Impact Behaviors of Cantilevered Nano-beams Based on the Nonlocal Theory

Abstract

Purpose

The present study is motivated by the shock response of a nano-cantilever component in nano-scaled electromechanical system (NEMS). For this purpose, the mechanical properties of a cantilevered nano-beams subjected to impact load are presented for the first time. Our work is concerned with the special moment that the impact deformation gets most and the impact velocity reduces to zero.

Methods

The nonlocal theory proposed by Eringen is applied to reveal the nonlocal effect involved in impact behaviors at nano-scale. To this end, the classical impact stress is determined first using the law of conservation of energy. Subsequently, the nonlocal impact stress is obtained from the nonlocal constitutive equation and clamped-free boundary constraints of cantilevered nano-beams, where an imaginary equivalent symmetrical nano-structure is constructed to solve the unknown coefficients in general solution of stress of nonlocal field. The bending moment in nonlocal theory is gained from the stress of nonlocal field and the deflection in nonlocal field under impact load is calculated. Finally, the nonlocal dynamical load coefficient is derived via the maximal nonlocal impact deflection divided by the corresponding traditional static deflection at free edge. In numerical examples, the maximal stress of nonlocal field, the maximal nonlocal impact deflection, and the nonlocal dynamical load coefficient are calculated, respectively, to show the significant nonlocal small-scale effect in cantilevered nano-beams subjected to impact load.

Results and Conclusions

The conclusion is that the nonlocality plays a remarkable role in nano-scaled impact behaviors and it cannot be neglected. The existence of scale coefficient decreases the nonlocal impact stress, deflection, and dynamical load coefficient. On the contrary, the impact velocity results in a higher nonlocal impact stress, deflection, and dynamical load coefficient. With increasing the scale coefficient, the nonlocal effect increases and the strengthening physical manifestation of nano-structural equivalent stiffness becomes even more pronounced. The results reported herein are expected to provide a useful reference for understanding the scale effect in dynamic impact that often occurs in the working process of NEMS.

Appendix

The traditional local stress can be recovered from the expression of stress of nonlocal field in Eq. (16) without the scale coefficient. Rewriting the stress of nonlocal field expression yields the following:

$$\begin{aligned} &\sigma_{\text{non}} = \frac{{6e_{0} av}}{{bh^{2} \left( {1 + e^{{2l/e_{0} a}} } \right)}}\sqrt {\frac{3EIm}{{l^{3} }}} e^{{x/e_{0} a}} \hfill \\ &\quad - \frac{{6e_{0} av}}{{bh^{2} \left( {1 + e^{{ - 2l/e_{0} a}} } \right)}}\sqrt {\frac{3EIm}{{l^{3} }}} e^{{ - x/e_{0} a}} + \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \left( {l - x} \right). \hfill \\ \end{aligned}$$

(28)

First, taking the first term into consideration yields the following:

$$\begin{aligned} &\mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{6e_{0} av}}{{bh^{2} \left( {1 + e^{{2l/e_{0} a}} } \right)}}\sqrt {\frac{3EIm}{{l^{3} }}} e^{{x/e_{0} a}} \\&\quad = \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e_{0} a}}{{\left( {1 + e^{{2l/e_{0} a}} } \right)}}e^{{x/e_{0} a}} \\ &\quad = \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e_{0} a}}{{\left( {e^{{ - x/e_{0} a}} + e^{{(2l - x)/e_{0} a}} } \right)}} \\ &\quad = \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e_{0} a}}{{\left( {e^{{ - x/e_{0} a}} + e^{{(2l - x)/e_{0} a}} } \right)}} \\ &\quad = \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{1}{{{{\left( {e^{{ - x/e_{0} a}} + e^{{(2l - x)/e_{0} a}} } \right)} \mathord{\left/ {\vphantom {{\left( {e^{{ - x/e_{0} a}} + e^{{(2l - x)/e_{0} a}} } \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} \\&\quad = \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{1}{{\frac{{e^{{ - x/e_{0} a}} }}{{e_{0} a}} + \frac{{e^{{(2l - x)/e_{0} a}} }}{{e_{0} a}}}}, \\ \end{aligned}$$

(29)

where

$$\begin{gathered} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e^{{ - x/e_{0} a}} }}{{e_{0} a}}\xrightarrow{{e_{0} a{\text{ = 1/t}}}}\mathop {\lim }\limits_{{t \to \infty }} \frac{t}{{e^{{xt}} }} = \mathop {\lim }\limits_{{t \to \infty }} \frac{1}{{xe^{{xt}} }} = 0 \hfill \\ \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e^{{(2l - x)/e_{0} a}} }}{{e_{0} a}}\begin{array}{*{20}c} {\underline{\underline{{{\text{let}}\;e_{0} a{\text{ = 1/t}}}}} } \\ {} \\ \end{array} \mathop {\lim }\limits_{{t \to \infty }} te^{{\left( {2l - x} \right)t}} = \infty \hfill \\ \end{gathered}$$

(30)

Consequently:

$$\mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e_{0} a}}{{\left( {e^{{{{ - x} \mathord{\left/ {\vphantom {{ - x} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} + e^{{{{\left( {2l - x} \right)} \mathord{\left/ {\vphantom {{\left( {2l - x} \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} } \right)}} = \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{1}{{\frac{{e^{{{{ - x} \mathord{\left/ {\vphantom {{ - x} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} }}{{e_{0} a}} + \frac{{e^{{{{\left( {2l - x} \right)} \mathord{\left/ {\vphantom {{\left( {2l - x} \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} }}{{e_{0} a}}}} = \frac{1}{0 + \infty } = 0.$$

(31)

Second, considering the second term of Eq. (28), one gets the following:

$$\begin{aligned} &\mathop {\lim }\limits_{{e_{0} a \to 0}} - \frac{{6e_{0} av}}{{bh^{2} \left( {1 + e^{{ - {{2l} \mathord{\left/ {\vphantom {{2l} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} } \right)}}\sqrt {\frac{3EIm}{{l^{3} }}} e^{{{{ - x} \mathord{\left/ {\vphantom {{ - x} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} \\&\quad= - \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e_{0} ae^{{{{ - x} \mathord{\left/ {\vphantom {{ - x} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} }}{{\left( {1 + e^{{ - {{2l} \mathord{\left/ {\vphantom {{2l} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} } \right)}} \hfill \\&\quad = - \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e_{0} a}}{{\left( {e^{{{x \mathord{\left/ {\vphantom {x {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} + e^{{{{\left( {x - 2l} \right)} \mathord{\left/ {\vphantom {{\left( {x - 2l} \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} } \right)}} \\&\quad= - \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e_{0} a}}{{\left( {e^{{{x \mathord{\left/ {\vphantom {x {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} + e^{{{{\left( {x - 2l} \right)} \mathord{\left/ {\vphantom {{\left( {x - 2l} \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} } \right)}} \hfill \\&\quad = - \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{1}{{{{\left( {e^{{{x \mathord{\left/ {\vphantom {x {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} + e^{{{{\left( {x - 2l} \right)} \mathord{\left/ {\vphantom {{\left( {x - 2l} \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} } \right)} \mathord{\left/ {\vphantom {{\left( {e^{{{x \mathord{\left/ {\vphantom {x {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} + e^{{{{\left( {x - 2l} \right)} \mathord{\left/ {\vphantom {{\left( {x - 2l} \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} } \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} \hfill \\&\quad = - \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{1}{{\frac{{e^{{{x \mathord{\left/ {\vphantom {x {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} }}{{e_{0} a}} + \frac{{e^{{{{\left( {x - 2l} \right)} \mathord{\left/ {\vphantom {{\left( {x - 2l} \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} }}{{e_{0} a}}}}, \hfill \\ \end{aligned}$$

(32)

where

$$\begin{aligned} \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e^{{{x \mathord{\left/ {\vphantom {x {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} }}{{e_{0} a}}\begin{array}{*{20}c} {\underline{\underline{{{\text{let}}\;e_{0} a{ = }{1 \mathord{\left/ {\vphantom {1 t}} \right. \kern-0pt} t}}}} } \\ {} \\ \end{array} \mathop {\lim }\limits_{t \to \infty } te^{xt} = \infty \hfill \\ \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e^{{{{\left( {x - 2l} \right)} \mathord{\left/ {\vphantom {{\left( {x - 2l} \right)} {e_{0} a}}} \right. \kern-0pt} {e_{0} a}}}} }}{{e_{0} a}}\begin{array}{*{20}c} {\underline{\underline{{{\text{let}}\;e_{0} a{ = }{1 \mathord{\left/ {\vphantom {1 t}} \right. \kern-0pt} t}}}} } \\ {} \\ \end{array} \mathop {\lim }\limits_{t \to \infty } te^{{\left( {x - 2l} \right)t}} \hfill \\ = \mathop {\lim }\limits_{t \to \infty } \frac{t}{{e^{{\left( {2l - x} \right)t}} }}\begin{array}{*{20}c} {\underline{\underline{\text{L'Hospital's rule}}} } \\ {} \\ \end{array} \mathop {\lim }\limits_{t \to \infty } \frac{1}{{\left( {2l - x} \right)e^{{\left( {2l - x} \right)t}} }} = 0. \hfill \\ \end{aligned}$$

(33)

Therefore

$$\mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{{e_{0} a}}{{\left( {e^{{x/e_{0} a}} + e^{{(x - 2l)/e_{0} a}} } \right)}} = \mathop {\lim }\limits_{{e_{0} a \to 0}} \frac{1}{{\frac{{e^{{x/e_{0} a}} }}{{e_{0} a}} + \frac{{e^{{(x - 2l)/e_{0} a}} }}{{e_{0} a}}}} = \frac{1}{\infty + 0} = 0.$$

(34)

In summary, the expression of stress of nonlocal field can degenerate into the traditional counterpart derived in this paper when $e_{0} a \to 0$, as follows:

$$\sigma_{\text{non}} = \frac{6v}{{bh^{2} }}\sqrt {\frac{3EIm}{{l^{3} }}} \left( {l - x} \right) \, \left( {0 \le x \le l} \right).$$

(35)

Hence, the stress of nonlocal field expression in Eq. (16) is reasonable and it can be reducible to the classical stress.