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Effective Coefficients and Local Fields of Periodic Fibrous Piezocomposites with 622 Hexagonal Constituents

  • Ransés Alfonso-Rodríguez
  • Julián Bravo-Castillero
  • Raúl Guinovart-Díaz
  • Reinaldo Rodríguez-Ramos
  • Renald Brenner
  • Leslie D. Pérez-Fernández
  • Federico J. Sabina
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)

Abstract

The asymptotic homogenization method is applied to a family of boundary value problems for linear piezoelectric heterogeneous media with periodic and rapidly oscillating coefficients.We consider a two-phase fibrous composite consisting of identical circular cylinders perfectly bonded in a matrix. Both constituents are piezoelectric 622 hexagonal crystal and the periodic distribution of the fibers follows a rectangular array. Closed-form expressions are obtained for the effective coefficients, based on the solution of local problems using potential methods of a complex variable. An analytical procedure to study the spatial heterogeneity of the strain and electric fields is described. Analytical expressions for the computation of these fields are given for specific local problems. Examples are presented for fiber-reinforced and porous matrix including comparisons with fast Fourier transform (FFT) numerical results.

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Notes

Acknowledgements

Support provided by the French Priority Solidarity Fund (Campus France FSP Cuba 29896TD), the Brazilian Coordination for the Improvement of Higher Education Personnel (Project CAPES 88881.030424/2013-01), and the CoNaCyT 129658 is gratefully acknowledged. The authors are also grateful to Ana Pérez Arteaga and Ramiro Chávez for computational support. JB acknowledges the Cátedra Extraordinaria IIMAS and PREI-DGAPA, UNAM.

References

  1. Aguiar AR, Bravo-Castillero J, Rodríguez-Ramos R, da Silva UP (2013) Effective electromechanical properties of 622 piezoelectric medium with unidirectional cylindrical holes. J App Mech 80(5):050,905Google Scholar
  2. Alfonso-Rodríguez R, Bravo-Castillero J, D PL (2017) Local fields in two-phase fibrous piezocomposites with 622 constituents. Math Methods App Sci 40:3221–3229Google Scholar
  3. Bakhvalov N, Panasenko GP (1989) Homogenisation: Averaging Processes in Periodic Media. Kluwer, OxfordGoogle Scholar
  4. Berger H, Gabbert U, Köppe H, Rodríguez-Ramos R, Bravo-Castillero J, Guinovart-Díaz R, Otero JA, Maugin GA (2003) Finite element and asymptotic homogenization methods applied to smart composite materials. Comp Mech 33:61–67Google Scholar
  5. Berger H, Kari S, Gabbert U, Rodríguez-Ramos R, Bravo-Castillero J, Guinovart-Díaz R, Sabina FJ, Maugin GA (2006) Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties. J Smart Mat Struct 15:451–458Google Scholar
  6. Bravo-Castillero J, Guinovart-Díaz R, Otero JA, Rodríguez-Ramos R (1997) Electromechanical properties of continuous fibre-reinforced piezoelectric composites. Mech Comp Mater 33:670–680Google Scholar
  7. Bravo-Castillero J, Otero JA, Rodríguez-Ramos R, Bourgeat A (1998) Asymptotic homogenization of laminated piezocomposite materials. Int J Solids Struct 35:527–541Google Scholar
  8. Bravo-Castillero J, Guinovart-Díaz R, Sabina FJ, Rodríguez-Ramos R (2001) Closed-form expressions for the effective coefficients of fibre-reinforced composite with transversely isotropic constituents - II. Piezoelectric and square symmetry. Mech Mater 33:237–248Google Scholar
  9. Brenner R (2009) Numerical computation of the response of piezoelectric composites using Fourier transform. Phys Rev B 79:184,106Google Scholar
  10. Brenner R (2010) Computational approach for composite materials with coupled constitutive laws. Smart Mater Struct 61:919–927Google Scholar
  11. Fukada E (1984) Piezoelectricity of natural biomaterials. Ferroelec 60:285–296Google Scholar
  12. Galka A, Telega JJ, Wojnar R (1992) Homogenization and thermopiezoelectricity. Mech Res Comm 19:315–324Google Scholar
  13. Galka A, Telega JJ,Wojnar R (1996) Some computational aspects of homogenization of thermopiezoelectric composites. Computer Assisted Mechanics and Engineering Sciences 3:133–154Google Scholar
  14. Kantorovich LV, Krylov VI (1964) Approximate Methods of Higher Analysis. Wiley, New YorkGoogle Scholar
  15. Lang S (1993) Complex Analysis, 3rd edn. Graduate Texts in Mathematics, Springer, New YorkGoogle Scholar
  16. López-López E, Sabina F, Bravo-Castillero J, Guinovart-Díaz R, Rodríguez-Ramos R (2005) Overall electromechanical properties of a binary composite with 622 symmetry constituents. Antiplane shear piezoelectric state. Int J Solids Struct 42:5765–5777Google Scholar
  17. Maugin GA, Turbé N (1991) Homogenization of piezoelectric composites via Bloch expansion. Int J Appl Electromagnetics in Mat 2:135–140Google Scholar
  18. Michel JC, Moulinec H, Suquet P (2001) A computational scheme for linear and non-linear composites with arbitrary phase contrast. Int J Numer Meth Engng 52:139–160Google Scholar
  19. Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Engrg 157:69–94Google Scholar
  20. Nava-Gómez GG, Camacho-Montes HF, Sabina FJ, Rodríguez-Ramos R, Fuentes L, Guinovart-Díaz R (2012) Elastic properties of an orthotropic binary fiber-reinforced composite with auxetic and conventional constituents. Mech Mater 48:1–25Google Scholar
  21. Nye JF (1957) Physical Properties of Crystals: Their Representation by Tensors and Matrices. Clarendon Press, OxfordGoogle Scholar
  22. Otero JA, Bravo-Castillero J, Guinovart-Díaz R, Rodríguez-Ramos R, Maugin GA (2003) Analytical expressions of effective constants for a piezoelectric composite reinforced with square crosssection fibers. Arch Mech 55:357–371Google Scholar
  23. Rodríguez-Ramos R, Otero JA, Bravo-Castillero J, Sabina FJ (1996) Electromechanical properties of laminated piezoelectric composites. Mech Comp Mater 32:410–417Google Scholar
  24. Sabina FJ, Rodríguez-Ramos R, Bravo-Castillero J, Guinovart-Díaz R (2001) Closed-form expressions for the effective coefficients of fibre-reinforced composite with transversely isotropic constituents - II. Piezoelectric and hexagonal symmetry. J Mech Phys Solids 49:1463–1479Google Scholar
  25. Sevostianov I, da Silva UP, Aguiar AR (2014) Green’s function for piezoelectric 622 hexagonal crystals. Int J Eng Sci 84:18–28Google Scholar
  26. Sixto-Camacho LM, Bravo-Castillero J, Brenner R, Guinovart-Díaz R, Mechkour H, Rodriguez- Ramos R, Sabina FJ (2013) Asymptotic homogenization of periodic thermo-magneto-electroelastic heterogeneous media. Comp Math App 66:2056–2074Google Scholar
  27. Telega JJ (1991) Piezoelectricity and homogenization. application to biomechanics. In: Maugin GA (ed) Continuum Models and Discrete Systems, Longman, Essex, vol 2, pp 220–229Google Scholar
  28. Turbé N, Maugin GA (1991) On the linear piezoelectricity of composite materials. Math Meth Appl Sci 14:403–412Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Ransés Alfonso-Rodríguez
    • 1
  • Julián Bravo-Castillero
    • 2
    • 3
  • Raúl Guinovart-Díaz
    • 2
  • Reinaldo Rodríguez-Ramos
    • 2
  • Renald Brenner
    • 4
  • Leslie D. Pérez-Fernández
    • 5
  • Federico J. Sabina
    • 3
  1. 1.Department of MathematicsUniversity of Central FloridaOrlandoUSA
  2. 2.Facultad de Matemática y ComputaciónUniversidad de La HabanaLa HabanaCuba
  3. 3.Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoDelegación Álvaro ObregónMéxico
  4. 4.Institut Jean Le Rond d’AlembertSorbonne Université, Centre National de la Recherche ScientifiqueParisFrance
  5. 5.Departamento de Matemática e Estatística, Instituto de Física e MatemáticaUniversidade Federal de PelotasPelotasBrazil

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