A boundary integral equation method for an inverse problem related to crack detection

This paper discusses an application of a boundary integral equation method (BIEM) to an inverse problem of determining the shape and the location of cracks by boundary measurements. Suppose that a given body contains an interior crack, the shape and the location of which are unknown. On the exterior boundary of this body one carries out measurements which are interpreted mathematically as prescribing Dirichlet data and measuring the corresponding Neumann data, or vice versa, for a field governed by Laplace's equation. The inverse problem considered here attempts to determine the geometry of the crack from these experimental data. We propose to solve this problem by minimizing the error of a certain boundary integral equation (BIE). The process of this minimization, however, is shown to require solutions of certain are proposed. Several 2D and 3D numerical examples are given in order to test the performance of the present method.     

The detection and measurement of symmetric corner cracks by the a.c. field method

An unfolding method, developed earlier for cracks in plane surfaces, is extended in this work to provide an approximate mathematical model for the perturbations produced by a symmetrical corner crack when interrogated by a surface electric field. The model disregards the nonuniform nature of the current distribution upstream of the crack which is produced by the presence of the corner, and this simplification leads to unfolded field problems of plane Laplacian form. The Schwarz-Christoffel transformation is applied here in their solution. Experimental measurements on slots cut on steel blocks to simulate corner cracks are found to be in good agreement with the predictions of the model.     

Analysis technique for interaction of high-frequency rhombicinducer field with cracks in metals

In the nondestructive evaluation (NDE) involving the ac field measurement (ACFM), a current carrying structure is required to induce the eddy current in the work-piece and a probe to sample the field. Due to its flat profile, slender shape, and other advantages, the rhombic wire loop is a suitable inducer for developing linear flexible arrays for the ACFM inspection of large surfaces of ferrous and nonferrous metals. This paper introduces an analysis technique for the evaluation of the interaction of the field of the rhombic inducer carrying a high-frequency current, with long surface cracks of uniform depth in flat metal plates. The technique is accurate and very efficient computationally. It uses the two-dimensional Fourier transform together with a special boundary condition at the metal surface. The boundary condition takes into account the thin-skin nature of the eddy current in the metal as well as the flux leakage at the crack mouth. The analysis technique benefits from the use of scalar potential functions and can be extended to simply or multiply connected wire loops. Also, it is applicable to high-frequency (thin-skin) eddy current problems. Using the analysis technique, the tangential field below a rhombic inducer along its long diagonal when the inducer is located above the surface of aluminum and steel is given in the presence and absence of a crack. This field was found to have a nonuniform phase distribution. Near a crack, the phase change is significant, even for shallow cracks. The role of the nonuniform phase in the detection sensitivity is addressed. Also, simulated ACFM crack responses using an inducer with a linear probe attached along the long diagonal are presented and discussed. To support the validity of the analysis technique, experimental results obtained for some of the simulations are also reported. In addition to its application in predicting crack responses, the technique can be used for model-based inversion of crack signals     

Vector potential integral formulation for eddy-current proberesponse to cracks

A boundary integral vector potential formulation has been developed to evaluate eddy-current interactions with three-dimensional finite cracks in conductors. The approach is compared with an electric field integral equation method also used for solving crack problems in eddy-current nondestructive evaluation. An important advantage of the vector potential integral formulation is that the kernel has a weak singularity, but a drawback is that two unknown functions must be found on the crack surface. One of these functions, the current dipole density, represents the effect of the crack in terms of an induced source, and the other function is a solution of the two-dimensional Laplace equation. By contrast, the source density alone is needed for a complete solution of the electric field integral equation. In order to determine the surface Laplacian for finite cracks of arbitrary shape, a general numerical solution utilizing the boundary element technique is introduced. Numerical predictions of the eddy-current probe response to a crack give good agreement with experimental measurements, supporting the validity of the formulation     

Considerations in the Validation and Application of Models for Eddy Current Inspection of Cracks Around Fastener Holes

To improve the detection and characterization of cracks around fastener holes in multilayer structures without removing the fastener, model-based approaches are proposed to support the design of advanced eddy current (EC) NDE systems. This work demonstrates the validation and application of models to simulate EC inspection as part of the design process. The volume integral method (VIM) and finite element method (FEM) are both used to simulate eddy current inspection of fastener sites for fatigue cracks. Convergence studies, validation with existing models, experimental validation studies and validation through inverse method demonstrations are presented, providing a continuum of methods to ensure the quality of measurement models. Consideration concerning convergence and validation is also given with features sensitive to the sample geometry and flaw characteristics. A novel calibration technique is also presented to practically evaluate the transformation between model-based impedance calculations and experimental voltage data. A series of studies are presented concerning the detection of cracks around fastener holes demonstrating the quality of the simulated data to represent experimental measurements.     

Finite-Element Analysis for Remote Field Eddy Current Responses from Near- and Farside Cracks

Results of three-dimensional finite-element analysis investigations of remote field eddy current (RFEC) signal responses from internal and external stress corrosion cracking (SCC) in steel pipes are presented. The fine shallow SCC cracks were simulated by double axial slits located on the near- and farside pipe surfaces under simulated RFEC excitation. Although it is normally a characteristic of RFEC testing that responses to interior or exterior defects are approximately equal, there is a considerable difference between the farside and nearside responses from very fine defects such as SCC cracks.     

Characterizing the performance of eddy current probes using photoinductive field-mapping

We present a new method for characterizing the performance of eddy current probes by mapping their electromagnetic fields. The technique is based on the photoinductive effect, the change in the impedance of an eddy current probe induced by laser heating of the material under the probe. The instrument we developed maps a probe's electric field distribution by scanning an infrared laser beam over a thin film of gold lying underneath the probe. Measurements of both photoinductive signals and flaw signals for a series of similar probes demonstrate that the impedance change caused by an electrical-discharge-machined (EDM) notch or a fatigue crack is proportional to the strength of the photoinductive signal. Thus, photoinductive measurements can supplant the use of artifact standards to calibrate eddy current probes. Furthermore, the shape and symmetry of the probe's field pattern can reveal defects in probe construction. By combining photoinductive measurements of a probe's field strength with a theoretical model, we are able to quantitatively predict the probe's performance under hypothetical conditions. To model commercial eddy current probes with ferrite cores, we developed a procedure to treat them as effective air-core probes. We obtained good agreement between the flaw signals calculated using this effective-coil approach and actual fatiguecrack signals measured with commercial probes. We also calculated probabilities of detection for target flaws in titanium alloys for a series of commercial probes. The results reveal how probe sensitivity can affect the reliability of an eddy current inspection.This article is dedicated to Professor Bertram A. Auld on the occasion of his 70th birthday and his retirement from Stanford University.     

Visualization and size estimation of fiber waviness in multidirectional CFRP laminates using eddy current imaging

This paper presents an eddy current method to visualize fiber waviness in multidirectional carbon fiber reinforced plastics. Since eddy currents induced by a driver coil flow along carbon fibers, waviness can be visualized if the eddy current path is visualized. We proposed a new complex plane analysis method to visualize an eddy current path from magnetic field measurements. The validity of the method was verified by finite element method analyses. Experiments were performed for multidirectional carbon fiber reinforced plastic specimens. In-plane waviness with misalignment angle from 6.9° to 24.9° were artificially induced in the specimens. An eddy current path was visualized from magnetic field data and it corresponded to the shape of induced waviness. The sizes of the wavy eddy current path were compared with waviness sizes measured by X-ray computed tomography and from optical images. Experimental results indicate that the surface waviness size can be estimated accurately, whereas the subsurface waviness size is underestimated.     

Boundary integral equation methods for two-dimensional models of crack-field interactions

An introduction to the application of surface integral equation methods to the calculation of eddy current-flaw interactions is presented. Two two-dimensional problems are presented which are solved by the boundary integral equation method. Application of collocation methods reduces the problems to systems of linear algebraic equations. The first problem is that of a closed surface crack in a flat slab with an AC magnetic field parallel to the plane of the crack. The second is that of av-groove crack in the AC field of a pair of parallel wires placed parallel to the vertex of the crack. In both cases, maps of the current densities at the surface are displayed, as well as the impedance changes due to the cracks.     

Review of Advances in Quantitative Eddy Current Nondestructive Evaluation

A comprehensive review of advancements in eddy current (EC) modeling is presented. This paper contains three main sections: a general treatise of EC theory, the thin skin EC forward modeling, and the EC inverse problem. (1) The general treatise of eddy current theory begins with an exposition of the reciprocity formulas for evaluating probe impedance changes, which are derivable from first principles. Two versions of the reciprocity formulas, one with a surface integral and the other with a volume integral, are given. Any particular type of defect, as well as both one-port and two-port probes, can be treated. Second, a brief account of analytical and numerical methods for calculating the field distributions is presented. Third, theory of probe/material interactions with various defect types is described. (2) The paper then proceeds to the forward modeling section, which contains a detailed treatment of the eddy current forward problem for surface breaking cracks and EDM notches in the thin skin approximation. (3) The inverse problem section begins with a general review of commonly used inversion methods, exemplified by selected references from the literature, followed by more detailed examinations of EC inversions for surface breaking cracks and slots. The last part of this section is devoted to the inverse problem for layered structures. Although being a review in nature, the paper contains a number of new accounts for time-domain eddy current interactions. In particular, a modification is proposed to the reciprocity formula in order to take a better account of pulsed eddy current signals.     

Calculation of 3D eddy current problems using a modifiedT-Ω method

A modified T-Ω method for computing 3D eddy current problems is proposed in which the exciting current distribution in the field domain is turned into an equivalent magnetized region so that the unknowns of the computation can be reduced as much as possible. With this model an equivalent variational formulation can be derived, and the finite-element method can be used to solve some eddy current problems more efficiently. Sample calculated and test results for 3-D laminated eddy current problems are given to verify the method     

Two simple models for analytical calculation of eddy currents inthin conducting plates

Two simple models are proposed for the analytical calculation of eddy currents induced by a time-varying magnetic field in thin conducting plates of various shapes. In the first model, it is assumed that the current paths are determined exclusively by the shape of the plate. This assumption makes evident a shape factor that characterizes the shape of the plate under investigation and is independent of the dimensions of the plate. Thus, several important parameters, such as the equivalent resistance of the plate, the total circulating current, etc., are expressed analytically as a function of this shape factor. The application of the model makes possible the estimation of the total eddy current, circulating in plates of any shape, but the accuracy of this estimate is strongly affected by the shape of the plate. This model is then improved for plates having symmetrical shapes by taking into consideration the principle of minimum energy dissipation. The results, obtained by the improved model for the total circulating current, are in good agreement with the results obtained numerically as well as with analytical results obtained by the use of variational methods     

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.     

Eddy current data for characterizing less volumetric stress corrosion cracking in nonmagnetic materials

The authors gather eddy current signals due to artificial stress corrosion cracking so that the data may be available to other researchers. Three cracks are introduced into an austenitic stainless steel plate, eddy current measurements are performed, and then the plate is destroyed to observe the cross-sectional profiles of the cracks. The authors will offer the data, as well as the true profile of the stress corrosion cracking revealed by destructive tests, to anybody who are interested in utilizing it in their studies.     

Finite-element calculations of remote field eddy current interactions with slit defects

Remote field eddy current (RFEC) excitation is a promising approach for detection of the very fine axial cracks typical of stress corrosion cracking (SCC) in pipelines. Interactions between adjacent cracks or slits can enhance responses in some cases. Detailed finite-element modeling was undertaken to establish the behavior and interactions of multiple slits such as those occurring in SCC. Three different field/slit configurations are considered, with anomalous source models used to aid interpretation of the results. The study noted that magnetic perturbations generated by ferromagnetic material tend to be vanishingly small, and that the interactions between multiple cracks give minimal enhancement, indicating that eddy current rather than magnetic field excitation is best for the detection of SCC. With eddy current excitation, field perturbations are generated by even very fine slits, and are larger in non-ferromagnetic material. For nonferromagnetic pipes, the perturbations tend to merge as a circumferential separation between parallel axial cracks decreases, resulting in significant interaction and signal enhancement.     

Effective magnetization and forces due to eddy currents

A simple method for evaluating the effective magnetization due to eddy currents excited by AC magnetic fields in finite conductive objects is presented. The values of magnetization enable the estimation of the forces exerted on the object by the effect of eddy currents. The method relies on assuming a similarity between eddy current magnetization and the magnetization due to diamagnetic effects, which is easier to evaluate. Its validity is checked by comparing results from the eddy current forces with data obtained from conventional but much more complicated methods or from experimental data. The approach may be useful for evaluating the eddy current losses in finite conductive objects excited by AC magnetic fields or the influence of small conductive objects on AC excitation coils. The method combines techniques related to the conventional evaluation of eddy currents in certain 1-D geometries with techniques that approximate behavior of the AC magnetic field in finite objects in a way similar to that followed when the magnetic forces acting on a diamagnetic object are calculated     

Bolt-Hole Corner Crack Inspection Using the Photoinductive Imaging Method

We applied a laser-excited eddy current (EC) imaging technique, or so-called photoinductive (PI) imaging, to characterize corner cracks at the edge of a bolt hole. Crack images with excellent signal-to-noise-ratios were obtained. The PI signals revealed the geometrical shape of the electrical-discharge-machined (EDM) notches that were either triangular or rectangular. The results show that this technique is promising to characterize the length, as well as possibly the depth and shape, of corner cracks. In this paper we present measurement results of 0.25-mm, 0.50-mm, and 0.75-mm rectangular and triangular EDM notches. We also show measurement results of a very small notch (<0.25 mm) which would be difficult to detect with conventional eddy current techniques. The dependencies of PI signals on laser chopping frequencies and eddy current frequencies are also examined. To demonstrate the photoinductive imaging capabilities to image actual cracks, we display images of fatigue cracks grown in a Ti-6Al-4V hole specimen. Finally, we present comparisons of the photoinductive imaging results with usual eddy current images obtained from a 0.75-mm triangular EDM notch using a rotating bolt-hole scanner. This article intends to verify experimentally that the photoinductive imaging technique has a potential to become a useful nondestructive testing method.Deceased.     

Finite-Element Calculations of Remote Field Eddy Current Interactions with Slit Defects

Abstract

Remote field eddy current (RFEC) excitation is a promising approach for detection of the very fine axial cracks typical of stress corrosion cracking (SCC) in pipelines. Interactions between adjacent cracks or slits can enhance responses in some cases. Detailed finite-element modeling was undertaken to establish the behavior and interactions of multiple slits such as those occurring in SCC. Three different field/slit configurations are considered, with anomalous source models used to aid interpretation of the results. The study noted that magnetic perturbations generated by ferromagnetic material tend to be vanishingly small, and that the interactions between multiple cracks give minimal enhancement, indicating that eddy current rather than magnetic field excitation is best for the detection of SCC. With eddy current excitation, field perturbations are generated by even very fine slits, and are larger in non-ferromagnetic material. For non-ferromagnetic pipes, the perturbations tend to merge as a circumferential separation between parallel axial cracks decreases, resulting in significant interaction and signal enhancement.     

飞机多层金属板隐藏缺陷远场涡流探头设计及试验研究

由于飞机多层金属板结构厚度大、复杂等特性,现有检测方法无法发现原位内部缺陷,对于多层板金属缺陷的检测一直都是航空无损检测的难题。远场涡流检测技术因打破趋肤效应的限制,涡流能量可穿透较厚的被测试件,对金属板结构中隐藏缺陷的检测具有潜在优势。该文针对飞机多层金属板隐藏裂纹的原位检测,建立多层金属板构件隐藏裂纹平面远场涡流检测有限元仿真模型,研究不同角度、不同深度裂纹检测幅值、相位的变化规律。为验证该仿真模型的正确性,开展远场涡流检测多层金属板的试验。试验结果表明:设计开发的远场涡流探头可检测埋深13 mm的裂纹缺陷,当裂纹倾斜角度为0°时,检测灵敏度最高,当裂纹倾斜角度为90°时,检测灵敏度最低,与仿真结果保持一致,且能够对缺陷进行精准定位,可为飞机多层金属板构件隐藏裂纹的定量检测提供依据。     

A combined finite element/strip element method for analyzing elastic wave scattering by cracks and inclusions in laminates

 A new numerical technique combining the finite element method and strip element method is presented to study the scattering of elastic waves by a crack and/or inclusion in an anisotropic laminate. Two-dimensional problems in the frequency domain are studied. The interior part of the plate containing cracks or inclusions is modeled by the conventional finite element method. The exterior parts of the plate are modeled by the strip element method that can deal problems of infinite domain in a rigorous and efficient manner. Numerical examples are presented to validate the proposed technique and demonstrate the efficiency of the proposed method. It is found that, by combining the finite element method and the strip element method, the shortcomings of both methods are avoided and their advantages are maintained. This technique is efficient for wave scattering in anisotropic laminates containing inclusions and/or cracks of arbitrary shape. Received 2 February 2001