NOTE TO USERS This reproduction is the best copy available. UM1* ANALYTICAL MODELING FOR TRANSIENT PROBE RESPONSE IN EDDY CURRENT TESTING MODELISATION ANALYTIQUE DE REPONSE TRANSITOIRE EN CONTROLE PAR COURANT DE FOUCAULT A Thesis Submitted to the Division of Graduate Studies of the Royal Military College of Canada by Daniel Desjardins, BSc Second Lieutenant In Partial Fulfillment of the Requirements for the Degree of Master of Physics 6 May 2011 © This thesis may be used within the Department of National Defence but copyright for open publication remains the property of the author. 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The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in this et des droits moraux qui protege cette these. Ni thesis. Neither the thesis nor la these ni des extraits substantias de celle-ci substantial extracts from it may be ne doivent etre imprimes ou autrement printed or otherwise reproduced reproduits sans son autorisation. without the author's permission. In compliance with the Canadian Conformement a la loi canadienne sur la Privacy Act some supporting forms protection de la vie privee, quelques may have been removed from this formulaires secondaires ont ete enleves de thesis. cette these. While these forms may be included Bien que ces formulaires aient inclus dans in the document page count, their la pagination, il n'y aura aucun contenu removal does not represent any loss manquant. of content from the thesis. 1+1 Canada Ill ABSTRACT Thesis completed by Desjardins, Daniel, in partial fulfillment of requirements for M.Sc. in Physics from the Royal Military College of Canada on this 26 April 2011 tided Analytical Modeling for Transient Probe Response in Eddy Current Testing under the direct supervision of Dr. Thomas Krause and Dr. Napoleon Gauthier. Fatigue cracks in multi-layer aluminum structures are prone to develop around the ferrous fasteners found on ageing Canadian Forces aircraft such as the F/A-18 Hornet, CC-130 Hercules and CP-140 Aurora. Normally, fastener removal is required for inspection by conventional eddy current testing (ET), since the conducting aluminum structure limits inspection to either surface breaking or near-surface cracks when fasteners are present. In this work the fundamental principles behind the application of transient eddy current as a means of inspecting fastened structures without fastener removal is investigated. In particular, the proposal of using ferrous fasteners as conduits to carry flux deeper into the multilayer structure is examined. Analytical models were derived from solutions to Maxwell's equations using the magnetic vector potential and appropriate boundary conditions. The models described the progression of magnetic flux, generated by a step function current applied to a coil, 1) down the length of ferrous and nonferrous conducting rods, and 2) a hole in an infinite aluminum plate (borehole). Analytical results were validated against finite element models of the same configuration. The analytical models were then expanded to model actual transient probe responses, including feedback effects, for transmit-receive probe configurations using measured driving currents and physical probe dimensions. The model results predicted a significant shift of signal peaks to later times for the ferromagnetic rod as the distance between transmitter and receiver was increased. A comparison of analytical and experimentally measured receive coil responses for air and conducting nonferrous rods showed excellent agreement with model predictions. Good qualitative agreement with experimental ferrous rod results was obtained, including the predicted shift in time of peaks with distance. The results are in support of experimental observations of greater depths of penetration achievable for the inspection of multilayer aluminum structures in the presence of ferrous fasteners as compared to multilayer structures with only aluminum. This work represents the first complete analytical model of transient eddy current coil response in the presence of ferrous and nonferrous conducting rods. IV RESUME Les structures a multiples couches en aluminium sur les aeronefs vieillissants des Forces Canadiennes tels le F/A-18 Hornet, CC-130 Hercules et CP-140 Aurora sont suscepdbles de developper des fissures dues a la tension autour des rivets de fixation. En temps normal, 1'enlevement des rivets est requis afin d'effectuer l'essai par courants de Foucault convendonnels, puisque l'influence des rivets ferrugineux presents dans ces structures restreint l'inspection aux fissures superficielles ou en proximite de la surface. Cette these developpe les principes fondamentaux qui permettraient l'application de la methode d'essai par courants de Foucault transitoires sans necessiter 1'enlevement des rivets de fixation. En particulier, l'hypothese que les rivets ferrugineux accroissent et acheminent le flux magnetique a de plus grandes profondeurs a I'interieur de la structure a multiples couches est etudiee. Des modeles analytiques sont developpes a partir des equations de Maxwell en utilisant le potentiel magnetique vectoriel et des conditions frontieres appropriees. Les modeles decrivent revolution du champ magnetique, suivant l'application soudaine d'un courant dans une bobine electromagnetique, 1) au long de tiges conductrices ferreuses et non ferreuses, et 2) dans un trou perce dans une plaque infinie en aluminium (trou de forage). Les resultats analytiques sont valides par des modeles d'element fini decrivant la meme geometric Les modeles analytiques sont ensuite utilises afin de modeliser des signaux transitoires, en incluant les effets de reaction, pour des configurations emetteur/recepteur en utilisant la fonction decrivant le courant applique ainsi que les dimensions physiques de la sonde. Dans le cas d'une tige ferromagnedque, les resultats de la modelisation prevoient que les maximums des signaux seraient deplaces plus tard dans le temps a mesure que la distance entre l'emetteur et le recepteur est accrue. Une comparaison effectuee entre les signaux experimentaux et modelises dans l'air et en la presence de tiges non ferreuses demontre un bon accord. Un accord qualitatif est etabli pour les tiges ferromagnetiques, qui demontre le deplacement anticipe des maximums en fonction de la distance. Ces resultats soutiennent qu'il serait possible d'utiliser les rivets ferrugineux afin de sonder plus profondement a I'interieur des structures a multiples couches en aluminium. Ce travail represente le premier modele analytique complet decrivant la reponse transitoire d'une sonde pour l'essai par courants de Foucault en la presence de tiges ferreuses et non ferreuses. V TABLE OF CONTENTS ACKNOWLEDGEMENTS ii ABSTRACT iii RESUME iv TABLE OF CONTENTS v LIST OF TABLES viii LIST OF FIGURES ix LIST OF ABREVIATIONS xiii LIST OF SYMBOLS xiv CHAPTER 1 -INTRODUCTION 1 1.1 The Ferrous Fastener Challenge 1 1.2 Objective 4 1.3 Research Survey 4 1.4 Scope and Methodology 7 1.5 Structure 8 CHAPTER 2 - THEORY AND BACKGROUND 9 2.1 Maxwell's Field Equations 9 2.2 Material-related Magnetic Quantities 10 2.3 Material Interface 11 2.4 Quasistationary Magnetic Fields 12 2.5 Magnetic Field Diffusion and Eddy Currents 14 2.6 Electrodynamics Problems 18 2.6.1 Problem Formulation 18 2.6.2 Solution Building 20 2.7 Feedback in Coupled Magnetic Circuits 25 2.8 Duhamel's Theorem 28 CHAPTER 3 - THEORETICAL MODEL 29 3.1 Long Rod 29 3.1.1 Stationary Solution 29 3.1.2 Transient Solution 41 3.2 Borehole 53 3.2.1 Stationary Solution 53 3.2.2 Transient Solution 55 VI CHAPTER 4 - FINITE ELEMENT MODEL 59 4.1 Long Rod Comparison 59 4.1.1 Stationary Current Loop Results 59 4.1.2 Stationary Finite Coil Results 64 4.1.3 Transient Results 68 4.2 Borehole Comparison 70 4.2.1 Stationary Results 70 4.2.2 Transient Results 72 CHAPTER 5 - TRANSIENT PROBE RESPONSE MODEL 75 5.1 Theoretical Response 75 5.2 Modeled Response in Air 79 5.3 Modeled Response of an Aluminum Rod 82 5.4 Modeled Response of a Brass Rod 84 5.5 Modeled Response of a Steel Rod 86 CHAPTER 6 - EXPERIMENTAL WORK 89 6.1 Experimental Technique 89 6.2 Long Rod 93 6.2.1 Air 94 6.2.2 Aluminum 97 6.2.3 Brass 101 6.2.4 Steel 104 6.3 Bored Aluminum Plate 109 6.4 Rod & Bored Plate 113 CHAPTER 7 - ANALYSIS AND DISCUSSION 117 7.1 Parameters Affecting Voltage Response 117 7.2 Channelling of Magnetic Flux 121 7.3 Electromagnetic Interactions in the Borehole 122 7.4 Dual Couple Diffusion Process 123 7.5 TEC and the Ferrous Fastener Challenge 124 CHAPTER 8 - CONCLUSION 127 CHAPTER 9 - FUTURE WORK 129 9.1 Complete Ferrous-Fastener Geometry 129 9.2 Calculating Feedback 130 REFERENCES 135 ANNEX A 137 vii ANNEX B 139 ANNEX C 141 ANNEX D 143 CURRICULUM VITAE 145 Vlll LIST OF TABLES Table 1: Sample Eigenvalues for A — 0, 10, 20 and 400, when^r = 66 and r0 = 4.8 mm 47 Table 2: Physical dimensions of the transmit and receive coils 141 Table 3: Inner wing intermediate spar lower cap specifications (source: F/A-18 Fleet Management repport) 143 IX LIST OF FIGURES Fig.