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


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.


Contact mechanics Boundary element method Adhesive contact Nanotribology 

List of Symbols


Contact half-width


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


Element in boundary element matrix

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

Matrix of coefficients in boundary element analysis


Element in boundary element matrix


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


Coordinate of the datum point


Young’s modulus


Equivalent Young’s modulus


Total load

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

Residue vector


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


Shear modulus


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


The gap between two surfaces at\(x\)

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

Jacobian matrix


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


The unit outward normal at the boundary


Point inside/on the cylinder


Point on the boundary surface


Radius of cylinder, radius of curvature

\(\bar {R}\)

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


Distance between p and Q


Horizontal coordinate


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


Area where boundary condition for traction is given


Area where traction is unknown

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

Boundary condition of traction


Traction kernel

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

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


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


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



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

Unknown displacement

\(\bar {u}\)

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


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

\(\bar {W}\)

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


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


Horizontal coordinate


Vertical coordinate


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


Approach distance


Intermolecular distance where zero force occurs between two infinite surfaces


Boundary in the boundary element analysis


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


Poisson’s ratio



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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