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: [email protected] (I. Sokolo
ng.tau.ac.il (S. Krylov), [email protected] (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).
References
[1] R.C. Batra, M. Porfiri, D. Spinello, Analysis of electrostatic MEMS usingmeshless local Petrov–Galerkin (MLPG) method, Eng. Anal. Bound. Elem. 30(11) (2006) 949–962. doi: http://dx.doi.org/10.1016/j.enganabound.2006.04.008.
[2] G. Li, N.R. Aluru, Efficient mixed-domain analysis of electrostatic MEMS, IEEETrans. Comput. Aided Des. 22 (9) (2002) 1228–1242. doi:http://dx.doi.org/10.1109/TCAD.2003.816210.
[3] E. Reissner, On one-dimensional finite-strain beam theory: the plane pro-blem, J. Appl. Math. Phys. 23 (5) (1972) 795–804. doi: http://dx.doi.org/10.1007/BF00944848.
[4] J.C. Simo, A finite strain beam formulation. The three-dimensional dynamicproblem. Part I, Comput. Methods Appl. Mech. Eng. 49 (1) (1985) 55–70.doi: http://dx.doi.org/10.1016/0045-7825(85)90050-7.
[5] L.A. Crivelli, C.A. Felippa, A three-dimensional non-linear Timoshenko beambased on the core-congruential formulation, Int. J. Numer. Methods Eng. 36(21) (1993) 3647–3673. doi: http://dx.doi.org/10.1002/nme.1620362106.
[6] P. Wriggers, Nonlinear Finite Element Methods, Springer-Verlag, Berlin, 2008.[7] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method, vol. 2, 5th ed.,
Butterworth-Heinemann, Oxford, 2000.
[8] A. Cardona, M. Geradin, A beam finite element non-linear theory withfinite rotations, Int. J. Numer. Methods Eng. 26 (11) (1988) 2403–2438.doi:http://dx.doi.org/10.1002/nme.1620261105.
[9] J. Makinen, Total Lagrangian Reissner’s geometrically exact beam elementwithout singularities, Int. J. Numer. Methods Eng. 70 (11) (2007) 1009–1048.
doi: http://dx.doi.org/10.1002/nme.1892.[10] J.C. Simo, R.L. Taylor, P. Wriggers, A note on finite-element implementation of
pressure boundary loading, Commun. Appl. Numer. Methods 7 (7) (1991)513–525. doi: http://dx.doi.org/10.1002/cnm.1630070703.
[11] N.R. Aluru, J. White, An efficient numerical technique for electrochemical
simulation of complicated microelectromechanical structures, Sensor. Actuat.A Phys. 58 (1) (1997) 1–11. doi: http://dx.doi.org/10.1016/S0924-4247(97)
80218-X.[12] A. Collenz, F. De Bona, A. Gugliotta, A. Som�a, Large deflections of microbeams
under electrostatic loads, J. Micromech. Microeng. 14 (3) (2004) 365–373doi: http://dx.doi.org/10.1088/0960-1317/14/3/008. URL /http://stacks.iop.org/JMM/14/365S.
[13] V. Rochus, D.J. Rixen, J.-C. Golinval, Monolithic modelling of electro-mechan-ical coupling in micro-structures, Int. J. Numer. Methods Eng. 65 (4) (2006)
461–493. doi: http://dx.doi.org/10.1002/nme.1450.[14] G. Li, N.R. Aluru, A Lagrangian approach for electrostatic analysis of deform-
able conductors, J. Microelectromech. Syst. 11 (3) (2002) 245–254.doi: http://dx.doi.org/10.1109/JMEMS.2002.1007403.
[15] K. Nabors, J. White, FastCap: a multipole accelerated 3-D capacitanceextraction program, IEEE Trans. Comput. Aided Des. 10 (11) (1991)1447–1459. doi:10.1109/43.97624.