Principal curvature

[yaston4]

Hi

I am trying to understand a theoretical problem involving the contact between two surfaces from Roarks Formulas for Stress and Strain. I have uploaded a screen shot of the 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 and how it is used to solve the contact problem.

I would really appreciate any help.

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

Thanks.

 

[rb1957]

isn't it just the reciporal of the radius ?

i'd have more questions about the comment of "max and min radius" for the body, then calling out only one radius ... i suspect the critcal combination is the smallest radius for both bodies (so why mention the max radius ?)

 

[yaston4]

Thanks rb1957,

I understood that it is the reciprocal of the radius, but I also don't really understand what the minimum radii of curvature is or how it is established for a contact problem?

The other aspect I am not sure on is how the planes of R₁ and R² are established?

Any ideas anyone?

Thanks.

 

[rb1957]

it makes sense to me that min radius produce the sharpest contact (least area) and highest stress.

i would consider a plane revolving around the line from the contact point to the centre of the body (however sense that makes ?)
so that the plane sections the body and you can see the radius near the contact area.

roark mentions the angle phi between the two planes on the two bodies.

 

[Cockroach]

But we're dealing with elastic bodies, hence a deformation in them upon contact. I've always used R as the principle radius of curavature and R' as that after contact, i.e. deformation. So you have some estimate of deformity in bodies after contact and due to the load P.

Or maybe I have been doing it wrong.

Regards,
Cockroach

 

[Denial]

Consider a point on a smooth curved surface, a tangent plane at that point, and a tangent line in that plane. Unless the curved surface is exactly spherical in the vicinity of the point, the radius of curvature (RoC) along the tangent line will depend upon the direction of the tangent line. Allow this direction to swing through 180 degrees, and there will be a direction for which the RoC is a maximum, and there will be a direction for which the RoC is a minimum. These two directions are the principal directions, and the reciprocals of their RoCs are the principal curvatures. The principal directions will always be 90 degrees apart.

 

[Andries]

Hi,

There is a good introduction to this type of contact problem in "Theory of Elasticity" by Timoshenko and Goodier.
All the terms are clearly defined

Andries

 

[chicopee]

At point of contact the surfaces should be flat, so does that mean that after contact R1 and R2 are not restored back to R'1 and R'2?

 

[prex]

An example of what Denial correctly states is the situation as seen by a ball in a ball bearing. The spherical ball of course has two equal and positive radii of curvature. The outer ring has two negative radii, the minimum is the radius of the throat, the maximum is the inner radius of the ring. In the contact of the ball to the inner ring, the latter has again a negative minimum radius as the radius of the throat, and a positive maximum radius as the outer radius of the ring.

prex
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