Nonlocal and Gradient Effects

  • Jiashi Yang


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length, classical continuum theories do not predict results consistent with lattice dynamics, and hence are no longer valid.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.


Lattice Dynamic Gradient Theory Circular Inclusion Nonlocal Theory Electromechanical Coupling Factor 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Landau DL, Lifshitz EM (1984) Electrodynamics of Continuous Media, 2nd edn. Butterworth-Heinemann, OxfordGoogle Scholar
  2. 2.
    Jackson JD (1990) Classical Electrodynamics. John Wiley & Sons, SingaporeGoogle Scholar
  3. 3.
    Eringen AC, Maugin GA (1990), Electrodynamics of Continua, vol. II. Springer, New YorkGoogle Scholar
  4. 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 YorkGoogle Scholar
  5. 5.
    Eringen AC, Kim BS (1977) Relations between nonlocal elasticity and lattice dynamics. Crystal Lattice Defects 7:51–57Google Scholar
  6. 6.
    Maugin GA (1979) Nonlocal theories or gradient-type theories: a matter of convenience? Arch Mech 31:15–26MATHMathSciNetGoogle Scholar
  7. 7.
    Eringen AC (1984) Theory of nonlocal piezoelectricity. J Math Phys 25:717–727MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Yang JS (1997) Thin film capacitance in case of a nonlocal polarization law. Int J Appl Electromag Mech 8:307–314Google Scholar
  9. 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 54:1281–1286CrossRefGoogle Scholar
  10. 10.
    Chopra KL (1969) Thin Film Phenomena. McGraw-Hill, New YorkGoogle Scholar
  11. 11.
    Mindlin RD (1972) Elasticity, piezoelectricity and crystal lattice dynamics. J Elasticity 2:217–282CrossRefGoogle Scholar
  12. 12.
    Mindlin RD (1968), Polarization gradient in elastic dielectrics. Int J Solids Struct 4:637–642MATHCrossRefGoogle Scholar
  13. 13.
    Mindlin RD (1969) Continuum and lattice theories of influence of electromechanical coupling on capacitance of thin dielectric films. Int J Solids Struct 5:1197–1208CrossRefGoogle Scholar
  14. 14.
    Askar A, Lee PCY, Cakmak AS (1970) A lattice dynamics approach to the theory of elastic dielectrics with polarization gradient. Phys Rev B 1:3525–3537CrossRefGoogle Scholar
  15. 15.
    Mindlin RD (1973) On the electrostatic potential of a point charge in a dielectric solid. Int J Solids Struct 9:233–235CrossRefGoogle Scholar
  16. 16.
    Mindlin RD (1971) Electromechanical vibrations of centrosymmetric cubic crystal plates. PMM-J Mech Appl Mathe 35:404–408CrossRefGoogle Scholar
  17. 17.
    Mindlin RD (1972) Coupled elastic and electromagnetic fields in a diatomic, electric continuum. Int J Solids Struct 8:401–408MATHCrossRefGoogle Scholar
  18. 18.
    Mindlin RD (1974) Electromagnetic radiation from a vibrating, elastic sphere. Int J Solids Struct 10:1307–1314MATHCrossRefGoogle Scholar
  19. 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 7:523–537CrossRefGoogle Scholar
  20. 20.
    Schwartz J (1969) Solutions of the equations of equilibrium of elastic dielectrics: stress functions, concentrated force, surface energy. Int J Solids Struct 5:1209–1220CrossRefGoogle Scholar
  21. 21.
    Chowdhury KL, Glockner PG (1977) Point charge in the interior of an elastic dielectric half space. Int J Engng Sci 15:481–493MATHCrossRefGoogle Scholar
  22. 22.
    Chowdhury KL, Glockner PG (1981) On a similarity solution of the Boussinesq problem of elastic dielectrics. Arch Mech 32:429–442MathSciNetGoogle Scholar
  23. 23.
    Collet B (1981) One-dimensional acceleration waves in deformable dielectrics with polarization gradients. Int J Engng Sci 19: 389–407MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Dost S (1983) Acceleration waves in elastic dielectrics with polarization gradient effects. Int J Engng Sci 21:1305–1311MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Collet B (1982) Shock waves in deformable dielectrics with polarization gradients. Int J Engng Sci 20:1145–1160MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Yang JS, Batra RC (1995) Conservation laws in linear piezoelectricity. Eng Fract Mech 51:1041–1047CrossRefGoogle Scholar
  27. 27.
    Suhubi ES (1969) Elastic dielectrics with polarization gradients. Int J Engng Sci 7:993–997MATHCrossRefGoogle Scholar
  28. 28.
    Chowdhury KL, Epstein M, Glockner PG (1979) On the thermodynamics of nonlinear elastic dielectrics. Int J Non-Linear Mech 13:311–322CrossRefGoogle Scholar
  29. 29.
    Chowdhury KL, Glockner PG (1976) Constitutive equations for elastic dielectrics. Int J Non-Linear Mech 11:315–324MATHCrossRefGoogle Scholar
  30. 30.
    Chowdhury KL, Glockner PG (1977) On thermoelastic dielectrics. Int J Solids Struct 13:1173–1182CrossRefGoogle Scholar
  31. 31.
    Tiersten HF, Tsai CF (1972) On the interaction of the electromagnetic field with heat conducting deformable insulators. J Math Phys 13:361–378CrossRefGoogle Scholar
  32. 32.
    Maugin GA (1977) Deformable dielectrics II. Voigt’s intramolecular force balance in elastic dielectrics. Arch Mech 29:143–151MATHGoogle Scholar
  33. 33.
    Maugin GA (1977) Deformable dielectrics III. A model of interactions. Arch Mech 29:251–258MATHGoogle Scholar
  34. 34.
    Maugin GA, Pouget J (1980) Electroacoustic equations for one-domain ferroelectric bodies. J Acoust Soc Am 68:575–587MATHCrossRefGoogle Scholar
  35. 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-HollandGoogle Scholar
  36. 36.
    Pouget J, Askar A, Maugin GA (1986) Lattice model for elastic ferroelectric crystals: microscopic approximation. Phys Rev B 33:6304–6319CrossRefGoogle Scholar
  37. 37.
    Pouget J, Askar A, Maugin GA (1986), Lattice model for elastic ferroelectric crystals: continuum approximation. Phys Rev B 33:6320–6325CrossRefGoogle Scholar
  38. 38.
    Pouget J, Maugin GA (1980) Coupled acoustic-optic modes in deformable ferroelectrics. J Acoust Soc Am 68:588–601MATHCrossRefGoogle Scholar
  39. 39.
    Pouget J, Maugin GA (1981) Bleustein-Gulyaev surface modes in elastic ferroelectrics. J Acoust Soc Am 69:1304–1318MATHCrossRefGoogle Scholar
  40. 40.
    Pouget J, Maugin GA (1981) Piezoelectric Rayleigh waves in elastic ferroelectrics. J Acoust Soc Am 69:1319–1325MATHCrossRefGoogle Scholar
  41. 41.
    Collet B (1984) Shock waves in deformable ferroelectric materials. In: Maugin GA (ed) Mechanical Behavior of Electromagnetic Solid Continua. Elsevier, North-HollandGoogle Scholar
  42. 42.
    Sahin E, Dost S (1988) A strain-gradient theory of elastic dielectrics with spatial dispersion. Int J Engng Sci 26:1231–1245CrossRefGoogle Scholar
  43. 43.
    Demiray H, Dost S (1989) Diatomic elastic dielectrics with polarization gradient. Int J Engng Sci 27:1275–1284MATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Askar A, Lee PCY (1974) Lattice dynamics approach to the theory of diatomic elastic dielectrics. Phys Rev B 9:5291–5299CrossRefGoogle Scholar
  45. 45.
    Maugin GA (1988) Continuum Mechanics of Electromagnetic Bodies. Elsevier, North-HollandGoogle Scholar
  46. 46.
    Maugin GA, Pouget J, Drouot JR et al (1992) Nonlinear Electromechanical Couplings. John Wiley and Sons, ChichesterGoogle Scholar
  47. 47.
    Li JY (2003) Exchange coupling in P(VDF-TrFE) copolymer based all-organic composites with giant electrostriction. Phys Rev Lett 90:17601Google Scholar
  48. 48.
    Kafadar CB (1971) Theory of multipoles in classical electromagnetism. Int J Engng Sci 9:831–853MATHCrossRefGoogle Scholar
  49. 49.
    Demiray H, Eringen AC (1973) On the constitutive relations of polar elastic dielectrics. Lett in Appl Engng Sci 1:517–527Google Scholar
  50. 50.
    Prechtl A (1980) Deformable bodies with electric and magnetic quadrupoles. Int J Engng Sci 18:665–680MATHCrossRefGoogle Scholar
  51. 51.
    Nelson DF (1979) Electric, Optic and Acoustic Interactions in Crystals. Wiley, New YorkGoogle Scholar
  52. 52.
    Kalpakides VK, Hadjigeorgiou EP, Massalas CV (1995) A variational principle for elastic dielectrics with quadruple polarization. Int J Engng Sci 33:793–801CrossRefMathSciNetGoogle Scholar
  53. 53.
    Kalpakides VK, Massalas CV (1993) Tiersten’s theory of thermoelectroelasticity: An extension. Int J Engng Sci 31:157–164CrossRefMathSciNetGoogle Scholar
  54. 54.
    Hadjigeorgiou EP, Kalpakides VK, Massalas CV (1999) A general theory for elastic dielectrics. II. The variational approach. Int J Non-Linear Mech 34:967–980CrossRefGoogle Scholar
  55. 55.
    Kalpakides VK, Agiasofitou EK (2002) On material equations in second order gradient electroelasticity. J Elasticity 67:205–227MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Maugin GA (1980) The principle of virtual power: Application to coupled fields. Acta Mech 35:1–70MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Yang XM, Hu YT, Yang JS (2004) Electric field gradient effects in anti-plane problems of polarized ceramics. Int J Solids Struct 41:6801–6811MATHCrossRefGoogle Scholar
  58. 58.
    Yang JS, Yang XM (2004), Electric field gradient effect and thin film capacitance. World J Eng 2:41–45Google Scholar
  59. 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 18:29–36Google Scholar
  60. 60.
    Li XF, Yang JS, Jiang Q (2005) Spatial dispersion of short surface acoustic waves in piezoelectric ceramics. Acata Mechanica 180:11–20MATHCrossRefGoogle Scholar
  61. 61.
    Bleustein JL (1968) A new surface wave in piezoelectric materials. Appl Phys Lett 13:412–413CrossRefGoogle Scholar
  62. 62.
    Gulyaev YuV (1969) Electroacoustic surface waves in solids. Sov Phys JETP Lett 9:37–38Google Scholar
  63. 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 462:3511–3522MATHCrossRefGoogle Scholar
  64. 64.
    Yang JS (2004) Effects of electric field gradient on an anti-plane crack in piezoelectric ceramics. Int J Fract 127:L111–L116MATHCrossRefGoogle Scholar
  65. 65.
    Zeng Y, Hu YT, Yang JS (2005) Electric field gradient effects in piezoelectric antiplane crack problems. J Huazhong Univ Sci Technol 22:31–35Google Scholar
  66. 66.
    Zeng Y (2005) Electric field gradient effects in anti-plane crack problems of piezoelectric ceramics. MS thesis, Huazhong University of Science and TechnologyGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Jiashi Yang
    • 1
  1. 1.Department of Engineering MechanicsUniversity of NebraskaLincolnUSA

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