 dB 5A15 12 548.0 Notch Nominal Yes 48 dB 5A16 12 548.0 Notch Nominal Yes 45 dB 5A17 8.5 388.2 Extensometer Nominal Yes 45 dB 5A18 8.5 388.2 Sensor Nominal Yes 45 dB 5A19 14 639.4 Notch Nominal Yes 45 dB 5A20 15 685.0 Notch Nominal Yes 45 dB 5A21 10 456.7 Notch Nominal Yes 45 dB 5A22 13 593.7 Notch Nominal Yes 45 dB 5A23 9 411.0 Notch Nominal Yes 45 dB 5A24 12 548.0 Notch Nominal Yes 45 dB 5A25 9 411.0 Notch Nominal Yes 45 dB 5A26 10 456.7 Notch Nominal Yes 45 dB 45 Chapter 4 – Post-Processing Methods 4.1 Noise Reduction Despite reducing the AE background noise to amplitudes below 45 dB with the mechanical damping method, it became evident during the 26 experiments that AE noise signals were still collected during some tests. Again, even though others have had success filtering AE signals based on frequency and arrival times [35, 51, 52, 54], this technique proved ineffective. Instead, another filtering technique based on applied load at the instant of AE signals was implemented. AE signals are emitted due to fatigue damage, and it is assumed that the AE signals will most likely occur when a structure is applied with maximum stress [28, 34, 36, 52]. For a fatigue cycle with minimum load of 1 kN and maximum load of 10 kN, this means that an AE signal would tend to occur between 7 and 10 kN rather than 1-7 kN. Five cycles with instances of potential AE hits are depicted in Figure 17 with hits marked in red to be likely AE hits and those in blue to be unlikely. Figure 17: Likely and unlikely AE hits based on applied load 46 Of the 11 experiments that had no errors in regards to extensometer performance, crack monitoring, or crack location, AE signals were inspected for unwanted noise signals based on load. The expected AE signal behavior can be described in accordance with the stages of fatigue damage. First, the initial few fatigue cycles are assumed to cause sudden dislocation realignment within the structure which emit a few AE events. These events, while likely to occur during the upper half of the loading cycle, are often sporadic and can occur at any point in a loading cycle. Then, fatigue damage continues to progress and is believed to cause sparse and erratic AE signals at various applied loads. Eventually, the accumulated microstructural damage will result in a crack initiating and growing at an unstable rate. During this stage, more AE signals are expected to occur especially at high loads. Finally, several hundreds of AE signals are anticipated as a crack grows towards final fracture. This final series of hits can occur at any load because a crack can grow at any applied stress once it reaches a certain length and crack surfaces rub against one another during unloading. Two of the 11 “good” experiments (5A25 and 5A26) exhibited the expected AE behavior with no apparent noise signals. The AE signals at the associated applied load are shown as red points in Figure 18 for one of these experiments, 5A26. AE signals occur sporadically and at various loads for the first three-quarters of fatigue life. These signals are most likely due to dislocation stacks unpinning and suddenly releasing stored strain energy. Damage continues to grow and more AE signals are recorded at high loads as a crack initiates. Then, AE signals occur at both high loads and middle-low loads during crack growth and specimen fracture. 47 Figure 18: AE signals at their associated loads versus time for 5A26. No apparent AE noise. The other 9 experiments of the 11 “good” tests showed both the expected AE behavior and another behavior that is believed to be unwanted noise. This unwanted behavior is characterized by AE signals continuously occurring at mid-range loads forming clusters when plotted against their associated load and arrival time. These clusters suggest either strain energy is released consistently away from the maximum load or that mechanical noise is generated at certain points of the loading cycle. The latter is assumed to be more likely, and therefore, these signals should be excluded. A possible cause of the mechanical noise could be the servo-hydraulic piston rubbing against another component or changes in piston acceleration at specific points during a loading cycle. This phenomenon could develop later in a fatigue test and then dissipate resulting in clusters or trends on AE associated applied load versus arrival time scatter plots. Of the 9 experiments tainted with AE noise signals, seven could be filtered (5A16, 5A19, and 5A21–5A24). The scatter plot of applied load at the instant of AE signals and their arrival times for one such test, 5A22, is pictured in Figure 19. Too many noise signals were recorded for the other three tests (5A7, 5A15, and 5A20) and 48 the expected AE behavior and noise could not be confidently differentiated. Figure 20 shows the applied load at AE hits versus time for 5A20 as an example of noise that proved impossible to separate from the assumed damage-related AE hits. Scatter plots of AE signal arrival time and associated applied for all 11 of these tests are given in Appendix A – Applied Load vs. Signal Arrival Time Scatter Plots. Figure 19: Applied load at instant of AE signals at their arrival times for 5A22. Filterable noise. Figure 20: Applied load at recorded AE signals and their arrival times for 5A20. Unfilterable AE noise. Table 10 lists the 11 tests with no previous experimental errors and summarizes the recorded AE signal behaviors. The table also notes the number of AE signals collected from the upper AE sensor for tests with no noise and tests after possible filtering. Each test with no or filterable noise has less than 2700 AE signals 49 while tests that were not filtered had more than 22,000 signals. Therefore, it is believed that at least 87% of the signals from the unfilterable tests were due to noise. Table 10: AE noise behavior for each test with an acceptable test setup Specimen Name Max. Applied Load (kN) Est. Notch Stress (MPa) AE Noise? No. of AE Signals Collected 5A7 9 411.0 Unfilterable noise 160024 5A15 12 548.0 Unfilterable noise 55445 5A16 12 548.0 Filterable noise 2108 5A19 14 639.4 Filterable noise 1915 5A20 15 685.0 Unfilterable noise 22087 5A21 10 456.7 Filterable noise 2617 5A22 13 593.7 Filterable noise 1222 5A23 9 411.0 Filterable noise 2311 5A24 12 548.0 Filterable noise 2403 5A25 9 411.0 No noise 1478 5A26 10 456.7 No noise 1865 Of course, significant assumptions were made during the filtering process. Other researchers may disagree with whether specific AE signals should be attributed to noise or are in fact damage related. However, the filtering method was kept consistent in that clusters of AE signals at low loads were removed. Tests that would have required questionable noise filtering were discarded. 4.2 Instantaneous Elastic Modulus Calculations Another post-processing step to perform is to estimate the elastic modulus for each fatigue cycle. Crack length is the common metric for fatigue damage. However, the goal of this work is to estimate damage prior to an observable crack where microscopic cracks are unnoticeable during fatigue loading even with the aid of an optical microscope. Microstructural damage could be accurately quantified if fatigue loading was repeatedly interrupted and the notch was examined under higher magnification. This is undesirable though because many testing conditions could differ including specimen alignment, precise AE sensor location, and AE background 50 noise behavior. Instead, it is assumed that structural degradation is reflected as a decrease in the structure’s elastic modulus. Dislocations move and microcracks grow such that the material becomes less stiff and will elongate more for the same applied load as fatigue loading progresses. Initial moduli values are expected to differ slightly between specimens but should all be near the anticipated elastic modulus of 67.8 GPa as noted in Table 4. Despite variations, a consistent modulus decline and degradation trend between specimens is expected. The modulus degradation is approximated by Equation (4.1). Here, E is the modulus, ∆P is the difference between the maximum and minimum applied load, Lo is the extensometer gauge length, ∆l is the difference between the extension at the maximum and minimum loads, and A is the cross-sectional area. The MATLAB code for calculating the cyclic modulus from the loading data recorded by the testing machine is given in Appendix B – Code for Modulus Evolution. 𝐸 = ∆𝜎 ∆𝜀 = ∆𝑃×𝐿𝑜 ∆𝑙×𝐴 (4.1) The measurement uncertainty in instantaneous elastic modulus can be estimated from the uncertainty in extension, load, and cross-sectional area and by utilizing error propagation equations, Equation (4.2) and Equation (4.3). The uncertainty in the extensometer and the load cell are taken from the calibration records while the uncertainty in the cross-sectional area comes from the uncertainty of the calipers used to measure the specimens’ thicknesses and widths. With this information, the elastic modulus uncertainty can be estimated with Equation (4.4). Table 11 provides a summary of the error propagation results while all intermediate steps are shown in Appendix C – Error Propagation. In the end, the initial modulus 51 values vary between 67.6 and 71.0 GPa with an uncertainty of between 2.6 and 3.7 GPa. 𝐹𝑜𝑟 𝑐 = 𝑎 + 𝑏 𝑜𝑟 𝑎 − 𝑏, 𝛿𝑐 = √(𝛿𝑎)2 + (𝛿𝑏)2 (4.2) 𝐹𝑜𝑟 𝑐 = 𝑎 ∗ 𝑏 𝑜𝑟 𝑎 𝑏 , 𝛿𝑐 = |𝑐|√( 𝛿𝑎 |𝑎| ) 2 + ( 𝛿𝑏 |𝑎| ) 2 (4.3) 𝐹𝑜𝑟 𝐸 = ∆𝑃 ∗ 𝐿𝑜 ∆𝑙 ∗ 𝐴⁄ , 𝛿𝐸 = |𝐸| √( 𝛿∆𝑃 |∆𝑃| ) 2 + ( 𝛿𝐿𝑜 |𝐿𝑜| ) 2 + ( 𝛿∆𝑙 |∆𝑙| ) 2 + ( 𝛿𝐴 |𝐴| ) 2 (4.4) Table 11: Measurement uncertainty for 11 experiments Specimen Name Area, A (mm^2) Change in Load, ∆P (kN) Change in extension, ∆l (mm) Original length, L0 (mm) Initial Elastic Modulus, E (GPa) 5A16 57.1 ± 0.18 10.8±0.037 0.070 ± 0.0026 25 ± 0.002 67.6 ± 2.6 5A19 57.3 ± 0.18 12.6±0.043 0.079 ± 0.0026 25 ± 0.002 69.6 ± 2.3 5A21 56.9 ± 0.18 9 ± 0.031 0.057 ± 0.0026 25 ± 0.002 69.4 ± 3.2 5A22 56.8 ± 0.18 11.7±0.040 0.073 ± 0.0026 25 ± 0.002 70.6 ± 2.6 5A23 57.1 ± 0.18 8.1 ± 0.028 0.052 ± 0.0026 25 ± 0.002 68.2 ± 3.2 5A24 57.1 ± 0.18 10.8±0.037 0.069 ± 0.0026 25 ± 0.002 68.5 ± 2.6 5A25 57.1 ± 0.18 8.1 ± 0.028 0.050 ± 0.0026 25 ± 0.002 71.0 ± 3.7 5A26 57.2 ± 0.18 9.0 ± 0.031 0.056 ± 0.0026 25 ± 0.002 70.2 ± 3.3 4.3 Discrete Information Entropy Formulations After AE noise has been filtered and elastic modulus degradation has been estimated, the disorder of AE signals representing fatigue damage can be quantified. Multiple methods of estimating AE signal disorder with information entropy will be presented and discussed. All of the formulations have one commonality; a probability distribution is formed based on voltage readings in individual AE waveforms. Again, this compares to AE features such as counts and energy that are acquisition system outputs that summarize AE waveforms. It is also believed that because information entropy is a measure of a random variable’s disorder as described by its probability distribution, the AE information entropy directly reflects microstructural changes due to cyclic loading. 52 There are two general steps to estimating information entropy from AE signals no matter the formulation. A probability distribution of a random variable describing the AE signals must first be formed followed by quantifying the distribution’s disorder via Shannon’s entropy equation. Most of the effort is devoted to developing the probability distribution. In this case, the chosen random variable is the signal amplitude expressed in units of voltage. The idea is that the AE voltage probability distribution represents the microstructural disorder, and the distribution is evolving as fatigue damage progresses. The underlying continuously-changing probability distribution is estimated based on received AE signals. During the beginning of a fatigue experiment, the “true” AE voltage probability distribution is completely unknown until a first AE signal is recorded. Let this AE signal be described by its 2048 voltage readings which are 2048 samples from the current underlying voltage probability distribution. Because we do not know the true AE voltage probability distribution, an imprecise distribution can be estimated from these 2048 voltage values. The disorder of this distribution can then be quantified by Shannon’s equation in Equation (2.4). When a second AE signal is received, the new set of 2048 voltage values provides new information about the voltage probability distribution and needs to be considered. Three different methods to account for this new information about the probability distribution have been investigated and yield three formulations of AE information entropy. 53 4.3.1 Feature Entropy First, one can assume that the underlying AE voltage distribution is constantly changing and independent between all time steps. Therefore, a new probability distribution of the second AE signal’s 2048 voltage values should be created to estimate the true distribution at that particular instant. With every new AE signal, a new probability distribution is constructed to represent the constantly-changing underlying distribution. The information entropy calculated from each of these distributions is referred to as feature entropy. The framework to calculate feature entropy is depicted in Figure 21. It can be seen that for each signal received, the voltage value is recorded at every microsecond for 2048 microseconds (Step a). The 2048 voltage values are then taken to form a normalized histogram or, in this case, a discrete probability distribution (Step b). Because the 16-bit data acquisition system can output values between -10V and +10V producing a resolution of 0.305 mV, the bin widths are set to 1 mV to encompass 3 or 4 possible voltage values. One could set the bin widths smaller or larger which would alter the entropy value but not the entropy trends. From this distribution, the entropy is calculated based on Equation (2.4) where xi are the 1 mV bin ranges between -10V and +10V and p(xi) is the probability of receiving a voltage value within the specific bins. The entropies from these distributions are then plotted against the signal arrival time (Step c). 54 Figure 21: Feature entropy calculation from AE signals The feature entropy of the three example signals in Figure 21 quantifies the signals’ disorders. One can predict that the second signal reflects an event during low microstructural disorder while the first and third signals are during higher disorder microstructural changes. Entropy quantifies this disorder in that the first signal has 2.29 bits, the second has 1.99 bits, and the third has 2.95 bits of entropy. In addition, 55 the cumulative form of feature entropy can be thought of as the total microstructural disorder as fatigue damage progresses. This form will be discussed in the results. 4.3.2 Updated Entropy The second formulation is based on the theory that the underlying probability distribution changes very slowly, and the second AE signal received is from approximately the same distribution as the first AE signal. In turn, the 2048 voltage values from the first and second signal are combined, and a distribution is formed from the total 4096 values. As more AE signals are received, the new and previous AE voltage values are concatenated and the estimated probability distribution of the true underlying distribution is essentially updated. Another way to explain this process is that every AE signal’s distributions are combined and have equal effect on the estimated distribution. The information entropy calculated from these updated distributions is referred to as updated entropy. Figure 22 shows the process to calculate the updated entropy from AE signals. Similar to the feature entropy procedure, AE signals are received and saved as voltage values per microsecond for 2048 microseconds (Step a). Then, rather than forming a normalized histogram for each individual signal, the current voltage values are first concatenated with all the previous voltage values (Step b). From this series of 2048*i voltage values for i AE signals currently received, a normalized histogram with bin widths of 1 mV are constructed (Step c). Finally, Equation (2.4) is employed to yield entropy and can be plotted against signal arrival time (Step d). 56 Figure 22: Updated entropy calculation from AE signals The updated entropy of the three example signals in Figure 22 quantifies the disorder of the approximated underlying AE voltage distribution. One can predict that the first and third signals advocate for a wider, more-disordered true voltage distribution while a thinner, less-disordered true distribution is more probable according to the second signal. Subsequently, the updated entropy values for the first, second, and third signals are 2.29, 2.15, and 2.45 bits, respectively. Compared to feature entropy, updated entropy values are dependent on the previous distributions 57 and are therefore more inertial rather than sporadic. Because of the dependency between current and previous signals, one cannot simply relate the cumulative updated entropy to cumulative fatigue damage and it is best to analyze the scatter plot of updated entropy versus arrival time. 4.3.3 Temporally Weighted Entropy Finally, the third formulation is based on a slowly-evolving true probability distribution of AE voltage values similar to updated entropy, but assumes recent AE signals are more representative of the underlying distribution than past AE signals. When a second AE signal is received, a distribution of the 2048 voltage values is formed similar to the procedure for feature entropy. Then, the first and second signal’s distributions are combined in such a way that the second signal’s distribution is weighted more than the first. When subsequent AE signals are received, new distributions are formed where the most recent AE signal has the greatest weight on the updated distribution. The specific weights are determined based on arrival time and are calculated using Equation (4.5) where the weight vector, w, and arrival time vector, AT, are 1 x n vectors and n is the current number of recorded AE signals. For example, if the first AE signal occurs at 2 seconds and the second is recorded at 4 seconds, then the first signal’s distribution will have one-half of the weight of the second signal’s distribution. The information entropy calculated from these weighted and updated distributions is referred to as temporally weighted entropy. It should be noted that references in literature to weighted entropy or temporal entropy are not related to this proposed formulation. 58 ?⃗⃗? = 𝐴𝑇 → ∑ 𝐴𝑇𝑖 𝑛 𝑖=1 (4.5) The procedure to find temporally weighted entropy from AE signals is shown in Figure 23. As with the other methods, the first step is to record AE signals as time and voltage data (Step a). Then, the probability distributions of each of the signals are found by creating normalized histograms with 1 mV bin widths (Step b). Next, the temporal weights are found based on arrival time following Equation (4.5) in which the sum of the weights is 1. The individual signal distributions are then multiplied by their appropriate weights and added together to form an estimate of the true voltage distribution (Step c). The entropy of this temporally weighted distribution is then calculated using Equation (2.4) and plotted against signal arrival times (Step d). Updated entropy equally weighs all signal distributions and combines them into an estimate of the underlying voltage distribution while temporally weighted entropy weighs current signals more than past signals. Of many possible methods to determined distribution weights, the weighting used in this configuration is simply linearly proportional to signal arrival time. The temporally weighted entropy of the three example signals in Figure 23 quantifies the disorder of the approximated underlying AE voltage distribution as signals are received. Subsequently, the temporally weighted entropy values for the first, second, and third signals are 2.29, 2.11, and 2.48 bits, respectively. Temporally weighted entropy seems to be very similar, yet slightly less inertial than updated entropy. This is expected because previous signals will have less of an effect on the estimated distribution. 59 Figure 23: Temporally weighted entropy calculation from AE signals 4.3.4 Summary of Entropy Formations Three methods to estimate information entropy from AE signals have been proposed; feature, updated, and temporally weighted entropy. All procedures seek to quantify the microstructural disorder due to fatigue damage as reflected in an evolving AE voltage distribution. While the exact underlying AE voltage distribution is unknown, estimates of this distribution are assumed to be the voltage distributions from collected AE signals. The particular procedure to incorporate newly received AE signals into the voltage distribution estimate is dictated by the specific entropy 60 formulation. The MATLAB code to derive the entropies from the AE signals is provided in Appendix D – Code for AE Entropy Formulations. An explicit comparison of the entropy results depending on formulation method is shown in Figure 24 for the three example signals. Feature entropy which produces independent entropy values is the most erratic, updated entropy has the most inertial trend, and temporally weighted entropy is similar to updated entropy with more significance placed on current signal’s feature entropy. Table 12 summarizes assumptions and best estimate for the AE voltage distribution for each formulation. Figure 24: Entropy evolution for three examples of AE signals for each formulation Table 12: Summary of entropy formulations Entropy Formulation Assumption about “true” AE voltage distribution How best to estimate “true” AE voltage distribution Feature entropy Ever-changing and independent between time steps Independent distributions from each received AE signal Updated entropy Slowly changing where changes are assumed to occur with received AE signals Combine all received AE signal distributions such that all signals have equal weight on estimate Temporally weighted entropy Slowly changing where changes are assumed to occur throughout fatigue life Combine received AE signal distributions such that recent signals have more weight on the estimate than previous signals 61 Chapter 5 – Results and Discussion 5.1 Crack Growth and Fracture Surface Images As mentioned in Section 2.1.1 beginning on page 7, fatigue crack growth is known to follow three stages of growth: crack initiation, stable crack growth, and unstable growth to fracture. Depending on the microstructure and loading conditions, cracks may grow through inclusions, grains, or grain boundaries [7]. Small cracks grow at persistent slip bands after crack initiation and form a brittle fracture surface with fatigue striations [8]. Once a crack is of a sufficient length, the crack will propagate rapidly and cause ductile fracture. These assumptions were investigated with high-magnification images. First, an optical microscope captured images of one of the specimen faces to determine how fatigue cracks grew through the microstructure. Figure 25 is an example of three stitched images that show a crack grew through the grains and often through inclusions. This phenomenon was found for all imaged specimens. Figure 25: Crack growth through grains and inclusions Next, fracture surface differences between brittle and ductile crack growth were captured with a scanning electron microscope. Figure 26 depicts the two different regions with 77x magnification. Even higher magnification images in these 250 µm 62 two different regions were then taken. Figure 27 in fact shows the fatigue striations in the brittle region at 5740x magnification while Figure 28 shows a more disordered fracture surface during final ductile fracture. Overall, these images proved that cracks generally grew through grains and inclusions and orderly and consistent fatigue striations are present in the brittle fracture region. Figure 26: Brittle and ductile regions of fatigue crack growth Figure 27: Fatigue striations in the brittle region Figure 28: Highly disordered ductile region with no striations Ductile region Brittle region Crack Growth Direction 63 5.2 Stress-Life Analysis The experiments performed were stress-controlled, constant-amplitude sinusoidal fatigue tests. The specific loading conditions varied between experiments in order to study microstructural damage for low-cycle and mid-cycle fatigue tests. The stress life can be summarized by plotting the nominal stress amplitude against the cycles to failure known as an S-N curve. As mentioned before, 26 tests were performed while issues with the extensometer, crack monitoring system, and AE noise reduced the number of analyzable tests down to eight. However, the cycles to failure were not affected by these errors other than the three tests that failed away from the notch at low loads. Therefore, all tests excluding the three that failed away from the notch can be used to form the experimental S-N curve. It is assumed that the S-N curve follows a power relationship between the cycles to failure and the stress amplitude given by Equation (5.1) where σf ’ is the stress amplitude for failure near 0 cycles, Nf are the cycles to failure, and m is the exponential parameter. Here, m is related to stress intensity factor (Kt), notch sensitivity (q), surface roughness (kSF), loading factor (kL), and effective diameter (ksize) of a notched specimen and is approximated by Equation (5.2) – Equation (5.6) [77]. 𝑆𝑡𝑟𝑒𝑠𝑠 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 = ∆𝜎 2 = 𝜎𝑓 ′(𝑁𝑓) 𝑚 (5.1) 𝜎𝑓 ′ = 1.62 ∗ 𝜎𝑢 (5.2) 𝑚𝑛𝑜𝑡𝑐ℎ = 𝑚𝑛𝑜 𝑛𝑜𝑡𝑐ℎ − log( 𝐾𝑓 𝑘𝑆𝐹𝑘𝐿𝑘𝑠𝑖𝑧𝑒 ) log (𝑁𝐹𝐿) (5.3) 𝐾𝑓 = 1 + (𝐾𝑡 − 1)𝑞 (5.4) 64 𝑞 = 1 1+ 0.5 𝑚𝑚 𝜌 (5.5) 𝑘𝑠𝑖𝑧𝑒 = ( 𝑑 7.62 𝑚𝑚 ) −0.1133 (5.6) The theoretical S-N curve can be determined for the particular specimen geometry and material used in this study. All parameters used to find the theoretical S-N curve are listed in Table 13 after implementing Equation (5.2) – Equation (5.6). Here, the theoretical curve has an intercept of 950.5 MPa and an exponential parameter of -0.242. Table 13: Parameters for theoretical S-N curve for particular Al7075-T6 specimen Parameter Name Parameter symbol Value Intercept 𝜎𝑓 ′ 950.5 MPa Exponential parameter with notch 𝑚𝑛𝑜𝑡𝑐ℎ -0.242 Exponential parameter without notch 𝑚𝑛𝑜 𝑛𝑜𝑡𝑐ℎ -0.176 Ultimate tensile strength 𝜎𝑢 587 MPa Stress concentration factor 𝐾𝑡 2.61 Notch radius 𝜌 1 mm Notch sensitivity factor 𝑞 0.667 Notch factor 𝐾𝑓 2.073 Effective diameter 𝑑 18 mm Size factor 𝑘𝑠𝑖𝑧𝑒 0.907 Surface roughness factor for polished surface 𝑘𝑆𝐹 1.000 Axial loading factor 𝑘𝐿 0.923 Assumed fatigue limit cycles 𝑁𝐹𝐿 10 6 The experimental S-N curve was determined by plotting the nominal stress amplitude against cycles to failure for the 23 experiments that had a crack grow at the notch. Then, a power relationship was fit to the data to yield an experimental S-N relationship. Figure 29 displays the experimental S-N plot along with theoretical and experimental curves. In the end, the experimental data agreed well with the theoretical curve and only slight differences are evident between the theoretical and experimental equations. 65 Figure 29: Experimental S-N plot with theoretical and experimental S-N curves 5.3 Measured Damage from Modulus Degradation 5.3.1 Modulus Trends The statistic assumed to quantify true fatigue damage and therefore the parameter that AE features and entropy will be correlated to is elastic modulus degradation. Unlike visible crack growth where AE counts and features have often been correlated to crack length or crack growth rate, elastic modulus is expected to reflect the fatigue damage prior to crack initiation. First, the relationship between elastic modulus and fatigue cycles should be understood. Figure 30 shows the trends for the 8 experiments that had consistent experimental setups with the exception of the loading conditions. The figure is divided into four different plots showing the 66 modulus degradation at crack initiation, when a crack grows to 0.25 mm, then to 1 mm and then when the specimen fractures. From this chart, three conclusions can be made. First, modulus does not decrease for the majority of life. In order to utilize elastic modulus as a measure of fatigue damage prior to a visible crack, the elastic modulus should change prior to crack initiation. However, this phenomenon is not apparent. In the first plot of Figure 30, it is proven that four experiments show a decrease in modulus at crack initiation while the other four do not. When a crack reaches 0.25 mm, the US Navy’s retirement crack length [1], still there are two experiments that do not show a decrease in modulus. It is not until a 1 mm crack, defined as the transition between small and large cracks as noted in Table 1, that all tests prove to have a decrease in modulus. Figure 30: Raw modulus degradation for 8 experiments at various damage levels 67 A more sensitive measure of modulus may be achieved if the strain was measured within a smaller gauge length near the notch rather than the 25 mm extensometer gauge length. Despite this drawback, the measured damage from modulus degradation will be used to correlate AE features after a 1 mm crack. The second finding, as expected, is modulus decreases near the cycles to failure as seen in the last plot of Figure 30. Of course, the particular number of cycles where modulus decreases is dependent on the applied load. For example, a 110 MPa stress amplitude was applied to 5A19 which experienced rapid damage near 10,000 cycles while 5A23 had a lower amplitude of 71 MPa with rapid damage occurring at a higher 44,000 cycles. Finally, one can compare the final value of modulus. Theoretically, modulus should decrease towards zero as a specimen fractures. However, the modulus depends on the final extensometer extension immediately prior to when the specimen breaks. With the exception of 5A25, it seems the final modulus value is related to the cycles to failure where tests that last longer will have a lower final modulus value. This result is attributed to the idea that a crack grows more slowly for lower stress amplitudes with more cycles to failure. In this case, more elongation and therefore lower modulus is apparent prior to sudden ductile fracture. Throughout the plots in Figure 30, 5A25 proved to have a greater change in modulus at all stages of fatigue prior to fracture than the other experiments. The reason to why the 5A25 modulus behaved in this manner can be explained by observing the fracture surface pictured in Figure 31. Typically, a fracture surface will be relatively flat and smooth at the site of crack initiation. However, two fracture 68 surface regions at the notch can be clearly seen. This suggests that a crack initiated at a notch corner but propagated along the specimen width rather than the thickness. In turn, this uneven crack propagation surface may have caused the specimen to elongate more prior to fracture. Figure 31: 5A25 fracture surface showing two displaced regions at notch 5.3.2 Normalizing Modulus to Produce Measured Damage The raw modulus trends can be transformed into a normalized metric of damage referred to as measured damage. The correlation between AE parameters and this statistic allows data from all tests to be combined and analyzed together. In this case, damage is calculated based on Equation (5.7) where Einstant is the modulus at each cycle, E0 is the initial modulus, and Ef is the final modulus. This equation ensures the variations between the initial and final modulus of the experiments are reduced to near zero. A measured damage value of 0 refers to a pristine specimen while 1 is associated with specimen fracture. 𝑀𝑒𝑎𝑠𝑢𝑟𝑒𝑑 𝐷𝑎𝑚𝑎𝑔𝑒 = 𝐸𝑖𝑛𝑠𝑡𝑎𝑛𝑡−𝐸0 𝐸𝑓−𝐸0 (5.7) Rather than plotting the raw measured damage data that has some scatter, curves can be fit to the data and approximate the trend between damage and cycles. Region 2 Region 1 Lower half of specimen Upper half 69 Because the measured damage stays approximately constant and then sharply increases near failure, the exponential model fits the data well. The general formula for the exponential model is given as Equation (5.8), where D is measured damage, N is cycles, and a and b are fitted parameters in this case. Fitted curves of each data series were found using MATLAB’s built-in curve fitting toolbox. Fitted model statistics are noted in Table 14. 𝐷 = 𝑎 ∗ 𝑒𝑏∗𝑁 (5.8) Table 14: Fitted models for measured damage Test Name a b SSE R-square 5A16 3.01E-18 0.00166 3.240 0.981 5A19 1.82E-13 0.00324 0.965 0.989 5A21 4.32E-40 0.00266 0.796 0.994 5A22 1.79E-30 0.00413 0.615 0.994 5A23 9.28E-36 0.00184 0.600 0.997 5A24 1.46E-30 0.00353 0.482 0.995 5A25 1.93E-12 0.00135 2.057 0.992 5A26 2.68E-34 0.00239 0.233 0.999 The damage trends against cycles can be plotted for all experiments from these exponential models. Rather than observing the raw modulus degradation as in Figure 30, Figure 32 shows the damage trends up until crack initiation, a 0.25 mm crack, a 1 mm crack, and then final fracture. Since the fatigue damage trends are scaled so that the initial and final damage values are the same for all tests, one can more clearly observe that it is not until a 1 mm crack when all experiments prove to have a measurable increase in damage. Another point of discussion is the value of damage at specific crack lengths. The measured damage should intuitively be the same for all tests at a consistent 70 damage level and crack length regardless of the paths (i.e., applied loading) to that damage. However, Figure 32 proves that the damage values vary between experiments at crack initiation, a 0.25 mm crack, and at a 1 mm crack. This unexpected outcome can potentially be attributed to measurement errors. Figure 32: Measured damage for 8 experiments at various damage levels There are two main variables that are uncertain; the cycles at which cracks reach 0.25 mm and 1 mm and the measured damage values at these cycles. First, accurately estimating the cycles is dependent on properly estimating the crack length. The exact points when a crack is at precisely 0.25 mm and 1 mm are most likely missed since crack monitoring images are taken every 25 cycles. Therefore, the nominal crack length measurement is assumed to be imprecise. Also, measuring the length of a crack from crack monitoring images has an uncertainty due to pinpointing the proper beginning and end of a crack as well as estimating the picture scale as introduced in Section 3.4. Table 15 shows a summary of the error propagation in 71 measuring the length of a 0.25 mm and 1 mm crack while more detailed steps are in Appendix C – Error Propagation. In the end, the average measurement for a 0.25 mm crack is about 0.258 mm ± 0.021 mm while the average measurement for a 1 mm crack is about 1.062 mm ± 0.083 mm from these inaccuracies. Table 15: Error for crack length measurements Specimen Name Picture Scale (pixels/mm) 0.25 mm Crack (mm) % Error at 0.25 mm 1 mm Crack (mm) % Error at 1 mm 5A16* 24.7 ± 2.0 0.268 ± 0.084 31.3 1.078 ± 0.120 11.1 5A19 672.8 ± 46.3 0.272 ± 0.020 7.2 1.036 ± 0.072 6.9 5A21 725.8 ± 57.6 0.239 ± 0.020 8.3 1.011 ± 0.080 8.0 5A22 714.3 ± 51.3 0.249 ± 0.019 7.5 1.100 ± 0.079 7.2 5A23 651.0 ± 43.4 0.258 ± 0.018 7.1 1.041 ± 0.070 6.7 5A24 682.3 ± 76.5 0.262 ± 0.030 11.4 1.065 ± 0.120 11.2 5A25 600.8 ± 43.5 0.256 ± 0.020 7.7 1.094 ± 0.080 7.3 5A26 713.2 ± 52.2 0.260 ± 0.020 7.6 1.074 ± 0.079 7.3 * The DIC was used to monitor the crack growth in 5A16 and resulted in higher uncertainty measurements With uncertainties in crack length, the cycles at which cracks reach such lengths are also uncertain and can be estimated from the approximate crack growth rate. For simplicity, the crack growth rate at a 0.25 mm crack is found by dividing the measured length by the difference in cycles between the 0.25 mm crack and initiation. Similarly, the growth rate at a 1 mm crack is found by dividing the difference between the measured 1 mm and 0.25 mm length by the difference in cycles between the 1 mm and 0.25 mm crack. Then, the uncertainty in cycles at the 0.25 mm and 1 mm crack are calculated by dividing the uncertainty in the measurement by the crack growth rate as shown in Equation (5.9). Table 16 displays the results of this process where the error in cycles at 0.25 mm is on average 110 cycles and the cycle uncertainty at 1 mm is about 170 cycles. Because the error in the cycles is relatively small compared to the failure cycles between 0.9x10 4 and 4.5x10 4 , the uncertainty in 72 crack length measurement in fact does not cause a significant change in measured damage. 𝛿𝑐𝑦𝑐𝑙𝑒𝑠 = 𝛿 𝑐𝑟𝑎𝑐𝑘 𝑙𝑒𝑛𝑔𝑡ℎ 𝑖𝑛 𝑚𝑚 𝑐𝑟𝑎𝑐𝑘 𝑔𝑟𝑜𝑤𝑡ℎ 𝑟𝑎𝑡𝑒 𝑖𝑛 𝑚𝑚/𝑐𝑦𝑐𝑙𝑒𝑠 (5.9) Table 16: Error in cycles at measured crack lengths Specimen Name Cycles at Initiation Cycles at 0.25 mm Cycles at 1 mm da/dN at 0.25 mm (mm/cyc) da/dN at 1 mm (mm/cyc) Error in Cycles at 0.25 mm Error in Cycles at 1 mm 5A16 21000 21150 22050 0.00179 0.00090 47.0 162.1 5A19 7910 8485 8785 0.00047 0.00255 41.5 29.1 5A21 24750 28060 31715 0.000072 0.00021 273.6 392.2 5A22 15650 15725 16155 0.00332 0.00198 5.6 41.1 5A23 31645 36380 40495 0.000054 0.00019 335.1 378.3 5A24 17100 17525 18505 0.00062 0.00082 48.6 150.4 5A25 18190 18410 19695 0.00116 0.00065 16.9 125.6 5A26 29110 30335 31110 0.00021 0.00105 93.5 77.4 Another instance where significant measurement error is introduced is monitoring the crack length from only one side of the specimens. In a pure axial loading condition, a crack would grow evenly such that the crack length is the same on both specimen faces. However, cracks often grow unevenly due to a slight bending moment when the specimen is not perfectly aligned and can initiate on the side of the specimen opposite from the optical microscope. In this case, the modulus may significantly decline due to a propagating crack while a smaller crack length is measured. This situation is believed to be the main reason why the damage value is not comparable between tests at similar crack length measurements based on the previous discussion on 5A25 fracture surfaces. Experiments with measured damage near zero at certain crack lengths (5A16, 5A21, 5A23, 5A24, and 5A26) suggest the crack grew evenly or on the microscope specimen side while those with higher measured damage at these crack lengths (5A19, 5A22, and 5A25) suggest the crack grew unevenly or on the opposite side. In the end, the measured damage level that is 73 most certain is at fracture. Thus, the measured damage was normalized so that all experiments had a consistent value here while the other damage values vary between experiments and are assumed to be uncertain. In summary, modulus was estimated for each cycle based on maximum and minimum values of stress and strain for all tests. The measured damage was then derived from normalizing the modulus data and fitted with exponential relationships using MATLAB’s curve fitting toolbox. Measurement error was investigated to reconcile differences between experiments in measured damage at specific crack lengths. In the end, it is believed that there is high uncertainty in measured damage due to monitoring the crack growth on only one side of the specimen. As such, AE parameters including AE counts, energy, and entropy statistics will be compared to this normalized metric at specimen fracture where there is less uncertainty. Comparisons will be drawn between the AE parameters’ correlations to measured damage. 5.4 AE Parameter Cyclic Trends A first step to understanding AE parameter behaviors is to observe the trends with respect to fatigue cycles. In this section, the time-dependent trends for AE cumulative counts, AE cumulative energy, cumulative feature entropy, updated entropy, and temporally weighted entropy will be presented and discussed. 5.4.1 AE Counts AE cumulative counts is an AE feature commonly correlated to fatigue damage. To reiterate, counts are the number of times an AE signal crosses the prescribed amplitude threshold. In this case, all tests had the same threshold of 45 dB 74 and specimens were fatigued until fracture producing comparable count results. Because of the uncertainty of measured damage during fatigue life, it is best to see the cumulative counts behavior until specimen fracture as reported in Figure 33. This graph shows that cumulative counts slightly increases initially, stays constant for the majority of life, and then begins to increases near 80% to 90% of fatigue life until final fracture. This trend proves to be similar to fatigue stages in which crack growth is relatively constant for almost the entirety of fatigue and then sharply increases once a crack has initiated. Figure 33: Cumulative AE counts versus fatigue cycles until fracture Another point to consider from cumulative counts is the total cumulative counts at fracture. Theoretically, measures of damage should be approximately the same between identical structures that have been exhibited to the same level of fatigue loading. In other words, any metric that represents the cumulative fatigue damage within a structure should be similar for all specimens with cracks of equal lengths and at fracture. This idea as first discussed in terms of final modulus degradation at fracture can also be applied to cumulative AE counts, cumulative energy, and cumulative feature entropy. 75 It appears that the final cumulative AE counts for the experiments are similar with values ranging between 9,000 and 16,000 counts. The spread of this value can be quantified by the coefficient of variation (CV). The CV measures the relative standard deviation of a set of random numbers assumed to have come from a normal distribution. Equation (5.10) shows how to calculate the CV in percent, where σ is the sample standard deviation and µ is the sample mean. A CV of 0 suggests all values are the same with no variability (i.e., no uncertainties) while wider data scatters will have larger CV values. In this case, the mean and standard deviation of the final cumulative counts are 12,360 and 2,245 producing a CV of 18.2%. %𝐶𝑉 = 𝜎 𝜇 ∗ 100% (5.10) There are several potential causes of this variation in the final cumulative counts as well as other AE parameters at fracture. First, as discussed throughout this work, AE is susceptible to background noise. Despite applying a novel mechanical damping apparatus to the specimens to damp background noise, noise signals were filtered based on applied load during post-processing. However, this procedure was somewhat subjective given that only clusters of AE signals on the applied load versus fatigue cycle graphs were removed rather than a more consistent set of criteria (e.g., load threshold, frequency band). Therefore, damage-related signals within these clusters could have been unintentionally discarded or stand-alone noise signals were accepted without indication they were due to noise. Thus, the AE features could vary due to inaccuracies in classifying damage and noise signals. Another reason why cumulative AE features could differ between specimens at failure is due to the sensor placement and attachment method. AE signals are 76 generated due to a sudden release of elastic strain energy which produces a propagating acoustic wave. Prior to reaching the AE sensor, the wave attenuates within the structure. If an AE sensor was placed closer to the fatigue notch, higher amplitude signals with generally more counts may be recorded since there would be less distance for the signal to travel and attenuate. In addition, the sensor is coupled to the specimen surface with an ultrasonic gel. If there were to have been any changes in the gel application method between experiments, the AE signal features may differ between specimens. Finally, inherent microstructural differences may cause differences in AE signals and their features. AE signals are emitted as a crack grows through grains and inclusions. As dislocations pile up at grain boundaries and a crack suddenly overcomes obstacles, more elastic energy is expected to be released causing a high count, high energy AE signal. Specimens with more crystallographic variation where slip system orientations often change may have more significant AE signals. Likewise, structures with smaller and therefore higher grain density could be expected to have more significant AE signals. Overall, cumulative counts as well as cumulative energy and cumulative feature entropy are expected to have similar final values at fracture with variation due to inaccurate AE noise reduction, variable sensor configuration, and microstructural differences. 5.4.2 AE Absolute Energy Another commonly used AE feature to detect fatigue damage is absolute energy. Absolute energy is derived from the squared voltage signal and the signal’s 77 duration. In addition, the cumulative form of absolute energy can be thought of as the total measureable acoustic energy expelled during cyclic fatigue. Figure 34 shows the cumulative absolute energy plotted against cycles until specimen fracture. Cumulative absolute energy, when plotted on a log scale, follows a similar trend to AE cumulative counts in that the energy initially increases, remains approximately constant, and then begins to increase once a crack initiates. Figure 34: Cumulative AE energy versus fatigue cycles until fracture Similar to cumulative counts, the final cumulative energy can be assessed. The mean of the final cumulative energy is 9.21x10 8 aJ and the standard deviation is 1.87x10 8 aJ resulting in a CV of 20.2% using Equation (5.10). This value suggests cumulative energy at the point of fracture is slightly more variable compared to the CV of cumulative counts at 18.2% meaning energy may be more susceptible to experimental inconsistencies that affect AE signals. 5.4.3 Feature Entropy Feature entropy proposed in this thesis is calculated from AE signals by forming independent and new distributions of each individual AE signal’s voltage 78 values and employing Shannon’s equation to estimate the distributions’ disorders. This entropy method is believed to reflect the disorder in each AE signal as well as the microstructural disorder due to fatigue damage. As such, the total disorder from AE signals is assumed to be measured by cumulative feature entropy. Cumulative feature entropy is plotted against cycles until fracture in Figure 35. Yet again, this statistic seems to be congruent with AE cumulative counts, AE cumulative energy, and fatigue damage stages where the disorder is found to increase initially, remain constant, and then increase sharply prior to a 1 mm crack and final fracture. Figure 35: Cumulative feature entropy versus fatigue cycles until fracture As with the two previous AE cumulative features, the final feature entropy at specimen fracture can be discussed. It is expected that the cumulative microstructural disorder at fracture is similar between specimens with some variation. The mean of the final cumulative feature entropy is 4221.6 bits and the standard deviation is 945.2 resulting in a CV of 22.4% from Equation (5.10). As such, this value suggests that cumulative feature entropy has the most variability at fracture as compared to cumulative counts and cumulative energy. Therefore, one could say feature entropy may be most susceptible to experimental differences compared to the other two 79 traditional AE features. While it may seem advantageous for a parameter to have a lower CV, the higher variation for feature entropy means that is it more sensitive to slight differences in AE signals. In other words, it could potentially be a better, more sensitive measure of damage if variations between experiments were reduced. 5.4.4 Updated Entropy Rather than estimating the individual and independent disorders from each AE signal, the updated entropy was introduced in this thesis as the disorder that describes the current and underlying AE voltage probability distributions. An estimate of the true AE voltage distribution is continuously updated as new AE signals are received by weighing all AE signal distributions evenly. The disorders of the estimated distributions are then calculated based on Shannon’s equation to yield updated entropy. Since updated entropy is found from a continuously updated distribution that utilizes all currently received AE signals, it is not necessarily logical to view updated entropy in a cumulative form. Instead, updated entropy should be plotted in its singular form. In turn, the final value of updated entropy is not expected to be consistent between experiments. Updated entropy is meant to estimate the current distribution of possible AE voltage values. If significant and unique microstructural damage occurs within the structure, then it is expected that AE signals with large voltage amplitudes will be received and therefore the probability distribution will grow wider. In contrast, when small microstructural damages occur and emit small amplitude AE signals, the estimated voltage distribution will become thinner and concentrate at 0 volts. 80 The trend of updated entropy can be predicted based on this logic. First, signals received during crack initiation are assumed to be infrequent due to dislocation pile up [47] and have high disorder from the sudden release of energy when dislocations break away [41]. In addition, AE signals are received from numerous locations around the notch since a crack has not yet initiated and the exact location of crack initiation is uncertain. Therefore, it is hypothesized that the updated entropy prior to crack initiation will be relatively high-valued and sporadic due to the high energy and inconsistent AE sources. After a crack begins to initiate, previous work [47] suggests sudden bursts of low amplitude AE signals due to micro-cleavage will then be received. It is believed that these AE signals are of low amplitude because most AE signals now are emitted from the particular microcrack causing initiation rather than from multiple potential crack initiation sources leading to frequent, low-disordered AE waveforms. Thus, the entropy trend will decrease as a crack initiates and grows in an orderly manner. Finally, high amplitude AE signals are expected to be received once the crack propagates to a critical length and causes fracture. In turn, updated entropy will then increase near fracture. To test this hypothesis, the updated entropy is plotted against cycles until both a 1 mm crack and until fracture in Figure 36. Results validate the hypothesized entropy trend. The updated entropy varies for the first few signals since the voltage distribution is not yet inertial and heavily dependent on these first initial signal distributions. Then, sporadic AE signals are received for the majority of life as dislocation pileups and breakaways occur at multiple potential crack initiation sources. After a crack initiates and continues to grow to 1 mm, the first plot of Figure 81 36 shows that the updated entropy sharply decreases. This finding confirms that entropy decreases after a crack has initiated and AE signals are more frequent and less disordered. The trend then increases as the crack grows towards fracture as seen in the second plot of Figure 36, again agreeing with the hypothesis. Thus, this entropy trend may be able to dictate when a crack initiates and grows to a small length characterized by a sharp decrease and when a specimen is near fracture described by a sharp increase. Figure 36: Updated entropy versus fatigue cycles until a 1 mm crack and fracture 5.4.5 Temporally Weighted Entropy Finally, the last entropy statistic introduced in this thesis as a possible measure of microstructural disorder for AE signals is the temporally weighted entropy. This entropy is calculated based on the same idea as updated entropy where the underlying AE voltage distribution is estimated from all measured AE signals. Rather than weighing the individual signal distributions evenly, the signal distributions from recent signals are weighed more heavily than past signals. The weights are linearly proportional to the signal arrival times. Then, once the estimated underlying 82 distribution is updated as new signals are collected, Shannon’s equation is employed to quantify the disorder. Figure 37: Temporally weighted entropy versus fatigue cycles until a 1 mm crack and fracture Temporally weighted entropy is expected to follow the same trend as updated entropy yet be more influence by individual AE signal disorder. This thought is validated when plotting the temporally weighted entropy versus cycles for a 1 mm crack and fracture in Figure 37. The weighted entropy has the same general trend as updated entropy in all graphs but is more erratic during initial fatigue cycles. This is because few AE signals are collected prior to crack initiation and a 1 mm crack. When a signal is received 200 cycles after the most recent signal, for example, the disorder of the current signal will have a greater influence on the estimated distribution that the signal 200 cycles before. When two signals received are separated by only a few cycles, their contributions to the estimated distribution will be approximately the same and weighted entropy will change very slightly. In the end, there are slight differences between updated entropy and temporally weighted entropy, but both show a noticeable decrease prior to a 1 mm crack followed by an increase as the specimen fractures. 83 5.5 Normalized AE Parameters Compared to Measured Damage The overall goal of this work is to investigate the ability of AE parameters to predict or correlate to fatigue damage. Now that each of the AE parameter trends with respect to cycles has been discussed, comparisons between the parameters with respect to damage can be drawn. The parameters will be normalized so that the variability between experiments is reduced and metrics are no longer scale- dependent. Parameters are normalized by Equation (5.11) where p is any AE parameter. As such, comparisons between the cumulative parameters (counts, energy, and feature entropy) with respect to measured damage will be first discussed followed by a comparison between updated and temporally entropy against damage. MATLAB code used to match the measured damage to the AE parameters at AE hits and to normalize AE parameters is given in Appendix E – Code for Matching Modulus and AE Hits Data. 𝑝𝑛𝑜𝑟𝑚𝑎𝑙𝑖𝑧𝑒𝑑 = 𝑝𝑖−𝑝𝑚𝑖𝑛 𝑝𝑚𝑎𝑥−𝑝𝑚𝑖𝑛 (5.11) 5.5.1 Cumulative Counts, Energy, and Feature Entropy Counts, energy, and feature entropy are assumed to reflect damage in a similar way. Progressing fatigue damage causes elastic strain energy to be suddenly released resulting in measured AE signals. When more significant microstructural changes occur, AE signals with higher counts, energy, and disorder are recorded. In turn, the cumulative forms of these parameters are expected to be metrics of cumulative fatigue damage. While these three parameters are similar, one may ask what the differences between them are and which one represents the damage “better”. These questions can 84 be answered by considering the normalized parameters against the normalized metric of modulus degradation, measured damage. Based on the cyclic trends, it is expected that each parameter increases more at the beginning and end of damage and increases less for the majority of damage. However, normalized cumulative AE parameters with a constant relationship with measured damage would be ideal and suggest “perfect correlation”. For every increase in damage, it is assumed in this case that there is an equal increase in cumulative AE damage parameters. In other words, it is assumed that cumulative AE features are best to predict damage if they have a one-to- one relationship on normalized scales in this specific context. Figure 38 presents these graphs for the 8 experiments until specimen fracture along with the desired one-to-one relationship. All features show a greater increase at the beginning and end of damage compared to the middle damage values and deviate from the ideal relationship. More specifically, about 60% of the total increase in counts occurs near the point of fracture while about 50% of the cumulative energy occurs at initial damage. This result suggests that counts may be more responsive at the end of fatigue life and energy may be more responsive during initial damage. In contrast, feature entropy seems to be more equally responsive for all fatigue damage stages. This means that feature entropy may be better correlated to measured damage and has an advantage over counts and energy. 85 Figure 38: Normalized cumulative counts, energy, and feature entropy with respect to measured damage where a one-to-one relationship is desired The differences between the cumulative AE parameters damage trends can be quantified by measuring the deviations from the one-to-one ideal relationship. The deviations are measured at each instance of an AE signal. Examples of several deviations are shown in Figure 39 for 5A16’s normalized cumulative counts versus damage. 86 Figure 39: Example of deviations between ideal and experimental trends The summation of these deviations, referred to as the deviation factor, can then be used as a goodness metric of the one-to-one model and compared between the AE parameters for each test. The AE parameter with the lowest deviation factor can be assumed to be a better representation of damage. It should be noted that the deviation factors can only be compared because all AE features are normalized to the same scale and the number of deviations are equal between parameters for each experiments. The deviation factors for each of the experiments and for each of the cumulative AE parameters were calculated and plotted in Figure 40. Results show the deviation factor for feature entropy is lowest for all but one experiment while deviation factors for counts and energy are inconsistently greater and less than one another. This means one could argue feature entropy may be a better statistic of measure damage compared to counts and energy. 87 Figure 40: Deviation factor for AE counts, energy, and feature entropy for all experiments Uncertainty in measured damage and the normalized AE parameters play a role in the experimental results and conclusions. However, if measurement errors were to be reduced in the future, feature entropy may remain a potentially better damage statistic than counts and energy. Overall, each of these cumulative AE parameters show different trends with measured damage meaning each feature contributes different information during various stages of fatigue damage. Therefore, improvements could be made if these three parameters were somehow combined into one metric in the future. 5.5.2 Updated and Temporally Weighted Entropy The final two AE parameters, updated and temporally weighted entropy, can also be plotted on a normalized scale and compared to measured damage. Because these two types of entropy do not have monotonically increasing trends, it is not expected for these parameters to have a one-to-one relationship with damage. Rather, these plots show a further comparison between the two entropies as well as their 88 behavior against a normalized damage scale. As such, the plots are shown in Figure 41. One conclusion is that both entropies decrease near zero damage. This decrease is associated with damage prior to a 1 mm crack. Both entropies then increase throughout most of the measured damage which reflect the periods between a 1 mm crack and fracture. In addition, one can see there is more variability between experiments for temporally weighted entropy than updated entropy. This is attributed to temporally weighted entropy being influenced more by the current AE signal voltage distributions and therefore is generally more erratic. Figure 41: Normalized updated and temporally weighted entropy against measured damage One could expect these trends to be consistent but more sensitive for future tests that have less measurement uncertainty. For example, if the extensometer was more sensitive to the strain around the notch during crack initiation, then a larger percentage of measured damage would be related to crack initiation rather than crack growth. Thus, the decrease in updated and temporally weighted entropies associated with fatigue damage prior to a 1 mm crack may be more apparent at low measured damage. In the end, updated and temporally weighted entropy are similar to one 89 another but provide a unique damage trend than the other previously discussed AE parameters. 5.6 Summary of Results Several conclusions have been made from the experimental results. The following list summarizes all points discussed throughout this chapter. 1. Crack grew through grains and inclusions and caused fatigue striations during stable crack growth. 2. The stress-life trend of the fatigue experiments was fairly similar to the theoretical trend. 3. The instantaneous modulus degradation and its normalized form proved to be a measure of the true fatigue damage, but its utility is limited due to the measurement uncertainty. 4. Cumulative counts, energy, and feature entropy displayed similar cyclic trends with variable final values at specimen fracture. Overall, feature entropy had the most variation which could mean it is a more sensitive damage parameter. 5. Updated and temporally weighted entropy provide unique damage trends and may potentially be used to differentiate between small and large cracks. 6. When normalized and plotted against measured damage, the cumulative feature entropy seems to be better correlated with damage compared to the cumulative counts and energy. 90 Chapter 6 – Conclusions 6.1 Summary Cyclic fatigue experiments were performed on Al7075-T6 to investigate potential damage indicators from AE signals prior to the presence of a crack. The experimental procedure was iteratively refined to reduce AE noise and ensure usable data. Two separate theories to measure fatigue damage prior to crack initiation were presented. First, it was assumed that elastic modulus degradation could reflect fatigue damage prior to crack initiation. Second, rather than correlating summary AE features such as AE counts and energy to fatigue damage, more formal measures of disorder inherent in the AE signals, known as information entropy, were developed. Results showed that the modulus degradation was not significantly responsive to early fatigue damage and was susceptible to measurement errors. However, because AE signals were collected prior to crack initiation, it is likely that AE parameters may act as damage precursors, but their utility is limited by the accuracy in quantifying damage before a visible crack appears. If a more sensitive and more accurate measure of true damage were to be used, it is believe that the presented AE parameters could in fact be damage precursors. Comparisons between information entropy metrics and traditional AE features concluded that the cumulative feature entropy is similar to cumulative counts and energy, but it may be a more sensitive parameter to experimental variations and be better correlated to damage throughout fatigue life. In addition, updated and temporally weighted entropy produce unique trends that can potentially differentiate between small and large cracks. Since all AE parameters proved to have unique 91 trends at various damage levels, combining all parameters into one damage metric in the future could be useful. In the end, all of these parameters should be considered in the future stage of STLP testing with more emphasis on feature entropy. 6.2 Contributions Through the numerous cyclic fatigue experiments performed in this work, several findings were concluded in regards to estimating damage with novel information entropy measures derived from AE signals. The contributions of this work are listed as the following: 1. A major obstacle with using AE signals as an NDE technique is it’s susceptibly to extraneous noise. As such, in this research a high-amplitude AE noise was effectively damped below a desired threshold by means of a mechanical damping apparatus and filtered with post-process noise reduction techniques. 2. The use of modulus degradation as a measure of damage prior to a crack was investigated and found to be unresponsive to damage prior to crack initiation. However, the sensitivity of elastic modulus depended on measuring notch strains with a 25 mm gauge length extensometer. As such, modulus degradation as a measure of damage remains a feasible option if a more sensitive method to measure strain was employed in the future. 3. Three different methods to estimate the information content or the disorder from AE signals were proposed. Rather than accepting AE summary statistics, such as counts and energy as fully informative damage measures, information entropy is calculated based on the raw voltage data from each recorded AE 92 signal. In turn, it is shown that the proposed feature entropy better correlates with the fatigue damage than the AE counts and energy, while the proposed updated and temporally weighted entropy methods provide a unique damage trend that could differentiate between small and large cracks. 6.3 Future Work Numerous directions are possible to extend this research pertaining to both conclusively determining AE damage precursors as well as the second stage of the project. These are outlined below. 1. Damage prior to crack initiation needs to be accurately and repeatedly measured or estimated. While still acting on the idea that modulus degrades prior to crack initiation, improvements could be made to measuring the strain around the notch. This can be done with an extensometer with a smaller gauge length, using strain gauges if they are ensured to not interact with AE signals or detach during the tests, or a highly accurate and fast DIC. The uncertainty in measured damage at certain damage levels would decrease if the crack length could be better monitored by monitoring the crack from multiple perspectives. 2. Other methods of estimating damage prior to a fatigue crack could be explored. The replica method employed by Newman [16,17,18] could be an option in which cyclic loading is repeatedly paused, the notch region is replicated with silicon-rubber, and then observed under high magnification. Another method, which was based on replicas, is to model fatigue damage with FASTRAN software. This software was developed 93 for small cracks, so while damage prior to a crack may not be able to be estimated, very small cracks can be modeled. Finally, simulations similar to those proposed in [25, 45, 46] could be used to model the atomic interactions during fatigue and those estimate damage prior to a crack. 3. Error in the measured damage at supposedly consistent damage levels at certain crack lengths is mainly attributed to measuring crack lengths on one side of the specimen as cracks propagating unevenly. This phenomenon is believed to be caused by a slight bending moment applied to the specimens due to misalignment of test grips. To limit the bending moment effect on crack growth behavior in axially-loaded fatigue tests, thicker specimens could be designed and tested in the future. 4. Once one or a few accurate measures of damage prior to a crack are found, then experiments should be performed while damage is measured and AE signals are collected. Then, the various information entropy metrics should be calculated and correlated to the more accurate measured damage. 5. Rather than using resonant sensors to detect AE signals, wideband sensors could be used in the future. Wideband sensors better reconstruct the received acoustic wave with respect to the frequency behavior. Therefore, more analysis could be performed in the frequency domain similar to those in previous work [35, 39, 52]. 6. Other damage precursors could be investigated along with AE signals including strain energy dissipation and thermodynamic entropy. These two 94 measures proved to be correlated to fatigue damage at crack initiation [78] and could be combined and compared with all AE damage parameters. 7. Fatigue damage precursors could be combined into one damage statistic in the future. One possible method is to use an extended Kalman filter which utilizes recursive Bayesian estimation to reduce uncertainty. It has been used successfully in combining AE counts and crack length inspections to better estimate crack growth rate [79]. 8. Finally, altering the current experimental setup used for coupon testing to be used for larger-scale testing should be considered. Most notably, the damage apparatus was designed to limit the mechanical noise from the servo-hydraulic testing machine. However, in a different experimental configuration, one should expect that the AE background noise to be different and would need to consider how to account for noisy AE signals. 95 Appendices Appendix A – Applied Load vs. Signal Arrival Time Scatter Plots This appendix shows the applied load and time for each AE hit. These plots were used to filter noise. For tests with no filtering or unfilterable noise, only one plot is presented. For tests with filtering, before and after graphs are provided. 5A7 – Unfilterable 5A15 – Unfilterable 96 5A16 – Before Filtering 5A16 – After Filtering 5A19 – Before Filtering 97 5A19 – After Filtering 5A20 – Unfilterable 5A21 – Before Filtering 98 5A21 – After Filtering 5A22 – Before Filtering 5A22 – After Filtering 99 5A23 – Before Filtering 5A23 – After Filtering 5A24 – Before Filtering 100 5A24 – After Filtering 5A25 – No Filtering Needed 5A25 – No Filtering Needed 101 Appendix B – Code for Modulus Evolution %% Strain and Modulus Calculation clear all %% Load Raw Data % Experiments to find modulus degradation from testLabels = [16 19 21 22 23 24 25 26]; % Find number of tests to be analyzed numOfTests = length(testLabels); % Pre allocate arrays MTS = cell(1,numOfTests); timeA = cell(1,numOfTests); cyclesA = cell(1,numOfTests); posA = cell(1,numOfTests); loadA = cell(1,numOfTests); extenA = cell(1,numOfTests); % The data from the MTS machine is set up so that the first 5 seconds, the % specimen is fully unloaded to 0 kN. Then the specimen is loaded to the % minimum load. Then fatigue begins. Therefore, need to find the exact time % when cyclic fatigue begins. This is done by plotting the data and finding % the start time. peakTimeMTS = [10.36 10.36 10.44 10.35 10.39 10.35 10.37 10.40]; % Load data from raw csv files for j = 1:numOfTests begName = 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5_StrainCalc\Test1.steps.tracking_5A'; fileIndex = num2str(testLabels(j)); fileName = strcat(begName,fileIndex,'.csv'); MTS{1,j} = csvread(fileName,1,0); timeA{1,j} = MTS{1,j}(:,1)-peakTimeMTS(j); cyclesA{1,j} = MTS{1,j}(:,6); posA{1,j} = MTS{1,j}(:,7); loadA{1,j} = MTS{1,j}(:,8); extenA{1,j} = MTS{1,j}(:,9); % Display the test as it has finished loading data disp(testLabels(j)) end %% Input Constants % Average thickness for each specimen found after three measurements around % the notch with calibers t = [3.157 3.157 3.15 3.15 3.163 3.16 3.157 3.167]; % mm % Average width of specimens after three measurements w = [18.07 18.137 18.057 18.027 18.053 18.067 18.073 18.063]; %mm % Maximum applied load for each test maxLoad = [12 14 10 13 9 12 9 10]; %kN % Extensometer gauge length localGaugeL = 25; %mm % Estimated times when a crack initiated, was 0.25 mm, and was 1 mm from % crack growth images. Went picture by picture until a crack of these % lengths were measured with ImageJ. crackInitTime = [4200 1582 4950 3130 6329 3420 3638 5822]; %s crack025mmTime = [4260 1697 5612 3145 7276 3505 3682 6067]; %s crack1mmTime = [4410 1757 6343 3231 8099 3701 3939 6222]; %s %% Calculate stress, strain, modulus, measured damage % Put specimen names into cell array Preallocate 102 maxim = cell(1,length(testLabels)); minim = cell(1,length(testLabels)); maxLoc = cell(1,length(testLabels)); minLoc = cell(1,length(testLabels)); posInv = cell(1,length(testLabels)); maxInd = zeros(1,length(testLabels)); minInd = zeros(1,length(testLabels)); firstMaxLoc = zeros(1,length(testLabels)); firstMinLoc = zeros(1,length(testLabels)); redPosMaxim = cell(1,length(testLabels)); redPosMinim = cell(1,length(testLabels)); redLoadMaxim = cell(1,length(testLabels)); redLoadMinim = cell(1,length(testLabels)); redExtenMaxim = cell(1,length(testLabels)); redExtenMinim = cell(1,length(testLabels)); graphTimeInd = cell(1,length(testLabels)); deltaPos = cell(1,length(testLabels)); deltaLoad = cell(1,length(testLabels)); deltaExten = cell(1,length(testLabels)); deltaStress = cell(1,length(testLabels)); localStrain = cell(1,length(testLabels)); modulus = cell(1,length(testLabels)); modTime = cell(1,length(testLabels)); changeInMod = cell(1,length(testLabels)); damage = cell(1,length(testLabels)); initModulus = zeros(1,length(testLabels)); indCrackInitTime = zeros(length(testLabels),1); indCrack025mmTime = zeros(length(testLabels),1); indCrack1mmTime = zeros(length(testLabels),1); for i = 1:length(testLabels) % Finding the max and min position values and their indices [maxim{1,i},maxLoc{1,i}] = findpeaks(posA{1,i}); posInv{1,i} = 1.01*max(posA{1,i}) - posA{1,i}; [minim{1,i},minLoc{1,i}] = findpeaks(posInv{1,i}); minim{1,i} = posA{1,i}(minLoc{1,i}); % Need to "line-up" the max and min position values with their cycles. % If there are a different number of maxs and mins, need to figure out % which ones should line up. This is what the next few lines and the if % loop do. % Getting the length of each max and min position vector maxInd(i) = length(maxLoc{1,i}); minInd(i) = length(minLoc{1,i}); % Getting the indice for the first max and min values firstMaxLoc(i) = maxLoc{1,i}(1); firstMinLoc(i) = minLoc{1,i}(1); % Ensure you get matching minimums and maximums during increasing part % of the cycle. Then calculate the deltaPosition and deltaLoads based % on the "matched-up" maxs and mins. Also identify the time for each % start of the cycle (time at the minimum values) if maxInd(i)> minInd(i) && firstMaxLoc(i) < firstMinLoc(i) redPosMaxim{1,i} = maxim{1,i}(2:end); redPosMinim{1,i} = minim{1,i}(1:end); redLoadMaxim{1,i} = loadA{1,i}(maxLoc{1,i}(2:end)); redLoadMinim{1,i} = loadA{1,i}(minLoc{1,i}(1:end)); redExtenMaxim{1,i} = extenA{1,i}(maxLoc{1,i}(2:end)); redExtenMinim{1,i} = extenA{1,i}(minLoc{1,i}(1:end)); graphTimeInd{1,i} = minLoc{1,i}(1:end); disp([i,1]) 103 elseif maxInd(i) == minInd(i) && firstMaxLoc(i) < firstMinLoc(i) redPosMaxim{1,i} = maxim{1,i}(2:end); redPosMinim{1,i} = minim{1,i}(1:end-1); redLoadMaxim{1,i} = loadA{1,i}(maxLoc{1,i}(2:end)); redLoadMinim{1,i} = loadA{1,i}(minLoc{1,i}(1:end-1)); redExtenMaxim{1,i} = extenA{1,i}(maxLoc{1,i}(2:end)); redExtenMinim{1,i} = extenA{1,i}(minLoc{1,i}(1:end-1)); graphTimeInd{1,i} = minLoc{1,i}(1:end-1); disp([i,2]) elseif maxInd(i) < minInd(i) && firstMaxLoc(i) > firstMinLoc(i) redPosMaxim{1,i} = maxim{1,i}(1:end); redPosMinim{1,i} = minim{1,i}(1:end-1); redLoadMaxim{1,i} = loadA{1,i}(maxLoc{1,i}(1:end)); redLoadMinim{1,i} = loadA{1,i}(minLoc{1,i}(1:end-1)); redExtenMaxim{1,i} = extenA{1,i}(maxLoc{1,i}(1:end)); redExtenMinim{1,i} = extenA{1,i}(minLoc{1,i}(1:end-1)); graphTimeInd{1,i} = minLoc{1,i}(1:end-1); disp([i,3]) elseif maxInd(i) == minInd(i) && firstMaxLoc(i) > firstMinLoc(i) redPosMaxim{1,i} = maxim{1,i}(1:end); redPosMinim{1,i} = minim{1,i}(1:end); redLoadMaxim{1,i} = loadA{1,i}(maxLoc{1,i}(1:end)); redLoadMinim{1,i} = loadA{1,i}(minLoc{1,i}(1:end)); redExtenMaxim{1,i} = extenA{1,i}(maxLoc{1,i}(1:end)); redExtenMinim{1,i} = extenA{1,i}(minLoc{1,i}(1:end)); graphTimeInd{1,i} = minLoc{1,i}(1:end); disp([i,4]) end deltaPos{1,i} = redPosMaxim{1,i}-redPosMinim{1,i}; deltaExten{1,i} = redExtenMaxim{1,i}-redExtenMinim{1,i}; deltaLoad{1,i} = redLoadMaxim{1,i}-redLoadMinim{1,i}; % From deltaPos, deltaExten, and deltaLoad, get stress, strain, % modulus, change in modulus, and measured damage deltaStress{1,i} = deltaLoad{1,i}./(t(i)*w(i))*1000; %MPa = kN/mm/mm*1000 localStrain{1,i} = deltaExten{1,i}./localGaugeL; %mm/mm modulus{1,i} = deltaStress{1,i}./localStrain{1,i}; modTime{1,i} = timeA{1,i}(graphTimeInd{1,i}); % Normalize modulus initModulus(1,i) = mean(modulus{1,i}(180:480))/1000; damage{1,i} = (modulus{1,i}(180:end-20)./1000- initModulus(i))./(modulus{1,i}(end-20)/1000-initModulus(i)); % Find the time indices for modulus where crack initiates, grows to 0.25 % and to 1 mm. indCrackInitTime(i) = find(modTime{1,i}>crackInitTime(i),1); indCrack025mmTime(i) = find(modTime{1,i}>crack025mmTime(i),1); indCrack1mmTime(i) = find(modTime{1,i}>crack1mmTime(i),1); stopInd4(i,1) = length(modTime{1,i}); %Fracture end 104 Appendix C – Error Propagation C.1 Initial Elastic Modulus 𝐹𝑜𝑟 𝐴 = 𝑡 ∗ 𝑤, 𝛿𝐴 = |𝐴|√( 𝛿𝑡 |𝑡| ) 2 + ( 𝛿𝑤 |𝑤| ) 2 (C.1) Test t (mm) δt (mm) w (mm) δw (mm) A (mm2) δA (mm2) 5A16 3.16 0.01 18.07 0.01 57.05 0.183 5A19 3.16 0.01 18.14 0.01 57.26 0.184 5A21 3.15 0.01 18.06 0.01 56.88 0.183 5A22 3.15 0.01 18.03 0.01 56.79 0.183 5A23 3.16 0.01 18.05 0.01 57.10 0.183 5A24 3.16 0.01 18.07 0.01 57.09 0.183 5A25 3.16 0.01 18.07 0.01 57.06 0.183 5A26 3.17 0.01 18.06 0.01 57.21 0.183 𝐹𝑜𝑟 ∆𝑃 = 𝑃𝑚𝑎𝑥 − 𝑃𝑚𝑖𝑛, 𝛿∆𝑃 = √(𝛿𝑃𝑚𝑎𝑥)2 + (𝛿𝑃𝑚𝑖𝑛)2 (C.2) Test Pmax (kN) δPmax (kN) Pmin (kN) δPmin (kN) ∆P (kN) δ∆P (kN) 5A16 12.0 0.036 1.20 0.0096 10.80 0.037 5A19 14.0 0.042 1.40 0.0112 12.60 0.043 5A21 10.0 0.03 1.00 0.008 9.00 0.031 5A22 13.0 0.039 1.30 0.0104 11.70 0.040 5A23 9.0 0.027 0.90 0.0072 8.10 0.028 5A24 12.0 0.036 1.20 0.0096 10.80 0.037 5A25 9.0 0.027 0.90 0.0072 8.10 0.028 5A26 10.0 0.03 1.00 0.008 9.00 0.031 Test Lo (mm) δLo (mm) lmax (mm) δlmax (mm) lmin (mm) δlmin (mm) 5A16 25 0.002 0 0.00186 -0.07 0.00186 5A19 25 0.002 0.084 0.00186 0.005 0.00186 5A21 25 0.002 -0.025 0.00186 -0.082 0.00186 5A22 25 0.002 -0.02 0.00186 -0.093 0.00186 5A23 25 0.002 0.034 0.00186 -0.018 0.00186 5A24 25 0.002 -0.009 0.00186 -0.078 0.00186 5A25 25 0.002 0.038 0.00186 -0.012 0.00186 5A26 25 0.002 0.014 0.00186 -0.042 0.00186 𝐹𝑜𝑟 ∆𝑙 = 𝑙𝑚𝑎𝑥 − 𝑙𝑚𝑖𝑛, 𝛿∆𝑙 = √(𝛿𝑙𝑚𝑎𝑥)2 + (𝛿𝑙𝑚𝑖𝑛)2 (C.3) 105 𝐹𝑜𝑟 ∆𝜀 = ∆𝑙 𝐿𝑜 , 𝛿∆𝜀 = |∆𝜀|√( 𝛿∆𝑙 |∆𝑙| ) 2 + ( 𝛿𝐿𝑜 |𝐿𝑜| ) 2 (C.4) 𝐹𝑜𝑟 𝐸 = ∆𝑃 ∗ 𝐿𝑜 ∆𝑙 ∗ 𝐴⁄ , 𝛿𝐸 = |𝐸| √( 𝛿∆𝑃 |∆𝑃| ) 2 + ( 𝛿𝐿𝑜 |𝐿𝑜| ) 2 + ( 𝛿∆𝑙 |∆𝑙| ) 2 + ( 𝛿𝐴 |𝐴| ) 2 (C.5) Test ∆l (mm) δ∆l (mm) ∆Strain (mm/mm) δDeltastrain (mm) E (GPa) δE (GPa) 5A16 0.070 0.00263 0.00280 0.000105 67.6 2.6 5A19 0.079 0.00263 0.00316 0.000105 69.6 2.3 5A21 0.057 0.00263 0.00228 0.000105 69.4 3.2 5A22 0.073 0.00263 0.00292 0.000105 70.6 2.6 5A23 0.052 0.00263 0.00208 0.000105 68.2 3.5 5A24 0.069 0.00263 0.00276 0.000105 68.5 2.6 5A25 0.050 0.00263 0.00200 0.000105 71.0 3.7 5A26 0.056 0.00263 0.00224 0.000105 70.2 3.3 C.2 Measured Crack Length *The DIC was used for crack monitoring of 5A16 where the notch was in full view and the picture scale could be directly estimated 𝐹𝑜𝑟 𝑐2, 𝛿𝑐2 = 2 ∗ 𝑐 ∗ 𝛿𝑐 (C.6) Test R (mm) δR (mm) c (pixels) δc (pixel) h (pixels) δh (pixels) c 2 (pixels/mm) δc2 (pixels) 5A16* 1 0.05 - - - - - - 5A19 1 0.05 774.7 4 122.7 4 600160 6198 5A21 1 0.05 712 4 93.3 4 506944 5696 5A22 1 0.05 768 4 112 4 589824 6144 5A23 1 0.05 786 4 132 4 617796 6288 5A24 1 0.05 546 4 57 4 298116 4368 5A25 1 0.05 695 4 110.7 4 483025 5560 5A26 1 0.05 754.7 4 108 4 569572 6038 S c h d R 106 𝐹𝑜𝑟 𝑐2 8ℎ , 𝛿 𝑐2 8ℎ = | 𝑐2 8ℎ | √( 𝛿𝑐2 |𝑐2| ) 2 + ( 𝛿ℎ |ℎ| ) 2 (C.7) 𝐹𝑜𝑟 𝑐2 8ℎ + ℎ 2 , 𝛿 ( 𝑐2 8ℎ + ℎ 2 ) = √(𝛿 𝑐2 8ℎ ) 2 + (𝛿ℎ)2 (C.8) 𝐹𝑜𝑟 𝑆𝑐𝑎𝑙𝑒 = ( 𝑐2 8ℎ + ℎ 2 ) 𝑝𝑖𝑥𝑒𝑙𝑠 𝑅 𝑚𝑚 , 𝛿𝑆𝑐𝑎𝑙𝑒 = |𝑆𝑐𝑎𝑙𝑒|√( 𝛿( 𝑐2 8ℎ + ℎ 2 ) |( 𝑐2 8ℎ + ℎ 2 )| ) 2 + ( 𝛿𝑅 |𝑅| ) 2 (C.9) Test c 2 /8h (pixels) δc2/8h (pixels) Numerator (pixel) δNumerator (pixels) Scale (pix/mm) δScale (pix/mm) 5A16* - - - - 24.65 2 5A19 611.4 20.9 672.8 31.9 672.8 46.3 5A21 679.2 30.1 725.8 44.8 725.8 57.6 5A22 658.3 24.5 714.3 36.8 714.3 51.3 5A23 585.0 18.7 651.0 28.7 651.0 43.4 5A24 653.8 46.9 682.3 68.4 682.3 76.5 5A25 545.4 20.7 600.8 31.5 600.8 43.5 5A26 659.2 25.4 713.2 38.1 713.2 52.2 𝐹𝑜𝑟 𝐶𝑟𝑎𝑐𝑘 𝐿𝑒𝑛𝑔𝑡ℎ = 𝐶𝐿 (𝑝𝑖𝑥𝑒𝑙𝑠) 𝑆𝑐𝑎𝑙𝑒 ( 𝑝𝑖𝑥𝑒𝑙𝑠 𝑚𝑚 ) , 𝛿𝐶𝐿 = |𝐶𝐿|√( 𝛿(𝐶𝐿) |𝐶𝐿| ) 2 + ( 𝛿𝑆𝑐𝑎𝑙𝑒 |𝑆𝑐𝑎𝑙𝑒| ) 2 (C.10) Test 0.25 mm crack (mm) δ0.25 mm crack (mm) 1 mm crack (mm) δ1 mm crack (mm) %error in 0.25 mm % error in 1 mm 5A16 0.268 0.084 1.078 0.120 31.3 11.1 5A19 0.272 0.020 1.036 0.072 7.2 6.9 5A21 0.239 0.020 1.011 0.080 8.3 8.0 5A22 0.249 0.019 1.1 0.080 7.5 7.2 5A23 0.258 0.018 1.041 0.070 7.1 6.7 5A24 0.262 0.030 1.065 0.120 11.4 11.2 5A25 0.256 0.020 1.094 0.080 7.7 7.3 5A26 0.26 0.020 1.074 0.080 7.6 7.3 107 Appendix D – Code for AE Entropy Formulations %% Entropy From AE Signals clear all clc % Used as a reference for chooseTestInd % 1 | 5A4 % 2 | 5A6 % 3 | 5A10 % 4 | 5A16 % 5 | 5A19 % 6 | 5A21 % 7 | 5A22 % 8 | 5A23 % 9 | 5A24 % 10 | 5A25 % 11 | 5A26 %% Setup % Choose which test to get entropy from chooseTestInd = 1; % out of 11 channel = 2; % 1 or 2 sizeOfWaveform = 2*1024; freq = 5; % Smallest division is 20V/65536 = 0.00030518, so choosing 0.001 binwidth = 0.0010; testsStr = {'5A4','5A6','5A10','5A16','5A19','5A21','5A22','5A23','5A24','5A25','5A26'} ; % Assumes waveform files are in one folder. Input the folder location testNames = {'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A4_102915\AE\Filtered_5A4_test1_'; 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A6_111915\AE\Waveforms\ampAndCountAndLoadF_5A6_test1_'; 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A10_120215\AE\test1_5A10_'; 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A16_040516\AE\Waveforms\F_Test1_5A16_'; 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A19_041916\AE\Waveforms 2\loadF_test1_5A19_'; 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A21_042516\AE\Waveforms\timeF_test1_5A21_'; 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A22_042616\AE\Waveforms\loadF_test1_5A22_' 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A23_042716\AE\Waveforms\loadF_test1_5A23_'; 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A24_042916\AE\Waveforms\loadF_test1_5A24_' 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A25_050216\AE\Waveforms\test1_5A25_'; 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\5A26_050216\AE\Waveforms\test1_5A26_'}; % Number of waveform files in the form of [ch1# ch2#; next test;...] chNumberOfFiles = [141 408; 137 164; 140 520; 1157 2108; 934 1795;... 653 2617; 958 1222; 888 2311; 863 2403; 658 1478; 418 1865]; % Specified threshold values thresholds = [52 46; 53 48; 52 47; 50 45; 50 45;... 50 45; 50 45; 50 45; 50 45; 50 45; 50 45]; 108 % Test name, label, number of waveform files, specified threshold % for particular test testName = testNames{chooseTestInd}; testStr = testsStr{chooseTestInd}; numberOfFiles = chNumberOfFiles(chooseTestInd,channel); thresholdDB = thresholds(chooseTestInd,channel); %Changes channel number to string for file name channelIndex = num2str(channel); %Text files fileType = '.txt'; % Preamp gain in dB preamp = 40; %Threshold gain in dB, equation from AEWin manual thresholdVolt = (10^((thresholdDB+preamp)./20))*(10^-6); % Preallocate entropy1 = zeros(numberOfFiles,2); entropy2 = zeros(numberOfFiles,2); entropy3 = zeros(numberOfFiles,2); maximum = zeros(numberOfFiles,2); countsFromWaveforms = zeros(numberOfFiles,2); time = zeros(1,numberOfFiles); mat1 = zeros(sizeOfWaveform,1); % Histogram bin edges and middle points for temporally weighted entropy edges = -10:0.001:10; mid = -10+0.0005:0.001:10-0.0005; mat2 = zeros(sizeOfWaveform,1); allProb = zeros(sizeOfWaveform,length(mid)); weightedProb = zeros(numberOfFiles,length(mid)); %% 3 different entropies as new signals are received for i = 1:numberOfFiles %%% Loading data from iterative text files % Indexes through waveform numbers fileIndex = num2str(i); % Entire file name fileName = strcat(testName,channelIndex,'_',fileIndex,fileType); % Open file fileID = fopen(fileName); % Imports file to strings wholeWaveform = textscan(fileID,'%s','Delimiter','\n'); timeWaveform = wholeWaveform{1,1}(11); workForTime= sscanf(timeWaveform{1,1},'%s%s%s%f'); % Signal arrival time time(i) = workForTime(12); % AE signal voltage data in strings waveformData = wholeWaveform{1,1}(13:end); % AE signal voltage data in values for k = 1:sizeOfWaveform mat1(k) = sscanf(waveformData{k}, '%f'); end %Close file fclose(fileID); %%% Indexing entropy1(i,1) = i; entropy2(i,1) = i; entropy3(i,1) = i; maximum(i,1) = i; 109 countsFromWaveforms(i,1) = i; % Maximum voltage value from AE signal maximum(i,2) = max(mat1); % Counts directly from raw AE signal countsFromWaveforms(i,2) = sum(mat1>thresholdVolt); %%% 1) Feature Entropy from each waveform % Finding probability distribution with auto binning rules [prob1, Edge1] = histcounts(mat1,'BinWidth',binwidth,'Normalization','probability'); % Preallocate S1 = zeros(length(prob1),1); for q = 1:length(prob1) % Calculate individual terms of Shannon's equation for each % possible outcome (in this case, histogram bar) if prob1(q)>0 S1(q) = -prob1(q)*log2(prob1(q)); end end % Sum terms to get entropy entropy1(i,2) = sum(S1); %%% 2) Updated entropy from waveforms % Want histogram to be mobile with constant bin width. This means the % waveforms need to be put into one array and then passed to % histcounts. % Making one very large array with all waveforms together mat2(sizeOfWaveform*(i-1)+1:sizeOfWaveform*i) = mat1; % Finding probability distribution of all received voltage values [prob2, Edge2] = histcounts(mat2,'BinWidth',binwidth,'Normalization','probability'); % Preallocate S2 = zeros(length(prob2),1); for j = 1:length(prob2) % Calculate individual terms of Shannon's equation for each % possible outcome (in this case, histogram bar) if prob2(j)>0 S2(j) = -prob2(j)*log2(prob2(j)); end end % Sum terms to get entropy entropy2(i,2) = sum(S2); %%% 3) Temporally weighted entropy from waveforms % Want to devise a way so that the probability distribution of the % current signal has a greater effect on the system probability % distribution than the previous signals % Making one very large array with all waveforms together [prob, edge1] = histcounts(mat1,'BinEdges',edges,'Normalization','probability'); allProb(i,:) = prob; % % Get the current vector of arrival times current = time(1:i); % Get the temporal linear weights weights = current./sum(current); % Multiply probably distributions by weights weightedProb(i,:) = weights*allProb(1:i,:); % Preallocate S3 = zeros(length(weightedProb(i,:)),1); for j = 1:length(weightedProb(i,:)) % Calculate individual terms of Shannon's equation for each % possible outcome (in this case, histogram bar) 110 if weightedProb(i,j)>0 S3(j) = -weightedProb(i,j)*log2(weightedProb(i,j)); end end % Sum terms to get entropy entropy3(i,2) = sum(S3); % Checking all pdf's always sum to 1 checkFor1_1 = sum(prob1); if checkFor1_1 > 1.0001 || checkFor1_1 < 0.9999 str = sprintf('ERROR: PDF for %d does not sum to 1', i); end checkFor1_2 = sum(prob2); if checkFor1_2 > 1.0001 || checkFor1_2 < 0.9999 str = sprintf('ERROR: PDF for %d does not sum to 1', i); end checkFor1_3 = sum(allProb(i,:)); if checkFor1_3 > 1.0001 || checkFor1_3 < 0.9999 str = sprintf('ERROR: PDF for %d does not sum to 1', i); end disp(i) end 111 Appendix E – Code for Matching Modulus and AE Hits Data %% Matching modulus to entropy and features at AE hits clear all clc % Load AE features and modulus data load('C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A1619212223242526_hitdata_061516.mat'); load('C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\modulus_062216.mat'); % Already in loaded data % testsStr = {'5A16','5A19','5A21','5A22','5A23','5A24','5A25','5A26'}; % testNumber = [16,19,21,22,23,24,25,26]; % numOfTests = length(testNumber); % crackInitTime = [4200 1582 4950 3130 6329 3420 3638 5822]; %s % crack025mmTime = [4260 1697 5612 3145 7276 3505 3682 6067]; %s % crack1mmTime = [4410 1757 6343 3231 8099 3701 3939 6222]; %s % peakTimeMTS = [10.36 10.36 10.44 10.35 10.39 10.35 10.37 10.40]; % peakTimeAE = [10 10.16 10.68 10.36 9.82 10.54 10.56 10.26]; % Coefficients for damage models stored in coeff % Preallocate featureEntropy = cell(numOfTests,1); updatedEntropy = cell(numOfTests,1); tempEntropy = cell(numOfTests,1); countsFromWaveformsA = cell(numOfTests,1); timeFromWaveforms = cell(numOfTests,1); % Load entropy data workspaceNames = {'C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A16_entropy_061316.mat';... 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A19_entropy_061316.mat';... 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A21_entropy_061316.mat';... 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A22_entropy_061316.mat';... 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A23_entropy_061316.mat';... 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A24_entropy_061316.mat';... 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A25_entropy_061316.mat';... 'C:\Users\Lab Admin\Documents\AAAFiles, 040816\00_Thesis\Results\5A26_entropy_061316.mat'}; for I = 1:length(testsStr) clearvars countsFromWaveforms entropy1 entropy2 maximum maxLoad time thresholdDB chooseTestInd testStr freq load(workspaceNames{I}); featureEntropy{I,1} = entropy1(:,2); updatedEntropy{I,1} = entropy2(:,2); tempEntropy{I,1} = entropy3(:,2); countsFromWaveformsA{I,1} = countsFromWaveforms(:,2); timeFromWaveforms{I,1} = time; end % Concatenate AE feature data 112 hitdata = {hitdata_5A16_mat;hitdata_5A19_mat;hitdata_5A21_mat; hitdata_5A22_mat;hitdata_5A23_mat;hitdata_5A24_mat;hitdata_5A25_mat;hitdata_ 5A26_mat}; labels = {'Time','Load','Exten','Risetime','Counts','Duration','Amplitude', 'Abs Energy','Updated Entropy','Feature Entropy','Temporally-Weighted Entropy'}; % Preallocate time = cell(numOfTests,1); load = cell(numOfTests,1); exten = cell(numOfTests,1); risetime = cell(numOfTests,1); counts = cell(numOfTests,1); duration = cell(numOfTests,1); amp = cell(numOfTests,1); energy = cell(numOfTests,1); channel1 = cell(numOfTests,1); channel2 = cell(numOfTests,1); cycles = cell(numOfTests,1); changeInModAtHit = cell(numOfTests,1); damage = cell(numOfTests,1); cumuFeat = cell(numOfTests,1); cumuEnergy = cell(numOfTests,1); cumuCounts = cell(numOfTests,1); normCounts = cell(numOfTests,1); normEnergy = cell(numOfTests,1); normFeat = cell(numOfTests,1); normUp = cell(numOfTests,1); normTemp = cell(numOfTests,1); diffCounts = cell(numOfTests,1); diffEnergy = cell(numOfTests,1); diffFeat = cell(numOfTests,1); stopInd1 = zeros(numOfTests,1); stopInd2 = zeros(numOfTests,1); stopInd3 = zeros(numOfTests,1); stopInd4 = zeros(numOfTests,1); endD = zeros(numOfTests,1); finalCounts = zeros(numOfTests,1); finalEnergy = zeros(numOfTests,1); finalFeat = zeros(numOfTests,1); devCounts = zeros(numOfTests,1); devEnergy = zeros(numOfTests,1); devFeat = zeros(numOfTests,1); for i = 1:numOfTests counter1 = 1; counter2 = 1; % Divide hit data into channel 1 and channel 2 for j = 1:length(hitdata{i}) if hitdata{i}(j,4) == 1 channel1{i}(counter1,:) = hitdata{i}(j,:); counter1 = counter1 + 1; else channel2{i}(counter2,:) = hitdata{i}(j,:); counter2 = counter2 + 1; end end % Get rid of hits that don't match up between entropy code and hit code hitLength2 = length(channel2{i}); waveformLength2 = length(updatedEntropy{i,1}); if hitLength2 > waveformLength2 113 channel2{i}(end,:) = []; elseif hitLength2 < waveformLength2 featureEntropy{i}(end) = []; updatedEntropy{i}(end) = []; tempEntropy{i}(end) = []; countsFromWaveformsA{i}(end) = []; timeFromWaveforms{i}(end) = []; end time{i,1} = channel2{i}(:,1)-peakTimeAE(i); load{i,1} = channel2{i}(:,2); exten{i,1} = channel2{i}(:,3); risetime{i,1} = channel2{i}(:,5); counts{i,1} = channel2{i}(:,6); duration{i,1} = channel2{i}(:,7); amp{i,1} = channel2{i}(:,8); energy{i,1} = channel2{i}(:,9); cycles{i,1} = time{i,1}*5; stopInd1(i,1) = find(time{i,1}>crackInitTime(i),1); stopInd2(i,1) = find(time{i,1}>crack025mmTime(i),1); stopInd3(i,1) = find(time{i,1}>crack1mmTime(i),1); stopInd4(i,1) = length(time{i,1}); %Fracture % Find estimated modulus at AE hits % Exp model: change in modulus = a*exp(b*Cycles) endD(i) = coeff(i,1)*exp(cycles{i,1}(end).*coeff(i,2)); % Need to divide by damage value of final AE signal so the final damage % value at fracture is 1 damage{i,1} = coeff(i,1)*exp(cycles{i,1}.*coeff(i,2))/endD(i); cumuCounts{i,1} = cumsum(counts{i,1}); cumuEnergy{i,1} = cumsum(energy{i,1}); cumuFeat{i,1} = cumsum(featureEntropy{i,1}); % Normalize all AE parameters normCounts{i,1} = (cumuCounts{i,1}-cumuCounts{i,1}(1))./ (cumuCounts{i,1}(end)-cumuCounts{i,1}(1)); normEnergy{i,1} = (log10(cumuEnergy{i,1})-log10(cumuEnergy{i,1}(1)))./ (log10(cumuEnergy{i,1}(end))-log10(cumuEnergy{i,1}(1))); normFeat{i,1} = (cumuFeat{i,1}-cumuFeat{i,1}(1))./(cumuFeat{i,1}(end)- cumuFeat{i,1}(1)); normUp{i,1} = (updatedEntropy{i,1}-min(updatedEntropy{i,1}(10:end- 200)))./(max(updatedEntropy{i,1})-min(updatedEntropy{i,1}(10:end-200))); normTemp{i,1} = (tempEntropy{i,1}-min(tempEntropy{i,1}(1:end- 200)))./(max(tempEntropy{i,1})-min(tempEntropy{i,1}(1:end-200))); % Difference between actual and ideal relationships diffCounts{i,1} = normCounts{i,1}-damage{i,1}; diffEnergy{i,1} = normEnergy{i,1}-damage{i,1}; diffFeat{i,1} = normFeat{i,1}-damage{i,1}; % Sum of the total deviations devCounts(i,1) = sum(abs(diffCounts{i,1})); devEnergy(i,1) = sum(abs(diffEnergy{i,1})); devFeat(i,1) = sum(abs(diffFeat{i,1})); % Final values for all parameter finalCounts(i,1) = cumuCounts{i,1}(end); finalEnergy(i,1) = cumuEnergy{i,1}(end); finalFeat(i,1) = cumuFeat{i,1}(end); end % Coefficients of variation covCounts = std(finalCounts)/mean(finalCounts); covEnergy = std(finalEnergy)/mean(finalEnergy); covFeat = std(finalFeat)/mean(finalFeat); 114 Bibliography [1] N. 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