Statistical analysis improves roughness measurement

Using statistical analysis, we have identified a potential source of error in the way automotive manufacturers measure surface roughness. Through our analysis, which we presented at the March SAE Show (Ref. 1), we developed an improved measurement technique that should help automakers more accurately predict how friction affects production of metal parts during forming.

Figure 1. These optical micrographs show two surfaces fwith the same average roughness values but dramatically different surface topographies. Courtesy of the National Institute of Standards and Technology.

The auto industry often tries to draw specific conclusions about the roughness of a surface with mathematically averaged data that are too general, a process that can produce the same roughness values for two surfaces with markedly different topographies (Figure 1). This can lead to inaccurate predictions of friction, because the surface-roughness measurement only quantifies the average peak heights and valley depths, and not the spacing between them.

Currently, most engineers measure surface roughness with contact stylus profilometry. This technique is common because the instruments are generally less expensive than other roughness measurement instruments.

The biggest source of error in measuring roughness with a contact stylus profilometer is the way in which the surface roughness is calculated from the measurement data. Most analyses of roughening behavior are derived from linear profiles. In addition, the assessments of the surface roughness in many analyses are based on the arithmetic mean of the height profile, or the Ra parameter:

where L is the length over which the profile is evaluated, and z(x) is the magnitude of the profile height.

The average profile height can also be represented by Rq, which is based on the root mean square (rms). The rms roughness is expressed as:

Both Ra and Rq collapse an entire roughness profile into a single value, and different surface morphologies can produce the same roughness value.

To see if we could improve the modeling process, we performed a series of both biaxial and uniaxial strain experiments using unpolished, commercially available cold-rolled AISI 1010 steel sheet with a 1-mm nominal thickness.

For the biaxial strain experiments, we used a modified Marciniak in-plane stretching test. The initial sheet size was 30x30 cm, and we tested the samples in the as-received condition. We stenciled a circle with a 28-mm diameter in the center of the sheet for strain determination. For the uniaxial strain experiments, we punched flat sheet, tensile specimens with the tensile axis perpendicular to the rolling direction of the sheet.

After straining the samples, we examined the specimens with a scanning laser confocal microscope (SLCM) to measure how much the surface topographies had changed.

We generated a map of the surface with x-y-z dimensions of 1000x800x20 µm. The z-spacing between focal planes in each image was nominally 0.1 µm. We generated topographic images from the surface intensity maps and linear roughness profiles of a nominal 750-µm length collected from those topographic images.

Figure 2. Surface roughness changes as a function of the strain induced on the samples.

Figure 2 shows the changes in surface roughness for the as-received surfaces, under both biaxial and uniaxial straining conditions. To correctly calculate the change in surface roughness, we subtracted the mean value of the initial roughness from all the subsequent roughness measurements. Note that the error bars in Figure 2 represent the statistical uncertainty in the roughness measurements, not the measurement uncertainty.

The variation in the roughness data shown in Figure 2 raises the question of how well the changes on the surface are represented by the Rq parameter. If the roughness profile is assumed to be a random distribution of heights, it becomes possible to describe the variation of the profile heights with simple statistics. Because the distribution of heights on most rough surfaces is approximately Gaussian, we used a statistical approach to represent all of the heights within a profile. We fitted the measured probability density data to a normal distribution function:

where p is the probability density of realizing a height (z), z is the magnitude of the height at a location along the profile length, and µ and s are the mean and standard deviation of all the height values in the profile, respectively.

Using this equation, we plotted the PDF distributions as a function of plastic strain level for both the uniaxial strain experiments and the biaxial strain experiments. The maximum probability density represents the number of "counts" that occurred at the mean value (i.e., the peak of the bell curves).

Given that the surfaces were not assumed to be perfectly Gaussian, deviations from the ideal condition are not surprising, but the relatively small magnitudes of the deviations demonstrate that the approach is valid and that the fit is robust. The trends exhibited by the PDF curves demonstrate that the range of probable heights—that is, the width of the distribution—increases substantially as a function of plastic strain.

This is consistent with the behavior observed in the height profile data. Since the PDFs have been normalized, their flattening suggests that the magnitude of the deviation from the mean increases monotonically with the surface roughness or plastic strain. This behavior also directly reflects an important characteristic about the Rq parameter. In a Gaussian profile, the region of the curve enclosed by one standard deviation of the mean (e.g., Rq) corresponds to 68% of the total distribution. The results of this PDF analysis demonstrate that the fit to a Gaussian form is quite accurate for the 1010 steel, and under these conditions, Rq is an accurate measure of the overall roughening behavior.

Figure 3. Probability density functions showing the relationship between plastic strain and the distribution of profile heights represented by the rms roughness Rq for a) a narrow Gaussian distribution and b) a broad Gaussian distribution.

As shown in Figure 3a, at low levels of roughness, the probability density at the peak is quite high, resulting in short tail regions. Now, look at Figure 3b. As the roughness increases, the probability density at the mean is considerably lower than that for the smooth surface, and the statistical significance of the tails of the PDF is much greater.

Neither the Ra nor the Rq parameter provides any information about the spatial or textural distribution of the features (peaks and valleys) on the surface. A simple spatial rearrangement of the order of the profile heights will have no influence on any of the parameters associated with the PDF. Other measures such as height autocorrelation functions, however, will exhibit substantial changes with such a rearrangement. Such parameters could also reveal distinctions in the roughness data resulting from the deformation mode. That is, the autocorrelation function is likely to be quite different for deformation produced under biaxial conditions than it is for the same strain level produced under uniaxial conditions.

The results of our evaluation indicate that a Gaussian height distribution provides an accurate characterization of the surface roughness for this material. But this type of analysis only reflects the behavior with respect to the mean and it does not provide any information about the occurrence or the distribution of the peaks and valleys in the profile data.

We acquired the roughness data for this evaluation at a resolution and length-scale where the slip steps and grain boundary effects were expected to be a larger fraction of the measurement, yet these individual contributions to the surface roughness did not appear to have any appreciable influence on the distribution analyses. This is probably due to an averaging effect produced by the relatively large sampling of the surface topography with respect to the small grain size of the 1010 steel, and combined with the large number of data points used in the fit, the result was a near-ideal Gaussian behavior. Consequently, other materials with larger grain sizes, such as aluminum, may not produce such ideal behavior, and greater caution must be exercised in both the construction of the height distribution and the use of a single average parameter to describe the overall surface roughness.