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# A new algorithm for solving the multi-indentation problem of rigid bodies of arbitrary shapes on a viscoelastic half-space

International Journal of Mechanical Sciences 01/2010; DOI:10.1016/j.ijmecsci.2009.10.015
Source: OAI

ABSTRACT In this paper the contact problem between rigid indenters of arbitrary shapes and a viscoelastic half-space is considered. Under the action of a normal force the penetration of the indenters changes and a few contact areas appeared. We wish to find the relations which link the pressure distribution, the resultant forces on the indenters and the penetration on the assumption that the surfaces are frictionless. For indenters of arbitrary shapes the problem may be solved numerically by using the matrix inversion method, extended to viscoelastic cases [1]. But when the problem involves a largenumber of points the matrix inversion method can become very time-consuming. Here the problem is solved using an alternative scheme, called the two-scale iterative method. In this method the local matrix inversion method is used at the micro-scale for each contact area to compute the pressure distribution taking into account interacting effect (the forces on the other contact areas which can be calculated at the macro-scale) between indenters. Two algorithms were proposed. The first algorithm takes into account the distribution of forces on the other contact areas and the second is the approximation of the first algorithm and takes into account the resultant forces on the other contact areas. The method was implemented for a simple configuration of seven spherical indenters, seven spherical-ended cylindrical indenters and seven flat-ended cylindrical indenters as well as for a more complex configuration of 12 randomly positioned indenters of arbitrary shapes: spherical-ended cylindrical, flat-ended cylindrical, conical and cylindrical indenters (finite cylindrical shape with its curved face). This last case is more difficult as the indenting geometry does not have an axisymmetric profile. For all these cases the two-scale iterative method permits to find the pressure distribution and the contact forces versus the penetration. It can be validated by comparing the numerical results to the numerical results obtained with the matrix inversion method.

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##### Article: Contact modeling — forces
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ABSTRACT: This paper reviews contact modeling with an emphasis on the forces of contact and their relationship to the geometrical, material and mechanical properties of the contacting bodies. Single asperity contact models are treated first. These models include simple Hertz contacts for spheres, cylinders, and ellipsoids. Further generalizations include the effects of friction, plasticity, adhesion, and higher-order terms which describe the local surface topography. Contact with a rough surface is generally represented by a multi-asperity contact model. Included is the well-known Greenwood–Williamson contact model, as well as a myriad of other models, many of which represent various modifications of the basic theory. Also presented in this review is a description of wavy surface contact models, with and without the effects of friction. These models inherently account for the coupling between each of the contacting areas. A brief review of experimental investigations is also included. Finally some recent work, which addresses the dynamics and associated instabilities of sliding contact, is presented and the implications discussed.
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