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Pull-in instability analysis of electrostatically actuated microplate with rectangular shape

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Abstract

As the size of the micro-electro-mechanical systems (MEMS) continues to decrease, the classical elasticity continuum theory may be inefficient to describe their mechanical behaviors. By introducing the strain gradient elasticity into the classical Kirchhoff plate theory, the size-dependent model for electrostatically actuated microplate-based MEMS is developed. The sixth-order partial differential equation (PDE), derived with the help of the principle of minimum potential energy, can be numerically solved by utilizing generalized differential quadrature (GDQ) method and pseudo arc-length algorithm. The model, with three material length scale parameters (MLSPs) included, can predict prominent size-dependent normalized pull-in voltage with the reduction of characteristic structural size, especially when the plate dimension is comparable to the MLSP (on the order of microns). This study may be helpful to characterize the mechanical properties of electrostatically actuated MEMS, or guide the design of microplate-based devices for a wide range of potential applications.

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Abbreviations

a :

length of the micro-plate

b :

width of the micro-plate

h :

thickness of the micro-plate

g :

gap between two electrodes

w(x,y) :

the transverse displacement along Z axis

\(\bar u\) :

the strain energy density

ɛ ij :

the strain tensor

ɛ′ ij :

the deviatoric strain

γ i :

the dilatation gradient tensor

η (1) ijk :

the deviatoric stretch gradient tensor

ξ s ij :

the symmetric rotation gradient tensor

∂ i :

the differential operator

u i :

the displace-ment vector

ɛ mm :

the dilatation strain

η s ijk :

the symmetric part of the second order displacement gradient tensor

δ ij :

Knocker delta

e ijk :

the permutation tensor

σ ij :

stress tensor

p i , τ (1) ijk , m s ij :

the higher-order stresses

k :

the bulk modulus

µ:

shear modulus

l 0, l 1, l 2 :

the additional independent MLSPs

ɛ :

the dielectric constant of the gap medium

x i , y i :

grid coordinates in X and Y directions

N, M :

grid number in X and Y directions

c (m) ik :

the weighting factors for the approximation of the m-th order derivative of the i-th point in the X direction

{ie1085-2}:

the weighting factors for the approximation of the m-th order derivative of the j-th point in the Y direction

s :

size effect

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Authors and Affiliations

  1. School of Civil Engineering, Shandong University, Jinan, Shandong, 250061, China

    Binglei Wang & Junfeng Zhao

  2. Department of Earth and Environmental Engineering, Columbia University, New York, NY, 10027, USA

    Binglei Wang & Xi Chen

  3. School of Mechanical Engineering, Shandong University, Jinan, Shandong, 250061, China

    Shenjie Zhou & Junfeng Zhao

  4. School of Aerospace, Xi’an Jiaotong University, Xi’an, 710049, China

    Xi Chen

  5. Department of Civil & Environmental Engineering, Hanyang University, Seoul, Korea, 133-791

    Xi Chen

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  1. Binglei Wang
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Correspondence to Shenjie Zhou.

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Wang, B., Zhou, S., Zhao, J. et al. Pull-in instability analysis of electrostatically actuated microplate with rectangular shape. Int. J. Precis. Eng. Manuf. 12, 1085–1094 (2011). https://doi.org/10.1007/s12541-011-0145-1

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