8/12/2019 aksjddnen

1/15

Bishakh Bhattacharya and Nachiketa Tiwari

Indian Institute of Technology, Kanpur

8/12/2019 aksjddnen

2/15

LECTURE 8

Modelling of Piezoelectric Material

8/12/2019 aksjddnen

3/15

Piezoelectric Property

Constitutive Relationship

Active Strain Evaluation

8/12/2019 aksjddnen

4/15

Input

Electric Conductivity Electromagnetic Reverse Ohmic Electro-Optic

e erm v y ec ezoe ec r c

Effect

SA

es s ance e ec

Field Permeability

Magnetostriction

SA

-

Effect

-

effect

Stress Direct Piezoelectr ic

Effect

SS

Villary Effect

SS

Elastic Modulus

Thermo-

Mechanical

Effect SS

Photo-elastic

Effect

SS

Heat Pyroelectric Effect Thermo-

magnetization

Thermal

Expansion/Phase Specific Heat

Thermo-

luminescence

SA

Light Photo-voltaic Effect Photo- Photostriction Photo-thermal Refractive

magnetization SA effect Index

8/12/2019 aksjddnen

5/15

Properties

important

for

Piezoelectric Material Magnetostrictive

Material

Phase-transi tion dependent

Material

Actuationezo-

ceramic

er eno - no

free strain

()

, ,

n m crons

Youngs

Modulus

60-70 2-3 48 27.5 M-

phase, 90 A-

0.45 0.82

(GPa) phase

Bandwidth 0.1 Hz-

GHz

0.1 Hz-GHz 0.1 Hz-10KHz 0-10 Hz 100 Hz

8/12/2019 aksjddnen

6/15

Fundamental equations of piezoelectricity

EEeSC

jjiklklii ESeD

where, the subscripts i,j, k, l = 1, 2, 3 denotes tensorial indices.

, ,

Eis the electric field intensity and D is the electric displacement field.

Elastic stiffness matrix is denoted by the symbol C

E

, where

under constant electric field;

e is the piezoelectric stress-charge matrix and the permittivity matrix,

- , .

8/12/2019 aksjddnen

7/15

Different Axes

x, 1

8/12/2019 aksjddnen

8/15

Di Hexagonal Crystal Symmetry

Greenockite Crystal

8/12/2019 aksjddnen

9/15

Tetragonal Wulfenite

crystal symmetry

8/12/2019 aksjddnen

10/15

Outcome of Symmetry

or 6mm symmetry. Following material symmetry conditions could be

applied to the constitutive relationship

klijjiklijkl CCC

kjikij

.jiij

8/12/2019 aksjddnen

11/15

31131211 00000 SeCCC xx

33331313

31131112

00000

00000

S

S

eCCC

eCCC

z

y

z

y

1544

1544

0000000

0000000

S

S

eC

eC

xz

yz

xz

yz

1115

66

1 0000000

00000000ES

eC

D

xyxy

3

2

3333131

215

3

2

00000

0000000

E

E

eee

e

D

D

The electro-mechanical coupling is shown inside the bordered boxes . Axes

1, 2 and 3 used for the electrical system are identical withx, yand z,

corresponding to the mechanical system.

8/12/2019 aksjddnen

12/15

-

Ignoring the normal stress zand the

shear stresses and for lane stressassumption:

310//1 dEES xppx

0/)1(200 ES xy

y

p

pp

xy

y

33332313

Ep is the modulus of elasticity of the piezoelectric material, is thePoissons ratio and dijare the piezoelectric strain-charge constants

8/12/2019 aksjddnen

13/15

If a piezoelectric thin slab is subjected to mechanical load, the total,

structural or elastic strain Ss and the piezoelectric strain Sa such that

s a

= - - T, a , , .

To generate strains along the direction of the thickness of the specimen,

ceramics with different cr stal-cuts are used which are commonlknown as Piezo-stacks. The electro elastic coupling components in the

3-3 directions, like d33 or e33, become important in such cases.

8/12/2019 aksjddnen

14/15

Gauenzi, P., Smart Structures, Wiley,

2009 Cady, W. G., Piezoelectricity, Dover

, Crawley, E. F., Intelligent Structures for

Aerospace: a technology overview and

-1699

8/12/2019 aksjddnen

15/15

END OF LECTURE 8