Using couple stress theory for modeling the size-dependent instability of double-sided beam-type nanoactuators in the presence of Casimir force

Abstract

While the pull-in instability of beam-type electromechanical nanoactuators with single actuating electrode has been widely addressed in literature, limited research works have been devoted to modeling the pull-in phenomenon in actuators with double-sided actuating electrodes. Herein, couple stress theory (CST) has been used to study the size-dependent instability of two double-sided actuators, i.e., nano-bridges and nano-cantilevers. The influence of Casimir force has been considered in the model. The analytical differential transformation method (DTM) has been applied to solve the governing equations as well as numerical method. Furthermore, a lumped parameter model has been developed to simply explain the physical performance of the systems without mathematical complexity. The critical deflection and pull-in voltage of the nanostructures as basic design parameters have been calculated. Effect of the Casimir attraction and the size dependency and the importance of coupling between them on the pull-in performance have been discussed for both nano-structures. The present work can be helpful to precise design and analysis of nano-cantilevers and nano-bridges.

Appendices

Appendix A: DTM

The basic idea and fundamental theorems of DTM are given in [47–49]. The differential transform of the kth derivative of arbitrary function f(x) is defined as:

$$ F\left( k \right) = \frac{1}{k!}\left[ {\frac{{d^{k} f(x)}}{{dx^{k} }}} \right]_{{x = x_{0} }} $$

(A.1)

where F(k) is the transformed function.

$$ f(x) = \sum\limits_{k = 0}^{\infty } {F(k)\left( {x - x_{0} } \right)^{k} } $$

(A.2)

The differential transformation relations for functional operations and boundary conditions are found in Table 3.

Table 3 DTM theorems used for equations and boundary conditions

By multiplying both sides of the governing Eq. (18) by $\left( { 1- \bar{w} (x )} \right)^{4} \left( {\Omega + \bar{w} (x )} \right)^{4}$, then substituting the relations of Table 3 in Eq. (18) and after some elaborations, one can found (A.3) for the nano-structures. For example, the transformation—of

$$ f\left( x \right) = \left( {1 - \bar{w}(x)} \right)^{2} $$

is calculated as $F(k) = \sum\limits_{\lambda = 0}^{k} {\left[ {\delta (\lambda ) - \bar{W}\left( \lambda \right)} \right]} \left[ {\delta (k - \lambda ) - \bar{W}\left( {k - \lambda } \right)} \right].$

$$ \begin{aligned} &\sum\limits_{R = 0}^{k} {\left( {\left[ {\sum\limits_{\lambda = 0}^{R} {\left\{ {\sum\limits_{m = 0}^{\lambda } {\sum\limits_{p = 0}^{m} {\sum\limits_{q = 0}^{p} {\left[ {\delta (p - q) - \bar{W}\left( {p - q} \right)} \right]\left[ {\delta (q) - \bar{W}(q)} \right]\left[ {\delta \left( {m - p} \right) - \bar{W}\left( {m - p} \right)} \right]\left[ {\delta \left( {\lambda - m} \right) - \bar{W}\left( {\lambda - m} \right)} \right]} } } } \right\}} } \right.} \right.} \hfill \\ & \quad \times\left. {\left\{ {\sum\limits_{t = 0}^{R - \lambda } {\sum\limits_{i = 0}^{t} {\sum\limits_{s = 0}^{i} {\left[ {\Omega \delta (i - s) + \bar{W}\left( {i - s} \right)} \right]\left[ {\Omega \delta (s) + \bar{W}(s)} \right]\left[ {\Omega \delta \left( {t - i} \right) + \bar{W}\left( {t - i} \right)} \right]\left[ {\Omega \delta \left( {R - \lambda - t} \right) + \bar{W}\left( {R - \lambda - t} \right)} \right]} } } } \right\}} \right] \hfill \\ &\quad \times\left[ {\left( {1 + \xi } \right)\left( {k - R + 4} \right)\left( {k - R + 3} \right)\left( {k - R + 2} \right)\left( {k - R + 1} \right)\bar{W}\left( {k - R + 4} \right)} \right. \hfill \\ & \quad - \eta \left\{ {\sum\limits_{r = 1}^{n} \frac{1}{r} \sum\limits_{L = 0}^{r - 1} {\left( {L + 1} \right)\bar{W}\left( {L + 1} \right)\left( {r - L} \right)\bar{W}\left( {r - L} \right)} } \right\}\left. {\left. {\left( {k - R + 2} \right)\left( {k - R + 1} \right)\bar{W}\left( {k - R + 2} \right)} \right]} \right) = \hfill \\ & \quad + \alpha \sum\limits_{t = 0}^{k} {\sum\limits_{i = 0}^{t} {\sum\limits_{s = 0}^{i} {\left[ {\Omega \delta (i - s) + \bar{W}\left( {i - s} \right)} \right]\left[ {\Omega \delta (s) + \bar{W}(s)} \right]} \left[ {\Omega \delta \left( {t - i} \right) + \bar{W}\left( {t - i} \right)} \right]\left[ {\Omega \delta \left( {k - t} \right) + \bar{W}\left( {k - t} \right)} \right]} } \hfill \\ & \quad - \alpha \sum\limits_{m = 0}^{k} {\sum\limits_{p = 0}^{m} {\sum\limits_{q = 0}^{p} {\left[ {\delta \left( {p - q} \right) - \bar{W}\left( {p - q} \right)} \right]\left[ {\delta (q) - \bar{W}(q)} \right]} \left[ {\delta \left( {m - p} \right) - \bar{W}\left( {m - p} \right)} \right]\left[ {\delta \left( {k - m} \right) - \bar{W}\left( {k - m} \right)} \right]} } \hfill \\ & \quad + \beta \sum\limits_{R = 0}^{k} {\left( {\left[ {\sum\limits_{\lambda = 0}^{R} {\left\{ {\sum\limits_{q = 0}^{\lambda } {\left[ {\delta (\lambda - q) - \bar{W}\left( {\lambda - q} \right)} \right]\left[ {\delta (q) - \bar{W}(q)} \right]} } \right\}} } \right.} \right.} \left. {\left\{ {\sum\limits_{s = 0}^{R - \lambda } {\left( {\Omega \delta \left( {R - \lambda - s} \right) + \bar{W}\left( {R - \lambda - s} \right)} \right)\left( {\Omega \delta (s) + \bar{W}(s)} \right)} } \right\}} \right] \hfill \\ &\quad \times\left. {\left[ {\sum\limits_{s = 0}^{k - R} {\left( {\Omega \delta \left( {k - R - s} \right) + \bar{W}\left( {k - R - s} \right)} \right)\left( {\Omega \delta (s) + \bar{W}(s)} \right)} - \sum\limits_{q = 0}^{k - R} {\left( {\delta \left( {k - R - q} \right) - \bar{W}\left( {k - R - q} \right)} \right)\left( {\delta (q) - \bar{W}(q)} \right)} } \right]} \right) \hfill \\ & \quad + \gamma \beta \sum\limits_{R = 0}^{k} {\left( {\left[ {\sum\limits_{\lambda = 0}^{R} {\left\{ {\sum\limits_{q = 0}^{\lambda } {\sum\limits_{h = 0}^{q} {\left[ {\delta \left( {q - h} \right) - \bar{W}\left( {q - h} \right)} \right]\left[ {\delta (h) - \bar{W}(h)} \right]\left[ {\delta \left( {\lambda - q} \right) - \bar{W}\left( {\lambda - q} \right)} \right]} } } \right\}} } \right.} \right.} \hfill \\ &\quad \times\left. {\left\{ {\sum\limits_{s = 0}^{R - \lambda } {\sum\limits_{j = 0}^{s} {\left[ {\Omega \delta \left( {s - j} \right) + \bar{W}\left( {s - j} \right)} \right]\left[ {\Omega \delta (j) + \bar{W}(j)} \right]\left[ {\Omega \delta \left( {R - \lambda - s} \right) + \bar{W}\left( {R - \lambda - s} \right)} \right]} } } \right\}} \right] \hfill \\ &\quad \times\left. {\left[ {\left( {\Omega \delta \left( {k - R} \right) + \bar{W}\left( {k - R} \right)} \right) - \delta (k - R) + \bar{W}\left( {k - R} \right)} \right]} \right) \hfill \\ \end{aligned} $$

(A.3)

The differential transformations of two boundary conditions is:

$$ \bar{W}\left( 0 \right) = \bar{W}\left( 1 \right) = 0\,\,\,\,\, $$

(A.4)

and the differential transformations of the remaining boundary conditions is:

$$ \sum\limits_{k = 0}^{n} {k\bar{W}\left( k \right)} = \sum\limits_{k = 0}^{n} {\bar{W}\left( k \right)} = 0\quad {\text{For nano-bridge}} $$

(A.5a)

$$ \sum\limits_{k = 0}^{n} {k\left( {k - 1} \right)\bar{W}\left( k \right)} = \sum\limits_{k = 0}^{n} {k\left( {k - 1} \right)\left( {k - 2} \right)\bar{W}\left( k \right)} = 0\quad {\text{For nano-cantilever}} $$

(A.5b)

By assuming $\bar{W}\left( 2 \right) = a$ and $\bar{W}\left( 3 \right) = b$, we obtain higher terms from Eq. (A.3) as:

$$ \bar{W}(4) = \frac{{\Omega^{4} \left( {8a^{3} \eta + 18a^{2} b\eta + 3\beta \gamma + 3\alpha + 3\beta } \right) - 3\beta \gamma \Omega^{3} - 3\beta \Omega^{2} - 3\alpha }}{{72(1 + \xi )\Omega^{4} }} $$

(A.6a)

$$ \bar{W}(5) = \frac{{\Omega^{4} \left[ {\left( {96a + 216b} \right)a^{3} b\eta^{2} + 36ab\eta \left( {\beta \gamma + \beta + \alpha } \right) + \eta b\left( {\xi + 1} \right)\left( {486b^{2} + 360a^{2} + 810ab} \right) - 36\eta ab\left[ {\Omega^{3} \gamma \beta + \Omega^{2} \beta + \alpha } \right]} \right]}}{{5400\Omega^{4} \left( {1 + \xi } \right)^{2} }} $$

(A.6b)

The constants a and b can be determined using Eq. (A.5a) for nano-bridge and (A.5b) for nano-cantilever.

It should be noted that the results of DTM series solution converge to that of numerical method by increasing the number of series terms.

Appendix B: Lumped parameter model

Using Rayleigh–Ritz principle (minimum potential energy approach), a LPM can be developed by minimizing the total energy of system. The total energy is a combination of bending elastic energy (U _b ), stretching elastic energy (U _s) and work of external force (W _e). These are determined as [45, 46, 50]:

$$ U_{b} = \frac{1}{2}\int\limits_{0}^{L} {\left( {EI + 4\mu Al^{2} } \right)\left( {\frac{{d^{2} w}}{{dX^{2} }}} \right)}^{2} dX $$

(B.1)

$$ U_{\text{s}} = \frac{BHE}{8L}\left[ {\int\limits_{0}^{L} {\left( \frac{dw}{dX} \right)}^{2} dX} \right]^{2} $$

(B.2)

$$ W_{\text{e}} = \int\limits_{0}^{L} {qw(X)dX} $$

(B.3)

where q is the external force per unite length of the beam which is assumed constant for the LPM.

To develop a LPM, trial solutions for deflection of the nano-structures are selected. For nano-cantilever and nano-bridge the trial solutions can be chosen as relations (B.4) and (B.5), respectively:

$$ w(X) = w_{\hbox{max} } \left[ {1 - \cos \left( {\frac{\pi X}{2L}} \right)} \right] $$

(B.4)

$$ w(X) = w_{\hbox{max} } \left[ {\frac{1}{2} - \frac{1}{2}\cos \left( {\frac{2\pi X}{L}} \right)} \right] $$

(B.5)

Now, for nano-cantilever the total energy of system can be written as:

$$ \varPi = U_{b} - W_{\text{e}} = \frac{{\pi^{4} \left( {BEH^{3} + 48\mu Al^{2} } \right)}}{{768L^{3} }}w_{\hbox{max} }^{2} - \frac{1}{\pi }Lqw_{\hbox{max} } \left( {\pi - 2} \right) $$

(B.6)

and for nano-bridge we have:

$$ \varPi = \varPi = U_{b} + U_{s} - W_{e} = \frac{{\pi^{4} \left( {BEH^{3} + 48\mu Al^{2} } \right)}}{{12L^{3} }}w_{\hbox{max} }^{2} + \frac{{BHE\pi^{4} }}{{32L^{3} }}w_{\hbox{max} }^{4} - \frac{1}{2}Lqw_{\hbox{max} } $$

(B.7)

By imposing $\frac{d\varPi }{{dw_{\hbox{max} } }} = 0$ we obtain relations (B.8) for nano-cantilever and (B.9) for nano-bridge:

$$ \frac{{\pi^{5} \left( {BEH^{3} + 48\mu Al^{2} } \right)}}{{384L^{4} (\pi - 2)}}w_{\hbox{max} } - q = 0 $$

(B.8)

$$ \frac{{\pi^{4} \left( {BEH^{3} + 48\mu Al^{2} } \right)}}{{3L^{4} }}w_{\hbox{max} } + \frac{{\pi^{4} }}{{4L^{4} }}HEBw_{\hbox{max} }^{3} - q = 0 $$

(B.9)

Substituting $q = \left. {f_{\text{elec}} } \right|_{{w = w_{\hbox{max} } }} +\ \left. {f_{Cas} } \right|_{{w = w_{\hbox{max} } }}$ in the above relations results in the equations (B.10) and (B.11) for nano-cantilever and nano-bridge, respectively:

$$ \begin{aligned} \frac{{\pi^{5} \left( {BEH^{3} + 48\mu Al^{2} } \right)}}{{384(\pi - 2)L^{4} }}w_{\hbox{max} } - \frac{{\varepsilon_{0} BV^{2} }}{{2\left( {g - w_{\hbox{max} } } \right)^{2} }}\left( {1 + 0.65\frac{{g - w_{\hbox{max} } }}{B}} \right) \hfill \\ \quad + \frac{{\varepsilon_{0} BV^{2} }}{{2\left( {\Omega g + w_{\hbox{max} } } \right)^{2} }}\left( {1 + 0.65\frac{{\Omega g + w_{\hbox{max} } }}{B}} \right) - \frac{{\pi^{2} \hbar cB}}{{240\left( {g - w_{\hbox{max} } } \right)^{4} }} + \frac{{\pi^{2} \hbar cB}}{{240\left( {\Omega g + w_{\hbox{max} } } \right)^{4} }} = 0 \hfill \\ \end{aligned} $$

(B.10)

$$ \begin{aligned} \frac{{\pi^{4} \left( {BEH^{3} + 48\mu Al^{2} } \right)}}{{3L^{4} }}w_{\hbox{max} } + \frac{{\pi^{4} HEB}}{{4L^{4} }}w_{\hbox{max} }^{3} - \frac{{\varepsilon_{0} BV^{2} }}{{2\left( {g - w_{\hbox{max} } } \right)^{2} }}\left( {1 + 0.65\frac{{\left( {g - w_{\hbox{max} } } \right)}}{B}} \right) \hfill \\ \quad + \frac{{\varepsilon_{0} BV^{2} }}{{2\left( {\Omega g + w_{\hbox{max} } } \right)^{2} }}\left( {1 + 0.65\frac{{\left( {\Omega g + w_{\hbox{max} } } \right)}}{B}} \right) - \frac{{\pi^{2} \hbar cB}}{{240\left( {g - w_{\hbox{max} } } \right)^{4} }} + \frac{{\pi^{2} \hbar cB}}{{240\left( {\Omega g + w_{\hbox{max} } } \right)^{4} }} = 0 \hfill \\ \end{aligned} $$

(B.11)

Using the definition of $\bar{w}_{\hbox{max} }^{{}} = {{w_{\hbox{max} } } \mathord{\left/ {\vphantom {{w_{\hbox{max} } } g}} \right. \kern-0pt} g}$, the above equations can be dimensionless to (B.12) for nano-cantilever and (B.13) for nano-bridge:

$$ \frac{{\pi^{5} }}{32(\pi - 2)}\left( {1 + \xi } \right)\bar{w}_{\hbox{max} } - \frac{{\beta \left( {\text{1} + } \right.\left. {\gamma \left( {1 - \bar{w}_{\hbox{max} } } \right)} \right)}}{{\left( {1 - \bar{w}_{\hbox{max} } } \right)^{2} }} + \frac{{\beta \left( {\text{1} + \left( {\Omega + \bar{w}_{\hbox{max} } } \right)} \right)}}{{\left( {\Omega + \bar{w}_{\hbox{max} } } \right)^{2} }} - \frac{\alpha }{{\left( {1 - \bar{w}_{\hbox{max} } } \right)^{4} }} + \frac{\alpha }{{\left( {\Omega + \bar{w}_{\hbox{max} } } \right)^{4} }} = 0 $$

(B.12)

$$ 4\pi^{4} \left( {1 + \xi } \right)\bar{w}_{\hbox{max} } + \frac{{\eta \pi^{4} }}{2}\bar{w}_{\hbox{max} }^{3} - \frac{{\beta \left( {\text{1} + } \right.\left. {\gamma \left( {1 - \bar{w}_{\hbox{max} } } \right)} \right)}}{{\left( {1 - \bar{w}_{\hbox{max} } } \right)^{2} }} + \frac{{\beta \left( {\text{1} + } \right.\left. {\gamma \left( {\Omega + \bar{w}_{\hbox{max} } } \right)} \right)}}{{\left( {\Omega + \bar{w}_{\hbox{max} } } \right)^{2} }} - \frac{\alpha }{{\left( {1 - \bar{w}_{\hbox{max} } } \right)^{4} }} + \frac{\alpha }{{\left( {\Omega + \bar{w}_{\hbox{max} } } \right)^{4} }} = 0 $$

(B.13)

Relations (B.12) and (B.13) can be rewritten in the form of Eq. (33).

Appendix C: Modified couple stress theory

The modified couple stress theory has been presented recently by Yang et al. [17] including only one higher-order material length scale parameter in the constitutive equations. They imposed some restrictions on the couple stress theory to develop a simpler theory. The strain energy density in $\bar{U}$, in the linear elastic isotropic material based on the modified couple stress theory can be written as [17]:

$$ \bar{U}\text{ = }\frac{\text{1}}{\text{2}}\left( {\sigma_{ij} \varepsilon_{ij} + m_{ij} \chi_{ij} } \right) $$

(C.1)

where

$$ \chi_{ij}^{s} = \frac{1}{2}e_{jkl} u_{l,ki} $$

(C.3)

$$ m_{ij} = \text{2}\mu l^{2} \chi_{ij} $$

(C.4)

In above equations, χ _ij and m _ij indicate components of symmetric rotation gradient tensor, Cauchy’s stress and high order stress tensors, respectively.

By substituting Eq. (3) in (C.3), (C.4) and then in (C.1) the total energy of the system based on the modified couple stress theory can be obtained as:

$$ \begin{aligned} U = & \frac{\text{1}}{\text{2}}\int\limits_{0}^{L} {\int\limits_{A} {\left[ {\varepsilon_{11} \sigma_{11} + \chi_{12} m_{12} + \chi_{21} m_{21} } \right]} } dAdX \\ = & \frac{\text{1}}{\text{2}}\int\limits_{0}^{L} {\left[ {\left( {EI + \mu Al^{2} } \right)\left( {\frac{{\partial^{2} w}}{{\partial X^{2} }}} \right)^{2} + EA\left( {\frac{\partial u}{\partial X} + \frac{1}{2}\left( {\frac{\partial w}{\partial X}} \right)^{2} } \right)^{2} } \right]} dX \\ \end{aligned} $$

(C.5)

Now, by utilizing Hamilton principle, i.e., δ(U–V) = 0, in which δ indicates variations symbol, the differential equation of longitudinal and lateral deflection of the system can be derived as the following:

$$ EA\frac{\partial }{\partial X}\left( {\frac{\partial u}{\partial X} + \frac{1}{2}\left( {\frac{\partial w}{\partial X}} \right)^{2} } \right) = 0 $$

(C.6)

$$ - EA\frac{\partial }{\partial X}\left[ {\left( {\frac{\partial u}{\partial X} + \frac{1}{2}\left( {\frac{\partial w}{\partial X}} \right)^{2} } \right)\frac{\partial w}{\partial X}} \right] + \left( {EI + \mu Al_{{}}^{2} } \right)\frac{{\partial^{4} w}}{{\partial X^{4} }} = f_{elec} + f_{Cas} $$

(C.7)

And the boundary conditions are obtained as:

$$ u(0) = u(L) = 0 $$

(C.8a)

$$ w\left( 0 \right) = \frac{\partial w\left( 0 \right)}{\partial X} = w\left( L \right) = \frac{\partial w\left( L \right)}{\partial X} = 0\quad {\text{For Nano-bridge}} $$

(C.8b)

$$ w\left( 0 \right) = \frac{\partial w\left( 0 \right)}{\partial X} = \frac{{\partial^{2} w\left( L \right)}}{{\partial X^{2} }} = \frac{{\partial^{3} w\left( L \right)}}{{\partial X^{3} }} = 0\quad {\text{For Nano-cantilever}} $$

(C.8c)

Figure 11 shows the normalized maximum deflection of the nano-bridge versus dimensionless input voltage using three different theories, i.e.: classic continuum theory (CCT), modified couple stress theory (MCST) and couple stress theory (CST). The geometry and material properties of the beams are identified in Table 4. It is obvious from this figure that for any given applied voltage the maximum deflection predicted by CST is lower than those of MCST and CCT. Furthermore, the pull-in voltage predicted by CST is greater than that of CCT and MCST

Fig. 11

Normalized maximum deflection of the nano-bridge versus dimensionless input voltage determined using classic continuum theory (CCT), modified couple stress theory (MCST) and couple stress theory (CST)

.

Table 4 Geometry and constitutive material properties used in Fig. 11