Iowa State University Capstones, Theses and Retrospective Theses and Dissertations Dissertations 1990 Theory for the eddy current characterization of materials and structures: applications to nondestructive evaluation and geophysics Satish M. Nair Iowa State University Follow this and additional works at: https://lib.dr.iastate.edu/rtd Part of the Applied Mechanics Commons, Electrical and Computer Engineering Commons, and the Geophysics and Seismology Commons Recommended Citation Nair, Satish M., "Theory for the eddy current characterization of materials and structures: applications to nondestructive evaluation and geophysics " (1990). Retrospective Theses and Dissertations. 9870. https://lib.dr.iastate.edu/rtd/9870 This Dissertation is brought to you for free and open access by the Iowa State University Capstones, Theses and Dissertations at Iowa State University Digital Repository. 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Ml 48106-1346 USA 313/761-4700 800 521-0600 Order Number 9110648 Theory for the eddy current characterization of materials and structures: Applications to nondestructive evaluation and geophysics Nair, Satish M., Ph.D. Iowa State University, 1990 UMI 300N.ZeebRd. Ann Aibor, MI 48106 Theory for the eddy current characterization of materials and structures: Applications to nondestructive evaluation and geophysics by Satish M. Nair A Dissertation Submitted to the Graduate Faculty in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY Department: Engineering Science and Mechanics Major Engineering Mechanics Approved: Signature was redacted for privacy. In Charge of Major Work Signature was redacted for privacy. For the Major Department Signature was redacted for privacy. the Graduate College Iowa State University Ames, Iowa 1990 ii TABLE OF CONTENTS Page GENERAL INTRODUCTION 1 Explanation of Dissertation Format 14 PART I. ELECTROMAGNETIC INDUCTION (EDDY CURRENTS) IN A CONDUCTING HALFSPACE IN THE ABSENCE AND PRESENCE OF INHOMOGENEITIES - A NEW FORMALISM 15 ABSTRACT 16 I. INTRODUCTION 17 IL FORMULATION OF BOUNDARY-VALUE PROBLEM 22 III. INTEGRAL REPRESENTATIONS FOR THE FIELDS DUE TO CURRENT SOURCES IN TWO CONJOINED CONDUCTING HALFSPACES 31 IV. CLOSED FORM EXPRESSIONS IN THE QUASISTATIC APPROXIMATION 38 V. INTEGRAL EQUATIONS FOR A 3D INHOMOGENEITY BSf A HALFSPACE 42 VL SMALL VARIATIONS IN CONDUCnVTTY 51 Vn. DISCUSSION AND SUMMARY 64 Vni. ACKNOWLEDGEMENTS 66 IX. REFERENCES 67 X. APPENDK A: DYADIC COMPONENTS FOR THE TENSOR GREEN'S FUNCTIONS 69 XI. APPENDIX B: SECOND-ORDER BORN APPROXIMATION FOR THE IMPEDANCE CHANGE DUE TO A CYLINDRICAL INHOMOGENEITY 73 iii PART IL LOW FREQUENCY ASYMPTOTICS FOR EDDY CURRENTS IN A CONDUCTING HALFSPACE IN THE ABSENCE AND PRESENCE OF INHOMOGENEITIES 75 ABSTRACT 76 I. INTRODUCTION 77 II. PRELIMINARIES 82 A. Introduction of the Problem 82 B. Current Source above a Homogeneous Conducting Halfspace 85 C. Current Source above a Halfspace Containing an Inhomogeneity 89 III. LOW FREQUENCY ASYMPTOTICS FOR THE ELECTRIC FIELD FIELD INSIDE A HOMOGENEOUS CONDUCTING HALFSPACE 93 rV. LOW FREQUENCY ASYMPTOTICS FOR THE ELECTRIC FIELD INSIDE A HALFSPACE CONTAINING AN INHOMOGENEITY 99 V. METHOD OF SOLUTION FOR THE LOW FREQUENCY ASYMPTOTICS 106 VI. LOW FREQUENCY ASYMPTOTICS FOR THE IMPEDANCE CHANGE 111 A. Examples where Low Frequency Fields are given by the Bom Approximation 113 1. Circular current loop over a spherical inhomogeneity in a conducting halfspace 113 2. Circular current loop over a circular cylindrical inhomogeneity 114 3. Uniform current sheet / Uniform field probe above a 2D / ID inhomogeneity in a halfspace 114 B. Examples where Low Frequency Fields ^ Bom Approximation 116 1. Semi-oblate spheroidal pit breaking the surface of a conducting halfspace 118 2. Semi-prolate spheroidal pit breaking the surface of a conducting halfspace 121 Vn. CONCLUSIONS AND SUMMARY 124 iv Vm. ACKNOWLEDGEMENTS 126 IX. REFERENCES 127 X. APPENDIX: ILLUSTRATION OF THE LOW FREQUENCY BEHAVIOR OF THE ELECTRIC FIELD 129 A. Current Source above a Homogeneous Conducting Halfspace 130 1. Circular current loop 130 2. Uniform current sheet 131 B. Current Source above a Conducting Halfspace Containing an InhomogeneiQr 132 1. Circular current loop over a conducting halfspace containing a spherical inhomogeneity 132 2. Unifomi current sheet over a conducting halfspace containing a circular cylindrical inhomogeneity 137 3. Uniform current sheet over a single-layered conducting halfspace 141 PART III. RECONSTRUCTION OF THREE- DIMENSIONAL CONDUCTIVITY VARIATIONS FROM EDDY CURRENT (ELECTROMAGNETIC INDUCTION) DATA 144 ABSTRACT 145 I. INTRODUCTION 146 II. PRELIMINARIES 151 III. INVERSION FOR A 3D VARIATION IN CONDUCTIVITY - FREQUENCY DOMAIN 157 IV. INVERSION FOR A 3D VARIATION IN CONDUCTT/ITY- TIME DOMAIN 164 V. NUMERICAL INVERSION OF THE LAPLACE TRANSFORM 172 VI. SPEOAL CASES: UNIFORM CURRENT SHEETS AND LAYERED HALFSPACES 176 A. Inversion Formulae for Uniform Current Sheets 177 B. Inversion for a Uniform Current Sheet over a Layered Halfspsce 178 C. Numerical Implementation of the ID Algorithm 181 Vn. CONCLUSIONS AND SUMMARY 193 Vni. REFERENCES 195 IX. APPENDIX: INVERSION FORMULAE FOR CERTAIN FREDHOLM INTEGRAL EQUATIONS OF THE FIRST KIND 197 GENERAL CONCLUSIONS 201 A. Highlights of Dissertation 201 B. Future Extensions 205 BIBLIOGRAPHY 209 ACKNOWLEDGEMENTS 213 1 GENERAL INTRODUCTION Eddy currents generated in conducting materials by electromagnetic induction have been used for various science and engineering applications. One such application arises in the area of nondestructive evaluation (NDE) where they are used to detect flaws in metal structures. Although flaw detection by means of eddy currents has existed for about half a century, flaw characterization has essentially been performed through a catalogue of eddy current responses from experimental models. The theoretical modeling of eddy current responses has only been carried out recentiy and has established a firm foundation for the quantitative characterization of defects. This dissertation establishes algorithms for the recovery of flaw parameters, viz., its type, shape, size, location, etc., from the direct inversion of the eddy current signal resonses. At the same time, the dissertation also enhances the existing research on the forward eddy current response for various probe-flaw models and establishes theories leading to the better understanding of the electromagnetic induction phenomenon in flawed metal structures. The generality of the flaw models permits wide application, ranging from the NDE of defects to the determination of material properties and to exploratory geophysics (where eddy currents are used to investigate and locate conductivity anomalies within the the crust of the earth). Eddy current testing was first introduced in NDE by Forster and Breitfeld [1] and is typically carried out through an eddy current probe, driven by a time-harmonic a.c. current source. See Figure 1. The eddy current probe usually consists of a coil wound in air, or around a ferrite core, placed over the flawed conducting structure. The time-varying Impedance A. C. Source Analyzer Probe Crack Test Plate Figure 1: Schematic diagram of an eddy current setup for NDE 3 magnetic flux generated by the eddy current probe induces eddy currents within the conducting structure through Faraday's law. Changes in the electrical and magnetic properties within the material disturb the flow of eddy currents and affect the electromagnetic fields within the conducting structure. These changes are generally registered indirectly as a change in the probe impedance and displayed through an impedance plane diagram in an oscilloscope. Both, coupled circuit theory [2] and reciprocity [3] have been used to relate this change in probe impedance to the electric and magnetic fields existing within the conducting structure.