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Tribology Letters

, 67:19 | Cite as

Boundary Element Analyses on the Adhesive Contact between an Elastic Cylinder and a Rigid Half-Space

  • Jiunn-Jong WuEmail author
Original Paper
  • 54 Downloads

Abstract

Boundary element method is used to analyze the adhesive contact between an elastic cylinder and a rigid half-space. Lennard-Jones potential is used for the surface traction. In the past, the simulation for the adhesive contact between cylinders usually used parabolic approximation for cylinder surface, and used line loading acting on a half-space. Since line loading may cause infinite deformation, only contact half-width/load relation and pull-off force can be obtained. In this paper, the adhesive contact between an exact elastic cylinder and a rigid half-space is investigated. The S-shaped load-approach curve and the whole solution are obtained. Using the load-approach curves, the pull-off force, pull-off distance and jump-in distance are obtained. The effects of Tabor parameter and radius are investigated. The result is compared with the numerical simulation for the adhesive contact between an elastic parabolically approximated cylinder and a rigid half-space and the two-dimensional JKR model. For large Tabor parameters, two-dimensional JKR model can approximate the adhesive contact. For small Tabor parameters, two-dimensional Bradley model can approximate the adhesive contact. The radii do affect the load-approach relation for large Tabor parameters, and have very small effects for small Tabor parameters. A semi-rigid cylinder model is proposed. This model can predict the load-approach curves for small Tabor parameters and can predict the jump-in distance for large Tabor parameters. In addition, a modified load-approach relation for two-dimensional JKR model is proposed. This relation can approximate the load-approach relation and predict the pull-off distance for large Tabor parameters. It is also found that the radius does not affect the pull-off force.

Keywords

Contact mechanics Boundary element method Adhesive contact Nanotribology 

List of Symbols

\(a\)

Contact half-width

\(A\)

Non-dimensional contact half-width, \(A=\frac{a}{{\sqrt {\mu \varepsilon R} }}\)

\(\bar {A}\)

Non-dimensional contact half-width, \(\bar {A}=\frac{a}{{\sqrt {\varepsilon R} }}\)

\(\underline{\underline {{\text{A}}}}\)

Matrix of coefficients in boundary element analysis

\({A_{ij}}\)

Element in boundary element matrix

\(\underline{\underline {{\text{B}}}}\)

Matrix of coefficients in boundary element analysis

\({B_{ij}}\)

Element in boundary element matrix

\(D\)

Non-dimensional coordinate of the datum point, \(D=\frac{d}{{\sqrt {\mu \varepsilon R} }}\)

\(d\)

Coordinate of the datum point

\(E\)

Young’s modulus

\(E^{\prime}\)

Equivalent Young’s modulus

\(F\)

Total load

\(\underline {\mathbf{F}}\)

Residue vector

\({F_i}\)

Element in \(\underline {\mathbf{F}}\)

\(G\)

Shear modulus

\(H\)

Non-dimensional gap, \(H=\frac{1}{\mu }\left( {\frac{h}{\varepsilon } - 1} \right)\)

\(h\)

The gap between two surfaces at\(x\)

\(\underline{\underline {\mathbf{J}}} (\underline {\mathbf{H}} )\)

Jacobian matrix

\({J_{ij}}\)

\({J_{ij}} \equiv \frac{{\partial {F_i}}}{{\partial {H_j}}}\), element of \(\underline{\underline {\mathbf{J}}} (\underline {\mathbf{H}} )\)

\(n\)

The unit outward normal at the boundary

\(p\)

Point inside/on the cylinder

\(Q\)

Point on the boundary surface

\(R\)

Radius of cylinder, radius of curvature

\(\bar {R}\)

Non-dimensional radius of cylinder, \(\bar {R}=\frac{R}{\varepsilon }\)

\(r(p,Q)\)

Distance between p and Q

\(s\)

Horizontal coordinate

\(S\)

Non-dimensional horizontal coordinate, \(S=\frac{s}{{\sqrt {\varepsilon R} }}\), surface

\({S_b}\)

Area where boundary condition for traction is given

\({S_t}\)

Area where traction is unknown

\({\bar {T}_i}\)

Boundary condition of traction

\({T_{ij}}\)

Traction kernel

\({\bar {T}_{ij}}\)

Non-dimensional traction kernel, \({\bar {T}_{ij}}={T_{ij}}\varepsilon\)

\(t\)

Traction, line load

\(\bar {t}\)

Non-dimensional traction, \(\bar {t}=\frac{{t\varepsilon }}{{\Delta \gamma }}\)

\({\bar {t}_i}\)

Unknown traction

\({\bar {U}_i}\)

Boundary condition of displacement

\({U_{ij}}\)

Displacement kernel

\({\bar {U}_{ij}}\)

Non-dimensional displacement kernel, \({\bar {U}_{ij}}={U_{ij}}E^{\prime}\)

\(\underline {{\text{U}}}\)

Vector of boundary conditions in boundary element analysis

\(\underline {\mathbf{u}}\)

Vector of unknowns in boundary element analysis

\(u\)

Displacement

\({\bar {u}_i}\)

Unknown displacement

\(\bar {u}\)

Non-dimensional displacement, \(\bar {u}=\frac{u}{\varepsilon }\)

\(W\)

Non-dimensional total load, \(W=\frac{F}{{{{(E^{\prime}R\Delta {\gamma ^2})}^{1/3}}}}\)

\(\bar {W}\)

Non-dimensional total load, \(\bar {W}=\frac{{F\sqrt \varepsilon }}{{\Delta \gamma \sqrt R }}\)

\(X\)

Non-dimensional horizontal coordinate, \(X=\frac{x}{{\sqrt {\mu \varepsilon R} }}\)

\(x\)

Horizontal coordinate

\(y\)

Vertical coordinate

\(\Delta\)

Non-dimensional approach distance, \(\Delta =\frac{\delta }{{\mu \varepsilon }}\)

\(\bar {\Delta }\)

Non-dimensional approach distance, \(\bar {\Delta }=\frac{\delta }{\varepsilon }\)

\({\Delta _e}\)

The approach for \(D=1\)

\(\Delta \gamma\)

Surface energy

\(\delta\)

Approach distance

\(\varepsilon\)

Intermolecular distance where zero force occurs between two infinite surfaces

\(\Gamma\)

Boundary in the boundary element analysis

\(\mu\)

Tabor parameter, \(\mu ={\left( {\frac{{R\Delta {\gamma ^2}}}{{{{E^{\prime}}^2}{\varepsilon ^3}}}} \right)^{1/3}}\)

\(\nu\)

Poisson’s ratio

Notes

Acknowledgements

The authors thank Ministry of Science and Technology, Taiwan for its financial support under Grant MOST 104-2221-R-182-080.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mechanical EngineeringChang Gung UniversityTaoyüanTaiwan

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