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Contact Mechanics

# An Efficient 3D Model of Heterogeneous Materials for Elastic Contact Applications Using Multigrid Methods

[+] Author and Article Information
Hugo Boffy1

LaMCoS, CNRS UMR 5259,  Université de Lyon, INSA-Lyon, Villeurbanne, F69621, Francehugo.boffy@insa-lyon.fr

Marie-Christine Baietto

LaMCoS, CNRS UMR 5259,  Université de Lyon, INSA-Lyon, Villeurbanne, F69621, Francemarie-christine.baietto@insa-lyon.fr

Philippe Sainsot

LaMCoS, CNRS UMR 5259,  Université de Lyon, INSA-Lyon, Villeurbanne, F69621, Francephilippe.sainsot@insa-lyon.fr

Antonius A. Lubrecht

LaMCoS, CNRS UMR 5259,  Université de Lyon, INSA-Lyon, Villeurbanne, F69621, Franceton.lubrecht@insa-lyon.fr

1

Corresponding author.

J. Tribol 134(2), 021401 (Apr 12, 2012) (8 pages) doi:10.1115/1.4006296 History: Received June 09, 2011; Revised February 17, 2012; Published April 10, 2012; Online April 12, 2012

## Abstract

A 3D graded coating/substrate model based on multigrid techniques within a finite difference frame work is presented. Localized refinement is implemented to optimize memory requirement and computing time. Validation of the solver is performed through a comparison with analytical results for (i) a homogeneous material and (ii) a graded material. The algorithm performance is analyzed through a parametric study describing the influence of layer thickness (0.01 < t/a < 10) and mechanical properties (0.005 < Ec /Es  < 10) of the coating on the contact parameters (Ph , a). Three-dimensional examples are then presented to illustrate the efficiency and the large range of possibilities of the model. The influence of different gradations of Young’s modulus, constant, linear and sinusoidal, through the coating thickness on the maximum tensile stress is analyzed, showing that the sinusoidal gradation best accommodates the property mismatch of two successive layers. A final case is designed to show that full 3D spatial property variations can be accounted for. Two spherical inclusions of different size made from elastic solids with Young’s modulus and Poisson’s ratio are embedded within an elastically mismatched finite domain and the stress field is computed.

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## Figures

Figure 1

Local refinement technique: (a) coarsen grid, (b) one-level local refinement, (c) two-level local refinement

Figure 2

Surface displacement error

Figure 3

Relative error in the maximum Hertzian pressure

Figure 4

Comparison between analytical (GS) and numerical (MG) dimensionless central pressure p¯ (0,0) using the exponential law

Figure 5

Comparison between analytical (GS) and numerical (MG) dimensionless contact radius a¯ using the exponential law

Figure 6

Rigid spherical punch pressed against (a) a coating/substrate system, (b) a linear graded layer/substrate system, and (c) a sinusoidal graded layer/substrate system

Figure 7

Normalized contact pressure as a function of normalized layer thickness for several Ec /Es values

Figure 8

σ¯VM for a coated material, Ec /Es  = 3, normal loading, t/a0  = 0.5

Figure 9

σ¯VM for a coated material, Ec /Es  = 3, normal and tangential loading f = 0.3, t/a0  = 0.5

Figure 10

Max(σ¯xx), t/a0  = 0.5 − f = 0

Figure 11

Max(σ¯xx), t/a0  = 0.5 − f = 0.3

Figure 12

Figure 13

σ¯VM for a linear graded layer, Ec /Es  = 3, normal and tangential loading f = 0.3, t/a0  = 0.5

Figure 14

Sinusoidal variation of the Young’s Modulus, E(z)=(Es-Ec)/2*sin(2π/l¯*(z/t-1))+(Ec+Es)/2

Figure 15

σ¯xx(x = y = 0), t/a0  = 0.5 − f = 0 - for a sinusoidal variation (Fig. 14)

Figure 16

σ¯VM in the case of a sinusoidal variation, t/a0=0.5-f=0-l¯=2

Figure 17

σ¯VM in a homogeneous bulk with a soft (Einc 1  = 0.01E) and a hard (Einc 2  = 5E) inclusion, Lx /a = Ly /a = 2 * Lz /a = 8, eight levels used: refinement strategy starting from the sixth one

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