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Special Topics in the Theory of Piezoelectricity pp 281–318Cite as

Nonlocal and Gradient Effects

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Abstract

Classical continuum theories such as elasticity, electrostatics, and piezoelectricity are the long-wave and low-frequency limits of the equations of lattice dynamics. The partial differential equations of these continuum theories can be obtained from the finite difference equations of lattice dynamics by Taylor expansions and truncations. The equations of classical continuum theories are accurate for phenomena with a characteristic length much larger than microscopic characteristic lengths, for example, the distance between neighboring atoms in a lattice. When the characteristic length of a problem is not much larger than the microscopic characteristic length, classical continuum theories do not predict results consistent with lattice dynamics, and hence are no longer valid. For example, lattice waves are dispersive but the theory of elasticity only predicts nondispersive plane waves which are the long wave limit of lattice waves. There are different ways to modify the classical continuum theories so that their range of applicability can be extended to problems with smaller characteristic lengths, with results closer to lattice dynamics in a wider range of wave lengths.

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References

  1. Landau DL, Lifshitz EM (1984) Electrodynamics of Continuous Media, 2nd edn. Butterworth-Heinemann, Oxford

    Google Scholar 

  2. Jackson JD (1990) Classical Electrodynamics. John Wiley & Sons, Singapore

    Google Scholar 

  3. Eringen AC, Maugin GA (1990), Electrodynamics of Continua, vol. II. Springer, New York

    Google Scholar 

  4. Eringen AC (1993), Vistas of nonlocal electrodynamics. In: Lee JS, Maugin GA, Shindo Y (ed) Mechanics of Electromagnetic Materials and Structures. American Society of Mechanical Engineers, New York

    Google Scholar 

  5. Eringen AC, Kim BS (1977) Relations between nonlocal elasticity and lattice dynamics. Crystal Lattice Defects 7:51–57

    Google Scholar 

  6. Maugin GA (1979) Nonlocal theories or gradient-type theories: a matter of convenience? Arch Mech 31:15–26

    MATH  MathSciNet  Google Scholar 

  7. Eringen AC (1984) Theory of nonlocal piezoelectricity. J Math Phys 25:717–727

    Article  MATH  MathSciNet  Google Scholar 

  8. Yang JS (1997) Thin film capacitance in case of a nonlocal polarization law. Int J Appl Electromag Mech 8:307–314

    Google Scholar 

  9. Yang JS, Mao SX, Yan K et al (2006) Size effect on the electromechanical coupling factor of a thin piezoelectric film due to a nonlocal polarization law. Scripta Materialia 54:1281–1286

    Article  Google Scholar 

  10. Chopra KL (1969) Thin Film Phenomena. McGraw-Hill, New York

    Google Scholar 

  11. Mindlin RD (1972) Elasticity, piezoelectricity and crystal lattice dynamics. J Elasticity 2:217–282

    Article  Google Scholar 

  12. Mindlin RD (1968), Polarization gradient in elastic dielectrics. Int J Solids Struct 4:637–642

    Article  MATH  Google Scholar 

  13. Mindlin RD (1969) Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Int J Solids Struct 5:1197–1208

    Article  Google Scholar 

  14. Askar A, Lee PCY, Cakmak AS (1970) A lattice dynamics approach to the theory of elastic dielectrics with polarization gradient. Phys Rev B 1:3525–3537

    Article  Google Scholar 

  15. Mindlin RD (1973) On the electrostatic potential of a point charge in a dielectric solid. Int J Solids Struct 9:233–235

    Article  Google Scholar 

  16. Mindlin RD (1971) Electromechanical vibrations of centrosymmetric cubic crystal plates. PMM-J Mech Appl Mathe 35:404–408

    Article  Google Scholar 

  17. Mindlin RD (1972) Coupled elastic and electromagnetic fields in a diatomic, electric continuum. Int J Solids Struct 8:401–408

    Article  MATH  Google Scholar 

  18. Mindlin RD (1974) Electromagnetic radiation from a vibrating, elastic sphere. Int J Solids Struct 10:1307–1314

    Article  MATH  Google Scholar 

  19. Askar A, Lee PCY, Cakmak AS (1971) The effect of surface curvature and discontinuity on the surface energy density and other induced fields in electric dielectrics with polarization gradient. Int J Solids Struct 7:523–537

    Article  Google Scholar 

  20. Schwartz J (1969) Solutions of the equations of equilibrium of elastic dielectrics: stress functions, concentrated force, surface energy. Int J Solids Struct 5:1209–1220

    Article  Google Scholar 

  21. Chowdhury KL, Glockner PG (1977) Point charge in the interior of an elastic dielectric half space. Int J Engng Sci 15:481–493

    Article  MATH  Google Scholar 

  22. Chowdhury KL, Glockner PG (1981) On a similarity solution of the Boussinesq problem of elastic dielectrics. Arch Mech 32:429–442

    MathSciNet  Google Scholar 

  23. Collet B (1981) One-dimensional acceleration waves in deformable dielectrics with polarization gradients. Int J Engng Sci 19: 389–407

    Article  MATH  MathSciNet  Google Scholar 

  24. Dost S (1983) Acceleration waves in elastic dielectrics with polarization gradient effects. Int J Engng Sci 21:1305–1311

    Article  MATH  MathSciNet  Google Scholar 

  25. Collet B (1982) Shock waves in deformable dielectrics with polarization gradients. Int J Engng Sci 20:1145–1160

    Article  MATH  MathSciNet  Google Scholar 

  26. Yang JS, Batra RC (1995) Conservation laws in linear piezoelectricity. Eng Fract Mech 51:1041–1047

    Article  Google Scholar 

  27. Suhubi ES (1969) Elastic dielectrics with polarization gradients. Int J Engng Sci 7:993–997

    Article  MATH  Google Scholar 

  28. Chowdhury KL, Epstein M, Glockner PG (1979) On the thermodynamics of nonlinear elastic dielectrics. Int J Non-Linear Mech 13:311–322

    Article  Google Scholar 

  29. Chowdhury KL, Glockner PG (1976) Constitutive equations for elastic dielectrics. Int J Non-Linear Mech 11:315–324

    Article  MATH  Google Scholar 

  30. Chowdhury KL, Glockner PG (1977) On thermoelastic dielectrics. Int J Solids Struct 13:1173–1182

    Article  Google Scholar 

  31. Tiersten HF, Tsai CF (1972) On the interaction of the electromagnetic field with heat conducting deformable insulators. J Math Phys 13:361–378

    Article  Google Scholar 

  32. Maugin GA (1977) Deformable dielectrics II. Voigt’s intramolecular force balance in elastic dielectrics. Arch Mech 29:143–151

    MATH  Google Scholar 

  33. Maugin GA (1977) Deformable dielectrics III. A model of interactions. Arch Mech 29:251–258

    MATH  Google Scholar 

  34. Maugin GA, Pouget J (1980) Electroacoustic equations for one-domain ferroelectric bodies. J Acoust Soc Am 68:575–587

    Article  MATH  Google Scholar 

  35. Askar A, Pouget J, Maugin GA (1984) Lattice model for elastic ferroelectrics and related continuum theories. In: Maugin GA (ed) Mechanical Behavior of Electromagnetic Solid Continua. Elsevier, North-Holland

    Google Scholar 

  36. Pouget J, Askar A, Maugin GA (1986) Lattice model for elastic ferroelectric crystals: microscopic approximation. Phys Rev B 33:6304–6319

    Article  Google Scholar 

  37. Pouget J, Askar A, Maugin GA (1986), Lattice model for elastic ferroelectric crystals: continuum approximation. Phys Rev B 33:6320–6325

    Article  Google Scholar 

  38. Pouget J, Maugin GA (1980) Coupled acoustic-optic modes in deformable ferroelectrics. J Acoust Soc Am 68:588–601

    Article  MATH  Google Scholar 

  39. Pouget J, Maugin GA (1981) Bleustein-Gulyaev surface modes in elastic ferroelectrics. J Acoust Soc Am 69:1304–1318

    Article  MATH  Google Scholar 

  40. Pouget J, Maugin GA (1981) Piezoelectric Rayleigh waves in elastic ferroelectrics. J Acoust Soc Am 69:1319–1325

    Article  MATH  Google Scholar 

  41. Collet B (1984) Shock waves in deformable ferroelectric materials. In: Maugin GA (ed) Mechanical Behavior of Electromagnetic Solid Continua. Elsevier, North-Holland

    Google Scholar 

  42. Sahin E, Dost S (1988) A strain-gradient theory of elastic dielectrics with spatial dispersion. Int J Engng Sci 26:1231–1245

    Article  Google Scholar 

  43. Demiray H, Dost S (1989) Diatomic elastic dielectrics with polarization gradient. Int J Engng Sci 27:1275–1284

    Article  MATH  MathSciNet  Google Scholar 

  44. Askar A, Lee PCY (1974) Lattice dynamics approach to the theory of diatomic elastic dielectrics. Phys Rev B 9:5291–5299

    Article  Google Scholar 

  45. Maugin GA (1988) Continuum Mechanics of Electromagnetic Bodies. Elsevier, North-Holland

    Google Scholar 

  46. Maugin GA, Pouget J, Drouot JR et al (1992) Nonlinear Electromechanical Couplings. John Wiley and Sons, Chichester

    Google Scholar 

  47. Li JY (2003) Exchange coupling in P(VDF-TrFE) copolymer based all-organic composites with giant electrostriction. Phys Rev Lett 90:17601

    Google Scholar 

  48. Kafadar CB (1971) Theory of multipoles in classical electromagnetism. Int J Engng Sci 9:831–853

    Article  MATH  Google Scholar 

  49. Demiray H, Eringen AC (1973) On the constitutive relations of polar elastic dielectrics. Lett in Appl Engng Sci 1:517–527

    Google Scholar 

  50. Prechtl A (1980) Deformable bodies with electric and magnetic quadrupoles. Int J Engng Sci 18:665–680

    Article  MATH  Google Scholar 

  51. Nelson DF (1979) Electric, Optic and Acoustic Interactions in Crystals. Wiley, New York

    Google Scholar 

  52. Kalpakides VK, Hadjigeorgiou EP, Massalas CV (1995) A variational principle for elastic dielectrics with quadruple polarization. Int J Engng Sci 33:793–801

    Article  MathSciNet  Google Scholar 

  53. Kalpakides VK, Massalas CV (1993) Tiersten’s theory of thermoelectroelasticity: An extension. Int J Engng Sci 31:157–164

    Article  MathSciNet  Google Scholar 

  54. Hadjigeorgiou EP, Kalpakides VK, Massalas CV (1999) A general theory for elastic dielectrics. II. The variational approach. Int J Non-Linear Mech 34:967–980

    Article  Google Scholar 

  55. Kalpakides VK, Agiasofitou EK (2002) On material equations in second order gradient electroelasticity. J Elasticity 67:205–227

    Article  MATH  MathSciNet  Google Scholar 

  56. Maugin GA (1980) The principle of virtual power: Application to coupled fields. Acta Mech 35:1–70

    Article  MATH  MathSciNet  Google Scholar 

  57. Yang XM, Hu YT, Yang JS (2004) Electric field gradient effects in anti-plane problems of polarized ceramics. Int J Solids Struct 41:6801–6811

    Article  MATH  Google Scholar 

  58. Yang JS, Yang XM (2004), Electric field gradient effect and thin film capacitance. World J Eng 2:41–45

    Google Scholar 

  59. Yang XM, Hu TY, Yang JS (2005) Electric field gradient effects in anti-plane problems of a circular cylindrical hole in piezoelectric materials of 6mm symmetry. Acta Mech Solida Sinica 18:29–36

    Google Scholar 

  60. Li XF, Yang JS, Jiang Q (2005) Spatial dispersion of short surface acoustic waves in piezoelectric ceramics. Acata Mechanica 180:11–20

    Article  MATH  Google Scholar 

  61. Bleustein JL (1968) A new surface wave in piezoelectric materials. Appl Phys Lett 13:412–413

    Article  Google Scholar 

  62. Gulyaev YuV (1969) Electroacoustic surface waves in solids. Sov Phys JETP Lett 9:37–38

    Google Scholar 

  63. Yang JS, Zhou HG, Li JY (2006) Electric field gradient effects in an anti-plane circular inclusion in polarized ceramics. Proc Royal Soc London A 462:3511–3522

    Article  MATH  Google Scholar 

  64. Yang JS (2004) Effects of electric field gradient on an anti-plane crack in piezoelectric ceramics. Int J Fract 127:L111–L116

    Article  MATH  Google Scholar 

  65. Zeng Y, Hu YT, Yang JS (2005) Electric field gradient effects in piezoelectric antiplane crack problems. J Huazhong Univ Sci Technol 22:31–35

    Google Scholar 

  66. Zeng Y (2005) Electric field gradient effects in anti-plane crack problems of piezoelectric ceramics. MS thesis, Huazhong University of Science and Technology

    Google Scholar 

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Authors and Affiliations

  1. Department of Engineering Mechanics, University of Nebraska, Str. Academiei 14, Lincoln, NE, 68588-0526, USA

    Jiashi Yang

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  1. Jiashi Yang
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  1. Department of Engineering Mechanics, University of Nebraska, Lincoln, Nebraska Hall, Lincoln, NE, 68588-0526, USA

    Jiashi Yang

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Yang, J. (2009). Nonlocal and Gradient Effects. In: Yang, J. (eds) Special Topics in the Theory of Piezoelectricity. Springer, New York, NY. https://doi.org/10.1007/978-0-387-89498-0_8

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