Hertz contact question

Mikael,

The parameters, m and n, are expressed in terms of complete elliptic integrals of the first and second kinds. See equations (4.25) through (4.32) in K. L. Johnson, Contact Mechanics, Cambridge University Press, 1985. You may also wish to consult the classic text by I. Ya. Shataerman, The Contact Problems of the Theory of Elasticity, Moscow, Gostekhizdat, 1949 (English translation by Foreign Technology Department of UK in 1970 available from the British Library MT-24-61-70).

Hope this helps.

 Mujibur Rahman

I looked in theory of elastictiy by Timoshenko and Goodier, 2nd edition on page 379.

The footnote at the bottom of the page referred to a document by Whittemore and Petrenko.

By a google search of their names I was led to a more recent pdf file, which I think answers your question on calculating your quantities, I just skimmed over it, but it looks like the elliptic integrals answer given above by Mujibur is correct.  The paper below discusses the Hertz contact problem and makes modern suggestions about how to solve the problem:

http://www.tycoelectronics.com/Documentation/whitepapers/pdf/2jot_2.pdf

Mikael,

Thanks for your query. The equations that I mentioned from Johnson's book are sufficient to be able to derive the expressions for the parameters m and n. Anyway, let me give the details. I don't know how to attach a file with this message. So, I will try to write the equations, etc. as much as I can. The equations for m and n are

m = [E(e)/k^2]^(1/3)

n = km = [k E(e)]^(1/3).                                                                        (1)

Here E(e) is the complete elliptic integral of the second kind, e = (1-k^2)^(1/2) is the eccentricity of the 'contact ellipse', and k=b/a, where a and b are respectively the major and minor semi-axes of the contact ellipse.

Now, here is the thing. The eccentricity e in the above equations needs to be determined from the following transcendental equation:

B/A=[k^(-2)*E(e)-K(e)]/[K(e)-E(e)],                                                             (2)

where K(e) is the complete elliptic integral of the first kind. So, the procedure is as follows: We know A and B. We then solve equation (2) for e. We then put e into equations (1) to get m and n. That's it!

A few words about the complete elliptic integrals E(e) and K(e). They are tabulated in many standard mathematical handbooks, e.g. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Reprint, 1964. But, there are also approximating formulas available in this handbook that you can use (see equations 17.2.34 and 17.2.34 on pages 591 and 592). Be careful though in using these equations (Abramowitz and Stegun have used the symbol m to denote e^2. So, keep this in mind). They are highly accurate. Using these equations, you can easily generate an excel program to solve equation (2).

I would also like to point out to you that this procedure is to some degree involved. Therefore, some researchers attempted to find an approximate closed-form solution to the equation (1). Here is one proposed by Solomon

e= [1-(B/A)^(4/3)]^(1/2).

Solomon claims that the maximum relative error from using this approximative approach does not exceed 2%, which might be adequate for many practical applications.

I encourage you to do the details by both exact and approximate approaches, and compare the results. It's worth doing!

Hope this helps,

Mujibur Rahman

P.S. I have noted the paper attached by Louis. It also contains all the necessary ingredients.

Hi,

We have two cylindrical rolls between which the product is pressed .one roll has small V shaped notches at both ends of 30 micron depth, in order to create an impression on the plastic sheet which rolls out with v shaped margins on 2 sides. 

how to determine the CONTACT PRESSURE and stiffness of this roll with notch?

how to modify the basic Hertzian equation

From your description, I am unable to form a clear picture of the problem that you would like to solve. Maybe it is not possible to adapt classical Hertzian solution to this problem? Or maybe it is possible to adapt it to this problem, but with simplifying assumptions so enormous that the estimates of contact pressure and stiffness derived from it will be grossly unrealistic! Anyway, I would recommend that you try to understand the underlying assumptions behind Hertzian contact problem. You can find them in Johnson's book on Contact Mechanics, for example, and then explore whether your problem is reducible to Hetzian by any means.

Mujibur Rahman

hi Rahman,
thankyou very much for your prompt response.Actually we have a die rolling machine which has an anvil and ring bearer on which there are small v shaped notches(indentations). When plastic sheet is passed between these steel rolls it forms the impression on the product. the ring bearer is a cylinder whose length is about 30 millimetres and there are v shaped notches on both sides whose widths are 3millimetre and depth is about 30 microns.
Now i would like to know what is the contact pressure that would be developed when both the surfaces come into contact I believe that when two cylinders come into line contact, the contact pressure can be determined by hertz equation but when the line contact is reduced because of the presence of notches on the surface can the same equation give accurate results.
If you need further details i ll try to send pictures of the machine so that you can get a clear picture of the manufacturing process