Hertz Stress II: Achieving the Right Degree of Resistance
By Dr. Robert S. Mroczkowski, Director Technology, Consultant, Bishop & Associates Inc.


In the first Hertz stress article, I cited two limitations of Hertz stress as a connector design parameter. First, a Hertz stress value can be realized by different combinations of contact force and contact geometry, each producing different values of connector contact resistance. Second, variations in contact force and geometry can result in significantly different connector durability (wear) and contact mating forces. This article will address these limitations. The figures and discussion in this article are taken from my paper “Concerning ‘Hertz Stress’ as a Connector Design Parameter.”

Consider contact resistance as a function of Hertz stress. In Figure 1 the phrase “apparent stress” is used instead of Hertz stress because some of the assumptions in Hertz stress calculations are not valid for connectors. Figure 1 includes two graphs of contact interface resistance versus apparent stress. The graph on the left is for a constant geometry interface with contact force as a variable and the graph on the right for a constant contact force with contact geometry as the variable parameter.

The graph on the left, with contact force as the variable, is the more commonly shown relationship, and contact interface resistance decreases with increasing force, as would be expected. This occurs because the contact area increases as force increases, so the contact resistance decreases with increasing force.

The graph on the right is more interesting. A constant force means a constant metal-to-metal contact area. But because on the microscale of the contact interface all surfaces are rough, multiple metal-to-metal contact spots, called a-spots, are created. The variation in geometry, in turn, produces a different distribution of the a-spots and, therefore, a variation in contact resistance. In this case, an increase in apparent stress is realized by decreasing the apparent contact area, resulting in an increase in contact interface resistance.

Figure 2 shows the distribution of a-spots in two different sphere-to-flat contact interfaces at the same apparent stress. The white circles indicate the distributional area within which the a-spots are contained. Notice the background roughness of the surface of the flat contact. The metal-to-metal contact areas are the dark spots inside the white circles. They appear dark optically because they are the flattened tops of the surface roughness peaks of the flat surface. 

The left contact was created by a smaller radius sphere and, therefore, has a smaller apparent contact area than that of the larger radius sphere contact on the right. To create a constant apparent stress, a lower force was used on the left contact than on the right. It is clear that the left contact has a smaller amount of metal-to-metal area than the contact on the right, as is expected.

In summary, the contact on the left has a smaller metal-to-metal area and a smaller distributional area, therefore a higher contact interface resistance than the contact on the right, even though they have the same apparent stress. In other words, the contact interface resistance depends on how the apparent stress was realized, and not its magnitude.

It was also noted in the previous article that Hertz stress considerations do not address the critical performance characteristics of mating durability (wear) and mating force. The discussion of mating force is straightforward. Mating force depends directly on the contact force, the coefficient of friction (which can vary with contact force), and the contact interface geometry. As just noted, a given Hertz stress value can be realized by different combinations of force and geometry, so has no correlation with mating force.

Wear mechanisms also vary with how Hertz stress is realized. Figure 3 demonstrates this dramatically. The wear tracks shown in Figure 3 are at a constant apparent stress of approximately 70,000 psi, about half the value recommended as a minimum in the previous article.

The wear tracks were generated by mechanical motion over an initial track length for 1,000 cycles. The length was then reduced after an additional 1,000, 2,000, and 4,000 cycles, giving the four intervals shown in the figure. The force and geometry parameters are shown. The geometry was again sphere-to-flat. The force range was 40 to 640 grams, 40 and 160 being low and high values for many connector systems, and the 640 a high, but occasionally used, value. The radii used range from typical to large. Note that at 640 grams the wear is severe due to the high force. Wear mechanisms shift from a “burnishing” low wear regime to an “adhesive” high wear regime as a function of contact force. The transition between the regimes depends on the state of lubrication of the contact and varies from tens to hundreds of grams. At the lower forces, wear is less dramatic, though at the high cycle end indications of fatigue-related wear are present. In sum, the wear mechanics depend on how the apparent stress is created and not its magnitude.

So it is clear that the correlation between apparent (Hertz) stress and the important connector parameters of contact resistance, durability (wear), and mating force is dependent on how the apparent stress is realized. Thus a given value of apparent stress is ambiguous and not useful as a connector design parameter or indicator of connector performance in the field.

When we review and confirm the important interrelationships between contact force and contact geometry, we see how those relationships complicate the determination of “easy” guidelines or requirements for connector parameters.

Contact force remains the most significant parameter because its effect on wear and mating force is a function of force. In addition to the direct relationship between force, wear kinetics can change from burnishing to adhesive as force increases. Coefficients of friction increase discontinuously with this change in wear kinetics with a resultant increase in mating force. The change in friction force, however, can have a positive effect on the mechanical stability of the contact interface that is a positive effect. The “appropriate” contact normal force depends on the connector application that will “prioritize” the relative importance of the effects of contact force on electrical resistance, mating force, and mechanical stability.

Bishop & Associates, Inc. © 2011