Electric Voltage Predictions and Correlation with Density. Measurements in Green-State Powder Metallurgy Compacts. Reinhold Ludwig, Georg Leuenberger, ...
Electric Voltage Predictions and Correlation with Density Measurements in Green-State Powder Metallurgy Compacts Reinhold Ludwig, Georg Leuenberger, Sergei Makarov and Diran Apelian* Powder Metallurgy Research Center (PMRC) Department of Electrical and Computer Engineering *Metal Processing Institute Worcester Polytechnic Institute Worcester, MA 01602 USA
ABSTRACT In this research we discuss an electrostatic measurement approach whereby electric current is injected into green-state compacts and the resulting surface voltages are recorded in an effort to determine density distributions. We present results for pure iron powder with and without lubricants compacted to various densities and their corresponding measured conductivity values clearly establishing correlation. The electric conductivity recordings will ultimately be utilized to infer density distributions throughout the green-state P/M compact. The constant current is injected through point and aperture electrodes and voltages are recorded along the surface of the compact. To gain confidence in the underlying physical concepts, the recorded voltages of the controlled cylindrical samples are compared with a mathematical Green’s function model involving an analytical electrostatic solution of Poisson’s equation.
Key words: Green-state powder metal compacts, density distributions, resistivity testing, cylindrical samples, Green’s function modeling.
1. Introduction A green-state compact is an intermediate step in the metallurgical manufacturing process of powder metallurgy parts. During compaction, metal powder is compressed to a particular density and geometric shape, known as the green-state compact. These compacts are subsequently heated in special furnaces, a process that typically lasts several hours. The heat treatment, or sintering, ensures that the sample attains its desired final metallurgical properties. From a quality assurance point of view, the process engineer desires the detection of material flaws and density variations early on in the manufacturing process, preferably immediately after compaction. For this purpose, an electric conductivity (or resistivity) test has been devised, whereby constant current is injected into the part in various directions and surface voltages are recorded in an effort to correlate these voltage deviations with abnormalities in the compact such as surface, even subsurface cracks and defects [1-6]. Electrostatic current flow and voltage recording is feasible because of the relatively high resistivity (or equivalently low conductivity) of the green-state compacts. Crack detection recording in this context has to be understood as a perturbation of a baseline voltage that has been established over several unflawed compacts. The purpose of this research is to investigate whether electric current flow can be utilized to detect density variations and the presence of lubricants. In particular, we are interested in •
developing a theoretical formulation that permits the prediction of voltage and current distributions throughout the compact for a cylindrical greenstate compact,
•
establishing a correlation between density and electric conductivity, and
•
examining the influence of type and amount of lubricant on the compact’s electric conductivity.
To address these three items, we develop in Chapter 2 a Green’s function approach for a cylindrical sample subject to a point current injection and reception at its top and bottom, respectively. Numerical predictions will show that non-uniform current and voltage distributions can be observed in the cylindrically shaped P/M compact.
2
Chapter 3
discusses practical voltage measurements of controlled green-state iron-powdered compacts of various densities under fixed lubricant mixtures. The goal is to establish a correlation between conductivity (which is the inverse of resistivity) and density. In addition, voltage measurements are compared with the theoretical predictions. Finally, the influence of different types and amounts of lubricants are investigated. Chapter 4 summarizes the results and points out future research directions.
2. Theoretical Foundation 2.1. Basic equations The electric current flow through a P/M compact can be cast in terms of an electrostatic model formulation of Laplace’s equation whereby the surface currents and voltages represent boundary conditions, and the conductivity σ is in general spatially nonuniformly distributed throughout the part. The voltage ϕ(r), as a function of the spatial observation vector r = r(x,y,z), is therefore given by the well-known Laplace equation
![# (r )!" (r )] = 0
(1)
The practically relevant boundary condition involves a prescribed current density input Jn = "
%! = $"#! %n
(2)
over an otherwise flux free surface. This current density is normal to the sample surface whose surface normal n is pointing outwards. An alternative approach involves Poisson’s equation where the current excitation is incorporated as part of the right hand side source term
"[% (r )"$ (r )] = ! I# (r ! r0 )
(3)
which for homogeneously conductive samples reduces to
3
% 2# (r ) = $ I" (r $ r0 ) / !
(4)
Here r0 denotes the location of the current source I. To develop a potential solution in a cylindrical (r,θ, z) coordinate system of total radius R and length L subject to flux free (or Neumann-type) boundary condition, we develop a spectral solution by utilizing eigenfunctions of the form
$ mnp (r ) = J m (# mn r / R) cos(m" ) cos( p!z / L)
(5)
Here J m ( !mn r / R ) is the Bessel function of order m. Index n denotes the zeros of the first derivative of the Bessel function, i.e. Jm’( !mn ) = 0, as required to satisfy the flux-free boundary condition. These functions can be expanded as part of a Green’s function expansion G (r | r0 ) =
I &
"
! ($
m,n , p
% mnp (r0 )% mnp (r ) mn
(6)
/ R) 2 + ( p# / L) 2
where the overbar is used to denote an orthonormal set. Function ! mnp is found by applying the orthonormality conditions R
) J m (+mn r / R) J m (+mn! r / R) rdr = 0
R2 2
' m2 %%1 ( 2 +mn &
$ 2 "" J m ( +mn )* n n! #
(7)
2#
! cos(m& ) cos(m"& )d& = 2#%
m m"
(8)
/$m
0
L
1
% cos( p#z / L) cos( p'#z / L) dz = L % cos(#px) cos(#p$x) dx = L" 0
0
In (8) and (9) we use the Neumann factor #1, m = 0, p = 0 $m ,$ p = " ! 2, otherwise
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p p$
/! p
(9)
This leads to the orthonormal eigenfunctions + ( % % $ m$ p % % ! mnp (r ) = * '! mnp (r ) 2 %# LR 2 1/1 2 m ., J 2 (" ) % / "2 , m mn % % mn 0 ) &
(10)
Substituting (5) and (10) into (6) permits us to develop a series expression in the form
G (r | r0 ) =
I &
"
!G
mnp
J m (% mn r / R) cos(m$ ) cos(#pz / L) J m (% mn r0 / R) cos(m$ 0 ) cos(#pz 0 / L) (11)
m ,n , p =0
where the coefficient Gmnp is a combination of the orthonormality condition (10) and the eigenfrequency expression in (6). Explicitly we obtain Gmnp =
# m# p ) m &) !2mn " 2 p 2 & 2 ' " LR '1 * 2 $$'' 2 + 2 $$ J m (! mn ) L % ( ! mn %( R 2
2
(12)
A simplification of expressions (11) and (12) can be achieved if the cylinder is axisymmetric. Since this implies independence of angle θ, we can write
G (r | r0 ) =
I #
%
$G
np
J 0 (" n r / R) cos(!pz / L) cos(!pz 0 / L)
(13)
n =0, p =0
where Gnp =
#p ) !2 " 2 p 2 & " LR 2 '' n2 + 2 $$ J 02 (! n ) L % (R
(14)
2.2. Current flow through a three-dimensional cylinder The electric voltage predictions are conducted based on the generic test arrangement shown in Figure 1. In particular, point and rod electrodes are employed to initiate the current flow distribution throughout the compact of uniform conductivity. If a uniform current flow is injected into the sample (Figure 1, right), we can expect a simple current-voltage relation based on Ohm’s law 5
V =
I L ! A
(15)
where A, L denote, respectively, sample surface area, and length. For the point electrode excitation (Figure 1, left) the theoretical Green’s function model (13) with flux-free boundary conditions solving (4) has to be utilized.
2.3. Numerical predictions The above derivation (13) in conjunction with (2) allows us to plot the magnitude of the total current density (Figure 2, left) and the associated voltage distribution (Figure 2, right) throughout the axis-symmetric interior of the compact. The current injection is a single point electrode applied normal to the surface, acting along the center of the cylinder. Clearly observable is the non-uniform field behavior, exhibiting a source region of maximum strength at the left hand side, and a sink region of minimum strength at the right hand side of the cylinder. Of particular interest, however, is the voltage along the surface since this ultimately forms the basis of the model comparison with measurements. The functional behavior of the voltage is non-linear as seen in Figure 3 for a 1-to-5 diameter-to-length (D/L) ratio. Based on the Green’s function model, equation (3), voltage predictions can be made for different diameter-to-length (D/L) aspect ratios. Specifically, Figure 4 provides voltage distributions along the outer surface of the sample as a function of D/L. As expected, for small D/L ratios the voltage distribution follows almost a lumped element distribution as predicted by Ohm’s law (15). As the ratio becomes larger, the electric field begins to show a flow pattern with less voltage gradients seen in the corners. If the voltage is recorded along the radial direction for the left hand end surface, we notice the rapid drop in magnitude as the (D/L) ratio increases. The practical implications of these simulations are such that long samples approach a one-dimensional behavior that can suitably be described by Ohm’s law. A different voltage prediction is obtained by plotting the radial voltage drop from the positive current injection point to the outside perimeter R of the upper cylinder surface. Figure 5 depicts the distribution as a function of R/L ratios.
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Finally, to test the analytical model against measurements, a generic test arrangement was configured as illustrated in Figure 6. The voltage controlled current source is capable of producing currents up to 2.5 A, and a precision bench voltmeter with microvolt resolution is employed to record the voltages. Comparing the voltage predictions with measurements, a green-state cylinder of pure iron compacted to 6.5 g/cm3 and with dimensions of D = 1.5 cm and L = 6 cm is chosen and subjected to a constant current of I = 1 A supplied through point electrodes on the top and bottom surfaces. Figure 7 compares actual voltage measurements with scaled theoretical predictions along the entire length of the sample. Voltages recorded in radial direction on the top or bottom surfaces drop off very quickly, rendering experimental voltage measurements impractical and thus preventing a reliable experimental verification. .
3. Experimental Measurements 3.1. Preparation of controlled samples To investigate the correlation between density and electric conductivity in more detail, controlled cylindrical samples (diameter D = 6cm, length L = 1.5 cm) were prepared of pure iron powder with and without lubricant (shown in Figure 8). The nominal compaction density of these parts ranges from 6 g/cm3 to 7.2 g/cm3. Each specimen density was replicated three times to take into account possible manufacturing variations. In a second batch, parts were produce from 1000B iron powder with densities ranging from 6.6 g/cm3 to 7.4 g/cm3. One set of these parts was produced with iron powder without lubricants, and the other sets with iron powder that was mixed with 0.3%, 0.5%, and 0.75% Acrowax lubricants, respectively.
3.2. Electrical Contact The measurement arrangement is again based on Figure 6 where the current is applied through full surface electrodes as illustrated in Figure 1 right. Several different voltage sensor configurations were chosen to measure the surface voltage. The simple 2-
7
pin sensor, shown in Figure 9 top left, provides the common setup used for recording the voltage drop along a fixed length. Unfortunately the measured parts were sensitive to placement variations as to where on the surface the voltage drop was recorded. In an effort to improve the repeatability of the measurements, the second sensor, as depicted in Figure 9 bottom left, with two conducting ring electrodes along the circumference was chosen. In an attempt to further increase reliability of the results and also to better accommodate the semi-automated measurement setup in the bench press, a third sensor was developed. This sensor employed a dual electrode sensor with an isolated center pin, surrounded by the current injecting outer circle. This sensor enables the voltage recording from top to bottom as seen in Figure 9 right.
3.3. Results The measurements evaluating the conductivity of green-state parts in relation to their density were undertaken on cylindrical discs, shown in Figure 8, with a diameter to length ratio of approximately 4:1. This geometry was chosen because the large diameter/length ratio provided the maximum achievable homogeneity of the part. Another reason was the direct applicability of Ohm’s law for conductivity reconstruction when using the measurement setup depicted in Figure 1 right, therefore reducing measurement uncertainties. The measurements were carried out for all parts at three different current levels. From these measurements it can be concluded that density and conductivity follow a linear correlation over the given density range for non-lubricated parts, as expected (see Figure 10). It is interesting to note that the amount and type of lubricants in the green-state samples not only influence the absolute value of the conductivity but also the nature of the density versus conductivity correlation. Figure 11 depicts the situation where the same sample is measured as shown in Figure 10, but with 0.75% Acrowax lubricant added. As seen, the conductivity exhibits a more complex behavior, which can best be approximated by a parabolic functional description. Contrary to the non-lubricated case,
8
we find a maximum in conductivity for a density of approximately 6.8 g/cm3. If the density is increased beyond this point, the conductivity begins to decrease again. This correlation between density and conductivity not only depends on the presence of a lubricant, but also on the type and the amount of lubricant as shown in Figure 12. Here the influence of Acrowax versus ZnSt and a difference in concentration ranging between 0.375% and 0.75% Acrowax are depicted.
4. Conclusions An electrostatic measurement approach is presented which yields a clear correlation between electric conductivity and sample density of green-state P/M parts. The technique is highly sensitive in that small changes in material density translate into large changes in electric conductivity. Furthermore, tests with controlled samples of various aspect ratios show good agreement with the classical electrostatic theory, whereby the current flowing through the compact can be treated as a curl-free electric field distribution scaled by the electric conductivity. Preliminary measurements indicate that the presence of lubricants significantly affect conductivity in a non-linear way. This warrants further investigations as to how various amounts of lubricants influence low-density to high-density transitions. Our measurements point to a marked change in the functional density-conductivity behavior for compaction densities exceeding 6.8 – 6.9 g/cm3.
Acknowledgement The researchers are grateful to Dr. L. Pease of Powder-Tech Inc., Andover, MA, for having delivered the controlled samples. Furthermore the authors express their appreciation to the Powder Metallurgy Resurch Center (PMRC) at WPI for having provided funding for this research.
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References [1]
G. Bogdanov, R. Ludwig, and W. Michalson, “A new apparatus for nondestructive evaluation of green-state powder metal compacts using the electrical-resistivity method,” Measurement Science and Technology, Vol. 11, pp. 157 – 166, 2000.
[2]
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. 16B, pp. 2005 - 2012, Plenum Press, 1997.
[3]
S. Makarov, R. Ludwig, and D. Apelian, “Numerical Solution Of A Direct 3D Electrostatic Resistivity Test Of Green-State Metal Powder Compacts,” in Review of Progress in Quantitative NDE, Vol. 17B, pp. 1462 - 1470, Plenum Press, 1998.
[4]
R. Ludwig, S. Makarov, and D. Apelian, “Three-dimensional solution and experimental confirmation for the electric resistivity testing of surface-breaking defects in green-state powder metallurgy compacts,” Journal of Nondestructive Evaluation, Vol. 17, No. 2, pp. 153 – 166, 1998.
[5]
R. Ludwig and D. Apelian, “A novel crack detection methodology for green-state powder metallurgy compacts using an array sensor electrostatic testing approach,” International PMTech Conference, Vancouver, BC, July 1999.
[6]
R. Ludwig et al. “Multi-probe impedance measurement system and method for detection of flaws in conductive articles,” US patent 6,218,846, April 17, 2001.
[7]
P. Morse, H. Fehrbach, „Methods of Theoretical Physics“, McGraw-Hill, 1953.
[8]
R. M. German, “Powder Metallurgy Science”, Metall Powder Industries Federation, 1984.
[9]
R. Ludwig, G. Bogdanov, and D. Apelian, “Nondestructive electrostatic determination of surface breaking and subsurface flaws in green-state P/M compacts,” International PMTech Conference, Las Vegas, NV, June 1998.
[10] K. Ikeda, M. Yoshimi, and C. Miki, “Electric potential drop method for evaluating crack depth, “ International Journal of Fracture, Vol. 47, pp. 25 – 38, 1991. [11] R. Ludwig, S. Makarov, G. Leuenberger, and D. Apelian, “Electric conductivity predictions and measurements in green-state powder metallurgy compacts,” to be submitted.
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Figure captions
Aluminum rod
V
I
V
I
!
-2e+004
-5e+004
-4e+004
-3e+004
-1e+004
Figure 1:
0.0000
1e+004
3e+004
4e+004
5e+004
I
! -2e+004
L
-1e+004
-5e+003
0.0000
5e+003
1e+004
2e+004
2e+004
Model development for controlled green-state samples. The compacts receive the current excitation either through point contact copper electrodes (left), or blocks of aluminum rods covering the entire surface of the sample (right).
r z
Figure 2:
Current density (left) and voltage (right) distribution for a cylindrical compact with a length of 3cm and a diameter of 2cm when excited by a point current source of 1A.
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Normalized Voltage
1 0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6 -0.8 -1 0
0.2
0.4
0.6
0.8
1
Position [z/L]
Figure 3: Predicted normalized surface voltage for the model shown in Figure 2.
Normalized Voltage
1 0.8 0.6
Ratio 1:0.1
0.4 0.2
Ratio 1:6
Ratio 1:1
0 -0.2 -0.4 -0.6 -0.8 -1 0
0.2
0.4
0.6
0.8
Position [z/L]
Figure 4:
Voltage distribution along the outside surface of the cylinder (normalized values) as a function of various D/L ratios.
12
1
0
0.2
0.4
0.6
0.8
1
Voltage Drop [dB]
0 -20 Ratio 1:0.5
-40
Ratio 1:1 Ratio 1:2 Ratio1:4 Ratio 1:6 Ratio 1:8
-60 -80 -100 -120 -140 -160 Position [r/R]
Figure 5:
Theoretical voltage predictions on the face of a cylindrical sample as measured from the center outwards. Calculations are shown for various D/L ratios.
Voltage Source
+
Precision Bench Voltmeter
Voltage Controlled Current Source
+
I+
Figure 6:
+
-
-
I-
DUT
Schematic block diagram of the measurement arrangement.
13
2000
Voltage [microV]
1500
theory measurement
1000 500 0 -500 -1000 -1500 -2000 0
0.2
0.4
0.6
0.8
1
Position [z/L]
Figure 7:
Comparison between theoretical voltage predictions and measurements along the surface of a cylindrical sample of L = 6 cm and D = 1.5 cm.
Figure 8:
Controlled green-state PM compacts used for the conductivity measurements.
14
V
V
I
Part under Test
V Figure 9:
Three different sensor types used to measure voltage on the surface of green state samples
60000
Conductivity [S/m]
50000 40000 30000 20000 10000 0 6
6.2
6.4
6.6
6.8
7
7.2
Density [g/cm3]
Figure 10: Electric conductivity versus density point electrode measurements averaged over three current strengths (0.5, 1, 2 A) and three green-state compacts for each density. The compacted powder is pure iron without any lubricants.
15
Conductivity [S/m]
15000
10000
5000
0 6
6.2
6.4
6.6
6.8
7
7.2
Density [g/cm3]
Figure 11: Conductivity versus density for measurements for the same samples as shown in Figure 5, but with 0.75% Acrowax lubricant added.
14000
Conductivity [S/m]
12000 10000 8000 6000 4000 2000 0 6
6.2
6.4
6.6
6.8
7
7.2
Density [g/cm3] 0.75%AW
0.375%AW
0.75%ZnSt
Figure 12: Comparison between the conductivity of green-state samples with different amounts and different types of lubricants.
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