Viscoelastic effective properties for composites with rectangular cross-section fibers using the asymptotic homogenization method

  • Oscar L. Cruz-González
  • Reinaldo Rodríguez-Ramos
  • José A. Otero
  • Julián Bravo-Castillero
  • Raúl Guinovart-Díaz
  • Raúl Martínez-Rosado
  • Federico J. Sabina
  • Serge Dumont
  • Frederic Lebon
  • Igor Sevostianov
Chapter
Part of the Advanced Structured Materials book series (STRUCTMAT, volume 89)

Abstract

The present work deals with the estimation of the linear viscoelastic effective properties for composites with periodic structure and rectangular cross-section fibers, using the two-scale asymptotic homogenization method (AHM). As a particular case, the effective properties for a layered medium with transversely isotropic properties are obtained. Two times the homogenization method, in different directions, according to the geometrical configuration of the composite material is applied for deriving the analytical expressions of the viscoelastic effective properties for a composite material with rectangular cross-section fibers, periodically distributed along one axis. In addition to that, models with different creep kernels, in particular, the Rabotnov’s kernel are analyzed. Finally, the numerical computation of the effective viscoelastic properties is developed for the analysis of the results. Moreover, a numerical algorithm using FEM is developed in the present work. Comparisons with other approaches are given as a validation of the present model.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The authors would like to be grateful to University of Matanzas and Tecnologico de Monterrey, Escuela de Ingenieria y Ciencias, Atizapan de Zaragoza, Estado de Mexico, for its support. France-Cuba project "Partenariat Hubert Curien franco-cubain Carlos J. Finlay" 2017-2018 and Proyecto Nacional de Ciencias Básicas 2017-2019 are gratefully acknowledged. Thanks to Departamento de Matemáticas y Mecánica IIMAS-UNAM and FENOMEC for their support and Ramiro Chávez Tovar and Ana Pérez Arteaga for computational assistance.

References

  1. Bakhvalov N, Panasenko GP (1989) Homogenisation: Averaging Processes in Periodic Media. Kluwer, DordrechtGoogle Scholar
  2. Berger H, Gabbert U, Köppe H, Rodriguez-Ramos R, Bravo-Castillero J, Guinovart-Diaz R, Otero JA, Maugin GA (2003) Finite element and asymptotic homogenization methods applied to smart composite materials. Computational Mechanics 33(1):61–67Google Scholar
  3. Berger H, Kari S, Gabbert U, Rodriguez-Ramos R, Bravo-Castillero J, Guinovart-Diaz R, Sabina FJ, Maugin GA (2006) Unit cell models of piezoelectric fiber composites for numerical and analytical calculation of effective properties. Smart Materials and Structures 15(2):451–458Google Scholar
  4. Beurthey S, Zaoui A (2000) Structural morphology and relaxation spectra of viscoelastic heterogeneous materials. European Journal of Mechanics - A/Solids 19(1):1–16Google Scholar
  5. Blair GWS, Coppen FMV (1939) The subjective judgment of the elastic and plastic properties of soft bodies; the “differential thresholds” for viscosities and compression moduli. Proceedings of the Royal Society of London Series B, Biological Sciences 128(850):109–125Google Scholar
  6. Blair GWS, Coppen FMV (1943) The estimation of firmness in soft materials. The American Journal of Psychology 56(2):234–246Google Scholar
  7. Brenner R, Masson R, Castelnau O, Zaoui A (2002) A quasi-elastic affine formulation for the homogenised behaviour of nonlinear viscoelastic polycrystals and composites. European Journal of Mechanics - A/Solids 21(6):943–960Google Scholar
  8. Chen M, Dumont S, Dupaigne L, Goubet O (2010) Decay of solutions to a water wave model with a nonlocal viscous dispersive term. Discrete and Continuous Dynamical Systems 27(4):1473–1492Google Scholar
  9. Christensen RM (1969) Viscoelastic properties of heterogeneous media. Journal of the Mechanics and Physics of Solids 17(1):23–41Google Scholar
  10. Christensen RM (1971) Theory of Viscoelasticity. Academic Press, New YorkGoogle Scholar
  11. Dormieux L, Kondo D, Ulm FJ (2006) Microporomechanics. John Wiley & Sons, ChichesterGoogle Scholar
  12. Dumont S, Duval JB (2013) Numerical investigation of asymptotical properties of solutions to models for waterways with non local viscosity. Int J Num Anal Modeling 10(2):333–349Google Scholar
  13. Hashin Z (1965) Viscoelastic behavior of heterogeneous media. Trans ASME J Appl Mech 32:630–636Google Scholar
  14. Hashin Z (1966) Viscoelastic fibre reinforced materials. AIAA Journal 4:1411–1417Google Scholar
  15. Hashin Z (1970a) Complex moduli of viscoelastic composites - I. General theory and application to particulate composites. Int J Solids Struct 6:539–552Google Scholar
  16. Hashin Z (1970b) Complex moduli of viscoelastic composites - II. Fibre reinforced materials. Int J Solids Struct 6:797–807Google Scholar
  17. Hollenbeck KJ (1998) Invlap.m: a Matlab function for numerical inversion of Laplace transforms by the Hoog algorithm URL http://www.mathworks.com
  18. Kachanov M (1992) Effective elastic properties of cracked solids: critical review of some basic concepts. Appl Mech Rev 45(8):304–335Google Scholar
  19. Lahellec N, Suquet P (2007) Effective behavior of linear viscoelastic composites: A time-integration approach. International Journal of Solids and Structures 44(2):507–529Google Scholar
  20. Lavergne F, Sab K, Sanahuja J, Bornert M, Toulemonde C (2016) Homogenization schemes for aging linear viscoelastic matrix-inclusion composite materials with elongated inclusions. International Journal of Solids and Structures 80:545–560Google Scholar
  21. Laws N, McLaughlin R (1978) Self-consistent estimates for the viscoelastic creep compliances of composite materials. Proceedings of the Royal Society of London Series A, Mathematical and Physical Sciences 359(1697):251–273Google Scholar
  22. Le QV, Meftah F, He QC, Le Pape Y (2007) Creep and relaxation functions of a heterogeneous viscoelastic porous medium using the Mori-Tanaka homogenization scheme and a discrete microscopic retardation spectrum. Mechanics of Time-Dependent Materials 11(3):309–331Google Scholar
  23. Lévesque M, Gilchrist MD, Bouleau N, Derrien K, Baptiste D (2007) Numerical inversion of the Laplace–Carson transform applied to homogenization of randomly reinforced linear viscoelastic media. Computational Mechanics 40(4):771–789Google Scholar
  24. Maghous S, Creus GJ (2003) Periodic homogenization in thermoviscoelasticity: case of multilayered media with ageing. International Journal of Solids and Structures 40(4):851–870Google Scholar
  25. 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 cross-section fibers. Arch Mech 55:357–371Google Scholar
  26. Persson LE, Persson L, Svanstedt N,Wyller J (1993) The Homogenization Method. An Introduction. Student litteratur, LundGoogle Scholar
  27. Pipkin AC (1986) Lectures on Viscoelastic Theory. Springer, New York, Berlin, HeidelbergGoogle Scholar
  28. Pobedria BE (1984) Mechanics of Composite Materials (in Russ.). Moscow State University Press, MoscowGoogle Scholar
  29. Rabotnov YN (1948) Equilibrium of an elastic medium with after-effect (in Russ.). Prikladnaya Matematika i Mekhanika (J Appl Math Mech) 12(1):53–62Google Scholar
  30. Rabotnov YN (1977) Elements of Hereditary Solid Mechanics. Mir, MoscowGoogle Scholar
  31. Rabotnov YN (2014) Equilibrium of an elastic medium with after-effect. Fractional Calculus and Applied Analysis 17(3):684–696Google Scholar
  32. Ricaud JM, Masson R (2009) Effective properties of linear viscoelastic heterogeneous media: Internal variables formulation and extension to ageing behaviours. International Journal of Solids and Structures 46(7):1599–1606Google Scholar
  33. Schapery RA (1964) Application of thermodynamics to thermomechanical, fracture, and birefringent phenomena in viscoelastic media. Journal of Applied Physics 35(5):1451–1465Google Scholar
  34. Schapery RA (1967) Stress analysis of viscoelastic composite materials. Journal of Composite Materials 1(3):228–267Google Scholar
  35. Sevostianov I, Levin V, Radi E (2015) Effective properties of linear viscoelastic microcracked materials: Application of Maxwell homogenization scheme. Mechanics of Materials 84:28–43Google Scholar
  36. Sevostianov I, Levin V, Radi E (2016) Effective viscoelastic properties of short-fiber reinforced composites. International Journal of Engineering Science 100:61–73Google Scholar
  37. Sokolnikoff IS, Redheffer RM (1968) Mathematics of Physics and Modern Engineering. McGraw-Hill Book Company, Inc, New York, Toronto, LondonGoogle Scholar
  38. Wang YM, Weng GJ (1992) The influence of inclusion shape on the overall viscoelastic behavior of composites. Trans ASME J Appl Mech 59(3):510–518Google Scholar
  39. Zhang J, Ostoja-Starzewski M (2015) Mesoscale bounds in viscoelasticity of random composites. Mechanics Research Communications 68:98–104Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Oscar L. Cruz-González
    • 1
  • Reinaldo Rodríguez-Ramos
    • 2
  • José A. Otero
    • 3
  • Julián Bravo-Castillero
    • 2
  • Raúl Guinovart-Díaz
    • 2
  • Raúl Martínez-Rosado
    • 3
  • Federico J. Sabina
    • 4
  • Serge Dumont
    • 5
  • Frederic Lebon
    • 6
  • Igor Sevostianov
    • 7
  1. 1.Facultad de Ciencias Técnicas, Departamento de MatemáticaUniversidad de MatanzasMatanzasCuba
  2. 2.Facultad de Matemática y ComputaciónUniversidad de La HabanaLa HabanaCuba
  3. 3.Tecnologico de Monterrey, Escuela de Ingenieria y CienciasAtizapan de ZaragozaMéxico
  4. 4.Instituto de Investigaciones en Matemáticas Aplicadas y en SistemasUniversidad Nacional Autónoma de MéxicoApartadoMéxico
  5. 5.Institut de Mathématiques Alexander Grothendieck, CNRS, UMR 5149Université de NîmesMontpellier Cedex 5France
  6. 6.CNRS, Centrale Marseille, LMAAix-Marseille UniversityMarseille Cedex 13France
  7. 7.Department of Mechanical and Aerospace EngineeringNew Mexico State UniversityLas CrucesUSA

Personalised recommendations