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axicontact - EN224 Linear Elasticity Division of...

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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 half-space. We seek with Generated by www.PDFonFly.com at 1/10/2010 9:49:16 PM URL: http://www.engin.brown.edu/courses/en224/axicontact/axicontact.html
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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 half - space 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 Generated by www.PDFonFly.com at 1/10/2010 9:49:16 PM URL: http://www.engin.brown.edu/courses/en224/axicontact/axicontact.html
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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 Generated by www.PDFonFly.com at 1/10/2010 9:49:16 PM URL: http://www.engin.brown.edu/courses/en224/axicontact/axicontact.html
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Note that the punch has a well -
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