The response of a rectangular micro-plate to mechanical shocks considering modified couple stress theory

ISAV2012_2162

Kaveh Rashvand, Ghader Rezazadeh, Mehrdad Sheikhlou, Siavash Kazemirad

Presenter Kaveh Rashvand

Urmia University / Mechanical Engineering Department

Outlines

Introduction Research Background Assumptions and Formulations Numerical Solution Results Conclusions

2

Introduction

• RF MEMS• Capacitive Rectangular Micro-plate• Pull-in Voltage• Modified Couple Stress Theory (MCT) • Material Length-scale parameter

3

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.

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.

• Younis et al. (2007) studied the dynamic response of an electrostatically deflected microbeam under

different shock types with different shock durations and amplitudes based on CT.

• Kimberley et al. (2009) investigated failure of Au RF MEMS beam subjected to dynamic loading

experimentally.

• Rashvand et al. (2012) studied an isotropic rectangular micro-plate statically via MCT without considering

the effect of the axial stress generated by the micro-plate stretching.

5

Assumptions and Formulations

2 2 2

42 2

2

s2

2

lxx xy yy

w w wD D w N N Nx yx y

wρh q ρha g(t)t

3

212 1

EhD

2

2

2 1

l El hD l Gh

20

20

,2 ,

Vq w V

g w x y

22

20 0

2 2

20 0

0 0

1 1 ,1 2 2

1 1 ,1 2 2

1 ,2 1 2

a bs

xx xx

b as

yy yy

b as

xy xy

Eh w wN σ h dx ν dyν a x b y

Eh w wN σ h dy ν dxν b y a x

Eh w wN σ h dxdyν ab x y

6

0,

0,

wwxwwy

at

at

0, ,

0, .

x a

y b

sin( ) πg t t H t H t dd

boundary conditions forfully clamped rectangular micro-plate:

Assumptions and Formulations

7

4 4 4 2 2 2

4 4 2 2

2 2

22

2 2

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ ˆ2 2ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ( ),ˆ ˆ1

l

xx xy yyD D w w w w w wN N ND x x y y x x y y

w V g tt w

0

4

ˆ ˆ ˆ ˆ, , , * .*

, w x y t aw x y t

a ah

tt

Dg

22 20

0 0

22 20

0 0

20

0 0

ˆ ˆˆ ˆ ˆ6 ,ˆ ˆ

ˆ ˆˆ ˆ ˆ6 ,ˆ ˆ

ˆ ˆˆ ˆ ˆ3 1ˆ ˆ

b a1

xx

b a 1

yy

b a1

xy

g w a wN dx ν dyh x b y

g a w wN dy ν dxh b y x

g a w wN ν dxdyh b x y

.

4

03

04

0

,2

sa aa hgg D

,D

Numerical Solution

8

N M

mn m nn 1 m 1

ˆ ˆ ˆˆˆ ˆ ˆw x, y, t q t x y

sklmn klmn klmn

n 1 m 1 n 1 m 1

N M N Me s

mn mn kl klˆ ˆt tM K K Fq Fq

2

mn

2

1 1ekl 2

s

0

0 0

1 1skl

0 0

F1

a t *

q

ˆF g(t)g

k l

m n

k l

Vφ α dxdyφ

φ dxdy

1 1 4

40 0

1 1 2 2l

klmn 2 20 0

1 1 4

40 0

D DK 2D

mk l n

m nk l

nk l m

φ φ dxdyx

φφ dxdyx y

φφ dxdyy

1 1sklmn

0 0

1 1 1 1

20 0 0

2

2

2

0

K

2

mk l n

m n nk l k l m

φ φ dxdyx

φ φφ dxdy φ dxdy

x y y

1 1

klmn0 0

M k l m nφ φ dxdy

k 1,...,Ml 1,..., N

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

9

m n2 2ˆ ˆsin (m x)siˆ ˆx n (ny y)

Results

10

Static pull-in voltage Dynamic response and dynamic pull-in voltage

Results

11

Applied voltage 0V for shock duration d=0.1ms in two gaps.

Difference between the linear and nonlinear model of the micro-plate based on MCT

Results

12

Applied voltage 1V and as=100g for shock duration d=0.5ms.

Dynamic response of the micro-plate based on MCT

shock amplitude as=100g shock amplitude as=1000gApplied voltage 1V and d=0.1 ms Applied voltage 1V and d=0.1 ms

Conclusions

• The numerical results revealed that the intrinsic size dependence of materials affects the stability region and the dynamic response.

• The results obtained from MCT showed that CT overestimates the deflection of the micro-plate in smaller thicknesses.

• Changes of the pulse duration cause the dynamic response to become similar to the quasi-static one; while changing the amplitude of the acceleration pulse moves the stable region of the micro-plate.

• In small shock amplitudes, the effect of stretching is negligible but in higher shocks this effect is considerable.

• The pulse duration and the amplitude of the acceleration pulse should be considered together with the pull-in voltage to prevent the failure of MEMS devices under mechanical shocks.

13

Thank you for your attention

The response of a rectangular micro-plate to mechanical shocks considering modified couple stress theory

Kaveh Rashvand, Ghader Rezazadeh, Mehrdad Sheikhlou, Siavash KazemiradMechanical Engineering Department