Finite Elements in Analysis and Design 49 (2012) 28–34

Contents lists available at SciVerse ScienceDirect

Finite Elements in Analysis and Design

0168-87

doi:10.1

n Corr

E-m

vadis@e

journal homepage: www.elsevier.com/locate/finel

Electromechanical analysis of micro-beams based on planarfinite-deformation theory

Igor Sokolov, Slava Krylov, Isaac Harari n

Faculty of Engineering, Tel Aviv University, 69978 Ramat Aviv, Israel

a r t i c l e i n f o

Article history:

Received 23 May 2011

Received in revised form

29 June 2011

Accepted 30 June 2011Available online 23 September 2011

Keywords:

MEMS devices

Nonlinear analysis

Geometrically exact beam

Electrostatic pressure

4X/$ - see front matter & 2011 Elsevier B.V.

016/j.finel.2011.08.018

esponding author.

ail addresses: sokolovi@eng.tau.ac.il (I. Sokolo

ng.tau.ac.il (S. Krylov), harari@eng.tau.ac.il (I

a b s t r a c t

In MEMS devices, solids, often slender in geometry, are in nonlinear interaction with complex three-

dimensional electrostatic fields. The computational cost of solving these coupled problems can be

reduced considerably by the use of structural models. A geometrically exact planar beam model is used

for the solid, with particular attention to normal tractions on the interface that arise from electrostatic

pressure distribution. The weakly coupled problem is solved with a staggered strategy. The resulting

scheme provides accuracy comparable to that obtained by full, three-dimensional representations of

the solid, at costs that may be reduced significantly.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

Electrostatically actuated micro- and nano-scale beams serveas a core part in many micro- and nanoelectromechanical systems(MEMS/NEMS) based applications including optical and electricalswitches, resonators, diffractive elements, and chemical andbiological sensors. Slender beam-like structures are attractivedue to their small size, high sensitivity and simplicity of fabrica-tion, while electrostatic actuation remains the most widely useddue to favorable scaling laws at the micro-scale, low powerconsumption and potential for integration in integrated circuits.

The common yet computationally intensive approach foranalysis of coupled electromechanical problems describing thebehavior of these structures is to consider three-dimensionalelastic and electrostatic continua. Alternatively, simplifiedapproaches based on order reduction of the elastic continuumto compact structural beam models are combined with approx-imate formulas for the electrostatic forces, often developed ona simplified basis. However, three-dimensional electrostaticfields may generate complex patterns of rapidly varying spatialdistributed loading on the solid, rendering such simplifiedapproaches ineffective. (For alternative approaches see [1,2].)

In the coupled problem electrostatic forces load the structure,and structural deformation in turn changes the geometry of theelectrostatic domain. Nonlinear effects become more pronouncedwith scale.

All rights reserved.

v),

. Harari).

In the present work, we implement rigorous structural reductionprocedures to consistently convert continuum electrostatic forces tothe form required by the structural representation, in the frameworkof finite deformation, geometrically exact, planar beam theory.

Geometrically exact beam theory [3–5] has, as its only kine-matic restriction, the classical assumption that plane sectionsremain plane [6, Chapter 9.2]. (See also a version without higher-order curvature [7, Chapter 11].) The conversion of continuumloadings to resultants appropriate for the beam model [8–10] is ofparticular relevance to the present work.

2. Electromechanical boundary-value problem

The electromechanical problem domain is an open region O [Oelec

�R3 (Fig. 1), made up of a bounded mechanical domain Oand an unbounded electrostatic domain Oelec. The electrostaticdomain has a fixed boundary Gelec representing a rigid electrode.The slender mechanical domain has a boundary @O. Part of thisboundary is the interface between the two domains Gint.

In a standard boundary-value problem of electrostatics weseek the electric potential f such that

�Df¼ 0 in Oelecð1Þ

f¼ V on Gelecð2Þ

f¼ 0 on @O ð3Þ

lim9x9-1

f¼ 0 ð4Þ

Ωelec

Γelec

Γint

Ω

Rigid

Fig. 1. Domain of electromechanical problem.

x = Φ (X)

Reference beam

Deformed beam

X3

w1

w3

θw1'

Fig. 2. Reference and current configurations of mechanical domain.

I. Sokolov et al. / Finite Elements in Analysis and Design 49 (2012) 28–34 29

Here V is the voltage applied to the electrode. The solution of thisproblem provides the electric charge density �e0f,n distributed atthe boundaries, where e0 is the given dielectric constant ofvacuum.

Consider a boundary-value problem of finite elastostatics inwhich displacements g are specified on parts of the boundary thatare not in interaction with the electrostatic field @O\Gint. In thecurrent configuration we seek the displacement vector u such that

�div r¼ f in O ð5Þ

rn¼�pn on Gintð6Þ

u¼ g on @O\Gintð7Þ

Here r is the Cauchy stress tensor which depends on the displace-ments (by the generalized Hooke’s law for linear elasticity) and f isthe given body force vector. The coupling load on the electro-mechanical interface is an ‘electrostatic pressure’

p¼e0ðf,nÞ

2

2ð8Þ

Due to the weak coupling, a staggered solution strategy is used[11,12]. (For alternative approaches see [13].) A load increment isdefined in terms of a voltage applied to the electrode. The solutionof the electrostatic problem is straightforward. This solutionprovides the loading of the mechanical problem on the interface.In the following we consider a reduced-order model of the finiteelastostatics problem based on a geometrically exact beamtheory. The resulting mechanical deformation changes the geo-metry of the electrostatic domain. The electrostatic problem issolved again in the updated electrical domain and its solution isthen used for the evaluation of an updated electrostatic pressure.This iterative procedure is repeated until convergence. Then,another load increment in terms of voltage is applied to theelectrode.

3. Planar beam kinematics

Consider a finite elastostatics problem in an initiallyunstressed straight slender body in reference configurationO0¼ A��0,L½ �R3 (Fig. 2). Here, L is the length of the reference

curve connecting a family of cross-sections occupying the openregion A¼ AðX3Þ �R2 through their geometric centroids. Theboundary of the domain is @O0, with outward pointing unitnormal N. The boundary of the cross-section is @A. The materialposition vector of a point in the reference configuration is XAO0.We assume that only natural conditions are given on the beamenvelope, �0,L½�@A, whereas either essential or natural boundaryconditions are specified on the ends X3 ¼ 0 and X3 ¼ L.

The deformation of the reference configuration O0 to thecurrent configuration is a one-to-one mapping O¼UðO0

Þ. Theboundary of the current configuration is @O, with outward

pointing unit normal n. The deformation, parallel to the X1–X3

plane, is described in terms of transverse and axial deflections,w1 ¼w1ðX3Þ and w3 ¼w3ðX3Þ, and rotation of the planar cross-section about the X2-axis, expressed by the angle y¼ yðX3Þ, seeFig. 2. By the assumptions of the geometrically exact beam theory,the spatial position vector x¼UðXÞ is

x¼

w1þX1 cos yX2

X3þw3�X1 sin y

8><>:

9>=>; ð9Þ

The beam displacements U¼UðXÞ�X are

U¼

w1�X1ð1�cos yÞ0

w3�X1 sin y

8><>:

9>=>; ð10Þ

The beam deflections are typically considered to be the elasticdisplacements on the beam axis (X1 ¼ 0 in this case). Alterna-tively, they may be viewed as cross-sectional averages

wi ¼1

A

ZA

Ui dA ð11Þ

This interpretation can be useful in determining inhomogeneousessential boundary conditions.

The deformation gradient is

F¼@U@X¼

cos y 0 w01�X1y0 sin y

0 1 0

�sin y 0 1þw03�X1y0 cos y

264

375 ð12Þ

Primes denote differentiation with respect to X3. The matrix of thedeformation gradient is the Jacobian matrix of the deformationmap U. The non-zero components of the Green strain

E¼ 12ðF

T F�1Þ ð13Þ

are

E33 ¼ E0þX1K1þX21 K2 ð14Þ

2E13 ¼G ð15Þ

The beam kinematic quantities are

E0 ¼w03þ12ððw

01Þ

2þðw03Þ

2Þ ð16Þ

G¼�ð1þw03Þsin yþw01cos y ð17Þ

K1 ¼Ly0 ð18Þ

K2 ¼12ðy0Þ2

ð19Þ

I. Sokolov et al. / Finite Elements in Analysis and Design 49 (2012) 28–3430

Here,

L¼�ð1þw03Þcos y�w01 sin y ð20Þ

The strains E0 and G are constant on the cross-section; K1 is thecurvature and K2 is the high-order curvature.

4. Body forces and boundary tractions

In the Lagrangian approach the equations are formulated withrespect to the reference configuration [14]. Consider the currentconfiguration with domain O and boundary @O¼Gg [ Gh . Thebody is under the action of prescribed body forces f : O-R3,boundary displacements g : Gg-R3, and boundary tractionsh : Gh-R3.

Defining body forces in the reference configuration such thatZO0

f0 dV ¼

ZO

f dv ð21Þ

leads, by change of variables, to

f0¼ Jf ð22Þ

here, J¼ det F. Similarly, for surface tractions, with Gh ¼UðG0hÞZ

G0h

h0 dA¼

ZGh

h da ð23Þ

For pressure loading,

h¼�pn ð24Þ

a consequence of Nanson’s formula yields

h0¼�pJF�T N ð25Þ

5. Virtual work

Virtual work in the reference configurationZO0

E : S dV ¼

ZO0

U � f0 dVþ

ZG0

h

U � h0 dA ð26Þ

is expressed in terms of the second Piola–Kirchhoff stress S,virtual displacements U satisfying homogeneous conditions onG0

g , with Gg ¼UðG0g Þ, and virtual strains E.

The beam internal work is expressed in terms of virtual beamkinematic quantities and force resultants. The non-zero compo-nents of the virtual Green strain are, see (14) and (15)

E33 ¼ E0þX1K 1þX21 K 2 ð27Þ

2E13 ¼G ð28Þ

Here, see (16)–(19)

E0 ¼w 01w01þw 03ð1þw03Þ ð29Þ

G ¼w 01 cos y�w03 sin yþyL ð30Þ

K 1 ¼�w 01y0 sin y�w 03y

0 cos y�yy0Gþy0L ð31Þ

K 2 ¼ y0y0 ð32Þ

The beam force resultants are

n¼

ZA

S33 dA ð33Þ

q¼

ZA

S13 dA ð34Þ

m1 ¼

ZA

X1S33 dA ð35Þ

m2 ¼

ZA

X21 S33 dA ð36Þ

For linear, isotropic elasticity, the beam constitutive relations are

n¼ EAE0þEIK2 ð37Þ

q¼ GAsG ð38Þ

m1 ¼ EIK1 ð39Þ

m2 ¼ EIE0þEI4K2 ð40Þ

The beam kinematic quantities are defined in (16)–(19), E and G

are Young’s modulus and the shear modulus, As is the effectiveshear area, and

I4 ¼

ZA

X41 dA ð41Þ

The beam external work is expressed in terms of virtualdeflections and rotation, and applied loads (per unit length). Thedistributed loads consist of an applied axial force

F3 ¼

ZA

f 03 dAþ

Z@A

h03 dS ð42Þ

an applied shear force

F1 ¼

ZA

f 01 dAþ

Z@A

h01 dS ð43Þ

and an applied couple

C2 ¼�

ZA

X1ðf01 sin yþ f 0

3 cos yÞ dA�

Z@A

X1ðh01 sin yþh0

3 cos yÞ dS

ð44Þ

Recall, @A is the boundary of cross-section A.End force resultants are

Q3 ¼

ZA

h03 dA ð45Þ

Q1 ¼

ZA

h01 dA ð46Þ

M2 ¼�

ZA

X1ðh01 sin yþh0

3 cos yÞ dA ð47Þ

To fix ideas, consider essential boundary conditions specifiedat X3 ¼ 0, and natural boundary conditions at X3 ¼ L. Other casesare formulated similarly. Thus, the beam deflections and rotationsatisfy the boundary conditions at X3 ¼ 0, and their variationssatisfy the homogeneous counterparts. Combining the precedingrelations, the equation of virtual work for the beam isZ L

0ðE0nþGqþK 1m1þK 2m2Þ dX3

¼

Z L

0ðw3F3þw1F1þyC2Þ dX3þðw3Q3þw1Q1þyM2Þ9X3 ¼ L

ð48Þ

6. Linearization

The statement of virtual work for the beam is a nonlinearrelation. The problem is addressed by incremental loading, ifneeded, and a Newton–Raphson iterative solution strategy withconsistent linearization. Given a configuration defined by w1, w3,and y, we seek admissible updates Dw1, Dw3, and Dy. Lineariza-tion of the beam virtual work (48) leads to a linear equation forthe unknown updates in terms of the effective tangent stiffness,

I. Sokolov et al. / Finite Elements in Analysis and Design 49 (2012) 28–34 31

driven by the out-of-balance forces, both of which depend on thegiven configurationZ L

0ðE0DnþGDqþK 1Dm1þK 2Dm2Þ dX3

þ

Z L

0ðDE0nþDGqþDK 1m1þDK 2m2Þ dX3

�

Z L

0ðw3DF3þw1DF1þyDC2Þ dX3

¼

Z L

0ðw3F3þw1F1þyC2Þ dX3þðw3Q3þw1Q1þyM2Þ9X3 ¼ L

�

Z L

0ðE0nþGqþK 1m1þK 2m2Þ dX3 ð49Þ

End force resultants are assumed to be independent of thedeformation.

Material stiffness arises from internal force resultant updates

Dn¼ EADE0þEIDK2 ð50Þ

Dq¼ GAsDG ð51Þ

Dm1 ¼ EIDK1 ð52Þ

Dm2 ¼ EIDE0þEI4DK2 ð53Þ

Here, the beam kinematic updates are

DE0 ¼Dw01w01þDw03ð1þw03Þ ð54Þ

DG¼Dw01 cos y�Dw03 sin yþDyL ð55Þ

DK1 ¼�Dw01y0 sin y�Dw03y

0 cos y�Dyy0GþDy0L ð56Þ

DK2 ¼Dy0y0 ð57Þ

Geometric stiffness arises from virtual kinematic updates

DE0 ¼w 01Dw01þw 03Dw03 ð58Þ

DG ¼�ðw 03 cos yþw 01 sin yþyGÞDy�yðDw03 cos yþDw01sin yÞð59Þ

DK 1 ¼�ðw03 cos yþw 01 sin yþyGÞDy0 þðw 03 sin y�w 01 cos yÞy0Dy

�yy0DGþy0DL ð60Þ

DK 2 ¼ y0Dy0 ð61Þ

Here

DL¼�Dw01 sin y�Dw03 cos y�DyG ð62Þ

Load stiffness arises from distributed load updates. For pres-sure loading

DF3 ¼�

Z@A

pð�N1Dw01þðN1X1y0 cos y�N3sin yÞDyþN1X1 sin yDy0Þ dS

ð63Þ

DF1 ¼�

Z@A

pðN1Dw03þðN1X1y0 sin yþN3 cos yÞDy�N1X1 cos yDy0Þ dS

ð64Þ

DC2 ¼

Z@A

pðN1 sin yDw03�N1 cos yDw01�N1LDyÞX1 dS ð65Þ

7. Computations

The numerical examples consider beams with rectangularcross-sections of thickness t and width b, under pressure loading.For pressure loading of the form (25), with the deformation

gradient (12), the beam distributed loads are

F3 ¼�

Z@A

pð�N1F13þN3F11Þ dS ð66Þ

F1 ¼�

Z@A

pðN1F33�N3F31Þ dS ð67Þ

C2 ¼�GZ@A

pðN1�N3ÞX1 dS ð68Þ

Note that pressure loading also results in an applied axial forceand couple in the total Lagrangian description with respect to thereference configuration.

Due to the beam geometry, N1 ¼ 71 on the top and bottomsurfaces respectively (X1 ¼ 7t=2), and N3 ¼ 0. It is convenient todefine

p ¼1

b

Z b=2

�b=2p dX2 ð69Þ

1fU¼ f 9X1 ¼ t=2�f 9X1 ¼ �t=2 ð70Þ

Then

F3 ¼1bpF13U ð71Þ

F1 ¼�1bpF33U ð72Þ

C2 ¼�G1bpX1U ð73Þ

Finite element structural calculations are performed withconversion of three-dimensional loadings to beam resultantsaccording to the preceding formulas. The elements are two-nodedbeams with linear interpolation of deflections and rotations.Uniformly reduced integration is used, one-point quadrature inthis case.

The underlying three-dimensional elasticity computations forcomparison are obtained with a commercial finite element code.The elasticity elements are eight-noded trilinear solids withincompatible modes. Highly refined meshes are used to obtainreference solutions.

7.1. Cantilever beam

Consider the mechanical problem of a slender body withsquare cross-section and aspect ratio of 1:20, and Poisson ration¼ 0:28. The body is fixed at one end and free at the other, withuniform pressure applied on the envelope. In the case beingconsidered, a pressure of p=E¼�5� 10�3 is applied to the topsurface, and p=E¼ 1� 10�3 to the bottom. The structural solutiondepends only on the difference in pressure, whereas the under-lying elasticity solution varies with the actual values of pressureapplied on the top and bottom surfaces.

Increasingly refined meshes are used to investigate conver-gence. The beam formulation is discretized with uniform meshes,starting with five elements and refined by a factor of two in eightstages to the finest mesh of 640 elements. The beam deflectionwith 20 elements, under four equal load increments, is shown inFig. 3. The continuum formulation is also discretized with uni-form meshes, starting with 1�1�20 elements, and refining by2�2�2 in four stages to the finest mesh of 8�8�160 elements.

Fig. 4 shows the convergence in the energy norm. The beamcomputation converges to an error below 5% with 100 degrees offreedom, and after that exhibits only model error. The elasticitycomputation needs almost 100 times more degrees of freedom toreach that accuracy.

0 0.2 0.4 0.6 0.8 1−2

0

2

4

6

8

10

12

14

x3/L

x 1/t

Fig. 3. Cantilever beam deflection under four equal load increments (20 beam

elements).

101 102 103 104 105

4

6

8

10

1214

Number of degrees of freedom

Rel

ativ

e er

ror i

n th

e en

ergy

nor

m, %

BeamElasticity

Fig. 4. Cantilever beam: relative error in the energy norm.

Electrode

LX2

X1

Beam

E,v

b

gt

VX3

Fig. 5. MEMS device: deformable solid actuated by electrostatic forces from rigid

electrode.

I. Sokolov et al. / Finite Elements in Analysis and Design 49 (2012) 28–3432

7.2. MEMS device

We now consider a coupled electromechanical problem in aconfiguration typical of a MEMS device. The MEMS device consistsof a deformable double-clamped slender solid over a rigidelectrode (Fig. 5). The solid has a length of L¼ 5000 mm, and arectangular cross-section of thickness t¼ 10 mm and widthb¼ 150 mm. The material is modeled as linear isotropic elasticwith Young’s modulus E¼188 GPa and Poisson’s ratio n¼ 0:18.

The beam is actuated by a distributed electrostatic force providedby a rigid electrode of dimensions 220� 150� 6000 mm (extend-ing beyond its edges) located at a distance g ¼ 20 mm. Fig. 6 showsthe rigid electrode and deformable double-clamped slender solid,along with a detail of the electromechanical mesh.

The electrostatic solution is computed by the boundary ele-ment method with multipole acceleration, based on a Fredholmintegral equation of the first kind in terms of the electric chargedensity, by means of the FastCap solver [15]. The surface of theelectrode and the interface are remeshed with boundary elementpanels (i.e. piecewise constant approximation of the electriccharge density) at each iteration, in response to the mechanicaldeformation. The electrostatic pressure is converted to loadresultants distributed along the beam axis, e.g. Fig. 7 for anapplied voltage of 10 V on the reference geometry.

The beam computation is performed with a uniform mesh of20 finite elements. The continuum computation is performedwith a uniform mesh of 5�15�500 elements (over 2500 timesmore degrees of freedom than in the beam computation). Theelectrostatic boundary element panels match the faces of thesolid finite elements in size, and the same boundary elementmesh division is kept for the beam computation. The electricalloading is incremented in steps of 10 V up to a level of 60 V, andthen incremented by 1 V.

Fig. 8 compares the displacement at the beam center obtainedby three-dimensional elasticity with incompatible modes and thegeometrically exact beam to a reference elasticity solution on ahighly refined mesh. There is excellent agreement as the electricalloading is increased. The pull-in voltage denotes a bifurcation atwhich the device collapses onto the electrode, a quantity ofinterest in the analysis of MEMS devices. Here, the pull-involtages are evaluated at 73 V for both the beam and elasticitycomputations, compared to 72 V for the reference elasticitysolution.

Fig. 9 shows the relative errors in the energy norm, andcapacitance. The reduced-order beam mechanical model providesgood accuracy at much lower computational cost (recall, a factorof over 2500 fewer degrees of freedom), particularly in thecapacitance.

8. Conclusions

The computational cost of solving coupled problems involvingslender solids in interaction with complex three-dimensionalfields can be reduced considerably by the use of structuralmodels. In the present work these ideas are used for electro-mechanical systems typical of MEMS devices. A geometricallyexact planar beam model is used for the solid, with particularattention to normal tractions on the envelope that arise fromelectrostatic pressure. The weakly coupled problem is solved witha staggered strategy of solving the electrostatic problem, comput-ing the structural response to the resulting electrostatic loading,and then solving the electrostatic problem in the modifieddomain. The resulting scheme provides accuracy comparable tothat obtained by full, three-dimensional representations of thesolid, at costs that may be reduced significantly.

This scheme is applicable to other coupled problems of solidmechanics, with more complex distributions of interaction forces.An interesting application that we intend to explore is theanalysis of beam-like structures fabricated from ionic polyelec-trolite hydrogels. In these materials, which are hydrophilic poly-mer networks responsive to electrical and chemical stimuli, theinteraction forces (chemoelectromechanical coupling) take theform of a body force distribution engendered by the electricallycontrolled osmotic pressure and swelling within the gel.

ZYX

XY Z

Fig. 6. MEMS device: double-clamped deformable slender solid over a rigid electrode. Detail (right) shows discretization of electromechanical domain.

0

2500

5000−1.68

−1.67

−1.66x 10−4

X3

[[p]]

Fig. 7. MEMS device: the electrostatic pressure on the top and bottom faces of the slender mechanical domain (left) is converted to resultant beam forces.

0 20 40 60 80

−0.5

−0.4

−0.3

−0.2

−0.1

0

Voltage

x 1/g

BeamElasticityReference

0 0.2 0.4 0.6 0.8 1

−0.5

−0.4

−0.3

−0.2

−0.1

0

x3/L

x 1/g

V = 20

V = 40

V = 60

V = 72 Vpull−in = 73

Fig. 8. MEMS device: center displacement (left) and deflection.

10 20 30 40 50 60 70 802224262830323436384042

Voltage

Rel

ativ

e er

ror i

n th

e en

ergy

nor

m, %

BeamElasticity

10 20 30 40 50 60 70 800

0.5

1

1.5

2

2.5

3

3.5

4

Voltage

Rel

ativ

e er

ror o

f cap

acita

nce,

%

BeamElasticity

Fig. 9. MEMS device: relative errors in the energy norm (left) and capacitance.

I. Sokolov et al. / Finite Elements in Analysis and Design 49 (2012) 28–34 33

I. Sokolov et al. / Finite Elements in Analysis and Design 49 (2012) 28–3434

Acknowledgment

The research was supported in part by the Israel ScienceFoundation (ISF, Grant No. 1426/08).

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