Hertzian Contact Stress and Material Hardness

[dans95]

I am designing an assembly which will incorporate a Maxwell Kinematic mount (3 balls, 3 V-grooves). The balls will be sapphire and the V-grooves some type of hardened SST (440C perhaps). The basic Hertzian spherical indentation equations use the Elastic modulus and Poisson's ratio to calculate the modified Elastic modulus. Does heat treating change the elastic modulus of a material? My local heat treater told me it does not, "modulus is a function of chemistry". Is this correct? Perhaps locally (contact) but not globally (deflection of a sample in tension or bending)???

 

[dimjim]

From Wikipedia:

Hertzian contact stress refers to the localized stresses that develop as two curved surfaces come in contact and deform slightly under the imposed loads. This amount of deformation is dependent on the elasticity of the the material in contact, i.e., its modulus of elasticity. It gives the contact stress as a function of the normal contact force,the radii of curvature of both bodies and the modulus of elasticity of both bodies.

 

[Gerry45]

The elastic modulus of steel is pretty independent of the type of steel or its heat treated condition. However, what is important in a hertzian contact situation is the stress level at which the material deforms plastically. So a piece of mild steel raceway will have the same INITIAL behavior as some hardened steel but as you apply increasing load it wont be long before the mild steel is permanently damaged. Hardening the raceways gives a useful load carrying capacity (plus some other benefits of course, like wear resistance fatigue etc.)

Just for my information, whats a kinematic mount ?

 

[Tmoose]

An INA catalog has a section on depth of steel race/ball/roller hardness required for varying stress levels. Can be 0.06 inch or greater, although the hardness at that depth is still much less than the surface.

Mis matched ball/race stiffness (hybrid ceramic bearings that use steel races and ceramic balls ) causes a generally greater race stress at a given load. I picture something like riding a narrow tired track bicycle on the beach

 

[dans95]

Gerry,
Thank you for the reply. If I understand this correctly the case with a mild steel race will deform the same amount (and thus the same stress) as with a hardened steel race but when the load is removed the mild steel race is left with a dent because the stress > yield strength. The hardened race did not dent because the stress < yield strength.

To summerize; the amount of indentation is a function of elastic modulus and poisson's ratio and is independent of hardness. Whether or not plastic deformation occurred is a function of applied stress and compressive yield strength of the materials. Since increased hardeness corresponds to increased compressive yield strength, the hardened steel can be used with higher loads without permanent deformation. Does that sound right?


A Kinematic mount is one that EXACTY contrains the component without over contraint. To constrain 6 degrees of freedom, 6 (and only 6) points of contact are needed. Kinematic mounts are commonly used in optical systems or anywhere extremely high repeatability is required. The downside is reduced load capacity because of only a few points of contact.
Check out

http://www.precisionballs.com/TDSKin-1/Micro_Inch_Positioning_with_Kinematic_Components.html

 

[Gerry45]

Yes, I think you have it about right ....but Im not quite so sure a pair of hardened and pair of soft component will show exactly the same deflection characteristics with only the level of recovery being different. I think that once the soft components are taken into their plastic region they may have a slightly different deflection curve to the hard. Both pairs follow hertzian contact rules initially (which assume fully elastic materials)but then the soft pair will start to follow a slightly different curve as an increasing amount of material is taken into the plastic region. It may not be a lot different - and it may be 2nd order stuff - but Im not aware anyone has ever bothered to investigate it.

Thanks for the explanation re kinematic mounts - very interesting stuff.