Numerical method for calculating the apparent

JOURNAL OF APPLIED PHYSICS

VOLUME 95, NUMBER 12

15 JUNE 2004

Numerical method for calculating the apparent eddy current conductivity loss on randomly rough surfaces Feng Yu and Peter B. Nagya) Department of Aerospace Engineering and Engineering Mechanics, University of Cincinnati, Cincinnati, Ohio 45221-0070

共Received 29 December 2003; accepted 12 March 2004兲 Because of their frequency-dependent penetration depth, eddy current measurements are capable of mapping the near-surface depth profile of the electrical conductivity. This technique is used to nondestructively characterize the subsurface residual stress and cold work distributions in shot-peened metal components. Unfortunately, the spurious surface roughness produced by the shot peening process causes an apparent loss of eddy current conductivity, thereby decreasing the accuracy of the measurements, especially in thermally relaxed specimens where the primary material effects are significantly reduced. In this paper, a numerical method is introduced based on the Rayleigh approximation for calculating the apparent eddy current conductivity loss exhibited by 1D randomly rough surfaces. The relevant boundary conditions are satisfied using the so-called point-by-point technique, and the results are first compared to the previously developed Rayleigh– Fourier technique for a 1D sinusoidal corrugation. Pseudorandom surface profiles of different autocorrelation functions are considered. It is found that the Gaussian model lends itself the best to numerical simulations, but it significantly underestimates the apparent eddy current conductivity loss expected on real shot-peened surfaces, which exhibit essentially exponential correlation function. It is also demonstrated that the Lorentzian model is numerically less stable, but physically closer to the exponential one. The latter could not be simulated reliably by the present numerical technique because of its slowly decaying high-frequency spectral components. © 2004 American Institute of Physics. 关DOI: 10.1063/1.1737474兴

I. INTRODUCTION

interplanar atomic spacing to determine the magnitude of the prevailing elastic strain 共stress兲. The disadvantage of XRD is that it only probes a shallow surface layer. X rays typically penetrate 5–20 ␮m below the surface, but the crucial compressive part of the residual stress profile ranges from 50 to 500 ␮m. Thus, residual stress assessment by XRD is nondestructive only in a shallow surface layer. To obtain detailed residual stress profiles at larger depths, successive layers should be removed, usually through etching or other destructive means. The removal of material will alter the stress field; therefore, corrections need to be applied to account for errors caused by stress release during the process. Recently, eddy current conductivity measurements have been suggested for nondestructive subsurface residual stresses assessment,1–9 and a few specialized instruments are already commercially available for this purpose. This technique is based on the subtle stress dependence of the electrical conductivity. However, it is clear from these published results that various factors, such as hardness, texture, and surface roughness, will have to be identified and separated from the overall measurement, while performing nondestructive evaluation of the residual stress in shot-peened components. One of the main problems for the practical utilization of eddy current inspection for near-surface residual stress assessment is the spurious electrical conductivity change caused by the rough surface, which increases the path length of the surface-hugging eddy currents, thereby producing a perceivable reduction in the observed conductivity. The resulting apparent eddy current conductivity loss substantially

It is well-known that surface properties play an important role in determining the performance of structural components, especially their fatigue resistance. In most cases, the failure of critical structural components originates in surface areas under tension, which leads to crack initiation and propagation. The existence of a near-surface compressive stress layer can cancel out service-induced tensile stresses, thus preventing crack nucleation and growth and ultimately increasing the service life of the component. A beneficial near-surface compressive stress layer can be generated by the application of shot peening, a cold working process in which the component surface is bombarded with small spherical projectiles called shots. Research has revealed that shot peening can improve the fatigue life by as much as ten times and increase the endurance limit by 20%. Unfortunately, shot peening is normally not regarded as a reliable quality control process in industry, mostly because of the expensive and time-consuming process needed to ensure high peening quality and the inherent instability of the produced residual stress during long-time service because of stress relaxation at high operational temperatures. It is therefore essential to directly assess the existing near-surface residual stress in order to predict the remaining structural life. The most widely used technology for measuring residual stress in metals is based on x-ray diffraction 共XRD兲, which measures the change in a兲

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0021-8979/2004/95(12)/8340/12/$22.00

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© 2004 American Institute of Physics

J. Appl. Phys., Vol. 95, No. 12, 15 June 2004

distorts or, even overshadows the small actual conductivity change caused by the presence of residual stress. For example, Lavrentyev et al. found that the electrical conductivity of 7075 aluminum was not sufficiently sensitive to stress to allow nondestructive residual stress assessment in shotpeened specimens, and that the measured conductivity loss at higher frequencies and increasing peening intensities could be, to a large degree, caused by the induced surface roughness.7 Blodgett et al. found that fully relaxed shot-peened copper specimens exhibited apparent loss of electrical conductivity as high as 10% to 30% at frequencies approaching 10 MHz.10 The copper specimens were thermally relaxed to remove the cold work and release the residual stress. As a result, surface roughness, and possibly unfused near-surface microcracks, were the only remaining factors to cause the measured conductivity loss relative to the intact specimens. The apparent reduction in electrical conductivity is due to the fact that the eddy current is forced to propagate along a more tortuous path as the penetration depth decreases with increasing inspection frequency than in the case of a smooth surface. We use the term ‘‘apparent’’ because the actual conductivity of the fully relaxed material is the same as that of the original intact material, and the lower value of conductivity measured by eddy current inspection is a geometrical artifact due to the presence of surface roughness. It should be mentioned that in high-temperature, high-strength engineering materials, such as titanium alloys and nickel-based superalloys, the expected apparent eddy current conductivity loss is much smaller since these materials exhibit two orders of magnitude lower conductivity, i.e., one order of magnitude larger eddy current penetration depth at a given inspection frequency. In addition, their hardness is also much higher; therefore, the surface roughness caused by a certain peening intensity is typically 2– 4 times lower. Still, the spurious surface roughness effect could play a significant role in the eddy current evaluation of engine components after long-term service when the primary material effects are significantly reduced by thermo-mechanical relaxation without a corresponding drop in the surface roughness effect, which could even increase due to erosion, corrosion, or fretting. Recently, Kalyanasundaram and Nagy used the Rayleigh–Fourier method to predict the apparent conductivity loss of eddy current conductivity due to the presence of a one-dimensional sinusoidal corrugation.11 Generally, a real rough surface is two-dimensional and can be regarded as a sum of sinusoidal corrugations of random phase and amplitude distributions to properly account for the statistical features of the surface topography. Therefore, simulation on one-dimensional sinusoidal corrugation can only qualitatively predict the absolute value and frequency dependence of the apparent conductivity loss. As the next step towards our objective of developing an accurate predictive model for randomly rough surfaces, in this paper we approximate the rough surface as a one-dimensional random corrugation. Our estimates for the apparent loss of eddy current conductivity will be based on the Rayleigh hypothesis and the relevant boundary conditions will be satisfied using the so-called point-by-point technique.

F. Yu and P. B. Nagy

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FIG. 1. Measured autocorrelation functions 共dashed line兲 in shot-peened copper and the best-fitting exponential curves 共solid line兲 for two different peening intensities.

First, we will describe the techniques used to generate 1D random surfaces with various autocorrelation functions. A unified definition of the correlation length will be introduced to facilitate the comparison of the root-mean-square 共rms兲 slopes, which will be shown to determine the highfrequency behavior of the apparent eddy current conductivity loss, of different 共sinusoidal, Gaussian, Lorentzian, and exponential兲 surfaces. Second, we will use a simple onedimensional sinusoidal corrugated profile to illustrate that the point-by-point method, which can be easily adapted to random surface profiles, yields results numerically identical to those previously obtained by the Rayleigh–Fourier method. Third, we will compare the apparent conductivity loss caused by different random surface profiles of nominally identical rms roughness and unified correlation length to show how the type of the autocorrelation function affects the apparent conductivity loss. We will show that the more highfrequency components the random surface exhibits, the more conductivity loss occurs as well as more numerical instability in the simulations. Because of this the Lorentzian model seems to be the best compromise between accuracy and numerical stability of the point-by-point method to predict the apparent conductivity loss caused by shot peening. Finally, the numerical results are compared with a simple ad hoc approximation developed in Ref. 11, and it is shown that the same approach can be extended for one-dimensional randomly rough corrugations with known rms roughness and correlation length. II. RANDOM SURFACE PROFILE SIMULATION

Randomly rough surfaces can be characterized by their autocorrelation function and certain numerical parameters, namely rms roughness h and correlation length L. Figure 1 shows the experimentally determined autocorrelation functions in shot-peened pure copper specimens of Almen inten-

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sity 2A and 8A and the best fitting exponential curves.12 It is clear that the two-dimensional random roughness on these surfaces can be well approximated by an exponential distribution. However, random surfaces of exponential autocorrelation functions present serious computational problems due to the presence of strong, high-frequency spectral components. Therefore, we will investigate the feasibility of modeling such surfaces by cosine, Gaussian, and Lorentzian distributions, which all lend themselves more easily to numerical simulations. First, we will discuss how to generate surface profiles with various autocorrelation functions and then consider the influence of their differences on the apparent conductivity loss. For simplicity, in each case, the actual 2D surface roughness will be modeled as a 1D random corrugation. We can use the so-called spectral method to realize 1D random surface roughnesses with different types of autocorrelation functions.13,14 Over a finite, but sufficiently long window (0⭐x⭐w, where LⰆw), a pseudorandom realization of the surface s(x) can be written as follows: ⬁

s共 x 兲⫽

兺

n⫽1

冋

册

2␲ x⫹ ␾ n , A n sin n w

共1兲

where ␾ n is a random phase parameter, which is uniformly distributed over the 关0–2␲兴 range and A n is a random amplitude distribution which depends on the autocorrelation function. However, we will demonstrate later that the randomization of the phase alone is sufficient to predict the expectation, i.e., ensemble average, value of the apparent eddy current conductivity loss on rough surfaces; therefore, for the sake of simplicity, random amplitude variations of the spectral components will be neglected in the following simplified analysis. In Eq. 共1兲, n⫽0 was excluded from the summation to assure that 具 s(x) 典 ⫽0 共the bracket indicates averaging over all values of x). The autocovariance 共or unnormalized autocorrelation function兲 of the surface is defined as C( ␰ ) ⫽ 具 s(x)s( ␰ ⫹x) 典 , so that C(0)⫽h 2 , where h denotes the rms roughness. Then, the 共normalized兲 autocorrelation function is simply c( ␰ )⫽C( ␰ )/h 2 . The rms roughness of a zeromean surface is ⬁

h 2 ⫽ 具 s 2 共 x 兲典 ⫽ 21

兺

n⫽1

A 2n .

共2兲

The width of the autocorrelation function is measured by the so-called correlation length. Generally, the correlation length has a different physical meaning for each type of autocorrelation function; therefore, its values cannot be easily compared. In order to facilitate the quantitative comparison of different autocorrelation functions, we will use a universal correlation length L defined as the half-width of the autocorrelation function at half of its maximum. The amplitude distribution A n will be calculated by exploiting the fact that the Fourier transform of the autocorrelation function is related to the power spectrum of the surface profile. We will consider four different autocorrelation functions, namely cosine, Gaussian, Lorentzian, and exponential. A cosine autocorrelation function is defined as

FIG. 2. Differences in the initial behavior of the various autocorrelation functions considered in this study.

c 共 ␰ 兲 ⫽cos

冉冊冉冊

2␲␰ ␲ ␰ ⫽cos . ⌳ 3 L

共3兲

where ⌳ is the period of the sinusoidal surface profile and L⫽⌳/6 is the previously mentioned universal correlation length, i.e., the half-width of the autocorrelation function at half maximum. A Gaussian autocorrelation function is defined as c 共 ␰ 兲 ⫽e⫺ ␰

2 /a 2

⫽2 ⫺ ␰

2 /L 2

共4兲

,

where a is the often-used half-width of the autocorrelation function at 1/e maximum and L⫽a 冑ln(2) is the universal correlation length. A Lorentzian autocorrelation function is defined as c共 ␰ 兲⫽

1 , 1⫹ ␰ 2 /L 2

共5兲

where L is the universal correlation length, which in this case coincides with the traditional definition of the correlation length. Finally, an exponential autocorrelation function is defined as c 共 ␰ 兲 ⫽e⫺ 兩 ␰ 兩 /a ⫽2 ⫺ 兩 ␰ 兩 /L ,

共6兲

where a is the half-width of the autocorrelation function at 1/e maximum and L⫽a ln(2) is the universal correlation length. Figure 2 illustrates the substantial differences in the initial behavior of the various autocorrelation functions considered in this study. The sharper the peak at ␰ ⫽0, the stronger the high-frequency components in the spectral representation are and the more difficult it is to simulate the surface because of numerical instabilities. In the following sections, we derive the amplitude distribution A n necessary to produce each of these surface types and calculate the variance of the surface slope ⬁

冉冊

2␲n 1 ␸ ⫽具s ⬘ 共 x 兲典⫽ 2 n⫽1 w 2

2

兺

2

A 2n ,

共7兲

which is the primary factor that determines the level of apparent eddy current conductivity loss exhibited by a conducting specimen of rough surface at high frequencies.11



A. Cosine autocorrelation function

This is the simplest case when w⫽⌳ and the amplitude distribution is trivially A n ⫽&h if n⫽1 and zero otherwise. The variance of the surface slope from Eq. 共7兲 is11

␸ 2⫽

冉冊 2␲ ⌳

2

h . L2

共8兲

B. Gaussian autocorrelation function

The Fourier transform of the Gaussian autocorrelation function given in Eq. 共4兲 is F共 ␻ 兲⫽

冕

⬁

⫺i ␻␰

⫺⬁

c共 ␰ 兲e

2 2 d ␰ ⫽ 冑␲ ae⫺ ␻ a /4,

A n ⫽Ch 冑F 共 ␻ 兲 ⫽Ch 冑␲ a e

2 ⫺ 共 ␲ 2 n 2 a 2 /2w 2 兲

4

C

⫽

冑␲ ah 2

,

共10兲

⬁

兺 e⫺共 ␲ n a /w 兲 n⫽1 2 2 2

2 C 2h 2w 2 冑␲

冕

⬁

0

2

e⫺ ␨ d ␨ ⫽

2

A n ⫽2 冑␲ h

冑

C 2h 2w , 4

共11兲

a ⫺ 共 ␲ 2 n 2 a 2 /2w 2 兲 e , w

⬁

⫽

冉冊

␲ na 8 冑␲ h 2 wa n⫽1 w

兺

8h 2

冑␲ a 2

冕␨ ⬁

2 ⫺␨2

e

0

2

e⫺ 共 ␲

d␨⫽

C 2h 2w 4

2 n 2 a 2 /w 2 兲

2h 2 h2 2 ⬇1.386 2 . a L

A n ⫽2h

C 2h 2w , 4

共16兲

冑

␲ L ⫺ ␲ nL/w . e w

共17兲

In order to determine the variance of the surface slope, Eq. 共7兲 can be written in an integral form 2␲h ␸ ⫽ wL 2

h ⫽ 2 L

2

冉冊

⬁

2 ␲ nL w

兺 n⫽1

冕␨ ⬁

2

e⫺2 ␲ nL/w 2

2 ⫺␨

e

0

h d ␨ ⫽2 2 . L

共18兲

D. Exponential autocorrelation function

F共 ␻ 兲⫽

␲a , 1⫹ ␻ 2 a 2

共13兲

共19兲

where again ␻ ⫽2 ␲ n/w. The amplitude distribution is A n ⫽Ch 冑F 共 ␻ 兲 ⫽

Ch 冑␲ a

冑冉冊 1⫹

2 ␲ na

2

,

共20兲

w

where C is an appropriate constant chosen so that the variance of the surface height distribution is h 2 . Like before, exploiting that aⰆw, we can rewrite Eq. 共2兲 by introducing ␨ ⫽2 ␲ na/w in the following integral form: C 2h 2w 4

⫽

冕

⬁

0

1 d␨ 1⫹ ␨ 2

␲ C 2h 2w C 2h 2w , 关 tan⫺1 ␨ 兴 ⬁0 ⫽ 4 8

共21兲

which yields C⫽ 冑8/( ␲ w). Therefore, the proper amplitude distribution for an exponential autocorrelation function is

The Fourier transform of the Lorentzian autocorrelation function given in Eq. 共5兲 is 共14兲

where ␻ ⫽2 ␲ n/w. The corresponding amplitude distribution is A n ⫽Ch 冑F 共 ␻ 兲 ⫽Ch 冑␲ Le⫺ ␲ nL/w ,

0

e⫺ ␨ d ␨ ⫽

which yields C⫽2/冑w. In conclusion, the proper amplitude distribution for a Lorentzian autocorrelation function is

h 2⫽

C. Lorentzian autocorrelation function

F 共 ␻ 兲 ⫽ ␲ Le⫺ 兩 ␻ 兩 L ,

冕

⬁

共12兲

which can be rewritten in terms of the universal correlation length by substituting a⫽L/ 冑ln(2). In order to determine the variance of the surface slope, Eq. 共7兲 can be written in the following integral form:15

␸ 2⫽

⫽

The Fourier transform of the exponential autocorrelation function given in Eq. 共5兲 is

which yields C⫽2/冑w. Therefore, the proper amplitude distribution for a Gaussian autocorrelation function is 4

兺

2

where C is an appropriate constant chosen so that the variance of the surface height distribution is h 2 . Exploiting that aⰆw, we can rewrite Eq. 共2兲 by introducing ␨ ⫽ ␲ na/w in the following integral form: h 2⫽

⬁

C 2␲ h 2L e⫺2 ␲ nL/w h ⫽ 2 n⫽1 2

共9兲

where ␻ ⫽2 ␲ n/w. The corresponding amplitude distribution is

2

before, exploiting that LⰆw, we can rewrite Eq. 共2兲 by introducing ␨ ⫽2 ␲ nL/w in the following integral form:

2

h 2 ⬇1.097

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共15兲

where, as before, C is an appropriate constant chosen so that the variance of the surface height distribution is h 2 . Like

冑8h A n⫽

冑

a w

冑冉冊 1⫹

2 ␲ na

2

,

共22兲

w

which can be rewritten in terms of the universal correlation length by substituting a⫽L/ln(2). In order to determine the variance of the surface slope, Eq. 共7兲 can be written in the following integral form:

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TABLE I. A summary of the principal surface parameters for various autocorrelation functions c( ␰ ) 共amplitude distribution A n , unified correlation length L, variance of the slope ␸ 2 ). c( ␰ )

Type Cosine

cos

冉冊 2␲␰ ⌳

e⫺ ␰

Gaussian Lorentzian

2 /a 2

1 1⫹␰2/L2

An

L

␸2

&h if n⫽1, 0 otherwise

⌳/6

1.097h 2 /L 2

冑冑冑

a ⫺ 共 ␲ 2 n 2 a 2 /2w 2 兲 a 冑ln(2) 1.386h 2 /L 2 e w

24冑␲ h

2 ␲h

冑8h Exponential

e⫺ 兩 ␰ 兩 /a

L ⫺ ␲ nL/w e w

冑

2h 2 /L 2

a ln(2)

Infinite

a w

冑冉冊 1⫹

L

2 ␲ na

2

w

FIG. 3. A schematic diagram of electromagnetic plane-wave reflection and transmission at a randomly corrugated surface between a dielectric and a conductive half-space at normal incidence.

␸ 2⫽

2h 2 ␲a2

冕

⬁

0

␨2 d␨, 1⫹ ␨ 2

共23兲

which is obviously not finite. Therefore, in sharp contrast with a periodic harmonic surface corrugation and randomly rough surfaces of Gaussian and Lorentzian autocorrelation functions, the variance of the surface slope for a randomly rough surface of exponential autocorrelation function is not finite; therefore, the apparent conductivity loss of such surfaces is not expected to exhibit a high-frequency plateau.11 This lack of convergence also indicates that the exponential distribution will present serious numerical challenges. Therefore, although the experimentally determined autocorrelation function of shot-peened surfaces was found to be essentially exponential, we are going to consider the Lorentzian autocorrelation function that mimics the main features of an exponential distribution, but remains convergent like the Gaussian distribution. It should be mentioned that the discontinuity of the exponential correlation function at ␰ ⫽0 is not physical. The actual correlation function of a real random surface must exhibit even symmetry, i.e., its slope at ␰ ⫽0 must vanish. As it is illustrated in Fig. 1, the idealized exponential correlation function appears to be a reasonable approximation for shotpeened surfaces at length scales comparable to the correlation length, but much less at small scales more than one order of magnitude below it. Later in this paper we will illustrate the lack of convergence of the numerical simulations at high frequencies in the case of a randomly rough surface of exponential correlation function. This numerical instability is due to the presence of strong, high spatialfrequency components that are associated with the behavior of the correlation function at very small arguments. We will show that further efforts are needed to increase the stability and accuracy of our simulations, e.g., by using low-pass filtered surface profiles of otherwise exponential correlation function, that more accurately describe the small-scale features of shot-peened surfaces. As a summary, Table I shows the amplitude distribution, universal correlation length, and slope variance of surface profiles with various autocorrelation functions, namely, co-

sine, Gaussian, Lorentzian, and exponential. The main conclusion we can draw from these results is that the sharper the initial decay of the autocorrelation function, the larger the variance of the surface slope. We will show later that the variance of the surface slope is the principal parameter that determines the loss of apparent eddy current conductivity at high frequencies; therefore, the more realistic the correlation function used to model a shot-peened surface, the more serious the resulting numerical problems during simulation. III. COMPUTATIONAL METHOD

In this section, we first briefly review the reflection of an electromagnetic plane wave of transverse magnetic 共TM兲 polarization from a conducting half-space with 1D surface roughness based on the Rayleigh hypothesis.11 The basic principle of the Rayleigh hypothesis is to write the reflected and transmitted fields as sums of outgoing diffracted plane waves and then determine the unknown amplitude coefficients by satisfying the boundary conditions on the surface.14,16,17 The main restrictions of this method are that the numerical convergence of the series is only achievable for ‘‘slightly rough’’ surfaces, because of the assumption that only outgoing plane waves contribute to satisfying the boundary conditions. Interestingly, it has been shown numerically on specific problems that the Rayleigh method yields accurate results well beyond the expected domain of validity of the Rayleigh hypothesis.18 It has also been shown that for analytical surface profiles the applicability of the numerical Rayleigh method is much more extensive than the validity of the Rayleigh hypothesis.19 Let us consider a 1D pseudorandom surface which is periodic over a window length w that is sufficiently large with respect to the autocorrelation length L to accurately simulate randomly rough surfaces s 共 x 兲 ⫽s 共 x⫹nw 兲 ,

共24兲

where n is an integer. Figure 3 shows a schematic diagram of the scattering of an electromagnetic plane wave at such a corrugated surface 共⌽ could be either the electric or mag-



netic field or, e.g., the vector potential兲. At the interface z ⫽s(x) between the conducting and dielectric media, the boundary conditions require that the incident ⌽ i , reflected ⌽ r and transmitted ⌽ t field satisfy the continuity of the tangential components of the electric and magnetic field intensity, which leads to the following two boundary conditions:11 N

e i ␬ s(x) ⫹

N

兺

n⫽⫺N

R n e i[nKx⫺ ␬ n s(x)] ⫽

兺

n⫽⫺N

T n e i[nKx⫹k n s(x)] , 共25兲

and N

␩ 0 e i ␬ s(x) ⫺

␩0 兺 R n 关 ␬ n ⫹nKs ⬘共 x 兲兴 e i[nKx⫺ ␬ ns(x)] ␬ n⫽⫺N

N

␨ ⫽ T 关 k ⫺nKs ⬘ 共 x 兲兴 e i[nKx⫹k n s(x)] , k n⫽⫺N n n

兺

共26兲

where the infinite summations were truncated at a sufficiently large number of N. R n and T n are the nth-order reflection and transmission coefficients, respectively, ␩ 0 ⫽ 冑␮ 0 /␧ 0 is the intrinsic impedance of free space, ␻ is the angular frequency, ␬ ⫽ ␻ /c⫽2 ␲ /␭ is the electromagnetic wave number in free space, c⫽1/冑␮ 0 ␧ 0 is the wave speed, ␭ is the wavelength, and ␮ 0 and ␧ 0 are the magnetic permeability and electric permittivity of free space, respectively. K is the wave number of the surface corrugation and can be expressed as K⫽2 ␲ /w, where w is the previously mentioned period of the surface corrugation. The wave number components are related through ␬ 2 ⫽n 2 K 2 ⫹ ␬ 2n , where ␨ ⫽ 冑⫺i ␻ ␮ / ␴ is the intrinsic impedance of the conductor, k 2 ⫽n 2 K 2 ⫹k 2n ⫽i ␻ ␮ ␴ , and ␮ and ␴ are the magnetic permeability and electrical conductivity of the material, respectively. Here, we applied the standard eddy current model for the Maxwell equations, which exploits the fact that at eddy current inspection frequencies the conduction current is much greater than the displacement current in metal conductors.20 One way to solve Eqs. 共25兲 and 共26兲 is the Rayleigh– Fourier method, which exploits the fact that in the case of a harmonic surface corrugation the Fourier decomposition of the boundary condition equations is explicitly known in terms of Bessel functions. Although the same technique can be extended to any periodic profile, the Fourier coefficients have to be calculated separately for each representation of the surface. A conceptionally simpler albeit numerically often less efficient, alternative approach is the so-called pointby-point method, which imposes the boundary conditions at a finite number of arbitrarily chosen points along the interface.16 This method can be more easily adapted to pseudorandom surfaces, but before we do so, we would like to illustrate that this method yields identical results to those previously obtained by the Rayleigh–Fourier method on a sinusoidal surface.11 Let us consider a corrugated surface of sinusoidal profile s(x)⫽&h cos(Kx), where K⫽2 ␲ /⌳, and ⌳⫽w is the period of the corrugation. In this case, since the harmonic profile is arbitrarily chosen to be symmetric to x⫽0, we can

8345

exploit that R n ⫽R ⫺n and T n ⫽T ⫺n to further reduce the number of linear equations to be solved by a factor of almost 2. We can approximately satisfy the previously given two boundary conditions of Eqs. 共25兲 and 共26兲 by exactly satisfying them at N⫹1 appropriately selected points over one period along the periodic surface. These points are chosen to be at x ᐉ ⫽ᐉ⌳/(N⫹1), where ᐉ⫽0,1,... ,N, so that the above equations represent 2N⫹2 equations for 2N⫹2 unknowns (R 0 ,R 1 ,...,R N ,T 0 ,T 1 ,...,T N ), which can be readily solved by numerical means. In eddy current inspection, the electrical conductivity of the specimen is usually determined from the impedance of a finite-size probe coil by suppressing the adverse lift-off effect based on the differences between the trajectories of the lift-off and conductivity curves in the complex impedance plane.21 Uzal et al. recently used perturbation methods to predict the impedance of a cylindrical coil over an infinite metallic half-space with shallow surface features.22 The observed change in coil impedance is consistent with a slight decrease in apparent conductivity and increase in apparent lift-off. However, the particular issue of interest, namely, the apparent eddy current conductivity loss at rough surfaces occurs only at the highest inspection frequencies when the standard penetration depth, ␦ ⫽1/冑 f ␲ ␮ ␴ , is comparable or smaller than the correlation length of the rough surface, which in turn is usually much smaller than the diameter of the probe coil. Under these conditions, finite coil effects can be neglected and the apparent conductivity of the specimen can be simply determined from the plane-wave reflection coefficient at normal incidence. Accordingly, regardless of the number of terms included in the numerical solution of the boundary conditions in order to achieve convergence, the apparent electrical conductivity of the specimen will be determined solely from the zeroth-order reflection coefficient R 0 by neglecting the higher-order diffraction terms of the reflected field, i.e., by evaluating R 0 as if it were produced by the smooth surface of a hypothetical material of reduced conductivity. Generally, the measured complex reflection coefficient can be written as23 R 0 ⫽ 关 1⫺ ␣ 共 1⫺i 兲兴 ei ␤ ⬇ 共 1⫺ ␣ 兲 ⫹i 共 ␣ ⫹ ␤ 兲 ,

共27兲

where ␣ and ␤ represent the effects of the surface irregularity on the reflection measurement, and they are both much smaller than unity

␣⫽

冑 ␮␮

2␻ ␧0 0␴ a

and

␤ ⫽2ᐉ a ␬ .

共28兲

Here, ␴ a and ᐉ a denote the ‘‘apparent’’ electrical conductivity and lift-off distance, respectively. The separation of variables represented by Eq. 共27兲 is analogous to the separation of lift-off and conductivity effects in the complex impedance of a probe coil. The normalized apparent conductivity of the specimen ␥ ⫽ ␴ a / ␴ can be directly calculated from the plane-wave reflection coefficient given by Eqs. 共27兲 and 共28兲 as follows:11

␥⬇

2␻␮␧0 . ␴ ␮ 0 共 Re兵 R 0 其 ⫺1 兲 2

共29兲

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FIG. 4. Normalized apparent conductivity versus frequency for a sinusoidal corrugation using the Rayleigh–Fourier and the point-by-point methods (N⫽30).

It should be mentioned that the point-by-point method is simpler, but less accurate and numerically less stable as well as slower than the Rayleigh–Fourier method due to the larger boundary condition matrix that needs to be inverted; therefore, we had to take a series of precautions during our calculations. Most importantly, we used quadruple precision 共32-digit兲 and increased the permittivity of the dielectric medium ␧ 0 five orders of magnitude relative to that of a vacuum to reduce the stringent requirements for the accuracy of the calculated reflection coefficient 共we verified that this procedure did not affect the apparent eddy current conductivity loss since ␣ remained sufficiently small兲. In the following sections we will present numerical examples to illustrate the apparent conductivity loss on onedimensional Lorentzian rough surfaces in the case of transverse magnetic 共TM兲 polarization, i.e., when the eddy current flows normal to the surface grooves. The relevant physical parameters and material properties used in the calculations are as follows: the rms roughness of the corrugation (h), the universal correlation length (L), the inspection frequency ( f ), the permittivity of the dielectric medium (␧ 0 ⫽8.854⫻10⫺12 As/Vm), and the permeability of a vacuum ( ␮ 0 ⫽4 ␲ ⫻10⫺7 Vs/Am), the electrical conductivity of the conductor ( ␴ ⫽5.8⫻107 A/Vm for copper兲, and the permeability of the conductor ( ␮ ⬇ ␮ 0 for copper兲. It was found that the period of the pseudorandom surface profile, i.e., the window length (w) used in the calculations, strongly affects the accuracy and numerical stability of the simulations and therefore will be given for each case when a pseudorandom surface profile is considered. The excellent agreement between the predictions of the Rayleigh–Fourier and the point-by-point methods for the apparent eddy current conductivity loss on a sinusoidal surface corrugation is illustrated in Fig. 4. These results strongly suggest that the simpler point-by-point method could be exploited to simulate the apparent conductivity loss on rough surfaces with random characteristics, which would be significantly more cumbersome to simulate using the Rayleigh–Fourier method. IV. RANDOMLY ROUGH SURFACES

A random corrugation does not have the same symmetry as the above-considered sinusoidal corrugation, i.e., R n

FIG. 5. Examples of 共a兲 a typical Lorentzian rough surface profile generated using input parameters h⫽20 ␮ m and L⫽300 ␮ m and 共b兲 the corresponding actual 共dots兲 and best fitting 共solid line兲 correlation functions.

⫽R⫺n and T n ⫽T ⫺n . As a result, the two boundary conditions given by Eqs. 共25兲 and 共26兲 represent 4N⫹2 equations and for 4N⫹2 unknowns (R ⫺N ,...,R 0 ,...,R N T ⫺N ,...,T 0 ,...,T N ), which can be readily solved using the point-by-point method described earlier. In other words, we need to satisfy the boundary conditions at 2N⫹1 appropriately selected points over the window along the randomly rough surface. These points are chosen to be at x ᐉ ⫽ᐉw/(2N⫹1), where ᐉ⫽0,1,...,2N, where w is the length of the window, i.e., the period of the pseudorandom surface profile. In order to obtain the apparent conductivity loss for onedimensional randomly rough surfaces, first we need to generate a representation of the surface profile and its slope at the 2N⫹1 selected points along the window using Eq. 共1兲. Then, we substitute these surface height and slope values into Eqs. 共25兲 and 共26兲 and invert the resulting (4N⫹2)-by(4N⫹2) matrix. Finally, we solve this set of linear equations for the zeroth-order reflection coefficient and calculate the normalized apparent eddy current conductivity from Eq. 共29兲. Figure 5 shows an example of a Lorentzian rough surface realized via Eqs. 共1兲 and 共17兲. The surface profile was generated using input parameters h⫽20 ␮ m, L⫽300 ␮ m, and w⫽30 mm. From the best-fitting analytical curve also shown in Fig. 5共b兲, the actual rms roughness and correlation length of this particular realization are h⬇19.7 ␮ m and L ⬇284 ␮ m, respectively. Figure 6 shows the apparent eddy current conductivity loss calculated by the point-by-point method for rough surfaces with four different autocorrelation functions, namely, cosine, Gaussian, Lorentzian, and exponential. Each profiles had the same rms roughness h⫽50 ␮ m and unified correlation length L⫽1000 ␮ m, though the window over which the



8347

FIG. 6. Comparison of the apparent eddy current conductivity for rough surfaces of four different autocorrelation functions with the same rms roughness h⫽50 ␮ m and universal correlation length L⫽1000 ␮ m 共the results for the exponential autocorrelation function show strong instability and are included only for illustration purposes兲.

pseudorandom profiles were periodic, w⫽10L⫽10 mm, was slightly longer than w⫽⌳⫽6L⫽6 mm for the harmonic profile. In each case the truncation was done at N⫽30. The simulation for the exponential autocorrelation function exhibited very strong numerical instabilities at high frequencies; therefore, no results are shown above 0.3 MHz at all and four different representation are presented below 0.3 MHz to indicate the significant scatter in the predicted apparent eddy current conductivity loss even at low frequencies. Of the numerically stable profiles, the Lorentzian and harmonic ones exhibit the largest and smallest highfrequency loss of apparent eddy current conductivity, respectively, which is consistent with the differences in the variance of their slope as listed in Table I. In spite of the quadruple precision used in our FORTRAN calculation, randomly rough surface profiles of exponential autocorrelation function could not be simulated reliably with the presently used point-by-point method and no further results will be presented in this paper for this case. Based on the relatively stable low-frequency results obtained for randomly rough surfaces with exponential correlation function, one can predict that the closer the simulated profile approximates the actual exponential distribution, the larger the high-frequency apparent conductivity loss becomes for a given combination of rms roughness and correlation length. Similar conclusions can be drawn from the fact that the Lorentzian profile produced a significantly stronger effect than the corresponding Gaussian profile. It should be mentioned that in order to generate these random surfaces we randomized only the phase distribution ␾ n and kept the amplitude term A n constant in Eq. 共1兲. Alternatively, we can also randomize only the amplitude distribution or both the amplitude and phase distributions to generate more realistic rough surfaces. Figure 7 shows examples of the normalized apparent conductivity for 30 statistically identical Lorentzian surface profiles (h⫽20 ␮ m, L ⫽300 ␮ m, and w⫽3 mm) generated by three different randomization methods. For rough surfaces generated by randomizing only the phase 关Fig. 7共a兲兴, the numerical results exhibit very small scattering over the whole frequency range.

FIG. 7. Examples of the normalized apparent conductivity for 30 statistically identical Lorentzian surface profiles generated by three different randomization techniques (h⫽20 ␮ m, L⫽300 ␮ m, and w⫽3 mm).

Scattering as much as five times bigger is observed in the high-frequency region for rough profiles generated by randomizing only the amplitude 关Fig. 7共b兲兴 or both the amplitude and the phase 关Fig. 7共c兲兴. In other words, the apparent eddy current conductivity is much more sensitive to the randomization of the amplitude than to that of the phase. Figure 8 shows a comparison between the ensemble average of 30 Lorentzian profiles generated by randomizing only the phase and that of 120 statistically identical surface profiles generated by randomizing both the phase and the amplitude (h ⫽20 ␮ m, L⫽300 ␮ m, and w⫽3 mm). The excellent agreement between these ensemble averages indicates that instead of averaging a large number of rough surface profiles gener-

FIG. 8. Comparison between the ensemble averages of 30 Lorentzian profiles generated by randomizing only the phase and that of 120 statistically identical surface profiles generated by randomizing both the phase and the amplitude (h⫽20 ␮ m, L⫽300 ␮ m, and w⫽3 mm).

8348


FIG. 9. Relative apparent conductivity change versus frequency for four different random profiles of Lorentzian autocorrelation function with the same rms roughness/correlation length ratio of L/h⫽30 (w/L⫽10).

ated by randomizing both the amplitude and the phase distributions, statistically reliable predictions can be obtained for the expectation value of the apparent eddy current conductivity loss by averaging a much smaller number of profiles generated by randomizing only the phase distribution. Actually, based on the very small scatter exhibited by surface profiles generated by randomizing the phase distribution only, even a single such sample of the rough surface can be regarded as an accurate representation of the whole ensemble, which obviously greatly reduces the computation time. Hereafter in this paper, any rough surface used as an example is generated by randomizing only the phase, which means the A n in Eq. 共1兲 is a constant for each spectral component that depends only on the rms roughness and the correlation length according to the previously derived rules of the given correlation function. Figure 9 shows the variation of the relative apparent conductivity change with frequency for a given rms roughness-to-correlation length ratio, h/L⫽1/30, for four different random profiles of Lorentzian autocorrelation function at a given window-to-correlation length ratio, w/L ⫽10. As expected, the measured conductivity monotonously decreases with increasing inspection frequency. This is consistent with the earlier explanation that at higher frequencies the eddy current loop becomes more tortuous in the presence of a rough surface relative to a smooth one, which leads to an apparent loss of electrical conductivity. The maximum loss of conductivity at high frequencies is related to the ‘‘sharpness’’ of the surface irregularity. For a randomly rough surface, this sharpness is determined by the ratio of the rms roughness h and the correlation length L of the surface; therefore, all curves in Fig. 9 approach the same highfrequency asymptotic value. The transition between the lowand high-frequency behaviors occurs when the eddy current penetration depth ␦ becomes smaller than the correlation length L of the surface. This behavior will be further investigated in Sec. VII. V. NUMERICAL CONVERGENCE

The 4N⫹2 linear equations obtained from Eqs. 共25兲 and 共26兲 can be readily solved for the zeroth-order reflection co-


FIG. 10. Numerical results obtained for the relative apparent conductivity change on a one-dimensional Lorentzian random surface profile of h ⫽20 ␮ m and L⫽300 ␮ m using 2N⫹1 points along the w⫽3 mm long window.

efficient R 0 by inverting the (4N⫹2)-by-(4N⫹2) boundary condition matrix. Then, the normalized apparent electrical conductivity of the material can be easily determined from R 0 . Unfortunately, severe numerical instability problems can occur in the calculation due to the inversion of the large complex matrix even when quadruple precision is used. Specifically, we have two contradictory constrains when choosing the size of the boundary condition matrix. First, we need to chose a sufficiently large number (2N⫹1) of points where the boundary conditions are exactly satisfied and we must truncate the number of diffraction orders included in the boundary conditions at the same number. Second, we have to limit the size of the boundary condition matrix to avoid numerical instabilities due to the finite precision of the calculations. As a result, we only have a finite window to run the simulation for a given correlation length, which in turn reduces the random nature of the pseudorandom surface profile. In order to illustrate the numerical instability problem in terms of the truncation value N, Fig. 10 shows the results obtained for a Lorentzian random surface profile with rms roughness h⫽20 ␮ m, correlation length L⫽300 ␮ m, and window length w⫽3 mm at four different truncation values. We found that the combination of a window-to-correlation length ratio of w/L⫽10 and a truncation level of N⫽30 assures good numerical stability; all the calculations presented elsewhere in this paper were obtained using this combination. Figure 10 also shows that at a higher truncation value, N⫽38, numerical instability appears at lower frequencies, while at a lower truncation value, N⫽15, the numerical results do not converge at high frequencies. For N⫽30, the size of the boundary condition matrix that must be inverted is 122⫻122. We should also mention that in order to simulate rough surfaces with the exponential correlation function, the discretization interval should be smaller than one-tenth of the correlation length.24 Such high sampling resolution combined with the minimum window-to-correlation length ratio of w/L⫽10 needed to assure the random nature of the generated surface profiles would require N to be higher than 50, which explains why we could not use the point-by-point method to predict the apparent eddy current conductivity loss



FIG. 11. Relative apparent conductivity change versus frequency curves showing the characteristic high-frequency plateau for one-dimensional Lorentzian random surface profiles of different rms roughnesses (L ⫽1 mm and w⫽10 mm).

on rough surfaces with exponential autocorrelation function. In the future, further efforts will be directed at overcoming this limitation in order to better predict the apparent conductivity loss observed on real shot-peened surfaces. VI. HIGH-FREQUENCY PLATEAU, KIRCHHOFF APPROXIMATION

As noted previously in connection with Fig. 9, above a certain transition frequency the apparent conductivity approaches a constant asymptotic value, ␥ ⬁ . This phenomenon is further demonstrated in Fig. 11, which shows that the high-frequency plateau is determined by the h/L ratio and the transition frequency is determined by the correlation length, L. Above this transition frequency, the penetration depth of the eddy current becomes negligible with respect to the correlation length of the randomly rough surface, and the apparent loss of conductivity can be approximated as a simple geometrical effect due to the tortuous path followed by the eddy current. From a physical point of view, this approximation is analogous to the so-called ‘‘thin-skin’’ limit, which is often used in eddy current inspection to evaluate the coil impedance in terms of surface crack properties.23,25–27 The high-frequency asymptotic value of the normalized apparent eddy current conductivity can be estimated in the well-known Kirchhoff approximation as follows:11

␥ ⬁ ⬇ 1⫺ ␸ 2 ,

共30兲

where ␸ is the previously introduced variance of the surface slope, which is listed in Table I for the different autocorrelation functions considered in this paper. Equation 共30兲 can be readily compared to numerical results obtained by the pointby-point method. For a rough surface with Lorentzian autocorrelation function, the high-frequency asymptotic value of the normalized apparent eddy current conductivity can be easily calculated from the rms surface roughness and the correlation length using Eq. 共18兲 as ␥ ⬁ ⫽1⫺2h 2 /L 2 . Figure 12 shows a comparison of the point-by-point method and the Kirchhoff approximation for the high-frequency asymptotic value of the relative apparent conductivity change for three different correlation lengths as a function of the rms rough2

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FIG. 12. The high-frequency asymptotic value of the relative apparent conductivity change for three different one-dimensional Lorentzian random corrugations as a function of the surface rms roughness. The solid lines were calculated from the numerical model at a sufficiently high frequency ( f ⫽10 MHz), while the dashed lines were calculated from the Kirchhoff approximation.

ness (w/L⫽10). The solid lines were calculated using the point-by-point method at a sufficiently high frequency of 10 MHz, whereas the dashed lines were calculated from the Kirchhoff approximation. The results show an excellent agreement between the two methods when the h/L ratio is relatively small. VII. EXPLICIT AD HOC APPROXIMATION

From a practical point of view, it would be highly beneficial if we could directly predict the apparent eddy current conductivity loss from the statistical parameters of the randomly rough surface, namely the rms surface roughness h and the correlation length L, which can be easily measured by optical means. We found that the apparent conductivity loss for one-dimensional randomly rough surfaces can be approximated by the following explicit ad hoc function, which was previously suggested for harmonic surface corrugations:11

␥共 f 兲⫽

1⫹ f / f 2 , 1⫹ f / f 1

共31兲

where f 1 and f 2 are characteristic frequencies that depend on the statistical surface parameters and the relevant material properties. At very low and very high frequencies, the results calculated from this ad hoc formula approach ␥ 0 ⫽1 and ␥ ⬁ ⫽ f 1 / f 2 , respectively. For a randomly rough surface profile of Lorentzian autocorrelation function, the transition from low-frequency to high-frequency behavior occurs when L/ ␦ ⬇1.2, where ␦ ⫽1/冑 f ␲ ␮ ␴ is the standard penetration depth of the eddy current distribution. Therefore, we can calculate f 1 from f 1⫽

1.44 , ␮ ␲␴ L 2

共32兲

and subsequently f 2 from f 2⫽

f1 . 2h 2 1⫺ 2 L

共33兲

8350



exponential autocorrelation function. However, these efforts will have to yield significantly improved numerical accuracy and stability because of the nonfinite nature of the slope variance on such surfaces.

VIII. CONCLUSIONS

FIG. 13. Comparison between the numerical solution 共solid lines兲 and the explicit ad hoc approximation 共dashed lines兲 for four one-dimensional Lorentzian random corrugations with different values of h at L⫽1000 ␮ m (w/L⫽10).

A comparison between the numerical solution 共solid line兲 and the explicit ad hoc approximation 共dashed line兲 is shown in Fig. 13. Fairly good agreement is observed except for some overestimation at large h/L ratios. In spite of this minor discrepancy, considering the expected uncertainty of eddy current conductivity measurements due to the random nature of rough surfaces, the proposed explicit ad hoc approximation is quite suitable for predicting the apparent conductivity loss in many practical situations. Our ultimate goal is to simulate the apparent conductivity loss caused by randomly rough surfaces of shot-peened specimens, which in turn is necessary to increase the accuracy of near-surface residual stress assessment, especially after thermomechanical relaxation. The Lorentzian autocorrelation function is closer to the exponential autocorrelation function exhibited by typical shot-peened surfaces than either the cosine or the Gaussian autocorrelation functions. This better similarity is also demonstrated by the considerably higher asymptotic value of the apparent eddy current conductivity loss on Lorentzian rough surfaces. Still, at high frequencies the Lorentzian model badly underestimates the apparent conductivity loss measured on shot-peened surfaces of exponential autocorrelation function because the latter one does not exhibit a finite high-frequency asymptote.10 In Ref. 11, the randomly rough surface was crudely modeled as a one-dimensional harmonic profile and the resulting discrepancy was bridged by overestimating the corrugation amplitude based on the rms roughness as a⬇3.3h and underestimating the period based on the correlation length as ⌳⬇2.5L, instead of the more realistic a⫽&h and ⌳⫽6L listed in Table I. A similar, albeit less severe, adjustment is still needed to bring the theoretical predictions based on assuming a Lorentzian autocorrelation function into reasonable agreement with experimental results obtained on shot-peened surfaces of exponential autocorrelation function. The improvement is due to the fact that the same rms roughness and unified correlation length result in roughly twice as much apparent eddy current conductivity loss for a Lorentzian rough surface as for a harmonic corrugation. Clearly, further efforts are needed to better simulate the strong highfrequency spectral contents of randomly rough surfaces of

The measured apparent eddy current conductivity change in shot-peened specimens is a combination of residual stress, cold work, and surface roughness effects; therefore, it is important to understand and correct for this geometric artifact in eddy current nondestructive assessment of near-surface residual stresses. We first considered various randomly rough surfaces with different autocorrelation functions, and in each case determined the variance of the surface slope. We found that a 1D randomly rough surface of exponential autocorrelation function does not possesses a finite slope variance; therefore, such a surface is not expected to exhibit a finite high-frequency asymptotic value in the apparent eddy current conductivity loss either. This conclusion is consistent with previously published experimental observations. We introduced a numerical method based on the Rayleigh approximation for calculating the apparent eddy current conductivity loss exhibited by 1D randomly rough surfaces. The relevant boundary conditions were satisfied using the so-called point-by-point technique. This technique was first verified on a harmonic corrugation, then extended to randomly rough surfaces of Gaussian and Lorentzian autocorrelation functions. Unfortunately, due to the slow decay of the high-frequency spectral components of a randomly rough surface of exponential autocorrelation function, we could not extend the point-by-point method to study this spurious effect on exponential surfaces. Therefore, we approximated the rough surface as a one-dimensional random corrugation with Lorentzian autocorrelation function, which is closer to the actual surfaces of shot-peened specimens than either Gaussian or harmonic surfaces. We have found that the best approach to generate one-dimensional rough surfaces is to randomize only the phase of the spectral components, which significantly reduces the computation time. We also found that the previously developed explicit ad hoc function11 could be adapted to predict the apparent conductivity loss on one-dimensional randomly rough surfaces, which allows us to directly estimate the expected apparent conductivity loss from independently measured surface parameters. In the future, this approach could lead to simple surface roughness corrections that can be used to increase the accuracy of residual stress measurements in shot-peened metals.

ACKNOWLEDGMENTS

This work was supported by the Air Force Research Laboratory partly under Contract No. F33615-03-2-5210 with the University of Cincinnati and partly through the Quantitative Inspection Technology for Aging Military Aircraft program with Iowa State University.

J. Appl. Phys., Vol. 95, No. 12, 15 June 2004 1

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