Nonlinear stress-strain relationships (physical nonlinearity) may have a large effect on the structural response of piezoelectric sensors and actuators. The paper addresses the subject concentrating on cylindrical piezoelectric rods that experience vibrations as a result of an alternating electric field applied in the axial direction. The problem considered in the paper is important in connection with design of piezoelectric transducers. It is shown that neglecting the non-linear stress-strain relationship of the piezoelectric material can result in a significant error in the predicted response of the transducer.
Volume Subject Area:
Aerospace
1.
Maugin, G.A., Pouget, J., Drouot, R. and Collet, B., Nonlinear Electromechanical Couplings, Wiley, Chichester, 1992.
2.
Berlington
D.
Krueger
H. H. A.
Domain Processes in Lead Titanate Zirconate and Barium Titanate Ceramics
,” Journal of Applied Physics
, 30
, pp. 1804
–1810
, 1959
.3.
Woolett
R. S.
Leblanc
G. L.
Ferroelectric Nonlinearities in Transducer Ceramics
,” IEEE Trans. on Sonics and Ultrasonics
, Vol. SU-20
, pp. 24
–31
, 1973
.4.
Krueger
H. H. A.
Mechanical Properties of Ceramic Barium Titanate
,” Physics Reviews
, 93
, p. 362
362
, 1954
.5.
Krueger
H. H. A.
Stress Sensitivity of Piezoelectric Ceramics: Part 1. Sensitivity to Compressive Stress Parallel to the Polar Axis
,” Journal of the Acoustical Society of America
, 42
, pp. 636
–645
, 1967
.6.
Krueger
H. H. A.
Stress Sensitivity of Piezoelectric Ceramics: Part 2. Heat Treatment
,” Journal of the Acoustical Society of America
, 43
, pp. 576
–582
, 1968
.7.
Krueger
H. H. A.
Stress Sensitivity of Piezoelectric Ceramics: Part 3. Sensitivity to Compressive Stress Perpendicular to the Polar Axis
,” Journal of the Acoustical Society of America
, 43
, pp. 583
–591
, 1968
.8.
Brown
R. F.
McMahon
G. W.
Material Constants of Ferroelectric Ceramics at High Pressure
,” Canadian Journal of Physics
, 40
, pp. 672
–674
, 1962
.9.
Brown
R. F.
McMahon
G. W.
Properties of Transducer Ceramics under Maintained Planar Stress
,” Journal of the Acoustical Society of America
, 38
, pp. 570
–575
, 1965
.10.
Fritz
I. J.
Uniaxial Stress Effects in a 95/5 Lead Zirconate Titanate Ceramic
,” Journal of Applied Physics
, 49
, pp. 4922
–4928
, 1978
.11.
Tiersten
H. F.
On the Nonlinear Equations in Thermoelectroelasticity
,” International Journal of Engineering Science
, 9
, pp. 587
–604
, 1971
.12.
Tiersten, H.F., “Equations for the Extension and Flexure of Relatively Thin Electroelastic Plates Undergoing Large Electric Fields,” Mechanics of Electromagnetic Materials and Structures, AMD-Vol. 161/MD-Vol. 42, ASME, New York, pp. 21–34, 1993.
13.
Tiersten
H. F.
Electroelastic Equations for Electroded Thin Plates Subject to Large Driving Voltages
,” Journal of Applied Physics
, 74
, pp. 3389
–3393
, 1993
.14.
Nakagawa
Y.
Yamanouchi
K.
Shibayama
K.
Third-Order Elastic Constants of Lithium Niobate
,” Journal of Applied Physics
, 44
, pp. 3969
–3974
, 1973
.15.
Cho
Y.
Yamanouchi
R.
Non-Linear Elastic, Piezoelectric, Electrostrictive and Dielectric Constants of Lithium Niobate
,” Journal of Applied Physics
, 61
, pp. 875
–887
, 1987
.16.
Chen
P. J.
Montgomery
S. T.
A Macroscopic Theory for Existence of the Hysteresis and Butterfly Loops in Ferroelectricty
,” Ferroelectrics
, 23
, pp. 199
–208
, 1980
.17.
Chen
P. J.
Madsen
M. M.
One Dimensional Polar Responses of the Electrooptic Ceramic PLZT 7/65/35 Due to Domain Switching
,” Acta Mechanica
, 41
, pp. 255
–264
, 1981
.18.
Bassiouny
E.
Ghaler
A. F.
Maugin
G. A.
Thermodynamical Formulation for Coupled Electromechanical Hysteresis Effects - I. Basic Equations
,” International Journal of Engineering Science
, 26
, pp. 1279
–1295
, 1988
.19.
Leigh, T. and Zimmermann, D., “An Implicit Method for the Non-linear Modeling and Simulation of Piezoceramic Actuators Displaying Hysteresis,” Smart Materials and Structures, AD-Vol. 24/ AMD-Vol. 123, ASME, ASME Press, New York, pp. 57–63, 1991.
20.
Ge
P.
Jovuaneh
M.
Modeling Hysteresis in Piezoceramic Actuators
,” Precision Engineering
, 17
, pp. 211
–221
, 1995
.21.
Beige
H.
Schmidt
G.
Electromechanical Resonances for Investigating Linear and Non-linear Properties of Dielectrics
,” Ferroelectrics
, 41
, pp. 39
–39
, 1982
.22.
Beige
H.
Elastic and Dielectric Non-linearities of Piezoelectric Ceramics
,” Ferroelectrics
, 51
, pp. 113
–119
, 1983
.23.
Von Wagner
U.
Hagedorn
P.
Piezo-Beam-Systems Subjected to Weak Electric Field: Experiments and Modeling of Non-linearities
,” Journal of Sound and Vibration
, 256
, pp. 861
–872
, 2002
.24.
Von Wagner
U.
Nonlinear Longitudinal Vibrations of Piezoceramics Excited by Weak Electric Fields
,” International Journal of Non-Linear Mechanics
, 38
, pp. 565
–574
, 2003
.25.
Von Wagner
U.
Non-linear Longitudinal Vibrations of Non-Slender Piezoelectric Rods
,” Journal of Non-Linear Mechanics
, 39
, pp. 673
–688
, 2004
.26.
Von Wagner
U.
Hagedorn
P.
Nonlinear Effects of Piezoceramics Excited by Weak Electric Fields
,” Journal of Nonlinear Dynamics
, 31
, pp. 133
–149
, 2003
.27.
Chattopadhyay
A.
Gu
H.
Liu
Q.
Modeling of Smart Composite Box Beams with Nonlinear Induced Strain
,” Composites
: Part B, 30
, pp. 603
–612
, 1999
.28.
Thornburgh
R. P.
Chattopadhyay
A.
Nonlinear Actuation of Smart Composites Using a Coupled Piezoelectric-mechanical Model
,” Smart Materials and Structures
, 10
, pp. 743
–749
, 2001
.29.
Sherrit, S., Stimpson, R.B., Wiederick, H.D. and Mukherjee, B.K., “Stress and Temperature Dependence of the Direct Piezoelectric Charge Coefficient in Lead Zirconate Titanate Ceramics,” Smart Materials, Structures and MEMS, Eds. Aarte, V.K., Varadan, V.K., and Varadan, V.V., Proceedings of SPIE 3321, pp. 104–113, 1996.
30.
Wiederick
H. S.
Sherrit
S.
Stimpson
R. B.
Mukherjee
B. K.
An Optical Lever Measurement of the Piezoelectric Charge Coefficient
,” Ferroelectrics
, 186
, pp. 25
–31
1996
.31.
Sherrit, S., Wiedenrick, H.D., Mukherjee, B.K. and Sayer, M., “Field Dependence of the Complex Piezoelectric, Dielectric, and Elastic Constants of Motorola PZT 3203 HD Ceramic,” Smart Materials Technologies, SPIE Proceedings 3040, Eds. Simmons, W.C. et al., SPIE, Bellingham, WA, pp. 99–109, 1997.
32.
Yang, G., Liu, S.-F., Ren, W., and Mukherjee, B.K., “Uniaxial Stress Dependence of the Piezoelectric Properties of Lead Zirconate Titanate Ceramics,” Active Materials: Behavior and Mechanics, SPIE Proceedings 3992, Ed. C.S. Lynch, SPIE, Bellingham, WA, pp. 103–1013, 2000.
33.
Mukherjee, B.K., Ren, W., Yang, G., Liu, S.F. and Masys, A.J., “Nonlinear Properties of Piezoelectric Coefficients,” Active Materials: Behavior and Mechanics, SPIE Proceedings 4333, Ed. C.S. Lynch, SPIE, Bellingham, WA, pp. 41–54, 2001.
34.
Ren, W., Masys, A.J., Yang, G. and Mukherjee, B.K., “The Field and Frequency Dependence of the Strain and Polarization in Piezoelectric and Electrostrictive Ceramics,” Presented at the 3rd Asian Meeting on Ferroelectrics (AMF 3), Hong Kong, December 12–15, 2000.
35.
Priya
S.
Viehland
D.
Carazo
A. V.
Ryu
J.
Uchino
K.
High-power Resonant Measurements of Piezoelectric Materials: Importance of Elastic Nonlinearities
,” Journal of Applied Physics
, 90
, pp. 1469
–1479
, 2001
.36.
Chaplya
P. M.
Carman
G. P.
Compression of Piezoelectric Ceramic at Constant Electric Field: Energy Absorption Through Non-180 Domain-wall Motion
,” Journal of Applied Physics
, 92
, pp. 1504
–1510
, 2002
.37.
Barrett, R., “Design and Manufacturing of Adaptive Composites for Active Flight Control Surfaces,” Presented at the Second International Conference on Composites Engineering, New Orleans, LA, August 21–24, 1995.
38.
Wang, D.P. and Carman, G.P., “Evaluating the Behavior of Piezoelectric Ceramics Subjected to Thermal Fields,” Adaptive Material Systems, Summer Symposium of ASME at Los Angeles, AMD-Vol. 206, MD-Vol. 58, pp. 33–47, 1995.
39.
Bert
C. W.
Birman
V.
Stress Dependency of the Thermoelastic and Piezoelectric Coefficients
,” AIAA Journal
, 37
, pp. 135
–137
, 1999
.40.
Bert
C. W.
Birman
V.
Effects of Stress and Electric Field on the Coefficients of Piezoelectric Materials: One-Dimensional Formulation
,” Mechanics Research Communications
, 25
, pp. 165
–169
, 1998
.41.
Joshi
S. P.
Non-linear Constitutive Relations for Piezoceramic Materials
,” Smart Materials and Structures
, 1
, pp. 80
–83
, 1992
.42.
Wang
Q.-M.
Zhang
Q.
Xu
B.
Liu
R.
Cross
E. L.
Nonlinear Piezoelectric Behavior of Ceramic Bending Mode Actuators Under Strong Electric Fields
,” Journal of Applied Physics
, 86
, pp. 3352
–3361
, 1999
.43.
Achuthan
A.
Keng
A. K.
Ming
W. C.
Shape Control of Coupled Nonlinear Piezoelectric Beams
,” Smart Materials and Structures
, 10
, pp. 914
–924
, 2001
.44.
Sherrit
S.
Catoiu
G.
Mukherjee
K. B.
The Characterization and Modeling of Electrostrictive Ceramics for Transducers
,” Ferroelectrics
, 228
, pp. 167
–196
, 1999
.45.
Williams, R.B., “Nonlinear Mechanical and Actuation Characterization of Piezoceramic Fiber Composites,” PhD thesis, Virginia Polytechnic Institute and State University, Blacksburg, Virginia, March 22, 2004.
46.
Li
L.
Sottos
N. R.
Predictions of Static Displacements in 1–3 Piezocomposites
,” Journal of Intelligent Material Systems and Structures
, 6
, pp. 169
–180
, 1995
.47.
Li
L.
Sottos
N. R.
A Design for Optimizing the Hydrostatic Performance of 1–3 Piezocomposites
.” Ferroelectric Letters
, 21
, 41
–46
, 1996
.48.
Li
L.
Sottos
N. R.
Measurement of Surface Displacements in 1–3 and 1–1–3 Piezocomposites
,” Journal of Applied Physics
, 79
, pp. 1707
–1712
, 1996
.49.
Sigmund
O.
Torquato
S.
Aksay
I. A.
On the Design of 1–3 Piezocomposites Using Topology Optimization
,” Journal of Material Research
, 13
, pp. 1038
–1048
, 1998
.50.
Tan
P.
Tong
L.
A One-dimensional Model for Non-linear Behaviour of Piezoelectric Composite Materials
,” Composite Structures
, 58
, pp. 551
–561
, 2002
.51.
Tan
P.
Tong
L.
Micromechanics Models for Non-linear Behavior of Piezo-Electric Fiber Reinforced Composite Materials
,” International Journal of Solids and Structures
, 38
, pp. 8999
–9032
, 2001
.52.
Simitses, G.J., An Introduction to the Elastic Stability of Structures, Robert Krieger Publishing Company, Malabar, Florida, 1986.
53.
Houbolt, J.C. and Brooks, G.W., “Differential Equations of Motion for Combined Flapwise Bending, Chordwise Bending, and Torsion of Twisted Nonuniform Rotor Blades,” NACA Report 1346, 1958.
54.
Whitney, J.M., Structural Analysis of Laminated Anisotropic Plates, Technomic, Lancaster, 1987.
55.
Bolotin, V.V., The Dynamic Stability of Elastic Systems, Holden-Day, San Francisco, 1964.
56.
Hsu
C. S.
Cheng
W.-H.
Steady State Response of a Dynamical System Under Combined Parametric and Forcing Excitations
,” Journal of Applied Mechanics
, 41
, pp. 371
–378
, 1974
.57.
Nguyen
D. V.
Interaction Between Parametric and Forced Oscillations in Multidimensional Systems
,” Journal of Technical Physics
, 16
, pp. 213
–225
, 1975
.58.
Trogerand
H.
Hsu
C. S.
Response of a Nonlinear System Under Combined Parametric and Forcing Excitation
,” ASME Journal of Applied Mechanics
, 44
, pp. 179
–181
, 1977
.59.
HaQuang
N.
Mook
D. T.
Plaut
R. H.
Non-linear Structural Vibrations Under Combined Parametric and External Excitations
,” Journal of Sound and Vibration
, 118
, pp. 291
–306
, 1987
.60.
HaQuang
N.
Mook
D. T.
Plaut
R. H.
A Non-linear Analysis of the Interactions Between Parametric and External Excitations
,” Journal of Sound and Vibration
, 118
, pp. 425
–439
, 1987
.61.
Plaut
R. H.
Gentry
J. J.
Mook
D. T.
Non-linear Structural Vibrations Under Combined Multi-frequency Parametric and External Excitations
,” Journal of Sound and Vibration
, 140
, pp. 381
–390
, 1990
.62.
Nguyen
P. H.
Ginsberg
J. H.
Vibration Control Using Parametric Excitation
,” ASME Journal of Vibration and Acoustics
, 123
, pp. 359
–364
, 2001
.
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