Principal curvature used in a contact problem

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The discussion focuses on understanding principal curvatures in the context of contact problems between two surfaces. The user seeks clarification on determining the principal radii of curvature (R1 and R2) and the corresponding planes. It is explained that the principal curvatures are derived from the curvature of the surface at a point, which is influenced by the normal vector at that point. The angle \Phi represents the relationship between the principal curvature planes of the two contacting bodies. Overall, the conversation emphasizes the geometric interpretation of curvature and its application in contact mechanics.
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Hi,

I am trying to understand a theoretical problem involving the contact between two surfaces. I have uploaded a screen shot of the mathematical formulations of the solution.

I understand most of the solution, except the principal curvatures. I have tried to look up principal curvature, but still not sure how is applied to this problem.

I would really appreciate any help for the mathematicians out there.

https://dl.dropbox.com/u/47274064/Prin_curve_1.JPG

Thanks.
 
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Curvature is the reciprocal of the radius. Is that what is confusing you?
 
Mandlebra said:
Curvature is the reciprocal of the radius. Is that what is confusing you?

Thanks Mandlebra, that much I understood.

I am just trying to understand how and why R1, R2, the planes of R1, R2 and the angle \Phi come into this. I guess my questions are more specifically:

1) How do you determine what R1 and R2 are?
2) How do you determine the plane of R1 and R2?


Would really appreciate some advice or explanation on this.

Thanks.
 
Not sure if this helps...
Take the normal N to the surface at point P. Now take any plane containing the normal. The intersection of the plane with the surface produces a line with some curvature at P. As you rotate the plane about N, the curvature reaches a minimum and a maximum, possibly of opposite sign. These are the principal curvatures. The planes containing them will be orthogonal. I believe the curvature in any intermediate plane through N can be computed from the principal curvatures and the angle this plane makes to the planes of principal curvature, but I don't the formula.
 
Forgot to mention phi. The other body will have the same tangent plane at P and thus the same normal. But the planes through it which give the min and max curvature for that body will not in general be the same. Phi is the angle between the two bodies' max curvature planes (and thus also the angle between their min curvature planes).
 
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