Talk:Contact mechanics

Too technical
In my opinion the page is too technical, I added the technical template to the top of the page. Edwinv1970 (talk) 09:20, 22 March 2011 (UTC)
 * 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.

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 $$z$$-axis pointing down. The results given in this article assume that the $$z$$-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

u_1 + u_2 = \delta - A x^2 - B y^2 $$
 * where $$x,y$$ are coordinates of the contact surfaces projected on to the $$x-y$$-plane. For a circular contact area with radius $$a$$,

A = B = \tfrac{1}{2}\left(\tfrac{1}{R_1} + \tfrac{1}{R_2}\right) $$
 * If the second surface is a half-plane, $$R_2 \rightarrow \infty$$ and we have

A = B = \tfrac{1}{2 R_1} = \tfrac{1}{2R} $$
 * Therefore,

u_1 + u_2 = \delta - \tfrac{1}{2 R} r^2 $$
 * where $$r$$ is the radial distance to a point in the contact region from the center of contact. The Hertzian pressure distribution

p = p_0 \left[ 1 - (\tfrac{r}{a})^2\right]^{1/2} $$
 * leads to the displacement field

u_1 = \left(\tfrac{1-\nu_1^2}{E_1}\right)\left(\tfrac{\pi p_0}{4a}\right)\left(2 a^2 - r^2\right) ~; u_2 = \left(\tfrac{1-\nu_2^2}{E_2}\right)\left(\tfrac{\pi p_0}{4a}\right)\left(2 a^2 - r^2\right) $$
 * Plugging these into the relation for $$u_1+u_2$$ gives

\left(\tfrac{1}{E^*}\right)\left(\tfrac{\pi p_0}{4a}\right)\left(2 a^2 - r^2\right) = \delta - \tfrac{1}{2 R} r^2 $$
 * At $$r = 0$$

\delta = \tfrac{\pi p_0 a}{2 E^*} $$
 * For $$r = a$$ plugging in the expression for $$\delta$$ gives

a = \tfrac{\pi p_0 R}{2 E^*} $$
 * Therefore

\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: $$ a=\frac{2}{\pi}d\tan\theta  $$. 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 (talk • contribs) 21:24, 8 September 2015 (UTC)

dxe
The description for "Adhesive surface forces" is "dxe" which, according to this article on wikipedia is related to animal rights. This should be corrected.