Frictionally excited thermoelastic dynamic instability of functionally graded materials

Abstract

The perturbation method is applied to investigate the frictionally excited thermoelastic dynamic instability (TEDI) of a functionally graded material (FGM) coating in half-plane sliding against a homogeneous half-plane. We assume that the thermoelastic properties of the FGM vary exponentially with thickness. We also examine the effects of the gradient index, sliding speed, and friction coefficient on the TEDI for various material combinations. The transverse normal stress for two different coating structures is calculated. Furthermore, the frictional sliding stability of two different coating structures is analyzed. The obtained results show that use of FGM coatings can improve the TEDI of this sliding system and reduce the possibility of interfacial failure by controlling the interfacial tensile stress.

Appendix

$$ \eta_{1j} = \sum\limits_{k = 1}^{4} {{{\left( {\gamma_{1k} - f\gamma_{2k} } \right)F_{kj} {\text{e}}^{{s_{0k} h}} } / {\delta_{k} }}} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, $$

(A.1)

$$ \begin{aligned} & \delta_{k} = \sum\limits_{k = 1}^{4} {\left( {f\gamma_{2k} - \gamma_{1k} } \right)F_{k3} {\text{e}}^{{s_{0k} h}} } \\ &\quad - f\beta_{0}^{2} {\text{e}}^{{\eta^{\prime}h}} \left( {d_{05} F_{53} {\text{e}}^{{s_{05} h}} + d_{06} F_{63} {\text{e}}^{{s_{06} h}} } \right) \\ & \quad {\kern 1pt} + \sum\limits_{k = 5}^{6} \left\{ f\left[ {\beta_{0}^{2} a_{0k} \left( {s_{0k} + \eta^{\prime}} \right) + {\text{i}}\left( {\beta_{0}^{2} - 2} \right)} \right]\right.\\ &\quad \left.- \left( {s_{0k} + \eta^{\prime} + {\text{i}}a_{0k} } \right) \right\}F_{k3} {\text{e}}^{{\left( {s_{0k} + \eta^{\prime}} \right)h}} , \\ \end{aligned} $$

(A.2)

$$ r_{1k} = s_{0k} + {\text{i}}a_{0k} ,\quad r_{2k} = \beta_{0}^{2} a_{0k} s_{0k} + {\text{i}}\left( {\beta_{0}^{2} - 2} \right), $$

(A.3)

$$ \begin{aligned} &l_{1j} = {\text{e}}^{{ - \varepsilon^{\prime}h}} \frac{{m_{1j} m_{24} - m_{2j} m_{14} }}{{m_{13} m_{24} - m_{23} m_{14} }},\\ & l_{2j} = {\text{e}}^{{ - \varepsilon^{\prime}h}} \frac{{m_{1j} m_{23} - m_{2j} m_{13} }}{{m_{14} m_{23} - m_{24} m_{13} }},{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \end{aligned} $$

(A.4)

$$ \begin{aligned} &m_{1j} = \sum\limits_{k = 1}^{4} {a_{0k} F_{kj} {\text{e}}^{{s_{0k} h}} }\\ &\quad + {\kern 1pt} \left[ {\sum\limits_{k = 1}^{4} {a_{0k} F_{k3} {\text{e}}^{{s_{0k} h}} } + a_{05} F_{53} {\text{e}}^{{\left( {s_{05} + \eta^{\prime}} \right)h}} + a_{06} F_{63} {\text{e}}^{{\left( {s_{06} + \eta^{\prime}} \right)h}} } \right]\\ &\quad\eta_{1j} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} j = 1,2, \end{aligned} $$

(A.5)

$$ \begin{aligned}& m_{13} = a_{21} {\text{e}}^{{s_{21} h}} + \eta_{21} a_{23} {\text{e}}^{{s_{23} h}} ,\\ & m_{14} = a_{22} {\text{e}}^{{s_{22} h}} + \eta_{22} a_{23} {\text{e}}^{{s_{23} h}}, \end{aligned} $$

(A.6)

$$ \begin{aligned} & m_{2j} = \left( {{{\mu_{0} } \mathord{\left/ {\vphantom {{\mu_{0} } {\mu_{2} }}} \right. \kern-0pt} {\mu_{2} }}} \right)\sum\limits_{k = 1}^{4} {r_{2k} F_{kj} {\text{e}}^{{s_{0k} h}} } + \left( {{{\mu_{0} } \mathord{\left/ {\vphantom {{\mu_{0} } {\mu_{2} }}} \right. \kern-0pt} {\mu_{2} }}} \right)\\ &\quad \left\{ \sum\limits_{k = 1}^{4} {r_{2k} F_{k3} {\text{e}}^{{s_{0k} h}} } + \sum\limits_{k = 5}^{6} \left[\beta_{0}^{2} a_{0k} \left( {s_{0k} + \eta^{\prime}} \right)\right. \right.\\ & \quad \left.\left. + \,{\text{i}}\left( {\beta_{0}^{2} - 2} \right) \right]F_{k3} {\text{e}}^{{\left( {s_{0k} + \eta^{\prime}} \right)h}} \right\}\eta_{1j} \\ & \quad - \beta_{0}^{2} \left( {{{\mu_{0} } \mathord{\left/ {\vphantom {{\mu_{0} } {\mu_{2} }}} \right. \kern-0pt} {\mu_{2} }}} \right)\left[ d_{05} F_{53} {\text{e}}^{{\left( {s_{05} + \eta^{\prime}} \right)}}\right.\\ &\quad \left. +\, d_{06} F_{63} {\text{e}}^{{\left( {s_{06} + \eta^{\prime}} \right)}} \right]\eta_{1j} ,\quad j = 1,2, \\ \end{aligned} $$

(A.7)

$$\begin{aligned} & m_{23} = \left[ {\beta_{2}^{2} a_{21} s_{21} + {\text{i}}\left( {\beta_{2}^{2} - 2} \right)} \right]{\text{e}}^{{s_{21} h}} \\ &\quad + \left[ {\beta_{2}^{2} a_{23} s_{23} + {\text{i}}\left( {\beta_{2}^{2} - 2} \right) - \beta_{2}^{2} d_{23} } \right]\eta_{21} {\text{e}}^{{s_{23} h}}, \end{aligned} $$

(A.8)

$$ \begin{aligned} &m_{24} = \left[ {\beta_{2}^{2} s_{22} a_{22} + {\text{i}}\left( {\beta_{2}^{2} - 2} \right)} \right]{\text{e}}^{{s_{22} h}} \\ &\quad + \left[ {\beta_{2}^{2} s_{23} a_{23} + {\text{i}}\left( {\beta_{2}^{2} - 2} \right) - \beta_{2}^{2} d_{23} } \right]\eta_{22} {\text{e}}^{{s_{23} h}}. \end{aligned} $$

(A.9)