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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References
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):050,905
Alfonso-Rodríguez R, Bravo-Castillero J, D PL (2017) Local fields in two-phase fibrous piezocomposites with 622 constituents. Math Methods App Sci 40:3221–3229
Bakhvalov N, Panasenko GP (1989) Homogenisation: Averaging Processes in Periodic Media. Kluwer, Oxford
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 33:61–67
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 15:451–458
Bravo-Castillero J, Guinovart-Díaz R, Otero JA, Rodríguez-Ramos R (1997) Electromechanical properties of continuous fibre-reinforced piezoelectric composites. Mech Comp Mater 33:670–680
Bravo-Castillero J, Otero JA, Rodríguez-Ramos R, Bourgeat A (1998) Asymptotic homogenization of laminated piezocomposite materials. Int J Solids Struct 35:527–541
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 33:237–248
Brenner R (2009) Numerical computation of the response of piezoelectric composites using Fourier transform. Phys Rev B 79:184,106
Brenner R (2010) Computational approach for composite materials with coupled constitutive laws. Smart Mater Struct 61:919–927
Fukada E (1984) Piezoelectricity of natural biomaterials. Ferroelec 60:285–296
Galka A, Telega JJ, Wojnar R (1992) Homogenization and thermopiezoelectricity. Mech Res Comm 19:315–324
Galka A, Telega JJ,Wojnar R (1996) Some computational aspects of homogenization of thermopiezoelectric composites. Computer Assisted Mechanics and Engineering Sciences 3:133–154
Kantorovich LV, Krylov VI (1964) Approximate Methods of Higher Analysis. Wiley, New York
Lang S (1993) Complex Analysis, 3rd edn. Graduate Texts in Mathematics, Springer, New York
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 42:5765–5777
Maugin GA, Turbé N (1991) Homogenization of piezoelectric composites via Bloch expansion. Int J Appl Electromagnetics in Mat 2:135–140
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 52:139–160
Moulinec H, Suquet P (1998) A numerical method for computing the overall response of nonlinear composites with complex microstructure. Comput Methods Appl Mech Engrg 157:69–94
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 48:1–25
Nye JF (1957) Physical Properties of Crystals: Their Representation by Tensors and Matrices. Clarendon Press, Oxford
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 55:357–371
Rodríguez-Ramos R, Otero JA, Bravo-Castillero J, Sabina FJ (1996) Electromechanical properties of laminated piezoelectric composites. Mech Comp Mater 32:410–417
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 49:1463–1479
Sevostianov I, da Silva UP, Aguiar AR (2014) Green’s function for piezoelectric 622 hexagonal crystals. Int J Eng Sci 84:18–28
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 66:2056–2074
Telega JJ (1991) Piezoelectricity and homogenization. application to biomechanics. In: Maugin GA (ed) Continuum Models and Discrete Systems, Longman, Essex, vol 2, pp 220–229
Turbé N, Maugin GA (1991) On the linear piezoelectricity of composite materials. Math Meth Appl Sci 14:403–412
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.
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Alfonso-Rodríguez, R. et al. (2018). Effective Coefficients and Local Fields of Periodic Fibrous Piezocomposites with 622 Hexagonal Constituents. In: Altenbach, H., Pouget, J., Rousseau, M., Collet, B., Michelitsch, T. (eds) Generalized Models and Non-classical Approaches in Complex Materials 1. Advanced Structured Materials, vol 89. Springer, Cham. https://doi.org/10.1007/978-3-319-72440-9_1
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