Research Papers

Atomically Informed Continuum Models for the Elastic Contact Properties of Hollow and Coated Rigid Cylinders at the Nanoscale

[+] Author and Article Information
Leon Gorelik, Dan Mordehai

Department of Mechanical Engineering,
Technion—Israel Institute of Technology,
Haifa 32000, Israel

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received October 20, 2016; final manuscript received December 1, 2016; published online January 25, 2017. Assoc. Editor: Harold S. Park.

J. Appl. Mech 84(3), 031009 (Jan 25, 2017) (12 pages) Paper No: JAM-16-1511; doi: 10.1115/1.4035365 History: Received October 20, 2016; Revised December 01, 2016

Understanding the mechanical properties of contacts at the nanoscale is key to controlling the strength of coated surfaces. In this work, we explore to which extent existing continuum models describing elastic contacts with coated surfaces can be extended to the nanoscale. Molecular dynamics (MD) simulations of hollow cylinders or coated rigid cylinders under compression are performed and compared with models at the continuum level, as two representative extreme cases of coating which is substantially harder or softer than the substrate, respectively. We show here that the geometry of the atomic-scale contact is essential to capture the contact stiffness, especially for hollow cylinders with high relative thicknesses and for coated rigid cylinders. The contact pressure profiles in atomic-scale contacts are substantially different than the one proposed in the continuum models for rounded contacts. On the basis of these results, we formulate models whose solution can be computed analytically for the contact stiffness in the two extreme cases, and show how to bridge between the atomic and continuum scales with atomically informed geometry of the contact.

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Fig. 1

Atomic structure and the crystallographic orientations of (a) a hollow cylinder and (b) a coated rigid cylinder

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Fig. 2

Atomic structure near the contact region. 2a is the size of the atomic step in contact with the indenter

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Fig. 3

Mesh of the FEM: (a) a hollow cylinder, (b) a truncated hollow cylinder, (c) a coated system, and (d) a truncated coated system. I–VI represent the different mesh zones.

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Fig. 4

Schematics of the FEM configurations and their boundary conditions: (a) a hollow cylinder, (b) a truncated hollow cylinder, (c) a coated system, and (d) a truncated coated system. The dotted lines represent the state before compression.

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Fig. 5

Indentation curve of a hollow cylinder (Ro = 120 Å, t = 90 Å). The straight line is a linear fit to the MD result in a region where full contact is established.

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Fig. 6

Hollow cylinders contact stiffness as a function of t/Ro: (a) at different outer radii in the MD simulations and (b) at an outer radius of 600 Å with different models. The different simulations and models are described in the text.

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Fig. 7

Dimensionless contact pressure distribution of hollow cylinders (Ro = 300 Å, t = 100 Å at δ = 26.2 Å). The contact pressure of the flat punch in contact with a half-space model (Eq. (6)) diverges at the edges of the contact. To demonstrate the singularity, the same plot but with a different scale is shown in the inset for the FEM of a truncated cylinder and the continuum model.

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Fig. 8

Indentation curve of a coated rigid cylinder (Ro = 300 Å, t = 30 Å)

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Fig. 9

Coated rigid cylinder contact stiffness: (a) as a function of t/Ro ratio for both outer radii in the MD simulations and (b) for an outer radius of 300 Å obtained with different simulation techniques and analytical solutions. The results are shown as a function of a dimensionless geometrical parameter that consists of both the step of terrace and the coating size.

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Fig. 10

Sliding displacements at the coating–substrate interface, divided by the lattice parameter (Ro = 300 Å, t = 100 Å at δ = 3.89 Å)

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Fig. 11

Maximum RSS fields of cylindrical coatings (Ro = 300 Å, t = 100 Å at δ = 3.89 Å) in (a) the MD simulation and (b)–(e) the FEM. The FEM differs in the geometry of the contact (truncated or rounded) and the adhesion at the coating–substrate interface (bonded or no-separation): (b) truncated/bonded, (c) truncated/no-separation, (d) rounded/bonded, and (e) rounded/no-separation. For clarity, the scaling bars in (b) and (c) are not linear, because of the singularity of stresses near the edge of the contact.

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Fig. 12

Dimensionless contact pressure distribution of cylindrical coatings (Ro = 300 Å, t = 100 Å at δ = 3.89 Å). For comparison, the dimensionless contact pressure distribution of a hollow cylinder (Ro = 300 Å, t = 100 Å at δ = 26.2 Å) in the MD simulation is also plotted.

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Fig. 14

Schematics of a rigid flat punch in contact with a thin layer. The dotted lines represent the state before indentation.

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Fig. 13

(a) A circular beam under diametrical compression. (b) The cross section of the beam perpendicular to θ axis.



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