On the Size-dependent Nonlinear Behavior of a Capacitive Rectangular Micro-plate Considering Modified Couple Stress Theory

بررسی رفتار غیرخطی وابسته به اندازه ی میکروصفحات مستطیلی خازنی با استفاده از تئوری تنش کوپل پیراستهICMEAT2012-2986

Kaveh Rashvand, Ghader Rezazadeh, Rasool Shabani, Mehrdad Sheikhlou

Presenter Kaveh Rashvand

Urmia University / Mechanical Engineering Department

Outlines

Introduction Research Background Assumptions and Formulations Numerical Solution Results Conclusions

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Introduction

• RF MEMS• Shapes of Actuators: Micro Beam, Rectangular or

Circular Micro-plate• Pull-in Voltage• Modified Couple Stress Theory (MCT) • Material Length-scale parameter

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Research Background of RF MEMS

• Français and Dufour (1999) proposed a complete normalized study of a diaphragm's behavior in MEMS and

compared the results with measurements made on silicon plates under electrostatic actuation.

• Senturia (2000) defined the structural instability phenomenon as “pull-in” in his book.

• Younis (2004) presented an analysis and simulation of RF MEMS switches employing resonant

microbeams and investigated the pull-in instability and formulated safety criteria for the design of MEMS sensors and RF filters in his Ph.D thesis.

• Talebian et al. (2010) investigated the effect of temperature, stretching and residual stresses on the static

instability and natural frequency of an electrostatically actuated Kirchhoff micro-plate using the classical theory (CT).

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Research Background of MCT

• Cosserat brothers (1909) The Cosserat theory of elasticity incorporates the local rotation of points as well as

the translation allowed in CT. Moreover, there is a torque per unit area, or couple stress, as well as the usual force per unit area, or stress.

• Yang et al. (2002) The symmetric part of curvature tensor is an additional measure of deformation

that conjugates to the couple stress. This means that the number of required material length scale parameters is only one in the modified couple stress theory (MCT) instead of two in the classical couple stress theory (CCT).

• Tsiatas (2009) A Kirchhoff plate model for the static analysis of isotropic micro-plates with

arbitrary shapes were derived based on MCT.

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Assumptions and FormulationsDistributed model

boundary conditions for fully clamped rectangular micro-plate:

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Assumptions and FormulationsLumped model

which are determined from the equalizing the static pull-in voltage and first natural frequency of the mass-spring model with those obtained by the distributed one.

and are the electrostatic and mass corrective coefficients, respectively,

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Assumptions and FormulationsNon-dimensional formulation

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Distributed model:

Lumped model:

Numerical Solution

Step by Step Linearization Method (SSLM):

Step i th:

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Numerical Solution

Step i+1 th:

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Subtracting step i th from step i+1 th yields the following linear residual:

Numerical SolutionGalerkin-based weighted reduced-order model:

Solving the set of algebraic equations, are calculated from:

is the matrix of unknown coefficients 11

Numerical Solution

The micro-plate is at rest at equilibrium points ( ), hence these points are obtained from

The order of this algebraic equation is three with respect to , which depends on the DC voltage.

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Physically fixed points exist in , but mathematically, these points might also exist in the range of .

ResultsShape functions satisfy the boundary conditions for the rectangular micro-plate:

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Results

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Results

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V=1.5 V

V=2.301 V V=3.864 V

V=0 V

Results

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Conclusions

• In the case of statically application of the voltage, the micro-plate goes towards an unstable condition through a local saddle node bifurcation.

• Applying MCT shifts the saddle-node bifurcation point to a greater value.• Based on the numerical results, the effect of material length-scale is

significant as the thickness becomes smaller. • A fixed value for the material length-scale is not always realistic and

different problems require different values.• Considering that the pull-in voltage and natural frequencies are directly

related to the plate rigidity, it is concluded that the obtained results for the fundamental frequency using MCT are more realistic than those obtained by CT.

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Thank you for your attention

On the Size-dependent Nonlinear Behavior of a Capacitive Rectangular Micro-plate Considering Modified Couple Stress Theory

Kaveh Rashvand, Ghader Rezazadeh, Rasool Shabani, Mehrdad SheikhlouMechanical Engineering Department