# Contact

## Contents

## Concentrated Force on a Half-Plane[edit]

From the *Flamant Solution*

and

If and, we obtain the special case of a concentrated force acting on a half-plane. Then,

or,

Therefore,

The stresses are

The stress is obviously the superposition of the stresses due to and , applied separately to the half-plane.

### Problem 1: Stresses and displacements due to [edit]

The tensile force produces the stress field

The stress function is

Hence, the displacements from Michell's solution are

At , (, ),

At , (, ),

where

Since we expect the solution to be symmetric about , we superpose a rigid body displacement

The displacements are

where

and on .

### Problem 2: Stresses and displacements due to [edit]

The tensile force produces the stress field

The displacements are

### Stresses and displacements due to [edit]

Superpose the two solutions. The stresses are

The displacements are

## Distributed Force on a Half-Plane[edit]

- Applied load is per unit length in the direction.
- We already know the stresses and displacements due to a concentrated force. The stresses and displacements due to the distributed load can be found by { superposition}.
- The Flamant solution is used as a Green's function, i.e., the distributed load is taken as the limit of a set of point loads of magnitude .

At the point

As , is unbounded. However, if we are interested in regions far from , we can apply the distributed force as a statically equivalent concentrated force and get displacements using the concentrated force solution.

The avoid the above issue, contact problems are often formulated in terms of the { displacement gradient}

If the point is inside , then the integral is taken to be the sum of the integrals to the left and right of .

## Indentation due to a Frictionless Rigid Flat Punch[edit]

- Start with uneven surface profile .
- Unsymmetric load , but sufficient for complete contact over the area .

Displacement in direction is

where is a rigid body translation and is a rigid body rotation.

Rigid body motions can be determined using a statically equivalent set of forces and moments

The displacement gradient is

Integral is a { Cauchy Singular Integral} that appears often and very naturally when the problem is solved using complex variable methods.

Note that the only thing we are interested in is the distribution of contact forces .If we change the variables so that and , then

If we write and as

and do some algebra, we get

### Flat Punch with Symmetric Load: [edit]

In this case,

Also, (origin at the center of ), hence . Therefore,

At , the load is infinite, i.e. there is a singularity.

## The Hertz Problem: Rigid Cylindrical Punch[edit]

- The contact length depends on the load .
- There is no singularity at .
- The radius of the cylinder () is large.

We have,

Hence,

and

Therefore,

and

Plug back into the expression for to get

This expression is singular at and , unless we choose

Plugging into the equation for ,

### Two deformable cylinders[edit]

If instead of the half-plane we have an cylinder; and instead of the rigid cylinder we have a deformable cylinder, then a similar approach can be used to obtain the contact length

and the force distribution

Many other problems are discussed in the texts by Timoshenko and Goodier (Elasticity) and K.L. Johnson (Contact Mechanics, 1985).