by J. Jäger, Blattwiesenstr. 7, D-76227 Karlsruhe, Germany

AMD-Vol. 205, 1995, Impact, Waves, and Fracture, ed. by R. C. Batra, A. K. Mal, G. P. MacSithigh, pp. 129-152


ABSTRACT: In this article, incremental contact laws for the analysis of collisions with friction between elastically similar bodies are studied. The contact conditions are satisfied by a superposition of special stress distributions. The solution for normal contact of axisymmetric surfaces is derived as a superposition of circular flat-ended punches. Similarly, the tangential torsion or translation of axi-symmetric surfaces in staticcontact under constant normal forces is presented as a superposition of flat punch displacements. As example, the solution for a surface profile of the form Akrk, with a positive real k, is derived as a series of hypergeometric functions. The static solution gives the basic function for a superposition technique, which solves arbitrary loading histories. The superposition method for three-dimensional translation can be presented geometrically in form of so-called yield cones in the force and displacement space. The  behaviour of the force- displacement curve depends on the loading  history. A short computer program for the contact law is presented. At the end of this paper, the impact of a spherical pendulum with double hinge suspension is analysed. Analytical solutions for special cases are presented.


The first solution for normal contact impact was published by H. Hertz (1882). A brief review on the development of contact mechanics is given in Johnson (1982). The problem of two bodies in contact under monotonically increasing tangential forces has been examined by Cattaneo (1938) and Mindlin (1949). Mindlin & Deresiewicz (1953) also examined special tangential loading histories, using differential compliances. This differential approach stems from plasticity theory and leads to complicated formulas. The first incremental procedure based on finite increments was published by Jaeger (1992, 1993a, 1993b). In that approach a  simplified model has been developed on the basis, that increments can erase each other. In later publi cations (Jaeger, 1994b, 1994d), incremental contact laws have been applied to the oblique and torsional impact and compared with experimental results. Numerical solutions for oblique impact have been published by Maw et al. (1976, 1981). Contact phenomena produced by the impact of elastic bodies are also studied in Goldsmith (1960). The first general theory of impact, that includes tangential effects, has been published by Brach (1991). Frictional contact laws are also important in granular materials, rock and soil mechanics, earthquake engineering, fracture mechanics, friction clamps etc.  In section two of this article, a superposition technique is used to derive the contact law for axi-symmetric bodies in normal contact. To simplify matters, the problem of one rigid body pressed on an elastic half-space  is examined, but the formulas for two bodies in contact can readily be calculated by summation of both displacements. The surface of the rigid indentor is modelled as a summation of infinitesimal flat-ended circular punches and the  stress distribution is calculated as a superposition of the corresponding flat punch stress distributions. Similarly, in sections three and four, the tangential and torsional problems are solved as a superposition of flat punch  displacements and torsions. It is shown, that the tangential problem can be reduced to the normal problem. In section five, the contact problem of ellipsoid surfaces is studied and the static solution of Hertz, Cattaneo and Mindlin are briefly summarized. A general contact law for load histories of arbitrary tangential displacements is outlined in section six, where the boundary conditions are satisfied by an adequate superposition of static solutions. The same method can be used for torsional loading histories. A short computer program is listed. Finally, in section seven, an example of a pendulum with double hinges is studied and special cases are calculated analytically.


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