Frictional moving contact problem for a layer indented by a rigid cylindrical punch

Abstract

In this study, frictional moving contact problem for a rigid cylindrical punch and an elastic layer is considered. The punch is subjected to concentrated normal and tangential force, and moves steadily with a constant subsonic velocity on the boundary. The problem is reduced to a singular integral equation of the second kind, in which the contact stress and the contact area are the unknowns, and it is treated using Fourier transforms and the boundary conditions for the problem. The numerical solution of the singular integral equation is obtained by using the Gauss–Jacobi integration formulas. Numerical results for the contact stress and the contact area are given. The results show that with increasing values of relative moving velocity, contact width between the moving punch and the layer increases, whereas contact stress decreases.

Appendix A

Expressions of $k_{1} (x,\xi )$, $k_2 (x,\xi )$, $\beta _1$ and $\beta _{2}$ appearing in (13) are given as follows.

$$\begin{aligned} k_1 (x,\xi )= & {} \int _0^\infty {\frac{K_1 (\alpha )-\beta _1 }{\beta _1 }} \sin \alpha (\xi -x)\mathrm{d}\alpha \end{aligned}$$

(A1)

$$\begin{aligned} k_2 (x,\xi )= & {} \int _0^\infty {\frac{K_2 (\alpha )-\beta _2 }{\beta _1 }} \cos \alpha (\xi -x)\mathrm{d}\alpha \end{aligned}$$

(A2)

where

$$\begin{aligned} K_1 \left( \alpha \right)= & {} \frac{-\left( {4\left( {-1+c_1^2 } \right) c_2 \left( {c_1 c_2 \left( {1+\hbox {e}^{2c_1 \alpha }} \right) \left( {-1+\hbox {e}^{2c_2 \alpha }} \right) -\left( {-1+\hbox {e}^{2c_1 \alpha }} \right) \left( {1+\hbox {e}^{2c_2 \alpha }} \right) } \right) } \right) }{\Delta ( \alpha )} \end{aligned}$$

(A3)

$$\begin{aligned} K_2 \left( \alpha \right)= & {} 4 \Big (\left( {-1+\hbox {e}^{2c_1 \alpha }} \right) \left( {-1+\hbox {e}^{2c_2 \alpha }} \right) +c_1^2 \left( {1+2d^{2}} \right) \left( {-1+\hbox {e}^{2c_1 \alpha }} \right) \left( {-1+\hbox {e}^{2c_2 \alpha }} \right) \nonumber \\&\quad -\,3c_1 c_2 \left( {1+\hbox {e}^{2c_1 \alpha }+\hbox {e}^{2c_2 \alpha }-4\hbox {e}^{\left( {c_1 +c_2 } \right) \alpha }+\hbox {e}^{2\left( {c_1 +c_2 } \right) \alpha }} \right) -c_1^3 c_2 (1+\hbox {e}^{2c_1 \alpha } \nonumber \\&\quad +\,\hbox {e}^{2c_2 \alpha }-4\hbox {e}^{\left( {c_1 +c_2 } \right) \alpha }+\hbox {e}^{2\left( {c_1 +c_2 } \right) \alpha })\Big )/\Delta ( \alpha ) \end{aligned}$$

(A4)

where

$$\begin{aligned} \Delta ( \alpha )= & {} \Big (\left( {1-\hbox {e}^{2c_1 \alpha }} \right) \left( {-1+\hbox {e}^{2c_2 \alpha }} \right) -c_1^4 \left( {-1+\hbox {e}^{2c_1 \alpha }} \right) \left( {-1+\hbox {e}^{2c_2 \alpha }} \right) -2c_1^2 (1+2c_2^2 )\nonumber \\&\quad \left( {-1+\hbox {e}^{2c_1 \alpha }} \right) \left( {-1+\hbox {e}^{2c_2 \alpha }} \right) +c_1^5 c_2 \left( {1+\hbox {e}^{2c_1 \alpha }} \right) \left( {1+\hbox {e}^{2c_2 \alpha }} \right) \nonumber \\&\quad +\,2c_1^3 c_2 \left( {1+\hbox {e}^{2c_1 \alpha }+\hbox {e}^{2c_2 \alpha }-8\hbox {e}^{\left( {c_1 +c_2 } \right) \alpha }+\hbox {e}^{2\left( {c_1 +c_2 } \right) \alpha }} \right) \nonumber \\&\quad +\,c_1 c_2 (5+5\hbox {e}^{2c_1 \alpha }+ 5\hbox {e}^{2c_2 \alpha }-16\hbox {e}^{\left( {c_1 +c_2 } \right) \alpha }5\hbox {e}^{2\left( {c_1 +c_2 } \right) \alpha })\Big ) \end{aligned}$$

(A5)

$$\begin{aligned} \beta _1= & {} \lim \nolimits _{\alpha \rightarrow \infty } K_1 \left( \alpha \right) =-\frac{4\left( {-1+c_1^2 } \right) c_2 }{1+2c_1^2 +c_1^4 -4c_1 c_2 } \end{aligned}$$

(A6)

$$\begin{aligned} \beta _2= & {} \lim \nolimits _{\alpha \rightarrow \infty } K_2 \left( \alpha \right) =-\frac{4\left( {1+c_1^2 -2c_1 c_2 } \right) }{1+2c_1^2 +c_1^4 -4c_1 c_2 } \end{aligned}$$

(A7)