Archive of Applied Mechanics

, Volume 84, Issue 9–11, pp 1565–1582 | Cite as

Micromechanical analysis of fibrous piezoelectric composites with imperfectly bonded adherence

  • R. Rodríguez-Ramos
  • R. Guinovart-Díaz
  • J. C. López-Realpozo
  • J. Bravo-Castillero
  • J. A. Otero
  • F. J. Sabina
  • H. Berger
  • M. Würkner
  • U. Gabbert
Special Issue


In this work, two-phase parallel fiber-reinforced periodic piezoelectric composites are considered wherein the constituents exhibit transverse isotropy and the cells have different configurations. Mechanical imperfect contact at the interface of the piezoelectric composites is studied via linear spring model. The statement of the problem for two-phase piezoelectric composites with mechanical imperfect contact is given. The local problems are formulated by means of the asymptotic homogenization method, and their solutions are found using complex variable theory. Analytical formulae are obtained for the effective properties of the composites with spring imperfect type of contact and different rhombic cells. Using the concept of a representative volume element (RVE), a finite element model is created, which combines the angular distribution of fibers and imperfect contact conditions (spring type) between the phases. Periodic boundary conditions are applied to the RVE, so that effective material properties can be derived. The fibers are distributed in such a way that the microstructure is characterized by a rhombic cell. The presented numerical homogenization technique is validated by comparing results with theoretical approach reported in the literature. Some studies of particular cases, numerical examples, and comparisons between the two aforementioned methods with other theoretical results illustrate that the model is efficient for the analysis of composites with presence of rhombic cells and the aforementioned imperfect contact.


Asymptotic homogenization Interfacial bonding Imperfect contact Piezoelectric composites Finite element method Fibrous composite 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Chopra I.: Review of state of art of smart structures and integrated systems. AIAA J. 40(11), 2145–2187 (2002)CrossRefGoogle Scholar
  2. 2.
    Timoshenko S.: Analysis of bi-metal thermostats. J. Opt. Soc. Am. 11, 233–255 (1925)CrossRefGoogle Scholar
  3. 3.
    Lighthill J., Bradshaw J.: Thermal stresses in turbine blades. Philos. Mag. 40, 770–780 (1949)MathSciNetMATHGoogle Scholar
  4. 4.
    Boley B.A., Weiner J.H.: Theory of Thermal Stresses. Dover, Mineola (1997)MATHGoogle Scholar
  5. 5.
    Bickford W.B.: A consistent higher-order beam. Theory Dev. Theor. 11, 137–142 (1982)Google Scholar
  6. 6.
    Kant T., Manjunath B.S.: Refined theories for composite and sandwich beams with C0 finite elements. Comput. Struct. 33, 755–764 (1992)CrossRefGoogle Scholar
  7. 7.
    Soldatos K.P., Elishakoff I.: A transverse shear and normal deformable orthotropic beam theory. J. Sound Vib. 154, 528–533 (1992)CrossRefGoogle Scholar
  8. 8.
    Reddy J.N.: Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, 2nd ed. CRC Press, New York (2003)Google Scholar
  9. 9.
    Berdichevsky V.L.: Variational-asymptotic method of constructing a theory of shells. J. Appl. Math. Mech. 43, 664–687 (1979)CrossRefGoogle Scholar
  10. 10.
    Hodges D.H., Atilgan A.R., Cesnik C.E.S., Fulton M.V.: On a simplified strain energy function for geometrically nonlinear behavior of anisotropic beams. Compos. Eng. 2, 513–526 (1992)CrossRefGoogle Scholar
  11. 11.
    Cesnik C.E.S., Hodges D.H.: Stiffness constants for initially twisted and curved composite beams. Appl. Mech. Rev. 46, 211–220 (1993)CrossRefGoogle Scholar
  12. 12.
    Cesnik C.E.S., Hodges D.H.: Variational-asymptotic analysis of initially twisted and curved composite beams. Int. J. Des. Eng. 1, 177–187 (1994)Google Scholar
  13. 13.
    Popescu B., Hodges D.H.: On asymptotically correct Timoshenko-like anisotropic beam theory. Int. J. Solids Struct. 37, 535–558 (2000)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Yu W., Hodges D.H., Volovoi V., Cesnik C.E.S.: On Timoshenko-like modeling of initially curved and twisted composite beams. Int. J. Solids Struct. 39, 5101–5121 (2002)CrossRefMATHGoogle Scholar
  15. 15.
    Maurini C., Pouget J., dell’Isola F.: On a model of layered piezoelectric beams including transverse stress effect. Int. J. Solids Struct. 41(16–17), 4473–4502 (2004)CrossRefMATHGoogle Scholar
  16. 16.
    Maurini C., Pouget J., dell’Isola F.: Extension of the Euler–Bernoulli model of piezoelectric laminates to include 3D effects via a mixed approach. Comput. Struct. 84(22–23), 1438–1458 (2006)CrossRefGoogle Scholar
  17. 17.
    Feng X.-Q., Li Y., Cao Y.-P., Yu S.-W., Gu Y.-T.: Design methods of rhombic tensegrity structures. Acta Mech. Sin. 26, 559–565 (2010)CrossRefMATHGoogle Scholar
  18. 18.
    Wegst U.G.K., Ashby M.F.: The mechanical efficiency of natural materials. Philos. Mag. 84(21), 2167–2186 (2004)CrossRefGoogle Scholar
  19. 19.
    Ingber D.E.: Cellular tensegrity: defining new rules of biological design that govern the cytoskeleton. J. Cell Sci. 104, 613–627 (1993)Google Scholar
  20. 20.
    Ingber D.E.: Tensegrity: the architectural basis of cellular mechanotransduction. Ann. Rev. Physiol. 59, 575–599 (1997)CrossRefGoogle Scholar
  21. 21.
    Bravo-Castillero J., Guinovart Díaz R., Sabina F.J., Rodríguez Ramos R.: Closed-form expressions for the effective coefficients of a fiber-reinforced composite with transversely isotropic constituents-II. Piezoelectric and square symmetry. Mech. Mater. 33(4), 237–248 (2001)CrossRefGoogle Scholar
  22. 22.
    Sabina F.J., Rodríguez Ramos R., Bravo Castillero J., Guinovart Díaz R.: 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 (2001)CrossRefMATHGoogle Scholar
  23. 23.
    Berger H., Kari S., Gabbert U., Rodriguez-Ramos R., Bravo-Castillero J., Guinovart-Diaz R., Sabina F.J., Maugin G.A.: Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties. Smart Mater. Struct. 15, 451–458 (2006)CrossRefGoogle Scholar
  24. 24.
    Guinovart-Díaz R., López-Realpozo J.C., Rodríguez-Ramos R., Bravo-Castillero J., Ramírez M., Camacho-Montes H., Sabina F.J.: Influence of parallelogram cells in the axial behaviour of fibrous composite. Int. J. Eng. Sci. 49, 75–84 (2011)CrossRefGoogle Scholar
  25. 25.
    Guinovart-Díaz R., Yan P., Rodríguez-Ramos R., López-Realpozo J.C., Jiang C.P., Bravo-Castillero J., Sabina F.J.: Effective properties of piezoelectric composites with parallelogram periodic cells. Int. J. Eng. Sci. 53, 58–66 (2012)CrossRefGoogle Scholar
  26. 26.
    Andrianov I.V., Bolshakov V.I., Danishevs’kyy V.V., Weichert D.: Asymptotic study of imperfect interfaces in conduction through a granular composite material. Proc. R. Soc. A 466, 2707–2725 (2010)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Molkov V.A., Pobedria B.E.: Effective elastic properties of a composite with elastic contact. Izvestia Akademia Nauk SSR, Mekh. Tverdovo Tela 1, 111–117 (1988)Google Scholar
  28. 28.
    Shodja H.M., Tabatabaei S.M., Kamali M.T.: A piezoelectric medium containing a cylindrical inhomogeneity: role of electric capacitors and mechanical imperfections. Int. J. Solids Struct. 44, 6361–6381 (2007)CrossRefMATHGoogle Scholar
  29. 29.
    Maugin G.A.: Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam (1988)MATHGoogle Scholar
  30. 30.
    Maugin G.A.: Material Inhomogeneities in Elasticity. Chapman & Hall, London (1993)CrossRefMATHGoogle Scholar
  31. 31.
    Telega, J.J.: Piezoelectric and homogenization. Application to biomechanics. In: Maugin, G.A. (ed.) Continuum Models and Discrete Systems. Logman, London 2:220–229 (1991)Google Scholar
  32. 32.
    Turbe N., Maugin G.A.: On the linear piezoelectricity of composite materials. Math. Method Appl. Sci. 14, 403–412 (1991)MathSciNetCrossRefMATHGoogle Scholar
  33. 33.
    Bensoussan A., Lions J.L., Papaicolaou G.: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978)MATHGoogle Scholar
  34. 34.
    Sanchez-Palencia E.: Non Homogeneous Media and Vibration Theory Lectures Notes in Physics. Springer, Berlin (1980)Google Scholar
  35. 35.
    Pobedria B.E.: Mechanics of Composite Materials. Moscow State University Press, Moscow (in Russian) (1984)Google Scholar
  36. 36.
    Bakhvalov, N.S., Panasenko G.P.: Homogenization Averaging Processes in Periodic Media. Kluwer, Kluwer Academic Publishers (1989)Google Scholar
  37. 37.
    Galka A., Telega J.J., Wojnar R.: Homogenization and thermopiezoelectricity. Mech. Res. Commun. 19, 315–324 (1992)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Galka A., Telega J.J., Wojnar R.: Some computational aspects of homogenization of thermopiezoelectric composites. Comput. Assist. Mech. Eng. Sci. 3, 133–154 (1996)Google Scholar
  39. 39.
    Bravo-Castillero J., Otero J.A., Rodriguez-Ramos R., Bourgeat A.: Asymptotic homogenization of laminated piezocomposite materials. Int. J. Solids Struct. 35(5–6), 527–541 (1998)MathSciNetCrossRefGoogle Scholar
  40. 40.
    Lopez-Realpozo J.C., Rodriguez-Ramos R., Guinovart-Diaz R., Bravo-Castillero J., Sabina F.J.: Transport properties in fibrous elastic rhombic composite with imperfect contact condition. Int. J. Mech. Sci. 53, 98–107 (2011)CrossRefGoogle Scholar
  41. 41.
    Sevostianov I., Rodríguez-Ramos R., Guinovart-Díaz R., Bravo-Castillero J., Sabina F.J.: Connections between different models describing imperfect interfaces in periodic fiber-reinforced composites. Int. J. Solids Struc. 49, 1518–1525 (2012)CrossRefGoogle Scholar
  42. 42.
    Ferretti, M., Madeo, A., dell’Isola, F., Boisse, P.: Modeling the onset of shear boundary layers in fibrous composite reinforcements by second-gradient theory. Zeitschrift fur Angewandte Mathematik und Physik, pp. 1–26 (2013, in press). doi:10.1007/s00033-013-0347-8
  43. 43.
    Berger H., Kari S., Gabbert U., Rodriguez-Ramos R., Rodriguez-Ramos R., Rodriguez-Ramos R., Rodriguez-Ramos R., Rodriguez-Ramos R.: Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties. J. Smart Mater. Struct. 15, 451–458 (2006)CrossRefGoogle Scholar
  44. 44.
    Berger H., Gabbert U., Köppe H., Rodriguez-Ramos R., Bravo-Castillero J., Guinovart-Diaz R., Otero J.A., Maugin G.A.: Finite element and asymptotic homogenization methods applied to smart composite materials. Comput. Mech. 33, 61–67 (2003)CrossRefMATHGoogle Scholar
  45. 45.
    Rodríguez-Ramos R., Guinovart-Diaz R., López J.C., Bravo-Castillero J., Sabina F.J.: Influence of imperfect elastic contact condition on the antiplane effective properties of piezoelectric fibrous composites. Arch. Appl. Mech. 80, 377–388 (2010)CrossRefMATHGoogle Scholar
  46. 46.
    Rodríguez-Ramos R., Guinovart-Díaz R., López-Realpozo J.C., Bravo-Castillero J., Otero J.A., Sabina F.J., Lebon F.: Analysis of fibrous electro-elastic composites with parallelogram cell and mechanic imperfect contact condition. Int. J. Mech. Sci. 73, 1–13 (2013)CrossRefGoogle Scholar
  47. 47.
    Royer, D., Dieulesaint, E.: Elastic Waves in Solids I. Springer, Berlin (2000)Google Scholar
  48. 48.
    Hashin Z.: Thin interphase/imperfect interface in elasticity with application to coated fiber composites. J. Mech. Phys. Solids 50, 2509–2537 (2002)MathSciNetCrossRefMATHGoogle Scholar
  49. 49.
    Würkner M., Berger H., Gabbert U.: On numerical evaluation of effective material properties for composite structures with rhombic fiber arrangement. Int. J. Eng. Sci. 49, 322–332 (2011)CrossRefMATHGoogle Scholar
  50. 50.
    Würkner M., Berger H., Gabbert U.: Numerical study of effective elastic properties of fiber reinforced composites with rhombic cell arrangements and imperfect interface. Int. J. Eng. Sci. 63, 1–9 (2013)CrossRefGoogle Scholar
  51. 51.
    Pastor J.: Homogenization of linear piezoelectric media. Mech. Res. Commun. 24, 145–50 (1997)Google Scholar
  52. 52.
    Hashin Z.: Analysis of properties of fibre composites with anisotropic constituents. J. Appl. Mech. 46, 543–550 (1979)CrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • R. Rodríguez-Ramos
    • 1
  • R. Guinovart-Díaz
    • 1
  • J. C. López-Realpozo
    • 1
  • J. Bravo-Castillero
    • 1
  • J. A. Otero
    • 2
    • 3
  • F. J. Sabina
    • 4
  • H. Berger
    • 5
  • M. Würkner
    • 5
  • U. Gabbert
    • 5
  1. 1.Facultad de Matemática y ComputaciónUniversidad de La HabanaHabanaCuba
  2. 2.Instituto de Cibernética, Matemática y Física (ICIMAF)HabanaCuba
  3. 3.Instituto Tecnologico de Estudios Superiores de Monterrey CEMEdo de MexicoMexico
  4. 4.Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoMéxico, D.F.México
  5. 5.Institute of MechanicsUniversity of MagdeburgMagdeburgGermany

Personalised recommendations