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EN224: Linear Elasticity
Division of Engineering
3.8 Axisymmetric Contact
One of the most successful applications of linear elasticity has been to predict the behavior of two solids in contact.
The results have provided a basis for designing gears, bearings, cams, wheels, continuously variable transmissions,
etc, etc.
We will use the results developed in the preceding section to solve some axisymmetric contact problems. Begin with
the simplest case.
Lubricated flat punch indenting a halfspace.
We seek
with
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View Full Document We will solve this problem using two approaches.
Solution via integral equations
First, note that we may use the results of Sect. 3.7 to compute the fields in a halfspace subjected to an arbitrary
distribution of pressure on its surface. We could ask: what pressure distribution should act on the surface of the half
space in order to satisfy the boundary conditions?
From the preceding section, we see that the surface displacement due to a unit point force at the origin is
Thus, we seek an axisymmetric contact pressure distribution
that satisfies
It is not a trivial exercise to solve this equation, unfortunately. One may readily verify that
satisfies the equation.
Solution via complex stress functions
A particularly elegant formulation for axisymmetric contact problems has been developed by Love, Green and
Zerna, Collins, and Hill. Recall that we are looking for a harmonic potential that satisfies
Consider the complex harmonic function
where
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It is straightforward to show that
The integrals and derivatives of this function are also harmonic.
Consider
On the surface:
also
Thus, choosing
will satisfy all our boundary conditions. This expression can be integrated to obtain the potential: Mathematica comes
up with the expression
which can probably be simplified further, but I am too lazy (any offers?)
The contact pressure is of particular interest:
The load applied to the punch follows as
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This note was uploaded on 01/10/2010 for the course EN 224 taught by Professor Allenbower during the Spring '05 term at Brown.
 Spring '05
 ALLENBOWER

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