Development of a Two- Dimensional Piezo Finite Element in an ANSYS Environment

7/28/2019 Development of a Two- Dimensional Piezo Finite Element in an ANSYS Environment

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DLR

Institut fr Strukturmechanik

Braunschweig

IB 131-2002/38

Development of a Two-Dimensional Piezo Finite Element

in an ANSYS Environment

Partha Bhattacharya, Michael Rose


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DEUTSCHES ZENTRUM FR

LUFT- UND RAUMFAHRT e.V. (DLR)

INSTITUT FR STRUKTURMECHANIK

Braunschweig, September 2002 Der Bericht umfat:

32 seiten4 Tabelle und

8 Bilder

Institutsdirektor: Verfasser:

Prof. Dr.-Ing. E. Breitbach Dr. Partha Bhattacharya, Dr. Michael Rose

Leiter der Organisationseinheit:

Dr.-Ing. H.P. Monner

IB 131-2002/38

Development of a Two-Dimensional Piezo Finite Element in

an ANSYS Environment


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Development of a two-dimensional piezo finite element in an ANSYS

environment

1. Introduction

In the last two decades there has been a flurry of activities in the area of application of

piezoelectric materials for the control of shape and structural vibration. Various theories

have been developed and a lot of efforts have been undertaken to demonstrate the

advantage of using these materials for the control application. In the course of

development of the theoretical aspects, usage of the computational mechanics and along

with it the finite element techniques is all but natural. In the present work an attempt has

been made to develop a FE code for the piezoelectric analysis in a general-purpose

software (e.g. ANSYS) which can then be implemented for modelling complex structures

with integrated piezo layers.

Before going into the review of some of the works carried by the previous

researchers, it is proper to explain a little into the behaviour of piezoelectric materials.

Piezoelectric effect is the two-way effect between stress/strain and electric field/ voltage

difference in materials without central symmetry. The anisotropy of a crystal structure

enables it to retain its polarisation in the absence of an external field. These materials if

integrated with structures as distributed sensors and actuators are found to be very

effective in controlling the flexible structures without much increase in weight and

spillover effects. In a piezoelectric material coupling of elastic and electric fields is

manifested through direct and converse piezoelectric effects. When a piezoelectric bodyis bonded or embedded in a structure (beam/plate/shell), it undergoes deformation along

with the deforming structure and a charge/electric potential is induced in it by virtue of

direct piezoelectric effect. The distributed measurement of this induced charge/potential

gives a distributed measure of the deformation of the flexible structures. On the other

hand if a charge or voltage is applied to a piezoelectric body attached to a flexible

structure, it undergoes deformation and in turn generates distributed forces (moments) on


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the parent structure. Hence both sensing and actuation of a structure are carried out using

piezoelectric materials.

The first reported piezoelectric effect of crystalline structure was made in 1880 by

Pierre Curie and his brother. Voigt in 1915 presented the fundamental equations on the

behaviour of piezoelectric materials. The discovery of PVDF in 1969 by Kawai was one

of the milestone achievement in the history of piezoelectric materials and its usage in the

electro-structural application.

Mindlin (1952, 1962 and 1972) presented a series of works deducing two-dimensional equations from the three-dimensional piezoelectric equations and thereafter

employing the variational principle to analyze low frequency vibration of anisotropic

plates. Tiersten (1969) derived the constitutive equations of the piezoelectric from energy

considerations. He also developed the three-dimensional linear differential equations, its

appropriate boundary conditions and presented the solution of pertinent three-

dimensional standing wave problem. Based on that he developed approximation

techniques to model the dynamic behaviour of piezoelectric plates.

One of the earliest works in using finite element technique that included piezoelectric

effects was done by Allik and Hughes (1970) for piezo-ceramic transducer design. They

proposed a tetrahedral (3-D) unit as the basic element for their FE model. Various

researchers have already worked in this area and a huge variety of Finite Elements are

already being theoretically developed and tested upon some simple geometry. These

elements being user specific, has got its limitation to model complicated structures or to

combine with other effects as well. Taking these backgrounds into consideration an

attempt has been made in this present work to develop an isoparametric 8-noded, 2D-

plate element in an ANSYS environment using the special USER feature provided by the

ANSYS. The element developed has got six mechanical degrees of freedom (five for 2D

case) and a single electrical degree of freedom. Although there exists a brick (3D)

element with piezoelectric features but it is limited to static cases only. In the presently

developed piezo finite element (USER102) the dynamic effects are also included. In the


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next section the detailed deduction of the governing equation is shown. The developed

matrices are then coded using FORTRAN. The developed element is then tested for its

convergence criteria. In the next step, results are obtained for the static and free vibration

cases considering only the mechanical degrees of freedom and they seem also to work

very good. Results are then obtained for piezoelectrically activated structures and are

compared with both theoretical and experimental results. Results are also obtained for

cylindrical piezo actuator and the results seem to follow the predicted physical behaviour.

The performance of the developed element is sufficiently good and the developed

element seems to fit well with the existing elements in the ANSYS. The results obtained

so far seem to be quite encouraging and in the future the developed element can be usedfor different structural geometry as well.

2. Constitutive Equations

2.1 Introduction

The analysis of an engineering system usually begins with the isolation and identification

of an idealised model of the system. The next step is to give a precise mathematical

statement to the static or dynamic behaviour of the model. This is done by applying

appropriate governing principles to the model to formulate differential equations of

motion. For mechanical systems the governing requirements can be divided into three

categories:

1. Geometric requirements, including kinematic relations.

2. Equilibrium requirements on the forces for static analysis and dynamic-force

requirements, including relations between forces and rates of change of momentum

for dynamic analysis.

3. Constitutive relation for forces in deformable elements and velocity-momentum

relations for inertial elements.

The main emphasis of this section is to develop the field equations governing a

deformable body, the interaction of the electrical field with the displacement (stress) field

and subsequently developing the material model with a focus towards achieving the

objective.


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The mechanical systems considered for the present analysis are laminated beams and

plates made from fibre reinforced plastic with piezoelectric patches/layers (active layers)

bonded to or embedded in them. The system is subjected to contact forces (e.g., surface

traction) and body forces (e.g., force exerted by the electrical field developed due to the

presence of the active layers).

The composite laminate is assumed to consist of n number of laminae, which

includes one or many active layers. The other layers are fibre-reinforced laminae, in each

one of them, the fibres may have arbitrary orientations, and the laminae may have

different thicknesses.

The bonded or embedded piezoelectric actuators/sensors are considered as an integral

part of the structure. Perfect bonding is assumed between the layers themselves and

between the piezo layers and the substrate. The lay-up details are shown in Figure 2.1.

One set of piezo layers may act as distributed actuators, whereas the other set, as

sensors. It is assumed that a piezo layer is much thinner compared to the thickness of the

structure substrate, and it is either distributed over the full structure or placed in patches.

A distributed sensor generates a voltage output when the structure is subjected to some

external disturbances due to the direct piezoelectric effect. In the actuator layer external

voltage is applied across the layer thickness and it is strained due to converse

piezoelectric effect which in turn induces stress and strain in the structure proper onto

which it is bonded.

2.2 Laminated Composite: Macro-Mechanical Behaviour

As has been discussed, the structural forms under consideration consist of two

distinctly different materials exhibiting two different behaviours. The composite layers

contribute to the stiffness characteristic and the piezo layer act as active elements

responding to the external excitation owing to their piezoelectric characteristics.

However, it is assumed that the active elements may also contribute to the stiffness

characteristic of the structure. In the following subsections, the constitutive relations for


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( )2.2

Q00000

0Q0000

00Q000

000QQQ

000QQQ

000QQQ

12

13

23

33

22

11

66

55

44

333231

232221

131211

12

13

23

33

22

11

=

( )2LTLTTTTT

L11EE21

1EQ

=

both the composite and piezo layers are discussed in details and an attempt is made to

bring the coupling characteristic to the fore.

The heterogeneity in a composite material is introduced due to not only the bi-phase

or in some cases, the multi-phase composition, but also laminations. This leads to

distinctly different stress-strain behaviour in the case of laminates. The anisotropy caused

due to fibre orientations and the resulting extension-shear as well as the bending-twisting

coupling and the extension-bending coupling developed due to an unsymmetric

lamination add to the complexities.

2.3 Lamina Constitutive Equations

Generally speaking, the elastic moduli Qij relating the cartesian components of stress

and strain depends on the orientation of the co-ordinate system. When the elastic moduli

Qij at a point remain invariant for every pair of co-ordinate systems that are mirror images

of each other in a plane, the plane is called the plane of elastic symmetry for the material

at that point. For a lamina, there exists three orthogonal planes of elastic symmetry and

for such a case the on-axis (orthotropic) stress-strain relationship for a unidirectional

composite in a three dimensional elastic domain can be written as

{ } (2.1)Q jiji =i.e.,

where [ ] [ ]T121323332211T

654321 =

and [ ] [ ]T121323332211T

654321 =

with


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( )2LTLTTTLT

T1312EE21

EQQ

==

( ) )1(EE21EE1

EQQTT

2LTLTTT

2LTLT

T3322+

==

Q44 = G23 ; Q55 = G31; Q66 = G12

where, EL = E1; ET = E2 =E3

and,LT =13 =12;TT =23

In the present case, a general laminate is made up of many such orthotropic layers

with arbitrarily oriented fibres. It is now of interest to relate the elastic moduli in one co-

ordinate system to those in another co-ordinate system. The elastic moduli Q ij of an

orthotropic material (with the material symmetry axes coinciding with the x1x2-axes) are

related to the elastic modulii ijQ in the xy-global co-ordinate system by the relations

(figure 2.1)

Figure 2.1. Laminated plate with arbitrarily located Piezo Laminae

[ ] [ ] [ ][ ] (2.3)TQTQ 3T

3=

Surface Bonded Piezo

h/2

z -1 z

h/2

x

z

Embedded Piezo

x1x2

x, u

y, v

z, w

a

b

( )( )TT2LTLTTT

2LTLTTT

T231EE21

EEEQ

++

=


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where, [T3] is the transformation matrix that allows one to transform the elastic constants

from material symmetry axes to a global (or problem) co-ordinate axis matrix is given by

[ ] (2.4)

)nm(000mn2mn2

0mn000

0nm000

000100

mn000mn

mn000nm

T

22

22

22

3

=

with m = cos and n = sin

The off-axis stiffness matrix [ ]Q for a lamina is defined in the following stress-strain

relationship:

where, 42222

66

22

12

4

1111 nQnmQ4nmQ2mQQ +++=

)nm(QnmQ4nm)QQ(Q 441222

66

22

221112 +++=

2

23

2

1313 nQmQQ +=

3

662212

3

66211116 mn)Q2QQ(nm)Q2QQ(Q ++=

4

22

22

66

22

12

4

1122 mQnmQ4nmQ2nQQ +++=

223

21323 mQnQQ +=

nm)Q2QQ(mn)Q2QQ(Q 36622213

66121126 ++=

mn)QQ(Q;QQ 3231363333 ==

mQnQQ;nQmQQ 2552

4455

2

55

2

4444 +=+=

mn)QQ(Q 445545 =

66

22222

22121166 Q)nm(nm)QQ2Q(Q ++=

( )2.5

Q00QQQ

0QQ000

0QQ000

Q00QQQ

Q00QQQ

Q00QQQ

xy

xz

yz

zz

yy

xx

66636261

5554

4544

36333231

26232221

16131211

xy

xz

yz

zz

yy

xx

=


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2.4 Laminate Behaviour with Displacement Model

In the present study two different types of displacement models are considered and

compared. The details of the models are presented in the following subsections.

Flat Plate Laminates

Let us consider a flat laminate of thickness h consisting of unidirectional laminae

bonded together to act as an integral part (figure 2.1). The bonds are infinitesimal and are

not shear deformable.

The assumptions made for two different models are stated as follows

CASE 1. zz = 0

(i) The material behaviour is linear and elastic.

(ii) The thickness of the laminate is small compared to other dimensions.

(iii) Displacement u, v, w are small compared to the laminate thickness.

(iv) Normal to the mid-surface before deformation remains straight but is not

necessarily normal to the mid-surface after deformation.

(v) Constant normal strain is present.

Employing a first order shear deformation theory the displacement u, v, w at any

point on the plate can be expressed as,

u = u0(x, y) + z y (x, y) (2.6a)

v = v0(x, y) - z x (x, y) (2.6b)

w = w0(x, y) + z z (x, y) (2.6c)

where, u0, v

0and w

0are the mid-surface displacements and x, y and z are the shear

rotations.


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2.5 Strain-Displacement Relations

The strains at any point of the laminate are given by,

xx0xxxx z+=

yy0yyyy z=

(2.7)0zzzz =

xz

0

xzxz z+=

yz

0

yzyz z+=

xy

0

xyxy z+=

The curvatures are expressed as,

x

y

xx

= ;

y

xyy

= ;

x

zxz

= ;

y

zyz

= ;

xy

xy

xy

+

=

and the mid-plane strains are expressed in terms of the mid-plane displacements as,

x

uo0xx

= ;

y

vo0yy

= ; z

0

zz=

x

w0

x

0

xz

+= ;y

w0

y

0

yz

+= ;x

v

y

u 0o0xy

+

=

2.6 Stress-Strain Relations

The stress-strain relation with respect to the global axes, xyz-system can be

expressed as

[ ] (2.8)0

zQ

xy

xz

yz

yy

xx

0

xy

0

xz

0

yz

0

zz

0

yy

0

xx

xy

xz

yz

zz

yy

xx

+

=

The stress resultants are given by


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12

=

2/h

2/h

xy

xz

yz

zz

yy

xx

xy

xz

yz

zz

yy

xx

(2.9)dz

N

S

S

N

N

N

and are computed as,

[ ] dz0

zQQQQdzN

zz

yy

xx

o

xy

0

zz

0

yy

0

xx

2/h

2/h

2/h

2/h

16131211xxxx

+

==

[ ] [ ]

+

=

xy

yy

xx

161211

0

xy

0

zz

0

yy

0

xx

16131211 BBBAAAA

and so on.

Similarly, the moment resultants are expressed as,

)10.2(zdz

M

R

R

M

M

2/h

2/h

xy

xz

yz

yy

xx

xy

xz

yz

yy

xx

+

=

and are computed as,

[ ] +

+

+

==2/h

2/h

h/2

h/2-

xy

yy

xx

2

0

xy

0

zz

0

yy

0

xx

16131211xxxx dz0zzQQQQzdzM


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[ ] [ ]

+

=xy

yy

xx

161211

0

xy

0zz

0

yy

0

xx

16131211 DDDBBBB

and so on.

From equations (2.9) and (2.10) the stress and moment resultants can be written as,

[ ] (2.11)D

M

R

R

M

M

N

S

S

N

N

N

xy

xz

yz

yy

xx

0

xy

0

xz

0

yz

0

zz

0

yy

0

xx

xy

xz

yz

yy

xx

xy

xz

yz

zz

yy

xx

=

where,

[ ] (2.12)

D00DDB00BBB

DD000BB000

D000BB000

DDB00BBB

DB00BBBA00AAA

AA000

A000

symAAA

AA

A

D

66626166636261

55545554

444544

222126232221

1116131211

66636261

5554

44

333231

2221

11

=


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( )2.13

Q0000

0Q000

00Q00

000QQ

000QQ

12

13

23

22

11

66

55

44

2221

1211

12

13

23

22

11

=

TLLT

L11

1

EQ

=

TLLT

TLT12

1

EQ

=

TLLT

T22

1EQ

=

CASE 2: zz = 0

In this particular case the stress strain relationship given in equation (2.2) is expressed

differently because of the assumption that the stress in the normal direction equals tozero.

Where,

Q44 = G23 ; Q55 = G31; Q66 = G12

The transformation of these coefficients from on-axis to the off axis is carried out using

the same procedure as described for the case 1.

The assumed displacement model for this case is expressed as

)t,y,x(w)t,z,y,x(w

(2.14))t,y,x(z)t,y,x(v)t,z,y,x(v

)t,y,x(z)t,y,x(u)t,z,y,x(u

0

x0

y0

==

+=


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The stress resultant strain relationship is expressed as follows

2.7 Piezoelectric constitutive equations

As has been discussed earlier, a piezoelectric material shows both direct and converse

effect and depending on the usage one can exploit the behaviour of the piezoelectric

material. The constitutive relationship that relates the piezoelectric, dielectric and the

structural properties are given as follows:

{ } [ ]{ } [ ] { } Effect)(ConverseEeQ T= (2.16a)

{ } [ ]{ } [ ]{ } Effect)(DirectEeD += (2.16b)

{ }{ }

[ ] [ ][ ] [ ]

{ }{ }

{ } [ ]{ } (2.15b)GQ

(2.15a)DB

BA

M

N 0

=

=

[ ]

=

662616

262212

161211

AAA

AAA

AAA

A

[ ]

=

662616

262212

161211

BBB

BBB

BBB

B

[ ]

=

662616

262212

161211

DDD

DDD

DDD

D

[ ]

=

5545

4544

GG

GGG


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3. Modelling the potential distribution through the piezoelectric layer

The following assumptions are made when modelling the potential distribution through

the thickness of the piezoelectric layer:

(a) The distribution of the potential across the piezoelectric thickness is linear.

(b) The surfaces of the piezoelectric layer in contact with the substrate is suitably

grounded such that the potential at the interface is zero.

(c) There is a perfect bond between the piezo layer and the elastic substrate.

Under such an assumption the applied voltage across the layer can be expressed as

)y,x(hh

hz

)z,y,x(

a

01nn

1na

=

(3.1)

The electrical field vector is defined as

,

,

,

E

E

E

z

y

x

z

y

x

=

(3.2)

At this point it is very important to mention that it is assumed that an electric field vectorperpendicular to the layers is assumed (Lammering)

Ex = Ey = 0

Therefore

4. Energy Formulation and FE Modelling

In this section the energy formulation of the piezoelectrically activated structure is

presented and the governing finite element equations are derived.

The total energy in the system can be contributed due to the potential and the kineticenergy. The potential energy is a combination due to mechanical strain and electrical

strain energy. The mechanical strain energy is expressed as,

(3.3))y,x(hh

1E 0

1nn

z =


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The electrical strain energy

{ }{ }=V

E dVDE2

1U (4.2)

Now replacing equation (2.11) in equation (4.1) we obtain the energy terms leading to the

mechanical potential energy

Potential Energy due to mechanical part alone

Kinetic Energy (Note: The deduction is shown for the 2D constitutive matrix case. For

the 3D case the development is similar)

with I1, I2, I3 =

2/h

2/h

2 dz)z,z,1)(z(

4.1 Finite Element Formulation

In the present work an 8-noded Isoparametric 2-dimensional plate element is developedand the displacement degrees of freedom as well as the actuator voltage is expressed

using the same shape functions. The details is given in the subsequent formulation,

{ }{ }[ ] [ ]

[ ] [ ] { }{ } { } [ ]{ } (4.3)dAGDBBA

21U T

0

A

T0

M +

=

(4.1)dV

2

1U

n

1k V

kkM

=

=

(4.4)dAw

v

u

I000I

0I0I0

00I00

0I0I0

I000I

w

v

u

2

1T

y

x

0

0

0

A

32

32

1

21

21

T

y

x

0

0

0

=

&

&

&

&

&

&

&

&

&

&

======

======8

1i

ziiz

8

1i

yiiy

8

1i

xiix

8

1i

ii

8

1i

ii

8

1i

ii N;N;N;wNw;vNv;uNu

(4.5)N8

1i

ii=

=


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( )( ) 5,7ifor112

1N i

2

i =+=

( )( )( ) 4to1ifor1114

1N iiiii =+++=

( )( ) 6,8ifor112

1N i

2

i =+=

Where

Now combining equations (2.6a) to (2.6c) and (4.5) we can express the generalised

strains as

Combining equations (2.16a,b), (3.3) and (4.2) we finally obtain the energy component

due to coupled field as follows

{ } [ ] [ ] [ ][ ][ ]{ } =V

aeop

ap

TTTe1E dVBZeZBd

2

1U (4.6)

{ } [ ] [ ] [ ] [ ][ ]{ }dVdBZeZB2

1U

V

eTTa

pT

p

Tae02E = (4.7)

and the energy component due to dielectric effect as given below

{ } [ ] [ ] [ ][ ][ ]{ } dVBZZB2

1U ae0p

ap

V

Tap

Tp

Tae03E = (4.8)

Similar procedure is carried out for the mechanical strain energy too and the finalexpression and now applying the Lagrangian on the total energy we obtain the final

governing finite element equations as follows

Here the mechanical Stiffness matrix [K] is given by

[ ]

=

u

B

[ ]{ } [ ]{ } { } { }

[ ]{ } [ ]{ } { } (4.10)QKuK(4.9)FKuKuM

aau

mau

aaa

a

=+

+=+

&&


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Similarly the coupling matrix is expressed as

[ ] [ ] [ ] [ ] [ ][ ]=V

PaP

TTTeu dVBZeZBK a (4.12)

and [ ] [ ]Teue u aa KK = and the stiffness due to electrical part alone is given by

[ ] [ ] [ ] [ ][ ][ ] =V

PaP

TaP

TaP

edVBZZBK

aa(4.13)

Finally the mass matrix is given by

Equation (4.9) and (4.10) is then modeled as a single matrix equation

The final equation is then modeled in the ANSYS USER102 routine (details in

Appendix) and results are obtained.

5. Results and Discussion

The verification of the developed element is carried out by comparing with both

theoretical and experimental results. The numerical results for different cases are

compared for both with piezoelectric layers and without piezoelectric layers. The results

for the cases without piezoelectric layers are compared with those obtained using

SHELL99, a standard ANSYS finite element. In this section the results are presented in

two sub-sections namely (a) Without Piezo Layer and (b) With Piezo Layer.

5.1 Without Piezo Layer

Convergence study

As a standard test for every developed element, a convergence test is carried out for the

developed element (SHELL102). As it has been pointed out earlier that the SHELL102

element has been developed using two different constitutive relationships, the results for

both the cases has been presented.

[ ] [ ][ ] [ ]

[ ] [ ]

[ ] [ ] [ ][ ] (4.11)dABGBB

DB

BABK S

TSB

T

A

B +

=

[ ] [ ] [ ][ ] (4.14)dVNNMV

T =

(4.15)Q

Fu

KK

KK

0

u

00

0M

u

u

=

+

&&


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Example 1:

Model Data

Length = Breadth = 1mThickness = 0.009 m ( 3 layers @ 0.003 m); Fiber Orientation = 0/90/0

All sides Clamped ( u = v = w = rotx = roty = rotz = 0)

Material Properties

E1 = 140 Gpa; E2 = E3 = 10 Gpa

12 = 0.3 =13 =23G12 = 7 Gpa; G23 = G13 = 6 Gpa

Table 1. Midpoint displacement ( x 10-5

m) due to a point load of 1 N applied at X = L/2

and Y = B/2

Element Type

Mesh Size

SHELL99 SHELL102

(2D)

SHELL102

(3D)

2 x 2

4 x 4

6 x 6

8 x 8

10 x 10

12 x 12

0.194

0.172

0.177

0.177

0.177

0.177

0.163

0.139

0.169

0.174

0.175

0.176

0.160

0.140

0.166

0.170

0.171

0.171

From the results it can be concluded that the developed SHELL102 element performs

quite well with the SHELL99 element which is an in-house ANSYS product. In the next

step a study is carried out to get an idea how the developed element performs for dynamic

cases compared to the SHELL99 element.

Comparison of the FrequencyIn the following two examples the comparison of the dynamic behavior of laminated

composite plates have been studied. The results are presented below.Example 2:Model Data 1

Structure: Square Plate clamped on all sides

Length = Breadth = 5m; Thickness = 0.03 m

Fiber orientation 0/90/0

Material PropertiesE1 = 140 Gpa; E2 = 15 Gpa

G12 = 6 Gpa; G13 = G23 = 5 Gpa

12 = 0.3


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21

= 1500 Kg/m3SHELL102 (2D) have been used.

Table 2. Comparison of the free vibration frequencies

Frequency Shell99 User102(2D) User102 (3D)

12

3

4

5

6

7

8

910

12.88118.058

28.479

32.755

36.240

44.086

44.107

57.663

62.83265.191

12.98218.270

28.915

33.272

37.217

44.810

46.155

61.530

64.38266.153

13.20218.828

29.991

33.669

37.850

46.559

47.275

63.238

65.05168.760

Example 3:

Model Data 2

Structure: Cantilever BeamL = 200 mm; B = 27 mm; Thickness = 2 mm

16 Layers (0/45/90/-45/0/45/90/-45)s

Material Properties

Similar to example 2.

Table 3 Comparison of the free vibration frequencies

Frequency All - Shell99 Shell102 (2D)

1 (B)

2 (B)

3 (T)

4 (D+T)

5 (B)

44.711

247.78

579.66

660.81

717.73

44.346

276.920

590.89

675.69

773.63

In both the examples it has been observed that the free vibration frequencies for

SHELL102 compares exceedingly well with the frequencies obtained from SHELL99,

which is a proven and tested ANSYS element. With this level of confidence numerical

experiments were undertaken to test the developed element with the results available for

piezo-structure coupled behaviour.


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22

Comparison of piezoelectric displacement

Example 1

As the first example an aluminium beam (Fig 5.1) with piezoelectric layer bonded on thetop is considered. The material properties of the beam is given as follows:

Aluminium: E1 = E2 = E3 = 68.9476 GPa, = 0.25

Adhesive : E1 = E2 = E3 = 6.894 Gpa, = 0.4

Piezoelectric: E1 = E2 = 68.9476 Gpa, E3 = 48.258 Gpa,13 = 0.25e31 = -10.126, e33 = 18.81 C/m

2

11 = 22 = 33 = 0.1153 x 10-7 F/mTo simulate the conditions as reported by Robbins and Reddy [13] and Saravanos [14], a

uniform voltage of 12.5 KV is applied on the piezoelectric layer. The result obtained isillustrated in Fig 5.2

Fig 5.1 Schematic diagram of an Aluminum beam with bonded PZT

The tip deflection as reported by Saravanos equals to 0.355 x 10-3

m and the tip

deflection as obtained by the present FE model in the ANSYS environment also equals to

0.360 x 10-3. The deflected pattern obtained from the ANSYS post-processing file is

illustrated in Fig 2.

15.24 cm

1.524 cm

0.1524 cm

0.0254 cm

Aluminium Beam

Piezoelectric LayerAdhesive


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23

Fig 5.2 Displacement pattern of a cantilever PZT-bonded Aluminum beam subjected to a

uniform voltage of 12.5 KV.

Example 2

In this example a comparison study is done with the experimental results on a laminated

composite beams with piezoelectric patch bonded on to the surface. The details of the

model is given as below

Geometry data

Length = 200 mm; Width = 27 mm16 layers @ 0.125 mm = 2 mm;

Lamination sequence is shown in Fig 5.3 and Fig 5.4


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25

The material properties for the different materials are as follows

Material 1 (Orthotropic)

E1 = 140 GPa, E2 = E3 = 9.784 GPa,12 =13 = 0.276;23 = 0.35; G12 = G13 = 5.310 GPaG12 = 1.313 GPa

Material 2 (Isotropic)

E1 = E2 = E3 = 22.3 GPa,12 =13 =23 = 0.38e31 = e32 = -15.3 C/m

2; e33 = 16.4 C/m

2

11 = 9.31 x 10-9 F/m = 22; 33 = 7.62 x 10-9 F/mMaterial 3 (Isotropic)

E1 = E2 = E3 = 19.02 GPa,12 =13 =23 = 0.25

The location of the piezoelectric patches is shown in Fig 5.5.

Fig 5.5 Schematic layout of the PZT bonded laminated composite beam

A uniform voltage of 100 volts is applied on the top piezo layer (material 2) and the

transverse deflection pattern obtained from ANSYS is illustrated in Fig 5.6. The results

are then compared with the experimental results. The results are shown in Fig 5.7. It has

to be noted that the FE results compare very well with the experimental results in the

lower voltage zone. From the experimental results it can be easily understood that the

piezoelectric behaviour does not show any more linearity and to capture the experimental

behaviour exactly the FE model has to be modified by incorporating a voltage dependent

function for the piezoelectric material property.

50 mm 50 mm 100 mm

27 mmPZT


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26

Fig 5.6 Plot of the transverse displacement (UZ) of the laminated beam subjected to a

uniform voltage of 100 volts applied on the top piezo layer

Fig 5.7 Comparison plot for the transverse displacement of the laminated composite

beam

0

0.1

0.2

0.3

0.4

0.5

0.6

0 50 100 150 200 250

Volts

Displacement(x10-

3m

)

Experimental

FE Results


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27

6. MATLAB Programming

In the present work a MATLAB program is also developed (Ref. IB 131-

2002/30). The binary files obtained from the ANSYS program are analyzed and the mass

and stiffness matrices are read. The resulting matrices are then used to obtain the free

vibration frequencies. The results are tabulated and are shown in table 4. From the results

it is clear that the MATLAB program works very well.

7. Future Plans

In the future some of the works that can be carried out are as follows

To obtain the modal matrices and use them for control application

To do some dynamic experimentation and comparing the results To include the pre-stress effects To include the radius of curvature effect

Acknowledgement: Mr. Johannes Riemenschneider & Mr. Christian Anhalt

8. References

1. Curie, J. and Curie, P., 1880, Piezo effect in Quartz and some other Materials, C. R.

Acad. Sci. Paris, 91,294(1880); 91,383(1880)

2. Voigt, W., 1915, Remarks on Some New Investigations on Pyro and Piezoelectricity

of Tourmaline, Anal. Physik, Vol. 46, pp. 221-230

3. Mindlin, R. D., 1952, Forced thickness-shear and flexural vibrations of piezoelectric

crystal plates, J. appl. Phys., Vol. 23, pp. 83-88

Table 4 Comparison of the frequencies obtained from ANSYS and from the

MATLAB program

Mode Number ANSYS MATLAB1

23

4

42.015

157.46268.23

427.98

42.0146

157.458268.230

437.97


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28

4. Mindlin, R. D., 1962, Forced vibrations of piezoelectric crystal plates, Q. appl.Math., Vol. 22, pp. 107-119

5. Mindlin, R. D., 1972, High Frequency Vibrations of Piezoelectric Crystal Plates,

Int. J. of Solids Structures, Vol. 8, pp. 895-906

6. Tiersten, H. F., 1969, Linear Piezoelectric Plate Vibration, Plenum Press, New York

7. Allik, H. and Hughes, T., 1970, Finite Element Method for Piezoelectric Vibration,J. Int. J. for Num. Methods in Engineering, Vol. 2, pp. 151-168.

8. Nailon, M., Coursant. H. and Besnier, F., 1983, Analysis of Piezoelectric Structures

by Finite Element Method, ACTA Electronica, Vol. 25, No. 4, pp. 341-362

9. Ray, M. C., Bhattacharya, R. and Samanta, B., 1994, Static Analysis of anIntelligent Structure by Finite Element Method, Comput. Struct., Vol. 52, No. 4, pp.

617-631

10. Tzou H. S. and M. Gadre, 1989, Theoretical Analysis of a Multi-layered Thin Shell

Coupled with Piezoelectric Shell Actuators for Distributed Vibration Controls, J.

Sound and Vib, Vol. 132, No. 3, pp. 433-451

11. Bhattacharya, P., H. Suhail and P. K. Sinha, Finite Element Free Vibration Analysisof Smart Laminated Composite Beams and Plates, Journal of Intelligent Material

Systems and Structures, Vol. 9, No. 1, pp. 20 29, January 1998

12. Lammering, R., 1991, The Application of a Finite Shell Element for Composites

Containing Piezo-Electric Polymers in Vibration Control, Computers and Structures,

Vol. 41, No.5, pp: 1101 1109

13. Reddy, J. N. and D. H. Robbins, 1991, Analysis of Piezoelectrically Actuated Bems

using a Layer-wise Displacement Theory, Computers & Structure, Vol. 41, No. 2,

pp-265 279

14. Saravanos, D. A. and P. R. Heyliger, 1995, Coupled Layerwise Analysis of

Composite Beams with Embedded Piezoelectric Sensors and Actuators, J. of

Intelligent Material Systems and Structures, Vol. 6, pp-350 363

15. Rose, Michael, 2002, Dokumentation einer Schnittstelle fur MATLAB zum

Einlesender Modell und Ergebnisdaten aus den binaren Ausgabedateien von

ANSYS, IB 131-2002/30, DLR, Braunschweig


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29

Appendix A

Important Features

USER102 element definition (et,*,102)

1. 8-noded element based on SHELL99.

2. Shear deformation and Rotary Inertia taken into consideration3. 5 degrees of mechanical degrees and one electrical degree at the element level (for the

3-dimensional constitutive matrix, 6 mechanical degrees taken into consideration.)

4. 6 mechanical degrees of freedom after transforming into global level.

5. Piezoelectric inputs should be given through TABLE, PIEZ option

Notes for the User

1. The 6th

degree (ROTZ) must be locked for the 2-dimensional case.

2. The input for the poissons ratio should be minor (NUXY, NUYZ, NUXZ).3. When defining a PIEZO layer, the material number should be always assigned as 2.

4. Even for Isotropic cases, all the material property inputs needs to be supplied.

Description of the programs

The user102 element comprises of the following *.f files:

uel102.f: This program is the heart of the element programs and all the ANSYS input areread in this program and passed onto the subroutines. The element stiffness and the massmatrices are calculated in this program and returned. The bending and the shear stiffness

for the mechanical part are calculated separately and then added up. The piezoelectric

coupling matrix and the dielectric matrix are calculated using 3-point (Gauss) integration.

The mass matrix is also calculated using 3-point integration. The parameters used in the

program is described below:

nl: Layer numbernj: material numberprop(100): Array to read the propertiesex, ey, ez: Modulus of Elasticity

anxy, anxz, anyz: Poissons Ratiothk: Thicknessgxy, gxz, gyz: Shear Modulustk: Fiber Orientationdens: Material Densityrvr: Array to read the real numbers for the layersbdi(48,48): [Kuu] matrix, Mechanical Stiffness (N.B: For 2-dimensional case the size ofthe matrix is 40 x 40)

bdf(48,48): Final form of the mechanical stiffness matrix after transformationftrans(48,48): Transformation matrix


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30

bdcouple(48,8): The piezo-mechanical coupling matrix (N.B: For 2-dimensional case thesize of the matrix is 40 x 8)

dd(7,7): ABD Matrix. Calculated in the dmat.f subroutine and passed on to uel102.f.gg(4,4): G Matrix. Calculated in the dmat.f subroutine and passed on to uel102.f.aplace(3,3), wgt(3,3): Matrices defined for Gauss Integrationen(8): Shape functions, defined in shape1.f routineanx(8), any(8): Derivative of the shape function, defined in shape1.f routinebb(7,48): Strain-displacement matrix defined in bmat.f. (N.B: For 2-dimensional case thesize of the matrix is 7 x 40)

amat(6,6): Inertia Matrixbdelec(8,8): [K] matrixapr(ip): Array for defining the piezoelectric coefficients.

uec102.f: See details in the ANSYS help menu. In this file the degrees of freedom aredefined.

Uex102.f & uep102.f: See details in the ANSYS help menubmat.f: The strain displacement matrix [B] is defined in this file.bpiezo.f: The relation between the mechanical displacement and the electrical voltage atthe nodes is described in this file.

coeftran.f: This file is important for the development of the reduced piezoelectricproperty in case of the 2-dimensional constitutive matrix. The material properties are readand are reduced according to the plane stress theory (Lammering, Computers and

Structures, Vol. 41, No. 5, pp. 1101-1109).

dmat.f: In this program the ABD matrix and the G matrix are calculated.shape1.f: The shape functions and their derivatives are calculated in this subroutinematden.f: The material density matrix amat is calculated in this subroutine.


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31

Appendix B

Transformation of the Element Matrices into Global Matrices

The element matrices as has already been described takes six degrees of mechanical

degrees and one electrical degree of freedom. Therefore there are 56 degrees of elemental

degrees of freedom per element. The voltage being a scalar quantity needs no rotation

and therefore the transformation matrix is a 48 x 48 matrix and the details of the matrix

are described in this section.

In the first step the local coordinates are transformed into the global coordinates using thefollowing transformation

[O] being orthogonal

[O]-1

= [O]T

Similarly the relation between the local and the global degrees of freedom are expressed

as

[ ] { }{ }{ }[ ]321 eeeO =

[ ]

=

l

l

l

g

g

g

z

yx

O

z

yx

{ } { } { }

{ } { } { } e,e;e

w

v

u

ew;

w

v

u

ev;

w

v

u

eu

zg

yg

xgT

3zl

zg

yg

xgT

2yl

zg

yg

xgT

1xl

g

g

gT

3l

g

g

gT

2l

g

g

gT

1l

=

=

=

=

=

=

{ }

{ }{ }

{ }

{ }

{ }

{ }

[ ]{ }g

zg

yg

xg

g

g

g

T3

T2

T1

T3

T2

T

1

zl

yl

xl

l

l

l

l Uew

vu

e

e

0

0

e0

0e

0e0e

w

vu

U =

=

=


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The electrical degree (volt) is a scalar quantity and therefore needs no transformation.Hence the transformation is carried out only on the mechanical degrees of freedom and

the transformation matrix [T] is as follows

The final matrices in the global coordinates before the assembly process can therefore be

written as

And the mass matrix in the global coordinates is expressed as

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

[ ]

=

e

e

e

e

e

e

e

e

T

[ ] [ ] [ ][ ]TKTK LuuT

Guu =

[ ] [ ] [ ] = LuT

Gu KTK

[ ] [ ] [ ][ ]TMTM TG =

Documents

Development of a Two- Dimensional Piezo Finite Element in an ANSYS Environment