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.

1.
Maugin, G.A., Pouget, J., Drouot, R. and Collet, B., Nonlinear Electromechanical Couplings, Wiley, Chichester, 1992.
2.
Berlington
D.
and
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.
and
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.
and
McMahon
G. W.
, “
Material Constants of Ferroelectric Ceramics at High Pressure
,”
Canadian Journal of Physics
,
40
, pp.
672
674
,
1962
.
9.
Brown
R. F.
and
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.
and
Shibayama
K.
, “
Third-Order Elastic Constants of Lithium Niobate
,”
Journal of Applied Physics
,
44
, pp.
3969
3974
,
1973
.
15.
Cho
Y.
and
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.
and
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.
and
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.
and
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.
and
Jovuaneh
M.
, “
Modeling Hysteresis in Piezoceramic Actuators
,”
Precision Engineering
,
17
, pp.
211
221
,
1995
.
21.
Beige
H.
and
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.
and
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.
and
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.
and
Liu
Q.
, “
Modeling of Smart Composite Box Beams with Nonlinear Induced Strain
,”
Composites
: Part B,
30
, pp.
603
612
,
1999
.
28.
Thornburgh
R. P.
and
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.
and
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.
and
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.
and
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.
and
Birman
V.
, “
Stress Dependency of the Thermoelastic and Piezoelectric Coefficients
,”
AIAA Journal
,
37
, pp.
135
137
,
1999
.
40.
Bert
C. W.
and
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.
and
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.
and
Ming
W. C.
, “
Shape Control of Coupled Nonlinear Piezoelectric Beams
,”
Smart Materials and Structures
,
10
, pp.
914
924
,
2001
.
44.
Sherrit
S.
,
Catoiu
G.
and
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.
and
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.
and
Sottos
N. R.
, “
A Design for Optimizing the Hydrostatic Performance of 1–3 Piezocomposites
.”
Ferroelectric Letters
,
21
,
41
46
,
1996
.
48.
Li
L.
and
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.
and
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.
and
Tong
L.
, “
A One-dimensional Model for Non-linear Behaviour of Piezoelectric Composite Materials
,”
Composite Structures
,
58
, pp.
551
561
,
2002
.
51.
Tan
P.
and
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.
and
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.
and
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.
and
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.
and
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.
and
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.
and
Ginsberg
J. H.
, “
Vibration Control Using Parametric Excitation
,”
ASME Journal of Vibration and Acoustics
,
123
, pp.
359
364
,
2001
.
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