On the sliding frictional nanocontact of an exponentially graded layer/substrate structure

Abstract

This paper analyzes the sliding frictional nanocontact of an exponentially graded layer perfectly bonded to a homogeneous half-plane substrate under the nanoindentation of a rigid cylinder. The punch is subjected to both normal and tangential loads, satisfying Coulomb’s friction law. The contact interface is modeled by the full version of Steigmann–Ogden surface mechanical theory, in which surface tension, surface membrane stiffness and surface flexural rigidity are all taken into account. The method of Fourier integral transforms was applied to convert the governing equations and nonclassical boundary conditions into a Fredholm integral equation. By separating a nonsingular term from the integrand of the kernel function of the integral equation, numerical integration of the kernel function can be significantly improved. After that, Gauss–Chebyshev numerical quadratures are further employed to discretize and collocate the integral equation and the indentation force equilibrium condition. An iterative algorithm is subsequently developed to solve the resultant nonlinear algebraic system regarding discretized contact pressures and the two asymmetric contact boundaries. Stresses and displacements in the layer/substrate structure are also determined for completeness. Extensive parametric studies clarify the relative importance among three surface parameters, all helping to partially carry on the indentation loading in addition to the conventional bulk portion of the layer/substrate system. The sliding frictional condition significantly affects the symmetry of the contact pressure distribution and contact zone. The property gradation of the layer is another important factor affecting contact properties. The results reported in the current work show a concrete means of tailoring nanocontact responses of graded layers in nanosized materials and devices.

Appendix A

The determinant M appearing in (7a, 7b) is defined by the coefficient matrix of the linear system

$$\begin{aligned}{} & {} \left( {\begin{array}{*{20}{c}} {{\Delta _1}}&{}{{\Delta _2}}&{}{{\Delta _3}}&{}{{\Delta _4}}&{}0&{}0\\ {{\Pi _1}}&{}{{\Pi _2}}&{}{{\Pi _3}}&{}{{\Pi _4}}&{}0&{}0\\ {{{ K}_1}}&{}{{{ K}_2}}&{}{{{ K}_3}}&{}{{{ K}_4}}&{}{ - 2{\mu _2}I\xi \cdot {e^{ - \left| \xi \right| h}}}&{}{ - 2{\mu _2}I\left( {\frac{{\left( {1 + \kappa } \right) \xi }}{{2\left| \xi \right| }} + \xi h} \right) \cdot {e^{ - \left| \xi \right| h}}}\\ {{\Lambda _1}}&{}{{\Lambda _2}}&{}{{\Lambda _3}}&{}{{\Lambda _4}}&{}{2{\mu _2}\left| \xi \right| \cdot {e^{ - \left| \xi \right| h}}}&{}{ - 2{\mu _2}\left( {\frac{{1 - \kappa }}{2} - \left| \xi \right| h} \right) \cdot {e^{ - \left| \xi \right| h}}}\\ {{e^{{m_1}h}}}&{}{{e^{{m_2}h}}}&{}{{e^{{m_3}h}}}&{}{{e^{{m_4}h}}}&{}{ - {e^{ - \left| \xi \right| h}}}&{}{ - h{e^{ - \left| \xi \right| h}}}\\ {{s_1}{e^{{m_1}h}}}&{}{{s_2}{e^{{m_2}h}}}&{}{{s_3}{e^{{m_3}h}}}&{}{{s_4}{e^{{m_4}h}}}&{}{I\frac{\xi }{{\left| \xi \right| }} \cdot {e^{ - \left| \xi \right| h}}}&{}{I\left( {\frac{\xi }{{\left| \xi \right| }}h + \frac{\kappa }{\xi }} \right) \cdot {e^{ - \left| \xi \right| h}}} \end{array}} \right) \left( {\begin{array}{*{20}{c}} {{B_1}}\\ {{B_2}}\\ {{B_3}}\\ {{B_4}}\\ {{A_5}}\\ {{A_6}} \end{array}} \right) \nonumber \\{} & {} \quad = \left( {\begin{array}{*{20}{c}} { - {{{\tilde{q}}}}}\\ { - {{{\tilde{p}}}}}\\ 0\\ 0\\ 0\\ 0 \end{array}} \right) , \end{aligned}$$

(38)

where

$$\begin{aligned}{} & {} {\Delta _j} = {\mu _1}{q_j} - {k_0}{\xi ^2}, \end{aligned}$$

(39)

$$\begin{aligned}{} & {} {\Pi _j} = \frac{{{\mu _1}}}{{\kappa - 1}}{p_j} - \left( {{\tau _0}{\xi ^2} + {d_0}{\xi ^4}} \right) {s_j}, \end{aligned}$$

(40)

$$\begin{aligned}{} & {} {{ K}_j} = {\mu _1}\frac{{{e^{(\beta + {m_j})h}}}}{{\kappa - 1}}{p_j}, \end{aligned}$$

(41)

$$\begin{aligned}{} & {} {\Lambda _j} = {\mu _1}{e^{(\beta + {m_j})h}}{q_j}, \end{aligned}$$

(42)

with $j = 1-4$ and

$$\begin{aligned}{} & {} {q_j} = {m_j} - I\xi {s_j}, \end{aligned}$$

(43)

$$\begin{aligned}{} & {} {p_j} = - I\xi \left( {3 - \kappa } \right) + \left( {1 + \kappa } \right) {m_j}{s_j}. \end{aligned}$$

(44)