A Novel Crack Detection Methodology for Green-State Powder Metallurgy Compacts using an Array Sensor Electrostatic Testing Approach

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1 A Novel Crack Detection Methodology for Green-State Powder Metallurgy Compacts using an Array Sensor Electrostatic Testing Approach ABSTRACT Reinhold Ludwig and Diran Apelian* Powder Metallurgy Research Center (PMRC) Department of Electrical and Computer Engineering *Metal Processing Institute Worcester Polytechnic Institute Worcester, MA 0609 This paper briefly reviews a new instrumentation approach developed for the electric resistivity testing of green-state P/M compacts. Rapid testing of the green-state specimens is made possible through a special-purpose sensor configuration, which incorporates a matrix of 0 by 0 spring-loaded needle contacts with pin spacings of 0. inches. The sensor permits the detection of hairline flaws with surface openings as small as 0 microns. A quantitative analysis of defect resolution is conducted to lay the foundations for the experimental and validation phase of the sensor development. In particular, a dipole model representation for flaws embedded in the compact is proposed. Further, by incorporating Gaussian random noise to the voltage recordings, measurement uncertainties can be explored. This model is utilized to investigate the depth a flaw can be detected based on a given signal-to-noise ratio of the instrument. Key words: Non-destructive evaluation, resistivity instrumentation, testing of green-state compacts, theoretical dipole modeling, sub-surface flaw detection. BACKGROUND In response to industrial quality assurance demands, an inexpensive, on-line nondestructive evaluation (NDE) instrument has been developed to inspect P/M compacts early in the manufacturing process when the samples are still in their greenstate prior to sintering. This project was funded through the PMRC consortium partners and CPMT. As recent research has shown [-4], electric resistivity testing is perhaps the most suitable NDE technique to inspect these green-state compacts. The relatively high electric resistivity, or conversely speaking low conductivity, of the metallurgical structures permits the creation of substantial electric field distributions inside and on the surface of the samples in response to an applied current. The resulting voltage patterns established on the surface carry sufficient information to detect both surface-breaking and sub-surface defects [5]. The system consists of a multitude of spring-loaded probes to contact the sample; DC current is injected through two of the probes, and voltages are measured using an array of probes that cover the entire surface area of the part. A flaw in the material produces a characteristic voltage response that can be detected. Multiple

current injection patterns are used to enhance the flaw discrimination. This method allows fast inspection because the sensor only has to be positioned once on the surface of the part.

2 current injection patterns are used to enhance the flaw discrimination. This method allows fast inspection because the sensor only has to be positioned once on the surface of the part. Although the method is most sensitive to surface breaking flaws, subsurface flaws can be detected to a depth limited by the instruments signal-to-noise ratio. The developed inspection apparatus of green-state P/M compacts is relatively inexpensive and allows for flexible sensor configurations for parts of different geometries. The physical origin of the electrical impedance NDE method is based on the fourwire resistivity measurement method illustrated in Figure. Four probes contact the surface of a material of unknown, but assumed constant conductivity. Current is injected though the two outer probes and voltage is measured on the inner probes. If the solid is large and thick so that boundary conditions can be neglected, we can model the solid as a half-space. For this simplified model, the relationship between the conductivity σ and the voltage and current measurements is given by: I σ = () πv r r r3 r4 where V is the measured surface voltage, I is the injected current and r through r 4 are the distances indicated in Figure [4]. I Current source V Voltmeter r r r 3 r 4 Figure : Four-probe resistivity measurement arrangement. The developed non-destructive testing apparatus uses a similar type of measurement, but the sensor contains many more probes that cover the entire area of the sample under test. Figure shows the basic structure of the testing apparatus. It consists of a press, a sensor and a part holder. The press lowers the sensor onto the surface of the part while the part rests in the part holder. The sensor, shown in Figure 3a, is a planar array of spring-loaded pins. The pins on the periphery of the sensor are used for current

3 injection. All of the remaining pins and even the current injection pins that do not carry current are used for voltage measurements. To conduct the testing, constant DC current is injected through a pair of outer probes in one of four directions shown in Figure 3b, and voltages are measured between adjacent pins in the direction of current flow. Sensor head Planar 0 by 0 pin sensor P/M compact Support platform Part positioning controls Figure. Diagram of the basic mechanical measurement instrument used for nondestructive testing of green-state P/M compacts. Current injection probes Voltage measurement probes (a) Figure 3. Sensor configuration: a) voltage and current contact points, b) four current flow directions. Additional sensor configurations for non-planar configurations are under development, which permit part contacts over several surfaces. (b) 3

4 The feasibility of this instrumentation approach was established by testing both controlled and production type green-state compacts [] as reported at last year s conference []. However, due to manufacturing difficulties of creating well-defined flaws of various sizes and locations within the compact, assessments of how deep a flaw can be reliably detected is difficult to obtain by experiments. For this reason a theoretical investigation based on the so-called dipole model is conducted to investigate the detection limits. This dipole model is one of the most useful approximations to represent the voltage response due to small flaw. From an intuitive point of view, we can argue that when an electric current is established in a metal, and there is a small cavity representing the flaw, charges accumulate on the surface of the cavity so as to cancel the current flow inside the defect. At a distance away from the flaw, this charge distribution creates a field pattern that has the appearance of a dipole field. The corresponding voltage signal measured on the surface of the P/M compact is a superposition of the dipole response and the voltage drop due to the unperturbed current flow in the material. The inverse method discussed in this paper analyzes the dipole field only since the surface voltage of the unflawed compact can be added without loss of information. PROBLEM DESCRIPTION The geometry for the inverse problem is generically shown in Figure 4. Cartesian coordinates represent the two-dimensional space. Point voltage measurements are made along the x-axis equally spaced in the normalized interval from to. Current Source M Voltage t Powder Metal Part Electric Field Lines Flaw with current sources on the surface, opposing the electric field Figure 4. Dipole model arrangement to describe a flaw in the P/M compact. The dipole is arbitrarily located inside the P/M compact described by the halfspace above the x-axis. The flaw is represented by four independent dipole-related parameters: x and y coordinates (vector r) defining the location, and x and y components 4

5 of the dipole moment (p x and p y components of vector p) defining strength and orientation. These parameters must be reconstructed from the measured voltage data. The relationship between the parameters of the dipole and the measured voltages is given as: ( r r) p Φ = K () r r where Φ is the electric dipole potential, K is a constant of proportionality (in MKS units: πσ ), r is the distance from the origin to the dipole location, r is the distance from the origin to the point where the measurement is recorded, p is the dipole moment. Since K is a constant scale factor we can neglect its influence. We also note that this relationship differs from the dipole response in three dimensions. The 3-D equivalent of this model is an infinitely long line of dipoles. Equation () must be solved for the parameters r and p given an arbitrary number of voltage data points. The number of data points will be chosen such that the problem is over-determined. This means that at least five distinct voltage measurements are required. An important part of the solution is the evaluation of data variation in the parameters due to noise in the measurements. Measurement noise is assumed to be Gaussian and uniform throughout the voltage measurements. SOLUTION APPROACH The problem is nonlinear, and standard linear inverse techniques do not apply directly. The problem must be solved using nonlinear parameter estimation techniques. We chose the method of nonlinear least squares because it is well suited for a data analysis with additive Gaussian noise. The noise is added to make the model reflect practical measurement conditions where a number of error sources adversely affect the data evaluation. The least squares solution to this over-determined problem lies in minimizing an error function of the form: m i= ( ) F( X ) = Φˆ X Φ (3) where X is the vector of dipole model parameters (x, y locations, and dipole moment), Φˆ i ( X) are functions denoting voltage predictions based on the model parameters, Φi are the actual voltages recording data, m is the number of data points. i i 5

6 Direct closed-form solutions cannot generally be found so the techniques that have been developed for solving these inverse problems are iterative. The simplest inversion techniques are similar to the well-known Newton s method. They require an initial guess and use the gradient to produce subsequent estimates of the solution. If the initial guess is sufficiently close to the solution, the gradient method is likely to converge to the least-squares solution. We have chosen the so-called Gauss-Newton method, also known as the damped Gauss-Newton method. This is an accurate and easy-to-implement gradient search method. However, it was found to converge sufficiently rapidly and almost always to the correct solution. Non-gradient methods, such as the error-field method or the Simplex method, always converge, and can replace the Gauss-Newton method if it fails to provide satisfactory results. Starting with an initial estimate of the model parameters X i we wish to produce a better estimate of the least squares solution X i+. This improvement can be achieved by solving the linear problem obtained by expanding Φˆ ( X) about the initial estimate X i using a Taylor series and truncating the second and higher order terms: J T [ ] i ( ) { ˆ i i i+ i X Φ( X ) + J( X ) ( X X ) + HOT Φ} = 0 where HOT denotes Higher Order Terms that will be dropped and where J T (X) is the transpose of the Jacobian matrix, which contains partial derivatives of all functions in Φˆ X with respect to the model parameters ( ) L x x xn L J ( X) = x x xn (5) M m m m L x x xn Here n is the number of model parameters. When rearranged to solve for X i+, Eq. (4) takes the form: X = X T i i T i [ J ( X ) J( X )] J ( X ) ( Φˆ ( X ) Φ) i+ i i which is the basic Gauss-Newton iteration. The convergence properties of this method can be improved by introducing a damping constant λ, (0 < λ ): i+ i T i i T i i X = X λ J X J X J X Φˆ X Φ (6) [ ( ) ( )] ( ) ( ( ) ) (4) (5) 6

7 This constant is chosen appropriately for each iteration by starting with and subsequent repeated divisions by until F(X) is less than the previous iteration. A mathematical explanation of this simple rule is not given. In essence, λ protects the algorithm from diverging by shortening the step size in order to ensure that the error function decreases with every iteration step. The addition of this constant λ results in the damped Gauss-Newton method. The Jacobian matrix (4) needed for this algorithm can be calculated directly. The gradient of Eq. () with respect to p is: ˆ ˆ ˆ Φ Φ pφ =, p x p y = r r r r (9) and with respect to r: Φ ˆ =, = r x r y r r (( r r) p)( r r) r r r p (0) Knowing that perfect voltage measurements can never be obtained, we would like to know how accurate we expect our predictions to be, given a certain level of noise in the measurements. In other words, once we obtain the least-squares solution, we would like to know the probability distribution of the reconstructed model parameters. Since the problem is non-linear, the probability distribution is unobtainable in closed-form; thus only estimates can be generated. The simplest method of estimating the distribution is to linearize the problem around the solution. Given the fact that the voltage data has a normal distribution, the model parameters will also have a jointly normal distribution centered around the least-squares solution with all other parameters contained in the covariance matrix. The linear approximation will only be close to reality when the noise in the voltage data is relatively small. Equation (6) is the linearization we require when X i = X i+. The covariance matrix takes the form: Cov( X i Φ T i i [ J ( X ) J( X )] where σ Φ is the standard deviation of the voltage data. ) = σ () IMPLEMENTATION RESULTS Variation in model parameters is approximated using the covariance matrix (). This matrix can be calculated for different values of model parameters to see the resulting uncertainty in the parameters while keeping the noise in the voltage measurements the same. It is important to note that the model is nonlinear in only two parameters: the two components of vector r. The dependence on p is linear. Therefore, the magnitude of p only scales the error in the model parameters. The effect of dipole orientation is more 7

8 0. important to study. The dependence on r is nonlinear and is most important to study because we would like to know how deep we can detect flaws. Therefore the magnitude of p is fixed at and the effects of r are studied for the horizontal orientation of p, i.e., p x 0 and p y = 0. The intent is to study the variations in all model parameters for given p, voltage noise, and variation in depth r. Figure 5 shows the standard deviations of the model parameters as contour plots. The dipole moment p is fixed and oriented horizontally with a magnitude of. The input data is collected from ten voltage measurement points evenly distributed on the normalized interval from to + along the x-axis. The voltage measurements are differential, meaning the voltage difference between adjacent points is recorded, and therefore only nine actual voltages are used in the reconstruction Figure 5. Standard deviations as a function of flaw locations x (horizontal axis) and y (vertical axis) for a fixed dipole moment aligned horizontally. We observe from Figure 5 that the standard deviations increase further away from the measurement surface. This implies that deeper flaw locations are detected less reliably. A numerical model [3] is used to simulate the voltage difference along the surface with no flaw present and with a small flaw (characteristic size: 00 microns) present at location r = (0,0.). Subtracting these two responses gives the pure flaw response that we expect to be represented by the dipole model. This voltage, which forms the data for the inverse solver Eq. (6), is shown in Figure 6. In this figure, input represents the simulated flaw response and output denotes the voltage response of the reconstructed dipole. The match is quite good considering that the simulated flaw is not spherical in shape but instead a narrow slit with 0. units in height oriented vertically and labeled original. 8

9 original input output Figure 6: Simulated flaw response (input) and reconstructed dipole response (output). The difference voltages in Figure 6 can be related to actual instrument measurements where the smallest voltage of approximately 0.4V is measured with 40dB Signal-to- Noise Ratio (SNR) (standard deviation is 0.004V). For the reconstructed model the solver reported a horizontally oriented dipole of strength of approximately This means that the noise standard deviation is 0.004/0.03 = 0.33 if the dipole strength is increased to. To obtain an appreciation of how the inverse algorithm (6) can handle embedded flaws at various depth locations, denoted by r x and r y, we illustrate in Figure 7 several such simulations with the dipole located deeper and deeper inside the compact. The curve labeled original is the differential voltage response predicted by the numerical solver, meaning the difference in voltage between adjacent measurement points. The curve labeled input is the original differential voltage plus added noise with standard deviation of 0.33, serving as the input to the solver. The curve labeled output is the voltage response of the reconstructed dipole based on equation (6). In Figure 7(a), the noise is very small compared to the voltage signal, and the reconstruction is highly accurate. As the depth increases the signal strength diminishes, and the reconstruction accuracy decreases until the noise completely consumes the signal and the reconstructed dipole becomes meaningless. In all simulations, the dipole moment p is aligned with the surface. 9

10 5 4 original input output original: p x = p y = 0 r x = 0. r y = 0. reconstructed: p x = 0.93 p y = r x = 0.8 r y = 0.89 (a).4. original input output original: p x = p y = 0 r x = 0. r y = 0.4 reconstructed: p x = 0.9 p y = r x = 0.0 r y = (b) 0

11 original input output original: p x = p y = 0 r x = 0. r y = 0.8 reconstructed: p x =.467 p y = r x = r y =.08 (c) Figure 7. Original voltage data, voltage data with noise (standard deviation of 0.33) added, and reconstructed voltage, for horizontally oriented dipole. Below graphs: original and reconstructed dipole parameters (p x, p y are components of p, and r x, r y are x and y locations of the dipole imbedded in the sample). The results presented in Figure 7 have a very practical implication. Based on our existing sensor configuration of 0 needle pins with a pin spacing of 0., we have a total length of 3 mm for the -unit length shown in Figure 7. If the depth resolution is set to r y =0.8 units (see Figure 7 (c)), we obtain a depth of 9.4 mm, or 0.36 for the instrument s SNR of 40 db. Practical tests of our instrument with controlled compacts have shown that hairline flaws can be reliably detected up to 0.3 below the P/M surface. CONCLUSIONS In this paper an inverse formulation is presented to determine the depth resolution of our electric resistivity instrument based on a four-parameter dipole model. The developed model gives the limit of flaw depth, which can be resolved for a recorded surface voltage distribution along the P/M compact, and for a given SNR. Flaw depth predictions with specifically prepared green-state compacts support the validity of this dipole model. The sensor configuration shown in Figure 3 consists of 0 linear measurement points spaced /9 units apart. Future research will focus on extending the analysis to

12 different numbers of recording pins as well as various pin arrangements to yield better noise immunity. In addition, the dipole model will be extended to 3D and a correlation of flaw size and dipole moment will be undertaken. ACKNOWLEDGEMENTS The authors would like to thank the PMRC consortium members, and CPMT for their support, encouragement, and valuable guidance. In particular, the authors would like to acknowledge the input of Mr. Ulf Gummeson and Mr. Howard Sanderow. REFERENCES. R. Ludwig, G. Bogdanov, and D. Apelian, Non-Destructive Electrostatic Determination Of Surface Breaking And Subsurface Flaws In Green State P/M Compacts, in PM TECH conference, Las Vegas, NV, May J. G. Stander, J. Plunkett, D. Zenger, J. McNeill, and R. Ludwig, Electric Resistivity Testing of Green-State Powdered Metallurgy Compacts, in Review of Progress in Quantitative NDE, Vol. 6B, pp , Plenum Press, R. Ludwig, S. Makarov, and D. Apelian, Theoretical and Practical Investigations of Electric Resistivity Testing of Green-State P/M Compacts, in Journal of Nondestructive Evaluation, Vol. 7, No. 3, pp , Nov B. van den Bos, The DC Potential Drop Method for Nondestructive Testing of P/M Components in Green- and Sintered States, Report, Division of Engineering Materials, Linkoping University, Stockholm, Sweden, June J. Stander, J. Plunkett, W. Michalson, J. McNeill, and R. Ludwig, A Novel Multi- Probe Resistivity Approach to Inspect Green-State Powder Compacts, Journal of Nondestructive Evaluation, Vol. 6, No. 4, pp. 05 4, A. Lewis and D. Bush, Resistivity Measurement for Evaluation of Coating Thickness, Materials Evaluation, Vol. 49, pp. 3-39, 99.

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

Journal of Nondestructive Evaluation, Vol. 16, No. 4. 1997 A Novel Multi-Probe Resistivity Approach to Inspect Green- State Metal Powder Compacts J. Stander, 1 J. Plunkett, 1 W. Michalson, 1 J. McNeill,