Asymmetric indentation of an elastic beam by a rigid cylinder

A simple method is proposed to analyze asymmetric indentation of a supported elastic beam by a rigid cylinder placed at an arbitrary position on the beam. A key advantage of our method over existing methods is that a single governing Kerr-type differential equation, which gives the relation between contact pressure and the normal deflection of the pressured surface of the beam, is applied both inside and outside the contact zone. With the present method, the two ends of contact zone will be determined as part of the solution, in contrast to the existing related studies which were all based on a simplifying assumption that the contact zone is geometrically symmetric about the tip of the indenter. Indeed, our results confirm that the contact zone is generally geometrically non-symmetric about the indenter tip, and even the whole contact zone can locate on one side of the indenter tip in some cases. Asymmetric indentation behaviors are demonstrated with numerical examples, with an emphasis on asymmetric contact pressure distribution inside the contact zone. Explicit formulas for the contact-zone width-displacement, force-displacement and moment-displacement relations are derived and illustrated with numerical examples. In particular, the present model predicts that as the indentation displacement exceeds a critical value, the indenter will lose contact with the beam inside the contact zone, which implies that the actual contact zone becomes two separate contact strips. Validity and accuracy of the present method are demonstrated by comparing its predictions with known results for some special cases. New features of asymmetric indentation as compared to symmetric indentation are summarized at the end of the paper.