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Research Papers

A Study of the Contact of an Elastic Layer–Substrate System Indented by a Long Rigid Cylinder Incorporating
Surface Effects

[+] Author and Article Information
Thamarai Selvan Vasu

Department of Mechanical Engineering,
Indian Institute of Technology Bombay,
Powai,
Mumbai 400076, India
e-mail: thamaraivasu@iitb.ac.in

Tanmay K. Bhandakkar

Assistant Professor
Department of Mechanical Engineering,
Indian Institute of Technology Bombay,
Powai,
Mumbai 400076, India
e-mail: tbhanda2@iitb.ac.in

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received January 14, 2016; final manuscript received March 13, 2016; published online April 6, 2016. Assoc. Editor: Shaoxing Qu.

J. Appl. Mech 83(6), 061009 (Apr 06, 2016) (10 pages) Paper No: JAM-16-1023; doi: 10.1115/1.4033079 History: Received January 14, 2016; Revised March 13, 2016

Contact problem of a layer–substrate system comprising of an elastic layer and an elastic substrate perfectly bonded to each other with surface effects based on Gurtin–Murdoch (GM) model indented by a long rigid cylinder is solved. The requisite Green's function relating surface displacement to surface load is obtained semi-analytically through the combination of the Airy stress function and Fourier transforms under the plane-strain condition. The contact solution is analyzed to study the influence of layer thickness, modulus mismatch between the layer and substrate, and surface parameters on contact size and contact pressure during indentation of a layer–substrate system. A map is presented which indicates whether during indentation by a rigid cylinder, a layer–substrate system is required or a homogeneous system based on layer properties is enough for a given shear modulus mismatch ratio and layer thickness. The map and the related analysis clearly indicate that whenever the contact size or layer thickness approaches intrinsic length scale based on the ratio of surface parameter and bulk elastic properties, surface effects should be considered.

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Figures

Grahic Jump Location
Fig. 1

Schematic of the layer–substrate system with the layer having thickness t and indented by a long rigid indentor of radius R with normal total load P, where a0 is the semicontact length. Layer and substrate are isotropic, homogeneous linear elastic solids with Lame's constant (λ1,μ1) and (λ0,μ0), respectively, and adhering perfectly to each other.

Grahic Jump Location
Fig. 2

Effect of shear modulus ratio α on the semicontact size a0 during indentation of the layer–substrate system by a rigid cylinder (Fig. 1) for varying classical Hertzian semicontact sizes aH=2(1−ν)RP/(πμ). The solid line with “∗” marker denotes result of the contact problem with surface effects, while the dashed line with “○” marker shows result of the contact problem without considering surface effects. Thickness t of the layer is t/aH  = 1. The Poisson's ratio for layer and substrate is ν0=ν1  = 0.4, shear modulus for layer μ1 = 1 MPa, residual surface energy τs=0.1J/m2, and the corresponding intrinsic length S1 = 60 nm.

Grahic Jump Location
Fig. 3

Effect of shear modulus ratio α on the contact pressure p(Y) during indentation of the layer–substrate system by a rigid cylinder (Fig. 1) for layer thickness: (a) t/aH  = 1 and (b) t/aH = 0.1 for aH = 61.8 μm. The solid lines with marker denote result of the contact problem with surface effects, while the dashed lines without marker show result of the contact problem without considering surface effects. pmid is the contact pressure at x=0,y=0, and a0 is the semicontact size. The Poisson's ratio for layer and substrate is ν0=ν1  = 0.4, shear modulus for layer μ1 = 1 MPa, residual surface energy τs=0.1J/m2, and the corresponding intrinsic length S1 = 60 nm.

Grahic Jump Location
Fig. 4

Map of shear modulus ratio α and normalized layer thickness t/aH which separates regions where response of the layer–substrate system during indentation of rigid cylinder is identical to the response of a homogeneous material made of layer material. The solid lines with “∗” marker and “○” marker correspond to S1=(1−ν1)aH and S1=(1−ν1)aH/10, respectively, while the dashed line without marker corresponds to the layer–substrate system where the surface effects are ignored. The Poisson's ratio for layer and substrate is ν0=ν1  = 0.4, shear modulus for layer μ1 = 1 MPa, and classical Hertzian semicontact length aH = 1 μm.

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