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Contact radius of sphere
Fri, 2011-06-10 05:40 - ColinGrant
Looking through books and papers I see an often quoted equation to show that the contact radius (a) of spherical indenter of radius (R) is related to the indentation depth (h):
a= √(R.h)
However, using simple trigonometry of a spherical cap it can be shown that:
a=√(2Rh-h2)
Contact area is very important for use in nanoindentation - however, if it is based on the wrong contact area calculation, then more errors become apparent.
On that note - I also want to ask if a rigid flat indenter press on a thin shell sphere, is the contact radius the same as a rigid spherical indenter pressing on a flat surface ?
Colin
Forums:
Hi You can find that
Hi
You can find that this formula
a=√(2Rh-h2)
is right if u draw a triangle.
Jason Zhu Best Regards
The answer depends on geometry and elasticity
Hi,
The contact radius between two objects depend on their material properties (e.g. elasticity) and on their geometry. If you press a rigid sphere into a fluid then after a while the wet area has radius a = sqrt(2Rh-h^2). However, if you press a rigid sphere onto an elastic block ("half-space") then the contact area will be smaller. And if the sphere is elastic as well, then the contact area will be in between. These latter two situations are described by Hertz' theory, which says that a = sqrt(Rh). See http://en.wikipedia.org/wiki/Frictional_contact_mechanics and http://en.wikipedia.org/wiki/Contact_mechanics and the books that are cited there.
The situation is different for plates and shells than for massive materials (half-spaces). I don't know much about shells.
Best wishes, Edwin
profile approximation
Based on my understanding, the mostly cited relationship
a= √(R.h) is based on the parabolic approximation of the sphere profile, which only applies to small contact deformation and area. This relationship is used in Hertz and JKR model.
In the case where contact area is relatively large, the relationship above is not accurate. a=√(2Rh-h2) should be the exact one. The extension of JKR model based on the exact expression of sphere profile can be found in a paper by D. Maugis (Langmuir 1995).