Mathematical modelization of electro, machanical coupling problems , application of finite element method and experiment for piezoelectric material

The piezoelectric materials form a complex system, which utilizes two physical phenomena, that is an interaction between an electric field and a field of mechanical displacement.. In thi

2

Introduction 8

Chapter 1 : Preliminary of piezoelectricity 18

1.1 Piezoelectric effect 18

1.2 Unidimensional piezoelectricity 18

1.3 Ferroelectric ceramics 20

1.4 PZT piezoelectric ceramic 21

1.5 PVDF piezoelectric polymers 22

Chapter 2 : Mathematical modeling and numerical method 24

2.1 Mathematical Modeling of Piezoelectricity 24

2.1.1 Static electromagnetism 24

2.1.2 Elements of mechanicals 27

2.1.3 The piezoelectric constitutive equations 28

2.1.4 Differential equations (strong form) and weak form 28

2.2 Finite element method 31

2.2.1 Discretization by finite elements 31

2.2.2 An error estimation of FEM 32

2.2.3 Matrix form of the weak form by FEM 33

2.3 Smoothed finite element method for 2D piezoelectricity 35

2.3.1 A smoothing operator on mechanical strains and electric field 36

2.3.2 Smoothed stiffness matrices for piezoelectricity problems 38

Chapter 3 : Numerical results 42

3.1 The traveling wave ultrasonic motor 43

3.2 Bimorph beam of PVDF 57

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3.4 PZT amplifier 67

3.5 Patch test for plane elements 71

3.6 Singer-player piezoelectric strip 74

3.7 Cook’s membrane 78

3.8 MEMS device 80

Chapter 4 : Experiments of piezoelectricity 84

4.1 Measuring equipments 84

4.2 PVDF bimorph beam Experiment 88

4.2.1 Prototyping of PVDF bimorph 88

4.2.2 Measure of PVDF bimorph 89

4.2.3 Remarks on experimental results of PVDF bimorph 93

4.3 PZT amplifier Experiment 94

4.3.1 Prototyping of amplifier 94

4.3.2 Measure of PZT amplifier 96

4.3.3 Remarks on Experimental Results 98

Chapter 5 : Conclusions 100

Publications and accomplishments of thesis 103

References 104

Appendix 116

4

for electromechanical quantities

c elastic coefficients when the

electric field is kept constant

U density of internal energy

i

x the cartesian coordinates

7

SFEM Smoothing finite element method

ES-FEM edge–based smoothed finite element method

PZT Lead Zirconate titanate (PbZryTi1-yO3)

PVDF Polyvinylidene Fluoride (-CH2-CF2-)n

PCM Meshless point collocation method

DCSELAB National key lab of digital control & system engineering

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Introduction General description of the problem

The phenomena of transforming directly the different forms of energy of intelligent materials leads to many impressive and interesting applications This has opened many new directions of research which have a great potential in the future This also helps to create high increased values for the products Therefore, researches and applications of intelligent materials are currently the hot topics These materials exist in many forms such as: piezoelectric materials, shape memory alloys , photoelastic materials, electrostrictive materials and relate to many type of energy convertions In particular, the domain of electromechanical energy conversion lives a strong growing in recently relating mainly to the progress of piezoelectric materials

The piezoelectric materials form a complex system, which utilizes two physical phenomena, that is an interaction between an electric field and a field of mechanical displacement It results in two effects The first one, called direct piezoelectric effect, corresponds with the appearance of an electric tension when a mechanical force is applied: the material then plays the role of a piezoelectric sensor Conversely, the reverse piezoelectric effect corresponds to the dilation or the contraction of piezoelectric material when one applies an electric tension to it The material then behaves like a piezoelectric actuator The comprehensive study of these phenomena, generally coupled between them, is a difficult task but necessary because of its vast technological applications

Subjected to electric fields, these materials are capable to produce high forces in a reduced volumes Inversely, subjected to external forces conducting to

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displacement and stress fields, the piezoelectric structure produces an elastic field Due to this special property, the piezoelectric material has an important role and has been applied widely in many fields of advanced technology such as: ultrasound technology, traveling wave ultrasonic motors, nano-positioning systems, microcontroller systems [25], actuators [9][12], sensors, biochip [13]

From a mathematic point of view, the piezoelectric domain is governed by the Maxwell’s equations [73] in the material and the general equation of the elasticity The coupling between two equilibrium of forces on the surface of interaction At low frequencies (up to hundreds of KHz) analytical solutions of this type of problems exits only for certain simple configurations For problems where geometry is complex, we must use methods of discretisation of fields, in particular the finite element method (FEM)

The researchs in this area

Nowadays, there are seldom research projects about this material in Vietnam While the inevitable exploitation of the piezoelectric materials boosts the study on piezolectric phenomenon in worldwide The most popular are piezoelectric ceramics (PZT) and piezopolymers (PVDF) It is used for a large number of applications Almost the research reports and applications of the piezoelectric material have been derived from the long-term and large-scale research projects with the cooperation of many leading experts and Ph D students, and these research reports have been considered as the technological secrets The research of piezolectric field relating to a wide range of many sciences In scope of thesis, we only mention the development in mathematical modeling, numerical methods and experiment of some well known groups such as:

-Tzou, H.S Dept of Mech Eng., Kentucky Univ Robotics and Automation Proceedings, 1989 IEEE International Conference In which the development of a light-weight robot end-effector using polymeric piezoelectric bimorph The

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development of a lightweight piezoelectric end-effector (multifinger type) for robotic applications is reported The fundamental configuration of each finger is a simple piezoelectric bimorph structure made of a piezoelectric polyvinylidene fluoride material Injection of a controlled high voltage into the finger can induce a deformation in a prescribed direction Depending on the applied voltage, the effector can hold an object from outside or inside The theoretical predictions are verified by the experimental and finite-element simulation results [97]

-Anderson, E H., Moore, D M., Fanson, J L & Ealey, M A., 1990,

‘Development of an active member using piezoelectric and electrostrictive actuation for control of precision structures’, in which the design and development of a zero stiction active member containing piezoelectric and electrostrictive actuator motors

is presented Experimental results are shown which illustrate actuator and device characteristics relevant to precision control applications [12]

-Ha, S K., Keilers, C & Chang, F K., 1992, ‘Finite element analysis of composite structures containing distributed piezoceramic sensors and actuators’, AIAA Journal A finite element formulation is presented The formulation was derived from the variational principle with consideration for energy of the piezoceramics An eight-node three-dimensional composite brick element was implemented for the analysis Experiments were also conducted to verify the analysis and the computer simulations [34]

-Hwang, W S & Park, H C., 1993, ‘Finite element modeling of

piezoelectric sensors and actuators’, AIAA Journal, Laminate theory with the

induced strain actuation and Hamilton's principle are used The equations of motion and the total charge are discretized The sensor is distributed, but is also integrated since the output voltage is dependent on the integrated strain rates over the sensor area The actuator induces the control moments at the ends of the actuator [35]

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-Peter L Levin [76], Finite Element Analysis of a Synthetically Loaded Stator for a Piezoelectrically Driven Ultrasonic Traveling ware Motor, Worcester Polytechnic Institute of Massachusetts, USA, 1996 In which, the research build

elements in FEM to simulate traveling ware ultrasonic motor

-J A Christman, R R Woolcott, A I Kingon, and R J Nemanich, 1998,

Piezoelectric measurements with atomic force microscopy, American Institute of

Physics In which, an atomic force microscope (AFM) is used to measure the magnitude of the effective longitudinal piezoelectric constant (d33) of thin films Measurements are performed with a conducting diamond tip in contact with a top electrode [39]

-Benjeddou A 2000, Advances in piezoelectric finite element modeling of adaptive structural elements: a survey, Comput Struct Focus is put on the development of adaptive piezoelectric finite elements [14]

-Sze K Y and Yao L Q 2000, Modelling smart structures with segmented piezoelectric sensors and actuators, In this paper, a number of finite element models have been developed for comprehensive modelling of smart structures with segmented piezoelectric sensing and actuating patches [90]

-Vincent Piefort , Finite Element Modeling of Piezoelectric Active Structure, Libre University of Bruxelles, 2001 [99] In which, constitutive equations is dedicated to the piezoelectricity and simplified for a Mindlin laminate embedding piezoelectric layers in piezoelectric actuation and sensing devices The Vincent Piefort research is implement by the commercial finite element package Samcef (Samtech s.a.) in FEM

-Le Dinh Tuan, modelization piezoelectric structures by finite element method, University of Liege, Belgium, 2002 [46] In which, the general thermopiezoelectric constitutive equations are established starting from

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thermodynamics principles In addition, the Le Dinh Tuan research build elements

in FEM to simulate ultrasonic motor by GetDP/Gmsh software

-Liu G R, Dai K Y, Lim K M and Gu Y T 2002, A mesh free method called point interpolation method (PIM) is presented for static and mode-frequency analysis of two-dimensional piezoelectric structures [50]

-Liu G R, Dai K Y, Lim K M and Gu Y T 2003, A radial point interpolation method for simulation of two-dimensional piezoelectric structures, Smart Mater Struct A meshfree, radial point interpolation method (RPIM) is presented for the analysis of piezoelectric structures, in which the fundamental electrostatic equations governing piezoelectric media are solved numerically without mesh generation [51]

-Andreas Gantner (2005) [13] , which the research presents mathematical modeling for piezoelectrical agitated surface acoustic waves

-Chih-Liang Chu and Sheng-Hao Fan, 2006, A novel long-travel piezoelectric-driven linear nanopositioning stage Precision Engineering Volume

30, Issue 1, January 2006, Journal of sciencedirect This study presents a novel long-travel piezoelectric-driven linear nanopositioning stage capable of operating in either a stepping mode or in a scanning mode [18]

-Long C S, Loveday P W and Groenwold A A 2006, Planar four node piezoelectric with drilling degrees of freedom, Int J Numer Methods Eng [55]

-Liu G R, Dai K Y and Nguyen T T 2007, A smoothed finite element method for mechanics problems, Comput Mech In this paper, they incorporate cell-wise strain smoothing operations into conventional finite elements and propose the smoothed finite element method (SFEM) for 2D elastic problems [49]

-Kenji Uchino (2009) [45], in which the research presents piezoelectric material and structure of piezoelectric motors

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-Liu G R, Nguyen-Thoi T and Lam K Y, 2009, An edge-based smoothed finite element method (ES-FEM) for static, free and force vibration analyses of solids, J Sound Vib This presents an edge-based smoothed finite element method (ES-FEM) to significantly improve the accuracy of the finite element method (FEM) without much changing to the standard FEM settings The ES-FEM can use different shape of elements but prefers triangular elements that can be much easily generated automatically for complicated domains In the ES-FEM, the system stiffness matrix is computed using strains smoothed over the smoothing domains associated with the edges of the triangles [53]

-Alexandre Pages, Majid Hihoud, Frank Claeyssen, Thomas Porchez, Ronan

Le Letty, Cedrat Technologies S.A 2010 Electrically-Tunable Low-Frequency Miniature Suspension In which optical instruments meet in Space, Aircraft or Military applications in general, or in the space experiment These requirements have driven the development of a new type of Amplified Piezo Actuator [10]

The above researchs have contributed greatly to the development of the field piezoelectricity but there are some unresolved limitations as:

-They skip the study of mathematical model to solve the electromechanical coupling problems closely

-Most of the above research present theory and use the commercial software

to simulations without develop the source code Numerical results of simulations have not improved because they still use the standard Finite Element Method (FEM) In addition, a few of which used the meshless methods but it is too complex and incomplete

- In the above researchs, edge-based smoothed finite element method FEM) is a good improvement of FEM but not yet used for electro-mechanical coupling problems

(ES-14

-The most of above researchs only have focused on numerical simulation, rarely invest in experiments to compare the data of experiment with the result of simulation therefore research process is not complete Otherwise a few of them have

to make mention of experiments with conventional measurements, which an idea suggest that need to find the non-contact measurements in high-tech

In this research, we are going to implement a series of studies in order to new supplement and improvement as: working closely with mathematical modeling, application innovative FEM, development source code to numerical simulation and perform experiments to verify the simulation results

These valuable resources, new ideas to perform and researched expectations

-The idea of the proposal to extend Lax-Milgram theorem [33] with four bilinear in variational problem and approximating problem, the estimation of error and convergence rate arenew theoritical supplements in mathematical modelization

of electromechanical coupling problems (present in 2.1 and 2.2) Which imply that the problem becomes more closely It is new research of thesis

-About smoothed finite element method (ES-FEM) [30][46][52]: it was recently proposed by [32] to significantly improve the accuracy and convergence rate of the standard finite element method for pure mechanical problems (only a mechanical field) In this thesis, ES-FEM is further extended for the electromechanical coupling problems with two fields, electric field and field of mechanical displacement (present in 2.3) It is new ideas of thesis

-This thesis will develop calculating programs in Matlab for FEM and FEM [53], that provides a convenient tool to predict behaviour of piezoelectric structures with 8 numerical examples (present in chapter 3) It is specific point of the research

ES-15

- In the research, we perform two experiments for piezoelectric materials, one made of PVDF and the other made of PZT, which prototyping and measuring systems will be tangible results of the research (are shown in chapter 4) All experiments are realized on high-tech equipments in Measurement Module, National key Lab at Digital Control & System Engineering (DCSELAB) The measurement of non-contact displacement by laser energy and optical microscopy

systems with resolution 10 nanometers are original contributions of the thesis

Objectives of the research

Generally speaking, this research aims at a mathematical modelization of electro-mechanical coupling problems, find effective solution to validate the finite element model and perform the experiments for comparison with simulation In detail, the research carried out the principal specifications as follows:

-Mathematical modeling of piezoelectricity in which proposal differential equations system, variational problem, Lax-Milgram extented theorem, approximating problem with error estimation and convergence rate (FE in Sobolev) That make process to solve a electromechanical coupling problem more closely

-Find effective solution and appropriate method to development of calculating programs concerns simulation of the piezoelectric structure which implies that applications of ES-FEM (edge-based smoothed FEM) to solve the electromechanical coupling problems It aims to improve the numerical results of simulation by a new method

-Perform experiments for two kind of piezoelectric ceramics (PZT) and piezopolymers (PVDF) in which the non-contact measurements by optical microscope, laser-energy systems are used and data of experiments in comparison with result of numerical simulations

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Structure of the thesis

The work composes of introduction and 5 chapters, the introduction by its current pages, gives us the general description of the research as well as the objectives of the thesis The chapter 1 is devoted to the preliminary of the piezoelectricity First of all, describe the piezoelectric effect and details the basic formula in the case of unidimentional piezoelectricity to show the nature of the electromechanical coupling phenomena The next is the behavior of PZT and PVDF material also considered under electric field

In chapter 2 concerns with mathematical modeling of piezoelectricity, standard finite element method (FEM) and smoothed finite element method for piezoelectric material

The chapter 3 includes a series of numerical test The first three examples in sections 3.1, 3.2, 3.3 are solved by Matlab code of thesis in standard finite element method (FEM) A more detailed the example 3.1 uses piezoelectric triangular element in 2D and the results in comparison with result of Peter L Levin’s paper [76], the example 3.2 uses piezoelectric plate element in 3D and the results in comparison with result of experiment in section 4.2 in chapter 4, the example 3.3 uses piezoelectric tetrahedral element in 3D and the results in comparison with result of COMSOL sotfware The next is the example 3.4, a problem of PZT amplifier with complex geometry boundary was simulated by COMSOL sotfware and the results in comparison with result of experiment in section 4.3 in chapter 4 The last four examples relate to the benchmarks problems to verify effect of the ES-FEM, in which the results in comparison with results of papers [24][71][75][92] They lead to the conclusion that ES-FEM is an effective solution to solve the electromechanical coupling problems

In the chapter 4, first of all the test facilities for piezoelectric systems is briefly introduced The next is presentation of two experiments One relating to

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design, fabrication the PVDF bimorph beam and design, assembly the non-contact displacement measurement by optical microscope system The other relating to design, fabrication the PZT amplifier and design, assembly the non-contact displacement measurement by laser-energy systems

In chapter 5, conclusions of thesis in which the main originalities of this research are described and key results are recalled, this chapter also mention to subsequent research of the thesis to extend the theory, additional technologies involves a complete fabrication process and commercialization of research products

After chapter 5 is the list of publication papers and references Finally, in Appendix we present contents of source codes

The piezoelectric effect is reversible: under an electric field of proper direction, a piezoelectric material deforms and may in particular be excited at its mechanical resonance, which is very acute This property finds its application in the production of genetic operator ultrasound filters and oscillators in the steering

These properties are here and the piezoelectric materials in which they occur are called piezoelectric PZT (PbZryTi1-yO3), and PVDF (-CH2-CF2-)n are commonly used piezoelectric materials [45]

absolute permittivity of the dielectric

Under a strain vector S applied into piezoelectric material induces a polarization eS, it is called the direct piezoelectric effect, from equation (1-2) we have the total polarization is given by:

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1.3 Ferroelectric ceramics

Now, aimed at understanding the piezoelectric material properties [77][78],

we must first consider the behavior of the material on a microscopic scale Above a certain temperature that is called temperature curie ( c), the crystal structure of a ferroelectric material does have a center of symmetry and therefore has not electric dipole moment, below this temperature which is non-central symmetric In here, the crystal presents a natural electric dipole; it may be reversed and also switched in certain allowed directions by the application of a high electric field The uniform polarization of regions in crystals called ferroelectric domain and the electric dipoles are aligned in the same direction There are many domains in a crystal separated by interfaces called domain walls A ferroelectric single crystal, when grown, has multiple ferroelectric domains in each of which the electric dipole is aligned in a specific allowed direction As each of the allowed direction has the same probability to appear, the net electric dipole summed over the whole crystal is zero [99] Figure 1-1 [45] shows schematically the domain reorientation in a multi-domain ferroelectric piezoceramic The material is initially poled along the negative direction (1) and an electric field is applied along the positive direction The crystal will the increase of the field as the field is opposite in direction to the polarization

Figure 1-1 Strain with electric fields & polarization reorientation

1.4 PZT piezoelectric ceramic

The PZT material is excellent piezoelectric performance under a high DC field as good as the state of remanent polarization Figure 1-2 [99] shows a typical electric field and strain curves (directions x3 and x1, parallel and perpendicular to the field) for a PZT based ferroelectric piezoceramic [98] In a cycle with a small maximum electric field, the field-induced strain curve is almost linear (a) The curve becomes distorted as the electrical field increases and shows a larger hysteresis (b, c, d) and finally transforms into a symmetric butterfly-shape when the electric field exceeds the coercive field; this is caused by the polarization due to dipole reorientation

Figure 1-2 Electric field / strains of PZT

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Under chemical point of view, the Lead Zirconate titanate (PZT) is a binary solid solution of PbZrO3 and PbTiO3 The PZT (PbZryTi1-yO3) is composed of a lattice of basic cells having the structure Figure 1-3 The electric properties of ferroelectric ceramics can be modified by substituting ions of different valence in the lattice

Figure 1-3 PZT structure

1.5 PVDF piezoelectric polymers

In many technical cases, one need a kind of piezoelectric materials is more flexible than PZT (it were known crispy materials) Professor Kawai discovered the piezoelectricity in polyvinylidene (PVDF) in the form of sheets shown as Figure 1-4

Figure 1-4 Chemical structure of polyvinylidene Fluoride

The PVDF material is a thin plastic polymer sheets, it is flexible and shows large compliance This feature makes is suitable for applications The performance

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of these materials greatly depends on the thermal and poling conditions A thin PVDF film was extended and poled at 1000 C under a high electric field such as 300- kVcm-1 [37] PVDF is a polymer, (-CH2-CF2-)n , which has crytallinity of 40 to

50 percent

There are two kind of PVDF form can be obtained, uni-axial and bi-axial, when applying high electric field by one or two perpendicular directions prior to polarization process shown as Figure 1-5 [99]

Figure 1-5 Uni-axial and Bi-axial film

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Chapter 2 : Mathematical modeling and

numerical method

2.1 Mathematical Modeling of Piezoelectricity

Piezoelectricity results from the interaction between electric field and mechanical displacement field It is presented in two forms The first rises to an electric tension when one applies a mechanical stress to the substrate; it is called the piezoelectric direct effect: the material then plays the role of a piezoelectric sensor Conversely, the piezoelectric inverse effect corresponds to the dilation of piezoelectric material when one applies an electric field to it The material then plays the role of a piezoelectric actuator From a mathematical point of view, the piezoelectric field is governed by the Maxwell's equations in the material and the general equations of elasticity

In this section, mathematical modeling of piezoelectricity is perform, which author formulated the variational problem and proved the unique existence of solution and unique solution depends continuously and stably on the mechanical loads and electric field in the standard Sobolev spaces [80]

2.1.1 Static electromagnetism

The framework of piezoelectricity places us in the field of static electromagnetism [61] where all the charges are fixed in a permanent way in space; if they are moving, they move like a DC current in a circuit (so that the volumetric density of free electric charge, e is constant) In this case, the electric displacement D i and the magnetic induction B i are constant and the Maxwell's equations [73] become then:

e i

t D

- Approach in potential scalar electric

The equation (2-3) expresses that the electric field [61] is irrotational and permits to introduce the electric scalar potential quantity such as

i i

D x

- Boundary conditions in electrostatics

There are two types of boundary conditions applicable to the piezoelectric field in electrostatics:

The inhomogeneous condition on the boundary v of  :

|

The case e  0 on vis called homogeneous condition

The research of the potential  inside a domain  bounded, of which the border

n x

- Equations governing the elastic domain

The mechanical schematization rests on the classical conservation law of the theory of the continuous medium

0

ij V i j

T

f x

ij

u u

The stressesTij and the strains Skl are related by the law of elastic

behavior (Hooke’s law):

ij ijkl kl

where cijkl is a tensor containing the elastic coefficients [3]

- Boundary conditions in mechanics

The principal boundary conditions usually met in the elastic domain are the following ones:

The condition on a part of the boundary u, given:

There is indeed three complementary data of displacement or surface force.

The equations (2-5) and (2-15) permit to write the linear piezoelectricity in the mathematical form To solve them, it is advisable to add the laws of behavior which express the physique of the medium in relations between stress, electric displacement, strain and electric field

2.1.3 The piezoelectric constitutive equations

In the general case, from energy conservation law by the first principle of thermodynamics the piezoelectric constitutive equations was find out in [3] under forms:

constants with unchanged strain field, ekij - Piezoelectric coupling constants, it has

minus sign with inverse effect

2.1.4 Differential equations (strong form) and weak form

We have an equation system by substituting equations (2-17), (2-18), (2-5) and (2-12) into equations (2-6) and (2-11) yielding

 satisfying

where f i v  ( L2 ( ) )n has given

We multiply our physical problem in (2-19) by vectorial test function

2.2 Finite element method

2.2.1 Discretization by finite elements

Let (V h )n be a finite-dimensional subspace, (V h )n  {H 1( ) }n with dimension n (n = 1 , 2 , 3 ) , let {N 1, , N n p} be a basic for (V h )n so that



  where n p is the total number of

nodes of discretized domain

1

n e e i

i 

    with  e i   e j   and

h  max{diameter of element}

2.2.2 An error estimation of FEM

Let (u , ) the exact solutions of the original problem from equations (2-19) and (2-20) If  is a convex polygonal domain and u h  (V h )n and  h  V h

are the finite element solutions using interpolation with k order continuous that satisfies the discrete weak statement, equations (2-38) and (2-39), then exist constants C1,C 2 , C 3,C 4 independent of u , and k such that

From equations (2-40), (2-41), (2-42) and (2-43), we have qualitative information that w L2 and w H1 approach zero when the size of element h approach zero

if the (k  1 ) order derivative of the exact solution(u , ) is bounded on domain  In addition, the power of h in equations (2-40), (2-41), (2-42) and (2-43) also shows the theoretical convergence rate of finite element solutions in the corresponding norms

2.2.3 Matrix form of the weak form by FEM

For the linear electroelastic problem, from equations (2-17) and (2-18) the constitutive equations have the following form

The strain – displacement and electric field – potential relationships are

x y

x z

I

I np

35

where

, , ,

, ,

0 0

in 2D : 0

I x

I y

I z uI

N N

, ,

I y

N N N

N N

2.3 Smoothed finite element method for 2D piezoelectricity

In a standard FEM model [94][102][84], we use directly the compatible strain field to evaluate the energy potential functional In an S-FEM model [26][54], however, we will modify compatible strain field based on the assumed displacement field The modified strain field is then used to evaluate the strain energy potential functional, and a proper energy weak form is used to construct the

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discretized model Such a strain modification must be done in a proper way to ensure stability, convergence and to obtain special property for the S-FEM models constructed

There are many kinds of smoothing domains for edges, node [7] and face

In this section, we will use the strain smoothing technique [21] for two dimensional piezoelectric problems base on edges [8]

B

C D

Figure 2-1 Division of domain into triangular element and smoothing cells (k)

connected to edge k of triangular elements

The above equations are the basic form for analyses of piezoelectric solids using the standard FEM Similar to the FEM; the ES-FEM also uses a mesh of elements When 3-node triangular elements are used, the shape functions used in the ES-FEM are also identical to those in the FEM, and hence the displacement field in the ES-FEM is also ensured to be continuous on the whole problem domain

2.3.1 A smoothing operator on mechanical strains and electric field

In the ES-FEM, we do not use the compatible strain fields (2-45) but strains

“smoothed” over local smoothing domains, and naturally the integration for the

stiffness matrix K is no longer based on elements, but these smoothing domains

k 

  and   ( )i ( )j   for i j, in which N e is the total number

of edges of all elements in the entire problem domain For triangular elements, the smoothing domain  ( )k associated with the edge k is created by connecting the

nodes at the ends of the edge to two centroids of two adjacent elements as shown

in Figure 2-1 Division of domain into triangular element and smoothing cells (k)

connected to edge k of triangular elements

Using the edge-based smoothing domains, smoothed strains and smoothed

electric fields over a smoothing domain ( )k

 associated with edge k are defined based on the compatible strains and electric fields as:

where ( )k

A is the area of the smoothing domain  ( )k :

( ) ( )

1

1 3

k k

N

e j

N  for the boundary edges and ( )

2

k e

N  for the inner edges) as shown in Figure 2-1, ( )j

e

A is

the area of the jth element sharing edge k

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Substituting Equation (2-60) into Equations (2-58) and (2-59) and applying the divergence theorem, the smoothed strains [27] and smoothed electric fields become

 defined by

( ) ( ) ( ) ( ) ( ) ( )

( ) ( )

0

k x

2.3.2 Smoothed stiffness matrices for piezoelectricity problems

We now introduce two simple ways to compute smoothed matrices in the ES-FEM By substituting Equation (2-48) into Equations (2-60)-(2-62), the smoothed strain and the smoothed electric field on the domain ( )k

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( )

( )

( ) ( )

( ) 1

( ) 1



where Buj,B j are the constant strain gradient matrices of the jth element around the

edge k when the triangular elements with the linear shape functions are used Note

that the matrices in Eq.(2-68) are directly constructed from the area and the usual

“compatible” strain matrices of the standard FEM using triangular elements However, these formulations are only suitable for approximations with the constant compatible strain matrices such as three–node triangular elements for 2D problems and four-node tetrahedral elements for 3D problems Therefore, to obtain

a general way that can be work well for high order elements such as four-node quadrilateral or six–node triangular elements, the smoothed strains and smoothed electric fields now should be computed along boundary of smoothing domains (cf Equations (2-62)&(2-63)) as

1

k

k n

k

k I k

( )1

x

Next, we assume that the boundary ( )k

 of an arbitrary smoothing domain

  , where nb is the total

number of the boundary segments of ( )k

 When a linear compatible displacement field along the boundary ( )k

 is used, one Gaussian point is sufficient for the accurate line integration along each segment of boundary ( )k

b

 of ( )k

 Hence the above equation can be further simplified to its algebraic form

A linear equations system is then obtained

uu u T u