Interfacial stress transfer of fiber pullout for carbon nanotubes with a composite coating

Abstract

An analytical approach has been established to evaluate the interfacial stress transfer characteristics of single- and multi-walled carbon nanotubes (CNTs) with composite coatings by means of fiber pullout model. According to the present model, the effects of several parameters such as coating thickness, layer numbers and dimension of CNTs on interfacial stress transfers were investigated and analyzed. The results suggested that the maximum interfacial shear stress occurred at the pullout end of CNTs and decreased with increasing coating thickness as well as CNT wall thickness (layer numbers). Moreover, the distribution of the interfacial shear and coating axial stress along the CNT length was found to be largely affected by the friction coefficient in the interface between the CNT and the coating layer.

Appendix A

For N = 2, the solution of Eqs (7)–(10) gives

$$ \Delta r_1 = \frac{{q\left( {{C \mathord{\left/ {\vphantom {C {r_2^2 }}} \right. \kern-\nulldelimiterspace} {r_2^2 }} + c_0 } \right) + \left[ {\frac{\nu } {{r_1 }}\left( {{C \mathord{\left/ {\vphantom {C {r_2^2 }}} \right. \kern-\nulldelimiterspace} {r_2^2 }} + c_0 } \right) + \frac{{c_0 \nu }} {{r_2 }}} \right]\;T_z }} {{\left( {{C \mathord{\left/ {\vphantom {C {r_1^2 }}} \right. \kern-\nulldelimiterspace} {r_1^2 }} + c_0 } \right)\,\left( {{C \mathord{\left/ {\vphantom {C {r_2^2 }}} \right. \kern-\nulldelimiterspace} {r_2^2 }} + c_0 } \right) - c_0^2 }} $$

(A1)

Substituting Eq. (A1) into Eq. (11), we have

$$ f_1 = - \frac{{\left[ {\left( {{C \mathord{\left/ {\vphantom {C {r_2^2 }}} \right. \kern-\nulldelimiterspace} {r_2^2 }} + c_0 } \right)\frac{{\nu _f }} {{\,\phi \,r_1^2 }} + \frac{{c_0 \nu _f }} {{\phi \,r_1 r_2 }}} \right]}} {{\left( {{C \mathord{\left/ {\vphantom {C {r_1^2 }}} \right. \kern-\nulldelimiterspace} {r_1^2 }} + c_0 } \right)\,\left( {{C \mathord{\left/ {\vphantom {C {r_2^2 }}} \right. \kern-\nulldelimiterspace} {r_2^2 }} + c_0 } \right) - c_0^2 }} $$

(A2)

$$ f_2 = - \frac{{\nu _m }} {{\phi \,E_m }} $$

(A3)

and

$$ \phi = \left( {\frac{{{{\left( {1 + n^2 } \right)} \mathord{\left/ {\vphantom {{\left( {1 + n^2 } \right)} {\left( {1 - n^2 } \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - n^2 } \right)}} + \nu _m }} {{E_m }} + \frac{{{{\left( {{C \mathord{\left/ {\vphantom {C {r_2^2 }}} \right. \kern-\nulldelimiterspace} {r_2^2 }} + c_0 } \right)} \mathord{\left/ {\vphantom {{\left( {{C \mathord{\left/ {\vphantom {C {r_2^2 }}} \right. \kern-\nulldelimiterspace} {r_2^2 }} + c_0 } \right)} {r_1 }}} \right. \kern-\nulldelimiterspace} {r_1 }}}} {{\left( {{C \mathord{\left/ {\vphantom {C {r_1^2 }}} \right. \kern-\nulldelimiterspace} {r_1^2 }} + c_0 } \right)\,\left( {{C \mathord{\left/ {\vphantom {C {r_2^2 }}} \right. \kern-\nulldelimiterspace} {r_2^2 }} + c_0 } \right) - c_0^2 }}} \right) $$

(A3)