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Microelectromechanical Systems for Nanomechanical Testing: Electrostatic Actuation and Capacitive Sensing for High-Strain-Rate Testing

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Abstract

There have been relatively few studies on mechanical properties of nanomaterials under high strain rates, mainly due to the lack of capable nanomechanical testing devices. Here we present a new on-chip microelectromechanical system (MEMS) for high strain-rate nanomechanical testing. The MEMS device consists of an electrostatic comb drive actuator, two capacitive displacement sensors and a load cell. The dynamic responses of the device in air and in vacuum are systematically modeled under both alternating and ramp forces. Two methods, capacitive readout and high-speed imaging, are used to measure the dynamic displacements, which agree well with the modeling results. While we demonstrate the maximum constant strain rate over 200 s−1 under ramp force, it is interesting to find that the capacitive readout used in this work can only measure strain rate up to 22 s−1 due to its limit in bandwidth. To demonstrate the utility of this new device, gold nanowires are tested at strain rates of 10−5 and 10 s−1 inside a scanning electron microscope. Increasing strain rate is found to yield higher yield strength and larger ductility.

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Acknowledgements

The authors gratefully acknowledge financial support from the National Science Foundation (NSF) through Award Nos. DMR-1410475 and CMMI-1929646. The authors would like to thank Dr. T. Fang for providing access to his high-speed camera and Dr. G. Richter for providing the Au nanowire samples.

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Appendix

Appendix

Dynamic Response of a Single-Mass-Spring System Under Ramp Force

For a single-mass-spring system as schematically shown in Fig. 14, if the input force is a ramp force F(t) = pt, the equation of motion can be written as

$$ m\ddot{x}+ kx= pt $$
(17)

Solution to this equation is \( x(t)=\frac{pt}{k}-\frac{p}{k}\sqrt{\frac{m}{k}}\mathit{\sin}\left(\sqrt{\frac{k}{m}t}\right) \). This solution contains two terms; the first is a ramp term \( \frac{pt}{k}=\frac{F(t)}{k} \), and the second is a sinusoidal term \( -\frac{p}{k}\sqrt{\frac{m}{k}}\mathit{\sin}\left(\sqrt{\frac{k}{m}t}\right) \). The sinusoidal term has a frequency \( \sqrt{\frac{k}{m}} \), which is the natural frequency of the system. Its amplitude is proportional to loading rate p and can be calculated if system parameters m and k are known.

Fig. 14
figure 14

Schematic of a single-mass-spring system

Dynamic Response of a Single-Mass-Spring-Damper System Under Ramp Force

For a single-mass-spring-damper system as schematically shown in Fig. 15, if the input force is a ramp force F(t) = pt, the equation of motion can be written as

$$ m\overset{\dot \dot }{x}+c\dot{x}+ kx= pt $$
(18)

If the system is an underdamped system, solution to this equation is \( x(t)=\frac{pt}{k}-\frac{cp}{k^2}-\frac{2 pm}{k\sqrt{4 mk-{c}^2}}\sin \left(\frac{\sqrt{4 mk-{c}^2}}{2m}t+\theta \right) \), where \( \theta =\arctan \left(\frac{c\sqrt{4 mk-{c}^2}}{c^2-2 km}\right) \). The solution contains three terms; the first is a ramp term \( \frac{pt}{k}=\frac{F(t)}{k} \), the second is a constant \( -\frac{cp}{k^2} \), and the third is a sinusoidal term \( -\frac{2 pm}{k\sqrt{4 mk-{c}^2}}\sin \left(\frac{\sqrt{4 mk-{c}^2}}{2m}t+\theta \right) \). The sinusoidal term has a frequency \( \frac{\sqrt{4 mk-{c}^2}}{2m} \), which is the damped natural frequency of the system.

Fig. 15
figure 15

Schematic of a single-mass-spring-damper system

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Li, C., Zhang, D., Cheng, G. et al. Microelectromechanical Systems for Nanomechanical Testing: Electrostatic Actuation and Capacitive Sensing for High-Strain-Rate Testing. Exp Mech 60, 329–343 (2020). https://doi.org/10.1007/s11340-019-00565-5

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