# Buckling Electrothermal NEMS Actuators: Analytic Design for Very Slender Beams

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methods

^{®}Multiphysics 4.4. A 2D model of a beam with a raised cosine pre-buckle and the dimensions above was built in the x–y plane. The heat transfer module was used to apply a temperature change and the solid mechanics module to apply boundary constraints. Structural and thermal physics were coupled by thermal expansion in the multiphysics modelling interface. The plane stress approximation was used to eliminate the out-of-plane components of the stress tensor, and a stationary study was carried out using a geometrically non-linear analysis. The number of iterations was set to 50, with the solver automatically determining the damping factor for Newton’s method. In addition to the previous parameters, a Poisson’s ratio of $\nu =0.28$ was assumed. The discrete points in Figure 2 show the results for the four largest beam widths. The results were in good agreement with the analytic model for $W=20\mathsf{\mu}\mathrm{m}\mathrm{and}W=10\mathsf{\mu}\mathrm{m}$ but diverged for $W=5\mathsf{\mu}\mathrm{m}\mathrm{and}W=1\mathsf{\mu}\mathrm{m}$; for smaller values of $W$, convergence was not reached after 50 iterations.

^{®}Core

^{TM}i7-2600 processor with a 3.4 GHz clock speed and 16 Gb RAM. The increase in the computation resource implied that the problem would rapidly become intractable, and these problems would worsen in 3D.

## 3. Results

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Variation of deflection with average temperature rise for a raised cosine actuator, with $E=170\times {10}^{9}{\mathrm{Nm}}^{-2}$, $\alpha =2.6\times {10}^{-6}{\mathrm{K}}^{-1}$, $L=1\mathrm{mm}$, $H=5\mathsf{\mu}\mathrm{m}$, $D=5\mathsf{\mu}\mathrm{m}$, and different values of $W$ in microns. Full lines show the analytic theory; points show the results from the FEA.

**Figure 3.**Comparison of the exact and approximate variations of deflection with the average temperature rise for raised cosine (RC) and V-beam actuators (V), with $E=170\times {10}^{9}{\mathrm{Nm}}^{-2}$, $\alpha =2.6\times {10}^{-6}{\mathrm{K}}^{-1}$, $L=1\mathrm{mm}$, $H=5\mathsf{\mu}\mathrm{m}$, $D=5\mathsf{\mu}\mathrm{m},$ and $W=0.1\mathsf{\mu}\mathrm{m}$. Also shown are the predictions of [6].

**Figure 4.**Comparison of the variations of the functions $f$ and ${f}_{4}$ and $g$ and ${g}_{4}$ with $\lambda $.

**Figure 5.**Comparison of the exact and two approximate variations of deflection with the average temperature rise for a raised cosine actuator with $E=170\times {10}^{9}\text{}{\mathrm{Nm}}^{-2}$, $\alpha =2.6\times {10}^{-6}\text{}{\mathrm{K}}^{-1}$, $L=1\text{}\mathrm{mm}$, $H=5\text{}\mathsf{\mu}\mathrm{m}$, $D=5\text{}\mathsf{\mu}\mathrm{m},$ and $W=5\text{}\mathsf{\mu}\mathrm{m}$.

**Figure 6.**Initial, deflected, and centrally loaded beam shapes ${y}_{0},y,\mathrm{and}y\prime $ of a raised cosine actuator for $kL/\pi =1$.

**Figure 7.**Variation of $\kappa =k{\prime}^{2}/{k}^{2}$ with $kL$ for a centrally loaded raised cosine actuator.

Beam Width (μm) | Physical Memory (Gb) | Virtual Memory (Gb) | Run Time (s) |
---|---|---|---|

20 | 1.23 | 1.32 | 155 |

10 | 1.37 | 1.49 | 342 |

5 | 1.72 | 1.86 | 816 |

1 | 4.37 | 4.69 | 5955 |

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**MDPI and ACS Style**

Syms, R.; Liu, D.
Buckling Electrothermal NEMS Actuators: Analytic Design for Very Slender Beams. *Micro* **2022**, *2*, 54-67.
https://doi.org/10.3390/micro2010003

**AMA Style**

Syms R, Liu D.
Buckling Electrothermal NEMS Actuators: Analytic Design for Very Slender Beams. *Micro*. 2022; 2(1):54-67.
https://doi.org/10.3390/micro2010003

**Chicago/Turabian Style**

Syms, Richard, and Dixi Liu.
2022. "Buckling Electrothermal NEMS Actuators: Analytic Design for Very Slender Beams" *Micro* 2, no. 1: 54-67.
https://doi.org/10.3390/micro2010003