Scalar potential model of eddy-current interactions with three-dimensional flaws

Magnetic scalar potential theory is applied to a model of eddy-current detection of a surface-breaking flaw in a conductor. A general boundary integral equation for the potential is derived first in a form suitable for numerical solution. The problem is then specialized to a flaw in a perfectly conducting half-space. A solution algorithm based on the boundary element method is outlined and demonstrated by application to a three-dimensional rectangular slot. Methods for accounting for the effects of nonvanishing skin depth are discussed.     

The volume integral method of eddy-current modeling: Verification

The volume integral method of eddy-current modeling represents a flaw in metal as a set of electric dipoles located within volume elements or cells defining the flaw volume. Given this dipole distribution, impedance changes may be computed. The electric field of the dipole distribution is determined by an integral equation relating, by means of the electric field Green's tensor, the electric field due to the source to the total electric field in the flaw. The integral equation is solved by assuming that the total electric field is constant in each volume element, resulting in a matrix equation. The method has been programmed for use on a microcomputer. The method and computer program are verified using the analytical solution for a small spherical flaw and three sets of measured impedance data, measured by air-core coils along profiles overlying both surface-breaking and buried simulated flaws of known dimensions. Operating frequencies ranged between 900 and 4000 Hz. Generally agreement is good at lower frequencies ( 1000 Hz). At higher frequencies ( 4000 Hz), the agreement is not as good. This is thought to be due to the inability of the constant electric field approximation to model the steep electric field gradients present in the host metal at high frequency. The results are also sensitive to the method of computation of the electric field due to the source. Some improvements can and should be made to the method.     

A hybrid finite element/method of moment formulation for singlefrequency eddy current inversion

Eddy-current inverse techniques using single-frequency currents have been applied with limited success to the reconstruction of crack width and thickness profiles primarily for one-dimensional and axisymmetric geometries. Because of the diffusive nature of the induced low-frequency eddy currents, the reconstruction process differs from high-frequency wave propagation methods. On the physical basis that both diffusive and wave phenomena can be described by the same Green's function with either a complex or real wave number, an integral formulation for the low-frequency magnetic vector potential is presented. By employing an iterative Born approximation algorithm and the method of moments, a reconstruction method for the conductivity profile in a metallic specimen is developed. To make this formulation amenable to complex geometries, finite-element analysis techniques are utilized to compute the integral kernel. The inversion process is tested with synthetic data generated by the numerical solution of a generic embedded flaw in a full-space and a surface-breaking defect     

Approximate model of eddy-current probe impedance for surface-breaking flaws

A theoretical model is derived for the prediction of eddy-current probe impedance changes caused by three-dimensional, surface-breaking flaws. Magnetic scalar potential theory and the surface impedance approximation are used to calculate fields on the flaw surface for arbitrary probe position and flaw geometry. Impedance changes are determined by a first-order perturbation calculation, with skin depth being the perturbation parameter. The end result is a relatively simple, three-dimensional model for simulating an eddy-current inspection. Numerical results for rectangular slots include maps of the impedance signals obtained in raster scan patterns and studies of skin-depth effects as a function of probe size, lift-off, and flaw dimensions.     

Techniques for depth-selective,low-frequency eddy current analysis for SQUID-based nondestructive testing

Conventional eddy current techniques are widely used for detection of surface-breaking cracks in metal structures. These techniques have limited success in the detection of deep, nonsurface-breaking flaws that require low frequency eddy currents, for which inductive pick-up probes have drastically reduced sensitivity. High resolution, Superconducting QUantum Interference Device (SQUID) magnetometers, which are very sensitive to do or low frequency magnetic fields, have been developed for detection of subsurface flaws. We have now extended SQUID NDE by utilizing a sheet inducer to produce an extended eddy current parallel to the surface in a conducting plate. The magnitude of the induced current density inside the plate reduces with the depth; however, the current component at a certain phase angle may increase with the depth. At a particular phase angle, the current density on the surface becomes zero, while the current inside the plate is large, so that the magnetic signal at that phase angle due to the surface structures can be minimized. With this method, we have detected simulated cracks in the sides of plugged holes in a thick plate, a hidden corrosion area in a specimen which consisted of two painted aluminum plates joined with sealant, as well as crack defects adjacent to fasteners in the second layer of lap joined aluminum plates. We present a theoretical model for simulation of the phase-related magnetic signal due to a flaw, which shows the relation between the phase angle and the depth of the flaw. The theoretical phase analysis is compared with the experimental results.     

A Novel Multi-Probe Resistivity Approach to Inspect Green-State Metal Powder Compacts

This paper describes a new instrumentation approach to the nondestructive testing of green-state powdered metallurgy components. These samples are likely to generate surface-breaking and subsurface defects prior to sintering. Exploiting the principles of electric resistivity or potential drop measurements in solids, a system is configured which is capable of recording surface voltage distributions due to impressed current inputs. At the heart of this novel testing procedure is a multiple-pin sensor which allows for flexible measurement conditions in order to cover wide surface areas. Practical tests with production samples compare well with both analytical and numerical modeling techniques in predicting surface voltage distributions. Furthermore, initial studies of surface-breaking flaws exhibit excellent defect detectability.     

Numerical analysis of the acoustic signature of a surface-breaking crack

The boundary element method is used to calculate the acoustic signature, produced by a line focus scanning acoustic microscope, of an elastic object containing a surface-breaking crack. The acoustic signature has a vertical (z) and horizontal (x) dependence. A model of the microscope developed earlier is used and extended to take account of the crack. The mathematical formulation of the scattering problem for the cracked object leads to hypersingular integral equations. A suitable technique is employed to solve such equations by the boundary element method. An electromechanical reciprocity identity is used to relate the received voltage to the acoustic wavefields collected by the lens. The acoustic wavefield scattered from the cracked object is investigated, and curves are presented that display the acoustic signature, as functions of (x ,z), for cracks of various depths and orientations. A method to measure the depth of a surface-breaking crack using the acoustic signature is suggested.     

Electrostatic tractions at crack faces and their influence on the fracture mechanics of piezoelectrics

The influence of electrostatic tractions acting upon crack faces on the fracture mechanical quantities in piezoelectric materials under electromechanical loading is investigated. The physical background are the mechanical and dielectric equilibria at an interface between two dielectric domains and related mechanical stresses. The model is applied to a crack problem, where a dielectric interface exists between the solid material and the insulating crack medium. The analytical solution for a crack in an infinite piezoelectric body accounting for intrinsic charges and electrostatic stresses on the crack faces gives insight into the influence of crack boundary conditions on the field intensity factors. Varying loading conditions and the dielectric permittivity of the flaw yields a parameter range in which induced crack surface tractions are relevant. Then, the calculation of the J-integral for thermodynamically consistent crack boundary conditions is discussed. The line integral along the crack faces is replaced by a simple jump term. This approach comes out to be exact only for a simplified model of the electrostatic tractions.     

Modeling of Conductivity Versus Density Behavior in Green-State Powder Metallurgy Compacts

Ongoing research at Worcester Polytechnic Institute (WPI) has recently resulted in the development of an electrostatic multipin instrument capable of testing green-state compacts directly after compaction. By monitoring a steady electric current flow through the sample and recording the voltages over the surface valuable information is gathered, leading to the prediction of the structural health of the green-state parts. Whereas our prior work concentrated on the detection of surface-breaking and subsurface defects, which requires the determination of large differences in material properties over small flaw sizes, the results presented in this paper aim at the density prediction throughout the volume of the sample. This requires the detection of small changes in material properties over large regions. A physical model and a mathematical formulation are reported, which are capable of relating green-state density changes to electric conductivity in the presence of various lubricant concentrations. Preliminary electrostatic measurements of cylindrical compacts have thus far confirmed the theoretical model assumptions, showing that the electric conductivity follows a complex graphical behavior that is determined by the type and concentration of the lubricant. Specifically, the green state conductivity increases as the sample density increases up to values of approximately 6.9 to 7.0 g/cm³. Any further density increase, however, results in a decrease in conductivity.     

Eddy-current signal analysis and inversion for semielliptical surface cracks

A scalar potential formulation of the Z formula for the change in impedance of an eddy-current probe caused by a surface-breaking flaw is developed. The resulting formula is evaluated using a finite-difference method, which permits calculation of Z for semielliptical flaws. The numerical results are checked by comparing calculations for rectangular-shaped flaws to previous calculations using an analytical solution for this geometry. Theoretical results are then verified by comparison with measurements on semielliptical fatigue cracks and EDM notches in aluminum alloy specimens using air-core eddy-current probes. An inversion method that compares features of the flaw profile, obtained by scanning the eddy-current probe along the length of the flaw, to a theoretical inversion chart (McFetridge chart) is demonstrated using the experimental data.     

Analytical versus boundary element modelling of horizontal ground electrode

Different approaches to the analysis of horizontal ground electrode, based on the wire antenna theory have been used in this work. The spatial distribution of the induced current along the electrode is obtained by solving the homogeneous Pocklington integro-differential equation. The spatial distribution of the scattered voltage along the electrode is computed using the generalized telegrapher's equation. The assessment of current and voltage induced along the electrode is carried out using both analytical and numerical approaches. The numerical solution of integral expressions arising from the antenna theory model is numerically handled via the Galerkin–Bubnov scheme of the Indirect Boundary Element Method (GB-IBEM). Some illustrative results for the current and voltage induced along the horizontal electrode are given in the paper.     

A New Approach for Restoration of Eddy Current Images

Eddy current images of defects are blurred due to convolution of point spread function of eddy current probe with defects. Disturbing variables such as lift-off, surface roughness, and material property variations influence the eddy current images. In order to restore the length, width, depth, and orientation of surface-breaking defects in the presence of disturbing variables, a new and comprehensive approach has been developed. This approach uses artificial neural network and image processing methods. Studies on austenitic steel plates confirm that through this approach it is possible to restore the spatial information of surface-breaking defects of uniform or slowly varying depth and also to form their accurate three-dimensional pictures. This approach is fast as well as amenable for automation.     

Slow torsional oscillations: The reissner-sagoci and crack problems at low frequencies

The axisymmetric elastic field produced by slowly forced torsional oscillations of a finite circular fiaw in a certain inhomogeneous medium is sought. The problem is reduced to a system of integral equations, which system is shown to cover intrinsically, the two separate cases of the flaw in the form of a penny-shaped crack and the flaw in the form of a rigid disc. The solutions are given in series of the frequency factor. Estimates of the radius of convergence are given as functions of the inhomogeneity parameter. For the flaw in the form of a rigid disc, the solution of the integral equations gives the normal displacement gradient just above the disc, from which simple integration gives the moment of the applied forces necessary to oscillate the disc. In the case of the flaw in the form of a penny-shaped crack, the solution gives the normal displacement over the crack region. This is then used to obtain the surface shear stress just outside the crack rim, from which is obtained the stress intensity factor. These physical results are all given as functions of the frequency factor and inhomogeneity parameter.     

Integral equation formulation for reflection by a mirror

When light is incident on a mirror, it induces a current density on its surface. This surface current density emits radiation, which is the observed reflected field. We consider a monochromatic incident field with an arbitrary spatial dependence, and we derive an integral equation for the Fourier-transformed surface current density. This equation contains the incident electric field at the surface as an inhomogeneous term. The incident field, emitted by a source current density in front of the mirror, is then represented by an angular spectrum, and this leads to a solution of the integral equation. From this result we derive a relation between the surface current density and the current density of the source. It is shown with examples that this approach provides a simple method for obtaining the surface current density. It is also shown that with the solution of the integral equation, an image source can be constructed for any current source, and as illustration we construct the images of electric and magnetic dipoles and the mirror image of an electric quadrupole. By applying the general solution for the surface current density, we derive an expression for the reflected field as an integral over the source current distribution, and this may serve as an alternative to the method of images.     

Algorithms for the Magnetic Assessment of Proton Exchange Membrane (PEM) Fuel Cells

An electromagnetic flaw model and imaging techniques are developed simulating the magnetic nondestructive evaluation of proton exchange membrane fuel cells. A small flaw model is introduced to simulate the perturbation in magnetic field due to pinholes in the membrane. An inversion scheme is demonstrated to reconstruct the ionic current distribution in the membrane. Methods of stray field removal are then discussed. The research objectives of the above techniques are to locate flaws and enable the determination of current density in the fuel cell membrane in the presence of stray fields produced by electrodes, current leads, and background noise sources.     

USE OF THE DISTRIBUTED DISLOCATIONS METHOD TO DETERMINE THE T-STRESS

Abstract— This paper demonstrates a method to determine the elastic T -stress for a semi-infinite half-plane containing a surface-breaking crack which is loaded by an arbitrarily distributed far-field tension. The method consists of representing the crack by a continuous distribution of edge dislocations and forming singular integral equations to determine the equilibrium dislocation distributions. By numerically solving the integral equations, stress intensity factors and T -stresses are obtained for the example case of a crack which is normal and inclined to the free-surface of a half-plane and loaded by a uniform far-field tension.     

Stress intensity factors for a surface-breaking crack in a half-plane subject to contact loading

Modes I and II stress intensity factors are derived for a crack breaking the surface of a half-plane which is subject to various forms of contact loading. The method used is that of replacing the crack by a continuous distribution of edge dislocations and assume the crack to be traction-free over its entire length. A traction free crack is achieved by cancelling the tractions along the crack site that would be present if the half-plane was uncracked. The stress distribution for an elastic uncracked half-plane subject to an indenter of arbitrary profile in the presence of friction is derived in terms of a single Muskhelishvili complex stress function from which the stresses and displacements in either the half-plane or indenter can be determined. The problem of a cracked half-plane reduces to the numerical solution of a singular integral equation for the determination of the dislocation density distribution from which the modes I and II stress intensity factors can be obtained. Although the method of representing a crack by a continuous distribution of edge dislocations is now a well established procedure, the application of this method to fracture mechanics problems involving contact loading is relatively new. This paper demonstrates that the method of distributed dislocations is well suited to surface-breaking cracks subject to contact loading and presents new stress intensity factor results for a variety of loading and crack configurations.     

Ultrasonic imaging by leaky Rayleigh waves

Leaky Rayleigh waves are used for the ultrasonic imaging of surface-breaking or near-surface defects. The phenomena which take place when a leaky Rayleigh wave propagates and interacts with a flaw are described in order to explain both how the ultrasonic image is built using a single transducer and what the main characteristics of these images are. In particular, the influence of the ultrasonic frequency is investigated in detail.     

Boundary Element Calculation of Eddy Currents in Cylindrical Structures Containing Cracks

Electromagnetic fields due to excitation coils in boreholes containing cracks have been calculated using a boundary element method formulated in terms of a scalar decomposition of the field. By using a Green's kernel that ensures continuity of the tangential electric and magnetic field at the cylindrical surface, it is only necessary to render the flaw region in a discrete form. The effect of an ideal crack, one with zero opening but acting as a perfect barrier to eddy currents, is here represented by a current dipole layer orientated normal to the crack surface. An integral equation determines the source density on the crack and a numerical solution is found using boundary elements. The coil impedance change due to the crack is then computed using an expression based on a reciprocity principle.      

Infrared Detection of Defects in Powder-Metallic Compacts

Electric Joule heating of low-conductivity media through direct current excitation can be used to generate a temperature profile throughout a powdermetallic (P/M) compact. When recording the surface temperature distribution with an infrared (IR) camera important information regarding the integrity of the sample can be gained. Unlike most existing IR techniques, this research concentrates on obtaining the temperature distribution and heat flow behavior in P/M parts when they are subjected to active electric current injection. The practical measurements are supported and complemented by a simple theoretical model that serves as a calibration tool to aid in the evaluation of the infrared signatures that are recorded over the sample surface and correlated with the detection of surface and subsurface flaws. In this paper we will report on the design of the active infrared detection system and a basic theoretical testbed that is suitable for calibration. Specifically, we state the governing equations and boundary conditions, followed by devising numerical solutions that enable a comparison to the measured thermal response. In addition, the numerical modeling approach can also serve as a method to model various flaw sizes and orientations in an effort to practically determine flaw resolution limits as a function of minimally detectable temperature distribution. Preliminary measurements with controlled and industrial samples indicate that this IR testing methodology can successfully be employed to inspect both green-state and sintered P/M compacts.