We analyse the oblique impact of linear-viscoelastic spheres by numerical models based on the Method of Dimensionality Reduction and the Boundary Element Method. Thereby we assume quasi-stationarity, the validity of the half-space hypothesis, short impact times and Amontons-Coulomb friction with a constant coefficient for both static and kinetic friction. As under these assumptions both methods are equivalent, their results differ only within the margin of a numerical error. The solution of the impact problem written in proper dimensionless variables will only depend on the two parameters necessary to describe the elastic problem and a sufficient set of variables to describe the influence of viscoelastic material behaviour; in the case of a standard solid this corresponds to two additional variables. The full solution of the impact problem is finally determined by comprehensive parameter studies and partly approximated by simple analytic expressions.
The paper presents a numerical study of the tangential contact between a rigid indenter and an elastomer with linear rheology. Occurring friction forces are caused exclusively by internal dissipative losses. Especially the combinations of a conical or paraboloid indenter and the Maxwell or standard body with fixed ratio of moduli are studied. One result is, that by the use of proper chosen dimensionless variables, for each of the combinations, the contact forces depend explicitly on the indentation depth only. Curves describing this dependencies are given as Bézier function fits. Application of these methods to an indenter with arbitrary shape and materials with a discrete relaxation spectrum described by Prony series is possible. Although only stationary solutions are analysed here, the method enables us to examine the transient process as well.
In the sliding contact of elastomer on a rigid substrate, the coefficient of friction may depend on a large number of system and loading parameters, including normal force, sliding velocity, shape of contacting bodies, surface roughness and so on. It was argued earlier that the contact configuration is determined more immediately through the indentation depth than the normal force, and thus the indentation depth can be considered as one of "robust governing parameters" of friction. Both models of friction of simple shapes and fractal surfaces demonstrate that the coefficient of friction of elastomers should be generally a function of dimensionless combinations of sliding velocity, surface gradient, relaxation time and size of micro-contacts. The relaxation time does depend only on temperature and the surface slope and the size of micro contacts mostly on the indentation depth. Based on this general structure of the law of friction, we propose a generalized master curve procedure for elastomer friction where the significant governing parameter -indentation depth (or normal force) was taken into account. Unlike the generation of the classical master curve by horizontal shifting of dependence "friction -logarithm of velocity" for different temperatures, in the case of various indentation depth the shifting in both horizontal and vertical direction is required. We experimentally investigated coefficient of friction of elastomer on sliding velocity for different indentation depths and temperatures, and generated a 'master curve' according to this hypothesis.
Abstract. The Boundary Element Method (BEM) for elastic materials is extended to deal with viscoelastic media. This is obtained by making use of a similar form of the fundamental solution for both the materials. Some considerations are attributed to the difference of the normal and the tangential contact problem. Both normal and tangential problems are furthermore assumed to be decoupled. Then the oblique impact of hard spheres with an incompressible viscoelastic half-space (linear standard-model) is studied. By assuming stick conditions during impact, one obtains the dependence of the two coefficients of restitution as functions of two input parameters. This result is expressed in an elegant and compact form of the fitting function.
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