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Contact between two spheres.

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 continuum mechanics. The original work in this field dates back to the publication of the paper "On the contact of elastic solids" ("Ueber die Berührung fester elastischer Körper") 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 mechanics is an area of physics in which the motion of two or more bodies in space is restricted by additional constraints. These so called unilateral constraints ensure that bodies once coming into contact do not penetrate each other. Once the general equations for a contact problem are set up, different solution schemes can be used to simulate the behaviour of bodies in contact and to compute displacement and stress fields. There are several possibilities to classify contact problems. Generally contact with and without friction is distinguished.

In case of analytical solution methods for contact problems the following classification was introduced. 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.

Such distinctions however do not have to be made when numerical solution schemes are employed to solve contact problems. These methods do not rely on further assumptions within the solution process since they base solely on the general formulation of the underlying equations. Besides the standard equations describing the deformation and motion of bodies to additional inequalities can be formulated. The first simply restricts the motion and deformation of the bodies by the assumption that no penetration can occur. Hence the gap $g N$ between two bodies can only be positive or zero

$g_N \ge 0$

where $g N = 0$ denotes contact. The second assumption in contact mechanics is related to the fact, that no tension force is allowed to occur within the contact area (contacting bodies can be lifted up without adhesion forces). This leads to an inequality which the stresses have to obey at the contact interface. It is formulated for the contact pressure $p_N = \mathbf{t} \cdot \mathbf{n}$

$p_N \le 0 \,.$

Since for contact, $g N = 0$ , the contact pressure is always negative, $p N < 0$ , and further for non contact the gap is open, $g N > 0$ , and the contact pressure is zero, $p N = 0$ , the so called Kuhn-Tucker form of the contact constraints can be written as

$g_N \ge 0\,, \quad p_N \le 0\,, \quad p_N\,g_N = 0\,.$

These conditions are valid in a general way. The mathematical formulation of the gap depends upon the kinematics of the underlying theory of the solid (e.g., linear or nonlinear solid in two- or three dimensions, beam or shell model),

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 principal shear stress is constant. From this stress field, the strain components and thence displacements 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:

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

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

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

These results may themselves be superposed onto those given above for normal loading.

[edit] References

Johnson, K.L. Contact Mechanics. (Cambridge University Press: Cambridge, UK), 1985.
Wriggers, P. Computational Contact Mechanics. 2nd ed. (Springer Verlag: Heidelberg), 2006.
Laursen, T. A. Computational Contact and Impact Mechanics: Fundamentals of Modeling Interfacial Phenomena in Nonlinear Finite Element Analysis, (Springer Verlag: New York), 2002.
Acary V. and Brogliato B. Numerical Methods for Nonsmooth Dynamical Systems. Applications in Mechanics and Electronics. Springer Verlag, LNACM 35, Heidelberg, 2008.

[edit] See also

Contents

[edit] Introduction

[edit] Loading on a Half-Plane

[edit] References

[edit] See also