Stress analysis in rolling contact problem of a finite thickness FGM layer

Abstract

The rolling contact problem of a non-homogeneous layer is considered here. The graded layer possesses a variable elastic modulus with an exponential distribution. The Poisons ratio is assumed to be constant. A rigid cylindrical indenter is rolling over the surface of the graded layer with a constant velocity. First, the Navier equations of equilibrium are solved in the Fourier domain. Later, the boundary and the continuity conditions are satisfied in order to extract the governing singular integral equations. The numerical solution of the integral equations is provided by means of the Gauss–Chebyshev integration method. Finally, the sensitivity of the solution is analyzed for the effective parameters namely: the stiffness ratio, the layer thickness and the coefficient of friction. The results indicate that a minimum value of the coating thickness is required to alleviate the severe stress gradients in the critical locations. If the coating thickness decreases by a 50% then the Von Mises stress will increases about 20%. Also, a softening graded layer can result in a lower stress level over the interface which may enhance the bonding strength.

Appendix 1: The integrand function of the Fredholm kernels

The integrand functions introduced in Eqs.18 and 20 are given as:

$$\begin{aligned} {{h}_{31}}\left( \alpha ,y \right)&= \frac{i\alpha }{({{r}_{6}}{{{\bar{r}}}_{8}}+{{{\bar{r}}}_{6}}{{r}_{8}})}\left( {{r}_{8}}{{{\bar{y}}}_{3}}+{{{\bar{r}}}_{8}}{{y}_{3}} \right) \end{aligned}$$

(49a)

$$\begin{aligned} {{h}_{32}}\left( \alpha ,y \right)&= \frac{i\alpha }{({{r}_{6}}{{{\bar{r}}}_{8}}+{{{\bar{r}}}_{6}}{{r}_{8}})}\left( {{{\bar{r}}}_{6}}{{y}_{3}}-{{r}_{6}}{{{\bar{y}}}_{3}} \right) \end{aligned}$$

(49b)

$$\begin{aligned} {{h}_{41}}\left( \alpha ,y \right)&= \frac{i\alpha }{({{r}_{6}}{{{\bar{r}}}_{8}}+{{{\bar{r}}}_{6}}{{r}_{8}})}\left( {{{\bar{r}}}_{6}}{{y}_{4}}+{{r}_{6}}{{{\bar{y}}}_{4}} \right) \end{aligned}$$

(49c)

$$\begin{aligned} {{h}_{42}}\left( \alpha ,y \right)&= -\frac{i\alpha }{({{r}_{6}}{{{\bar{r}}}_{8}}+{{{\bar{r}}}_{6}}{{r}_{8}})}\left( {{r}_{8}}\bar{y}{}_{4}-{{{\bar{r}}}_{8}}{{y}_{4}} \right) \end{aligned}$$

(49d)

$$\begin{aligned} {{\Phi }_{31}}\left( \alpha \right)&= \frac{\alpha }{{{\Delta }_{5}}}\left( {{y}_{3}}{{{\bar{r}}}_{8}}+{{{\bar{y}}}_{3}}{{r}_{8}} \right) -\frac{\kappa +1}{4} \end{aligned}$$

(50a)

$$\begin{aligned} {{\Phi }_{32}}\left( \alpha \right)&= \frac{i\alpha }{{{\Delta }_{5}}}\left( {{y}_{3}}{{{\bar{r}}}_{8}}-{{{\bar{y}}}_{3}}{{r}_{6}} \right) +\frac{\kappa -1}{4} \end{aligned}$$

(50b)

$$\begin{aligned} {{\Phi }_{41}}\left( \alpha \right)&= -\frac{\alpha }{{{\Delta }_{5}}}\left( {{y}_{4}}{{{\bar{r}}}_{6}}+{{{\bar{y}}}_{4}}{{r}_{6}} \right) -\frac{\kappa +1}{4} \end{aligned}$$

(50c)

$$\begin{aligned} {{\Phi }_{42}}\left( \alpha \right)&= -\frac{i\alpha }{{{\Delta }_{5}}}\left( {{{\bar{y}}}_{4}}{{r}_{6}}-{{y}_{4}}{{{\bar{r}}}_{8}} \right) -\frac{\kappa +1}{4} \end{aligned}$$

(50d)

here

$$\begin{aligned} {{y}_{3}}&= {{e}^{\left( h+y \right) {{n}_{5}}}}+{{e}^{\left( h+y \right) {{n}_{6}}}}{{r}_{5}}+{{e}^{\left( h+y \right) {{{\bar{n}}}_{6}}}}{{r}_{7}} \end{aligned}$$

(51a)

$$\begin{aligned} {{y}_{4}}&= {{e}^{\left( h+y \right) {{n}_{5}}}}{{a}_{5}}+{{e}^{\left( h+y \right) {{n}_{6}}}}{{a}_{6}}{{r}_{5}}-{{e}^{\left( h+y \right) {{{\bar{n}}}_{6}}}}{{r}_{7}}{{\bar{a}}_{6}} \end{aligned}$$

(51b)

$$\begin{aligned} {{r}_{6}}&= {{z}_{55}}+{{z}_{56}}{{r}_{5}}{{e}^{-h({{n}_{5}}-{{n}_{6}})}}+{{\bar{z}}_{56}}{{r}_{7}}{{e}^{-h({{n}_{5}}-{{{\bar{n}}}_{6}})}} \end{aligned}$$

(52a)

$$\begin{aligned} {{r}_{8}}&= {{z}_{65}}+{{z}_{66}}{{r}_{5}}{{e}^{-h({{n}_{5}}-{{n}_{6}})}}+{{\bar{z}}_{66}}{{r}_{7}}{{e}^{-h({{n}_{5}}-{{{\bar{n}}}_{6}})}} \end{aligned}$$

(52b)

and:

$$\begin{aligned} {{z}_{55}}&= -i\alpha {{a}_{5}}\left( -3+\kappa \right) +{{n}_{5}}\left( 1+\kappa \right) \end{aligned}$$

(53a)

$$\begin{aligned} {{z}_{56}}&= -i\alpha {{a}_{6}}\left( -3+\kappa \right) +{{n}_{6}}\left( 1+\kappa \right) \end{aligned}$$

(53b)

$$\begin{aligned} {{z}_{65}}&= i\alpha +{{a}_{5}}{{n}_{5}} \end{aligned}$$

(53c)

$$\begin{aligned} {{z}_{66}}&= i\alpha +{{a}_{6}}{{n}_{6}} \end{aligned}$$

(53d)

In all mentioned equations, the symbol “ $\bar{}$ ” stands for the complex conjugate of the corresponding variable.