Contact mechanics

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Contact mechanics is the study of the deformation of solids that touch each other at one or more points. The physical and mathematical formulation of the subject is built upon the mechanics of materials and theory of elasticity. The original work in this field dates back to the publication of the paper "On the contact of elastic solids" by Heinrich Hertz in 1882. Hertz was attempting to understand how the optical properties of multiple, stacked lenses might change with the force holding them together. Results in this field have since been extended to all branches of engineering, but are most essential in the study of tribology and indentation hardness.

[edit] Introduction

Contact may occur between bodies in two distinct ways. A conforming contact is one in which the two bodies touch at multiple points before any deformation takes place (i.e. they just "fit together"). The opposite is non-conforming contact, in which the shapes of the bodies are dissimilar enough that, under no load, they only touch at a point (or possibly along a line). In the non-conforming case, the contact area is small compared to the sizes of the objects and the stresses are highly concentrated in this area.

Complex forces and moments are transmitted between the bodies where they touch, so problems in contact mechanics can become quite sophisticated. Typically, a frame of reference is defined in which the objects (possibly in motion relative to one another) are static. They interact through surface tractions (or pressures/stresses) at their interface. As an example, consider two objects which meet at some surface $S$ in the ( $x$ , $y$ )-plane. One of the bodies will experience a (normally-directed) pressure $p = p (x, y)$ and (in-plane) surface traction $q = q (x, y)$ over the region $S$ . In terms of a Newtonian force balance, the forces:

$P_z=\int\!\int_S p(x,y)\, dS$

and:

$Q_x=\int\!\int_S q_x(x,y)\, dS$

$Q_y=\int\!\int_S q_y(x,y)\, dS$

must be equal and opposite to the forces established in the other body. The moments corresponding to these forces:

$M_x=\int\!\int_S p(x,y)y\, dS$

$M_y=-\int\!\int_S p(x,y)x\, dS$

$M_z=\int\!\int_S (q_y(x,y)x-q_x(x,y)y)\, dS$

are also required to cancel between bodies so that they are kinematically immobile.

[edit] Loading on a Half-Plane

Loading at a Point - Objects in contact will deform under the influence of the tractions mentioned above and there are a number of elasticity solutions that are applicable to determining these deformations. The starting point is understanding the effect of a "point-load" applied to an elastic half-plane, shown in the figure to the right. Like all problems in elasticity, this is a boundary value problem subject to the conditions:

Schematic of the loading on a plane by force P at a point (0,0).

$σ x z (x,0) = 0$

$σ z (x, z) = - P δ(x, z)$

(i.e. there are no shear stresses on the surface and singular normal force P applied at (0,0)). Applying these conditions to the governing equations of elasticity produces the result:

$\sigma_x=-\frac{2P}{\pi}\frac{x^2z}{(x^2+z^2)^2}$

$\sigma_z=-\frac{2P}{\pi}\frac{z^3}{(x^2+z^2)^2}$

$\sigma_{xz}=-\frac{2P}{\pi}\frac{xz^2}{(x^2+z^2)^2}$

for some point $(x, y)$ in the half-plane. The circle shown in the figure indicates a surface on which the prinicipal shear stress is constant. From this stress field, the strain components and thence displacments of all material points may be determined.

Loading over a Region (a,b) - This above is an important result that can be built upon. Suppose, rather than a point load $P$ , a distributed load $p (x)$ is applied to the surface instead, over the range $a < x < b$ . The principle of linear superposition can be applied to determine the resulting stress field as the solution to the integral equations:

$\sigma_x=-\frac{2z}{\pi}\int_a^b\frac{p(x')(x-x')^2\, dx'}{[(x-x')^2+z^2]^2}$

$\sigma_z=-\frac{2z^3}{\pi}\int_a^b\frac{p(x')(x-x')^2\, dx'}{[(x-x')^2+z^2]^2}$

$\sigma_{xz}=-\frac{2z^2}{\pi}\int_a^b\frac{p(x')(x-x')\, dx'}{[(x-x')^2+z^2]^2}$

Shear Loading over a region $(a, b)$ - The same principle applies for loading on the surface in the plane of the surface. These kinds of tractions would tend to arise as a result of friction. The solution is similar the above (for both singular loads $Q$ and distributed loads $q (x)$ ) but altered slightly:

$\sigma_x=-\frac{2}{\pi}\int_a^b\frac{q(x')(x-x')^3\, dx'}{[(x-x')^2+z^2]^2}$