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Materials for Pivots and Knife EdgesIntroductionMiddle-aged readers may remember expensive watches with the words "17 Sapphires" printed on the face, roughly where the word "Quartz" now appears. The message was that the mechanism had bearings made of sapphire; (Al203) - which was good. A really expensive watch had, not sapphires, but diamonds.Diamond and sapphire are examples of good materials for knife edges and pivots. These are bearings in which two members are loaded together in nominal line contact or point contact, and can tilt relative to one another or rotate freely about the load axis (Figure 1). The main requirements of materials for such bearings are high hardness - to carry the contact pressures; and high modulus - to give positional precision and to minimise frictional losses. The desired combination of these properties depends on the objective: maximum precision or maximum load-carrying capacity. Table 1 summarises the requirements.
Figure 1 A knife edge and a pivot. For precision, a small contact area is essential; for load-carrying capacity, a high resistance to indentation pressure is requiredDesign Requirements
Table 1The ModelConsider maximising precision first. The positional accuracy of a pivot or knife edge increases, and its frictional losses decrease, as the area of contact A is made smaller and smaller. To maximise precision, we seek to minimise A, but as we do so, the contact pressure rises. For a given load, F, the peak contact pressure p is given by Hertzian contact theory; it is:
If both surfaces are to remain elastic, this pressure must not exceed their hardness, H, divided by a safety factor, which we ignore. (It does not affect the outcome of the analysis.) Thus
Thus the area of contact is minimised, and the precision maximised, by selecting materials with large values of the performance index
In addition, change of temperature will influence the positional accuracy of a pivot. To minimise thermal distortion we must simultaneously seek materials with small values of the thermal expansion coefficient, a. Now consider the alternative objective, that of maximising the load that the pivot can support at fixed geometry. Hertzian contact theory gives the maximum pressure in the contact zone of a hemispherical pivot-tip pressed against a flat block by a force F as
where R is the radius of curvature of the tip of the pivot, E is its Young's modulus and C is a constant close to unity. The constraint remains the same: this pressure must not exceed the hardness H of the pivot or the block. Thus
The load is maximised, for a given geometry R, by choosing materials with large values of the performance index
Robustness additionally requires that the pivot or knife-edge can survive rough handling - and for this a degree of toughness (i.e. a high value of KIc) is desirable. The SelectionThe performance indices involve H and E. Figure 2 shows the appropriate chart. For precision you want high M1, and then ceramics are definitely the best choice: Al2O3 (sapphire), silicon carbide and nitride and - above all - diamond. Cermets (composites of tungsten carbide and cobalt) are the only other class of solid which competes with them.
Figure 2 Chart of hardness H against modulus E, showing the index M1For load-bearing capacity, you want high M2 instead; then high-carbon steels, tool steels and high-strength titanium alloys become possibilities, although ceramics and cermets remain the most attractive candidates. This chart is shown in Figure 3.
Figure 3 Chart of hardness H against modulus E, showing the index M2But there is more to it than that. Change of temperature will influence the positional accuracy of a pivot. Precision is maintained by choosing materials with low thermal expansion. Robustness, on the other hand, requires that the pivot or knife-edge survive rough handling - and for this a degree of toughness is desirable. Figure 4 shows a chart of fracture toughness plotted against thermal expansion coefficient. Using this as a second stage, gives the results in Table 2.
Figure 4 A chart of fracture toughness, KIC, against thermal expansion coefficient, l, showing the selection stage for robust, precise, pivots
Results
Table 2 Materials for knife edges and pivots |
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