Stress-driven nonlocal Timoshenko beam model for buckling analysis of carbon nanotubes constrained by surface van der Waals interactions

Abstract

In this work, the stress-driven nonlocal integral elasticity model is developed for the size-dependent buckling analysis of single-walled carbon nanotubes (SWCNT) in an array constrained by surface van der Waals (vdW) interactions. Using the Lennard–Jones (LJ) potential and Gaussian quadrature method, the vdW force of the continuum model of parallel adjacent CNTs is calculated. According to Hooke's law, the nonlinear vdW interaction is equivalent to a spring constant. Based on the stress-driven nonlocal model and Timoshenko beam theory, the governing equations and the natural boundary conditions of the CNT are derived using Hamilton's principle. Two extra constitutive boundary conditions are generated by the transformation of the Fredholm-type integral constitutive equation into an equivalence of a differential form. Using the Laplace transform technique and the eigenvalue method, the axial critical buckling loads of an SWCNT are analytically solved and compared with the results without surface vdW interaction under pre-pressure processors. In numerical simulations, the influences of the nonlocal elastic parameter, length-to-radii ratio, and type of the SWCNT on the critical buckling load are also examined.

Appendix A: The inverse Laplace transform expressions of \(f_{i} (x)\) and \(g_{i} (x)\)

$$\begin{gathered} f_{0} \left( x \right) = \frac{{bc\left( {b^{2} - c^{2} } \right)\sinh \left[ {ax} \right] + ac\left( { - a^{2} + c^{2} } \right)\sinh \left[ {bx} \right] + ab\left( {a^{2} - b^{2} } \right)\sinh \left[ {cx} \right]}}{{abc\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)}} \hfill \\ f_{1} \left( x \right) = \frac{{\left( {b^{2} - c^{2} } \right)\cosh \left[ {ax} \right] + \left( { - a^{2} + c^{2} } \right)\cosh \left[ {bx} \right] + \left( {a^{2} - b^{2} } \right)\cosh \left[ {cx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)}} \hfill \\ f_{2} \left( x \right) = \frac{{a\left( {b^{2} - c^{2} } \right)\sinh \left[ {ax} \right] + b\left( { - a^{2} + c^{2} } \right)\sinh \left[ {bx} \right] + \left( {a^{2} - b^{2} } \right)c\sinh \left[ {cx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)}} \hfill \\ f_{3} \left( x \right) = \frac{{a^{2} \left( {b^{2} - c^{2} } \right)\cosh \left[ {ax} \right] + b^{2} \left( { - a^{2} + c^{2} } \right)\cosh \left[ {bx} \right] + \left( {a^{2} - b^{2} } \right)c^{2} \cosh \left[ {cx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)}} \hfill \\ f_{4} \left( x \right) = \frac{{a^{3} \left( {b^{2} - c^{2} } \right)\sinh \left[ {ax} \right] + b^{3} \left( { - a^{2} + c^{2} } \right)\sinh \left[ {bx} \right] + \left( {a^{2} - b^{2} } \right)c^{3} \sinh \left[ {cx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)}} \hfill \\ f_{5} \left( x \right) = \frac{{a^{4} \left( {b^{2} - c^{2} } \right)\cosh \left[ {ax} \right] + b^{4} \left( { - a^{2} + c^{2} } \right)\cosh \left[ {bx} \right] + \left( {a^{2} - b^{2} } \right)c^{4} \cosh \left[ {cx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)}} \hfill \\ \hfill \\ \end{gathered}$$

(A.1)

$$\begin{aligned} g_{0} \left( x \right) & = \frac{{\sinh \left[ {ax} \right]}}{{a\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {a^{2} - d^{2} } \right)}} - \frac{{\sinh \left[ {bx} \right]}}{{b\left( {a^{2} - b^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {b^{2} - d^{2} } \right)}} \\ & \quad + \frac{{\sinh \left[ {cx} \right]}}{{c\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {c^{2} - d^{2} } \right)}} - \frac{{\sinh \left[ {dx} \right]}}{{d\left( {a^{2} - d^{2} } \right)\left( {b^{2} - d^{2} } \right)\left( {c^{2} - d^{2} } \right)}} \\ g_{1} \left( x \right) & = \frac{{\cosh \left[ {ax} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {a^{2} - d^{2} } \right)}} - \frac{{\cosh \left[ {bx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {b^{2} - d^{2} } \right)}} \\ & \quad + \frac{{\cosh \left[ {cx} \right]}}{{\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {c^{2} - d^{2} } \right)}} - \frac{{\cosh \left[ {dx} \right]}}{{\left( {a^{2} - d^{2} } \right)\left( {b^{2} - d^{2} } \right)\left( {c^{2} - d^{2} } \right)}} \\ g_{2} \left( x \right) & = \frac{{a\sinh \left[ {ax} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {a^{2} - d^{2} } \right)}} - \frac{{b\sinh \left[ {bx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {b^{2} - d^{2} } \right)}} \\ & \quad + \frac{{c\sinh \left[ {cx} \right]}}{{\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {c^{2} - d^{2} } \right)}} - \frac{{d\sinh \left[ {dx} \right]}}{{\left( {a^{2} - d^{2} } \right)\left( {b^{2} - d^{2} } \right)\left( {c^{2} - d^{2} } \right)}} \\ g_{3} \left( x \right) & = \frac{{a^{2} \cosh \left[ {ax} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {a^{2} - d^{2} } \right)}} - \frac{{b^{2} \cosh \left[ {bx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {b^{2} - d^{2} } \right)}} \\ & \quad + \frac{{c^{2} \cosh \left[ {cx} \right]}}{{\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {c^{2} - d^{2} } \right)}} - \frac{{d^{2} \cosh \left[ {dx} \right]}}{{\left( {a^{2} - d^{2} } \right)\left( {b^{2} - d^{2} } \right)\left( {c^{2} - d^{2} } \right)}} \\ g_{4} \left( x \right) & = \frac{{a^{3} \sinh \left[ {ax} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {a^{2} - d^{2} } \right)}} - \frac{{b^{3} \sinh \left[ {bx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {b^{2} - d^{2} } \right)}} \\ & \quad + \frac{{c^{3} \sinh \left[ {cx} \right]}}{{\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {c^{2} - d^{2} } \right)}} - \frac{{d^{3} \sinh \left[ {dx} \right]}}{{\left( {a^{2} - d^{2} } \right)\left( {b^{2} - d^{2} } \right)\left( {c^{2} - d^{2} } \right)}} \\ g_{5} \left( x \right) & = \frac{{a^{4} \cosh \left[ {ax} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {a^{2} - d^{2} } \right)}} - \frac{{b^{4} \cosh \left[ {bx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {b^{2} - d^{2} } \right)}} \\ & \quad + {\kern 1pt} \frac{{c^{4} \cosh \left[ {cx} \right]}}{{\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {c^{2} - d^{2} } \right)}} - \frac{{d^{4} \cosh \left[ {dx} \right]}}{{\left( {a^{2} - d^{2} } \right)\left( {b^{2} - d^{2} } \right)\left( {c^{2} - d^{2} } \right)}} \\ g_{6} \left( x \right) & = \frac{{a^{5} \sinh \left[ {ax} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {a^{2} - d^{2} } \right)}} - \frac{{b^{5} \sinh \left[ {bx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {b^{2} - d^{2} } \right)}} \\ & \quad + \frac{{c^{5} \sinh \left[ {cx} \right]}}{{\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {c^{2} - d^{2} } \right)}} - \frac{{d^{5} \sinh \left[ {dx} \right]}}{{\left( {a^{2} - d^{2} } \right)\left( {b^{2} - d^{2} } \right)\left( {c^{2} - d^{2} } \right)}} \\ g_{7} \left( x \right) & = \frac{{a^{6} \cosh \left[ {ax} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {a^{2} - c^{2} } \right)\left( {a^{2} - d^{2} } \right)}} - \frac{{b^{6} \cosh \left[ {bx} \right]}}{{\left( {a^{2} - b^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {b^{2} - d^{2} } \right)}} \\ & \quad + \frac{{c^{6} \cosh \left[ {cx} \right]}}{{\left( {a^{2} - c^{2} } \right)\left( {b^{2} - c^{2} } \right)\left( {c^{2} - d^{2} } \right)}} - \frac{{d^{6} \cosh \left[ {dx} \right]}}{{\left( {a^{2} - d^{2} } \right)\left( {b^{2} - d^{2} } \right)\left( {c^{2} - d^{2} } \right)}} \\ \end{aligned}$$

(A.2)