# Talk:Contact mechanics

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## Too technical

In my opinion the page is too technical, I added the technical template to the top of the page.

• The introduction is quite long, and already contains a lot of details. It might try to focus more on the essential ideas.
• The distinction between non-adhesive and adhesive contact might be introduced separately.
• Classical solutions could be an entire top-level section by itself.
• Analytical and numerical solution techniques could also be discussed separately.
• The purposes, strengths and weaknesses of the various adhesive contact theories could be introduced in more general terms, before the theories are discussed in detail.

Edwinv1970 (talk) 09:20, 22 March 2011 (UTC)

## Line contact on a plane section

I think the integral formulas given in line contact on a plane section are incorrect. The dimensions don't match. Can someone confirm? I was reading contact mechanics by johnson and the formulas look a little different there. User:Blooneel 24 June, 2010

Johnson's book assumes a left-handed coordinate system with the ${\displaystyle z}$-axis pointing down. The results given in this article assume that the ${\displaystyle z}$-axis points up. That leads to the different relations. See Barber's book on elasticity for the form given in this article. Bbanerje (talk) 03:45, 25 June 2010 (UTC)
There seems to be an inconsitency between the (x,y) directions shown on the diagram and the use of z in the formulas. It needs to be clear what the directions are.Eregli bob (talk) 04:37, 30 August 2010 (UTC)

## Coordinate system

I am wondering about the coordinate system in the Chapter "Loading on a Half-Plane". The coordinate z seems to be the direction normal to the surface (as also in the chapter before). Does this chapter present a 3D solution for a point load given in the plane y=0? Than the term "Loading on a Half space" would be better. Or is a plane strain (plane stress) solution presented?

In any case: the appearance of the y coordinate in the figure ( (x,y) and σy ) is misleading. For the same reason y should also be replaced by z in the sentence following the formulae  : "for some point, (x,y), in the half-plane. " B Sadden (talk) 14:57, 30 May 2009 (UTC)

## Error in sphere on half-space?

I may be wrong, but I believe that there is a mistake here; the radius of the contact area is quoted as being sqrt (R * d), I think (from a bit of cursory mathematics) that is should actually be sqrt (2 * R * d), can anyone confirm this, I may be mistaken so I won't change this unless someone else confirms...

thanks,

Mike Strickland —Preceding unsigned comment added by 152.78.178.59 (talk) 16:59, 27 July 2010 (UTC)

The Hertz solution for the elastic displacements in the region of contact is
${\displaystyle u_{1}+u_{2}=\delta -Ax^{2}-By^{2}}$
where ${\displaystyle x,y}$ are coordinates of the contact surfaces projected on to the ${\displaystyle x-y}$-plane. For a circular contact area with radius ${\displaystyle a}$,
${\displaystyle A=B={\tfrac {1}{2}}\left({\tfrac {1}{R_{1}}}+{\tfrac {1}{R_{2}}}\right)}$
If the second surface is a half-plane, ${\displaystyle R_{2}\rightarrow \infty }$ and we have
${\displaystyle A=B={\tfrac {1}{2R_{1}}}={\tfrac {1}{2R}}}$
Therefore,
${\displaystyle u_{1}+u_{2}=\delta -{\tfrac {1}{2R}}r^{2}}$
where ${\displaystyle r}$ is the radial distance to a point in the contact region from the center of contact. The Hertzian pressure distribution
${\displaystyle p=p_{0}\left[1-({\tfrac {r}{a}})^{2}\right]^{1/2}}$
${\displaystyle u_{1}=\left({\tfrac {1-\nu _{1}^{2}}{E_{1}}}\right)\left({\tfrac {\pi p_{0}}{4a}}\right)\left(2a^{2}-r^{2}\right)~;~~u_{2}=\left({\tfrac {1-\nu _{2}^{2}}{E_{2}}}\right)\left({\tfrac {\pi p_{0}}{4a}}\right)\left(2a^{2}-r^{2}\right)}$
Plugging these into the relation for ${\displaystyle u_{1}+u_{2}}$ gives
${\displaystyle \left({\tfrac {1}{E^{*}}}\right)\left({\tfrac {\pi p_{0}}{4a}}\right)\left(2a^{2}-r^{2}\right)=\delta -{\tfrac {1}{2R}}r^{2}}$
At ${\displaystyle r=0}$
${\displaystyle \delta ={\tfrac {\pi p_{0}a}{2E^{*}}}}$
For ${\displaystyle r=a}$ plugging in the expression for ${\displaystyle \delta }$ gives
${\displaystyle a={\tfrac {\pi p_{0}R}{2E^{*}}}}$
Therefore
${\displaystyle {\tfrac {a}{\delta }}={\tfrac {R}{a}}\Leftrightarrow a^{2}=R\delta \implies a={\sqrt {R\delta }}\quad \square }$
Bbanerje (talk) 00:00, 28 July 2010 (UTC)

## Error in rigid conical indenter and an elastic half-space?

The German Wikipedia has a and d switched in this formula: ${\displaystyle a={\frac {2}{\pi }}d\tan \theta }$. And indeed, if one lets theta get towards 90° then only the switched version makes sense (radius gets towards 0). Peterthewall (talk) 17:55, 28 February 2013 (UTC)

## Hertz Model for Sphere on Plane is Parabola Approximation

I would like to point out that the sphere on a plane section is for a parabola. Many make the no-slip assumption for a spherical indenter so they can approximate the sphere for a parabola. JPK instruments has a decent read on this in terms of AFM on cells: www.jpk.com/jpk-app-elastic-modulus4.download.5fb2f841667674176fd945e65f073bad

They have the sphere on

force=E/(1-v^2)*(((a^2+R^2)/2)*ln((R+a)/(R-a))-a R)


where a=(R*d)^1/2 (I think) E is Young's Modulus v is Poisson's Ratio d is indentation of plane I think it would be good to at least state somewhere that it is an approximation. — Preceding unsigned comment added by EvanN90 (talkcontribs) 21:24, 8 September 2015 (UTC)