Variational method for non-conservative instability of a cantilever SWCNT in the presence of variable mass or crack

Abstract

In the present paper, the non-conservative instability of a cantilever single-walled carbon nanotube (SWCNT) through nonlocal theory is investigated. The nanotube is modeled as clamped-free beam carrying a concentrated mass, located at a generic position, or in the presence of crack, and subjected to a compressive axial load, at the free end. Nonlocal Euler–Bernoulli beam theory is used in the formulation and the governing equations of motion and the corresponding boundary conditions are derived using an extended Hamilton’s variational principle. The governing equations are solved analytically. In order to show the sensitivity of the SWCNT to the values of an added mass, or crack and the influence of the nonlocal parameter and nondimensional crack severity coefficient on the fundamental frequencies values, some numerical examples have been performed and discussed. Also, the validity and the accuracy of the proposed analysis have been confirmed by comparing the results with those obtained from the literature.

Appendix A—Integration by parts and boundary conditions

A series of integration by part can be conducted on the terms of Eq. (7), leading to:

$$\begin{aligned}&\int _0^{\gamma \text {L}}\left[ \int _{t_1}^{t_2}\rho \text {A}\frac{\partial v_1(z,t)}{\partial t} \delta \frac{\partial v_1(z,t)}{\partial t}\text {dt}\right] \text {dz}\nonumber \\&\quad =\int _0^{\gamma \text {L}}\left[ \rho \text {A} \frac{\partial v_1(z,t)}{\partial t} \delta \text {v}_1(z,t)\right] _{t_1}^{t_2}\text {dz}-\int _0^{\gamma \text {L}}\left[ \int _{t_1}^{t_2}\rho \text {A}\frac{\partial ^2v_1(z,t)}{\partial t^2}\delta \text {v}_1(z,t) \text {dt}\right] \text {dz}\nonumber \\&\int _{\gamma \text {L}}^\text {L}\left[ \int _{t_1}^{t_2}\rho \text {A}\frac{\partial v_2(z,t)}{\partial t} \delta \frac{\partial v_2(z,t)}{\partial t}\text {dt}\right] \text {dz}\nonumber \\&\quad =\int _{\gamma \text {L}}^{\gamma \text {L}}\left[ \rho \text {A}\frac{\partial v_2(z,t)}{\partial t} \delta \text {v}_2(z,t)\right] _{t_1}^{t_2}\text {dz}-\int _{\gamma \text {L}}^\text {L}\left[ \int _{t_1}^{t_2}\rho \text {A}\frac{\partial ^2v_2(z,t)}{\partial t^2}\delta \text {v}_2(z,t) \text {dt}\right] \text {dz}; \end{aligned}$$

(A1)

and

$$\begin{aligned} \int _{t_1}^{t_2}\text {M}\frac{\partial v_1(\gamma \text {L},t)}{\partial t} \delta \frac{\partial v_1(\gamma \text {L},t)}{\partial t}\text {dt}=\left[ \text {M} \frac{\partial v_1(\gamma \text {L},t)}{\partial t} \delta \text {v}_1(\gamma \text {L},t)\right] _{t_1}^{t_2}-\int _{t_1}^{t_2}\text {M}\frac{\partial ^2v_2(\gamma \text {L},t)}{\partial t^2}\delta \text {v}_2(\gamma \text {L},t) \text {dt}; \end{aligned}$$

(A2)

and

$$\begin{aligned}&\int _{t_1}^{t_2}\left[ \int _0^{\gamma \text {L}}\text {p}\frac{\partial v_1(z,t)}{\partial z} \delta \frac{\partial v_1(z,t)}{\partial z}\text {dz}\right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \text {p} \frac{\partial v_1(z,t)}{\partial z} \delta \text {v}_1(z,t)\right] _0^{\gamma \text {L}}\text {dt}-\left[ \int _0^{\gamma \text {L}}\text {p}\frac{\partial ^2v_1(z,t)}{\partial z^2}\delta \text {v}_1(z,t) \text {dz}\right] \text {dt};\nonumber \\&\int _{t_1}^{t_2}\left[ \int _{\gamma \text {L}}^\text {L}\text {p}\frac{\partial v_2(z,t)}{\partial z} \delta \frac{\partial v_2(z,t)}{\partial z}\text {dz}\right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \text {p} \frac{\partial v_2(z,t)}{\partial z} \delta \text {v}_2(z,t)\right] _{\gamma \text {L}}^\text {L}\text {dt}-\left[ \int _{\gamma \text {L}}^\text {L}\text {p}\frac{\partial ^2v_2(z,t)}{\partial z^2}\delta \text {v}_2(z,t) \text {dz}\right] \text {dt}; \end{aligned}$$

(A3)

and

$$\begin{aligned}&\int _{t_1}^{t_2}\left[ \int _0^{\gamma \text {L}}\left( -\text {EI}+\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^2v_1(z,t)}{\partial z^2}\delta \frac{\partial ^2v_1(z,t)}{\partial z^2}\text {dz} \right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \left( -\text {EI}+\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^2v_1(z,t)}{\partial z^2}\delta \frac{\partial v_1(z,t)}{\partial z}\right] _0^{\gamma \text {L}}\text {dt}\nonumber \\&\qquad -\,\int _{t_1}^{t_2}\left[ \int _0^{\gamma \text {L}}\left( -\text {EI} +\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^3v_1(z,t)}{\partial z^3} \delta \frac{\partial v_1(z,t)}{\partial z}\text {dz} \right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \left( -\text {EI}+\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^2v_1(z,t)}{\partial z^2}\delta \frac{\partial v_1(z,t)}{\partial z}\right] _0^{\gamma \text {L}}\text {dt}\nonumber \\&\qquad -\,\int _{t_1}^{t_2}\left[ \left( -\text {EI} +\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^3v_1(z,t)}{\partial z^3} \delta \text {v}_1(z,t)\right] _0^{\gamma \text {L}}\text {dt}\nonumber \\&\qquad +\,\int _{t_1}^{t_2}\left[ \int _0^{\gamma \text {L}}\left( -\text {EI}+\left( e_0a\right) ^2\text {p} \right) \frac{\partial ^4v_1(z,t)}{\partial z^4} \delta v_1(z,t) \text {dz} \right] \text {dt};\nonumber \\&\int _{t_1}^{t_2}\left[ \int _{\gamma \text {L}}^\text {L}\left( -\text {EI}+\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^2v_2(z,t)}{\partial z^2}\delta \frac{\partial ^2v_2(z,t)}{\partial z^2}\text {dz} \right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \left( -\text {EI}+\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^2v_2(z,t)}{\partial z^2}\delta \frac{\partial v_2(z,t)}{\partial z}\right] _{\gamma \text {L}}^\text {L}\text {dt}\nonumber \\&\qquad -\,\int _{t_1}^{t_2}\left[ \int _{\gamma \text {L}}^\text {L}\left( -\text {EI} +\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^3v_2(z,t)}{\partial z^3} \delta \frac{\partial v_2(z,t)}{\partial z}\text {dz} \right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \left( -\text {EI}+\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^2v_2(z,t)}{\partial z^2}\delta \frac{\partial v_2(z,t)}{\partial z}\right] _{\gamma \text {L}}^\text {L}\text {dt}\nonumber \\&\qquad -\,\int _{t_1}^{t_2}\left[ \left( -\text {EI} +\left( e_0a\right) ^2\text {p}\right) \frac{\partial ^3v_2(z,t)}{\partial z^3} \delta \text {v}_2(z,t)\right] _{\gamma \text {L}}^\text {L}\text {dt}\nonumber \\&\qquad +\,\int _{t_1}^{t_2}\left[ \int _{\gamma \text {L}}^\text {L}\left( -\text {EI}+\left( e_0a\right) ^2\text {p} \right) \frac{\partial ^4v_2(z,t)}{\partial z^4} \delta v_2(z,t) \text {dz} \right] \text {dt}; \end{aligned}$$

(A4)

and

$$\begin{aligned}&\int _{t_1}^{t_2}\left[ \int _0^{\gamma \text {L}}\left( e_0a\right) ^2\rho \text {A}\frac{\partial ^2v_1(z,t)}{\partial t^2} \delta \frac{\partial ^2v_1(z,t)}{\partial z^2}\text {dz}\right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \left( e_0a\right) ^2\rho \text {A}\frac{\partial ^2v_1(z,t)}{\partial t^2}\delta \frac{\partial v_1(z,t)}{\partial z}\right] _0^{\gamma \text {L}}\text {dt}\nonumber \\&\qquad -\,\int _{t_1}^{t_2}\left[ \int _0^{\gamma \text {L}}\left( e_0a\right) ^2\rho \text {A} \frac{\partial ^3v_1(z,t)}{\partial t^2\partial z}\delta \frac{\partial v_1(z,t)}{\partial z} \text {dz} \right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \left( e_0a\right) ^2\rho \text {A}\frac{\partial ^2v_1(z,t)}{\partial t^2}\delta \frac{\partial v_1(z,t)}{\partial z}\right] _0^{\gamma \text {L}}\text {dt}\nonumber \\&\qquad -\,\int _{t_1}^{t_2}\left[ \left( e_0a\right) ^2\rho \text {A}\frac{\partial ^3v(z,t)}{\partial t^2\partial z}\delta \text {v}_1(z,t)\right] _0^{\gamma \text {L}}\text {dt}\nonumber \\&\qquad +\,\int _{t_1}^{t_2}\left[ \int _0^{\gamma \text {L}}\left( e_0a\right) ^2\rho \text {A} \frac{\partial ^4v_1(z,t)}{\partial t^2\partial z^2}\delta v_1(z,t) \text {dz} \right] \text {dt};\nonumber \\&\int _{t_1}^{t_2}\left[ \int _{\gamma \text {L}}^\text {L}\left( e_0a\right) ^2\rho \text {A}\frac{\partial ^2v_2(z,t)}{\partial t^2} \delta \frac{\partial ^2v_2(z,t)}{\partial z^2}\text {dz}\right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \left( e_0a\right) ^2\rho \text {A}\frac{\partial ^2v_2(z,t)}{\partial t^2}\delta \frac{\partial v_2(z,t)}{\partial z}\right] _{\gamma \text {L}}^\text {L}\text {dt}\nonumber \\&\qquad -\,\int _{t_1}^{t_2}\left[ \int _{\gamma \text {L}}^\text {L}\left( e_0a\right) ^2\rho \text {A} \frac{\partial ^3v_2(z,t)}{\partial t^2\partial z}\delta \frac{\partial v_2(z,t)}{\partial z} \text {dz} \right] \text {dt}\nonumber \\&\quad =\int _{t_1}^{t_2}\left[ \left( e_0a\right) ^2\rho \text {A}\frac{\partial ^2v_2(z,t)}{\partial t^2}\delta \frac{\partial v_2(z,t)}{\partial z}\right] _{\gamma \text {L}}^\text {L}\text {dt}\nonumber \\&\qquad -\,\int _{t_1}^{t_2}\left[ \left( e_0a\right) ^2\rho \text {A}\frac{\partial ^3v_2(z,t)}{\partial t^2\partial z}\delta \text {v}_2(z,t)\right] _{\gamma \text {L}}^\text {L}\text {dt}\nonumber \\&\qquad +\,\int _{t_1}^{t_2}\left[ \int _{\gamma \text {L}}^\text {L}\left( e_0a\right) ^2\rho \text {A} \frac{\partial ^4v_2(z,t)}{\partial t^2\partial z^2}\delta v_2(z,t) \text {dz} \right] \text {dt}; \end{aligned}$$

(A5)