Theory of electroelasticity accounting for biasing fields: Retrospect, comparison and perspective

WU Bin; ZHANG Chunli; ZHANG Chuanzeng; CHEN Weiqiu

doi:10.6052/1000-0992-15-020

Volume 46 Issue 1

May 2016

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WU Bin, ZHANG Chunli, ZHANG Chuanzeng, CHEN Weiqiu. Theory of electroelasticity accounting for biasing fields: Retrospect, comparison and perspective[J]. Advances in Mechanics, 2016, 46(1): 201601. doi: 10.6052/1000-0992-15-020

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WU Bin, ZHANG Chunli, ZHANG Chuanzeng, CHEN Weiqiu. Theory of electroelasticity accounting for biasing fields: Retrospect, comparison and perspective[J]. Advances in Mechanics, 2016, 46(1): 201601. doi: 10.6052/1000-0992-15-020

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Theory of electroelasticity accounting for biasing fields: Retrospect, comparison and perspective

doi: 10.6052/1000-0992-15-020

1 Department of Engineering Mechanics, Zhejiang University, Hangzhou 310027, China
2 Soft Matter Research Center, Zhejiang University, Hangzhou 310027, China
3 Department of Civil Engineering, University of Siegen, Siegen D57068, Germany

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Corresponding author: CHEN Weiqiu
Received Date: 2015-05-07
Rev Recd Date: 2015-08-31
Publish Date: 2016-05-20

Abstract

Abstract

The nonlinear continuum theory of solids with electromechanical coupling was first developed in the 1950s, and matured in the 1970s. In the late 1980s and early 1990s, it gained new impetus for further elaboration and drawn wider attention due to the rapid development of intelligent materials and structures. However, research priority was given to linear analysis in applications. Since the early twentieth century, electromechanical soft ma-terials have inspired many research interests because of their potential application prospect. On account of the large deformation that is inevitably involved, the mathematical model-ing of problems and the subsequent quantitative analysis must be carried out within the general framework of nonlinear continuum mechanics. Consequently, the nonlinear theory of solids with electromechanical coupling has received great attention and many new and seemingly different versions of the theory have appeared. Based on the general framework of the nonlinear continuum theory, the aim of this paper is to review in detail the theory of electroelasticity that accounts for biasing fields by adopting both Lagrangian description and the updated Lagrangian description based on three configurations. We attempt to iden-tify the similarities and differences between different versions of the theory in order to clear the confusions in the current literature and provide a theoretical guidance for the related research in the future. The current and future research focus and development trend of the electromechanical biasing field theory in different areas are also brie°y summarized and discussed.
- electromechanical coupling,
- nonlinear continuum mechanics,
- biasing field the-ory,
- formulations,
- comparison

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References

[1]	苏益品, 陈伟球. 2014. 偏场作用下不可压缩软电弹性圆柱壳的轴对称波动. 应用力学学报, 31: 7-13 (Su Y P, Chen W Q. 2014. Axisymmetric waves in incompressible soft electroactive cylindrical shells subject to a biasing field. Chinese Journal of Applied Mechanics, 31: 7-13).
[2]	锁志刚. 2011. 介电高弹聚合物理论. 力学进展, 41: 730-750 (Suo Z G. 2010. Theory of dielectric elastomers. Acta Mechanica Solida Sinica, 23: 549-578).
[3]	杨庆生, 魏巍, 马连华. 2014. 智能软材料热-电-化-力学耦合问题的研究进展. 力学进展, 44: 201404. (Yang Q S, Wei W, Ma L H. 2014. Research advances in thermo-electro-chemo-mechanical coupling problem for intelligent soft materials. Advances in Mechanics, 44: 201404).
[4]	赵晓鹏, 尹剑波. 2011. 电场调控的智能软材料. 北京: 科学出版社(Zhao X P, Yin J B. 2011. Smart Soft Materials Tuned by Electric Fields. Beijing: Science Press).
[5]	周伟建, 陈伟球. 2015. 表面效应对偏场下介电高弹体表面波传播的影响. 应用数学和力学, 36: 119-127 (Zhou W J, Chen W Q. 2015. Surface effect on propagation of surface waves in a dielectric elastomer half space subject to biasing fields. Applied Mathematics and Mechanics, 36: 119-127).
[6]	Arruda E M, Boyce M C. 1993. A three-dimensional constitutive model for the large stretch behavior of rubber elastic materials. Journal of the Mechanics and Physics of Solids, 41: 389-412.
[7]	Baesu E, Fortune D, Soós E. 2003. Incremental behaviour of hyperelastic dielectrics and piezoelectric crystals. Zeitschrift fÄur angewandte Mathematik und Physik, 54: 160-178.
[8]	Bassett C A L, Becker R O. 1962. Generation of electric potentials by bone in response to mechanical stress. Science, 137: 1063-1064.
[9]	Bauer S. 2006. Piezo-, pyro-and ferroelectrets: Soft transducer materials for electromechanical energy conversion. IEEE Transactions on Dielectrics and Electrical Insulation, 13: 953-962.
[10]	Baumhauer J C, Tiersten H F. 1972. Nonlinear electroelastic equations for small fields superposed on a bias.Journal of the Acoustical Society of America, 54: 1017-1034.
[11]	Bayat A, Gordaninejad F. 2015. Bandgaps of a soft magneitorheological phononic crystal. Journal of Vibration and Acoustics, 137: 011011.
[12]	Bazant Z. 1971. A correlation study of formulations of incremental deformation and stability of continuous bodies. Journal of Applied Mechanics, 38: 919-928.
[13]	Bercoff J, Tanter M, Fink M. 2004. Supersonic shear imaging: A new technique for soft tissue elasticity mapping. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 51: 396-409.
[14]	Bertoldi K, Boyce M C. 2008a. Mechanically triggered transformations of phononic band gaps in periodic elastomeric structures. Physical Review B, 77: 052105.
[15]	Bertoldi K, Boyce M C. 2008b. Wave propagation and instabilities in monolithic and periodically structured elastomeric materials undergoing large deformations. Physical Review B, 78: 184107.
[16]	Bertoldi K, Gei M. 2011. Instabilities in multilayered soft dielectrics. Journal of the Mechanics and Physics of Solids, 59: 18-42.
[17]	Bigoni D. 2012. Nonlinear Solid Mechanics: Bifurcation Theory and Material Instability, Cambridge: Cam-bridge University Press.
[18]	Biot M A. 1965. Mechanics of Incremental Deformations: Theory of Elasticity and Viscoelasticity of Initially Stressed Solids and Fluids, Including Thermodynamic Foundations and Applications to Finite Strain. New York: Wiley.
[19]	Boyce M C, Arruda E M. 2000. Constitutive models of rubber elasticity: A review. Rubber Chemistry and Technology, 73: 504-523.
[20]	Brun M, Colquitt D J, Jones I S, Movchan A B, Movchan N V. 2014. Transformation cloaking and radial approximations for fiexural waves in elastic plates. New Journal of Physics, 16: 093020.
[21]	Bustamante R. 2009. Transversely isotropic non-linear electro-active elastomers. Acta Mechanica, 206: 237-259.
[22]	Bustamante R, Dorfmann A, Ogden R W. 2009a. Nonlinear electroelastostatics: A variational framework. Zeitschrift fÄur angewandte Mathematik und Physik, 60: 154-177.
[23]	Bustamante R, Dorfmann A, Ogden R W. 2009b. On electric body forces and Maxwell stresses in nonlinearly electroelastic solids. International Journal of Engineering Science, 47: 1131-1141.
[24]	Bustamante R, Ogden R W. 2013. Nonlinear magnetoelastostatics: Energy functionals and their second variations. Mathematics and Mechanics of Solids, 18: 760-772.
[25]	Camacho-Lopez M, Finkelmann H, Palffy-Muhoray P, Shelley M. 2004. Fast liquid-crystal elastomer swims into the dark. Nature Materials, 3: 307-310.
[26]	Campbell C. 1998. Surface Acoustic Wave Devices for Mobile and Wireless Communications. Boston: Academic Press.
[27]	Carpi F, Rossi D, Kornbluh R, Pelrine R, Sommer-Larsen P. 2011. Dielectric Elastomers as Electromechani-cal Transducers: Fundamentals, Materials, Devices, Models and Applications of an Emerging Electroactive Polymer Technology. New York: Elsevier.
[28]	Chai J F, Wu T T. 1996. Propagation of surface waves in a prestressed piezoelectric material. Journal of the Acoustical Society of America, 100: 2112-2122.
[29]	Chee C Y, Tong L, Steven G P. 1998. A review on the modelling of piezoelectric sensors and actuators incorporated in intelligent structures. Journal of Intelligent Material Systems and Structures, 9: 3-19.
[30]	ChenWQ, Dai H H. 2012. Waves in pre-stretched incompressible soft electroactive cylinders: Exact solution. Acta Mechanica Solida Sinica, 25: 530-541.
[31]	Cho Y, Yamanouchi K. 1987. Nonlinear, elastic, piezoelectric, electrostrictive, and dielectric constants of lithium niobate. Journal of Applied Physics, 61: 875-887.
[32]	Colquitt D J, Brun M, Gei M, Movchan A B, Movchan N V, Jones I S. 2014. Transformation elastodynamics and cloaking for fiexural waves. Journal of the Mechanics and Physics of Solids, 72: 131-143.
[33]	Deng Q, Liu L, Sharma P. 2013. Flexoelectricity and electrets in soft materials and biological membranes. Journal of the Mechanics and Physics of Solids, 62: 209-227.
[34]	Destrade M, Gilchrist M D, Ogden R W. 2010a. Third-and fourth-order elasticities of biological soft tissues. Journal of the Acoustical Society of America, 127: 2103-2106.
[35]	Destrade M, Gilchrist M D, Saccomandi G. 2010b. Third-and fourth-order constants of incompressible soft solids and the acousto-elastic effect. Journal of the Acoustical Society of America, 127: 2759-2763.
[36]	Diao J, Gall K, Dunn M L, Zimmerman J A. 2006. Atomistic simulations of the yielding of gold nanowires. Acta Materialia, 54: 643-653.
[37]	Díaz-Calleja R, Riande E, Sanchis M J. 2008. On electromechanical stability of dielectric elastomers. Applied Physics Letters, 93: 101902.
[38]	Dingreville R. 2007. Modeling and characterization of the elastic behavior of interfaces in nanostructured materials: From an atomistic description to a continuum approach. [PhD Dissertation]. Atlanta: Georgia Institute of Technology.
[39]	Dingreville R, Hallil A, Berbenni S. 2014. From coherent to incoherent mismatched interfaces: A generalized continuum formulation of surface stresses. Journal of the Mechanics and Physics of Solids, 72: 40-60.
[40]	Domenjoud M, Lematre M, Gratton M, Lethiecq M, Tran-Huu-Hue L-P. 2013. Theoretical and experimen-tal study of the electroacoustic behavior of lithium niobate under an initial mechanical stress. IEEE
[41]	Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 60: 2219-2224.
[42]	Dorfmann A, Ogden R W. 2005. Nonlinear electroelasticity. Acta Mechanica, 174: 167-183.
[43]	Dorfmann A, Ogden R W. 2006. Nonlinear electroelastic deformations. Journal of Elasticity, 82: 99-127.
[44]	Dorfmann A, Ogden R W. 2010a. Electroelastic waves in a finitely deformed electroactive material. IMA Journal of Applied Mathematics, 75: 603-636.
[45]	Dorfmann A, Ogden R W. 2010b. Nonlinear electroelastostatics: Incremental equations and stability. In-ternational Journal of Engineering Science, 48: 1-14.
[46]	Dorfmann L, Ogden R W. 2014a. Instabilities of an electroelastic plate. International Journal of Engineering Science, 77: 79-101.
[47]	Dorfmann L, Ogden R W. 2014b. Nonlinear Theory of Electroelastic and Magnetoelastic Interactions. New
[48]	York: Springer.
[49]	Doyley M M. 2012. Model-based elastography: A survey of approaches to the inverse elasticity problem.
[50]	Physics in Medicine and Biology, 57: R35-R73.
[51]	Du J, Jin X, Wang J. 2007a. Love wave propagation in layered magneto-electro-elastic structures with initial stress. Acta Mechanica, 192: 169-189.
[52]	Du J, Jin X, Wang J, Zhou Y. 2007b. SH wave propagation in a cylindrically layered piezoelectric structure with initial stress. Acta Mechanica, 191: 59-74.
[53]	Du J, Xian K, Wang J, Yong Y-K. 2008. Propagation of Love waves in prestressed piezoelectric layered structures loaded with viscous liquid. Acta Mechanica Solida Sinica, 21: 542-548.
[54]	Duquennoy M, Devos D, Ouaftouh M, Lochegnies D, Roméro E. 2006. Ultrasonic evaluation of residual stresses in fiat glass tempering: Comparing experimental investigation and numerical modeling. Journal of the Acoustical Society of America, 119: 3773-3781.
[55]	Duquennoy M, Ouaftouh M, Deboucq M, Lefebvre J-E, Jenot F, Ourak M. 2012. Infiuence of a superficial field of residual stress on the propagation of surface waves-Applied to the estimation of the depth of the superficial stressed zone. Applied Physics Letters, 101: 234104.
[56]	Duquennoy M, Ouaftouh M, Deboucq M, Lefebvre J-E, Jenot F, Ourak M. 2013. Characterization of micrometric and superficial residual stresses using high frequency surface acoustic waves generated by interdigital transducers. Journal of the Acoustical Society of America, 134: 4360-4371.
[57]	Duquennoy M, Ouaftouh M, Ourak M. 1999a. Determination of stresses in aluminium alloy using optical detection of Rayleigh waves. Ultrasonics, 37: 365-372.
[58]	Duquennoy M, Ouaftouh M, Ourak M. 1999b. Ultrasonic evaluation of stresses in orthotropic materials using Rayleigh waves. NDT&E International, 32: 189-199.
[59]	Duquennoy M, Ouaftouh M, Ourak M, Jenot F. 2002. Theoretical determination of Rayleigh wave acous-toelastic coe±cients: Comparison with experimental values. Ultrasonics, 39: 575-583.
[60]	Eliseev E A, Morozovska A N, Glinchuk M D, Blinc R. 2009. Spontaneous fiexoelectric/fiexomagnetic effect in nanoferroics. Physical Review B, 79: 165433.
[61]	Eliseev E A, Morozovska A N, Glinchuk M D, Zaulychny B Y, Skorokhod V V, Blinc R. 2010. Surface-induced piezomagnetic, piezoelectric, and linear magnetoelectric effects in nanosystems. Physical Review
[62]	B, 82: 085408.
[63]	Ericksen J L. 2007. Theory of elastic dielectrics revisited. Archive for Rational Mechanics and Analysis, 183: 299-313.
[64]	Eringen A, Maugin G. 1990. Electrodynamics of Continua, Vol. 1: Foundations and Solid Media. New
[65]	York: Springer-Verlag.
[66]	Foo C C, Cai S Q, Koh S J A, Bauer S, Suo Z. 2012. Model of dissipative dielectric elastomers. Journal of
[67]	Applied Physics, 111: 034102.
[68]	Fox J W, Goulbourne N C. 2008. On the dynamic electromechanical loading of dielectric elastomer mem-branes. Journal of the Mechanics and Physics of Solids, 56: 2669-2686.
[69]	Fukada E. 1968. Piezoelectricity in polymers and biological materials. Ultrasonics, 6: 229-234.
[70]	Fung Y C. 1990. Biomechanics: Motion, Flow, Stress, and Growth. New York: Springer-Verlag.
[71]	Gafka D, Tani J. 1993. Sensitivity of surface acoustic wave velocity in lithium niobate to electric field or biasing stress. Journal of Applied Physics, 73: 7145-7151.
[72]	Gao L, Parker K J, Lerner R M, Levinson S F. 1996. Imaging of the elastic properties of tissue\|A review.
[73]	Ultrasound in Medicine & Biology, 22: 959-977.
[74]	Gei M, Movchan A B, Bigoni D. 2009. Band-gap shift and defect-induced annihilation in prestressed elastic structures. Journal of Applied Physics, 105: 063507.
[75]	Gei M, Roccabianca S, Bacca M. 2011. Controlling bandgap in electroactive polymer-based structures.
[76]	IEEE/ASME Transactions on Mechatronics, 16: 102-107.
[77]	Gent A N. 1996. A new constitutive relation for rubber. Rubber Chemistry and Technology, 69: 59-61.
[78]	Gennisson J L, Rénier M, Catheline S, Barriµere C, Bercoff J, Tanter M, Fink M. 2007. Acoustoelasticity in soft solids: Assessment of the nonlinear shear modulus with the acoustic radiation force. Journal of the
[79]	Acoustical Society of America, 122: 3211-3219.
[80]	Gennisson J L, De±eux T, Macé E, Montaldo G, Fink M, Tanter M. 2010. Viscoelastic and anisotropic mechanical properties of in vivo muscle tissue assessed by supersonic shear imaging. Ultrasound in
[81]	Medicine & Biology, 36: 789-801.
[82]	Goncu F, Luding S, Bertoldi K. 2012. Exploiting pattern transformation to tune phononic band gaps in a two-dimensional granular crystal. Journal of the Acoustical Society of America, 131: 475-480.
[83]	Hashimoto K Y. 2000. Surface Acoustic Wave Devices in Telecommunications. Berlin: Springer.
[84]	He X, Yong H, Zhou Y. 2011. The characteristics and stability of a dielectric elastomer spherical shell with a thick wall. Smart Materials and Structures, 20: 055016.
[85]	Hoger A. 1993. The constitutive equation for finite deformations of a transversely isotropic hyperelastic material with residual stress. Journal of Elasticity, 33: 107-118.
[86]	Holzapfel G A. 2000. Nonlinear Solid Mechanics: A Continuum Approach for Engineering. Chichester:
[87]	Wiley.
[88]	Hong W. 2011. Modeling viscoelastic dielectrics. Journal of the Mechanics and Physics of Solids, 59: 637-650.
[89]	Hu Y T, Yang J S, Jiang Q. 2002. On modeling of extension and fiexure response of electroelastic shells under biasing fields. Acta Mechanica, 156: 163-178.
[90]	Hu Y T, Yang J S, Jiang Q. 2004. Surface waves in electrostrictive materials under biasing fields. Zeitschrift fÄur angewandte Mathematik und Physik, 55: 678-700.
[91]	Huang J S, Lu T Q, Zhu J, Clarke D R, Suo Z. 2012. Large, uni-directional actuation in dielectric elastomers achieved by fiber stiffening. Applied Physics Letters, 100: 211901.
[92]	Huang W J, Sun R, Tao J, Menard L D, Nuzzo R G, Zuo J M. 2008. Coordination-dependent surface atomic contraction in nanocrystals revealed by coherent diffraction. Nature Materials, 7: 308-313.
[93]	Huang Y, Shen X D, Zhang C L, Chen W Q. 2014. Mechanically tunable band gaps in compressible soft phononic laminated composites with finite deformation. Physics Letters A, 378: 2285-2289.
[94]	Huang Y, Zhang C L, Chen W Q. 2014. Tuning band structures of two-dimensional phononic crystals with biasing fields. Journal of Applied Mechanics, 81: 091008.
[95]	Jacob X, Catheline S, Gennisson J L, Barriµere C, Royer D, Fink M. 2007. Nonlinear shear wave interaction in soft solids. Journal of the Acoustical Society of America, 122: 1917-1926.
[96]	Jang J H, Koh C Y, Bertoldi K, Boyce M C, Thomas E L. 2009. Combining pattern instability and shape-memory hysteresis for phononic switching. Nano Letters, 9: 2113-2119.
[97]	Jiang Y, Li G Y, Qiang L X, Hu X D, Liu D, Liang S, Cao Y P. 2015. Characterization of the nonlinear elastic properties of soft tissues using the supersonic shear imaging (SSI) technique: Inverse method, ex vivo and in vivo experiments. Medical Image Analysis, 20: 97-111.
[98]	Junge M, Qu J, Jacobs L J. 2006. Relationship between Rayleigh wave polarization and state of stress.Ultrasonics, 44: 233-237.
[99]	Khaled W, Ermert H. 2008. Ultrasonic strain imaging and reconstructive elastography for biological tissue.In: Artmann G M, Chien S. Bioengineering in Cell and Tissue Research, Springer, New York, 103-132.
[100]	Kosinski J, Pastore R A, Yang J S, Yang X M, Turner J A. 2002. Second-order frequency shifts in crystal resonators under relatively large biasing fields. In: Proc. of IEEE International Frequency Control Symposium, New Orleans LA, 103-110.
[101]	Kovetz A. 2000. Electromagnetic Theory. Oxford: Oxford University Press.
[102]	Latorre-Ossa H, Gennisson J L, De Brosses E, Tanter M. 2012. Quantitative imaging of nonlinear shear mod-ulus by combining static elastography and shear wave elastography. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 59: 833-839.
[103]	Lehmann W, Skupin H, Tolksdorf C, Gebhard E, Zentel R, KrÄuger P, LÄosche M, Kremer F. 2001. Giant lateral electrostriction in ferroelectric liquid-crystalline elastomers. Nature, 410: 447-450.
[104]	Lematre M, Domenjoud M, Tran-Huu-Hue L P. 2011. Exact second order formalism for the study of electro-acoustic properties in piezoelectric structures under an initial mechanical stress. Ultrasonics, 51: 898-910.
[105]	Lematre M, Feuillard G, Clézio E L, Lethiecq M. 2006a. Modeling of the infiuence of a prestress gradient on guided wave propagation in piezoelectric structures. Journal of the Acoustical Society of America, 120: 1964-1975.
[106]	Lematre M, Feuillard G, Delaunay T, Lethiecq M. 2006b. Modeling of ultrasonic wave propagation in inte-grated piezoelectric structures under residual stress. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 53: 685-696.
[107]	Leng J, Liu L, Liu Y, Yu K, Sun S. 2009. Electromechanical stability of dielectric elastomer. Applied Physics Letters, 94: 211901.
[108]	Li B, Chen H, Qiang J, Hu S, Zhu Z, Wang Y. 2011. Effect of mechanical pre-stretch on the stabilization of dielectric elastomer actuation. Journal of Physics D: Applied Physics, 44: 155301.
[109]	Li B, Liu L, Suo Z. 2011. Extension limit, polarization saturation, and snap-through instability of dielectric elastomers. International Journal of Smart and Nano Materials, 2: 59-67.
[110]	Liang H, Upmanyu M, Huang H. 2005. Size-dependent elasticity of nanowires: Nonlinear effects. Physical Review B, 71: 241403.
[111]	Lines M E, Glass A M. 1977. Principles and Applications of Ferroelectrics and Related Materials. Oxford: Clarendon Press.
[112]	Liu H, Kuang Z B, Cai Z M. 2003a. Propagation of Bleustein-Gulyaev waves in a prestressed layered piezoelectric structure. Ultrasonics, 41: 397-405.
[113]	Liu H, Kuang Z B, Cai Z M, Wang T J, Wang Z K. 2003b. Propagation of surface acoustic waves in prestressed anisotropic layered piezoelectric structures. Acta Mechanica Solida Sinica, 16: 16-23.
[114]	Liu H, Lee J J, Cai Z M. 2004. Analysis of nonlinear acoustoelastic effect of surface acoustic waves in laminated structures by transfer matrix method. Mechanics Research Communications, 31: 667-675.
[115]	Liu H, Shin K C, Lee J J, Cai Z M. 2004. Nonlinear acoustoelastic interactions of Lamb waves with LiNbO3 films deposited on sapphire substrates. Key Engineering Materials, 261: 263-268.
[116]	Liu H, Wang T J, Wang Z K, Kuang Z B. 2002a. Effect of a biasing electric field on the propagation of antisymmetric Lamb waves in piezoelectric plates. International Journal of Solids and Structures, 39: 1777-1790.
[117]	Liu H, Wang T J, Wang Z K, Kuang Z B. 2002b. Effect of a biasing electric field on the propagation of symmetric Lamb waves in piezoelectric plates. International Journal of Solids and Structures, 39: 2031-2049.
[118]	Liu L P. 2013a. An energy formulation of continuum magneto-electro-elasticity with applications. Journal of the Mechanics and Physics of Solids, 63: 451-480.
[119]	Liu L P. 2013b. On energy formulations of electrostatics for continuum media. Journal of the Mechanics and Physics of Solids, 61: 968-990.
[120]	Liu L W, Liu Y J, Li B, Yang K, Li T F, Leng J S. 2011. Thermo-electro-mechanical instability of dielectric elastomers. Smart Materials and Structures, 20: 075004.
[121]	Liu L W, Liu Y J, Luo X J, Li B, Leng J S. 2012. Electromechanical instability and snap-through instability of dielectric elastomers undergoing polarization saturation. Mechanics of Materials, 55: 60-72.
[122]	Liu Y M, Zhang Y H, Chow M J, Chen Q N, Li J Y. 2012. Biological ferroelectricity uncovered in aortic walls by piezoresponse force microscopy. Physical Review Letters, 108: 078103.
[123]	Mannsfeld S C B, Tee B C K, Stoltenberg R M, Chen C V H, Barman S, Muir B V O, Sokolov A N, Reese C, Bao Z. 2010. Highly sensitive fiexible pressure sensors with microstructured rubber dielectric layers. Nature Materials, 9: 859-864.
[124]	Maugin G A. 1988. Continuum Mechanics of Electromagnetic Solids. Amsterdam: North-Holland.
[125]	McCarty L S, Whitesides G M. 2008. Electrostatic charging due to separation of ions at interfaces: Contact electrification of ionic electrets. Angewandte Chemie International Edition, 47: 2188-2207.
[126]	McMeeking R M, Landis C M. 2005. Electrostatic forces and stored energy for deformable dielectric mate-rials. Journal of Applied Mechanics, 72: 581-590.
[127]	Mironov M A, Pyatakov P A, Konopatskaya I I, Clement G T, Vykhodtseva N I. 2009. Parametric excitation of shear waves in soft solids. Acoustical Physics, 55: 567-574.
[128]	Mooney M. 1940. A theory of large elastic deformation. Journal of Applied Physics, 11:582-592.
[129]	Narayanaswamy O S. 1978. Stress and structural relaxation in tempering glass. Journal of the American Ceramic Society, 61: 146-152.
[130]	Nelson D F. 1978. Theory of nonlinear electroacoustics of dielectric, piezoelectric, and pyroelectric crystals. Journal of the Acoustical Society of America, 63: 1738-1748.
[131]	Nelson D F. 1979. Electric, Optic, and Acoustic Interactions in Dielectrics. New York: Wiley.
[132]	Nelson D F, Lax M. 1976. Linear elasticity and piezoelectricity in pyroelectrics. Physical Review B, 13: 1785-1796.
[133]	Norris A N, Amirkulova F A, ParnellWJ. 2014. Active elastodynamic cloaking. Mathematics and Mechanics of Solids, 19: 603-625.
[134]	Norris A N, Parnell W J. 2012 Hyperelastic cloaking theory: Transformation elasticity with pre-stressed solids. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 468: 2881-2903.
[135]	Norris A N, Shuvalov A L. 2011. Elastic cloaking theory. Wave Motion, 48: 525-538.
[136]	O'Brien W D. 2007. Ultrasound-biophysics mechanisms. Progress in Biophysics and Molecular Biology, 93: 212-255.
[137]	O'Halloran A, O'Malley F, McHugh P. 2008. A review on dielectric elastomer actuators, technology, appli-cations, and challenges. Journal of Applied Physics, 104: 071101.
[138]	Ogden R W. 1997. Nonlinear Elastic Deformations. New York: Dover.
[139]	Ogden R W. 2009. Incremental elastic motions superimposed on a finite deformation in the presence of an electromagnetic field. International Journal of Non-Linear Mechanics, 44: 570-580.
[140]	Ophir J, Alam S K, Garra B, Kallel F, Konofagou E, Krouskop T, Varghese T. 1999. Elastography: Ul-trasonic estimation and imaging of the elastic properties of tissues. Proceedings of the Institution of
[141]	Mechanical Engineers, Part H: Journal of Engineering in Medicine, 213: 203-233.
[142]	Ophir J, Cespedes I, Ponnekanti H, Yazdi Y, Li X. 1991. Elastography: A quantitative method for imaging the elasticity of biological tissues. Ultrasonic Imaging, 13: 111-134.
[143]	Palma A, Palmieri L, Socino G, Verona E. 1985a. Acoustic Lamb wave-electric field nonlinear interaction in YZ LiNbO3 plates. Applied Physics Letters, 46: 25-27.
[144]	Palma A, Palmieri L, Socino G, Verona E. 1985b. Lamb-wave electroacoustic voltage sensor. Journal of Applied Physics, 58: 3265-3267.
[145]	Palmieri L, Socino G, Verona E. 1986. Electroelastic effect in layer acoustic mode propagation along ZnO films on Si substrates. Applied Physics Letters, 49: 1581-1583.
[146]	Palmieri L, Socino G, Verona E, Tran H T, Marini A. 1988. Nonlinear electroacoustic interaction between a bias electric field and acoustic Lamb modes in LiNbO3 plates. Journal of Applied Physics, 64: 1033-1039.
[147]	Pan X H, Yu S W, Feng X Q. 2011. A continuum theory of surface piezoelectricity for nanodielectrics. Science China Physics, Mechanics and Astronomy, 54: 564-573.
[148]	Pao Y H. 1978. Electromagnetic forces in deformable continua. In: Nemat-Nasser S, ed. Mechanics Today, Vol. 4, Pergamon Press, Oxford, 209-306.
[149]	Pao Y H, Sachse W, Fukuoka H. 1984. Acoustoelasticity and ultrasonic measurements of residual stresses. In: Mason W P, Thurston R N, ed. Physical Acoustics, Vol. XVII, Academic Press, New York, 61-143.
[150]	Parnell W J. 2012. Nonlinear pre-stress for cloaking from antiplane elastic waves. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science, 468: 563-580.
[151]	Parnell W J, Norris A N, Shearer T. 2012. Employing pre-stress to generate finite cloaks for antiplane elastic waves. Applied Physics Letters, 100: 171907.
[152]	Parnell W J, Shearer T. 2013. Antiplane elastic wave cloaking using metamaterials, homogenization and hyperelasticity. Wave Motion, 50: 1140-1152.
[153]	Pelrine R, Kornbluh R, Pei Q B, Joseph J. 2000. High-speed electrically actuated elastomers with strain greater than 100%. Science, 287: 836-839.
[154]	Qian Z, Jin F, Kishimoto K, Wang Z. 2004a. Effect of initial stress on the propagation behavior of SH-waves in multilayered piezoelectric composite structures. Sensors and Actuators A: Physical, 112: 368-375.
[155]	Qian Z, Jin F, Wang Z, Kishimoto K. 2004b. Love waves propagation in a piezoelectric layered structure with initial stresses. Acta Mechanica, 171: 41-57.
[156]	Rao S S, Sunar M. 1994. Piezoelectricity and its use in disturbance sensing and control of fiexible structures: A survey. Applied Mechanics Reviews, 47: 113-123.
[157]	Rénier M, Gennisson J L, Barrire C, Royer D, Fink M. 2008. Fourth-order shear elastic constant assessment in quasi-incompressible soft solids. Applied Physics Letters, 93: 101912.
[158]	Rinaldi C, Brenner H. 2002. Body versus surface forces in continuum mechanics: Is the Maxwell stress tensor a physically objective Cauchy stress? Physical Review E, 65: 036615.
[159]	Rivlin R S. 1948. Large elastic deformations of isotropic materials. IV. further developments of the gen-eral theory. Philosophical Transactions of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 241: 379-397.
[160]	Rodriguez E K, Hoger A, McCulloch A D. 1994. Stress-dependent finite growth in soft elastic tissues. Journal of Biomechanics, 27: 455-467.
[161]	Rosset S, Shea H R. 2013. Flexible and stretchable electrodes for dielectric elastomer actuators. Applied Physics A, 110: 281-307.
[162]	Rudykh S, Boyce M C. 2014. Transforming wave propagation in layered media via instability-induced interfacial wrinkling.Physical Review Letters, 112: 034301.
[163]	Sandrin L, Tanter M, Catheline S, Fink M. 2002a. Shear modulus imaging with 2-D transient elastography. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 49: 426-435.
[164]	Sandrin L, Tanter M, Gennisson J L, Catheline S, Fink M. 2002b. Shear elasticity probe for soft tissues with 1-D transient elastography. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 49: 436-446.
[165]	Saravanan U. 2008. Representation for stress from a stressed reference configuration. International Journal of Engineering Science, 46: 1063-1076.
[166]	Sarvazyan A, Hall T J, Urban M W, Fatemi M, Aglyamov S R, Garra B S. 2011. An overview of elastography-An emerging branch of medical imaging. Current Medical Imaging Reviews, 7: 255-282.
[167]	Shams M, Destrade M, Ogden R W. 2011. Initial stresses in elastic solids: Constitutive laws and acoustoe-lasticity. Wave Motion, 48: 552-567.
[168]	Shams M, Ogden R W. 2014. On Rayleigh-type surface waves in an initially stressed incompressible elastic solid. IMA Journal of Applied Mathematics, 79: 360-376.
[169]	Shan S, Kang S H, Wang P, Qu C, Shian S, Chen E R, Bertoldi K. 2014. Harnessing multiple folding mechanisms in soft periodic structures for tunable control of elastic waves. Advanced Functional Materials, 24: 4935-4942.
[170]	Shmuel G. 2013a. Axisymmetric wave propagation in finitely deformed dielectric elastomer tubes. Proceed-ings of the Royal Society A: Mathematical, Physical and Engineering Science, 469: 20130071.
[171]	Shmuel G. 2013b. Electrostatically tunable band gaps in finitely extensible dielectric elastomer fiber com-posites. International Journal of Solids and Structures, 50: 680-686.
[172]	Shmuel G, deBotton G. 2012. Band-gaps in electrostatically controlled dielectric laminates subjected to incremental shear motions. Journal of the Mechanics and Physics of Solids, 60: 1970-1981.
[173]	Shmuel G, Gei M, deBotton G. 2012. The Rayleigh-Lamb wave propagation in dielectric elastomer layers subjected to large deformations. International Journal of Non-Linear Mechanics, 47: 307-316.
[174]	Simionescu P O. 2000. The infiuence of initial fields on wave propagation in piezoelectric crystals. Interna-tional Journal of Applied Electromagnetics and Mechanics, 12: 241-251.
[175]	Simionescu P O. 2001. Progressive wave propagation in the meridian plane of a 6 mm-type piezoelectric crystal subject to initial fields. Mathematics and Mechanics of Solids, 6: 661-670.
[176]	Simionescu P O. 2002. Wave propagation in cubic crystals subject to initial mechanical and electric fields. Zeitschrift fÄur angewandte Mathematik und Physik, 53: 1038-1051.
[177]	Simionescu P O. 2005. Attenuated wave propagation on a face of a cubic crystal subject to initial electro-mechanical fields. International Journal of Applied Electromagnetics and Mechanics, 22: 111-120.
[178]	Simionescu P O. 2007. Propagation of attenuated waves along an edge of a cubic crystal subject to initial electro-mechanical fields. Mathematics and Mechanics of Solids, 12: 107-118.
[179]	Simionescu P O, Ana I. 2009. On the coupling of guided waves propagation in piezoelectric crystals subject to initial fields. Mathematics and Mechanics of Solids, 14: 502-513.
[180]	Simionescu P O, Sos E. 2001. Wave propagation in piezoelectric crystals subjected to initial deformations and electric fields. Mathematics and Mechanics of Solids, 6: 437-445.
[181]	Singh B. 2010. Wave propagation in a prestressed piezoelectric half-space. Acta Mechanica, 211: 337-344.
[182]	Sinha B K, Tiersten H F. 1979. On the infiuence of a fiexural biasing state on the velocity of piezoelectric surface waves. Wave Motion, 1: 37-51.
[183]	Smith G F, Smith M M, Rivlin R S. 1963. Integrity bases for a symmetric tensor and a vector \| The crystal classes. Archive for Rational Mechanics and Analysis, 12: 93-133.
[184]	Son M S, Kang Y J. 2011. The effect of initial stress on the propagation behavior of SH waves in piezoelectric coupled plates. Ultrasonics, 51: 489-495.
[185]	Sos E. 1996. Stability, resonance and stress concentration in prestressed piezoelectric crystals containing a crack. International Journal of Engineering Science, 34: 1647-1673.
[186]	Spencer A J M. 1971. Theory of invariants. In: Eringen A C, ed. Continuum Physics, Vol. 1, Academic Press, New York, 292-307.
[187]	Su J, Kuang Z B, Liu H. 2005. Love wave in ZnO/SiO2/Si structure with initial stresses. Journal of Sound and Vibration, 286: 981-999.
[188]	Sunar M, Rao S S. 1999. Recent advances in sensing and control of fiexible structures via piezoelectric materials technology. Applied Mechanics Reviews, 52: 1-16.
[189]	Suo Z, Zhao X, Greene W H. 2008. A nonlinear field theory of deformable dielectrics. Journal of the Mechanics and Physics of Solids, 56: 467-486.
[190]	Tagarielli V L, Hildick-Smith R, Huber J E. 2012. Electro-mechanical properties and electrostriction response of a rubbery polymer for EAP applications. International Journal of Solids and Structures, 49: 3409-3415.
[191]	Thomsen D L, Keller P, Naciri J, Pink R, Jeon H, Shenoy D, Ratna B R. 2001. Liquid crystal elastomers with mechanical properties of a muscle. Macromolecules, 34: 5868-5875.
[192]	Thurston R, Brugger K. 1964. Third-order elastic constants and the velocity of small amplitude elastic waves in homogeneously stressed media. Physical Review, 133: A1604.
[193]	Tiersten H F. 1969. Linear Piezoelectric Plate Vibrations. New York: Plenum.
[194]	Tiersten H F. 1971. On the nonlinear equations of thermo-electroelasticity. International Journal of Engi-neering Science, 9: 587-604.
[195]	Tiersten H F. 1975. Nonlinear electroelastic equations cubic in the small field variables. Journal of the Acoustical Society of America, 57: 660-666.
[196]	Tiersten H F. 1981. Electroelastic interactions and the piezoelectric equations. Journal of the Acoustical Society of America, 70: 1567-1576.
[197]	Tiersten H F. 1995. On the accurate description of piezoelectric resonators subject to biasing deformations.International Journal of Engineering Science, 33: 2239-2259.
[198]	Tiersten H F, Sinha B K, Meeker T R. 1981. Intrinsic stress in thin films deposited on anisotropic substrates and its infiuence on the natural frequencies of piezoelectric resonators. Journal of Applied Physics, 52: 5614-5624.
[199]	Toupin R A. 1956. The elastic dielectric. Journal of Rational Mechanics and Analysis, 5: 849-915.
[200]	Toupin R A. 1963. A dynamical theory of elastic dielectrics. International Journal of Engineering Science, 1: 101-126.
[201]	Treloar L R G. 1975. The Physics of Rubber Elasticity. Oxford: Oxford University Press.
[202]	Trimarco C. 2009. On the Lagrangian electrostatics of elastic solids. Acta Mechanica, 204: 193-201.
[203]	Uchino K. 1997. Piezoelectric Actuators and Ultrasonic Motors. Boston: Kluwer Academic Publishers.
[204]	Vertechy R, Frisoli A, Bergamasco M, Carpi F, Frediani G, De Rossi, D. 2012. Modeling and experimental validation of buckling dielectric elastomer actuators. Smart Materials and Structures, 21: 094005.
[205]	Voltairas P, Fotiadis D, Massalas C. 2003. A theoretical study of the hyperelasticity of electro-gels. Pro-ceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 459: 2121-2130.
[206]	Wang J S, Evans A G. 1998. Measurement and analysis of buckling and buckle propagation in compressed oxide layers on superalloy substrates. Acta Materialia, 46: 4993-5005.
[207]	Wang P, Casadei F, Shan S, Weaver J C, Bertoldi K. 2014. Harnessing buckling to design tunable locally resonant acoustic metamaterials. Physical Review Letters, 113: 014301.
[208]	Wang P, Shim J, Bertoldi K. 2013. Effects of geometric and material nonlinearities on tunable band gaps and low-frequency directionality of phononic crystals. Physical Review B, 88: 014304.
[209]	Wang S D, Xiao J L, Song J Z, Ko H C, Hwang K C, Huang Y G, Rogers J A. 2010. Mechanics of curvilinear electronics. Soft Matter, 6: 5757-5763.
[210]	Wang X Q, Gu Y S, Sun X, Wang H, Zhang Y. 2014. Third-order elastic constants of ZnO and size effect in ZnO nanowires. Journal of Applied Physics, 115: 213516.
[211]	Wang Y Z, Zhang C L, Dai H H, Chen W Q. 2015. Adjustable solitary waves in electroactive rods. Journal of Sound and Vibration, DOI: 10.1016/j.jsv.2015.04.023.
[212]	Wang Z J, Liu C, Li Z G, Zhang T Y. 2010. Size-dependent elastic properties of Au nanowires under bending and tension-surfaces versus core nonlinearity. Journal of Applied Physics, 108: 083506.
[213]	Xu Y, Shen S P. 2012. Surface electric Gibbs free energy and its effect on the electromechanical behavior of nano-dielectrics. Computers Materials and Continua, 28: 81-95.
[214]	Yang J S. 2001. Bleustein-Gulyaev waves in strained piezoelectric ceramics. Mechanics Research Commu-nications, 28: 679-683.
[215]	Yang J S. 2003. Equations for small fields superposed on finite biasing fields in a thermoelectroelastic body.
[216]	IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 50: 187-192.
[217]	Yang J S. 2004. Variational formulation of the equations for small fields superposed on finite biasing fields in an electroelastic body. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 51: 1030-1034.
[218]	Yang J S. 2005a. An Introduction to the Theory of Piezoelectricity. New York: Springer.
[219]	Yang J S. 2005b. Free vibrations of an electroelastic body under biasing fields. IEEE Transactions on Ultrasonics Ferroelectrics, and Frequency Control, 52: 358-364.
[220]	Yang J S. 2009. Special Topics in the Theory of Piezoelectricity. New York: Springer.
[221]	Yang J S, Hu Y T. 2004. Mechanics of electroelastic bodies under biasing fields. Applied Mechanics Reviews, 57: 173-189.
[222]	Yang W, Suo Z. 1994. Cracking in ceramic actuators caused by electrostriction. Journal of the Mechanics and Physics of Solids, 42: 649-664.
[223]	Yeoh O H. 1990. Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber Chemistry and Technology, 63: 792-805.
[224]	Yong H D, He X Z, Zhou Y H. 2012. Electromechanical instability in anisotropic dielectric elastomers. International Journal of Engineering Science, 50: 144-150.
[225]	Zhang Q, Bharti V, Zhao X. 1998. Giant electrostriction and relaxor ferroelectric behavior in electron-irradiated poly (vinylidene fiuoride-trifiuoroethylene) copolymer. Science, 280: 2101-2104.
[226]	Zhang S H, Huang C, Klein R J, Xia F, Zhang Q M, Cheng Z Y. 2007. High performance electroactive poly-mers and nano-composites for artificial muscles. Journal of Intelligent Material Systems and Structures, 18: 133-145.
[227]	Zhang T Y, Luo M, Chan W K. 2008. Size-dependent surface stress, surface stiffness, and Young's modulus of hexagonal prism [111] fi-SiC nanowires. Journal of Applied Physics, 103: 104308.
[228]	Zhang T Y, Wang Z J, Chan W K. 2010. Eigenstress model for surface stress of solids. Physical Review B, 81: 195427.
[229]	Zhao X, Hong W, Suo Z. 2007. Electromechanical hysteresis and coexistent states in dielectric elastomers. Physical Review B, 76: 134113.
[230]	Zhao X, Suo Z. 2008. Electrostriction in elastic dielectrics undergoing large deformation. Journal of Applied Physics, 104: 123530.
[231]	Zhao X, Suo Z. 2010. Theory of dielectric elastomers capable of giant deformation of actuation. Physical Review Letters, 104: 178302.
[232]	Zhao X, Wang Q M. 2014. Harnessing large deformation and instabilities of soft dielectrics: Theory, exper-iment, and application. Applied Physics Reviews, 1: 021304.
[233]	Zheng Q S. 1994. Theory of representations for tensor functions. A unified invariant approach to constitutive equations. Applied Mechanics Reviews, 47: 545-587.
[234]	Zhou Y Y, Chen W Q, Yang J S. 2011. Thickness-shear vibration of a quartz plate connected to piezoelectric plates and electric field sensing. Ultrasonics, 51: 131-135.
[235]	Zhou Y Y, LÄu C F, Chen W Q. 2012. Bulk wave propagation in layered piezomagnetic/piezoelectric plates with initial stresses or interface imperfections. Composite Structures, 94: 2736-2745.