Hello.
I have begun studying Contact Mechanics very recently, and I am having trouble in understanding the Reduced Young's Modulus (E*).
I have looked at several different books, and I haven't been able to understand why they define E* differently.
When two bodies of Young's Moduli E1 and E2, and Poisson's ratios of v1 and v2 are in contact, I have seen the following definitions:
1) as seen in the attachment 1.PNG:
1/E* = (1-v1²)/E1 + (1-v2²)/E2
2) as seen in the attachment 12.PNG:
1/E* = (1/2)*((1-v1²)/E1 + (1-v2²)/E2)
The first approach would result in a value of E* that is lower than both E1 and E2, while the second approach yields a value of E* that is between E1 and E2.
Another thing that makes me curious is that the first approach is more frequent in books, while I have only seen the second one in a couple of books and papers. But, for some reason, most of the online calculators for contact mechanics agree with the second approach.
Certainly, I am failing to understand this matter properly and, if possible, I would like to know the difference between both approaches with your assistance.
Thank you very much.
Sincerely,
Diogo
Please consider that the formulas where E* appears must have been adapted.
Final result is the same.
Formulas for a, b, delta have additional coefficients 2 inside.
Article Contact stiffness of spindle-tool holder based on fractal th...
https://www.sciencedirect.com/search/advanced?qs=Reduced%20Young%27s%20Modulus%20in%20contact%20mechanics&authors=&pub=&cid=&volume=&issue=&page=&offset=0&show=25&sortBy=relevance&origin=home&zone=qSearch
Hi;
Please see the reference below:
Michel Yetna N’Jock, Francine Roudet, I. Idriss, Olivier Bartier, Didier Chicot.
Work-of-indentation coupled to contact stiffness for calculating elastic modulus by instrumented indentation. Mechanics of Materials, Elsevier, 2016, 94, 170-179. DOI: 10.1016/j.mechmat.2015.12.003.
With my best regards
Hi Diogo,
It backs to how you use E* and how the contacting curvatures are defined. You need both equivalent radius (Re) and Equivalent stiffness (E*) to calculate the contact pressure.
however, as Johnson has mentioned in "Contact Mechanics" or Article Analysis of Elliptical Hertzian Contacts
:an equivalent radius Re defined by
Re = (R'R")^0.5 = 1/2 (AB)^-0.5
If the two bodies are solids of revolution, then
R', = R ; = R 1 and R; = R i = R 2 , whereupon A = B =1/2 ( l / R 1 + 1/R 2).
Thus contours of constant separation between the surfaces before loading
are circles centred at 0. After loading, it is evident from the circular symmetry
that the contact area will also be circular, Two cylindrical bodies of radii R 1
and R 2 in contact with their axes parallel to the y-axis have R ; = R 1 , R1'' ; = inf,
R2 = R 2 , R 2'' = inf, so that A =1/2 ( l / R 1 + 1/ R 2), B = 0. Contours of
constant separation are straight lines parallel to the y-axis and, when loaded,
the surfaces will make contact over a narrow strip parallel to they-axis. A special case arises when two equal cylinders both of radius
R are in contact with their axes perpendicular. Here R 1 = R , R ; = inf, R ; = R ,
R2'' = inf) a = n/2, from which A = B = 1/2R. In this case the contours of constant
separation are circles and identical to those due to a sphere of the same radius
R in contact with a plane surface (R2´ = R2´´ = inf).
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