This research deals with the effect of the intrinsic material length-scale parameter on the stability of a fully clamped rectangular micro-plate, which can be used as a RF MEMS resonator. A modified couple stress theory is utilized to model the micro-plate, considering the variable material length-scale parameter. The nonlinear governing equation of motion for static analysis is solved using the step-by-step linearization method (SSLM) and the static pull-in parameters, limiting the stable regions of capacitive resonators, are determined and compared to those obtained by the classical theory. The numerical results reveal that the intrinsic size dependence of materials leads to an increase in the pull-in voltage and natural frequency depending on the thickness of the micro-plate. Comparing these results to the experimental outcomes shows that utilizing the fixed material length-scale leads to unrealistic results in some manner.

1. On the Size-dependent Nonlinear Behavior of a Capacitive Rectangular Micro-
plate Considering Modified Couple Stress Theory
‫است‬‫ر‬‫پی‬‫کوپل‬‫تنش‬‫تئوری‬‫از‬ ‫استفاده‬ ‫با‬ ‫نی‬‫ز‬‫خا‬‫مستطیلی‬ ‫میکروصفحات‬‫ی‬‫ه‬‫ز‬‫اندا‬‫به‬‫وابسته‬ ‫غیرخطی‬ ‫فتار‬‫ر‬‫ی‬ ‫بررس‬‫ه‬
ICMEAT2012-2986
Kaveh Rashvand, Ghader Rezazadeh, Rasool Shabani,
Mehrdad Sheikhlou
Presenter Kaveh Rashvand
Urmia University / Mechanical Engineering Department

2. Outlines
 Introduction
 Research Background
 Assumptions and Formulations
 Numerical Solution
 Results
 Conclusions
2

3. 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
3

4. 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).
4

5. 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.
5

6. Assumptions and Formulations
Distributed model
boundary conditions for fully clamped
rectangular micro-plate:
6

7. Assumptions and Formulations
Lumped 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,
7

8. Assumptions and Formulations
Non-dimensional formulation
8
Distributed model:
Lumped model:

9. Numerical Solution
Step by Step Linearization Method (SSLM):
Step i th:
9

10. Numerical Solution
Step i+1 th:
10
Subtracting step i th from step i+1 th yields the following linear residual:

11. Numerical Solution
Galerkin-based weighted reduced-order model:
Solving the set of algebraic equations, are calculated from:
is the matrix of unknown coefficients
11

12. 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.
12
Physically fixed points exist in , but mathematically, these
points might also exist in the range of .

13. Results
Shape functions satisfy the boundary conditions for the rectangular
micro-plate:
13

14. Results
14

15. Results
15
V=1.5 V
V=2.301 V V=3.864 V
V=0 V

16. Results
16

17. 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.
17

18. 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 Sheikhlou
Mechanical Engineering Department