A new algorithm for computing the indentation of a rigid body of arbitrary shape on a viscoelastic half-space

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Abstract

In this paper the contact problem between a rigid indenter of arbitrary shape and a viscoelastic half-space is considered. Under the action of a normal force the penetration of the indenter and the distribution of contact pressure change. We wish to find the relations which link the pressure distribution, the resultant force on the indenter and the penetration on the assumption that the surfaces are frictionless. For indenters of arbitrary shape the problem may be solved numerically by using the Matrix Inversion Method (MIM), extended to viscoelastic case. In this method the boundary conditions are satisfied exactly at specified “matching points” (the mid-points of the boundary elements). It can be validated by comparing the numerical results to the analytic solutions in cases of a spherical asperity (loading and unloading) and a conical asperity (loading only). Finally, the method was implemented for a finite cylindrical shape with its curved face indenting the surface of the half-space. This last example shows the efficiency of the method in case of a prescribed penetration as well as a given normal load history.

Introduction

Many contact problems are influenced by the viscoelastic behaviour of the materials. This influence is difficult to analyse for practical problems. For example, the contact between a tyre and a road is mainly modelled in the frame of the elastic theory. The viscoelastic effects have not been taken into account for computing the pressure distribution. In fact the dynamic modulus of rubber is frequency dependent and more important than the elastic modulus. The aim of our work is to compare the viscoelastic model to the elastic model for a single asperity of arbitrary shape for an increasing and then decreasing vertical loading.

The normal contact between a perfectly rigid indenter and a plane elastic half-space was first investigated by Boussinesq [1] using the potential theory. Hertz [2], [3] also gave an analytical solution of the contact problem in the case of two elastic bodies with smooth and quadratic contacting surfaces. Numerous analytical or semi-analytical solutions were then derived from Boussinesq's theory for a rigid indenter of arbitrary shape on an elastic half-space, especially in axisymmetric contact cases by Sneddon [4]. The paper of Gauthier et al. [5] is concerned with the indentation of an elastic half-space by an axisymmetric indenter under a monotonically applied normal force and under the assumption of Coulomb friction in the region of contact.

The problem of a rigid indenter pressed into contact with a viscoelastic solid was also investigated by many authors. The simplest approach to this problem follows a suggestion by Radok [6] for finding the stresses and deformations in cases where the corresponding solution for an elastic material is known. It consists in replacing the elastic constant in the elastic solution by the corresponding integral operator from the viscoelastic stress–strain relations. This approach can be applied to the contact problem provided that the loading program is such that the contact area is increasing. Radok's technique breaks when the contact area decreases. This complication has been studied by Ting [7] for a rigid axisymmetric indenter. Recently, Vandamme and Ulm [8] showed that for a conical indenter the suggestion of Radok remains valid at the very beginning of the unloading phase as well.

Analytical, numerical and experimental studies have been made in case of spherical, conical and pyramid indenters. Ball (Brinell) and flat (Boussinesq) punch indentation was analysed theoretically in the work of Larsson and Carlsson [9]. The elliptic indenter has been investigated by Yang [10]. The sharp indentation tests (the standard shapes of the Vickers, Berkovich and Knoop pyramids) are frequently used to examine hard materials like ceramics. The advantage of these tests is the simplicity of the experimental procedure. The work of Giannakopoulos [11] presents the results of frictionless and adhesionless contact of flat surfaces by pyramid indenters. One can also note the numerical works of Murakami et al. [12] and Larsson et al. [13] for the analysis of Berkovich indentation; the works of Rabinovich and Savin [14] for the analysis of Knoop indentation. Cheng et al. [15] analyse the indentation of viscoelastic solids by a spherical-tip indenter. Their solutions can apply to the response of compressible as well as incompressible coated layers to a spherical-tip indentation.

In this paper we propose a new algorithm for computing the indentation of a rigid body of arbitrary shape on a viscoelastic half-space. The Matrix Inversion Method (MIM) [16], which is described in the book of Johnson [17], is used in Ref. [18] for the analysis of the elastic tyre–road contact. We are extending this method to the viscoelastic problem to compute the pressure distribution history for any load or penetration history.

The paper is structured as follows. The indentation by a rigid indenter of arbitrary shape will be considered first. Then the discretization of the contact problem will be made. Next it will be possible to solve the general contact problem by the MIM. The general methodology will be applied to the spherical, conical and cylindrical indenters. The MIM will be compared first to the analytical results for a single spherical indenter (loading and unloading) and secondly to the analytical results for a single conical indenter (loading only). Then the indentation by a rigid cylindrical indenter will be considered. The results will be discussed before concluding remarks.

Section snippets

Algorithm for viscoelastic contacts

The stress–strain relations for an incompressible elastic solid may be written as eithersij=2μeeijandeij=12μesijwhere μe is the elastic shear modulus, sij is the deviatoric components of stress and eij is the deviatoric components of strain. The corresponding relationships for a linear viscoelastic material can be expressed by the creep and relaxation functions (Volterra equation):sij(t)=0tG(t-τ)deij(τ)dτdτwhere G(t) is the relaxation function, which specifies the stress response to a unit

The indentation of a viscoelastic half-space by a rigid spherical indenter

The MIM was first compared to the analytical results for a single spherical indenter of radius R acting on a viscoelastic half-space. The analytical result of Ting [7] and Graham [20] which was used for comparisons is described in the book of Christensen [21] and can be written asδ(t)=a2(t)R-1RtmtJ(t-θ)ddθt1(θ)θG(θ-τ)da2(τ)dτdτdθp(x,y,t)=2π(1-ν)R0t1(t)G(t-τ)da2(τ)-x2-y2dτdτwhere a(t) is the time dependent contact radius. The resultant normal contact force F(t) on the indenter acting on a

Conclusions

The Matrix Inversion Method (MIM) introduced by Kalker and used for the analysis of elastic contact has been extended to the equivalent viscoelastic problem so that the problem can be solved for different loading histories and for indenters of arbitrary shape. The numerical results were first compared to the analytical results for a single spherical indenter acting on a viscoelastic half-space. Then the comparison was made in case of a single conical indenter when the contact radius increases

Acknowledgements

This work is supported by ADEME in the frame of the French and German Cooperation P2RN project.

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