In any optomechanical system the ultimate goal is to hold the optic(s) in place, many times during shifts in the environment (but we won't get into that now).  Obviously to do this there needs to be some kind of force being placed on the lens. A side-effect of this force is a stress in the optic and the mount. In many cases this stress can be ignored because the holding force (and with it the stress) is small.

 

However, if the forces are not small due to some environmental requirements (i.e. temperature changes and/or shock) the stress needs to be quantified to determine an appropriate  mounting schemes. If the calculation is done by hand then the normal approach is to use Hertz Contact Theory.

 

Before getting into the details of this method a couple of things need to be defined first. The three types of interfaces normally used in mounting optics are "sharp corner", "toroidal", and the "tangential" interfaces. These are illustrated in figures 1,2, and 3 respectively. The meaning of the terms in the illustrations will be dealt with later.

 

This will only deal with the cases involving spherical optics.

Types

Toroidal

This is the most general case. In a toroidal interface the radius of the edge in contact with the lens has a radius somewhere between infinity and zero. You can use the equations for this to determine the stresses for both the sharp corner and the tangential interfaces.

Sharp Corner

If you assume that the radius of the edge is zero (r=0) then you get the sharp corner interface. For this interface the edge contacting the lens surface is assumed to be absolutely sharp (i.e. r=0). As will be seen later this results in an unrealistically high stress. In reality this edge is not infinitely sharp but will typically have a slight radius. Even if it really was infinitely sharp once it came into contact with the lens surface the high stress would plastically deform it to conform to the lens surface, so either way you look at it this is not a practical assumption.

 

This limitation aside it is a very common way of estimating the stress in the lens. If using the equations for the toroidal interface calculations this type of interface can be estimated  by setting r=0.05mm (converted to whatever units you are using).²

Tangential

The tangential interface is a another specific case of the toroidal. That being the radius of the edge is infinite, resulting in the face contacting the lens is flat. This results in  the largest contact area and so the lowest contact stress. In systems that I have worked on it is routine for the stress to drop by orders of magnitude when going from a sharp corner to a tangential.

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Equations and Calculations

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Toroidal

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Linear Preload

where

F=Applied Force

Y_contact= Semi-Diameter of the contact

 

Material Constant

Configuration

where

Half-Contact Width

Maximum Compressive Stress

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Sharp Corner

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F=Applied Force

Y_contact= Semi-Diameter of the contact

Linear Preload

Configuration

where

Contact Width

Maximum Compressive Stress

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Tangential

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F=Applied Force

Y_contact= Semi-Diameter of the contact

Linear Preload

Configuration

where

Contact Width

Maximum Compressive Stress

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References

Contact Stress

¹Young, Budyna's Roark's Formulas for Stress and Strain, 7th Edition

²Yoder, P.R., Opto-Mechanical Systems Design, 3rd Edition, 2005

³Wikipedia contributors, "Contact mechanics," Wikipedia, The Free Encyclopedia, http://en.wikipedia.org/w/index.php?title=Contact_mechanics&oldid=445578149

 

Other Useful Resources

Hertz, H. R., 1882, Ueber die Beruehrung elastischer Koerper (On Contact Between Elastic Bodies), in Gesammelte Werke (Collected Works), Vol. 1, Leipzig, Germany, 1895