On the Deformation of Chiral Piezoelectric Plates

  • Dorin Ieşan
  • Ramon Quintanilla
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)


The paper is concerned with the linear theory of piezoelectricity for isotropic chiral Cosserat elastic solids. The behavior of chiral bodies is of interest for the investigation of auxetic materials, carbon nanotubes, bones, honeycomb structures, as well as composites with inclusions. First, we establish the basic equations which govern the behavior of thin plates. It is shown that, in contrast with the theory of achiral plates, the stretching and flexure cannot be treated independently of each other. Then, we present a uniqueness result with no definiteness assumption on elastic constitutive coefficients. A reciprocity theorem is also established. Then, we present the conditions on the constitutive coefficients which guarantee that the energy of the system is positive definite and we give a continuous dependence result. In the case of stationary theory we derive a uniqueness result for the Neumann problem. Finally, the effects of a concentrated charge density in an unbounded plate are investigated.


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R.Q. is supported by the Project “Análisis Matemático de Problemas de la Termomecánica“(MTM2016-74934-P) (AEI/FEDER, UE) of the Spanish Ministry of Economy and Competitiveness.


  1. Altenbach H, Eremeyev VA (2013) Cosserat-type shells. In: Altenbach H, Eremeyev VA (eds) Generalized Continua from the Theory to Engineering Applications, Springer Vienna, Vienna, pp 131–178Google Scholar
  2. Batra RC, Vidoli S (2002) Higher-order piezoelectric plate theory derived from a three-dimensional variational principle. AIAA J 40:91–104Google Scholar
  3. Bisegna P, Maceri F (1996) A consistent theory of thin piezoelectric plates. Journal of Intelligent Material Systems and Structures 7(4):372–389Google Scholar
  4. Brun L (1965) Sur l’unicité en thermoélasticité dynamique et diverses expressions analogues à la formule de Clapeyron. C R Acad Sci 261 A:2584–2587Google Scholar
  5. Chandraseker K, Mukherjee S, Paci JT, Schatz GC (2009) An atomistic-continuum Cosserat rod model of carbon nanotubes. Journal of the Mechanics and Physics of Solids 57(6):932–958Google Scholar
  6. De Cicco S, Ieşan D (2013) A theory of chiral Cosserat elastic plates. Journal of Elasticity 111(2):245–263Google Scholar
  7. Eringen AC (1967) Theory of micropolar plates. Zeitschrift für angewandte Mathematik und Physik ZAMP 18(1):12–30Google Scholar
  8. Eringen AC (1999) Microcontinuum Field Theories, vol I. Foundations and Solids. Springer, New YorkGoogle Scholar
  9. Eringen AC (2004) Electromagnetic theory of microstretch elasticity and bone modeling. International Journal of Engineering Science 42(3):231–242Google Scholar
  10. Eringen AC, Maugin GA (1990) Electrodynamics of Continua. Springer, New YorkGoogle Scholar
  11. Ha CS, Plesha ME, Lakes RS (2016) Chiral three-dimensional lattices with tunable Poisson’s ratio. Smart Materials and Structures 25(5):054,005Google Scholar
  12. Healey TJ (2002) Material symmetry and chirality in nonlinearly elastic rods. Mathematics and Mechanics of Solids 7(4):405–420Google Scholar
  13. Ieşan D, Quintanilla R (2007) Some theorems in the theory of microstretch thermopiezoelectricity. International Journal of Engineering Science 45(1):1–16Google Scholar
  14. Irschik H (2002) A review on static and dynamic shape control of structures by piezoelectric actuation. Engineering Structures 24(1):5–11Google Scholar
  15. Krommer M, Irschik H (2000) A Reissner-Mindlin-type plate theory including the direct piezoelectric and the pyroelectric effect. Acta Mechanica 141(1):51–69Google Scholar
  16. Lakes R (1987) Foam structures with a negative poisson’s ratio. Science 235(4792):1038–1040Google Scholar
  17. Lakes R (2001) Elastic and viscoelastic behavior of chiral materials. International Journal of Mechanical Sciences 43(7):1579–1589Google Scholar
  18. Maugin GA (1988) Continuum Mechanics of Electromagnetic Solids, North-Holland Series in Applied Mathematics and Mechanics, vol 33. North-Holland, Amsterdam, New York, TokioGoogle Scholar
  19. Maugin GA, Attou D (1990) An asymptotyc theory of thin piezoelectric plates. The Quarterly Journal of Mechanics and Applied Mathematics 43(3):347–362Google Scholar
  20. Mindlin RD (1951) Influence of rotatory inertia and shear on the flexural motions of isotropic elastic plates. Trans ASME J Appl Mech 18:31–38Google Scholar
  21. Mindlin RD (1974) Equations of high frequency vibrations of thermopiezoelectric crystal plates. International Journal of Solids and Structures 10(6):625–637Google Scholar
  22. Mindlin RD (1984) Frequencies of piezoelectrically forced vibrations of electroded, doubly rotated, quartz plates. International Journal of Solids and Structures 20(2):141–157Google Scholar
  23. Naghdi PM (1972) Theory of shells and plates. In: Truesdell C (ed) Handbuch der Physik, Springer, Berlin, vol VIa/2, pp 425–640Google Scholar
  24. Natroshvili D, Giorgashvili L, Stratis I (2006) Representation formulae of general solutions in the theory of hemitropic elasticity. The Quarterly Journal of Mechanics and Applied Mathematics 59:451–474Google Scholar
  25. Park HC, Lakes RS (1986) Cosserat micromechanics of human bone: Strain redistribution by a hydration sensitive constituent. Journal of Biomechanics 19(5):385–397Google Scholar
  26. Ray MC, Batra RC (2007) A single-walled carbon nanotube reinforced 1-3 piezoelectric composite for active control of smart structures. Smart Materials and Structures 16(5):1936–1947Google Scholar
  27. Telega JJ, Wojnar R (2002) Piezoelectric effects in biological tissues. J Theoretical Appl Mech 40:723–759Google Scholar
  28. Tiersten HF (1993) Equations for the extension and flexure of relatively thin electrostatic plates undergoing electric fields. In: Lee JS, Maugin GA, Shindo Y (eds) Mechanics of Electromagnetic Materials and Structures, ASMA, New York, pp 21–34Google Scholar
  29. Toupin RA (1963) A dynamical theory of elastic dielectrics. International Journal of Engineering Science 1(1):101–126Google Scholar
  30. Yang J (2006) The Mechanics of Piezoelectric Structures. World Scientific, SingaporeGoogle Scholar

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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsAl. I. Cuza University and Octav Mayer Institute of Mathematics (Romanian Academy)IaşiRomania
  2. 2.Department of MathematicsESEIAAT, Polytechnic University of CataloniaTerrassa, BarcelonaSpain

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