Influences of Boundary Slip Coating on Elastohydrodynamically Lubricated Contacts

Abstract

Recent research has identified the novel lubrication scheme brought by boundary slip effect in lubrication. However, the implementation of an endurable slip coating on elastohydrodynamic lubrication (EHL) is challenging. One plausible solution is to mount a slip surface coating onto a substrate with an interlayer, forming a multi-layer structure. This study presents a robust framework for analysing the effects of the thickness and strength of the coating and interlayer. A general algorithm for investigating the mechanics of boundary slip EHL contact that utilises a linear elasticity equation is developed. Besides the influence of boundary slip on the lubrication film, the effect on the mechanical response of the contacting bodies is emphasised.

Appendix: Governing Equations for the Boundary slip EHL Contact

The dimensionless parameters of the EHL model are derived following the scaling of:

$$H=h\frac{R}{{a}^{2}};\;P=\frac{p}{{P}_{h}};\;X=\frac{x}{a};\;Y=\frac{y}{a};Z=\left\{\begin{array}{c}\frac{z}{{t}_{\mathrm{c}}}\;\mathrm{ coating }\\ \frac{z}{a} \;\mathrm{other parts}\end{array} \right.;\;\overline{\eta }=\frac{\eta }{{\eta }_{0}};\;\overline{\rho }=\frac{\rho }{{\rho }_{0}};$$

$$\lambda = \frac{{6\eta _{0} R^{2} }}{{a^{3} P_{h} }};\,\varepsilon = \frac{{\bar{\rho }H^{3} }}{{\lambda \bar{\eta }}};\,U = \frac{{uR}}{{a^{2} }};\,V = \frac{{vR}}{{a^{2} }};\,W = \frac{{wR}}{{a^{2} }}$$

(16)

where h is film thickness, R is the radius of Body II, p is pressure, a is the half Hertzian contact width, ${P}_{h}$ is maximum Hertzian contact stress, t_c is the thickness of the surface coating, ${\eta }_{0}$ is the ambient viscosity of the lubricant, ${\rho }_{0}$ is the ambient density of the lubricant u, v and w are the elastic deformations. According to [25], to enable a stabilized solution, the weak form for the Reynolds equation and its stabilization terms are:

$$-{\int }_{{\Omega }_{L}}\varepsilon \frac{\partial P}{\partial X}\frac{\partial {P}_{t}}{\partial X}\mathrm{d}X-{\int }_{{\Omega }_{L}}\varepsilon \frac{\partial P}{\partial Y}\frac{\partial {P}_{t}}{\partial Y}\mathrm{d}Y+{\int }_{{\Omega }_{L}}\overline{\rho }H\left({u}_{1x}+{u}_{2x}\right)\frac{\partial {P}_{t}}{\partial X}\mathrm{d}X+{\int }_{{\Omega }_{L}}\overline{\rho }H\left({u}_{1y}+{u}_{2y}\right)\frac{\partial {P}_{t}}{\partial Y}\mathrm{d}X+{\int }_{{\Omega }_{L}}R\left(P\right)\frac{{h}_{e}}{2\left|{\varvec{\beta}}\right|l}\xi \left({P}_{e}\right)\left({\varvec{\beta}}\nabla {P}_{t}\right)\mathrm{d\Omega }+{\int }_{{\Omega }_{L}}{\rho }_{ID}\frac{\left|{\varvec{\beta}}\right|{h}_{e}}{2{l}_{o}}\nabla P\nabla {P}_{t}\mathrm{d\Omega }+{\int }_{{\Omega }_{L}}{p}_{f}{h}_{e}^{2}{P}^{-}{P}_{t}\mathrm{d\Omega }=0$$

(17)

where P_t is the shape function,

$${\varvec{\beta}}={\left({\beta }_{x},\;{\beta }_{y} \right)}^{T}, \;{ \beta }_{x}=H\left({u}_{\mathrm{I}x}+{U}_{\mathrm{II}x}\right)\frac{\partial \overline{\rho }}{\partial P}, \;{ \beta }_{y}=H{(u}_{\mathrm{I}y})\frac{\partial \overline{\rho }}{\partial P} ,$$

$$R\left(P\right)=-\nabla \left(\varepsilon \nabla P\right)+{\varvec{\beta}}\nabla P+\overline{\rho }\frac{\partial \left(H\left({u}_{\mathrm{I}x}+{U}_{\mathrm{II}x}\right)\right)}{\partial X}+\overline{\rho }\frac{\partial \left(H{(u}_{\mathrm{I}y})\right)}{\partial Y},$$

$${P}_{e}=\frac{\left|{\varvec{\beta}}\right|{h}_{e}}\,{2\varepsilon l}, \xi \left({P}_{e}\right)=\mathrm{coth}\left({P}_{e}\right)-\frac{1}{{P}_{e}} , \,{P}^{-}=\mathrm{min}\left(P, 0\right)$$

${\rho }_{ID}$ and ${p}_{f}$ are the coefficients, h_e is the characteristic length of the mesh, l_o is the order of the shape function, and ${\Omega }_{L}$ is the lubrication domain. The penalty term: ${\int }_{{\Omega }_{L}}{p}_{f}{h}_{e}^{2}{P}^{-}{P}_{t}\mathrm{d\Omega }$, is added to restrain the negative pressure in the converged results with very large p_f, e.g., p_f ~ 10⁶. When solving Eq. 16, we apply zero pressure boundary condition to the boundary of lubrication domain, and zero pressure gradient boundary condition to the axisymmetric boundary.

To obtain elastic deformation, we apply the dimensionless linear elasticity equation for Bodies I and II as:

$$\left\{\begin{array}{c}\begin{array}{c}\frac{\partial }{\partial X}\left({c}_{1}\frac{\partial U}{\partial X}+{c}_{2}\left(\frac{\partial V}{\partial Y}+\frac{\partial W}{\partial Z}\right)\right)+\frac{\partial }{\partial Y}\left({c}_{3}\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)\right)+\frac{\partial }{\partial Z}\left({c}_{3}\left(\frac{\partial U}{\partial Z}+\frac{\partial W}{\partial X}\right)\right)=0\\ \end{array}\\ \begin{array}{c}\frac{\partial }{\partial X}\left({c}_{3}\left(\frac{\partial U}{\partial Y}+\frac{\partial V}{\partial X}\right)\right)+\frac{\partial }{\partial Y}\left({c}_{1}\frac{\partial V}{\partial Y}+{c}_{2}\left(\frac{\partial U}{\partial X}+\frac{\partial W}{\partial Z}\right)\right)+\frac{\partial }{\partial Z}\left({c}_{3}\left(\frac{\partial V}{\partial Z}+\frac{\partial W}{\partial Y}\right)\right)=0\\ \end{array}\\ \frac{\partial }{\partial X}\left({c}_{3}\left(\frac{\partial U}{\partial Z}+\frac{\partial W}{\partial X}\right)\right)+\frac{\partial }{\partial Y}\left({c}_{3}\left(\frac{\partial V}{\partial Z}+\frac{\partial W}{\partial Y}\right)\right)+\frac{\partial }{\partial Z}\left({c}_{1}\frac{\partial W}{\partial Z}+{c}_{2}\left(\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}\right)\right)=0\end{array}\right.$$

(18)

where ${c}_{1}=\frac{E\left(1-\nu \right)}{\left(1+\nu \right)\left(1-2\nu \right)}$, ${c}_{2}=\frac{E\nu }{\left(1+\nu \right)\left(1-2\nu \right)}$, ${c}_{3}=\frac{E}{2\left(1+\nu \right)}$, $E$ is the Young’s modulus and $\nu$ is the Poisson’s ratio. Similarly, the dimensionless linear elasticity equation for the surface coating is:

$$\left\{ \begin{gathered} \frac{\partial }{{\partial X}}\left( {c_{{c1}} \frac{{t_{c} }}{a}\frac{{\partial U}}{{\partial X}} + c_{{c2}} \frac{{t_{c} }}{a}\frac{{\partial V}}{{\partial Y}} + c_{{c2}} \frac{{\partial W}}{{\partial Z}}} \right) + \frac{\partial }{{\partial Y}}\left( {c_{{c3}} \frac{{t_{c} }}{a}\left( {\frac{{\partial U}}{{\partial Y}} + \frac{{\partial V}}{{\partial X}}} \right)} \right) \hfill \\ + \frac{\partial }{{\partial Z}}\left( {c_{{c3}} \left( {\frac{a}{{t_{c} }}\frac{{\partial U}}{{\partial Z}} + \frac{{\partial W}}{{\partial X}}} \right)} \right) = 0 \hfill \\ \frac{\partial }{{\partial X}}\left( {c_{{c3}} \left( {\frac{{t_{c} }}{a}\frac{{\partial U}}{{\partial Y}} + \frac{{t_{c} }}{a}\frac{{\partial V}}{{\partial X}}} \right)} \right) + \frac{\partial }{{\partial Y}}\left( {c_{{c1}} \frac{{t_{c} }}{a}\frac{{\partial V}}{{\partial Y}} + c_{{c2}} \left( {\frac{{t_{c} }}{a}\frac{{\partial U}}{{\partial X}} + \frac{{\partial W}}{{\partial Z}}} \right)} \right) \hfill \\ + \frac{\partial }{{\partial Z}}\left( {c_{{c3}} \left( {\frac{a}{{t_{c} }}\frac{{\partial V}}{{\partial Z}} + \frac{{\partial W}}{{\partial Y}}} \right)} \right) = 0 \hfill \\ \frac{\partial }{{\partial X}}\left( {c_{{c3}} \left( {\frac{{\partial U}}{{\partial Z}} + \frac{{t_{c} }}{a}\frac{{\partial W}}{{\partial X}}} \right)} \right) + \frac{\partial }{{\partial Y}}\left( {c_{{c3}} \left( {\frac{{\partial V}}{{\partial Z}} + \frac{{t_{c} }}{a}\frac{{\partial W}}{{\partial Y}}} \right)} \right) \hfill \\ + \frac{\partial }{{\partial Z}}\left( {c_{{c1}} \frac{a}{{t_{c} }}\frac{{\partial W}}{{\partial Z}} + c_{{c2}} \left( {\frac{{\partial U}}{{\partial X}} + \frac{{\partial V}}{{\partial Y}}} \right)} \right) = 0 \hfill \\ \end{gathered} \right.$$

(19)

where c_c1, c_c2 and c_c3 are the elasticity constants of the coating. The boundary conditions for Eqs. 18 and 19 are:

$$\left\{ \begin{gathered} c_{{c1}} \frac{a}{{t_{c} }}\frac{{\partial W}}{{\partial Z}} + c_{{c2}} \left( {\frac{{\partial U}}{{\partial X}} + \frac{{\partial V}}{{\partial Y}}} \right) = - \frac{{Pp_{h} R}}{a}\,{\text{coating}} \hfill \\ c_{1} \frac{{\partial W}}{{\partial Z}} + c_{2} \left( {\frac{{\partial U}}{{\partial X}} + \frac{{\partial V}}{{\partial Y}}} \right) = \frac{{Pp_{h} R}}{a}\,{\text{Body}}\,{\text{II}} \hfill \\ U = V = W = 0\,{\text{bottom}}\,{\text{of}}\,{\text{body}}\,{\text{I}}\,{\text{and}}\,{\text{top}}\,{\text{of}}\,{\text{body}}\,{\text{II}} \hfill \\ \end{gathered} \right.$$

(20)

We also implement deformation continuity at the interface of the surface coating and the substrate in the model. After solving Eqs. 18, 19, the dimensionless von Mises stress is obtained:

$${\tilde{\sigma }}_{\text{von}}=\sqrt{\frac{1}{2}\left({\left({\tilde{\sigma }}_{xx}-{\tilde{\sigma }}_{yy}\right)}^{2}+{\left({\tilde{\sigma }}_{xx}-{\tilde{\sigma }}_{zz}\right)}^{2}+{\left({\tilde{\sigma }}_{zz}-{\tilde{\sigma }}_{yy}\right)}^{2}+6\left({\tilde{\sigma }}_{xy}^{2}+{\tilde{\sigma }}_{xz}^{2}+{\tilde{\sigma }}_{yz}^{2}\right)\right)}$$

(21)

where ${\tilde{\sigma }}_{xx}$, ${\tilde{\sigma }}_{yy}$ and ${\tilde{\sigma }}_{zz}$ are the dimensionless normal stresses inside the contacting bodies:

$$\left\{\begin{array}{c}\begin{array}{c}\left.\begin{array}{c}\begin{array}{c}{\tilde{\sigma }}_{xx}=\frac{a}{R}\left({c}_{c1}\frac{\partial U}{\partial X}+{c}_{c2}\frac{\partial V}{\partial Y}+{c}_{c2}\frac{a}{{t}_{c}}\frac{\partial W}{\partial Z}\right)/{p}_{h}\\ \end{array}\\ \begin{array}{c}{\tilde{\sigma }}_{yy}=\frac{a}{R}\left({c}_{c2}\frac{\partial U}{\partial X}+{c}_{c1}\frac{\partial V}{\partial Y}+{c}_{c2}\frac{a}{{t}_{c}}\frac{\partial W}{\partial Z}\right)/{p}_{h}\\ \end{array}\\ {\tilde{\sigma }}_{zz}=\frac{a}{R}\left({c}_{c2}\frac{\partial U}{\partial X}+{c}_{c2}\frac{\partial V}{\partial Y}+{c}_{c1}\frac{a}{{t}_{c}}\frac{\partial W}{\partial Z}\right)/{p}_{h}\end{array}\right\}\, \mathrm{coating}\\ \end{array}\\ \left.\begin{array}{c}\begin{array}{c}{\tilde{\sigma }}_{xx}=\frac{a}{R}\left({c}_{c1}\frac{\partial U}{\partial X}+{c}_{c2}\frac{\partial V}{\partial Y}+{c}_{c2}\frac{\partial W}{\partial Z}\right)/{p}_{h}\\ \end{array}\\ \begin{array}{c}{\tilde{\sigma }}_{yy}=\frac{a}{R}\left({c}_{c2}\frac{\partial U}{\partial X}+{c}_{c1}\frac{\partial V}{\partial Y}+{c}_{c2}\frac{\partial W}{\partial Z}\right)/{p}_{h}\\ \end{array}\\ {\tilde{\sigma }}_{zz}=\frac{a}{R}\left({c}_{c2}\frac{\partial U}{\partial X}+{c}_{c2}\frac{\partial V}{\partial Y}+{c}_{c1}\frac{\partial W}{\partial Z}\right)/{p}_{h}\end{array}\right\} \,substrate\end{array}\right.$$

(22)

${\tilde{\sigma }}_{xy}$, ${\tilde{\sigma }}_{xz}$ and ${\tilde{\sigma }}_{yz}$ are the dimensionless tangential stresses inside the contacting bodies.

$$\left\{ \begin{gathered} \left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\tilde{\sigma }_{{xy}} = \frac{a}{R}\left( {c_{{c3}} \left( {\frac{{\partial U}}{{\partial Y}} + \frac{{\partial V}}{{\partial X}}} \right)} \right)/p_{h} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\tilde{\sigma }_{{yz}} = \frac{a}{R}\left( {c_{{c3}} \left( {\frac{a}{{t_{c} }}\frac{{\partial U}}{{\partial Z}} + \frac{{\partial W}}{{\partial X}}} \right)} \right)/p_{h} } \\ \end{array} } \\ {\tilde{\sigma }_{{xz}} = \frac{a}{R}\left( {c_{{c3}} \left( {\frac{a}{{t_{c} }}\frac{{\partial V}}{{\partial Z}} + \frac{{\partial W}}{{\partial Y}}} \right)} \right)/p_{h} } \\ \end{array} } \right\}\,{\text{coating}} \hfill \\ \left. {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\tilde{\sigma }_{{xy}} = \frac{a}{R}\left( {c_{3} \left( {\frac{{\partial U}}{{\partial Y}} + \frac{{\partial V}}{{\partial X}}} \right)} \right)/p_{h} } \\ \end{array} } \\ {\begin{array}{*{20}c} {\tilde{\sigma }_{{yz}} = \frac{a}{R}\left( {c_{3} \left( {\frac{{\partial U}}{{\partial Z}} + \frac{{\partial W}}{{\partial X}}} \right)} \right)/p_{h} } \\ \end{array} } \\ {\tilde{\sigma }_{{xz}} = \frac{a}{R}\left( {c_{3} \left( {\frac{{\partial V}}{{\partial Z}} + \frac{{\partial W}}{{\partial Y}}} \right)} \right)/p_{h} } \\ \end{array} } \right\}\,{\text{substrate/Body}}\,{\text{II}} \hfill \\ \end{gathered} \right.$$

(23)