Tribology Letters

, 67:19

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

• Jiunn-Jong Wu
Original Paper

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

$$F$$

$$\underline {\mathbf{F}}$$

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

$$R$$

$$\bar {R}$$

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, $${\bar {T}_{ij}}={T_{ij}}\varepsilon$$

$$t$$

$$\bar {t}$$

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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